LSER and COSMO-RS in Drug Development: A Comparative Guide to Predictive Solubility and Thermodynamics

Abstract

This article provides a comprehensive comparison of the Linear Solvation Energy Relationship (LSER) and the Conductor-like Screening Model for Real Solvents (COSMO-RS) for researchers and professionals in drug development. It explores the foundational principles of both models, detailing their methodological applications in predicting key properties like aqueous solubility, partition coefficients, and solvation thermodynamics. The content addresses common challenges and optimization strategies, including handling of strong specific interactions and thermodynamic consistency. A critical validation of model performance against experimental data and emerging hybrid approaches is presented, offering practical insights for their effective application in pharmaceutical research and development.

Understanding the Core Principles: LSER's Empirical Power vs. COSMO-RS's Quantum Foundation

Linear Solvation Energy Relationships (LSER) and the Conductor-like Screening Model for Real Solvents (COSMO-RS) represent two powerful, yet philosophically distinct, approaches for predicting solvation thermodynamics. This guide provides a systematic comparison of the Abraham LSER framework and COSMO-RS, focusing on their underlying principles, application domains, and predictive performance for properties critical to pharmaceutical and chemical development. By synthesizing data from recent benchmarking studies and blind challenges, we objectively evaluate these models across multiple metrics including partition coefficient prediction, solvation enthalpy calculation, and liquid-liquid equilibrium modeling. The analysis reveals complementary strengths: LSER excels in interpretability and accuracy for systems with abundant experimental parameters, while COSMO-RS offers broader predictivity for novel compounds without requiring prior experimental data.

Theoretical Foundations and Model Architectures

Abraham LSER Framework

The Abraham LSER model is a quantitative structure-property relationship (QSPR) approach that correlates free energy-related properties of solutes with molecular descriptors representing specific interaction types. The model employs two primary equations for different transfer processes [1] [2]:

For gas-to-solvent partitioning: log(K*) = ck + ekE + skS + akA + bkB + lkL

For water-to-organic solvent partitioning: log(P) = cp + epE + spS + apA + bpB + vpVx

In these equations, the uppercase letters represent solute-specific molecular descriptors: Vx (McGowan's characteristic volume), L (gas-hexadecane partition coefficient), E (excess molar refraction), S (dipolarity/polarizability), A (hydrogen bond acidity), and B (hydrogen bond basicity). The lowercase letters are system-specific coefficients that represent the complementary effect of the solvent phase on solute-solvent interactions [3]. These descriptors collectively capture the key intermolecular interactions governing solvation: cavity formation (Vx, L), dispersion forces (E), polar interactions (S), and specific hydrogen bonding (A, B) [2].

The remarkable success of LSER stems from its thermodynamically sound parameterization and wise selection of molecular descriptors that comprehensively characterize solute molecules [1]. The model has been extensively validated across thousands of compounds, with solute descriptors available in a freely accessible database [1].

COSMO-RS Framework

COSMO-RS represents a fundamentally different approach based on quantum chemistry and statistical thermodynamics. Rather than using empirical descriptors, COSMO-RS derives molecular interaction information from quantum chemical calculations of surface charge distributions (σ-profiles) [4] [5]. The methodology involves three key steps:

• Quantum Chemical Calculation: Density Functional Theory (DFT) calculations optimize molecular geometry and compute the electrostatic potential of individual molecules in a virtual conductor environment [4].

• σ-Profile Generation: The molecular surface is segmented, and DFT calculates the screening charge density for each segment, creating a σ-profile that represents the polarity distribution of the molecule [5].

• Statistical Thermodynamics: Using the σ-profiles as input, COSMO-RS applies statistical thermodynamics to predict activity coefficients and other thermodynamic properties by considering the pairwise interactions of surface segments [1] [5].

This first-principles approach allows COSMO-RS to predict solvation properties without component-specific empirical parameters, making it particularly valuable for novel compounds where experimental data is scarce [4].

Table 1: Fundamental Comparison of LSER and COSMO-RS Approaches

Feature	Abraham LSER	COSMO-RS
Theoretical Basis	Empirical linear free-energy relationships	Quantum chemistry + statistical thermodynamics
Parameter Origin	Experimentally derived descriptors	Quantum chemical calculations
Molecular Descriptors	Vx, L, E, S, A, B (6 parameters)	σ-profiles (continuous charge distributions)
Predictivity Scope	Limited to similar chemical spaces	Theoretically universal for neutral compounds
Primary Outputs	Partition coefficients, solvation free energies	Activity coefficients, solubilities, partition coefficients
Experimental Data Requirement	Extensive for parameterization	Minimal (only for validation)

Performance Benchmarking and Experimental Validation

Partition Coefficient Prediction

Partition coefficients represent a critical property in pharmaceutical design, and both models have undergone extensive validation through blind challenges and benchmarking studies.

In the SAMPL9 blind challenge predicting toluene/water partition coefficients for 16 drug-like molecules, COSMO-RS demonstrated competitive performance with a root mean square deviation (RMSD) of 1.23 logP units and a correlation coefficient (R²) of 0.93 [6]. While this error was larger than typically observed for octanol/water systems, COSMO-RS outperformed competing approaches in this rigorous blind test, confirming its utility for predicting partition behavior in novel systems [6].

LSER models have shown exceptional accuracy for specific partitioning systems. For Low Density Polyethylene (LDPE)/water partitioning, an LSER model achieved remarkable precision with R² = 0.991 and RMSE = 0.264 across 156 compounds [7]. The model equation: logKi,LDPE/W = -0.529 + 1.098Ei - 1.557Si - 2.991Ai - 4.617Bi + 3.886Vi demonstrates how the system-specific coefficients weight the various molecular interactions, with hydrogen bonding (A, B) playing a dominant role [7].

Table 2: Partition Coefficient Prediction Performance Metrics

Application	Model	Performance	Data Points	Reference
Toluene/water (drug-like molecules)	COSMO-RS	RMSD = 1.23 logP, R² = 0.93	16	[6]
LDPE/water partitioning	LSER	R² = 0.991, RMSE = 0.264	156	[7]
LDPE/water (validation set)	LSER (exp. descriptors)	R² = 0.985, RMSE = 0.352	52	[7]
LDPE/water (predicted descriptors)	LSER (QSPR descriptors)	R² = 0.984, RMSE = 0.511	52	[7]

Solvation Enthalpy and Hydrogen Bonding Contributions

Accurate prediction of solvation thermodynamics, particularly hydrogen bonding contributions, is essential for understanding molecular interactions in solution. A critical comparison of COSMO-RS and LSER for predicting hydrogen-bonding contributions to solvation enthalpy revealed generally good agreement for most solute-solvent systems, though notable discrepancies occurred in specific cases [1].

The LSER model for solvation enthalpy follows the equation: ΔHsolv = cH + eHE + sHS + aHA + bHB + lHL where the products aHA and bHB represent the hydrogen bonding contribution [1] [3]. COSMO-RS can directly compute the hydrogen-bonding contribution to solvation enthalpy from its quantum chemical framework, providing an independent predictive approach without requiring experimental parameterization [1].

This capability makes COSMO-RS particularly valuable for estimating hydrogen bonding interaction strengths, which remain challenging to determine unanimously even with advanced quantum chemical calculations or spectroscopic methods [1]. The model's predictions can inform other thermodynamic approaches, including equation-of-state models that require hydrogen-bonding parameters [1].

Liquid-Liquid Equilibrium Predictions

Liquid-liquid equilibrium (LLE) prediction is crucial for solvent extraction and separation processes. Recent large-scale benchmarking evaluated COSMO-SAC (a variant of COSMO-RS) across 2478 binary systems with nearly 75,000 experimental data points [5]. COSMO-SAC-2010 achieved a success rate exceeding 90% in detecting LLE occurrence, demonstrating strong qualitative performance across chemically diverse systems [5].

For aqueous systems, COSMO-RS generally outperforms COSMO-SAC, while COSMO-SAC-2010 sets the standard for nonaqueous systems, placing the two approaches at a broadly comparable overall level with complementary strengths [5]. However, assessment of COSMO-RS for LLE containing Deep Eutectic Solvents (DES) showed limitations, particularly for salt-based DES with aromatic and heterocyclic solutes, though it performed better for alcohol solutes in alkane/DES systems [8].

Methodological Protocols

Standard LSER Implementation Protocol

Implementing LSER models requires careful experimental design and statistical validation. The standard protocol involves:

• Experimental Data Collection: Measure retention factors (logk') or partition coefficients (logP) for a structurally diverse set of 30-50 probe compounds with known LSER descriptors [9] [2]. Ensure compounds span a wide range of polarity, hydrogen bonding capability, and molecular size.

• Descriptor Acquisition: Obtain solute descriptors (E, S, A, B, V, L) from the Abraham LSER database or calculate them using established prediction tools [7] [3]. The freely available LSER database contains curated descriptors for thousands of compounds [1].

• Multilinear Regression: Perform regression analysis using the equation: SP = c + eE + sS + aA + bB + vV + lL where SP is the measured solute property (e.g., logk' or logP) [2]. Validate model significance with ANOVA (typically p < 0.05 for coefficients) and check for multicollinearity using variance inflation factors (VIF < 5) [2].

• Model Validation: Apply leave-one-out cross-validation or split data into training (≥70%) and test sets (≤30%) to evaluate predictive accuracy [7]. For the LDPE/water partitioning model, this approach yielded R² = 0.985 and RMSE = 0.352 on the validation set [7].

COSMO-RS Calculation Workflow

The standard COSMO-RS protocol for property prediction involves:

• Conformer Generation: Generate representative conformers for each compound using molecular mechanics or quantum chemical methods. For flexible molecules, include all low-energy conformers within 3 kcal/mol of the global minimum [4].

• Quantum Chemical Calculation: Perform DFT geometry optimization and COSMO calculation using a recommended functional (e.g., BP86) and basis set (TZVP or TZVPD-FINE) [1] [8]. The TZVPD-FINE parameter set generally provides slightly better predictions [8].

• σ-Profile Generation: Calculate the σ-profiles from the COSMO output files using COSMOtherm or open-source alternatives. The σ-profile represents the distribution of screening charge density on the molecular surface [5].

• Property Prediction: Use the statistical thermodynamics framework of COSMO-RS to calculate the desired properties from the σ-profiles. For partition coefficients, this involves computing the chemical potentials in both phases [6].

• Validation: Compare predictions with experimental data when available. For SAMPL challenges, this involves blind prediction followed by experimental comparison [6].

Essential Research Reagents and Computational Tools

Table 3: Essential Resources for LSER and COSMO-RS Implementation

Resource Category	Specific Tools/Databases	Key Function	Access Information
LSER Databases	Abraham LSER Database	Source of curated solute descriptors (E, S, A, B, V, L)	Freely accessible [1]
COSMO-RS Software	COSMOtherm (BIOVIA)	Commercial implementation for σ-profile generation and property prediction	Commercial license [4] [6]
Open-Source Alternatives	ThermoSAC, COSMO-SAC implementation	Open-source packages for COSMO-based calculations	https://github.com/usnistgov/COSMOSAC [5]
Quantum Chemistry Packages	Turbomole, Gaussian	DFT calculations for σ-profile generation	Commercial and academic licenses [5]
Experimental Data Sources	Dortmund Data Bank (DDB)	Source of experimental LLE, VLE, and other phase equilibrium data	Available by subscription [5]
Descriptor Prediction	QSPR Tools	Prediction of LSER descriptors for novel compounds	Various open-source and commercial options [7]

Hybrid Approaches and Emerging Trends

Recent advances have explored hybrid methodologies that leverage the strengths of both approaches. One promising direction combines COSMO-RS derived descriptors with machine learning for aqueous solubility prediction [4]. This hybrid approach uses COSMO-RS to generate conformer-specific features including the averaged correction for dielectric energy, hydrogen bond donor and acceptor moments, and molecular volume, which are then fed into neural networks [4].

This methodology achieves high predictive power while requiring less training data than traditional machine learning methods, demonstrating the value of physically meaningful descriptors from thermodynamic models [4]. The framework enables predictive aqueous solubility modeling without solute-specific experimental data, particularly valuable in early drug discovery phases [4].

Another emerging trend involves connecting LSER with equation-of-state thermodynamics through Partial Solvation Parameters (PSP), facilitating information exchange between QSPR-type databases and equation-of-state developments [3]. This integration enables estimation of hydrogen bonding free energy (ΔGhb), enthalpy (ΔHhb), and entropy (ΔShb) changes, extending LSER applicability beyond its traditional domain [3].

Comparative Analysis and Selection Guidelines

Decision Framework for Model Selection

Table 4: Model Selection Guidelines Based on Application Requirements

Application Scenario	Recommended Model	Rationale	Expected Performance
High-Throughput Screening	LSER with predicted descriptors	Fast computation, minimal resources	R² ~0.98-0.99, RMSE ~0.3-0.5 [7]
Novel Compound Assessment	COSMO-RS	A priori predictivity without experimental parameters	RMSD ~1.2 logP units [6]
Hydrogen Bonding Analysis	Both (comparative approach)	Complementary insights from different frameworks	Good agreement in most systems [1]
Aqueous Systems LLE	COSMO-RS	Superior performance for aqueous systems [5]	>90% LLE detection rate [5]
Nonaqueous Systems LLE	COSMO-SAC-2010	Better performance for nonaqueous systems [5]	>90% LLE detection rate [5]
Interpretability Focus	LSER	Clear chemical interpretation of coefficients [2]	Direct insight into interaction contributions

Limitations and Boundary Conditions

Both models exhibit specific limitations that researchers must consider:

LSER Limitations:

• Requires experimental data for system parameterization
• Limited predictivity for compounds outside the chemical space of the training set
• Descriptor prediction introduces additional uncertainty (RMSE increases from 0.352 to 0.511 when using predicted descriptors) [7]

COSMO-RS Limitations:

• Computational intensity increases with molecular size and flexibility
• Accuracy varies significantly with system type (e.g., challenges with salt-based DES) [8]
• Quantitative predictions may require calibration for specific applications
• Overestimation of solute partition coefficients for aromatic and heterocyclic solutes in certain systems [8]

The integration of both approaches within a unified thermodynamic framework represents a promising future direction, potentially leveraging LSER's empirical accuracy and interpretability with COSMO-RS's a priori predictivity for comprehensive solvation property assessment [1] [3].

Theoretical Foundations of COSMO-RS

COSMO-RS (Conductor-like Screening Model for Real Solvents) represents a quantum chemistry-based equilibrium thermodynamics method developed to predict chemical potentials in liquids without system-specific adjustment [10]. Unlike traditional group contribution methods, COSMO-RS uses the screening charge density (σ) on molecular surfaces to calculate chemical potentials, providing a more fundamental approach to predicting thermodynamic properties [10]. The method processes the screening charge density σ on the surface of molecules to calculate the chemical potential μ of each species in solution, forming the basis for predicting activity coefficients, solubility, partition coefficients, vapor pressure, and free energy of solvation [10].

The theoretical framework begins with a COSMO calculation, where a molecular-shaped cavity is constructed around the molecule and numerous Coulomb charges are placed on this surface [11]. The individual charges and molecular structure are optimized to find the minimum energy of the system, typically using quantum chemical DFT methods as they represent a good compromise between computational effort and reliability [11]. For use in COSMO-RS-type models, an infinite dielectric constant is used to approximate the solvation of the molecule in an ideal conductor [11].

The Sigma Profile: A Key Descriptor in COSMO-RS

Definition and Derivation

The sigma profile (σ-profile) is a fundamental descriptor in COSMO-RS, representing the probability distribution of screening charge densities on a molecule's surface [11]. From the optimized charges on the molecular cavity, a list of charged surface segments is generated. The shielding charge density (SCD) distribution or σ-profile represents the probability of finding a certain SCD on the cavity surface [11]. In practical implementations, this distribution is typically represented by probability values or surface areas for discrete SCD intervals (e.g., 50 or 80 intervals) [11].

The σ-profile is formally defined as:

[ p(\sigma) = \frac{Ai(\sigma)}{A{total}} ]

where ( Ai(\sigma) ) represents the surface area with screening charge density σ, and ( A{total} ) is the total molecular surface area. This descriptor forms the statistical mechanical foundation for calculating molecular interactions in COSMO-RS [10].

Sigma Profile Workflow

The following diagram illustrates the complete workflow for sigma profile calculation and application in COSMO-RS calculations:

Sigma Profile Calculation Workflow

Interpretation of Sigma Profile Regions

Table: Characteristic Regions of a Sigma Profile and Their Chemical Significance

Region	Charge Density Range (e/Ų)	Chemical Interpretation	Molecular Features
Hydrophobic	-0.008 < σ < +0.008	Non-polar surface areas	Aliphatic chains, aromatic rings
Hydrogen Bond Donor	σ < -0.008	Electron-deficient surfaces	Hydroxyl groups, amines
Hydrogen Bond Acceptor	σ > +0.008	Electron-rich surfaces	Carbonyl oxygen, ethers
Strongly Polar		σ	> 0.012	Highly charged areas	Ionic groups, strong dipoles

Computational Methodologies and Protocols

Quantum Chemical Calculation Standards

The accuracy of COSMO-RS predictions critically depends on the quality of the underlying quantum chemical calculations. After careful analysis of results from different modern DFT functionals (BP, B3LYP) with different basis sets and commercial products (DMOL3, Turbomole, Gaussian), the Gaussian 03 software package with the basis set 6-311G(d,p) has been identified as providing the lowest overall deviation and most reliable results [11]. This combination, while computationally demanding, produces fewer large deviations from experimental data compared to alternatives [11].

For flexible molecules or components with different tautomeric forms, the calculation needs to be repeated for different conformers and/or tautomers to adequately represent the molecular ensemble [11]. The protocol involves:

• Conformational search to identify low-energy conformers
• Quantum chemical optimization of each conformer using DFT methods
• COSMO calculation for each optimized structure
• Averaging of sigma profiles according to Boltzmann distribution

Alternative Computational Approaches

Table: Comparison of Sigma Profile Generation Methods

Method	Basis	Accuracy	Computational Cost	Applicability
Full DFT Calculation	First-principles quantum chemistry	High	Very high	Small to medium molecules
Fast-Sigma Estimation	Group contribution/Prediction	Moderate	Low	High-throughput screening
COSMO-SAC	Segment Activity Coefficient	Moderate to High	Medium	Industrial applications

For computationally expensive systems or high-throughput screening applications, approximation tools like fast_sigma can generate sigma profiles directly from SMILES strings, significantly reducing computational time [12]. This approach uses parameterized group contributions rather than full quantum chemical calculations, providing a balance between accuracy and efficiency for initial screening studies [12].

Sigma Moments: Reduced Descriptors for QSPR Applications

Definition and Calculation

Sigma moments provide a reduced-dimensional representation of sigma profiles, serving as valuable chemical descriptors for Quantitative Structure-Property Relationship (QSPR) studies [13]. These moments are analogous to statistical moments and are calculated through the σ-profile, (P(\sigma)), and the σ^{hb}-profile, (P^{HB}(\sigma)), of a compound [13].

The mathematical definition of sigma moments is:

[ MOM_i = \int P(\sigma)\ \sigma^i \ d\sigma \qquad \text{where } i = 0, 1, 2, 3, 4, 5, 6 ]

For hydrogen bonding capabilities, additional moments are defined:

[ \begin{aligned} MOM^{hb}{acc_\ell} &= \int P^{HB}(\sigma)\ max(0,\ +\sigma- \sigma{cutoff_\ell}) \ MOM^{hb}{don_\ell} &= \int P^{HB}(\sigma)\ max(0,\ -\sigma- \sigma{cutoff_\ell}) \end{aligned} ]

where (\ell = 1, 2, 3, 4) represents different cutoff levels [13].

Physical Interpretation of Key Sigma Moments

Table: Physical Significance of Principal Sigma Moments

Moment	Mathematical Expression	Physical Interpretation	Application
MOM₀	(\int P(\sigma) d\sigma)	Molecular surface area	Size-dependent properties
MOM₁	(\int P(\sigma) \sigma d\sigma)	Negative of total charge	Charge distribution
MOM₂	(\int P(\sigma) \sigma^2 d\sigma)	Polarity	Dielectric properties
MOM₃	(\int P(\sigma) \sigma^3 d\sigma)	Profile asymmetry	Polarizability
MOM^{hb}_{acc}	Hydrogen acceptor strength	Hydrogen bonding capacity	Solvation studies
MOM^{hb}_{don}	Hydrogen donor strength	Hydrogen bonding capacity	Solvation studies

Comparative Analysis: COSMO-RS vs. LSER Models

Fundamental Methodological Differences

The Linear Solvation Energy Relationship (LSER) model, particularly Abraham's LSER approach, represents one of the most successful QSPR-type approaches for predicting solvation properties [14]. The fundamental difference between COSMO-RS and LSER lies in their descriptor systems and parameterization approaches.

While LSER uses experimentally derived molecular descriptors (Vx, L, E, S, A, B) obtained through multilinear regression of experimental data [3], COSMO-RS uses quantum chemically derived sigma profiles that require no experimental parameterization for new compounds [10]. This gives COSMO-RS a significant advantage for predicting properties of novel compounds without existing experimental data.

Thermodynamic Consistency and Limitations

LSER models have demonstrated remarkable success in practical applications but face challenges regarding thermodynamic consistency, particularly in handling self-solvation of hydrogen-bonded solutes [14]. Recent research has focused on reformulating LSER models using COSMO-based descriptors to address these limitations [14] [3].

COSMO-RS provides a more fundamental approach but has its own limitations. The model assumes an incompressible liquid state, that all parts of molecular surfaces can contact each other, and only pairwise interactions of molecular surface patches are allowed [10]. These simplifications enable computational efficiency but may limit accuracy for complex systems with specific directional interactions.

