Configurational Partition Function Research Articles

Computation of rate coefficients in solutions based on transition state theory combined with a heuristically corrected polarizable continuum model: intermolecular Diels-Alder reactions as case studies.

Transition state theory (TST) based on activation parameters computed using quantum mechanics calculations combined with the polarizable continuum model (QM/PCM) is a fundamental tool for investigating reaction rates in the liquid phase. In conventional QM/PCM methods, thermodynamic data and partition functions for a solute are often derived from a quasi-ideal gas treatment (IGT) widely implemented in commercially available computation packages. This approach tends to overestimate entropy because calculations of thermodynamic parameters in the liquid phase ignore hindered translational and rotational modes in real solutions. The present work formulated partition functions for more realistic solutes hindered by surrounding solvent molecules in conjunction with the basic QM/PCM concept. In addition, a configuration partition function for solute molecules at a standard concentration of 1 mol dm-3 was incorporated using a simple lattice model. The canonical partition function and thermodynamic functions were derived based on statistical thermodynamics for localized systems. Expressions for rate coefficients within TST were also derived with a consistent formalism based on the standard state selected in partition function calculations. The performance of the proposed method was assessed by predicting rate coefficients for three different Diels-Alder reactions and comparing these with experimental results. QM/PCM calculations at the G4//ωB97X-D/6-311++G(d,p)/IEF-PCM level of theory with corrections for the dispersion and repulsion energies were performed to obtain the electronic structures of stationary points on potential energy surfaces as a means of finding activation enthalpy, entropy and Gibbs energy values based on revised partition functions as well as predicting rate coefficients. The activation Gibbs energies obtained from our proposed method were lower than those obtained from the IGT method due to reasonable entropy computations. The proposed method overestimated the rate coefficients by one to two orders of magnitude compared to the experimental values, whereas the IGT method underestimated them by the same amount. This discrepancy arises because the proposed method calculates the partition function from the viewpoint of a localized system, whereas the IGT method calculates it from the viewpoint of a non-localized system. Given that actual liquids exist in a state between non-localized and localized systems, it is essential to formulate the partition function in a way that more accurately represents the liquid state.

Shannon information entropy, soliton clusters and Bose-Einstein condensation in log gravity

We give a probabilistic interpretation of the configurational partition function of the logarithmic sector of critical cosmological topologically massive gravity, in which the Hurwitz numbers considered in our previous works assume the role of probabilities in a distribution on cycles of permutations. In particular, it is shown that the permutations are distributed according to the Ewens sampling formula which plays a major role in the theory of partition structures and their applications to diffusive processes of fragmentation, and in random trees. This new probabilistic result together with the previously established evidence of solitons in the theory provide new insights on the instability originally observed in the theory. We argue that the unstable propagation of a seed soliton at single particle level induces the generation of fragments of defect soliton clusters with rooted tree configuration at multiparticle level, providing a disordered landscape. The Shannon information entropy of the probability distribution is then introduced as a measure of the evolution of the unstable soliton clusters generated. Finally, based on Feynman’s path integral formalism on permutation symmetry in the λ-transition of liquid helium, we argue that the existence of permutation cycles in the configurational log partition function indicates the presence of Bose-Einstein condensates in log gravity.

Combinatorics of the double-dimer model

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to a problem in Donaldson-Thomas and Pandharipande-Thomas theory. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.

Adsorption of binary mixtures on two-dimensional triangular lattices

The adsorption of interacting binary mixtures on triangular lattices is studied by combining theory and Monte Carlo (MC) simulations. Two theoretical approximations are used in the present work: (i) the cluster approximation (CA), based on exact counting of adsorption states on small lattices; and (ii) an extension of the standard quasi-chemical approximation (QCA) that includes two adsorbed species (a and b). In the case of CA, an own algorithm is developed to obtain the configurational grand partition function for small cells. Repulsive lateral couplings between adsorbate-adsorbate species are incorporated in the lattice-gas framework. Theoretical (CA and QCA) total and partial adsorption isotherms are compared with MC simulations. Quantitative and qualitative differences are shown and discussed, being CA the more accurate approach in all cases.

Stochastic dynamics of quantized vortices: Continuum and discrete approaches

The thermodynamic equilibrium in a system of randomly quantized vortices in superfluid helium with counterflow between the normal and superfluid components is considered. Both continuum and discrete approaches are studied. Even when using the continuum approach for the system as a whole, the partition function for the various vortex loop configurations can only be calculated by involving the discrete approach. It is obvious that discretization is important to numerical studies. Numerical simulation is the main tool for solving the stochastic dynamics of quantum vortex filaments subject to random force, due to their complexity. Some physical consequences of the results are discussed.

