Continuous medium theory for nonequilibrium solvation: I. How to correctly evaluate solvation free energy of nonequilibrium

Considering the influences of electrostatic potential Phi upon the change of solute charge distribution deltarho and rho upon the change deltaPhi at the same time, a more reasonable integral formula of dG = (1/2) integral (V) (rhodeltaPhi + Phideltarho)dV is used to calculate the change of the electrostatic free energy in charging the solute-solvent system to a nonequilibrium state, instead of the one of dG = integral (V) PhideltarhodV used before. This modification improves the expressions of electrostatic free energy and solvation free energy, in which no quantity of the intermediate equilibrium state is explicitly involved. Detailed investigation reveals that the solvation free energy of nonequilibrium only contains the interaction energy between the field due to the solute charge in vacuum, and the dielectric polarization at the nonequilibrium state. The solvent reorganization energies of forward and backward electron transfer reactions have been redefined because the derivations lead to a remarkable feature that these quantities are direction-dependent, unlike the theoretical models developed before. The deductions are given in the electric field-displacement form. Relevant discussions on the reliability of theoretical models suggested in this work have also been presented.     

Thermodynamics for nonequilibrium solvation and numerical evaluation of solvent reorganization energy

This work presents a thermodynamic method for treating nonequilibrium solvation. By imposing an extra electric field onto the nonequilibrium solvation system, a virtual constrained equilibrium state is prepared. In this way, the free energy difference between the real nonequilibrium state and the con-strained equilibrium one is simply the potential energy of the nonequilibrium polarization in the extra electronic field, according to thermodynamics. Further, new expressions of nonequilibrium solvation energy and solvent reorganization energy have been formulated. Analysis shows that the present formulations will give a value of reorganization energy about one half of the traditional Marcus theory in polar solvents, thus the explanation on why the traditional theory tends to overestimate this quantity has been found out. For the purpose of numerical determination of solvent reorganization energy, we have modified Gamess program on the basis of dielectric polarizable continuum model. Applying the procedure to the well-investigated intramolecular electron transfer in biphenyl-androstane-naphthyl and biphenyl-androstane-phenanthryl systems, the numerical results of solvent reorganization energy have been found to be in good agreement with the experimental fittings.     

Continuous medium theory for nonequilibrium solvation: III. Solvation shift by monopole approximation and multipole expansion in spherical cavity

According to the classical electrodynamics, a new and reasonable method about electrostatic energy decomposition of the solute-solvent system has been proposed in this work by introducing the concept of spring energy. This decomposition in equilibrium solvation gives the clear comprehension for different parts of total electrostatic free energy. Logically extending this cognition to nonequilibrium leads to the new formula of electrostatic free energy of nonequilibrium state. Furthermore, the general solvation shift for light absorption/emission has been reformulated and applied to the ideal sphere case with the monopole approximation and multipole expansion. Solvation shifts in vertical ionizations of atomic ions of some series of main group elements have been investigated with monopole approximation, and the variation tendency of the solvation shift versus atomic number has been discussed. Moreover, the solvation shift in photoionization of nitrate anion in glycol has been investigated by the multipole expansion method.     

丙酮分子n→π%*跃迁光谱的平均溶剂静电势/分子动力学方法研究

采用量子力学/分子动力学方法研究了具体溶剂分子结构对溶质光谱行为的静电影响. 通过拟合溶质所处外电场和引入溶剂分子极化率, 考虑了溶质溶剂分子之间的相互极化效应, 得到合理的溶质和溶剂分子的电荷分布. 经过严格推导发现, 在传统的显溶剂模型中, 平衡和非平衡溶剂化能表达式均未考虑溶剂分子永久偶极弹簧能的贡献. 因此, 在正确计算永久偶极弹簧能的基础上, 重新建立了溶剂化能的表达式和新的吸收/发射光谱移动公式. 采用修改后的ASEP/MD程序, 计算得到了与实验值比较吻合的丙酮在水溶液中n→π*跃迁的光谱移动值, 验证了新公式的合理性.     