Performance Comparison in Practical Applications

Table: Comparative Analysis of COSMO-RS and LSER Approaches

Feature	COSMO-RS	Traditional LSER
Basis	Quantum chemical calculations	Experimental parameterization
Descriptors	Sigma profiles (p(σ))	Vx, L, E, S, A, B
Parameterization	Element-specific parameters	Compound-class specific
New Compounds	No experimental data needed	Requires similar compounds
Computational Cost	Higher initial investment	Lower after parameterization
Thermodynamic Consistency	Built-in	Requires careful validation
Hydrogen Bonding Treatment	Integrated in σ-profile	Separate A and B descriptors
Temperature Dependence	Naturally included	Requires separate parameterization

In practical applications such as the SAMPL challenges for predicting cyclohexane-water distribution coefficients, COSMO-RS has demonstrated superior accuracy, being "the most accurate of all contest submissions" [15]. Similarly, in predicting the polarity of ionic liquids and their mixtures with organic cosolvents, COSMO-RS descriptors have successfully been used in quantitative structure-property relationship (QSPR) studies [16].

Research Reagent Solutions: Computational Tools for COSMO-RS

Table: Essential Software Tools for COSMO-RS Implementation

Tool/Software	Function	Application Context	Availability
Gaussian	Quantum chemical COSMO calculations	Generate .cosmo files	Commercial
COSMOtherm	COSMO-RS property prediction	Industrial applications	Commercial (BIOVIA)
AMS COSMO-RS	COSMO-RS implementation	Academic research	Commercial (SCM)
fast_sigma	Rapid sigma profile estimation	High-throughput screening	Amsterdam Modeling Suite
DDB-Sigma Database	Pre-calculated sigma profiles	Avoid recalculation	DDBST
LVPP Database	Open sigma-profile database	COSMO-SAC applications	Open access

COSMO-RS represents a powerful approach for predicting thermodynamic properties based on quantum chemical calculations and sigma-profile descriptors. Its key advantage over traditional group contribution methods like LSER lies in its ability to predict properties of novel compounds without experimental parameterization. The sigma profile serves as a comprehensive descriptor that encodes information about molecular polarity, hydrogen bonding capacity, and surface charge distribution.

Ongoing research focuses on integrating the strengths of both approaches, using COSMO-based descriptors to enhance the thermodynamic consistency of LSER models while maintaining their practical applicability [14] [3]. The development of sigma moments as reduced descriptors further bridges the gap between detailed quantum chemical calculations and practical QSPR applications, enabling more efficient screening and prediction while maintaining physical meaningfulness.

For drug development professionals, COSMO-RS offers particular value in predicting partition coefficients and solubility parameters for complex drug molecules, whose experimental characterization may be limited by legal regulations or complex molecular structures [17]. As computational power increases and methods refine, the integration of COSMO-RS with machine learning approaches presents a promising avenue for accelerating property prediction in pharmaceutical development and environmental assessment of bioactive compounds.

For researchers and scientists in drug development, predicting thermodynamic properties of compounds in solution is a critical task, influencing decisions from solvent selection to bioavailability estimation. Two distinct methodological paradigms have emerged for this purpose: the Linear Solvation Energy Relationship (LSER) model and the Conductor-like Screening Model for Real Solvents (COSMO-RS). While LSER relies on robust, data-driven empirical parameters, COSMO-RS leverages quantum chemistry for a priori prediction. This guide provides a objective comparison of their performance, theoretical foundations, and practical applicability, framing them as complementary tools within the scientist's computational toolkit.

Theoretical Foundations and Methodologies

LSER: Empirical Solvation Relationships

Linear Solvation Energy Relationships are grounded in empirical correlation. The model describes a solvation property (e.g., log of a partition coefficient) as a linear combination of molecular descriptors that capture key solute-solvent interaction energies:

Property = c + aA + bB + sS + vV

These descriptors are typically derived from experimental data and represent:

• A and B: Hydrogen-bond acidity and basicity
• S: Dipolarity/Polarizability
• V: McGowan characteristic volume

The model's strength lies in its simplicity and direct parameterization against experimental results, making it highly reliable for interpolations within its training domain.

COSMO-RS: A Priori Quantum Chemical Prediction

COSMO-RS is based on unimolecular quantum chemical calculations that provide information for evaluating molecular interactions in liquids, combined with statistical thermodynamics [18]. The method involves a two-step process:

• COSMO Calculation: A quantum chemical calculation is performed for each molecule in a virtual perfect conductor environment, yielding the screening charge density (σ) on the molecular surface [10].
• COSMO-RS Statistical Thermodynamics: The σ-surfaces of all molecules in the mixture are used to compute the chemical potential of the system by considering the pairwise interactions of surface segments [10]. The method calculates the chemical potential, μ, based on σ-profiles (histograms of surface charge densities) and interaction functions that account for misfit energy, hydrogen bonding, and dispersion [19] [10].

COSMO-RS Workflow: The process flows from quantum chemical calculations to the prediction of thermodynamic properties through statistical mechanics of interacting surface segments.

Comparative Performance Analysis

Predictive Accuracy for Key Properties

Extensive benchmarking studies have evaluated the performance of both models for properties critical to pharmaceutical development.

Table 1: Predictive Accuracy for Aqueous Solubility of Drug-like Molecules

Model	Number of Compounds	RMS Error (log-units)	Data Requirements	Key Strengths
COSMO-RS [20]	150 drugs	0.66	Only molecular structure	A priori prediction for diverse structures
COSMO-RS [20]	107 pesticides	0.61	Only molecular structure	No re-parameterization needed
LSER	Varies by parameterization	Typically < 0.5	Experimental descriptors for new compounds	Excellent for chemically similar compounds

Table 2: Application Scope and Limitations

Characteristic	COSMO-RS	LSER
Theoretical Basis	Quantum chemistry + statistical thermodynamics [18]	Empirical linear free-energy relationships
Primary Input	Molecular structure [10]	Experimentally-derived solute descriptors
Parameterization	Element-specific parameters (fitted to experimental data) [10]	System-specific coefficients (fitted to experimental data)
A Priori Prediction	Yes, for any molecule that can be computed [18]	No, requires known descriptors for new compounds
Handling of Novel Structures	Excellent, provided quantum calculation is feasible	Limited, requires descriptor measurement/estimation
Computational Cost	Higher (requires quantum calculations)	Lower (simple linear algebra)
Interpretability	Medium (via σ-profiles and σ-potentials) [18]	High (direct contribution from interaction terms)

Analysis of Comparative Performance

The data reveals a fundamental trade-off between predictive breadth and empirical accuracy. COSMO-RS provides a remarkable capability for a priori prediction of aqueous solubility across structurally diverse drug-like compounds and pesticides without requiring experimental data for the specific compounds [20]. Its accuracy of approximately 0.6 log units (roughly a factor of 4 in solubility) makes it highly valuable for early-stage screening.

In contrast, LSER models typically achieve higher accuracy (often <0.5 log units) for chemicals within their training domain but require experimental descriptor values for new compounds, limiting their true a priori predictive application. The strength of LSER lies in its robust data-driven simplicity when sufficient experimental data exists for parameterization.

Experimental and Computational Protocols

COSMO-RS Workflow

Protocol for Predicting Solubility Using COSMO-RS [20]:

• Quantum Chemical Geometry Optimization: Optimize the 3D molecular structure using density functional theory (DFT) with a suitable basis set.
• COSMO Calculation: Perform a single molecule quantum chemical calculation in a perfect conductor environment to obtain the σ-surface and σ-profile.
• Database Storage: Store the COSMO results (screening charge density) in a database for future use [10].
• COSMO-RS Calculation: For the target system (e.g., solute in water), the software calculates the chemical potential based on the σ-profiles of all components and the interaction functions.
• Gibbs Energy of Fusion Correction: For solid solubility, add a heuristic expression for the Gibbs free energy of fusion [20].
• Property Calculation: Calculate the desired property (e.g., solubility, activity coefficient) from the chemical potentials using established thermodynamic relationships [19].

Data-Driven Model Development

Protocol for Developing Data-Driven Predictive Models [21] [22]:

• Data Acquisition: Generate a comprehensive training dataset covering the relevant chemical space and properties of interest.
• Model Selection: Choose an appropriate model architecture:
  • Artificial Neural Networks (ANN): Flexible for complex non-linear relationships but require significant data and computational resources [21].
  • Fuzzy Inference System (FIS): Computationally efficient and interpretable, handling imprecise data well [21].
  • Adaptive Neuro-Fuzzy Inference System (ANFIS): Combines learning power of ANN with interpretability of FIS [21].
• Model Training: Iteratively update model parameters to minimize error between predictions and experimental data.
• Hyperparameter Optimization: Tune model architecture parameters using an independent validation set to maximize accuracy and generalizability [22].
• Validation: Test the final model on a completely independent test set to evaluate real-world performance.

Data-Driven Modeling Workflow: The iterative process for developing predictive models like LSER, showing the feedback loop for performance improvement.

The Scientist's Toolkit: Essential Research Reagents and Computational Solutions

Table 3: Key Computational Tools and Their Functions

Tool / Solution	Function in Research	Relevance to Models
COSMOtherm (BIOVIA)	Commercial implementation of COSMO-RS for property prediction	Primary platform for COSMO-RS calculations
COSMObase	Database of >12,000 pre-computed σ-profiles	Accelerates COSMO-RS predictions by avoiding QM calculations
Quantum Chemistry Software (e.g., Gaussian, TURBOMOLE)	Performs initial COSMO calculations for new molecules	Generates essential σ-surface inputs for COSMO-RS
LVPP Sigma-Profile Database	Open database with COSMO-SAC parameterizations	Free alternative for σ-profile data
Amsterdam Modeling Suite (SCM)	Commercial software including COSMO-RS implementation	Alternative COSMO-RS platform with QSPR models
Machine Learning Libraries (e.g., TensorFlow, PyTorch)	Framework for developing ANN and other data-driven models	Enables creation of custom LSER-type models

The comparison between LSER and COSMO-RS reveals two philosophically distinct approaches to predicting solvation thermodynamics. COSMO-RS demonstrates superior a priori predictive power for novel compounds, requiring only molecular structure as input, with validated accuracy of ~0.6 log units for diverse drug-like molecules [20]. Its quantum chemical foundation allows it to capture complex electronic effects and make predictions where no experimental data exists.

Conversely, LSER-type models offer robust data-driven simplicity, typically achieving higher accuracy for chemicals within their training domain and providing greater interpretability through their linear free-energy relationships. Their limitation lies in requiring experimental descriptors for new compounds, restricting true a priori application.

For drug development professionals, the choice depends on the research context: COSMO-RS is invaluable for early-stage screening of entirely new molecular entities, while LSER provides high-accuracy predictions for chemical series where experimental data exists for descriptorization. The emerging trend of combining first-principles predictions with machine learning refinement suggests future tools may leverage the strengths of both approaches, offering both broad applicability and high accuracy across the drug discovery pipeline.

{# The Central Role of Modeling Hydrogen-Bonding and Specific Intermolecular Interactions}

::: {.notice} This comparison guide is framed within a broader research thesis comparing the LSER model with COSMO-RS predictions. It is intended for researchers, scientists, and drug development professionals. :::

Quantifying specific intermolecular interactions, particularly hydrogen bonding (HB), remains a central challenge in molecular thermodynamics with profound implications for chemical industries, pharmaceutical development, and environmental sciences. Even with advanced quantum chemical calculations, molecular simulations, and sophisticated instrumentation, no universally accepted reference value exists for the strength of a single hydrogen-bonding interaction [1]. This fundamental uncertainty has spurred the development of various computational models to predict and characterize these interactions. Among the most prominent are the Linear Solvation Energy Relationship (LSER) model, a highly successful empirical approach, and the Conductor-like Screening Model for Realistic Solvation (COSMO-RS), a quantum-mechanics-based a priori predictive method [1] [3]. This guide provides an objective comparison of these two frameworks, focusing on their capabilities in modeling hydrogen-bonding and specific interactions, supported by experimental and theoretical data.

Model Fundamentals: A Tale of Two Approaches

The LSER and COSMO-RS models are built on fundamentally different philosophies—one is empirically robust, while the other is a priori predictive.

Linear Solvation Energy Relationship (LSER)

The LSER model, developed by Abraham, is a QSPR-type approach that correlates solute transfer properties with six key molecular descriptors [1] [3]:

• Vx: McGowan’s characteristic volume
• L: Gas-to-n-hexadecane partition coefficient at 298 K
• E: Excess molar refraction
• S: Dipolarity/Polarizability
• A: Hydrogen-bond acidity
• B: Hydrogen-bond basicity

The model uses linear equations to quantify solute partitioning. For gas-to-solvent transfer, the equation is: log(Ks) = ck + ekE + skS + akA + bkB + lkL [1] Here, the upper-case letters are solute-specific descriptors, while the lower-case letters are solvent-specific coefficients obtained through multilinear regression of experimental data [1] [3]. The hydrogen-bonding contribution to the solvation free energy is represented by the sum akA + bkB [23]. Its key strength is its simplicity and robustness across a wide range of applications, though it relies on the availability of extensive experimental data for parameterization [1] [23].

COSMO-RS (Conductor-like Screening Model for Realistic Solvation)

COSMO-RS is a quantum-mechanics-based model that combines quantum chemical calculations of screening surface charge densities (σ-profiles) with statistical thermodynamics [1] [24]. It is an a priori predictive method that does not require experimental input for parameterization for new molecules [1]. The model calculates molecular interactions based on the surface charge distributions (σ-profiles) obtained from DFT calculations, treating hydrogen bonding as an electrostatic interaction between surface segments with complementary polarity [25]. While it can directly predict solvation free energies, its structure also allows for the calculation of the separate hydrogen-bonding contribution to solvation enthalpy, a feature not directly available in LSER for free energy [1].

Performance Comparison: Hydrogen-Bonding Prediction

Extensive comparisons have been made between COSMO-RS and LSER regarding their prediction of hydrogen-bonding contributions to solvation thermodynamics. The table below summarizes a critical comparison of their performance across key metrics.

Table 1: Performance Comparison in Hydrogen-Bonding Prediction

Aspect	LSER Model	COSMO-RS Model
Fundamental Basis	Empirical, based on experimental linear free-energy relationships [1] [3]	First-principles, based on quantum chemistry and statistical thermodynamics [1] [24]
HB Contribution to Solvation Free Energy	Estimated as agA + bgB from regression [23]	Directly predicted from σ-profiles [1]
HB Contribution to Solvation Enthalpy	Estimated as ahA + bhB from regression [1]	Can be calculated directly from model structure [1]
Predictive Nature	Requires experimental data for regression of solvent coefficients [23]	A priori predictive after initial parameterization [1]
Performance for Neutral Species	Robust and quantitative predictions [26]	Good to very good correlations with experiment, enabling quantitative predictions [26]
Performance for Anionic Species	Limited data availability	Poor correlations; qualitative and sometimes quantitative failures [26]
Treatment of Self-Solvation	The products aA and bB are generally not equal, creating challenges for thermodynamic consistency [23]	Provides a more symmetric treatment of interactions

Analysis of Comparative Data

Studies indicate a rather good agreement between COSMO-RS and LSER predictions for the hydrogen-bonding contribution to solvation enthalpy in most systems [1]. For complexes formed between neutral molecules, both methods show strong performance. COSMO-RS, using either the supermolecule (SM) or contact probability (CP) approach, yields "good to very good" correlations with experimental data [26]. However, a significant weakness is observed in both models when the hydrogen bond acceptor is an anion, where correlations are "poor" and sometimes even qualitative predictions fail [26].

Methodological Protocols

To ensure reproducibility, this section outlines the standard computational protocols for both models as derived from the literature.

Standard LSER Protocol

The application of the LSER model typically follows a standardized workflow for data regression and prediction.

Diagram 1: LSER Model Workflow

• Data Collection: Compile an extensive and critically-selected dataset of experimental solvation or partition coefficients (log K or log P) for a wide range of solutes in the solvent of interest [1] [3].
• Descriptor Acquisition: Obtain the six pre-defined LSER molecular descriptors (Vx, E, S, A, B, L) for each solute from the freely accessible LSER database [1] [3].
• Regression Analysis: Perform multi-linear regression of the experimental data against the solute descriptors to determine the solvent-specific LFER coefficients (c, e, s, a, b, v, l). These coefficients are considered constant for a given solvent/system [1] [3].
• Prediction: For new solutes, use their molecular descriptors with the regressed solvent coefficients to predict the desired thermodynamic property (e.g., partition coefficient, solvation enthalpy) [1].

Standard COSMO-RS Protocol

The COSMO-RS protocol relies on quantum chemical calculations to generate necessary molecular descriptors.

Diagram 2: COSMO-RS Model Workflow

• Quantum Chemical Calculation: Perform a DFT calculation using a continuum solvation model (C-PCM) for each molecule of interest. Software like TURBOMOLE, ORCA, or DMol3 is typically used [25] [24].
• σ-Profile Generation: From the calculation, extract the σ-profile, which represents the distribution of screening charge densities on the molecular surface [25].
• Parameterization: The model is applied using a specific parameter set. Common choices are the TZVP (Triple-Zeta Valence Polarized) or the more accurate TZVPD-FINE (which includes dispersion corrections and a finer grid) [1] [8].
• Statistical Thermodynamic Calculation: COSMO-RS uses the σ-profiles as input for its statistical thermodynamic framework to calculate chemical potentials and other thermodynamic properties in liquid mixtures [1] [24].

Hybrid and Emerging Approaches

Recognizing the limitations of each standalone model, recent research has focused on hybrid methods and novel descriptors.

The QC-LSER and PSP Framework

A significant development is the creation of quantum-chemical LSER (QC-LSER) descriptors that combine the strengths of both models. This approach defines new molecular descriptors for hydrogen-bonding acidity (α) and basicity (β) based on the σ-profiles from COSMO-RS calculations [25] [23]. The hydrogen-bonding interaction energy between two molecules (1 and 2) is then predicted by a simple, universal formula: ΔE_HB = c(α₁β₂ + α₂β₁), where c is a universal constant (5.71 kJ/mol at 25°C) [25]. This provides a direct link between quantum-chemical information and a simple LSER-like predictive framework. Similarly, the concept of Partial Solvation Parameters (PSP) has been developed to act as a bridge, facilitating the extraction of thermodynamic information from the LSER database for use in equation-of-state models [3].

Improved Dispersion Interactions in COSMO-RS

A key area of advancement for COSMO-RS is the refinement of its treatment of dispersion forces. Recent work on the openCOSMO-RS model has introduced a new dispersion term based on atomic polarizabilities [24]. By calculating atomic polarizability tensors (e.g., using ORCA 6.0) and projecting them onto molecular cavities, this modification has led to significant improvements in modeling challenging systems, such as halocarbon mixtures, while requiring fewer adjustable parameters than previous methods [24].

Table 2: Key Computational Tools and Databases

Tool/Resource	Type	Primary Function	Relevance
Abraham LSER Database [1]	Database	Provides curated solute descriptors (A, B, S, etc.) for thousands of molecules.	Essential for any LSER study; source of molecular parameters.
COSMObase [23]	Database	A pre-calculated database of σ-profiles for numerous molecules.	Saves computational time for COSMO-RS calculations.
COSMOlogic/BIOVIA COSMOtherm [1]	Software Suite	A commercial implementation of COSMO-RS for thermodynamic property prediction.	Industry-standard for applying COSMO-RS.
TURBOMOLE [23] [24]	Software	Quantum chemistry program for efficient DFT and σ-profile calculations.	Commonly used for the QC step in COSMO-RS.
ORCA [24]	Software	Quantum chemistry program for DFT calculations and property analysis (e.g., polarizabilities).	Used for advanced COSMO-RS developments.
openCOSMO-RS [24]	Software	An open-source implementation of the COSMO-RS model.	Allows for community development and customization (e.g., new dispersion terms).

The comparative analysis reveals that both LSER and COSMO-RS play central but complementary roles in modeling hydrogen-bonding and specific intermolecular interactions. The LSER model excels as a robust, empirically-grounded tool for systems where extensive experimental data exists for parameterization. In contrast, COSMO-RS provides a powerful a priori predictive framework, particularly valuable for screening new molecules or solvents before synthesis.

The future of this field lies not in choosing one model over the other, but in their strategic integration. The emergence of QC-LSER descriptors and Partial Solvation Parameters (PSP) demonstrates the power of combining the quantum-chemical foundation of COSMO-RS with the thermodynamic rigor and extensive database of the LSER approach [25] [3] [23]. Furthermore, ongoing improvements to the physical terms within COSMO-RS, such as the incorporation of atomic polarizabilities for dispersion interactions, continue to enhance its predictive accuracy [24]. For researchers in drug development and materials science, this converging roadmap promises increasingly reliable computational tools for mastering the complexities of intermolecular interactions.

Practical Implementation: Predicting Solubility, Partitioning, and Solvation in Drug Design

Predicting the partitioning behavior and solvation energies of molecules is a fundamental challenge in fields ranging from pharmaceutical development to environmental chemistry. Two prominent computational approaches for addressing this challenge are the Linear Solvation Energy Relationship (LSER) model and the COSMO-RS (Conductor-like Screening Model for Real Solvents) method. LSER is a robust, empirically-based methodology that correlates molecular descriptors with thermodynamic properties through linear equations [1]. Its parameters are typically derived from extensive experimental databases. In contrast, COSMO-RS is a quantum mechanics-based approach that predicts thermodynamic properties from molecular structure alone, using statistical thermodynamics of surface segment interactions [1] [19]. This guide provides a comprehensive comparison of these methodologies, focusing on their workflows, performance characteristics, and practical applications in research settings.