Analytical ab initio-Based Modeling of the Adsorption Isotherm

An analytical ab initio-based model for adsorption isotherm is presented. As input data, this model takes the adsorption Gibbs free energies of different possible coverages within a reasonably large unit cell. In this way, the lateral interactions between adsorbates and their dependence on temperature are included. The self-dependence of surface coverage is taken into account via the introduction of heterogeneity of adsorption and the use of an appropriate configurational partition function. The model reproduces the isotherm and selectivity (for binary gas mixtures) of ab initio-based grand-canonical Monte Carlo simulations. The presented isotherm has a moderated dependence on the width of the chosen ab initio unit cell, particularly for high surface loading. This isotherm seems to be a very good approximation for the grand-canonical solution, depending on the accuracy and completeness of the ab initio data.

Temperature extrapolation of multicomponent grand canonical free energy landscapes.

We derive a method for extrapolating the grand canonical free energy landscape of a multicomponent fluid system from one temperature to another. Previously, we introduced this statistical mechanical framework for the case where kinetic energy contributions to the classical partition function were neglected for simplicity [N. A. Mahynski et al., J. Chem. Phys. 146, 074101 (2017)]. Here, we generalize the derivation to admit these contributions in order to explicitly illustrate the differences that result. Specifically, we show how factoring out kinetic energy effects a priori, in order to consider only the configurational partition function, leads to simpler mathematical expressions that tend to produce more accurate extrapolations than when these effects are included. We demonstrate this by comparing and contrasting these two approaches for the simple cases of an ideal gas and a non-ideal, square-well fluid.

Investigation of strain effects on phase diagrams in the ternary nitride alloys (InAlN, AlGaN, InGaN)

In this work, we used a modified Stillinger–Weber potential and a methodology of free energy calculation based on numerical computation of the configuration partition function of an alloy, to make a comprehensive study of the properties of group-III nitride ternary compounds (InxGa1−xN; InxAl1−xN; AlxGa1−xN). The wurtzite structure was used; and the critical temperatures for the random ternary alloys are determined as 2717 K for InxAl1−xN, 1718 K for InxGa1−xN, and 177 K for AlxGa1−xN, respectively. Therefore, AlxGa1−xN has no unstable mixing region at typical growth temperatures around 1100 °C. In contrast, InxAl1−xN and InxGa1−xN exhibit a wide unstable region, which means that being thick layers, their stability as homogeneous alloys is probably limited. In agreement with other reports, it is also pointed out that the critical temperature Tc may be decreased when the layers are grown under strain. Although the compression and extension have the same effect below 1.5% strain, it is shown, for the first time, that when the compressive strain goes beyond, Tc abruptly increases in contrast to the case of tensile strain where it continues to decrease.

Interplay of the sign problem and the infinite volume limit: Gauge theories with a theta term

QCD and related gauge theories have a sign problem when a $\theta$-term is included; this complicates the extraction of physical information from Euclidean space calculations as one would do in lattice studies. The sign problem arises in this system because the partition function for configurations with fixed topological charge $Q$, $\mathcal{Z}_Q$, are summed weighted by $\exp(i Q \theta)$ to obtain the partition function for fixed $\theta$, $ \mathcal{Z}(\theta)$. The sign problem gets exponentially worse numerically as the space-time volume is increased. Here it is shown that apart from the practical numerical issues associated with large volumes, there are some interesting issues of principle. A key quantity is the energy density as a function of $\theta$, $\varepsilon(\theta) = -\log \left( \mathcal{Z}(\theta) \right )/V$. This is expected to be well defined in the large 4-volume limit. Similarly, one expects the energy density for a fixed topological density $\tilde{\varepsilon}(Q/V) = -\log \left(\mathcal{Z}_Q \right )/V$ to be well defined in the limit of large 4-volumes. Intuitively, one might expect that if one had the infinite volume expression for $\tilde{\varepsilon}(Q/V)$ to arbitrary accuracy, that one could reconstruct $\varepsilon(\theta)$ by directly summing over the topological sectors of the partition function. We show here that there are circumstances where this is not the case. In particular, this occurs in regions where the curvature of $\varepsilon(\theta)$ is negative.