Spectral shift of the n → π* transition for acetone and formic acid with an explicit solvent model

In our recent work, a new form of the electrostatic solvation energy for the nonequilibrium polarization has been derived by introducing the method of constrained equilibrium state in the framework of continuous medium theory. Up until now, the idea of the constrained equilibrium state method has not been introduced into the explicit solvent model by others; therefore this nonequilibrium energy form was further equivalently extended to the explicit solvent model in this work based on the discrete representation of the solvent permanent charges and induced dipoles. Making use of this expression in explicit solvent model, we modified the nonequilibrium module in the averaged solvent electrostatic potential/molecular dynamics program to implement numerical calculations. Subsequently, the new codes were applied to study the solvatochromic shifts of the n → π* absorption spectra for acetone and trans-formic acid in aqueous solution. The calculation results show a good agreement with the experimental observations. When our results of spectral shift are compared with those achieved directly from the continuum model, it can be seen that both the explicit solvent model and continuum model derived based on the constrained equilibrium approach can give reasonable predictions. The hydrogen bond effect was also discussed and deemed to be a dominant contribution to the spectral shift by calculating the n → π* absorption spectra of acetone-water complexes.     

Differential geometry based solvation model. III. Quantum formulation

Solvation is of fundamental importance to biomolecular systems. Implicit solvent models, particularly those based on the Poisson-Boltzmann equation for electrostatic analysis, are established approaches for solvation analysis. However, ad hoc solvent-solute interfaces are commonly used in the implicit solvent theory. Recently, we have introduced differential geometry based solvation models which allow the solvent-solute interface to be determined by the variation of a total free energy functional. Atomic fixed partial charges (point charges) are used in our earlier models, which depends on existing molecular mechanical force field software packages for partial charge assignments. As most force field models are parameterized for a certain class of molecules or materials, the use of partial charges limits the accuracy and applicability of our earlier models. Moreover, fixed partial charges do not account for the charge rearrangement during the solvation process. The present work proposes a differential geometry based multiscale solvation model which makes use of the electron density computed directly from the quantum mechanical principle. To this end, we construct a new multiscale total energy functional which consists of not only polar and nonpolar solvation contributions, but also the electronic kinetic and potential energies. By using the Euler-Lagrange variation, we derive a system of three coupled governing equations, i.e., the generalized Poisson-Boltzmann equation for the electrostatic potential, the generalized Laplace-Beltrami equation for the solvent-solute boundary, and the Kohn-Sham equations for the electronic structure. We develop an iterative procedure to solve three coupled equations and to minimize the solvation free energy. The present multiscale model is numerically validated for its stability, consistency and accuracy, and is applied to a few sets of molecules, including a case which is difficult for existing solvation models. Comparison is made to many other classic and quantum models. By using experimental data, we show that the present quantum formulation of our differential geometry based multiscale solvation model improves the prediction of our earlier models, and outperforms some explicit solvation model.     

A case against large cluster-induced nonequilibrium solvent effects in supercritical water

Using a partially compressible continuum solvation model, we have shown that solvent compression in just the first two solvation shells (or thereabouts) is all that is required to gain the bulk of the compression-induced enhancement to the solvation energy of ions in supercritical water. This result is found to hold even when the direct, equilibrium solvent-solute cluster involves well over a hundred solvent molecules. We argue that, for charge variation reactions in supercritical water, the observed short-range behavior of the compression-induced solvation free energy precludes the existence of any anomalously large nonequilibrium solvent effects which might be expected on the basis of the very large size of the equilibrium clusters. Received: 8 January 1997 / Accepted: 17 January 1997     

Geometry optimization based on linear response free energy with quantum mechanical/molecular mechanical method: applications to Menshutkin-type and Claisen rearrangement reactions in aqueous solution

