Thermodynamic perturbation theory of simple liquids

It was proven that after averaging over the canonical Gibbs ensemble, the mean perturbation energy was singled out of the classical partition function before the expansion in a series of perturbation theory. Therefore, the term that formally coincides with first order perturbation theory in a decomposition of the Helmholtz free energy bears no relationship to perturbation theory. Then the proper series of the thermodynamic perturbation theory always starts with a second order infinitesimal. Therefore, the wellknown condition of applicability of the thermodynamic perturbation theory, “...the requirement that the perturbation energy per particle be small compared with T...” (L. D. Landau and E. M. Livshits, Statistical Physics, Vol. V, Pt. I), can be substantially weakened. The most important factor for applicability of thermodynamic perturbation theory is the value of many-particle correlations in an unperturbed system, but not the smallness of the perturbation potential.     

Solvation thermodynamics: theory and applications

Potential distribution and coupling parameter theories are combined to interrelate previous solvation thermodynamic results and derive several new expressions for the solvent reorganization energy at both constant volume and constant pressure. We further demonstrate that the usual decomposition of the chemical potential into noncompensating energetic and entropic contributions may be extended to obtain a Gaussian fluctuation approximation for the chemical potential plus an exact cumulant expansion for the remainder. These exact expressions are further related to approximate first-order thermodynamic perturbation theory predictions and used to obtain a coupling-parameter integral expression for the sum of all higher-order terms in the perturbation series. The results are compared with the experimental global solvation thermodynamic functions for xenon dissolved in n-hexane and water (under ambient conditions). These comparisons imply that the constant-volume solvent reorganization energy has a magnitude of at most approximately kT in both experimental solutions. The results are used to extract numerical values of the solute-solvent mean interaction energy and associated fluctuation entropy directly from experimental solvation thermodynamic measurements.     

Some practical aspects of free energy calculations from molecular dynamics simulation

In this work, we address two critical aspects of calculation of the free energy differences in molecular systems from molecular simulations. The first aspect involves checking whether the calculated free energy difference depends significantly on the extent of perturbation used for accomplishment of a given transformation. The second aspect of interest is to verify if the sampling errors in calculating the free energy differences between the wild-type molecule and a mutated one in its free state and in a complex are similar, or not, for a finite-length dynamic simulation. The reliability of the free energy estimates obtained from molecular simulations using thermodynamic cycles depends in part on this fact. For investigating these aspects, we use a self-transformation scheme in which a transformation of a part of a molecular system into itself is considered. We perform MD simulations of DNA fragments in which a part of a specific base is subjected to such a self-transformation. Results indicate that the estimated free energy differences do not depend significantly on the extent of perturbation used to achieve the transformation. Interestingly, the variation in the cumulative free energy difference, ΔA, with the coupling parameter, λ, depends significantly on the extent of perturbation. We examine the physical basis of the observed nature of the variation of the accumulated free energy difference, ΔA, against the λ value in the case of a self-transformation. In a thermodynamic cycle, the sampling errors due to the finite-length simulation for the molecular system are found to be similar to each other for the two perturbations (free and in a complex) justifying the use of such approach in calculating ΔΔA in molecular complexes. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 877–885, 1999     

On the choice of a reference state for one‐step perturbation calculations between polar and nonpolar molecules in a polar environment

One‐step perturbation is an efficient method to estimate free energy differences in molecular dynamics (MD) simulations, but its accuracy depends critically on the choice of an appropriate, possibly unphysical, reference state that optimizes the sampling of the physical end states. In particular, the perturbation from a polar moiety to a nonpolar one and vice versa in a polar environment such as water poses a challenge which is of importance when estimating free energy differences that involve entropy changes and the hydrophobic effect. In this work, we systematically study the performance of the one‐step perturbation method in the calculation of the free enthalpy difference between a polar water solute and a nonpolar “water” solute molecule solvated in a box of 999 polar water molecules. Both these polar and nonpolar physical reference states fail to predict the free enthalpy difference as obtained by thermodynamic integration, but the result is worse using the nonpolar physical reference state, because both a properly sized cavity and a favorable orientation of the polar solute in a polar environment are rarely, if ever, sampled in a simulation of the nonpolar solute in such an environment. Use of nonphysical soft‐core reference states helps to sample properly sized cavities, and post‐MD simulation rotational and translational sampling of the solute to be perturbed leads to much improved free enthalpy estimates from one‐step perturbation. © 2012 Wiley Periodicals, Inc.     