Exact Solution to the Extended Zwanzig Model for Quasi-Sigmoidal Chemically Induced Denaturation Profiles: Specific Heat and Configurational Entropy

Temperature and chemically induced denaturation comprise two of the most characteristic mechanisms to achieve the passage from the native state N to any of the unstructured states Dj in the denatured ensemble in proteins and peptides. In this work we present a full analytical solution for the configurational partition function 𝒵qs of a homopolymer chain poly-X in the extended Zwanzig model (EZM) for a quasisigmoidal denaturation profile. This solution is built up from an EZM exact solution in the case where the fraction α of native contacts follows exact linear dependence on denaturant’s concentration ζ; thus an analytical solution for 𝒵L in the case of an exact linear denaturation profile is also provided. A recently established connection between the number ν of potential nonnative conformations per residue and temperature-independent helical propensity ω complements the model in order to identify specific proteinogenic poly-X chains, where X represents any of the twenty naturally occurring aminoacid residues. From 𝒵qs, equilibrium thermodynamic potentials like entropy 𝒮 and average internal energy 〈E〉 and thermodynamic susceptibilities like specific heat C𝓋 are calculated for poly-valine (poly-V) and poly-alanine (poly-A) chains. The influence of the rate at which native contacts denature as function of ζ on thermodynamic stability is also discussed.

Thermodynamics of ideal proteinogenic homopolymer chains as a function of the energy spectrum E, helical propensity ω and enthalpic energy barrier

A reformulation and generalization of the Zwanzig model (ZW model) for ideal homopolymer chains poly-X, where X represents any of the twenty naturally occurring proteinogenic amino acid residues is presented. This reformulation and generalization provides a direct connection between coarse-grained parameters originally proposed in the ZW model with variables from the Lifson–Roig (LR) theory, such as the helical propensity per residue ω, and new variables introduced here, such as the energy gap Δ between unfolded and folded structures, as well as the ratio f of the energy scales involved. This enables us to discover the relevance of the energy spectrum E to the onset of configurational phase transitions. From the configurational partition function \U0001d4ac, thermodynamic properties such as the configurational entropy S, specific heat Cv and average energy 〈E〉 are calculated in terms of the number of residues K, temperature T, helical propensity ω and energy barrier ΔH for different poly-X chains in vacuo. Results obtained here provide substantial evidence that configurational phase transitions for ideal poly-X chains correspond to first-order phase transitions. An anomalous behavior of the thermodynamic functions 〈E〉, Cv, S with respect to the number K of residues is also highlighted. On-going methods of solution are outlined.

Statistical description of thermoreversible chemical bonding in metallic alloys

A general statistical model, developed in our previous works for describing network-forming systems, have applied to study of metallic alloys with chemical bonding. The model is based on the representation of the partition function for all possible configurations of a thermoreversible network in the form of a functional integral over a scalar field. The main object of our investigation was the solution of free atoms A, B and the elements of chemical short-range order with the stoichiometry A2 B that could form thermoreversible chemical bonds. The thermodynamic and structural properties of this system is analyzed. It is revealed that at some temperatures and concentrations, geometry percolation of the network of chemical bonds and thus a sol-gel transition may take place. The critical percolation line is plotted in common with phase diagram. Three different scenarios of sol-gel transition is considered in light of competition between gelation and phase separation phenomena.

Statistical description of the sol-gel transition in systems with thermoreversible chemical bonds

A general statistical model is proposed for describing network-forming systems. The model is based on the representation of the partition function for all possible configurations of a thermoreversible network in the form of a functional integral over a scalar field. According to this model, two types of first-order phase transitions can occur in the systems under consideration: macroscopic phase separation with the structural phase transition due to the change in the configuration of the spatial network and the sol-gel transition due to the formation of a thermoreversible percolation cluster consisting of bound structural units. A detailed analysis is performed of the thermodynamic and structural properties of a solution of monomers that have f functional groups and can form thermoreversible chemical bonds. The influence of specific features of the chemical and volume interactions on the phase diagram of the system is investigated. The mutual position of the sol-gel transition line and the phase diagram is determined for different model parameters. It is revealed that two substantially different regimes of the behavior of the sol-gel transition line in the “temperature-volume fraction of structural units” plane are observed with a change in the rigidity of chemical bonds.

Density functional theory, molecular dynamics, and differential scanning calorimetry study of the RbF–CsF phase diagram

A multiscale modeling approach is developed to compute the phase diagram of the RbF-CsF binary system. The mixing enthalpies of the (Rb,Cs)F solid and liquid solutions are evaluated using density functional theory and classical molecular dynamics calculations, respectively. For the solid solution, 18 different configurations are studied with density functional theory and the surrounded atom model is applied in order to compute the configurational partition function. We also measure the solidus and liquidus equilibria using differential scanning calorimetry. Finally the RbF-CsF phase diagram is constructed using the calculated excess free enthalpies of the solid and liquid solutions and a very good agreement with our experimental data is found.