The authors present a method based on a linear response theory that allows one to optimize the geometries of quantum mechanical/molecular mechanical (QM/MM) systems on the free energy surfaces. Two different forms of linear response free energy functionals are introduced, and electronic wave functions of the QM region, as well as the responses of electrostatic and Lennard-Jones potentials between QM and MM regions, are self-consistently determined. The covariant matrix relating the QM charge distribution to the MM response is evaluated by molecular dynamics (MD) simulation of the MM system. The free energy gradients with respect to the QM atomic coordinates are also calculated using the MD trajectory results. They apply the present method to calculate the free energy profiles of Menshutkin-type reaction of NH3 with CH3Cl and Claisen rearrangement of allyl vinyl ether in aqueous solution. For the Menshutkin reaction, the free energy profile calculated with the modified linear response free energy functional is in good agreement with that by the free energy perturbation calculations. They examine the nonequilibrium solvation effect on the transmission coefficient and the kinetic isotope effect for the Claisen rearrangement.     

A study on orientation and absorption spectrum of interfacial molecules by using continuum model

In this work, a numerical procedure based on the continuum model is developed and applied to the solvation energy for ground state and the spectral shift against the position and the orientation of the interfacial molecule. The interface is described as a sharp boundary separating two bulk media. The polarizable continuum model (PCM) allows us to account for both electrostatic and nonelectrostatic solute-solvent interactions when we calculate the solvation energy. In this work we extend PCM to the interfacial system and the information about the position and orientation of the interfacial molecule can be obtained. Based on the developed expression of the electrostatic free energy of a nonequilibrium state, the numerical procedure has been implemented and used to deal with a series of test molecules. The time-dependent density functional theory (TDDFT) associated with PCM is used for the electron structure and the spectroscopy calculations of the test molecules in homogeneous solvents. With the charge distribution of the ground and excited states, the position- and orientation-dependencies of the solvation energy and the spectrum have been investigated for the interfacial systems, taking the electrostatic interaction, the cavitation energy, and the dispersion-repulsion interaction into account. The cavitation energy is paid particular attention, since the interface portion cut off by the occupation of the interfacial molecule contributes an extra part to the stabilization for the interfacial system. The embedding depth, the favorable orientational angle, and the spectral shift for the interfacial molecule have been investigated in detail. From the solvation energy calculations, an explanation has been given on why the interfacial molecule, even if symmetrical in structure, tends to take a tilting manner, rather than perpendicular to the interface.     

非平衡溶剂化的连续介质模型和超快过程溶剂效应

在连续介质理论基础上, 根据Jackson的能量积分公式导出非平衡态静电自由能和溶剂化能的正确表达式. 引入“弹簧能”概念, 对平衡态和非平衡态的静电能构成给出了合理解释, 即此能量由溶质自由电荷和溶剂极化电荷的自能、 两者之间的相互作用能和极化电荷的“弹簧能”构成. 对目前几种代表性的非平衡溶剂化理论进行了论证和比较, 指出其中存在的基本理论问题. 根据新的非平衡溶剂化能建立了电子转移反应溶剂重组能的双球模型、 光谱移动的单球孔穴点偶极模型, 多级展开方法和非平衡溶剂效应的数值解方法.在Poisson方程求解中引入类导体屏蔽模型, 建立了任意孔穴极化电荷数值解方法并应用到Closs-Miller电子转移体系, 得到与实验值吻合的溶剂重组能, 解决了传统非平衡溶剂化理论高估溶剂重组能的问题.     