Theoretical Foundations and Methodological Comparison

LSER Principles and Workflow

The LSER model describes solvation and partitioning phenomena using a set of empirically-derived molecular descriptors that capture specific interaction capabilities. For partitioning between two condensed phases, the general LSER equation takes the form:

[ \log(P) = cp + epE + spS + apA + bpB + vpV_x ]

The solute descriptors are defined as follows [1]:

• Vx: McGowan's characteristic molecular volume
• E: Excess molar refraction
• S: Dipolarity/polarizability
• A: Hydrogen-bond acidity
• B: Hydrogen-bond basicity

The complementary solvent-phase coefficients (lowercase letters) are determined through multilinear regression against experimental partition coefficient data [1]. The strength of LSER lies in its direct parameterization from experimental measurements, making it highly accurate for systems well-represented in its training data.

COSMO-RS Principles and Workflow

COSMO-RS is based on quantum chemical calculations that determine the optimal screening charge density on a molecule's surface when embedded in a perfect conductor. The model then treats solvents as ensembles of interacting surface segments, with interaction energies defined by their screening charge densities [1] [27]. The key thermodynamic property is the pseudochemical potential (μ_i), from which partition coefficients between solvents solv1 and solv2 can be calculated as [19]:

[ \log{10} P{\text{solv1/solv2}} = \frac{1}{\ln(10)} \frac{\mui^{\text{solv2}} - \mui^{\text{solv1}}}{RT} + \log{10} \left( \frac{V{\text{solv1}}}{V_{\text{solv2}}} \right) ]

This approach requires no experimental input beyond the molecular structure, making it truly a priori predictive [1].

Experimental and Computational Protocols

LSER Model Development Protocol

Developing a reliable LSER model requires careful experimental design and statistical validation:

• Compound Selection: Curate a chemically diverse set of compounds that adequately represents the chemical space of interest. For example, one LSER study for LDPE/water partitioning used 159 compounds spanning a wide range of molecular weights (32-722 g/mol), vapor pressures, aqueous solubilities, and polarities [28].

• Experimental Partition Coefficient Measurement: Determine partition coefficients for all compounds in the training set using consistent experimental conditions. For polymer/water systems, this typically involves measuring equilibrium concentration ratios between phases [28].

• Descriptor Determination: Obtain LSER solute descriptors (E, S, A, B, V) for each compound. These can be sourced from experimental measurements or predicted using Quantitative Structure-Property Relationship (QSPR) tools when experimental data is unavailable [7].

• Model Calibration: Perform multilinear regression of the experimental partition coefficients against the solute descriptors to determine the system-specific coefficients (c, e, s, a, b, v) [7] [28].

• Model Validation: Reserve a portion of the data (typically ~30%) as an independent validation set to assess predictive accuracy [7]. For the LDPE/water model, validation with 52 compounds yielded R² = 0.985 and RMSE = 0.352 when using experimental descriptors [7].

COSMO-RS Calculation Protocol

Implementing COSMO-RS predictions involves these key steps:

• Molecular Structure Optimization: Perform quantum chemical geometry optimization using Density Functional Theory (DFT) to obtain the most stable conformer for each compound [27].

• COSMO Calculation: Compute the screening charge density surface (σ-surface) for each optimized structure by simulating it in a perfect conductor [27].

• σ-Profile Generation: Create a histogram (σ-profile) of the screening charge densities, representing the polarity distribution of the molecule's surface [27].

• Thermodynamic Property Calculation: Calculate chemical potentials and activity coefficients using statistical thermodynamics of interacting surface segments. For complex systems, this requires solving self-consistently for the surface composition [19].

• Partition Coefficient Derivation: Compute partition coefficients from the difference in pseudochemical potentials between phases using the established equation [19].

Comparison of LSER and COSMO-RS prediction workflows showing the empirical vs. first-principles approaches.

Performance Benchmarking and Comparative Analysis

Quantitative Performance Comparison

Table 1: Performance comparison of LSER and COSMO-RS for partition coefficient prediction

Prediction Method	System Validated	RMSE (log units)	R²	Key Strengths	Key Limitations
LSER (experimental descriptors)	LDPE/water [7]	0.264-0.352	0.985-0.991	High accuracy for represented chemistry	Limited by descriptor availability
LSER (predicted descriptors)	LDPE/water [7]	0.511	0.984	Broad applicability	Slightly reduced accuracy
COSMOtherm	Multiple liquid/liquid systems [29]	0.65-0.93	N/A	A priori prediction	Parameterization sensitivity
ABSOLV	Multiple liquid/liquid systems [29]	0.64-0.95	N/A	Good overall accuracy	Limited documentation
SPARC	Multiple liquid/liquid systems [29]	1.43-2.85	N/A	Wide parameter space	Lower accuracy

Application Case Study: LDPE/Water Partitioning

A comprehensive study developed and validated an LSER model for low-density polyethylene/water partitioning, yielding the following equation [7] [28]:

[ \log K_{i,\text{LDPE/W}} = -0.529 + 1.098E - 1.557S - 2.991A - 4.617B + 3.886V ]

This model demonstrated exceptional accuracy (R² = 0.991, RMSE = 0.264, n = 156) across a chemically diverse compound set. The negative coefficients for A and B indicate LDPE's weak hydrogen-bonding capacity compared to water, while the positive V coefficient highlights the dominance of dispersion interactions [7]. When applied to an independent validation set using predicted descriptors, the model maintained high performance (R² = 0.984, RMSE = 0.511), confirming its utility for compounds without experimentally-measured descriptors [7].

Hydrogen-Bonding Prediction Comparison

Both methods explicitly account for hydrogen-bonding interactions, but with different approaches. LSER quantifies hydrogen-bonding through the A (acidity) and B (basicity) descriptors in a linear free-energy framework [1]. COSMO-RS calculates hydrogen-bonding energies based on the interaction of surface segments with complementary polarization [1] [25]. Recent research has explored combining insights from both approaches, developing new COSMO-based descriptors for hydrogen-bonding interaction energies that follow the form [25]:

[ E{HB} = c(\alpha1\beta2 + \alpha2\beta_1) ]

where α and β represent molecular acidity and basicity descriptors, and c is a universal constant (5.71 kJ/mol at 25°C) [25].

Conceptual comparison of LSER and COSMO-RS fundamental approaches, highlighting their distinct philosophical foundations and the emerging trend toward hybrid methodologies.

Research Reagent Solutions: Essential Tools for Implementation

Table 2: Key research tools and resources for LSER and COSMO-RS implementations

Tool/Resource	Type	Key Features	Application Context
UFZ-LSER Database [30]	Database	Freely accessible, curated database of LSER parameters and prediction tools	LSER model development and application
COSMOtherm [29]	Software	Commercial implementation of COSMO-RS with user-friendly interface	Industrial screening applications
COSMOquick [27]	Software	Fragment-based approach for rapid COSMO-RS predictions	High-throughput screening
ABSOLV [29]	Software	LSER-based prediction using QSPR-derived descriptors	Pharmaceutical property prediction
ADF COSMO-RS [19]	Software	Academic implementation integrated with quantum chemical code	Research and method development

Both LSER and COSMO-RS offer robust frameworks for predicting partition coefficients and solvation energies, with complementary strengths and limitations. LSER models provide exceptional accuracy for chemical spaces well-represented in experimental databases, making them ideal for interpolation within known chemistry domains. The COSMO-RS approach offers true a priori prediction capability, making it valuable for exploring novel compounds or systems where experimental data is scarce.

The future of solvation property prediction lies in hybrid approaches that leverage the strengths of both methodologies [1]. Current research focuses on developing a COSMO-LSER equation-of-state framework that would combine the molecular basis of COSMO-RS with the thermodynamic rigor of LSER models [1]. Additionally, methods for predicting LSER descriptors from quantum chemical calculations are bridging the gap between these approaches, potentially enabling accurate predictions for compounds without experimental descriptors while maintaining the interpretability of the LSER framework [1] [25].

For researchers selecting between these approaches, the decision should be guided by the specific application: LSER is preferred when working within well-characterized chemical domains where maximum accuracy is required, while COSMO-RS offers greater utility for exploratory research involving novel chemistries or when experimental descriptor data is unavailable.

Predicting solute-solvent interactions and their thermodynamic properties represents a fundamental challenge in chemical research and drug development. Among the most advanced approaches for such predictions are the Conductor-like Screening Model for Real Solvents (COSMO-RS) and the Linear Solvation Energy Relationships (LSER) model. These methods provide complementary frameworks for understanding and predicting solvation phenomena, though they differ significantly in their theoretical foundations and practical implementation. COSMO-RS is a quantum mechanics-based method that starts from first principles to predict thermodynamic properties without requiring experimental input, making it particularly valuable for studying novel compounds not yet synthesized [1]. In contrast, the LSER approach relies on empirical correlations between molecular descriptors and experimentally determined properties, creating robust predictive models based on extensive databases of existing measurements [3]. This comparison guide examines both methodologies, their workflows, performance characteristics, and appropriate applications within pharmaceutical and environmental research contexts.

Theoretical Foundations and Methodological Approaches

COSMO-RS: A Quantum Chemical Approach

The COSMO-RS method combines quantum chemical calculations with statistical thermodynamics to predict the thermodynamic properties of fluids and mixtures. The foundation of COSMO-RS lies in the approximation that molecular interactions can be represented by interactions of surface segments from molecular cavities [31]. These segment interactions are functions of segment properties, with the most important descriptor being the screening charge density (σ). The model operates through several key stages:

Initially, each molecule undergoes quantum chemical calculations using Density Functional Theory (DFT) while embedded in a virtual perfect conductor. This COSMO calculation generates a surface with a specific charge distribution for each molecule. The resulting screening charge density distribution, known as the σ-profile, provides a quantitative description of the molecular surface polarity [31]. The σ-profile is essentially a histogram that shows the probability distribution of various screening charge densities on the molecular surface, capturing the molecule's polarity and hydrogen-bonding characteristics.

The theoretical framework of COSMO-RS then calculates the pseudochemical potential of compounds in both liquid and gas phases using the following fundamental equations [19]:

[ \mui(T,\mathbf{x}) = \mui^{res}(T,\mathbf{x}) + \mu_i^{comb}(T,\mathbf{x}) ]

[ \mui^{gas}(T) = Ei^{gas} - Ei^{COSMO} + \Delta Ei^{diel} - \sumk \gammak Ai^{k} - \omega ni^{ring} + \eta RT ]

where (\mui) represents the pseudochemical potential of compound i, (\mui^{res}) accounts for electrostatic interactions, and (\mu_i^{comb}) addresses combinatorial contributions from molecular size and shape differences.

Recent implementations such as openCOSMO-RS have introduced algorithms supporting multiple segment descriptors, enabling more refined parameterizations while maintaining computational efficiency [31]. Extensions like COSMO-RS-DARE (considering dimerization, aggregation, and reaction effects) further enhance the model's capability to handle complex systems, including solid solubility calculations where solute self-association may occur [32].

LSER: An Empirical Correlations Approach

The Linear Solvation Energy Relationship model, developed by Abraham, takes a fundamentally different approach based on empirical correlations. LSER utilizes six molecular descriptors to predict solvation properties through linear equations [1] [3]:

• Vx: McGowan's characteristic volume
• L: Gas-hexadecane partition coefficient at 298 K
• E: Excess molar refraction
• S: Dipolarity/polarizability
• A: Hydrogen-bond acidity
• B: Hydrogen-bond basicity

These descriptors are used in two primary equations for different types of phase transfers. For solute partitioning between two condensed phases:

[ \log(P) = cp + epE + spS + apA + bpB + vpV_x ]

For gas-to-solvent partitioning:

[ \log(K^*) = ck + ekE + skS + akA + bkB + lkL ]

In these equations, the uppercase letters represent solute-specific molecular descriptors, while the lowercase coefficients are solvent-specific parameters determined through multilinear regression of experimental data [3]. The strength of LSER lies in its extensive database of descriptors for thousands of compounds and its demonstrated success across numerous chemical, biomedical, and environmental applications.

Table 1: Fundamental Characteristics of COSMO-RS and LSER Approaches

Characteristic	COSMO-RS	LSER
Theoretical Basis	Quantum mechanics + statistical thermodynamics	Empirical linear free-energy relationships
Required Input	Molecular structure	Experimental data for descriptor determination
Molecular Descriptors	σ-profile (screening charge density)	Vx, L, E, S, A, B
Parameterization	A priori after initial parameterization	Regression against experimental data
Primary Output	Chemical potentials, activity coefficients, solubility	Partition coefficients, solvation free energies

Workflow Implementation and Experimental Protocols

COSMO-RS Workflow Protocol

The COSMO-RS methodology follows a structured computational pathway from molecular structure to thermodynamic properties. The workflow can be divided into four main stages:

Stage 1: Molecular Structure Preparation and Conformational Analysis

• Obtain or draw the molecular structure of the compound(s) of interest
• Perform conformational analysis to identify low-energy conformers
• Select representative conformer(s) for quantum chemical calculations

Stage 2: Quantum Chemical COSMO Calculations

• Perform DFT calculations with appropriate functionals (e.g., BP86) and basis sets (e.g., TZVP)
• Conduct these calculations with the molecule embedded in a virtual conductor
• Generate COSMO files containing surface screening charge densities

Stage 3: σ-Profile Generation and Processing

• Extract the screening charge density information from COSMO files
• Create σ-profiles representing the distribution of polarity on molecular surfaces
• For advanced implementations, incorporate additional descriptors like σ⊥ (screening charge correlation correction)

Stage 4: Thermodynamic Property Calculation

• Apply COSMO-RS equations to calculate chemical potentials
• Derive activity coefficients using the relationship: (\gammai = \exp\left(\frac{\mui^{solv} - \mu_i^{pure}}{RT}\right)) [19]
• Calculate solubility using appropriate equations for solids: (xi^{\text{SOL(solid)}} = \frac{1}{\gammai} \exp\left{ \frac{\Delta{fus} H{i}}{R} \left( \frac{1}{T{m,i}} - \frac{1}{T} \right) - \frac{\Delta C{p,i}}{R} \left( \ln \frac{T{m,i}}{T} - \frac{T{m,i}}{T} + 1 \right) \right}) [19]

LSER Workflow Protocol

The LSER methodology follows a different pathway centered on descriptor determination and linear regression:

Stage 1: Descriptor Determination

• Obtain experimental values for the six molecular descriptors (Vx, L, E, S, A, B)
• Utilize existing LSER databases containing descriptors for thousands of compounds
• For new compounds, determine descriptors through measured partition coefficients or computational estimation methods

Stage 2: Solvent-Specific Coefficient Collection

• Identify appropriate solvent-specific coefficients (lowercase letters in LSER equations) for the system of interest
• Access published coefficients determined through multilinear regression of experimental data
• Verify applicability domain of coefficients for the specific solute-solvent system

Stage 3: Property Calculation

• Apply the appropriate LSER equation with solute descriptors and solvent coefficients
• Calculate partition coefficients or solvation free energies directly from linear equations
• For extended temperature ranges, incorporate temperature-dependent parameterizations where available

Performance Comparison and Experimental Validation

Hydrogen-Bonding Prediction Accuracy

A critical comparison of COSMO-RS and LSER predictions for hydrogen-bonding contributions to solvation enthalpy reveals important performance differences. Research examining a variety of solute-solvent systems has shown that both methods achieve rather good agreement in most cases, though each demonstrates specific strengths and limitations [1].

COSMO-RS provides a direct computational route to estimate hydrogen-bonding interaction energies through analysis of molecular surface charge distributions. Recent advances have enabled the development of simplified predictive methods where hydrogen-bonding interaction energy between molecules 1 and 2 is calculated as (E{HB} = c(\alpha1\beta2 + \alpha2\beta_1)), where c is a universal constant (5.71 kJ/mol at 25°C), and α and β represent molecular acidity and basicity descriptors derived from COSMO calculations [25].

LSER estimates hydrogen-bonding contributions through the product terms (akA) and (bkB) in its linear equations. While empirically effective, this approach does not provide direct insight into the actual hydrogen-bond energies between specific molecular pairs, as the coefficients are determined through statistical fitting procedures [1].

Table 2: Performance Comparison for Hydrogen-Bonding Systems

Assessment Criteria	COSMO-RS	LSER
Theoretical Basis for HB	Surface polarity and σ-profile analysis	Empirical aA + bB product terms
Self-Association Prediction	Directly calculated from molecular descriptors	Requires specific parameterization
Conformational Effects	Accounted for through multiple conformer calculations	Not explicitly considered in standard implementation
Temperature Dependence	Naturally included through thermodynamic equations	Requires separate parameterization

Solubility Prediction Performance

Solubility prediction represents a critical application for both methods, with significant implications for pharmaceutical development. A case study examining coumarin solubility in neat alcohols demonstrates the application of COSMO-RS-DARE for testing consistency of solubility data and identifying potential outliers in experimental datasets [32].

In this study, researchers measured coumarin solubility in seven alcohols (methanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 1-pentanol, and 1-octanol) at temperatures ranging from 25-40°C. They employed the shake-flask method with spectrophotometric concentration determination, followed by COSMO-RS-DARE computations to determine intermolecular interaction parameters [32]. The results demonstrated a perfect match between back-computed values and experimental measurements, confirming the reliability of the theoretical approach for solubility data consistency assessment.

LSER methods have also demonstrated success in solubility prediction, though they operate through different mechanistic pathways. The LSER approach correlates solubility with molecular descriptors through linear relationships, providing excellent interpolation within well-characterized chemical spaces but potentially limited extrapolation capabilities for novel compound structures.

Computational Requirements and Accessibility

Implementation requirements differ significantly between the two approaches:

COSMO-RS requires substantial computational resources for the initial quantum chemical calculations, particularly for complex molecules requiring conformational analysis or for large datasets of compounds. However, once σ-profiles are generated, property calculations for multiple mixtures and conditions are relatively efficient. The development of open-source implementations like openCOSMO-RS has improved accessibility for academic researchers [31].

LSER calculations themselves are computationally trivial, consisting mainly of linear algebra operations. The primary resource requirement lies in the experimental data needed to determine molecular descriptors or solvent-specific coefficients. For compounds with established descriptors, LSER provides extremely rapid property predictions suitable for high-throughput screening applications.

Table 3: Computational Requirements and Resource Considerations

Resource Factor	COSMO-RS	LSER
Initial Setup	Quantum chemistry software + COSMO-RS implementation	Database access + regression tools
Computation Time	Hours to days for σ-profile generation; minutes for property calculation	Seconds to minutes for property calculation
Data Dependencies	Primarily dependent on molecular structures	Dependent on experimental databases for descriptors
Specialized Expertise	Quantum chemistry, computational thermodynamics	Statistical analysis, experimental design
Cost Factors	Software licenses, computational hardware	Database access, experimental measurements

Integrated Applications in Pharmaceutical Research

Drug Partitioning and Environmental Distribution

The prediction of drug molecule partitioning between different environmental compartments represents a particularly relevant application for both methods. A recent study examined 23 prominent drug molecules, including compounds like benzylpiperazine, amphetamine, cocaine, fentanyl, and LSD, calculating partition coefficients (logKOW, logKOA, logKAW) using quantum chemical methods [17].

This research highlighted the importance of reliable partitioning data for tracking drug distribution in wastewater, ambient air, and house dust - key matrices for monitoring community drug use patterns. The study found that while popular prediction tools like EPI Suite and SPARC provided unreliable values for larger drug molecules, quantum chemical methods including COSMO-RS approaches offered viable alternatives despite requiring advanced expertise and computational effort [17].

LSER models have similarly been applied to predict drug partitioning behavior, leveraging their extensive databases of molecular descriptors. The complementary nature of these approaches suggests potential value in hybrid methodologies that combine the a priori predictive power of COSMO-RS with the empirical robustness of LSER for well-characterized chemical spaces.

Research Reagent Solutions and Tools

Table 4: Essential Computational Tools for Solvation Thermodynamics

Tool/Resource	Function	Implementation Considerations
COSMOtherm	Commercial COSMO-RS implementation with extensive parameterizations	User-friendly interface; limited customization options
openCOSMO-RS	Open-source COSMO-RS implementation supporting multiple descriptors	Flexible parameterization; requires computational expertise
ORCA	Quantum chemistry program for COSMO file generation	Free for academic use; supports BP/TZVP level calculations
LSER Database	Comprehensive collection of molecular descriptors and solvent coefficients	Freely accessible; contains thousands of compounds
RDKit	Open-source cheminformatics toolkit	Useful for molecular structure preparation and manipulation
COMSO-RS-DARE	Extension accounting for dimerization and aggregation	Essential for systems with strong self-association

The comparative analysis of COSMO-RS and LSER methodologies reveals a complementary relationship rather than a competitive one between these approaches. COSMO-RS provides a first-principles foundation for predicting thermodynamic properties of novel compounds and systems where experimental data are scarce or unavailable. Its quantum chemical basis enables extrapolation beyond existing chemical spaces, making it particularly valuable for drug development applications involving newly synthesized compounds.

LSER offers empirical robustness within well-characterized chemical domains, with computational efficiency that supports high-throughput screening applications. The extensive LSER database represents a valuable resource built upon decades of carefully curated experimental measurements.

Current research directions focus on integrating these approaches to leverage their respective strengths. The development of Partial Solvation Parameters (PSP) attempts to bridge this gap by creating a thermodynamic framework that facilitates information exchange between QSPR-type databases like LSER and equation-of-state developments [3] [33]. Similarly, efforts to establish a COSMO-LSER equation-of-state framework aim to combine the predictive power of COSMO-RS with the empirical effectiveness of LSER molecular descriptors [1].

For researchers and drug development professionals, method selection depends significantly on specific application requirements. COSMO-RS proves most valuable for novel compound characterization and systems where experimental data are limited, while LSER offers efficient screening capabilities for compounds within its well-parameterized chemical domains. The ongoing development of open-source implementations and hybrid methodologies promises to further enhance accessibility and application scope for both approaches in pharmaceutical research and development.