Quantum free-energy differences from nonequilibrium path integrals. I. Methods and numerical application

In this paper, the imaginary-time path-integral representation of the canonical partition function of a quantum system and nonequilibrium work fluctuation relations are combined to yield methods for computing free-energy differences in quantum systems using nonequilibrium processes. The path-integral representation is isomorphic to the configurational partition function of a classical field theory, to which a natural but fictitious Hamiltonian dynamics is associated. It is shown that if this system is prepared in an equilibrium state, after which a control parameter in the fictitious Hamiltonian is changed in a finite time, then formally the Jarzynski nonequilibrium work relation and the Crooks fluctuation relation hold, where work is defined as the change in the energy as given by the fictitious Hamiltonian. Since the energy diverges for the classical field theory in canonical equilibrium, two regularization methods are introduced which limit the number of degrees of freedom to be finite. The numerical applicability of the methods is demonstrated for a quartic double-well potential with varying asymmetry. A general parameter-free smoothing procedure for the work distribution functions is useful in this context.

Residual Entropy, the Third Law and Latent Heat

A novel thermodynamic treatment of residual entropy in crystals, involving the configurational partition function, is suggested, which is consistent with both classical and statistical thermodynamics. It relates residual entropy to the inherent latent heat which would be released upon cooling if the reversible path were available. The nature of this heat is that if the crystal possessing residual entropy freezes above its Boltzmann’s characteristic temperature of molecular alignment, the difference in energy between different molecular arrangements is overcome by the kT heat bath to form a nearly-ideal solution. However, upon cooling below this characteristic temperature, they would separate with a concomitant release of the corresponding energy, provided the reversible path were available.

String Matching and 1d Lattice Gases

We calculate the probability distribution for the number of occurrences n of a given l letter word x inside a random string of k letters, whose letters have been generated by a known stationary stochastic process. Denoting by p(x) the probability of occurrence of the word, it is well-known that the distribution of occurrences in the asymptotic regime k → ∞ such that kp(x) ≫ 1 is Gaussian, while in the limit k→ ∞, and p(x) → 0 , such that kp(x) is finite, the distribution is Compound Poisson. It is also known that these limiting forms do not work well in the intermediate regime when kp(x) >rsim 1 and k is finite. We show that the problem of calculating the probability of occurrences is equivalent to determining the configurational partition function of a 1d lattice gas of interacting particles, with the probability distribution given by the n-particle terms of the grand-partition function and the number of particles corresponding to the number of occurrences on the string. Utilizing this equivalence, we obtain the probability distribution from the equation of state of the lattice gas. Our result reproduces rather well the behavior of the distribution in the asymptotic as well as the intermediate regimes. Within the lattice gas description, the asymptotic forms of the distribution naturally emerge as certain low density approximations. Thus our approach which is based on statistical mechanics, also provides an alternative to the usual statistics based treatments employing the central limit and Chen–Stein theorems.

Statistical mechanics of two hard spheres in a box

We investigate some statistical mechanical properties of a system consisting of two hard spheres in a D-dimensional rectangular box (D=1, 2, ...). We give a theoretical method for computing a configurational partition function Zc,D of this system and compare the equation of state obtained from Zc,D with molecular dynamics simulations. Especially in D=3, we give a fully analytic expression for the pressure which turns out to have one or more negative compressibility regions when the box size is small.

An Alternative Approach to Nonrandomness in Solution Thermodynamics

An alternative formulation of the configurational partition function of mixtures is proposed, which leads to analytical expressions for the basic thermodynamic quantities, including the local compo...

Molecular theory of dendritic liquid crystals: self-organisation and phase transitions

We present tractable molecular theory descriptions of liquid crystalline dendrimers based on clearly defined approximations and in terms of the dominant interactions underlying the self-organisation of these large and complex supermolecular entities. We formulate the configurational partition function for dendrimers, taking explicit account of their conformations and segmental interactions. Two approximate schemes are presented: the first is based on the effective interactions of the dendrimers as a whole while the second scheme is based on the interactions among the mesogenic units contained in the dendrimers. Results of lattice calculations for phase transitions in the context of the first scheme are presented and they show that the minimal inclusion of shape anisotropy and of sub-molecular partitioning into chemically distinct parts is sufficient to reproduce the variety of phases and phase sequences observed experimentally and provides insights into the conformational aspect of these transitions. In the second scheme, the description of the dendromesogenic system reduces to that of an ensemble of mesogenic dimers. This scheme can be readily extended to the description liquid crystalline oligomers and polymers consisting of mesogenic units connected by flexible spacers in various architectures. It thus provides a unified approach for treating mesomorphic phase transitions of supermolecular and macromolecular systems that can be built by connecting the same submolecular units in topologically different ways.