Solvation Free Energy Calculations: The Combination between the Implicitly Polarized Fixed-charge Model and the Reference Potential Strategy

The IPolQ-Mod charges, which are the average of two charge sets fitted in vacuum state and condensed phase, take account of polarization effect implicitly in the solvation free energy calculation. However, the performance of the IPolQ-Mod charges sensitively depends on the QM levels used to generate the electrostatic potential from which the charges are fitted. In addition, the forces on atoms are not accurate theoretically in the molecular dynamics (MD) simulation as the solvent only feels the electrostatic potential of a half-polarized density of the solute according to the derivation of the IPolQ-Mod charges. To study these issues in detail, the IPolQ-Mod charges are combined with the reference potential (RP) strategy to predict the solvation free energies in the present study. It is found that the thermodynamic perturbation (TP) corrections utilizing total energy difference and interaction energy difference are almost the same and free of bias. The solvation free energies estimated by the RP method match very well with those obtained by applying IPolQ-Mod charges into MD simulation directly. By means of the RP strategy, the performances of the IPolQ-Mod charges with the change of the strength of the exact HF exchange in several DFT functionals are determined effectively. Although the “optimal” strengths are found in B3LYP and LC-ωPBE, the improvements over the default strength are not too much. In addition to the IPolQ-Mod charges, other charge models like bond charge correction (BCC) charges could also be combined with the RP strategy to study the thermodynamic properties like solvation free energy. © 2019 Wiley Periodicals, Inc.     

Using non-Gaussian density functional fits to improve relative free energy calculations

The accurate and reliable computation of relative free energy differences remains an important long-term goal. Major stumbling blocks for achieving this goal reflect the difficulty of sampling in a known fashion along the reaction coordinate and of maximally combining information that has been collected from the simulation along the reaction coordinate. In this paper we examine the utility of a probability density functional type fit to the distribution of work events collected during a nonequilibrium sample along the reaction coordinate. This approach can readily be generalized to equilibrium sampling and has the potential to estimate the quality of a relative free energy estimate as data are being collected. The method may have the greatest utility for nonequilibrium sampling where non-Gaussian work distributions are generally present that are strongly dominated by rare event sampling in the tail region. We believe that the approach can be used to augment the design and the error analysis of relative free energy computations thus improving the ability to reliably and with known accuracy compute a relative free energy.     

Breaking the polar‐nonpolar division in solvation free energy prediction

Implicit solvent models divide solvation free energies into polar and nonpolar additive contributions, whereas polar and nonpolar interactions are inseparable and nonadditive. We present a feature functional theory (FFT) framework to break this ad hoc division. The essential ideas of FFT are as follows: (i) representability assumption: there exists a microscopic feature vector that can uniquely characterize and distinguish one molecule from another; (ii) feature‐function relationship assumption: the macroscopic features, including solvation free energy, of a molecule is a functional of microscopic feature vectors; and (iii) similarity assumption: molecules with similar microscopic features have similar macroscopic properties, such as solvation free energies. Based on these assumptions, solvation free energy prediction is carried out in the following protocol. First, we construct a molecular microscopic feature vector that is efficient in characterizing the solvation process using quantum mechanics and Poisson–Boltzmann theory. Microscopic feature vectors are combined with macroscopic features, that is, physical observable, to form extended feature vectors. Additionally, we partition a solvation dataset into queries according to molecular compositions. Moreover, for each target molecule, we adopt a machine learning algorithm for its nearest neighbor search, based on the selected microscopic feature vectors. Finally, from the extended feature vectors of obtained nearest neighbors, we construct a functional of solvation free energy, which is employed to predict the solvation free energy of the target molecule. The proposed FFT model has been extensively validated via a large dataset of 668 molecules. The leave‐one‐out test gives an optimal root‐mean‐square error (RMSE) of 1.05 kcal/mol. FFT predictions of SAMPL0, SAMPL1, SAMPL2, SAMPL3, and SAMPL4 challenge sets deliver the RMSEs of 0.61, 1.86, 1.64, 0.86, and 1.14 kcal/mol, respectively. Using a test set of 94 molecules and its associated training set, the present approach was carefully compared with a classic solvation model based on weighted solvent accessible surface area. © 2017 Wiley Periodicals, Inc.     