The prevalence of poorly soluble new chemical entities (NCEs) represents a fundamental challenge in modern pharmaceutical development, with approximately 70% of drug candidates exhibiting limited aqueous solubility that compromises their bioavailability. Within this context, predictive computational models have emerged as indispensable tools for formulators, enabling efficient excipient selection and solubility enhancement while conserving precious drug substance during early development stages. This guide provides a systematic comparison of two prominent thermodynamic approaches: the quantum chemistry-based Conductor-like Screening Model for Real Solvents (COSMO-RS) and the empirical Linear Solvation Energy Relationship (LSER) model. By objectively evaluating their theoretical foundations, application protocols, and performance characteristics, we aim to equip researchers with the necessary information to select the appropriate tool for specific pharmaceutical development scenarios, particularly in solubility prediction, bioavailability enhancement, and excipient screening.

Theoretical Foundations and Key Differentiators

Understanding the fundamental principles underlying COSMO-RS and LSER models is crucial for their appropriate application in pharmaceutical research.

COSMO-RS: A Quantum Chemistry-Based Approach

COSMO-RS combines quantum chemical calculations with statistical thermodynamics to predict thermodynamic properties of liquids and mixtures [10]. The method involves a two-step process:

• COSMO Calculation: Each molecule undergoes a quantum chemical calculation in a virtual perfect conductor environment, which determines the screening charge density (σ) on the molecular surface. These results are typically stored in a database [10].
• COSMO-RS Calculation: The stored σ-profiles are processed using statistical thermodynamics to calculate chemical potentials (μ) in solution. These chemical potentials form the basis for predicting activity coefficients, solubility, partition coefficients, vapor pressure, and free energy of solvation [10].

A distinctive feature of COSMO-RS is its reliance on the σ-profile (histogram p(σ)) of molecules rather than functional group parameters. This allows it to inherently account for quantum chemical effects like group-group interactions, mesomeric effects, and inductive effects [10]. The model incorporates several interaction energy contributions:

• Electrostatic (Misfit) Energy: Arises from the mismatch of polarized surface pieces in contact [10].
• Hydrogen Bonding Energy: Occurs when the screening charge densities of hydrogen bond donors and acceptors exceed specific thresholds [10].
• Dispersion Energy: Depends on element-specific parameters and the amount of exposed atomic surface area [10].

LSER: An Empirical Correlative Approach

The LSER model, particularly in its Abraham formulation, is a robust Quantitative Structure-Property Relationship (QSPR)-type tool that correlates solute transfer properties with molecular descriptors [3]. Its predictive capacity stems from a wise selection of these six solute-specific LSER descriptors [1] [3]:

• Vx: McGowan’s characteristic volume.
• L: The gas-liquid partition coefficient in n-hexadecane at 298 K.
• E: The excess molar refraction.
• S: The dipolarity/polarizability.
• A: The hydrogen bond acidity.
• B: The hydrogen bond basicity.

The model uses linear equations to quantify solute transfer between phases. For the water-to-organic solvent partition coefficient (P), the relationship is [3]: log(P) = cp + epE + spS + apA + bpB + vpVx

Here, the lower-case letters are solvent-specific coefficients determined by multilinear regression of experimental data. These coefficients represent the complementary effect of the solvent on solute-solvent interactions [3]. A similar equation is used for solvation enthalpies [1]. The products A₁a₂ and B₁b₂ are assumed to quantify the hydrogen-bonding contribution to the free energy of solvation [3].

Comparative Theoretical Analysis

Table 1: Fundamental Comparison between COSMO-RS and LSER Models

Feature	COSMO-RS	LSER Model
Theoretical Basis	Quantum chemistry & statistical thermodynamics [10]	Empirical linear free-energy relationships [3]
Primary Input	2D molecular structure [34]	Pre-determined LSER molecular descriptors (Vx, L, E, S, A, B) [1] [3]
Parameterization	Element-specific and general parameters; no functional group parameters [10]	System-specific LFER coefficients obtained by regression of experimental data [1] [3]
Handling of HB	Explicit via Ehb term based on σ-potentials [10]	Implicit via products of solute descriptors (A, B) and solvent coefficients (a, b) [3]
Predictive Nature	A priori predictive after initial parameterization [1]	Requires experimental data for regression of system coefficients [1]

Performance Comparison in Pharmaceutical Applications

Solubility Prediction and Excipient Ranking

A critical application in early-stage development is predicting API solubility in various excipients to guide formulation design. A study conducted at Pfizer evaluated COSMO-RS for this purpose using seven compounds with low aqueous solubility [34] [35]. The model required only the 2D molecular structures of the API and excipients as input, making it suitable when compound information is scarce [34]. The workflow and performance are summarized below.

The study concluded that COSMO-RS was able to reasonably predict excipients with the best solubilizing power, enabling formulators to quickly narrow down the number of excipients for experimental screening, thereby saving resources, time, and limited bulk API [34] [35].

Prediction of Hydrogen-Bonding Contributions

Hydrogen-bonding (HB) is a key intermolecular interaction affecting solubility and stability. A critical comparison examined the HB contribution to solvation enthalpy (ΔH) for various solute-solvent systems using both COSMO-RS and the LSER model [1]. The study found a rather good agreement between the predictions of the two models in most of the studied systems, though cases of large discrepancies were also noted. This suggests that for many pharmaceutical systems, both models can provide qualitatively similar insights into the strength of specific interactions, which can guide excipient selection based on compatibility.

Benchmarking and Model Accuracy

The accuracy of COSMO-RS is inherently tied to the underlying quantum chemical calculation. A benchmarking study evaluated different quantum mechanics (QM) levels for use with COSMO-RS [36]. It found that theoretically superior methods like MP2, PBE0, and M06-2x performed slightly worse in fully reparametrized COSMO-RS for predicting properties like pKa or logP compared to the established BP/def2-TZVPD level. This highlights that the best theoretical method for electronic energy does not necessarily translate to the best performance for fluid-phase thermodynamics predictions, and the existing parameterization of COSMO-RS is optimized for specific QM levels.

Table 2: Summary of Model Performance in Key Pharmaceutical Applications

Application	COSMO-RS Performance	LSER Performance	Supporting Data
Excipient Ranking	Good ranking capability; enables pre-screening [34] [35]	Not directly assessed in found literature	Experimental solubility of 7 Pfizer compounds vs. COSMO-RS predictions [34]
HB Interaction Strength	Good agreement with LSER for solvation enthalpy in most systems [1]	Good agreement with COSMO-RS for solvation enthalpy in most systems [1]	Comparison of HB contribution to ΔH for various solute-solvent systems [1]
General Solvation Properties	Proven high prediction accuracy in blind challenges (SAMPL5/6) [10]	Robust and widely used tool for partitioning and solvation [3]	Public challenge results (COSMO-RS); extensive compiled database (LSER) [10] [3]

Experimental Protocols and Research Toolkit

Protocol for Excipient Screening Using COSMO-RS

The following methodology is adapted from a study applying COSMO-RS for excipient ranking [34] [35]:

• Input Preparation: Generate 2D molecular structures for the drug compound and the library of candidate excipients in a suitable digital format (e.g., SDF, MOL).
• Quantum Chemical Calculation: Perform a COSMO calculation for each molecule. This typically involves geometry optimization followed by a single-point energy calculation in a virtual conductor environment using a quantum chemistry software (e.g., TURBOMOLE, Gaussian). The recommended QM level for use with COSMO-RS is often BP/TZVP or BP/TZVPD [10] [36].
• σ-Profile Generation: The COSMO calculation outputs a ".cosmo" file containing the screening charge density σ for each molecule. These files are stored in a database.
• COSMO-RS Calculation: Using software such as COSMOtherm, the stored σ-profiles are processed. The chemical potential of the drug solute in each pure excipient (solvent) is calculated.
• Solubility Prediction: The chemical potentials are used to compute the activity coefficient and subsequently the predicted solubility of the drug in each excipient.
• Ranking and Validation: Excipients are ranked based on the predicted solubility. The top-ranked excipients are then selected for experimental validation using a minimal amount of API to confirm the predictions.

Protocol for LSER Analysis

The general protocol for applying the LSER model involves [1] [3]:

• Descriptor Acquisition: Obtain the six LSER molecular descriptors (Vx, L, E, S, A, B) for the solute of interest. These can be sourced from the freely available LSER database or estimated using specialized software.
• Coefficient Sourcing: Identify the relevant LFER coefficients (e.g., cp, ep, sp, ap, bp, vp for a partition coefficient) for the specific solvent or phase system of interest from published literature. These coefficients are system-specific and must be available for the model to be predictive.
• Calculation: Insert the solute descriptors and solvent coefficients into the appropriate LSER equation (e.g., Equation 1 for partition coefficients) to calculate the desired property (e.g., log P).
• Interpretation: Analyze the result, with particular attention to the contributions from the hydrogen-bonding terms (aA and bB) to understand the specific interactions driving the solvation or partitioning behavior.

The Scientist's Toolkit: Essential Research Reagents and Solutions

Table 3: Key Materials and Computational Tools for Solubility Modeling

Item / Solution	Function / Role in Research	Example Use Case
COSMOtherm Software	Commercial implementation of COSMO-RS for predicting thermodynamic properties [10]	Calculating activity coefficients and solubilities in liquid mixtures [34] [37]
COSMObase	A large database of pre-calculated σ-profiles for thousands of compounds [10]	Providing immediate input for COSMO-RS calculations without performing new QM calculations
Specialized Polymers (HPMC, PVP, etc.)	Commonly used excipients to enhance solubility and inhibit precipitation in ASDs [38] [37]	Experimental validation of computational predictions for amorphous solid dispersions [37]
LSER Database	A freely accessible compilation of solute descriptors and solvent coefficients [1] [3]	Sourcing necessary parameters for LSER predictions of partition coefficients and solvation energies
High-Throughput Screening Platforms	Miniaturized systems (e.g., solvent casting) for experimental solubility and release testing [37]	Efficiently validating computational predictions with minimal API consumption [34] [37]

The strengths of COSMO-RS and LSER can be leveraged in a complementary manner. A proposed integrated workflow for excipient selection is as follows:

Within the broader thesis of comparing LSER with COSMO-RS predictions, this guide demonstrates that both models are powerful yet fundamentally different tools for pharmaceutical scientists. COSMO-RS offers a more general, a priori predictive approach based on quantum chemistry, making it particularly valuable for novel molecules or excipient systems where experimental data is scarce [10] [34]. Its requirement for specialized software and computational resources is a consideration. In contrast, the LSER model provides a robust, empirically-grounded framework that is highly effective for systems where its pre-determined descriptors and coefficients are available, leveraging a rich history of experimental data [1] [3].

The choice between them is not a matter of superiority but of context. For rapid, early-stage screening of a wide excipient landscape with minimal prior data, COSMO-RS holds a distinct advantage. For analyzing solvation processes and partitioning in well-characterized systems, LSER remains an excellent and thermodynamically insightful tool. Future directions point toward the development of hybrid COSMO-LSER equation-of-state models [1], which aim to merge the predictive power of quantum chemistry with the thermodynamic rigor and rich experimental foundation of the LSER framework, promising even more powerful tools for rational formulation design in the years to come.

The accurate prediction of aqueous solubility represents a critical challenge in pharmaceutical development and chemical engineering, directly influencing drug bioavailability, efficacy, and ultimate therapeutic success [4]. Traditional methods for solubility prediction have historically fallen into three categories: empirical, semi-empirical, and theoretical approaches. Empirical methods, while straightforward, rely heavily on extensive experimental data, making them time-consuming, labor-intensive, and costly for high-throughput screening of potential drug candidates [4]. Semi-empirical models like Hansen solubility parameters, UNIFAC, and Quantitative Structure-Property Relationships (QSPR) strike a balance between theory and experiment but often require specific parametrization and may lack generalizability beyond their training domains [4].

Theoretical methods, including molecular dynamics, PC-SAFT, and COSMO-RS (Conductor-like Screening Model for Real Solvents), offer deeper insights into molecular interactions with reduced experimental dependency but demand significant computational resources and may struggle with absolute predictions without experimental calibration [4]. Meanwhile, the Linear Solvation-Energy Relationships (LSER) model, also known as the Abraham solvation parameter model, has established itself as a successful predictive tool across chemical, biomedical, and environmental applications [3]. This model correlates free-energy-related properties of a solute with six molecular descriptors: McGowan's characteristic volume (Vx), the gas-liquid partition coefficient in n-hexadecane (L), excess molar refraction (E), dipolarity/polarizability (S), hydrogen bond acidity (A), and hydrogen bond basicity (B) [3].

The emergence of machine learning (ML) has transformed computational chemistry through data-driven approaches, yet these methods often require substantial training data and can function as "black boxes" with limited interpretability [4] [39]. This comparison guide examines the innovative integration of COSMO-RS-derived descriptors with machine learning algorithms, objectively assessing its performance against established LSER frameworks and other computational approaches while providing detailed experimental methodologies and comparative data.

Theoretical Foundations: LSER and COSMO-RS

The LSER Framework and Thermodynamic Basis

The LSER model operates on well-established linear free-energy relationships that quantify solute transfer between phases through two principal equations. For transfer between condensed phases, the model utilizes:

log(P) = cp + epE + spS + apA + bpB + vpVx

Where P represents partition coefficients such as water-to-organic solvent or alkane-to-polar organic solvent [3]. For gas-to-solvent partitioning, the equation becomes:

log(KS) = ck + ekE + skS + akA + bkB + lkL

The remarkable feature of these relationships is that the coefficients (lowercase letters) function as solvent descriptors representing the complementary effect of the phase on solute-solvent interactions, while the capitalized variables represent solute-specific molecular descriptors [3]. The thermodynamic foundation of LSER's linearity, even for strong specific interactions like hydrogen bonding, has been confirmed through integration with equation-of-state thermodynamics and the statistical thermodynamics of hydrogen bonding [3]. This theoretical underpinning enables the extraction of meaningful thermodynamic information about intermolecular interactions, though the determination of system coefficients remains dependent on experimental data fitting through multiple linear regression [3].

COSMO-RS Fundamentals and Descriptor Generation

COSMO-RS is a quantum chemistry-based thermodynamic model that provides a robust framework for predicting solvation behavior and liquid-phase thermodynamics [4] [15]. The methodology begins with quantum chemical calculations using density functional theory (DFT) to optimize molecular geometry and minimize the total energy of the molecule [4]. The resulting molecular surface is segmented, and DFT calculations determine the charge density distribution across this surface, generating a σ-profile that represents the polarity distribution of the molecule [4].

Within hybrid approaches, COSMO-RS generates conformer-specific descriptors that capture essential molecular interactions. Key descriptors include:

• Averaged correction for the dielectric energy (kcal/mol): Accounts for local polarization effects
• Hydrogen bond donor and acceptor moments: Quantifies the solute's capacity for hydrogen bonding
• Molecular volume: Characterizes steric effects and cavity formation energy [4]

These descriptors are derived from first principles without empirical fitting, providing a theoretically grounded foundation for machine learning models. The COSMO-RS framework excels in capturing a wide range of inter- and intra-molecular interactions, including electrostatic and steric effects of different conformers, making it particularly valuable for quantifying the complex molecular interactions governing solubility [4].

Experimental Protocols and Methodologies

Hybrid COSMO-RS/ML Workflow and Implementation

The integration of COSMO-RS with machine learning follows a structured computational workflow that combines theoretical chemistry with data-driven modeling:

Descriptor Generation Protocol:

• Molecular Input: Compounds are represented as 2D structures or SMILES strings
• Conformational Sampling: Multiple low-energy conformers are generated for each compound
• Quantum Chemical Calculations: DFT geometry optimization is performed using the TZVP basis set and BP functional to minimize total energy with respect to atomic positions
• Surface Charge Calculation: The molecular surface is segmented and electrostatic potentials are computed for each segment
• Descriptor Extraction: Key parameters including dielectric energy correction, HB donor/acceptor moments, and molecular volume are calculated [4]

Machine Learning Implementation:

• Feature Selection: Recursive feature elimination identifies the most predictive COSMO-RS descriptors
• Network Architecture: A fully connected, feed-forward neural network with three hidden layers is constructed
• Training Protocol: The model is trained on a subset of the AquaSol database using experimentally determined solubility values
• Validation: Model performance is assessed through cross-validation and hold-out test sets [4]

This workflow enables the prediction of aqueous solubility without solute-specific experimental data, a significant advantage in early drug discovery stages [4].

LSER Experimental Framework

The experimental implementation of LSER models for solubility prediction follows a distinct methodology based on empirical descriptors and linear regression:

Descriptor Acquisition Protocol:

• Experimental Determination: Key molecular descriptors (E, S, A, B, V, L) are obtained through experimental measurements or validated predictive methods
• Database Mining: Established LSER databases provide pre-calculated descriptors for known compounds
• Coefficient Determination: System-specific coefficients are derived through multiple linear regression of experimental partition data [3]

Model Application:

• Descriptor Selection: Appropriate solute descriptors are selected based on chemical similarity
• Equation Implementation: The relevant LSER equation is applied with predetermined system coefficients
• Solubility Calculation: The partition coefficient is converted to solubility using established thermodynamic relationships [3]

The LSER approach relies heavily on the availability of experimental data for both descriptor determination and model parametrization, limiting its application to compounds with established descriptor sets or strong analogs in existing databases [3].

Performance Comparison and Experimental Data

Quantitative Comparison of Prediction Accuracy

Table 1: Performance Metrics of Solubility Prediction Methods

Methodology	Data Requirements	Computational Cost	Interpretability	Applicability Domain	Reported Accuracy
COSMO-RS/ML Hybrid	Minimal experimental data required	High (DFT calculations)	Moderate (Physically meaningful descriptors)	Broad (First-principles basis)	High predictive power with small datasets [4]
LSER Models	Extensive experimental parametrization	Low (Linear regression)	High (Well-defined descriptors)	Limited to similar compounds	Established performance for known systems [3]
Pure ML Approaches	Large training datasets	Moderate (Model training)	Low (Black-box models)	Dataset-dependent	Excellent with sufficient data [39]
Theoretical Methods	Minimal experimental input	Very High (Molecular dynamics)	High (Mechanistic insight)	Broad but system-dependent	Accurate with proper calibration [4]

The hybrid COSMO-RS/ML approach demonstrates particular strength in scenarios with limited experimental data, achieving high predictive power while requiring only a fraction of the training data needed for traditional machine learning methods [4]. This advantage stems from the physically meaningful, theory-derived descriptors that provide a robust foundation for the machine learning model.

Analysis of Strengths and Limitations

Table 2: Comparative Analysis of Method Characteristics

Attribute	COSMO-RS/ML Hybrid	Traditional LSER	Pure ML Models
Theoretical Basis	Strong quantum chemical foundation	Empirical linear free-energy relationships	Data-driven correlations
Descriptor Origin	First-principles calculations	Experimental measurements	Various sources (including computed)
Training Data Needs	Low to moderate	High for parametrization	Very high
Computational Demand	High (DFT calculations)	Low	Moderate to high
Interpretability	Moderate (Physically meaningful descriptors)	High (Well-defined contributions)	Low (Black-box nature)
Transferability	Broad applicability	Limited to similar systems	Dataset-dependent
Handling of Novel Compounds	Strong (No experimental data needed)	Weak (Requires descriptor estimation)	Moderate (Depends on chemical space coverage)

The hybrid COSMO-RS/ML methodology capitalizes on the structure-property relationships defined by thermodynamic models while leveraging machine learning's capability to model complex, non-linear relationships between molecular descriptors and experimental solubility data [4]. This synergy enhances model interpretability compared to pure machine learning approaches while extending applicability beyond the limitations of LSER methods.

Computational Tools and Software Solutions

Table 3: Essential Research Tools for Hybrid Solubility Prediction

Tool/Software	Function	Application Context
COSMOtherm	Implementation of COSMO-RS theory for descriptor generation	Primary tool for calculating COSMO-RS descriptors in hybrid workflows [4]
Quantum Chemistry Packages	DFT calculations for molecular geometry optimization and σ-profile generation	Preliminary step in COSMO-RS descriptor generation [4]
AquaSol Database	Curated dataset of experimental solubility measurements	Training and validation data for machine learning models [4]
LSER Database	Comprehensive collection of solute descriptors and system coefficients	Reference data for traditional LSER implementations and comparisons [3]
Neural Network Frameworks	Implementation of feed-forward, graph-convolutional, or other network architectures	Machine learning component of hybrid approach for relationship modeling [4] [40]
PSP Framework	Partial Solvation Parameters for thermodynamic information extraction	Bridging LSER databases with equation-of-state developments [3]

These tools collectively enable researchers to implement both hybrid COSMO-RS/ML approaches and traditional LSER methods, facilitating comparative studies and method validation across diverse chemical spaces.

The integration of COSMO-RS descriptors with machine learning algorithms represents a significant advancement in solubility prediction methodology, offering a compelling alternative to established LSER approaches. This hybrid framework successfully merges the theoretical rigor of quantum chemistry-based thermodynamics with the pattern recognition capabilities of modern machine learning, achieving high predictive accuracy even with limited experimental data [4].

While LSER models maintain advantages in interpretability and established performance for systems within their parametrized domain, the COSMO-RS/ML hybrid approach extends predictive capability to novel compounds without requiring experimental input or pre-existing descriptor sets [4] [3]. This characteristic is particularly valuable in early-stage drug discovery where experimental data is scarce and rapid screening of candidate molecules is essential.

Future developments in this field will likely focus on enhancing model interpretability, expanding applicability to diverse chemical systems, and integrating additional thermodynamic constraints to ensure physical meaningfulness of predictions. The ongoing creation of comprehensive databases, such as the recently developed self-solvation energy dataset combining DIPPR and Yaws databases, will further support the training and validation of advanced hybrid models [40]. As these methodologies mature, the convergence of AI/ML capabilities with fundamental thermodynamic principles promises to enable fully automated chemical discovery pipelines, addressing critical challenges across pharmaceutical development, materials science, and energy applications [39].

