Molecular thermodynamic model for equilibria in solution IV. Macroscopic partition functions

The molecular ensembles statistically distributed according to internal specific characteristics and distinguished for the different exchanges with the surroundings are represented on the macroscopic scale by appropriate partition functions. The partition function for osmotic non-reacting ensemble is a function of concentration or activity of the ligand and is suited to the definition of thermodynamic potential μ. The partition function for thermal non-reacting ensemble shows the dependence upon the temperature and that for thermo-osmotic non-reacting ensemble shows the dependence upon both concentration and temperature.

The reaction partition function is suited to show the distribution of the different species over the different enthalpy levels of the reacting ensemble. The dispersion of the distributions are represented by second derivatives of the partition function.

The information contained in the entropy axis of the thermodynamic space for reacting ensembles concerning the induced dilution of the bound ligand and final dilution of the free ligand can be spanned to a formation function diagram where free energy of reaction can be graphically represented.     

Molecular thermodynamic model for equilibria in solution: I. Reacting and non-reacting ensembles

The partition functions of solution thermodynamics are mathematical representations of the properties of molecular ensembles statistically distributed according to specific characteristics. The ensembles are classified as non-reacting or reacting. The non-reacting ensembles are characterized by one mean enthalpy level with dispersion around the mean. The reacting ensembles are characterized by two or more distinctly separated enthalpy levels over which the different species are variably distributed, depending on concentration and/or temperature.

The non-reacting ensembles can be distinguished into microcanonical, thermal, osmotic, thermo-osmotic ensembles, depending on the type of exchange with the surroundings which is connected to the fluctuations of the ensemble variables. The reacting ensembles can be distinguished into thermal, osmotic, thermo-osmotic, electrochemical, electro-osmotic, electro-thermal, electro-thermo-osmotic ensembles, depending on the type of reaction and of exchange with the surroundings.     

Molecular energy parameters by solution thermodynamics

The relationships between statistical thermodynamics and equilibrium constants, either cumulative, stepwise or specific site constants are investigated. In order to show the link between equilibrium constants and statistical thermodynamic microscopic properties, a distinction has been introduced between non-reacting and reacting systems. The non-reacting systems are those for which continuous statistical distributions of enthalpies can be assumed. The distribution function can be obtained as an integral of the interparticle potential extended to the whole ensemble. Molecular partition functions ζ_A, are used to describe the properties of the ensembles. The reacting ensemble is represented by distinct distributions of enthalpies, each distribution being grouped around a mean value. Each level is representative of one species. Around each mean level the distribution is continuous as in non-reacting ensembles. The reacting ensemble of particles is described by a grand canonical molar partition function Z_M=(1+kγ(i)[A])^t where k is the specific site constant, γ(i) is the cooperativity function, [A] is the concentration of free ligand, and the power t indicates the maximum number of i sites in one class. The specific site constant k is proportional to the affinity of binding and is related to the depth of the minimum of the potential function. The factor γ_i is the cooperativity factor given by the value of the cooperativity function γ(i) at the ith level and indicates how the depth of the potential function is affected by previous binding of a ligand. The values of the stability constant and cooperativity factor can be optimized by a computer program. The derivatives of the partition function Z_M with respect to ln[A] correspond to the formation function 〈n〉 (first derivative) and to the buffer capacity B_C (second derivative). The derivatives of the partition function Z_M with respect to temperature are reaction enthalpy ΔH (first derivative with −(1/T)) and apparent heat capacity ΔC_p,app (second derivative with ln T). The denaturation heat obtained by integration of ΔC_p,app dT for many proteins explains why the denaturation enthalpy depends linearly upon T.     

Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. I. Thermodynamic properties in the bulk and at the liquid-vapor phase boundary

The Wang-Landau sampling is a powerful method that allows for a direct determination of the density of states. However, applications to the calculation of the thermodynamic properties of realistic fluids have been limited so far. By combining the Wang-Landau method with expanded grand-canonical simulations, we obtain a high-accuracy estimate for the grand-canonical partition function for atomic and molecular fluids. Then, using the formalism of statistical thermodynamics, we are able to calculate the thermodynamic properties of these systems, for a wide range of conditions spanning the single-phase regions as well as the vapor-liquid phase boundary. Excellent agreement with prior simulation work and with the available experimental data is obtained for argon and CO(2), thereby establishing the accuracy of the method for the calculation of thermodynamic properties such as free energies and entropies.     