A size-modified poisson–boltzmann ion channel model in a solvent of multiple ionic species: Application to voltage-dependent anion channel

We present a new size-modified Poisson–Boltzmann ion channel (SMPBIC) model and use it to calculate the electrostatic potential, ionic concentrations, and electrostatic solvation free energy for a voltage-dependent anion channel (VDAC) on a biological membrane in a solution mixture of multiple ionic species. In particular, the new SMPBIC model adopts a membrane surface charge density and a natural Neumann boundary condition to reflect the charge effect of the membrane on the electrostatics of VDAC. To avoid the singularity difficulties caused by the atomic charges of VDAC, the new SMPBIC model is split into three submodels such that the solution of one of the submodels is obtained analytically and contains all the singularity points of the SMPBIC model. The other two submodels are then solved numerically much more efficiently than the original SMPBIC model. As an application of this SMPBIC submodel partitioning scheme, we derive a new formula for computing the electrostatic solvation free energy. Numerical results for a human VDAC isoform 1 (hVDAC1) in three different salt solutions, each with up to five different ionic species, confirm the significant effects of membrane surface charges on both the electrostatics and ionic concentrations. The results also show that the new SMPBIC model can describe well the anion selectivity property of hVDAC1, and that the new electrostatic solvation free energy formula can significantly improve the accuracy of the currently used formula. © 2019 Wiley Periodicals, Inc.     

Relative proton transfer abilities of acids and alcohols in gas phase and solutions

The present paper applies a method, based on SCF CNDO MO charge densities and assumption of the validity of the virial theorem, for calculation of relative proton affinities of alcohols, and substituted and unsubstituted carboxylic acids in the gas phase. The paper also chalks out a path for calculation of their relative acidities in solution phase by utilising a solvation energy equation and binding energy data in the gas phase. The results obtained by the present method are mostly in agreement with ion cyclotron resonance mass spectrometric experimental studies. The method has also been applied to cover the cases of amines in the gas phase.     

Theoretical investigation of electron transfer transition in tetracyanoethylene-contained organic complexes

In this work, the authors use complete active space self-consistent field method to investigate the photoinduced charge-separated states and the electron transfer transition in complexes ethylene-tetracyanoethylene and tetramethylethylene-tetracyanoethylene. Geometries of isolated tetracyanoethylene, ethylene, and tetramethylethylene have been optimized. The ground state and the low-lying excited states of ethylene and tetracyanoethylene have been optimized. The state energies in the gas phase have been obtained and compared with the experimentally observed values. The torsion barrier of tetracyanoethylene has been investigated through the state energy calculation at different conformations. Attention has been particularly paid to the charge-separated states and the electron transfer transition of complexes. The stacked conformations of the donor-acceptor complexes have been chosen for the optimization of the ground and low-lying excited states. Equilibrium solvation has been considered by means of conductor-like screening model both in water and in dichloromethane. It has been found that the donor and tetracyanoethylene remain neutral in complexes in ground state (1)A(1) and in lowest triplet state (3)B(1), but charge separation appears in excited singlet state (1)B(1). Through the correction of nonequilibrium solvation energy based on the spherical cavity approximation, pi-->pi* electron transfer transition energies have been obtained. Compared with the experimental measurements in dichloromethane, the theoretical results in the same solvent are found higher by about 0.5 eV.     

Exact density functional theory for ideal polymer fluids with nearest neighbor bonding constraints

We present a new density functional theory of ideal polymer fluids, assuming nearest-neighbor bonding constraints. The free energy functional is expressed in terms of end site densities of chain segments and thus has a simpler mathematical structure than previously used expressions using multipoint distributions. This work is based on a formalism proposed by Tripathi and Chapman [Phys. Rev. Lett. 94, 087801 (2005)]. Those authors obtain an approximate free energy functional for ideal polymers in terms of monomer site densities. Calculations on both repulsive and attractive surfaces show that their theory is reasonably accurate in some cases, but does differ significantly from the exact result for longer polymers with attractive surfaces. We suggest that segment end site densities, rather than monomer site densities, are the preferred choice of "site functions" for expressing the free energy functional of polymer fluids. We illustrate the application of our theory to derive an expression for the free energy of an ideal fluid of infinitely long polymers.