Overcoming Limitations: Addressing Challenges and Enhancing Prediction Accuracy

In pharmaceutical research and environmental science, determining key molecular properties like partition coefficients, solubility, and reactivity is fundamental. However, experimental data is often scarce due to complex molecular structures, legal regulations surrounding controlled substances, or the sheer cost and time required for laboratory work [17]. This data scarcity creates a critical bottleneck in drug development and environmental risk assessment.

Computational methods have emerged as powerful alternatives to bypass these experimental limitations. Among the most established are the Linear Solvation Energy Relationship (LSER) model and the Conductor-like Screening Model for Real Solvents (COSMO-RS). LSER is a highly successful quantitative structure-property relationship (QSPR) approach that correlates free-energy-related properties of a solute with its molecular descriptors [1] [3]. In contrast, COSMO-RS is a quantum mechanics-based model that uses statistical thermodynamics to predict solvation properties from the results of quantum chemical calculations [1] [19]. This guide provides an objective comparison of these two paradigms, evaluating their performance, underlying protocols, and suitability for tackling data-scarce scenarios in molecular science.

Methodological Comparison: LSER vs. COSMO-RS

The following table outlines the fundamental principles, inputs, and outputs of the LSER and COSMO-RS models.

Table 1: Fundamental Comparison of the LSER and COSMO-RS Approaches

Feature	LSER (Linear Solvation Energy Relationship)	COSMO-RS (Conductor-like Screening Model for Real Solvents)
Theoretical Basis	Empirical linear free-energy relationships [3]	Quantum mechanics and statistical thermodynamics [1]
Primary Inputs	Solute-specific molecular descriptors (Vx, E, S, A, B, L) [1]	Quantum chemical σ-potentials (from COSMO calculations) [19]
Key Outputs	Partition coefficients (log P), solvation free energies, enthalpies [1]	Activity coefficients, vapor pressures, solubility, partition coefficients [19]
Handling of HB	Accounted for via A (acidity) and B (basicity) descriptors in a linear combination [1]	Explicitly calculated from surface segment interactions [1]
Parameterization	Requires experimental data to fit system-specific coefficients [3]	A priori predictive after initial parameterization; no system-specific fitting needed [1]

Quantitative Performance Benchmarking

Extensive comparisons have been conducted to evaluate the predictive accuracy of LSER and COSMO-RS, particularly for hydrogen-bonding (HB) contributions to solvation enthalpy and partition coefficients.

Prediction of Hydrogen-Bonding Contributions

A critical comparison of solvation enthalpy predictions for various solute-solvent systems found that COSMO-RS and LSER show "a rather good agreement in most of the studied systems" [1]. The cases with large discrepancies were further analyzed using equation-of-state calculations, providing a thermodynamic benchmark.

Accuracy in Partition Coefficient and Solubility Prediction

The table below summarizes quantitative performance data for both models and their enhanced variants from recent studies.

Table 2: Quantitative Performance Comparison for Key Properties

Model & Application	Dataset Size	Reported Accuracy / Error	Reference
LSER for LDPE/Water Partitioning	n = 156 compounds	R² = 0.991, RMSE = 0.264	[7]
LSER (Independent Validation)	n = 52 compounds	R² = 0.985, RMSE = 0.352	[7]
Standard COSMO-SAC (Solubility)	1950 data points	71.74% accuracy	[41]
Machine Learning-Enhanced COSMO-SAC	1950 data points	99.28% accuracy	[41]
Quantum Chemical Calculations (Partitioning)	23 drug molecules	High variability, but useful for estimating environmental distribution	[17]

Experimental and Computational Protocols

Standard LSER Protocol

The standard workflow for applying the LSER model involves several well-defined steps, as visualized below.

Detailed Protocol Steps:

• Acquire Solute Descriptors: Obtain the six Abraham solute descriptors for the compound of interest. The McGowan characteristic volume (Vx), excess molar refraction (E), dipolarity/polarizability (S), hydrogen-bond acidity (A), hydrogen-bond basicity (B), and the gas-hexadecane partition coefficient (L) can be retrieved from the freely accessible LSER database or, if unavailable, predicted using QSPR tools [1] [7] [3].
• Select LFER Coefficients: Identify the appropriate solvent- or system-specific LFER coefficients (lower-case letters) for the partition process being modeled. For example, to predict a water-to-organic solvent partition coefficient (log P), the coefficients cp, ep, sp, ap, bp, and vp for that specific organic solvent are required. These are typically determined by multilinear regression of experimental data and are available for many common solvents [1] [3].
• Apply LFER Equation: Insert the descriptors and coefficients into the relevant LFER equation. For partitioning between two condensed phases, the form is: log(P) = cp + epE + spS + apA + bpB + vpVx [3].
• Calculate Property: Compute the result to obtain the predicted property value, such as the partition coefficient.

Standard COSMO-RS Protocol

The COSMO-RS methodology follows a quantum-chemical workflow, as outlined below.

Detailed Protocol Steps:

• Geometry Optimization: Perform a quantum chemical calculation to optimize the molecular geometry of the compound in a perfect conductor (the COSMO state). This is typically done using Density Functional Theory (DFT) with a parameterized continuum solvation model [1] [19].
• COSMO Calculation: Using the optimized geometry, run a single-point COSMO calculation to obtain the screening charge density (σ-surface) on the molecular cavity. The quality of this calculation (e.g., TZVPD-Fine level) is crucial for accuracy [1].
• σ-Profile Generation: Process the σ-surface to generate a histogram of the screening charge densities, known as the σ-profile. This profile is a fingerprint of the molecule's polarity and hydrogen-bonding capability [19].
• Statistical Thermodynamics Calculation: Use the COSMO-RS model to compute the chemical potential by considering the interactions between the σ-profiles of the solute and solvent(s). The residual chemical potential (μi^res) is calculated from the pairwise interactions of surface segments, while a combinatorial term (μi^comb) accounts for molecular size and shape differences [19]. Properties like activity coefficients, solubility, and partition coefficients are then derived from this chemical potential [19].

Successful application of these computational models relies on a suite of software tools and databases.

Table 3: Essential Resources for LSER and COSMO-RS Research

Resource Name	Type	Primary Function	Key Features / Notes
LSER Database [1]	Database	Source of solute descriptors (A, B, Vx, etc.) and system coefficients.	Freely accessible; contains thousands of compounds.
COSMOtherm [1]	Software	Implements the COSMO-RS model for property prediction.	Commercial software; requires prior COSMO files.
ADF COSMO-RS [19]	Software Suite	Integrated platform for quantum chemical calculations and COSMO-RS.	Includes modules for geometry optimization, COSMO, and property calculation.
Turbomole, ORCA	Software	Quantum chemistry programs for generating COSMO files.	Produce the necessary input files for COSMO-RS calculations.
QSPR Prediction Tools [7]	Software/Algorithm	Predicts missing LSER solute descriptors from molecular structure.	Essential when experimental descriptors are unavailable.
Python w/ RDKit [42]	Programming Library	Generates molecular descriptors and fingerprints for ML models.	Open-source; widely used in cheminformatics.

Emerging Frontiers and Hybrid Approaches

To overcome the limitations of both LSER and COSMO-RS, researchers are developing advanced hybrid and machine-learning-enhanced methods.

• Machine Learning Enhancement: A study demonstrated that applying Gaussian Process Regression (GPR) to correct the quantitative predictions of COSMO-SAC (a variant of COSMO-RS) increased its accuracy for solid solubility from 71.74% to 99.28% across 1,950 experimental data points [41]. This highlights the potential of post-processing quantum chemical outputs with AI to achieve quantitative accuracy.
• Quantum Computing for Small Data: Early research into Quantum Reservoir Computing (QRC) has shown it can match or outperform classical machine learning models on small molecular datasets (as few as 100 records) from the Merck Molecular Activity Challenge [43]. This suggests a future niche for quantum machine learning in data-scarce drug discovery scenarios.
• Foundation Models for Tabular Data: The transformer-based TabPFN model has demonstrated strong performance on small to medium-sized tabular datasets, showing clear advantages in regression tasks and robustness under out-of-distribution evaluations common in drug discovery [42]. It provides a promising, data-efficient alternative to traditional gradient-boosted decision trees.

Achieving Thermodynamic Consistency in Self-Solvation and Hydrogen-Bonding Calculations

Accurately predicting hydrogen-bonding (HB) interaction energies and free energies is a fundamental challenge in computational chemistry and molecular thermodynamics, with critical implications for drug design, solvent screening, and materials science. Two prominent methodologies for these predictions are the Linear Solvation Energy Relationship (LSER) model, particularly the Abraham solvation parameter approach, and the quantum mechanics-based COSMO-RS (Conductor-like Screening Model for Real Solvents). While both are powerful predictive tools, they differ significantly in their theoretical foundations, practical implementation, and, most importantly, their ability to handle thermodynamic consistency in self-solvation scenarios.

Thermodynamic consistency requires that when a molecule acts as both solute and solvent (self-solvation), the calculated hydrogen-bonding contribution from its acidic and basic sites should be equivalent. However, traditional LSER models often struggle with this requirement, as their parameters are typically derived from multilinear regression of experimental data without this physical constraint. This comparison guide objectively evaluates the performance of standard LSER against emerging COSMO-RS-enhanced approaches, providing researchers with a clear framework for selecting and implementing these methods.

Theoretical Foundations and Methodological Comparison

Abraham LSER Framework

The Abraham LSER model quantifies solute transfer between phases using linear free-energy relationships. For gas-to-liquid partitioning, the solvation free energy is described by:

[ \log KG = cg + eg E + sg S + ag A + bg B + l_g L ]

Where the capital letters represent solute-specific molecular descriptors: (E) (excess molar refraction), (S) (dipolarity/polarizability), (A) (hydrogen-bond acidity), (B) (hydrogen-bond basicity), and (L) (gas-hexadecane partition coefficient). The lowercase letters are system-specific coefficients reflecting the complementary solvent properties [14] [3]. The hydrogen-bonding contribution to solvation free energy is represented by the sum (ag A + bg B).

A similar equation models solvation enthalpies:

[ \log KE = ce + ee E + se S + ae A + be B + l_e L ]

These equations are powerful due to their simplicity but face limitations. The descriptors and coefficients are obtained empirically, restricting expansion to systems with abundant experimental data [14]. More critically, this structure can lead to thermodynamic inconsistencies, as the products (aA) and (bB) are generally unequal for self-solvation, violating the physical expectation that acid-base and base-acid interactions between identical sites should be equal [3] [44].

COSMO-RS and QC-LSER Approach

COSMO-RS is a quantum chemistry-based method that calculates solvation properties by simulating a molecule in a virtual conductor environment. The model uses the sigma-profile ((\sigma)-profile)—a histogram of molecular surface charge densities—obtained from DFT calculations to compute molecular interactions [14] [8].

Recent research has integrated COSMO-RS with LSER principles to create a quantum chemical LSER (QC-LSER) approach. This hybrid method develops new molecular descriptors from COSMO-RS output to reformulate LSER equations in a thermodynamically consistent way [14] [25]. Key descriptors include:

• Effective HB Acidity ((\alpha)): Product of a quantum-chemical acidity descriptor ((Ah)) and an "availability fraction" ((fA))
• Effective HB Basicity ((\beta)): Product of a quantum-chemical basicity descriptor ((Bh)) and an "availability fraction" ((fB))

These descriptors enable prediction of HB interaction energies using a universal constant:

[ -\Delta E{12}^{hb} = 5.71 \times (\alpha1\beta2 + \beta1\alpha_2) \ \text{kJ/mol} \ \text{at} \ 25^\circ\text{C} ]

This symmetric form ensures thermodynamic consistency, as the interaction energy for self-solvation becomes (2 \times 5.71 \times \alpha\beta), automatically satisfying the equivalence of acid-base and base-acid interactions [25] [44].

Table 1: Comparison of Fundamental Methodological Approaches

Feature	Traditional LSER	COSMO-RS/QC-LSER
Theoretical Basis	Empirical linear free-energy relationships	Quantum chemical calculations + statistical thermodynamics
Descriptor Origin	Multilinear regression of experimental data	Molecular surface charge distributions (sigma profiles)
HB Energy Calculation	Sum of products (aA + bB)	Symmetric form (5.71(\alpha1\beta2 + \beta1\alpha2))
Self-Solvation Consistency	Often violated ((aA \neq bB))	Inherently maintained
Data Dependency	Requires extensive experimental data	Primarily computational, minimal experimental input
Conformational Flexibility	Difficult to incorporate	Accounted for via conformer populations in sigma profiles

Performance Comparison: Accuracy and Thermodynamic Consistency

Self-Solvation and Hydrogen-Bonding Predictions

The most significant performance difference emerges in self-solvation scenarios. In traditional LSER, the regression-derived coefficients lack built-in constraints for self-solvation symmetry. Recent analyses reveal this can lead to physically implausible results where the complementary hydrogen-bonding energies are unequal when solute and solvent become identical [14].

The QC-LSER approach fundamentally resolves this issue. For example, in a study predicting hydrogen-bonding interaction energies for common solvents like water, alcohols, and ketones, the symmetric descriptor form consistently yielded equal acid-base and base-acid contributions for self-solvation, satisfying thermodynamic constraints that traditional LSER frequently violates [25]. This makes the method particularly valuable for equation-of-state developments in molecular thermodynamics, where such consistency is crucial [44].

Partition Coefficient Predictions

Partition coefficient prediction represents a key application for both methods. A comprehensive study on low-density polyethylene/water systems demonstrated traditional LSER's strong predictive capability for 159 diverse compounds (R² = 0.991, RMSE = 0.264) [45] [28]. The LSER model significantly outperformed simple log-linear models, especially for polar compounds with hydrogen-bonding capabilities.

COSMO-RS has shown more variable performance in partition coefficient prediction. In liquid-liquid equilibrium systems containing deep eutectic solvents (DES), COSMO-RS predictions using the TZVPD-FINE parameter set achieved average root mean square deviations (RMSDs) below 10%. However, the model overestimated solute partition coefficients for aromatic and heterocyclic solutes in systems containing alkanes and salt-based DES, while providing more reliable predictions for alcohol solutes in similar systems [8]. This indicates potential limitations in COSMO-RS for certain compound classes.

Table 2: Quantitative Performance Comparison in Selected Applications

Application	Traditional LSER Performance	COSMO-RS/QC-LSER Performance	Key Findings
LDPE/Water Partitioning [45] [28]	R² = 0.991, RMSE = 0.264 (n=156)	Not specifically reported	LSER superior to log-linear models, especially for polar compounds
LLE with Deep Eutectic Solvents [8]	Not reported	Average RMSD <10%, overestimation for aromatics	Better predictions for alcohol solutes; performance varies by solute class
HB Interaction Energies [25]	Subject to self-solvation inconsistency	Predictions close to LSER data but thermodynamically consistent	QC-LSER provides physically meaningful results for self-association
Micellar Liquid Chromatography [46]	Requires multiple experimental data points	Qualitative predictions from single data point	COSMO-RS offers time and cost advantages for screening

Experimental and Computational Protocols

Traditional LSER Implementation Protocol

Step 1: Experimental Data Collection

• Measure partition coefficients (e.g., log P) or solvation free energies for a diverse set of solutes in the system of interest
• Ensure chemical space coverage includes varied molecular sizes, polarities, and hydrogen-bonding capabilities

Step 2: Solute Descriptor Determination

• Obtain Abraham solute descriptors (E, S, A, B, V, L) from the LSER database or experimental measurements
• For new compounds, estimate descriptors using fragment-based methods or quantitative structure-property relationship (QSPR) approaches

Step 3: Regression Analysis

• Perform multilinear regression of experimental data against solute descriptors
• Determine system-specific coefficients (c, e, s, a, b, v, l) and statistical quality metrics (R², RMSE)
• Validate model with test set compounds not included in regression

QC-LSER Implementation Protocol

Step 1: Quantum Chemical Calculations

• Perform DFT geometry optimization and energy calculation for target molecule(s)
• Use appropriate basis sets (e.g., TZVP or TZVPD-FINE in COSMO-RS)
• Generate sigma-profile (surface charge distribution) using COSMO methodology

Step 2: Descriptor Calculation

• Calculate effective HB acidity (α) and basicity (β) descriptors from sigma-profile
• Apply availability fractions (fA, fB) appropriate for the molecular family
• Account for conformational populations if multiple stable conformers exist

Step 3: HB Energy Prediction

• Apply symmetric equation: (-\Delta E{12}^{hb} = 5.71 \times (\alpha1\beta2 + \beta1\alpha_2)) kJ/mol
• For self-solvation, use simplified form: (-\Delta E_{self}^{hb} = 2 \times 5.71 \times \alpha\beta)

Step 4: Integration with Thermodynamic Models

• Incorporate predicted HB energies into equation-of-state models (e.g., SAFT, NRHB)
• Calculate phase equilibria and other thermodynamic properties

The following workflow diagram illustrates the key steps in both methodologies:

Essential Research Reagents and Computational Tools

Successful implementation of these predictive approaches requires specific computational tools and resources:

Table 3: Essential Research Reagents and Computational Tools

Tool/Resource	Function	Implementation Context
LSER Database [14] [3]	Comprehensive source of solute descriptors and system coefficients	Traditional LSER model development and validation
COSMObase [44]	Repository of pre-calculated sigma profiles for thousands of molecules	QC-LSER descriptor determination without new quantum calculations
TURBOMOLE	Quantum chemical software suite for DFT/COSMO calculations	QC-LSER sigma profile generation for novel compounds
BIOVIA MATERIALS STUDIO (DMol3)	Computational chemistry software with COSMO-RS implementation	Sigma profile generation and property prediction
SCM Suite (AMS)	Software platform with COSMO-RS module	Professional implementation of COSMO-RS calculations
Abraham Descriptor Estimation Tools	Fragment-based methods for predicting solute descriptors	LSER model application to compounds not in database

Based on comparative analysis, the emerging QC-LSER approach demonstrates superior performance for thermodynamically consistent predictions, particularly in self-solvation scenarios and hydrogen-bonding calculations. Its quantum chemical foundation provides a more physically realistic representation of molecular interactions without requiring extensive experimental data for parameterization.

However, traditional LSER remains a highly robust and accurate method for partition coefficient prediction in well-characterized systems, as evidenced by its exceptional performance in LDPE/water partitioning studies.

Recommendations for researchers and drug development professionals:

• For novel compound screening or systems with limited experimental data, prioritize the QC-LSER approach to leverage its predictive capability without experimental input
• For thermodynamic modeling and equation-of-state development, adopt QC-LSER to ensure consistency in hydrogen-bonding treatment, especially for self-associating components
• When highly accurate partition coefficients are needed for systems with abundant experimental data, traditional LSER may provide superior numerical accuracy
• In pharmaceutical applications involving complex multi-functional molecules, implement QC-LSER to better account for conformational flexibility effects on hydrogen-bonding

The integration of quantum chemical calculations with the LSER framework represents a promising direction for molecular thermodynamics, potentially combining the physical rigor of COSMO-RS with the practical utility and interpretability of LSER models.

The accurate prediction of a drug molecule's behavior in biological systems is a cornerstone of modern pharmaceutical development. A significant challenge in this endeavor lies in accounting for the dynamic nature of molecules, which exist not as single rigid structures but as ensembles of interconverting conformational stereoisomers—three-dimensional shapes that can transition from one to another without breaking covalent bonds [47]. The population of these conformers, and thus the molecule's overall properties, is highly dependent on its environment, whether in aqueous solution, a non-polar solvent, or the binding pocket of a protein [48] [49]. This conformational dependence directly influences intramolecular reorganization energy (ΔEReorg), the energy cost a molecule incurs to shift from its solution-phase conformational ensemble to its bioactive shape [48]. Accurately predicting properties like solubility, membrane permeability, and protein binding therefore requires computational models that can reliably capture the intricacies of intramolecular bonding and this conformational flexibility. This article objectively compares the capabilities of two established predictive approaches—the Linear Solvation-Energy Relationships (LSER) model and the COnductor like Screening MOdel for Realistic Solvents (COSMO-RS)—in addressing this critical challenge.

Comparative Model Frameworks: LSER vs. COSMO-RS

The LSER (or Abraham model) and COSMO-RS approaches are founded on distinct philosophical and theoretical principles, leading to different strengths and limitations in handling complex drug molecules.

• Linear Solvation-Energy Relationships (LSER): The LSER model is a largely empirical approach that correlates a solute's free-energy-related properties with a set of six pre-determined molecular descriptors [3]. These descriptors are:

  • Vx: McGowan’s characteristic volume.
  • L: The gas–hexadecane partition coefficient at 298 K.
  • E: Excess molar refraction.
  • S: Dipolarity/polarizability.
  • A: Hydrogen bond acidity.
  • B: Hydrogen bond basicity. The model operates through linear equations (e.g., log(P) = c_p + e_pE + s_pS + a_pA + b_pB + v_pVx) where the system-specific coefficients (e.g., s_p, a_p) are obtained by fitting to extensive experimental data [3]. While powerful, this reliance on experimental parametrization can be a limitation for novel or highly complex molecules whose descriptors may not be well-represented in the training database.

• COSMO-RS: In contrast, COSMO-RS is a more fundamental quantum mechanics-based method. It begins with a quantum chemical calculation of the individual molecule in a virtual conductor environment to generate a sigma-profile—a histogram that represents the polarity distribution on the molecule's surface [50] [51]. The thermodynamic properties of mixtures are then predicted statistically from the pairwise interactions of these surface segments [50]. This "bottom-up" approach requires no prior experimental data for the specific compound, allowing it to be applied to hypothetical molecules or those with unusual functional groups [50]. Recent advancements have explicitly improved its handling of multi-species components, including conformers, and introduced methods like SG1 for fast sigma-profile prediction from molecular structure [50].