Bridging continuum and statistical thermodynamics via equations of state and the density of states

The connection between molecular force fields and equations of state (EoS) is typically established at the level of predicted quantities, e.g., by comparing simulation data and EoS data. In this paper we show how an EoS can be used to extract the density of states (Omega) of a system thus establishing a deeper connection between EoSs and statistical thermodynamics. We also show how such an EoS Omega can be used to aid molecular simulation methods designed to map out Omega (like the multicanonical approach). Central to the implementation of these ideas is the fact that the configurational Omega is related to thermodynamic properties accessible by an EoS via Boltzmann's equation. Sample calculations are presented for the Omega relevant to isothermal-isobaric and grand canonical ensemble simulations using the hard-sphere system and the Lennard-Jones system as model fluids, and the Carnahan-Starling EoS and a cubic EoS, respectively, as thermodynamic models.     

子的配分函数——物理意义与作用

讨论怎样理解子的配分函数的物理意义,以及它和系统的配分函数在计算热力学函数中的作用,指出它既不是系统的广延性质,也不是系统的强度性质,只是联系系统热力学函数与微观信息的纽带。     

Complexation thermodynamics of free and graft oligomers with complementary polymers

The dilute solution complexation equilibrium between linear macromolecules and smaller complementary oligomers is considered when: (1) the oligomers are free in solution; and (2) the oligomers are covalently attached at one end to the polymer. A general statistical mechanical framework is developed and is illustrated using a simple random walk model for polymer conformation. The statistical mechanical partition functions are formulated using a generating function technique, allowing thermodynamic averages in the complexed state to be calculated. Loops, trains, and tails of all possible length are allowed in the conformation of a complexed oligomer. Simulation results for the free oligomer case are compared with those obtained for oligomers covalently attached to the polymeric molecular. The model provides a theoretical explanation for the experimentally observed enhancement of complexation of oligomers grafted to the complementary polymers.     

Multiple minima hypersurfaces of water clusters for calculations of association energy

Multiple minima of water cluster hypersurfaces are explored to find thermodynamic properties by means of the corresponding partition functions of their canonical distributions. The combination of semiempirical quantum chemical procedures for calculating the cluster energies in local minima of supermolecules and the statistical thermodynamics approach for both the evaluation of macroscopic association energies and the possible reduction by average of absolute errors intrinsic to the parametrized Hamiltonian are shown to provide an appropriate model for comparison between experimental and theoretical results. The method can explicitly take into account environmental effects due to intermolecular interaction. Water trimer and tetramer association energies of ?10.9 and ?14.1 kJ/mol obtained from virial coefficient calculations compare very well to the values of ?10.5 and ?16.4 kJ/mol, respectively, found for the theoretical association energies in this paper. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 8–16, 2000     

Estimating the temperature dependence of peptide folding entropies and free enthalpies from total energies in molecular dynamics simulations

The temperature dependence of thermodynamic quantities, such as heat capacity, entropy and free enthalpy, may be obtained by using well-known equations that relate these quantities to the enthalpy of the molecular system of interest at a range of temperatures. In turn, the enthalpy of a molecular system can be estimated from molecular dynamics simulations of an appropriate model. To demonstrate this, we have investigated the temperature dependence of the enthalpy, heat capacity, entropy and free enthalpy of a system that consists of a beta-heptapeptide in methanol and have used the statistical mechanics relationships to describe the thermodynamics of the folding/unfolding equilibrium of the peptide. The results illustrate the power of current molecular simulation force fields and techniques in establishing the link between thermodynamic quantities and conformational distributions.     

Comparison of two simulation methods to compute solvation free energies and partition coefficients

The thermodynamic integration (TI) and expanded ensemble (EE) methods are used here to calculate the hydration free energy in water, the solvation free energy in 1‐octanol, and the octanol‐water partition coefficient for a six compounds of varying functionality using the optimized potentials for liquid simulations (OPLS) all‐atom (AA) force field parameters and atomic charges. Both methods use the molecular dynamics algorithm as a primary component of the simulation protocol, and both have found wide applications in fields such as the calculation of activity coefficients, phase behavior, and partition coefficients. Both methods result in solvation free energies and 1‐octanol/water partition coefficients with average absolute deviations (AAD) from experimental data to within 4 kJ/mol and 0.5 log units, respectively. Here, we find that in simulations the OPLS‐AA force field parameters (with fixed charges) can reproduce solvation free energies of solutes in 1‐octanol with AAD of about half that for the solute hydration free energies using a extended simple point charge (SPC/E) model of water. The computational efficiency of the two simulation methods are compared based on the time (in nanoseconds) required to obtain similar standard deviations in the solvation free energies and 1‐octanol/water partition coefficients. By this analysis, the EE method is found to be a factor of nine more efficient than the TI algorithm. For both methods, solvation free energy calculations in 1‐octanol consume roughly an order of magnitude more CPU hours than the hydration free energy calculations. © 2012 Wiley Periodicals, Inc.     