Finite-temperature full configuration interaction

The exact basis-set values of various thermodynamic potentials of a molecule are evaluated by the finite-temperature full configuration-interaction (FCI) method using ab initio molecular integrals over Gaussian-type orbitals. The thermodynamic potentials considered are the grand partition function, grand potential, internal energy, entropy, and chemical potential in the grand canonical ensemble as well as the partition function, Helmholtz energy, internal energy, and entropy in canonical ensemble. Approximations to FCI that are accurate at low and high temperatures are proposed, implemented, and tested. The results of finite-temperature FCI and its approximations are compared with one another as well as with the results of finite-temperature zeroth-order many-body perturbation theory, in which the Fermi–Dirac statistics is exact. Analytical asymptotic properties in the low- or high-temperature limits of some of these thermodynamic potentials are also given.     

Assessment of enveloping distribution sampling to calculate relative free enthalpies of binding for eight netropsin-DNA duplex complexes in aqueous solution

The performance of enveloping distribution sampling (EDS) simulations to estimate free enthalpy differences associated with seven alchemical transformations of A-T into G-C base pairs at the netropsin binding site in the minor groove of a 13-base pair DNA duplex in aqueous solution is evaluated. It is demonstrated that sufficient sampling can be achieved with a two-state EDS Hamiltonian even for large perturbations such as the simultaneous transformation of up to three A-T into three G-C base pairs. The two parameters required to define the EDS reference state Hamiltonian are obtained automatically using a modified version of a scheme presented in earlier work. The sensitivity of the configurational sampling to a variation of these parameters is investigated in detail. Although for relatively small perturbations, that is, one base pair, the free enthalpy estimate depends only weakly on the EDS parameters, the sensitivity is stronger for the largest perturbation. Yet, EDS offers various convenient measures to evaluate the degree of sampling and thus the reliability of the free enthalpy estimate and appears to be an efficient alternative to the conventional thermodynamic integration methodology to obtain free energy differences for molecular systems.     

Free energy derivatives: A new method for probing the convergence problem in free energy calculations

The convergence behavior of free energy calculations has been explored in more detail than in any previously reported work, using a model system of two neon atoms in a periodic box of water. We find that for thermodynamic integration-type free energy calculations as much as a nanosecond or more molecular dynamics sampling is required to obtain a fully converged value for a single λ point of the integrand. The concept of “free energy derivatives” with respect to the individual parameters of the force field is introduced. This formalism allows the total convergence of the simulation to be deconvoluted into components. A determination of the statistical “sampling ratio” from these simulations indicates that for window-type free energy calculations carried out in a periodic waterbox of typical size at least 0.6 ps of sampling should be performed at each window (0.7 ps if constraint contributions to the free energy are being determined). General methods to estimate and reduce the error in thermodynamic integration and free energy perturbation calculations are discussed. We show that the difficulty in applying such methods is determining a reliable estimate of the correlation length from a short series of data. © 1994 by John Wiley & Sons, Inc.     