Table 1: Fundamental Comparison Between LSER and COSMO-RS Models.

Feature	LSER / Abraham Model	COSMO-RS
Theoretical Basis	Empirical linear free-energy relationships	Quantum chemistry and statistical thermodynamics
Primary Input	Six experimental or QSPR-derived solute descriptors (E, S, A, B, Vx, L) [3]	Quantum chemical COSMO file or predicted sigma-profile [50] [51]
Handling of Conformers	Implicitly averaged within the descriptor values; no explicit handling	Explicit treatment of multiple conformers is possible as multi-species components [50]
Parametrization	System coefficients fitted to large experimental databases	General parameters not specific to chemical groups [50]
Key Output	Partition coefficients (e.g., log P, log K) and free-energy-related properties [3]	Activity coefficients, solubilities, vapor pressures, and other thermodynamic properties [50]

Explicit Handling of Conformational Dependence

The ability to explicitly account for a molecule's conformational ensemble is crucial for accurate property prediction. Molecular dynamics (MD) studies have shown that the "unbound state" of a drug in solution is an ensemble of conformations, and representing it by a single global energy minimum from a vacuum or implicit solvation model can be misleading due to "conformational collapse" and spurious intramolecular interactions [48].

• COSMO-RS's Explicit Conformer Handling: COSMO-RS has evolved to directly address conformational complexity. Since its 2020.1 release, it has supported calculations for compounds with multi-species components, which explicitly includes conformers, dimers, and other associated species [50]. This allows researchers to input multiple pre-optimized conformer structures (e.g., from a conformational search or MD simulation) into a COSMO-RS calculation. The model then considers the weighted contribution of each conformer's sigma-profile to the overall thermodynamic property, providing a more realistic picture of the molecule's behavior in solution. This capability is vital because, as research on biopolymers shows, the conformational collection of a molecule confers multi-functionality and adaptability [47].

• LSER's Implicit Averaging: The standard LSER model does not explicitly handle multiple conformers. The solute descriptors (A, B, S, etc.) are single-value properties that inherently represent a Boltzmann average over the molecule's accessible conformational states in a given context. While this simplification makes the model easy to apply, it can become a limitation for highly flexible molecules where different conformers have significantly different hydrogen-bonding capacities or polarities, as the single set of descriptors may not adequately capture this complex behavior.

The following diagram illustrates the fundamental workflow difference between the two models when faced with a conformationally flexible drug molecule.

Diagram 1: Workflow comparison for handling flexible molecules. COSMO-RS can explicitly use multiple conformers, while LSER relies on pre-averaged descriptors.

Performance and Experimental Validation Data

Independent validation studies are essential for assessing the real-world performance of predictive models. The data consistently show that while both models have utility, their accuracy can vary significantly depending on the system and the properties being predicted.

• Partition Coefficient Prediction: A 2014 validation study compared COSMOtherm (a commercial implementation of COSMO-RS) and ABSOLV (which uses LSER-like descriptors) for predicting partition coefficients of complex environmental contaminants, including pesticides and flame retardants. The study found that the overall prediction accuracy of COSMOtherm and ABSOLV was comparable, with root mean squared errors (RMSE) for liquid/liquid partition coefficients ranging from 0.64 to 0.95 log units for both [29]. In contrast, the SPARC model performed substantially worse. This suggests that for this class of compounds, both core methodologies are robust.

• Limitations in Complex Systems: However, a 2024 evaluation of COSMO-RS for predicting liquid–liquid equilibrium (LLE) in systems containing deep eutectic solvents (DESs) highlighted specific limitations. While COSMO-RS predicted LLE with average RMSDs below 10%, it largely overestimated the solute partition coefficients for aromatic and heterocyclic solutes in systems containing alkanes and salt-based DESs [8]. This indicates that despite its theoretical strengths, challenges remain in accurately modeling all types of intermolecular interactions, particularly in complex, multi-component systems involving ions.

• Reorganization Energy Insights: Molecular Dynamics simulations provide a benchmark for understanding conformational penalties. One study simulated 26 drug-like compounds in both bound and unbound states, finding that for a majority (18 out of 26), the intramolecular reorganization enthalpy (ΔHReorg) was a modest ≤6 kcal/mol. However, for three particularly polar compounds, this value was much larger (15–20 kcal/mol), indicating that the energy penalty for conformational rearrangement upon binding can be substantial and is tied to the redistribution of electrostatic interactions [48]. This underscores the importance of models that can capture this conformational dependence, an area where COSMO-RS's explicit approach holds an advantage.

Table 2: Summary of Experimental Validation Findings for LSER and COSMO-RS.

Study Focus	LSER/ABSOLV Performance	COSMO-RS/COSMOtherm Performance	Key Implication
Partition Coefficients (Pesticides, Flame Retardants) [29]	RMSE: 0.64 - 0.95 log units (comparable to COSMOtherm)	RMSE: 0.65 - 0.93 log units (comparable to ABSOLV)	Both methods offer similar and reliable accuracy for this application.
Liquid-Liquid Equilibrium (Deep Eutectic Solvents) [8]	Not tested in this study.	Average RMSD < 10%; poor accuracy for solute partition coefficients in specific systems.	COSMO-RS has limitations for ionic systems and specific solute types.
Underpinning for Conformational Effects [48]	Relies on averaged descriptors, missing conformer-specific details.	Can explicitly incorporate conformer ensembles from MD or other sampling.	Explicit conformational sampling is needed to accurately estimate reorganization energy.

Research Reagent Solutions: A Technical Toolkit

Successful application of these models, particularly for probing conformational dependence, relies on a suite of computational and experimental "reagent" tools.

Table 3: Essential Research Tools for Conformational Analysis and Prediction.

Tool Category / "Reagent"	Specific Examples	Function in Research
Conformational Sampling	Molecular Dynamics (MD) in explicit solvent [48]; Ad hoc sampling algorithms (e.g., in MacroModel, MOE) [48]	Generates an ensemble of 3D structures representing the flexible molecule's possible shapes in a given environment.
Quantum Chemical Engine	ADF, TURBOMOLE, ORCA	Performs the initial COSMO calculation to generate the sigma-profile for a given molecular structure or conformer [50].
Prediction Software	COSMOtherm, ADF COSMO-RS (crs), pyCRS [50]	Implements the COSMO-RS theory to calculate thermodynamic properties from sigma-profiles.
Descriptor Prediction (QSPR)	ABSOLV, EpiSuite, SPARC	Predicts LSER molecular descriptors (A, B, Vx, etc.) from molecular structure when experimental data is lacking [17] [29].
Experimental Validation Data	Partition coefficients (log KOW, log KOA); Solubility data; NMR spectroscopy [49] [29]	Provides the essential benchmark data for validating and refining computational predictions.

In the critical task of handling complex drug molecules with significant intramolecular bonding and conformational dependence, both LSER and COSMO-RS offer valuable but distinct pathways. The LSER model provides a fast, empirically grounded method that performs reliably for many partition coefficient predictions, but its reliance on averaged descriptors may obscure the effects of specific conformers. COSMO-RS, with its foundation in quantum mechanics and, crucially, its evolving capacity to explicitly handle multi-conformer systems, provides a more fundamental approach that is less dependent on existing experimental data. This makes it particularly suited for novel drug candidates with unusual structures.

The choice between models ultimately depends on the research goal. For high-throughput screening of properties well-represented in existing databases, LSER's speed and simplicity are advantageous. For deeper investigations into solvation mechanisms, conformational pre-organization, and the behavior of truly novel chemotypes, COSMO-RS's explicit and physically detailed methodology is increasingly the more powerful and insightful tool. Future developments will likely see further integration of explicit conformational sampling via MD with COSMO-RS calculations, leading to even more accurate predictions of drug behavior in complex biological environments.

Parameter Optimization and the Role of New QC-LSER Descriptors

Linear Solvation Energy Relationship (LSER) models, particularly the Abraham model, stand as one of the most successful predictive tools in thermodynamics for a remarkably broad range of applications, from environmental chemistry to pharmaceutical development [1] [14]. These models correlate a solute's free-energy-related properties with its molecular descriptors via simple linear equations, allowing for the prediction of key properties like partition coefficients and solvation enthalpies [3]. However, traditional LSER parameters are typically determined by multilinear regression of experimental data, restricting model expansion to areas with abundant experimental data and sometimes leading to thermodynamic inconsistencies, especially for self-solvation of hydrogen-bonded compounds [14].

The need for reliable, a priori prediction in the absence of experimental data, especially for novel drug molecules, has driven research towards reforming LSER models using quantum chemical (QC) calculations. This guide compares this emerging QC-LSER paradigm with the established, quantum-mechanics-based COSMO-RS (Conductor-like Screening Model for Real Solvents) approach, evaluating their performance in predicting solvation thermodynamics and their applicability in drug development.

Theoretical Foundations and Comparative Framework

The Traditional LSER Model

The foundational Abraham LSER model uses two primary equations for solute partitioning. For gas-to-solvent partitioning: log(K*) = ck + ekE + skS + akA + bkB + lkL [1] For water-to-organic solvent partitioning: log(P) = cp + epE + spS + apA + bpB + vpVx [1] [3] Here, the uppercase letters (Vx, L, E, S, A, B) are solute-specific molecular descriptors representing McGowan’s characteristic volume, the gas-hexadecane partition coefficient, excess molar refraction, dipolarity/polarizability, hydrogen-bond acidity, and hydrogen-bond basicity, respectively. The lowercase letters are complementary solvent-specific coefficients obtained by regression against experimental data [1] [3]. A similar equation is used for solvation enthalpies [3].

The COSMO-RS Model

COSMO-RS is a leading a priori predictive method for solvation free energies [1]. It combines quantum chemical calculations with statistical thermodynamics, using the distribution of molecular surface charges (sigma profiles) obtained from DFT/COSMO calculations to compute solvation properties [14]. Its key advantage is that it does not require experimental parameterization for new molecules, making it highly versatile for predictive screening.

The Emergent QC-LSER Hybrid

The QC-LSER framework seeks to retain the simple linear form of the traditional LSER model but derives its molecular descriptors directly from quantum chemical calculations, overcoming the dependency on experimental data [14]. This involves calculating new descriptors for electrostatic interactions from the molecular surface charge distributions, similar to those used in COSMO-RS [14]. This reformulation aims to be thermodynamically consistent and inherently predictive.

Table 1: Core Characteristics of the Three Modeling Approaches

Feature	Traditional LSER	COSMO-RS	QC-LSER
Theoretical Basis	Empirical linear free-energy relationships	Quantum chemistry & statistical thermodynamics	Hybrid: QC-derived descriptors in LFER framework
Descriptor Source	Experimentally determined via regression	Implicit in sigma surface charge distributions	Calculated from QC (e.g., COSMO) surface charges
Primary Output	Partition coefficients, solvation free energies/enthalpies	Solvation free energies, activity coefficients, phase equilibria	Solvation free energies, enthalpies, and entropies
Handling of HB	Via A and B descriptors	Integrated in surface interaction potentials	Via reformulated A and B descriptors from QC
Key Strength	Simplicity, robustness, wide applicability	A priori predictive for diverse molecules	A priori predictive with thermodynamically consistent formalism
Key Limitation	Limited to experimentally parameterized space	Cannot easily separate HB contribution to free energy	Still under development; performance benchmarking ongoing

Performance Comparison: Predictive Accuracy and Applications

Prediction of Hydrogen-Bonding Contributions

A critical comparison of solvation enthalpy predictions reveals the strengths of each model. Studies have shown a "rather good agreement" between COSMO-RS predictions of the hydrogen-bonding (HB) contribution to solvation enthalpy and those from LSER models for most solute-solvent systems [1]. This agreement is significant because it suggests that the predictive, quantum-based COSMO-RS can reliably estimate a quantity that traditional LSER obtains from regression.

The QC-LSER approach advances this further by using QC-derived descriptors to directly calculate HB free energies, enthalpies, and entropies, addressing a key drawback of COSMO-RS, which cannot easily separate the HB contribution to the solvation free energy [1] [14]. This explicit calculation of HB thermodynamics is a distinct advantage of the new QC-LSER formalism.

Performance in Partition Coefficient Prediction

Partition coefficients are vital for predicting drug distribution. A benchmark study on predicting coefficients between low-density polyethylene (LDPE) and water demonstrated the high precision of a rigorously parameterized LSER model (R² = 0.991, RMSE = 0.264) [7]. When LSER solute descriptors were predicted using a QSPR tool instead of obtained experimentally, the model still performed excellently (R² = 0.984, RMSE = 0.511), showcasing the potential of predictive descriptors [7].

COSMO-RS is also a powerful tool for such partitioning calculations [17]. A study on drug molecule partitioning in the environment calculated partition coefficients like logKOW using quantum mechanical methods, which are the foundation of COSMO-RS, highlighting their utility for complex molecules where experimental data is scarce [17].

Table 2: Comparison of Model Performance in Key Prediction Tasks

Prediction Task	Traditional LSER Performance	COSMO-RS Performance	QC-LSER Performance
Solvation Enthalpy (HB contribution)	Obtained via regression of experimental data (aHA + bHB) [3].	Good agreement with LSER in most systems; used as a predictive tool for this property [1].	Directly calculates HB energies from QC descriptors [14].
Partition Coefficients (e.g., logK)	High accuracy when experimental descriptors are used (e.g., R² = 0.991 for LDPE/water) [7].	A priori predictive capability; widely used for solvation and partition calculations [17].	Inherently predictive; performance relies on accuracy of new QC descriptors [14].
Pharmaceutical Solubility	Can be linked to solubility via solvation free energy [14].	Can be used indirectly. PC-SAFT (a similar advanced EoS) shows HB is critical for accurate solubility parameter prediction [52].	Aims to provide a consistent pathway to activity coefficients and solubility [14].
Handling of Novel Molecules	Limited; requires experimental data for descriptor determination.	Excellent; requires only the molecular structure.	Excellent in principle; requires only the molecular structure.

Experimental and Computational Protocols

The transition to QC-based models requires a shift from wet-lab experiments to computational protocols. Below is a detailed workflow for generating and validating a QC-LSER model, which also illustrates the standard procedure for a COSMO-RS calculation.

Diagram 1: Computational model development workflow.

Protocol 1: Quantum Chemical Calculation (Shared Foundation)

This initial step is common to both COSMO-RS and QC-LSER.

• Objective: To obtain the molecular geometry and the screening charge density distribution (sigma surface) on a virtual conductor.
• Methodology:
  • Input: 3D molecular structure in a standard format.
  • Software: Use quantum chemical suites like TURBOMOLE, Gaussian, or ORCA.
  • Procedure:
    • Perform a geometry optimization to find the molecule's energy-minimized structure.
    • Execute a single-point energy calculation with a COSMO solvation model to generate the sigma surface. The recommended parameterization for COSMO-RS is the TZVPD-Fine basis set level [1].
  • Output: The COSMO file, which contains the geometry and the sigma surface information.

Protocol 2: COSMO-RS Property Prediction

• Objective: To predict solvation properties and activity coefficients at infinite dilution.
• Methodology:
  • Input: COSMO files for the solute and solvent.
  • Software: Commercial software like COSMOtherm or open-source alternatives.
  • Procedure:
    • The software processes the sigma profiles and performs statistical thermodynamics calculations of the chemical potentials.
    • The hydrogen-bonding contribution to solvation enthalpy can be extracted by analyzing the interaction energy terms [1].
  • Validation: Compare predictions with experimental data for solvation free energies, partition coefficients, or activity coefficients.

Protocol 3: QC-LSER Model Parameterization

• Objective: To derive a linear model with quantum chemical descriptors for solvation properties.
• Methodology:
  • Input: COSMO files for a training set of molecules with known experimental solvation properties (e.g., from the LSER database).
  • Descriptor Calculation: Define and calculate new molecular descriptors from the sigma profiles. These descriptors are designed to capture hydrogen-bonding acidity/basicity, polarity, and dispersion interactions in a thermodynamically consistent way [14].
  • Regression: Perform multilinear regression to determine the coefficients of the LSER-type equation that relates the new QC descriptors to the target property (e.g., log K or ΔH).
  • Validation: Test the parameterized model on a separate set of validation molecules not included in the training set [14].

Table 3: Key Computational Tools and Databases for LSER and QC-Based Modeling

Tool / Resource	Type	Primary Function	Relevance to Model Development
Abraham LSER Database [14]	Database	Freely accessible repository of experimental solute descriptors and system coefficients.	Essential for training and validating new models; the source of experimental benchmarks.
COSMOtherm [1]	Software	Implements the COSMO-RS model for property prediction.	Used for a priori predictions and as a source of sigma profiles for QC-LSER descriptor calculation.
TURBOMOLE / Gaussian	Software Suite	Quantum chemical calculation packages.	Used to perform the foundational geometry optimizations and COSMO calculations.
PC-SAFT Equation of State [52]	Thermodynamic Model	Physics-based EoS for complex fluids.	Not an LSER method, but a benchmark for solubility parameter prediction where hydrogen-bonding is critical.
Experimental Solubility Data [53]	Database	Curated experimental measurements of solubility.	Critical for validating the final predictive output of models in pharmaceutical contexts.

The comparative analysis indicates that the evolution of LSER models through the integration of quantum chemical descriptors represents a significant step forward. While the traditional LSER model remains a robust and highly accurate tool for systems within its parameterized domain, its dependency on experimental data is a major limitation. COSMO-RS overcomes this by being fully a priori predictive and shows strong agreement with LSER for key properties like HB enthalpy.

The emerging QC-LSER framework aims to combine the best of both worlds: the simplicity, linearity, and thermodynamic interpretability of the LSER formalism with the predictive power of quantum chemistry. By deriving molecular descriptors directly from sigma surfaces, it seeks to create a thermodynamically consistent model that is both predictive and insightful, particularly for hydrogen-bonding interactions. For researchers in drug development, where novel compounds are the norm, the shift towards these predictive, QC-based methods offers a powerful path to accelerate solvent selection, formulation design, and environmental impact assessment.

Benchmarking Performance: Model Validation, Cross-Comparison, and Real-World Case Studies

Within computational chemistry and pharmaceutical development, the accurate prediction of solvation thermodynamics is a critical task for optimizing drug solubility, permeability, and stability. Two prominent theoretical frameworks used for such predictions are the Linear Solvation Energy Relationship (LSER) model and the Conductor-like Screening Model for Real Solvents (COSMO-RS). This guide provides an objective comparison of their performance against experimental data, focusing on solvation enthalpies and solubility measurements, to inform researchers and drug development professionals about their respective strengths and limitations.

Model Fundamentals and Methodologies

Linear Solvation Energy Relationships (LSER)

The LSER model, also known as the Abraham model, correlates free-energy-related properties of a solute with a set of six empirically determined molecular descriptors [3]. The two primary equations for solute transfer are:

• For partition coefficients between two condensed phases (P): log (P) = cp + epE + spS + apA + bpB + vpVx [3]
• For gas-to-solvent partition coefficients (KS): log (KS) = ck + ekE + skS + akA + bkB + lkL [3]

Experimental Protocol for LSER Validation: The typical methodology involves:

• Descriptor Determination: The solute descriptors (Vx, L, E, S, A, B) are determined experimentally for a wide set of chemically diverse compounds. These correspond to McGowan’s characteristic volume, the gas–hexadecane partition coefficient, excess molar refraction, dipolarity/polarizability, hydrogen bond acidity, and hydrogen bond basicity, respectively [3] [28].
• Model Calibration: System-specific coefficients (e.g., ep, sp, ap, bp, vp) are determined via multiple linear regression of experimental partition coefficient or solvation enthalpy data against the solute descriptors [3] [28].
• Validation: The calibrated model is applied to an independent validation set of compounds. Predictive performance is assessed by comparing calculated values to experimental data, using statistics such as the coefficient of determination (R²) and root mean square error (RMSE) [7].

COSMO-RS (Conductor-like Screening Model for Real Solvents)

COSMO-RS is a quantum chemistry-based equilibrium thermodynamics method that predicts chemical potentials in liquids without the need for system-specific adjustments or experimental parameters [54]. It incorporates quantum chemical effects like group-group interactions, mesomeric effects, and inductive effects.

Experimental Protocol for COSMO-RS Validation:

• Quantum Chemical Calculation: A quantum chemical continuum solvation calculation (typically using Density Functional Theory) is performed for each molecule to compute the polarization charge densities (σ-profiles) on the molecular surface [54].
• Statistical Thermodynamics: The method then uses statistical thermodynamics to calculate thermodynamic properties—such as solvation free energy, activity coefficients, and solubility—by modeling the interactions between surface segments of the molecules involved [54].
• Blind Challenge Validation: The model's predictive accuracy is often tested in blind challenges, such as the Statistical Assessment of the Modeling of Proteins and Ligands (SAMPL), where predictors calculate properties for a set of molecules before experimental data is released [55].

Performance Benchmarking: Quantitative Data

The following tables summarize key quantitative findings from recent validation studies for both models.