基于光微热量-荧光光谱联用技术研究光催化热力学和动力学的温度效应

利用光微热量-荧光光谱联用技术,对光催化过程的热谱和光谱信息同步监测,获取了五个温度下,g-C_3N_4@Ag@Ag_3PO_4光催化降解罗丹明B的原位热动力学、光谱动力学信息,探究了温度对相关参数的影响。结果表明,催化降解反应分为三个阶段:(i)污染物和催化剂的光响应过程;(ii)光响应吸热和污染物降解放热的竞争过程;(iii)保持稳定的放热率。吸热和放热的竞争过程符合一级动力学,降解速率随着温度的升高而增大;稳定放热阶段为拟零级反应,在283.15 K、288.15 K、293.15 K、298.15 K、303.15 K下的放热速率分别为0.4668±0.3875μJ·s~(-1)、0.5314±0.3379μJ·s~(-1)、0.5064±0.3234μJ·s~(-1)、0.5328±0.3377μJ·s~(-1)、0.5762±0.3452μJ·s~(-1)。本研究为探究光催化过程的原位热力学、热动力学及光谱信息及机理的推测提供科学依据。     

Monte Carlo method for computing density of states and quench probability of potential energy and enthalpy landscapes

The thermodynamics and kinetics of a many-body system can be described in terms of a potential energy landscape in multidimensional configuration space. The partition function of such a landscape can be written in terms of a density of states, which can be computed using a variety of Monte Carlo techniques. In this paper, a new self-consistent Monte Carlo method for computing density of states is described that uses importance sampling and a multiplicative update factor to achieve rapid convergence. The technique is then applied to compute the equilibrium quench probability of the various inherent structures (minima) in the landscape. The quench probability depends on both the potential energy of the inherent structure and the volume of its corresponding basin in configuration space. Finally, the methodology is extended to the isothermal-isobaric ensemble in order to compute inherent structure quench probabilities in an enthalpy landscape.     

Statistical mechanical theory for steady state systems. V. Nonequilibrium probability density

The phase space probability density for steady heat flow is given. This generalizes the Boltzmann distribution to a nonequilibrium system. The expression includes the nonequilibrium partition function, which is a generating function for statistical averages and which can be related to a nonequilibrium free energy. The probability density is shown to give the Green-Kubo formula in the linear regime. A Monte Carlo algorithm is developed based upon a Metropolis sampling of the probability distribution using an umbrella weight. The nonequilibrium simulation scheme is shown to be much more efficient for the thermal conductivity of a Lennard-Jones fluid than the Green-Kubo equilibrium fluctuation method. The theory for heat flow is generalized to give the generic nonequilibrium probability densities for hydrodynamic transport, for time-dependent mechanical work, and for nonequilibrium quantum statistical mechanics.     

The High Temperature Vibrational Partition Function of Stretching of Triatomic Systems

Plasma Chemistry and Plasma Processing - The statistical thermodynamic model for the vibrational partition function with separated stretching and bending is developed. The model is studied on the...     

Inverse Monte Carlo procedure for conformation determination of macromolecules

A novel numerical method for determining the conformational structure of macromolecules is applied to idealized biomacromolecules in solution. The method computes effective inter-residue interaction potentials solely from the corresponding radial distribution functions, such as would be obtained from experimental data. The interaction potentials generate conformational ensembles that reproduce thermodynamic properties of the macromolecule (mean energy and heat capacity) in addition to the target radial distribution functions. As an evaluation of its utility in structure determination, we apply the method to a homopolymer and a heteropolymer model of a three-helix bundle protein [Zhou, Y.; Karplus, M. Proc Natl Acad Sci USA 1997, 94, 14429; Zhou, Y. et al. J Chem Phys 1997, 107, 10691] at various thermodynamic state points, including the ordered globule, disordered globule, and random coil states.     

Considerations on the temperature dependence of the gas-liquid chromatographic retention

A discussion on the temperature dependence of the partition coefficient K is developed. This discussion embraces topics such as the limitations of conventional thermodynamic approaches followed in the chromatographic literature, qualitative theoretical notions arising from molecular thermodynamics and the experimental information that is accessible through modern capillary gas chromatography. It is shown that the heat capacity difference of solute transfer for flexible molecules has at least one maximum in the chromatographic range of temperature. As a consequence, a great amount of experimental data is required for a correct thermodynamic interpretation of the chromatographic retention.     

缔合溶液热力学性质与谱学性质的关联

缔合溶液具有与理想溶液显著不同的热力学和谱学性质,对于热力学和谱学的研究,有助于我们理解缔合溶液的特殊行为.谱学技术中核磁共振(NMR)、红外(IR)和拉曼(Raman)光谱是研究分子间相互作用和溶液结构等微观性质的有效方法,谱学已成为分子热力学研究体系"四面体结构"中的第四个顶点.本文对缔合溶液中热力学(汽液平衡和焓)和谱学(NMR,IR和Raman)联系的最新研究进展进行了综述,着重介绍相关的模型,如化学缔合模型、局部组成(LC)、格子流体氢键(LFHB)理论以及统计缔合流体理论(SAFT).