On the Temperature–Pressure Free‐Energy Landscape of Proteins

We studied the thermodynamic stability of a small monomeric protein, staphylococcal nuclease (Snase), as a function of both temperature and pressure, and expressed it as a 3D free‐energy surface on the p,T‐plane using a second‐order Taylor expansion of the Gibbs free‐energy change ΔG upon unfolding. We took advantage of a series of different techniques (small‐angle Xray scattering, Fourier‐transform infrared spectroscopy, differential thermal analysis, pressure perturbation calorimetry and densitometry) in the evaluation of the conformation of the protein and in evaluating the changes in the thermodynamic parameters upon unfolding, such as the heat capacity, enthalpy, entropy, volume, isothermal compressibility and expansivity. The calculated results of the free‐energy landscape of the protein are in good agreement with experimental data of the p,T‐stability diagram of the protein over a temperature range from 200 to 400 K and at pressures from ambient pressure to 4000 bar. The results demonstrate that combined temperature–pressure‐dependent studies can help delineate the free‐energy landscape of proteins and hence help elucidate which features and thermodynamic parameters are essential in determining the stability of the native conformational state of proteins. The approach presented may also be used for studying other systems with so‐called re‐entrant or Tamman loop‐shaped phase diagrams.     

Knots “Choke Off” Polymers upon Stretching

Long polymer chains inevitably get tangled into knots. Like macroscopic ropes, polymer chains are substantially weakened by knots and the rupture point is always located at the “entry” or “exit” of the knot. However, these phenomena are only poorly understood at a molecular level. Here we show that when a knotted polyethylene chain is tightened, most of the stress energy is stored in torsions around the curved part of the chain. The torsions act as “work funnels” that effectively localize mechanical stress in the immediate vicinity of the knot. As a result, the knot “chokes” the chain at its entry or exit, thus leading to bond rupture at much lower forces than those needed to break a linear, unknotted chain. Our work not only explains the weakening of the polymer chain and the position of the rupture point, but more generally demonstrates that chemical bonds do not have to be extensively stretched to be broken.     

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.     

Towards a “Chemical” Hamiltonian

An analysis of the LCAO Hamiltonian is performed in terms of a “mixed” formulation of the second quantization for nonorthogonal orbitals, compressing the different interactions to one- and two-center terms as far as possible by performing appropriate projections. For this purpose an operator of atomic charge is also introduced, the expectation values of which are the Mulliken gross atomic populations on the individual atoms. The LCAO Hamiltonian is decomposed into terms having different physical meaning and significance: (i) sum of effective atomic Hamiltonians; (ii) the electrostatic interactions in the point-charge approximation; (iii) the electrostatic effects connected with the deviation of the actual charge distribution from the pointlike one; (iv) two-center overlap effects; (v) finite basis (“counterpoise”) correction terms related to the individual atoms; and (vi) similar finite basis correction terms with respect to the two-center interactions. Only terms of types (i) to (iv), containing no three- or four-center integrals, are considered as having physical significance. Based on the analysis of the Hamiltonian, an energy partitioning scheme is developed, and explicit expressions are given for one- and two-center (and basis extension) components of the SCF energy. The approach is also applied to the problem of intermolecular interactions, and an explicit formula is given permitting calculation of the “counterpoise” part of the supermolecule energy by properly taking into account that it depends not only on the extension of the basis, but also on the occupation of the additional orbitals in the intervening molecule—a factor completely overlooked in the usual scheme of calculations.     

Use of numerical methods in the prediction of polymer adsorption

Average polymer segment densities and thermodynamic properties of polymer adsorbed at liquid–solid interfaces were computed by extension of the polymer adsorption theory of Forsman and Hughes. Expressions were derived for the total free energy of adsorbed polymer chains by using the Flory-Huggins theory to represent free energy of mixing. A square-well potential was used to represent segment–surface interaction, and configurational entropy was calculated from the probability density function for the radius of gyration of random-flight chains. For each specified amount of surface coverage the free energy of the adsorbed polymer was minimized by varying the density of segments normal to the adsorbing surface and using a modified gradient search algorithm on a digital computer. Two different segment densities were considered, and they both gave qualitatively the same results. The two densities were (1) the sum of two Gaussian distributions and (2) a two-step density distribution. Isotherms were then calculated by equating the partial molal free energy of polymer at the surface to that of polymer in bulk solution for each specified amount of surface coverage. The results showed that for the initial region of the isotherms the distribution of polymer segments normal to the surface consisted of a high-density layer adjacent to the surface and a low-density “tail” extending far out into the solution. At higher amounts of adsorbed polymer, i.e., in the general concentration range of the pseudo-plateau, the tail of the polymer density distribution was predicted to thicken, and a single Gaussian distribution best described the segment density. Predicted adsorptions were in good agreement with reported experimental values.     