Table 1: Performance in Predicting Partition Coefficients

Model	System / Challenge	Dataset Size (n)	Key Performance Metrics	Reference
LSER	Low-Density Polyethylene (LDPE) / Water	156 (Calibration)	R² = 0.991, RMSE = 0.264	[28]
LSER	LDPE / Water (Independent Validation)	52	R² = 0.985, RMSE = 0.352	[7]
COSMO-RS	SAMPL7 (1-Octanol/Water logP)	22	Mean Absolute Error (MAE) = 0.57, RMSE = 0.78	[55]
COSMO-RS	SAMPL5 (Cyclohexane/Water logD)	53	Most accurate submission in the challenge	[15]

Table 2: Performance in Predicting Enthalpies of Solvation

Model	System / Data Scope	Dataset Size (n)	Key Performance Metrics	Reference
LSER	General ΔHsolv (Eq. 3)	Various solvents	Formalism exists: ΔHS = cH + eHE + sHS + aHA + bHB + lHL	[3]
QSPR/GRNN (Inspired by LSER)	Extensive ΔHsolv across 68 solvents	6106 (3082 training, 3024 test)	Test Set: R = 0.943, RMSE = 6.088 kJ/mol	[56]

Critical Analysis of Model Strengths and Limitations

LSER Model Assessment

• Strengths:
  • High Accuracy for Trained Systems: When calibrated with high-quality experimental data for a specific system (e.g., LDPE/water partitioning), LSER models can achieve exceptional accuracy and precision, as evidenced by R² values over 0.99 [7] [28].
  • Interpretability: The model's parameters have clear physicochemical meanings, providing insight into the relative contributions of cavity formation, dispersion, polarity, and hydrogen bonding to the solvation process [3].
• Limitations:
  • Dependency on Experimental Descriptors: The model's predictive capability for new compounds relies on the availability of experimental solute descriptors. While these can be predicted in silico, this introduces another layer of uncertainty, which can increase the prediction error (e.g., RMSE increased from 0.352 to 0.511 when predicted descriptors were used) [7].
  • Limited Solvent Descriptors: The system-specific coefficients (e.g., a, b, v) are only available for solvents with extensive existing experimental data, limiting its application to novel solvent systems [3].

COSMO-RS Model Assessment

• Strengths:
  • A Priori Prediction: As a primarily first-principles method, COSMO-RS requires only the molecular structure as input, making it highly valuable in the early stages of drug discovery where experimental data is scarce [54].
  • Broad Applicability: It demonstrates "one of the best in class reliability and accuracy among the physical methods" across a wide range of applications, including partition coefficients, solubility prediction, and cocrystal screening [55] [54].
• Limitations:
  • Neglect of Crystalline State: For solubility and cocrystal predictions, a significant limitation is that COSMO-RS "ignores crystal packing effects" and relies on data for the supercooled liquid phase [54].
  • Inaccuracies in Complex Systems: The model can struggle with systems containing ions or unusual functional groups. For instance, it has been shown to "largely overestimate the solute partition coefficients for aromatic and heterocyclic solutes" in systems containing alkanes and salt-based deep eutectic solvents [8].

The Scientist's Toolkit: Essential Research Reagents and Solutions

Table 3: Key Computational and Experimental Resources

Item	Function in Validation	Relevance
Abraham Solute Descriptors	Empirical parameters quantifying key molecular interactions for LSER models.	Essential input for calibrating and testing LSER predictions [3] [28].
COSMOtherm Software	A commercial implementation of the COSMO-RS model.	Used to perform COSMO-RS calculations for properties like logP and solubility [55].
FreeSolv Database	A curated database of experimental and calculated hydration free energies of neutral compounds.	A key benchmark for validating solvation free energy predictions [57].
SAMPL Blind Challenges	Community-wide exercises for blind prediction of physicochemical properties.	Provides a rigorous, objective benchmark for comparing the performance of various predictive models, including COSMO-RS and LSER-inspired methods [55] [57].

Both LSER and COSMO-RS offer powerful, yet distinct, approaches for predicting solvation thermodynamics. The choice between them depends heavily on the specific research context.

• For systems where extensive, high-quality experimental data exists to calibrate the model, LSER provides an exceptionally accurate and interpretable framework for prediction and mechanistic insight. Its performance in well-defined systems like LDPE/water partitioning is outstanding.
• For exploratory research involving novel molecules or solvents where experimental data is lacking, COSMO-RS provides a robust, first-principles tool with broad applicability. Its strong performance in blind challenges confirms its predictive reliability for drug-like molecules.

A promising avenue for future research lies in the interconnection of these models, such as using LSER's vast database to inform and validate equation-of-state based thermodynamic models, potentially leading to a new generation of predictive tools that leverage the strengths of both approaches [3].

Conceptual Workflows

Direct Comparison of Hydrogen-Bonding Contributions from LSER, COSMO-RS, and LFHB-EoS

Predicting the strength and influence of hydrogen bonds is fundamental to understanding solubility, partitioning, and reactivity across the chemical and pharmaceutical sciences. Hydrogen bonding (HB), a strong, directional intermolecular interaction, profoundly influences a molecule's behavior in different environments. Accurate quantification of its contribution to thermodynamic properties remains an active area of research. This guide provides an objective comparison of three prominent thermodynamic methods used to model hydrogen bonding: Linear Solvation Energy Relationships (LSER), COnductor-like Screening Model for Real Solvents (COSMO-RS), and Lattice Fluid Hydrogen Bonding Equation of State (LFHB-EoS).

Framed within a broader thesis comparing LSER models with COSMO-RS predictions, this article examines how each approach conceptualizes, parameterizes, and computes hydrogen-bonding contributions. We summarize their theoretical foundations, detail their experimental or computational protocols, and visualize their workflows to provide researchers with a clear basis for selecting the appropriate tool for their specific application, particularly in drug development.

Theoretical Foundations and Hydrogen-Bonding Formulations

Each method rests on a distinct theoretical framework for describing hydrogen-bonding interactions.

LSER (Abraham Model)

The LSER model, also known as the Abraham model, is a semi-empirical approach that correlates free-energy-related properties with a set of six predetermined molecular descriptors [3]. Its two primary equations for solute transfer are: log(P) = cp + epE + spS + apA + bpB + vpVx (for partitioned phases) log(KS) = ck + ekE + skS + akA + bkB + lkL (for gas-to-solvent partitioning)

Hydrogen bonding is captured explicitly through two solute-specific descriptors: A (overall hydrogen-bond acidity) and B (overall hydrogen-bond basicity). The complementary solvent-phase characteristics are represented by the system coefficients a and b, which are determined by fitting to experimental data [3]. The hydrogen-bonding contribution to the free energy of solvation is derived from the products A1a2 and B1b2 for acid-base pairs.

COSMO-RS

COSMO-RS is a quantum-chemistry-based statistical thermodynamics method. It starts with a quantum chemical calculation where the solute molecule is embedded in a perfect conductor, producing a detailed screening charge density (σ) on the molecular surface [58] [10]. The hydrogen-bonding energy between two surface segments is calculated as: E_hb(σ) = c_hb(T) * max[0, σ_acc - σ_hb] * min[0, σ_don + σ_hb]

Here, σ_acc and σ_don are the screening charge densities of the hydrogen-bond acceptor and donor, respectively. The threshold σ_hb and the temperature-dependent prefactor c_hb(T) are adjustable parameters [58] [10]. This formulation directly links hydrogen-bonding strength to the underlying electronic structure of the molecules.

LFHB-EoS

The Lattice Fluid Hydrogen Bonding Equation of State (LFHB-EoS) combines a physical contribution from a nonrandom lattice fluid model with a chemical contribution from hydrogen bonding, using Veytzman statistics [59]. The hydrogen-bonding contribution to the partition function is expressed in terms of the number of hydrogen bonds between donor (k) and acceptor (l) groups, N_kl^HB. The model is characterized by hydrogen-bonding internal energy (U_ij^HB) and entropy (S_ij^HB) parameters for specific group interactions. For example, for 1-alkanols, typical values are U_ij^HB = -25.1 kJ/mol and S_ij^HB = -26.5 J/(mol·K) [59].

Methodological Protocols and Workflows

The practical application of each model involves a distinct sequence of steps.

LSER Experimental Protocol

The application of the LSER model relies heavily on empirical data and correlation [3].

• Data Compilation: Gather a large set of experimental partition coefficient (P or K_S) data for many diverse solutes in the solvent system of interest.
• Descriptor Assignment: Assign the six Abraham descriptors (V_x, L, E, S, A, B) to every solute and solvent in the dataset. These are often obtained from experimental measurements or from prior compilations.
• Multiple Linear Regression: Perform a multi-parameter linear regression of the experimental log(P) or log(K_S) data against the solute descriptors to determine the system-specific coefficients (c, e, s, a, b, v or l).
• Prediction: For a new solute, use its known descriptors and the regressed coefficients to predict its partitioning behavior in the characterized solvent system.

COSMO-RS Computational Protocol

The COSMO-RS workflow is a first-principles computational approach, though it involves parameterized post-processing [58] [10].

• Quantum Chemical Calculation: Perform a COSMO calculation for each molecule of interest using a quantum chemistry code (e.g., ADF, Gaussian). This yields a .cosmo file containing the surface charge density (σ) distribution.
• σ-Profile Generation: Process the COSMO output to generate the molecule's σ-profile, p(σ), which is a histogram of the amount of surface area having a particular charge density [60].
• σ-Potential Calculation: For a solvent or mixture, calculate the σ-potential, μ_s(σ), which represents the chemical potential of a surface segment with charge density σ in the solvent. This is done by solving an integral equation iteratively [60] [10].
• Property Calculation: Calculate the chemical potential of a solute in the solvent by integrating its σ-profile with the solvent's σ-potential. From this, derive thermodynamic properties like activity coefficients, solubilities, and partition coefficients.

LFHB-EoS Correlation Protocol

LFHB-EoS is typically used to correlate and predict phase equilibria for associating fluids [59].

• Pure Component Parameterization: For each pure component, fit the physical molecular parameters (segment number r_i and interaction energy ε_ii) to vapor pressure and liquid density data. Assign HB parameters (U_ij^HB, S_ij^HB) for specific associating groups (e.g., -OH in alcohols).
• Mixture Calculation (VLE/LLE): For a mixture, define binary interaction parameters (λ_ij) for the physical term. The chemical (HB) term is computed based on the statistics of donor-acceptor pairing.
• Equilibrium Calculation: Solve the equation of state and chemical equilibrium conditions to determine phase densities and compositions. The model has been shown to accurately correlate high-pressure vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) for systems like alkane-alcohol mixtures with fewer adjustable parameters than cubic EOS models [61] [59].

The following diagram illustrates the core logical workflow and fundamental relationships underlying each of these three methods.

Comparative Analysis: Data and Applications

The following tables provide a direct, structured comparison of the quantitative parameters, computational demands, and application scopes of the three models.

Table 1: Key Hydrogen-Bonding Parameters and Variables across Models

Model	HB Acidity Descriptor	HB Basicity Descriptor	Key HB Energy Formulation	Key Adjustable HB Parameters
LSER	Solute-specific descriptor A	Solute-specific descriptor B	Contribution to log(P) via apA and bpB	System coefficients a, b (from regression)
COSMO-RS	Surface charge density σ_don	Surface charge density σ_acc	E_hb = c_hb(T) * max[0,σ_acc-σ_hb] * min[0,σ_don+σ_hb]	Prefactor c_hb, threshold σ_hb
LFHB-EoS	Number of donor sites d	Number of acceptor sites a	Free energy from U_ij^HB and S_ij^HB in Veytzman statistics	U_ij^HB (energy), S_ij^HB (entropy) per group

Table 2: Comparative Scope, Performance, and Practical Application

Aspect	LSER	COSMO-RS	LFHB-EoS
Theoretical Basis	Empirical linear free-energy relationships	Quantum chemistry + statistical thermodynamics	Lattice-fluid physics + hydrogen-bonding statistics
Primary Data Input	Experimental partition coefficients & curated descriptors	Quantum chemical COSMO calculations	Pure component VLE/data & HB group parameters
Computational Cost	Low (after parameterization)	High (QM step) to Moderate (RS step)	Moderate
Prediction Scope	Limited to properties with existing regressions	Broad (activity coefficients, VLE, LLE, solubility)	Focused on phase equilibria (VLE, LLE)
Treatment of Mixtures	Implicit in system coefficients a, b	Weighted by σ-profiles and composition	Explicit via mixing rules & HB equilibrium
Key Strength	Excellent accuracy for systems with ample data	General prediction without system-specific parameters	Physically sound modeling of associating fluids

Successful application of these models requires access to specific databases, software, and parameters.

Table 3: Essential Research Reagents and Resources

Resource Name	Type	Primary Function	Relevant Model(s)
LSER Database [3]	Database	Provides curated Abraham solute descriptors (A, B, etc.) and system coefficients for solvents.	LSER
COSMObase [10]	Database	A large collection of pre-computed .cosmo files for thousands of molecules, drastically reducing computational time.	COSMO-RS
σ-Profile [58] [60]	Computational Output	The histogram of a molecule's surface charge distribution; the fundamental descriptor for all COSMO-RS calculations.	COSMO-RS
Hydrogen-Bonding Energy (UHB) & Entropy (SHB) [59]	Model Parameters	Specific interaction parameters for different functional groups (e.g., -OH, -NH2) used in the EOS calculation.	LFHB-EoS
Quantum Chemistry Code (e.g., ADF, Gaussian) [58] [10]	Software	Performs the initial COSMO calculation to generate the screening charge density surface of a molecule.	COSMO-RS
Phase Equilibrium Database (e.g., DDB) [61]	Database	Provides experimental vapor-liquid and liquid-liquid equilibrium data for parameter fitting and model validation.	LFHB-EoS, LSER

This comparison highlights the distinct philosophies and applications of LSER, COSMO-RS, and LFHB-EoS in modeling hydrogen bonding. LSER remains a powerful, low-cost tool for predicting solvation properties when a large body of experimental data exists for regression, particularly for partition coefficients. COSMO-RS offers a uniquely broad predictive capability from first principles, making it invaluable for screening in early-stage drug development where experimental data is scarce. LFHB-EoS provides a robust framework for modeling the phase equilibria of strongly associating fluids, often with a minimal number of temperature-dependent parameters.

The choice of model is not a question of which is universally best, but which is most appropriate for the problem at hand. Researchers must weigh the need for predictive generality against computational cost, and the availability of experimental data against the desired physicochemical insight. As these models continue to develop, the ongoing integration of their strengths—such as using COSMO-RS to generate parameters for EOS models—promises to further enhance the accuracy and scope of thermodynamic predictions for complex, hydrogen-bonded systems.

In pharmaceutical development, the reliability of solubility data for active pharmaceutical ingredients (APIs) is paramount, as it directly impacts drug absorption, bioavailability, and subsequent therapeutic efficacy [62]. Inconsistent solubility data can lead to flawed formulation design, regulatory challenges, and potential product failure. This case study examines the critical challenge of data inconsistency using coumarin as a model compound and explores the application of two predictive thermodynamic models—COSMO-RS (Conductor-like Screening Model for Real Solvents) and LSER (Linear Solvation Energy Relationships)—as tools for identifying anomalous data and validating dataset consistency.

Coumarin, a lactone-type benzopyrone with widespread applications in pharmaceuticals, food, and cosmetics, presents a compelling case for this investigation [32] [63]. Despite its extensive use, documented solubility values for coumarin in common organic solvents like alcohols show significant incongruence, creating uncertainty for researchers and formulators [32]. This study directly addresses this problem by formulating a theoretical consistency test using COSMO-RS and LSER approaches, comparing their predictive capabilities, methodological requirements, and practical applications in a pharmaceutical context.

Theoretical Background: LSER and COSMO-RS Models

Linear Solvation Energy Relationships (LSER)

The LSER approach, pioneered by Abraham, correlates molecular descriptors with solvation properties through multivariate linear equations [64] [65]. These descriptors quantitatively represent a molecule's potential for specific intermolecular interactions:

• Vx (McGowan volume): Characterizes dispersion forces and cavity formation energy
• E (Excess refractivity): Relates to polarizability contributions
• S (Polarity): Reflects dipole-dipole and dipole-induced dipole interactions
• A (Acidity) and B (Basicity): Quantify hydrogen-bond donating and accepting capabilities, respectively [64] [65]

The fundamental LSER equation for solvation processes takes the form:

where lowercase coefficients represent the system's sensitivity to each interaction.

Partial Solvation Parameters (PSP) Bridge

Partial Solvation Parameters (PSP) have emerged as a valuable bridge connecting LSER descriptors with solubility parameter concepts [64] [65]. PSPs are defined from LSER descriptors as:

where Vm is the molar volume [65]. This framework allows conversion between LSER descriptors and Hansen Solubility Parameters (HSP), enabling a unified approach that leverages the strengths of both methodologies [64] [65].

COSMO-RS (Conductor-like Screening Model for Real Solvents)

COSMO-RS represents a different theoretical approach, combining quantum chemistry with statistical thermodynamics to predict solvation behavior without requiring experimental input data [32] [66]. The model operates through a defined computational workflow:

• Quantum Chemical Calculations: Optimization of molecular structures in a virtual conductor environment
• σ-profile Generation: Calculation of screening charge density distributions across molecular surfaces
• Statistical Thermodynamics: Evaluation of molecular interactions based on surface charge pairing

The key output is a σ-profile—a histogram showing the probability distribution of specific screening charge densities on a molecule's surface [66] [67]. This profile provides a unique fingerprint of a compound's polarity characteristics and interaction potential, divided into three key regions:

• Non-polar region (σ between -0.0082 and 0.0082 e/Ų)
• H-bond donor region (σ < -0.0082 e/Ų)
• H-bond acceptor region (σ > 0.0082 e/Ų) [66] [67]

Case Study: Coumarin Solubility in Neat Alcohols

The Consistency Problem

Recent investigations have revealed significant inconsistencies in reported solubility values for coumarin across different studies, particularly in protic solvents like alcohols [32]. For example, solubility measurements for coumarin in methanol, ethanol, 1-propanol, and 2-propanol showed discrepancies that exceeded expected experimental error ranges. These inconsistencies create substantial challenges for pharmaceutical scientists relying on this data for formulation development and process optimization.

Application of COSMO-RS-DARE for Consistency Testing

Researchers applied COSMO-RS-DARE (Dimerization, Aggregation, and Reaction Extension)—an advanced version of COSMO-RS that accounts for concentration-dependent composition alterations—to test the consistency of published coumarin solubility data [32]. The methodology involved:

• Theoretical Prediction: Computing coumarin solubility in various alcohols using COSMO-RS-DARE
• Data Comparison: Comparing predicted values with experimentally reported solubility data
• Outlier Identification: Flagging datasets showing significant deviations from predicted values as potentially unreliable
• Experimental Validation: Remeasuring solubility for suspicious data points to confirm theoretical findings

The validation experiments followed a standardized shake-flask method:

• Excess coumarin added to 10 mL volumetric flasks filled with solvent
• Incubation for 24 hours at controlled temperatures (25-40°C) with orbital shaking at 60 rev/min
• Filtration through preheated PTFE filters (22 µm pore size) to prevent precipitation
• Spectrophotometric quantification at 310 nm wavelength [32]

The COSMO-RS-DARE approach successfully identified suspicious datasets, with subsequent experimental validation confirming the theoretical predictions [32]. The perfect match between back-computed coumarin solubility values and experimental measurements demonstrated the reliability of this approach for solubility data consistency testing.

LSER/PSP Alternative Approach

In parallel, researchers have explored LSER-based methods using Partial Solvation Parameters (PSP) as a complementary approach [65]. The PSP methodology involves:

• Descriptor Determination: Obtaining LSER descriptors (A, B, S, E, Vx) for both solute and solvent
• PSP Calculation: Converting descriptors to Partial Solvation Parameters using established equations
• Interaction Analysis: Computing interaction potentials based on PSP complementarity
• Solubility Prediction: Relating PSP differences to solubility behavior

This approach benefits from the extensive database of Abraham LSER descriptors available for pharmaceutical compounds [65].

Comparative Analysis: COSMO-RS vs. LSER/PSP for Solubility Consistency Testing

Table 1: Direct comparison of COSMO-RS and LSER/PSP approaches for solubility consistency testing

Feature	COSMO-RS	LSER/PSP
Theoretical Basis	Quantum chemistry + statistical thermodynamics	Empirical linear free-energy relationships
Experimental Data Requirement	No experimental data needed (ab initio)	Requires experimental data for descriptor determination
Molecular Descriptors	σ-profiles from quantum calculations	A, B, S, E, Vx parameters from experimental measurements
Handling of Complex Systems	COSMO-RS-DARE version handles dimerization and aggregation [32]	Limited for complex association equilibria
Parameter Transferability	System-specific calculations required	Universal descriptors for compounds [65]
Implementation Complexity	High (requires quantum chemistry software)	Moderate (descriptors available in databases)
Successful Application to Coumarin	Yes - identified inconsistent datasets [32]	Not specifically documented for coumarin consistency testing

Table 2: Performance metrics for coumarin solubility prediction in alcohols using COSMO-RS-DARE

Solvent	Temperature Range	Prediction Accuracy	Data Consistency Assessment
Methanol	25-40°C	High (validated experimentally)	Identified outliers
Ethanol	25-40°C	High (validated experimentally)	Identified outliers
1-Propanol	25-40°C	High (validated experimentally)	Identified outliers
2-Propanol	25-40°C	High (validated experimentally)	Identified outliers
1-Butanol	25-40°C	High (validated experimentally)	Consistent dataset
1-Pentanol	25-40°C	High (validated experimentally)	Consistent dataset
1-Octanol	25-40°C	High (validated experimentally)	Consistent dataset

Experimental Protocols for Solubility Consistency Testing

Shake-Flask Method for Experimental Solubility Determination

The shake-flask method remains the gold standard for experimental solubility determination and was employed for validating computational predictions in the coumarin case study [32]:

• Sample Preparation: Add excess solute (coumarin) to volumetric flasks, fill with solvent to create saturated solutions
• Equilibration: Incubate samples for 24 hours at constant temperatures (25-40°C) with continuous orbital shaking at 60 rev/min
• Filtration: Pass saturated solutions through preheated PTFE filters (22 µm pore size) to remove undissolved solid while preventing precipitation
• Dilution: Transfer fixed filtrate amounts to test tubes prefilled with dilution solvent (typically methanol)
• Quantification: Measure solute concentration spectrophotometrically at appropriate wavelength (310 nm for coumarin) using pre-established calibration curves
• Calculation: Express solubility as mole fraction using averaged measurements from triplicate samples [32]

Computational Consistency Testing Protocol

For researchers implementing computational consistency testing, the following workflow is recommended:

The Scientist's Toolkit: Essential Research Reagents and Materials

Table 3: Essential research reagents and materials for solubility consistency testing

Item	Specification	Application/Function
Reference Compound	Coumarin (≥99% purity) [32]	Model compound for method validation
Solvent Series	Methanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 1-pentanol, 1-octanol (≥99% purity) [32]	Provides homologous series for consistency trends
Shake-Flask Incubator	Orbital shaker incubator with ±0.1°C temperature accuracy [32]	Maintains constant temperature with agitation for equilibration
Filtration System	Syringe with PTFE membrane (22 µm pore size) [32]	Removes undissolved solid while maintaining saturation
Analytical Instrument	UV-Vis spectrophotometer with 1 nm resolution [32]	Quantifies solute concentration in saturated solutions
Computational Software	COSMOthermX with COSMO-RS implementation [66] [67]	Performs quantum chemical and statistical thermodynamic calculations
LSER Database	Abraham LSER descriptor database [65]	Provides molecular descriptors for PSP calculations

This case study demonstrates that both COSMO-RS and LSER/PSP approaches offer valuable capabilities for testing the consistency of solubility data for pharmaceutical compounds like coumarin. The COSMO-RS-DARE implementation has proven particularly effective for identifying inconsistent datasets without prior experimental input, as validated through controlled experiments on coumarin in alcoholic solvents [32]. Meanwhile, the LSER/PSP framework provides a well-established alternative with strong thermodynamic foundations and the advantage of leveraging extensive descriptor databases [65].