Temperature-dependent approach to chemical reactivity concepts in density functional theory

The chemical reactivity concepts of density functional theory are studied through a unified view in the temperature-dependent approach provided by the grand canonical ensemble. This procedure leads to a more general perspective that enriches our understanding of the behavior of the average energy and its derivatives with respect to the average number of electrons, provides alternative definitions for those quantities that are “ill defined” at zero temperature, and allows one to determine the relationships among reactivity concepts at any temperature. In particular, it has been found that at high temperatures the parabolic model for reactivity indicators may be justified through the electronic entropy term in the Helmholtz free energy, and that at nonzero temperatures there is an electronic heat capacity contribution to the average energy. In summary, the unified view of the temperature-dependent approach is an important complement to the zero-temperature formulation that clarifies fundamental issues therein.     

The changes in free energies and entropies from analytical radial distribution functions

The changes in Helmholtz free energies and entropies in dense fluids have been evaluated using three known analytical expressions for radial distribution functions (RDFs) of Lennard–Jones (L-J) fluid. This method provides a simpler and a more expeditious way for the calculation of free energy and entropy in L-J dense fluids through statistical mechanics. Previously, integral equations or perturbation theories were used for this purpose. Such approach not only tests the power of analytical distribution functions in predicting the changes in Helmholtz free energies and entropies, but also specifies better expressions in determining these properties. The results are compared with experimental data and an accurate analytic equation of state for the L-J fluid. It is shown if an expression properly presents RDFs as a function of interparticle distance, density and temperature, it is possible to calculate the changes in Helmholtz free energies and entropies from analytical distribution functions.     

Use of numerical methods in the prediction of polymer adsorption

Average polymer segment densities and thermodynamic properties of polymer adsorbed at liquid–solid interfaces were computed by extension of the polymer adsorption theory of Forsman and Hughes. Expressions were derived for the total free energy of adsorbed polymer chains by using the Flory-Huggins theory to represent free energy of mixing. A square-well potential was used to represent segment–surface interaction, and configurational entropy was calculated from the probability density function for the radius of gyration of random-flight chains. For each specified amount of surface coverage the free energy of the adsorbed polymer was minimized by varying the density of segments normal to the adsorbing surface and using a modified gradient search algorithm on a digital computer. Two different segment densities were considered, and they both gave qualitatively the same results. The two densities were (1) the sum of two Gaussian distributions and (2) a two-step density distribution. Isotherms were then calculated by equating the partial molal free energy of polymer at the surface to that of polymer in bulk solution for each specified amount of surface coverage. The results showed that for the initial region of the isotherms the distribution of polymer segments normal to the surface consisted of a high-density layer adjacent to the surface and a low-density “tail” extending far out into the solution. At higher amounts of adsorbed polymer, i.e., in the general concentration range of the pseudo-plateau, the tail of the polymer density distribution was predicted to thicken, and a single Gaussian distribution best described the segment density. Predicted adsorptions were in good agreement with reported experimental values.     

Molecular dynamics simulations and free energy calculations on the enzyme 4‐hydroxyphenylpyruvate dioxygenase

4‐Hydroxyphenylpyruvate dioxygenase is a relevant target in both pharmaceutical and agricultural research. We report on molecular dynamics simulations and free energy calculations on this enzyme, in complex with 12 inhibitors for which experimental affinities were determined. We applied the thermodynamic integration approach and the more efficient one‐step perturbation. Even though simulations seem well converged and both methods show excellent agreement between them, the correlation with the experimental values remains poor. We investigate the effect of slight modifications on the charge distribution of these highly conjugated systems and find that accurate models can be obtained when using improved force field parameters. This study gives insight into the applicability of free energy methods and current limitations in force field parameterization. © 2011 Wiley Periodicals, Inc. J Comput Chem 2011