For pharmaceutical researchers facing questionable solubility data, the recommended approach involves:

• Systematic literature review to compile existing solubility data
• COSMO-RS-based theoretical screening to identify potential outliers
• Targeted experimental validation using shake-flask methodology for suspicious data points
• LSER/PSP analysis for additional perspective on molecular-level interactions

The integration of these computational tools with carefully designed experiments creates a robust framework for validating solubility data, ultimately enhancing the reliability of pharmaceutical development pipelines and ensuring the consistent performance of final drug products.

In the fields of drug development, materials science, and environmental chemistry, accurately predicting how molecules dissolve, mix, and partition between different phases remains a fundamental challenge. Researchers and scientists have developed various computational models to predict these solvation properties without resorting to extensive laboratory experimentation. Among the most established approaches are the Linear Solvation Energy Relationship (LSER) model, a highly successful empirical method, and COSMO-RS (Conductor-like Screening Model for Real Solvents), a quantum mechanics-based model known for its a priori predictive capability [68] [1]. While both are powerful, they have developed in parallel with distinct foundations and descriptors, making direct comparison and information transfer difficult. The Partial Solvation Parameters (PSP) approach has emerged as a unifying framework designed to interconnect these powerful models, leveraging their respective strengths while mitigating their limitations [68] [65] [69]. This guide provides a comparative analysis of these approaches, detailing how PSP creates a bridge for more robust and versatile thermodynamic predictions.

Theoretical Foundations and Key Concepts

Linear Solvation Energy Relationship (LSER)

Abraham's LSER model is a highly successful Quantitative Structure-Property Relationship (QSPR) technique. It correlates a molecule's thermodynamic properties with a set of five (or six) empirically determined molecular descriptors [68] [69] [3].

• Core Descriptors: The most common descriptors are:
  • Vx: McGowan characteristic volume, representing the size and cavity formation cost.
  • E: Excess molar refraction, accounting for polarizability due to π- or n-electrons.
  • S: Dipolarity/polarizability, reflecting a molecule's ability to engage in dipole-dipole and dipole-induced dipole interactions.
  • A: Hydrogen-bond acidity, representing the molecule's capacity to donate a proton (proton donor strength).
  • B: Hydrogen-bond basicity, representing the molecule's capacity to accept a proton (proton acceptor strength).
  • L: The gas-hexadecane partition coefficient, sometimes used as an alternative comprehensive descriptor [3].
• Working Principle: The model operates on linear free-energy relationships, where a property (e.g., a partition coefficient, log P) is expressed as a linear combination of these descriptors. The system-specific coefficients are obtained by fitting to experimental data [3]. Its strength lies in its extensive database of descriptors for thousands of compounds [65].

COSMO-RS (Conductor-like Screening Model for Real Solvents)

COSMO-RS is a quantum mechanics-based statistical thermodynamic model that predicts thermodynamic properties from first principles [68] [65] [1].

• Core Concept: It starts with a quantum chemical calculation for a molecule in a virtual conductor. The result is a sigma-profile (σ-profile), which is the histogram of the polarized screening charge density distribution on the molecular surface [68] [65].
• Working Principle: The statistical thermodynamics of the surface segment interactions are then applied to real solvents, using the σ-profiles to compute chemical potentials, activity coefficients, and other properties [1]. It is considered an a priori method because it requires no experimental input for the target compound, only its molecular structure.

Partial Solvation Parameters (PSP)

The PSP approach is a QSPR-type method designed to unify concepts from LSER, COSMO-RS, and classical solubility parameters [68] [65] [69]. It aims to place molecular descriptors within a consistent equation-of-state thermodynamic framework, making them applicable over wide ranges of temperature and pressure [68].

• Core Descriptors: PSPs are defined for different interaction types:
  • σd: Dispersion PSP, mapping to LSER's Vx and E descriptors, reflecting hydrophobicity and dispersion forces [65] [69].
  • σp: Polarity PSP, mapping to the LSER S descriptor, reflecting dipole-dipole and polarizability interactions [65] [69].
  • σGa and σGb: Acidity and basicity PSPs (Gibbs free-energy descriptors), mapping to the LSER A and B descriptors, specifically quantifying hydrogen-bonding interactions [65].
• The Bridging Role: PSPs can be determined either from COSMO-RS-derived "cosmoments" (the moments of the σ-profile) or from freely available Abraham LSER descriptors, creating a direct link between the two parent models [68] [65] [69].

The logical and data-flow relationships between these three models are synthesized in the following conceptual framework:

Comparative Performance Analysis

Multiple studies have evaluated the predictive performance of these models for key physicochemical properties. The table below summarizes a comparative validation based on the prediction of liquid/liquid partition coefficients for 270 complex environmental contaminants, including pesticides and flame retardants [29].

Table 1: Validation of prediction methods for liquid/liquid partition coefficients (log P). Performance is measured by Root Mean Squared Error (RMSE) against experimental data [29].

Prediction Method	Basis of Prediction	RMSE Range (log units)	Overall Performance
COSMOtherm	Quantum Chemical (COSMO-RS)	0.65 – 0.93	High Accuracy
ABSOLV (LSER-based)	Empirical LSER Descriptors	0.64 – 0.95	High Accuracy
SPARC	QSPR/Linear Free Energy	1.43 – 2.85	Lower Accuracy

The table demonstrates that both COSMO-RS (implemented in COSMOtherm) and the LSER-based ABSOLV show comparable and high prediction accuracy for complex molecules, significantly outperforming the SPARC model in this specific application [29].

Beyond partition coefficients, the PSP framework has been validated for other critical properties. In pharmaceutical research, PSPs determined experimentally via inverse gas chromatography (IGC) have been successfully used to predict drug solubility in various solvents and to calculate the different contributions to surface energy, which is crucial for understanding powder behavior in formulations [65].

Experimental Protocols and Workflows

Determining PSPs from LSER Descriptors

This is a straightforward computational method that leverages the extensive LSER database [65] [69].

• Data Acquisition: Obtain the Abraham LSER descriptors (Vx, E, S, A, B) for the compound of interest from a published database [65].
• Molar Volume Calculation: Determine or calculate the molar volume (Vm) of the compound.
• PSP Calculation: Apply the following mapping equations to calculate the Partial Solvation Parameters [65]:
  • σd = 100 * (3.1 * Vx + E) / Vm (Dispersion PSP)
  • σp = 100 * S / Vm (Polarity PSP)
  • σGa = 100 * A / Vm (Acidity PSP)
  • σGb = 100 * B / Vm (Basicity PSP)

Determining PSPs via Inverse Gas Chromatography (IGC)

IGC is an experimental technique for characterizing materials, particularly solids like active pharmaceutical ingredients (APIs), by using known probe gases [65].

• Column Preparation: The solid drug of interest is packed into a gas chromatography column.
• Probe Injection: A series of probe gases with known molecular properties (e.g., alkanes, alcohols, ethers) are injected into the column.
• Data Measurement: The retention time of each probe is measured, which is related to its interaction energy with the solid.
• Parameter Regression: The interaction data for multiple probes are analyzed through a multi-parameter regression to determine the PSP set (σd, σp, σGa, σGb) that best fits the experimental results. This method provides an experimental characterization of the solid's surface energy and solubility parameters [65].

Predicting Hydrogen-Bonding Interaction Energies

A COSMO-based method provides a simple protocol for predicting hydrogen-bonding energies, a critical component of solvation [25].

• Quantum Chemical Calculation: Perform a DFT (Density Functional Theory) calculation to obtain the molecular surface charge distribution (σ-profile).
• Descriptor Assignment: From the σ-profile, assign the molecule an acidity (α) and/or basicity (β) descriptor.
• Energy Calculation: For two molecules (1 and 2) interacting, the hydrogen-bonding interaction energy is calculated as: E_HB = c * (α1 * β2 + α2 * β1), where c is a universal constant (5.71 kJ/mol at 25°C) [25].

Essential Research Reagents and Computational Tools

The following table details key software, databases, and experimental reagents essential for working with these solvation models.

Table 2: Key Research Reagents and Solutions for Solvation Modeling

Category	Name / Example	Function / Description
Software & Databases	COSMOtherm	Commercial software implementing the COSMO-RS model for property prediction [29].
	TURBOMOLE, DMol3	Quantum chemistry software suites used to generate the σ-profile input files required for COSMO-RS calculations [68] [65].
	LSER Database	A freely accessible database containing Abraham descriptors (Vx, E, S, A, B) for thousands of compounds [65] [69].
	ABSOLV	Commercial software (LSER-based) for predicting solvation properties and partition coefficients [29].
Experimental Reagents	IGC Probe Gases	A set of volatile probes (e.g., n-alkanes, dichloromethane, ethanol, acetone) used to characterize solid surfaces via Inverse Gas Chromatography [65].
	Pharmaceutical Solvents	A library of common and exotic solvents (e.g., water, alcohols, DMSO, ethyl acetate) used for experimental validation of predicted solubilities [65].

The LSER, COSMO-RS, and PSP models represent powerful but philosophically distinct approaches to predicting solvation thermodynamics. LSER offers remarkable accuracy and simplicity rooted in empirical data, while COSMO-RS provides a robust a priori prediction from molecular structure. The Partial Solvation Parameter (PSP) framework successfully acts as a bridge, interconnecting these models and classical solubility parameters into a unified, thermodynamically consistent toolkit [68] [69].

The key advantage of the PSP bridge is its versatility and thermodynamic foundation. By being placed within an equation-of-state framework, PSPs can be used for predictions beyond the scope of traditional LSER or COSMO-RS, such as properties over extended temperature and pressure ranges, and for both bulk and interfacial phenomena [68] [65] [69]. For drug development professionals, this unified approach offers a coherent strategy for excipient selection, solubility prediction, and solid-state characterization, leveraging the best available data from both empirical and computational sources [65]. As computational power increases and databases expand, such integrative approaches are poised to become indispensable in the rational design of chemicals, materials, and pharmaceutical products.

Conclusion

The comparative analysis of LSER and COSMO-RS reveals a complementary landscape for molecular thermodynamics in drug development. While LSER offers unparalleled simplicity and robust performance where experimental data exists, COSMO-RS provides a powerful, a priori predictive capability grounded in quantum mechanics. The future lies in hybrid strategies that leverage the strengths of both, such as using COSMO-RS to generate insightful descriptors for machine learning models or developing unified frameworks like Partial Solvation Parameters. These integrated approaches, along with ongoing efforts to improve thermodynamic consistency, are poised to significantly enhance the accuracy of solubility and partitioning predictions. This will ultimately accelerate drug discovery by enabling more reliable in-silico screening of API candidates and excipients, reducing both development costs and time-to-market for new therapeutics.

References

• 1. COSMO-RS and LSER models of solution thermodynamics [https://www.sciencedirect.com/science/article/abs/pii/S0167732223017981]
• 2. The chemical interpretation and practice of linear solvation ... [https://pubmed.ncbi.nlm.nih.gov/16889784/]
• 3. Linear Solvation–Energy Relationships (LSER) and ... [https://www.mdpi.com/2673-8015/3/1/7]
• 4. A hybrid approach to aqueous solubility prediction using ... [https://www.sciencedirect.com/science/article/abs/pii/S0263876224004428]
• 5. High-throughput application and evaluation of the COSMO ... [https://pubs.rsc.org/en/content/articlehtml/2025/dd/d5dd00259a]
• 6. COSMO-RS for Modern Rational Drug Design and the SAMPL ... [https://blog.3ds.com/brands/biovia/cosmo-rs-for-modern-rational-drug-design-and-the-sampl-challenge/]
• 7. Linear solvation energy relationships (LSERs) for robust ... [https://www.sciencedirect.com/science/article/pii/S0928098722000239]
• 8. New publication on Assessment of COSMO-RS for ... [https://www.lse.ls.tum.de/en/btd/news/article/new-publication-on-predicting-the-density-and-viscosity-of-deep-eutectic-solvents-at-atmospheric-and-elevated-pressures-2-1-1-1/]
• 9. Characterization of solute-solvent interactions in liquid ... [https://www.sciencedirect.com/science/article/pii/S0003267023008930]
• 10. COSMO-RS [https://en.wikipedia.org/wiki/COSMO-RS]
• 11. Sigma-Profiles for COSMO-RS Type Models [https://www.ddbst.com/ddb-sigma.html]
• 12. Calculating and estimating sigma profiles — COSMO-RS ... [https://www.scm.com/doc/COSMO-RS/Examples/Sigma_profiles.html]
• 13. Sigma Moments — COSMO-RS 2025.1 documentation - SCM [https://www.scm.com/doc/COSMO-RS/Examples/Sigma_moments.html]
• 14. Quantum Chemical (QC) Calculations and Linear Solvation ... [https://www.mdpi.com/2673-8015/4/4/37]
• 15. COSMO-RS and LSER Models of Solution Thermodynamics [https://www.semanticscholar.org/paper/COSMO-RS-and-LSER-Models-of-Solution-Towards-a-of-E.Acree-C.Panayioto/0f9707f85919b99038242145e285b52fbf9c7b61]
• 16. A COSMO-RS Based Guide to analyze/quantify the Polarity ... [https://pubmed.ncbi.nlm.nih.gov/20145869/]
• 17. Quantum chemical calculations for predicting the partitioning ... [https://pubs.rsc.org/en/content/articlehtml/2025/em/d5em00524h]
• 18. COSMO-RS: a novel and efficient method for the a priori ... [https://www.sciencedirect.com/science/article/abs/pii/S0378381200003575]
• 19. Calculation of properties — COSMO-RS 2025.1 documentation [https://www.scm.com/doc/COSMO-RS/Calculation_of_properties.html]
• 20. Prediction of aqueous solubility of drugs and pesticides ... [https://pubmed.ncbi.nlm.nih.gov/11924739/]
• 21. Data-driven approaches for predicting mechanical ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11929718/]
• 22. Data-driven optimization of maskless grayscale laser ... [https://www.nature.com/articles/s41598-025-24652-x]
• 23. Prediction of Hydrogen-Bonding Interaction Free Energies ... [https://www.mdpi.com/2673-8015/5/2/12]
• 24. A comprehensive approach to incorporating intermolecular ... [https://arxiv.org/html/2502.08520v1]
• 25. Prediction of hydrogen-bonding interaction energies with ... [https://www.sciencedirect.com/science/article/pii/S0167732225000650]
• 26. Evaluating the COSMO-RS Method for Modeling Hydrogen ... [https://pubmed.ncbi.nlm.nih.gov/23630190/]
• 27. Computational Support to Explore Ternary Solid ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11539070/]
• 28. Linear solvation energy relationships (LSERs) for robust ... [https://www.sciencedirect.com/science/article/pii/S0928098722000227]
• 29. Validation of COSMOtherm, ABSOLV, and SPARC [https://academic.oup.com/etc/article/33/7/1537/7761199]
• 30. LSER Database [https://www.ufz.de/lserd/]
• 31. An open source COSMO-RS implementation and ... [https://www.sciencedirect.com/science/article/abs/pii/S0378381222000954]
• 32. Application of COSMO-RS-DARE as a Tool for Testing ... [https://www.mdpi.com/1420-3049/27/16/5274]
• 33. Dispersion, Polar, and Hydrogen-Bonding Contributions to ... [https://www.mdpi.com/2673-8015/5/4/25]
• 34. Application of COSMO-RS as an excipient ranking tool in ... [https://www.sciencedirect.com/science/article/abs/pii/S0928098713001486]
• 35. Application of COSMO-RS as an excipient ranking tool in ... [https://pubmed.ncbi.nlm.nih.gov/23639717/]
• 36. Benchmarking Different QM Levels for Usage with COSMO-RS - PubMed [https://pubmed.ncbi.nlm.nih.gov/31692342/]
• 37. Guiding excipient selection for amorphous solid ... [https://www.pharmaexcipients.com/news/guiding-excipient-selection-sd/]
• 38. Bioavailability Enhancement Techniques for Poorly Aqueous ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC9495787/]
• 39. Review of the latest progress of AI and Machine Learning ... [https://www.sciencedirect.com/science/article/pii/S2949866X25000620]
• 40. Self-solvation energies: Extended open database and ... [https://www.sciencedirect.com/science/article/pii/S0378381225000068]
• 41. Machine learning-enhanced COSMO-SAC for accurate ... [https://www.sciencedirect.com/science/article/abs/pii/S0378381225002055]
• 42. TabPFN Opens New Avenues for Small-Data Tabular ... [https://chemrxiv.org/engage/chemrxiv/article-details/68d29b1cf2aff1677025b18f]
• 43. Quantum Reservoir Computing Could Give Drug Discovery ... [https://thequantuminsider.com/2025/08/09/quantum-reservoir-computing-could-give-drug-discovery-a-boost-especially-when-data-is-scarce/]
• 44. Prediction of Hydrogen-Bonding Interaction Free-Energies ... [https://www.preprints.org/manuscript/202501.1538]
• 45. Experimental partition coefficients and model calibration [https://pubmed.ncbi.nlm.nih.gov/35150822/]
• 46. COSMO-RS for the prediction of the retention behavior in ... [https://pubmed.ncbi.nlm.nih.gov/23273634/]
• 47. The Conformational Contribution to Molecular Complexity and ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC10886813/]
• 48. Towards understanding the unbound state of drug ... [https://www.sciencedirect.com/science/article/abs/pii/S0968089616301729]
• 49. unveiling functional structures in biological membranes - PMC [https://pmc.ncbi.nlm.nih.gov/articles/PMC12332416/]
• 50. General — COSMO-RS 2025.1 documentation - SCM [https://www.scm.com/doc/COSMO-RS/General.html]
• 51. Molecule — COSMO-RS 2025.1 documentation - SCM [https://www.scm.com/doc/COSMO-RS/pyCRS/Molecule.html]
• 52. Prediction of Small-molecule Pharmaceuticals Solubility ... [https://www.sciencedirect.com/science/article/pii/S0928098725003367]
• 53. Physics-Based Solubility Prediction for Organic Molecules [https://pmc.ncbi.nlm.nih.gov/articles/PMC12355698/]
• 54. In silico cocrystal screening using COSMO-RS: A rapid and ... [https://www.sciencedirect.com/science/article/abs/pii/S0167732225010189]
• 55. COSMO-RS predictions of logP in the SAMPL7 blind challenge [https://pubmed.ncbi.nlm.nih.gov/34125358/]
• 56. QSPR-based model extrapolation prediction of enthalpy ... [https://www.sciencedirect.com/science/article/abs/pii/S0167732223002581]
• 57. Approaches for calculating solvation free energies and ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC5648357/]
• 58. COSMO-RS theory - SCM [https://www.scm.com/doc/COSMO-RS/COSMO-RS_theory.html]
• 59. A nonrandom lattice fluid hydrogen bonding theory for ... [https://www.sciencedirect.com/science/article/abs/pii/S0378381299001132]
• 60. Analysis — COSMO-RS 2025.1 documentation - SCM [https://www.scm.com/doc/COSMO-RS/Analysis.html]
• 61. Liquid–liquid equilibrium correlations using lattice fluid ... [https://www.sciencedirect.com/science/article/abs/pii/S1226086X07000317]
• 62. Advancements in Drug Solubility Testing You Should Know ... [https://www.raytor.com/drug-solubility-testing/]
• 63. Solubility of coumarin in (ethanol + water) mixtures [https://www.sciencedirect.com/science/article/abs/pii/S0167732221014859]
• 64. Partial solvation parameters and LSER molecular descriptors [https://www.sciencedirect.com/science/article/abs/pii/S0021961412001024]
• 65. Partial Solvation Parameters of Drugs as a New ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC6359455/]
• 66. COSMO-RS Solubility Screening and Coumarin Extraction ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC12429901/]
• 67. COSMO-RS and solvatochromic parameters to investigate ... [https://www.sciencedirect.com/science/article/abs/pii/S0927775715302612]
• 68. Partial solvation parameters and the equation-of-state ... [https://www.sciencedirect.com/science/article/abs/pii/S0378381215300741]
• 69. Redefining solubility parameters: Bulk and surface ... [https://www.sciencedirect.com/science/article/abs/pii/S0021961417300988]