Eigenvalues of a statistical mechanics matrix

We consider the problem of finding the spectrum of an n × n matrix which arises in the study of a certain model of long-range interactions in a one-dimensional statistical mechanics system. Our analysis exhibits a curious resemblance of the suitably normalized distribution of eigenvalues to the Marčenko–Pastur law in the limit n → ∞.     

Practical applicability of the Jarzynski relation in statistical mechanics: a pedagogical example

We suggest and discuss a simple model of an ideal gas under the piston to gain an insight into the workings of the Jarzynski identity connecting the average exponential of the work over the nonequilibrium trajectories with the equilibrium free energy. We show that the identity is valid for our system, due to the very rapid molecules belonging to the tail of the Maxwell distribution. For the most interesting extreme, when the system volume is large, while the piston is moving with great speed (compared to thermal velocity) for a very short time, the necessary number of independent experimental runs to obtain a reasonable approximation for the free energy from averaging the nonequilibrium work grows exponentially with the system size.     

Molecular chains with fixed bond lengths and angles: a study based on quantum statistical mechanics

This work studies large three-dimensional open molecular chains at thermal equilibrium in which bond lengths and angles are fixed (hard variables), based upon quantum statistics. A model for a chain formed by N particles interacting through harmonic-like vibrational potentials is treated in the high-frequency limit in which all bond lengths and angles become constrained, while other N angles (soft variables) remain unconstrained. The associated quantum partition function is bounded rigorously, using a variational inequality (related to the Born-Oppenheimer approximation), by another quantum partition function, Z. The total vibrational zero-point energy is shown to be independent of the soft variables thereby solving for this model a generic difficulty in the elimination of hard variables. Z depends only on soft variables and, under certain conditions, it can be approximated by a classical partition function Z_c. The latter satisfies the equipartition principle and it differs from other classical partition functions for related molecular chains. The extension of the model when only part of the bond angles become fixed in the high-frequency limit is outlined. As another generalization, a systematic study of macromolecules, as composed of electrons and heavy particles with Coulomb interactions, is also presented. Its exact quantum partition function is bounded, supposing that the effective molecular potential also tends to constrain all bond lengths and angles, and under suitable assumptions, by another quantum partition function. The latter depends only on the remaining soft variables and it generalizes the one obtained for the first model.     

The statistical mechanics of two-dimensional vesicles

Recent work on the statistical mechanics of model membranes and vesicles in two dimensions by Leibler, Singh and the author [l,2] is briefly summarized.     

Ordering of fluoro-mesogens: a statistical approach based on quantum mechanics and computer simulation

Computational analysis of the molecular ordering of nematic p -phenylene-4-methoxy benzoyl 4-trifluoromethylbenzoate (FLUORO1) and smectic 4-propyloxyphenyl 4-(4-trifluoromethylbenzoyloxy) benzoate (FLUORO2) mesogens has been carried out with respect to translatory and orientational motions. The net atomic charge and atomic dipole components at each atomic centre of the molecule have been evaluated using the CNDO/2 method. Rayleigh-Schrodinger perturbation theory, along with the multicentred-multipole expansion method, has been employed to evaluate long range intermolecular interactions while a '6-exp' potential function has been assumed for short range interactions. The total interaction energy values obtained through these computations were used to calculate the probability of each configuration at the phase transition temperature using the Maxwell-Boltzmann formula. The flexibility of various configurations has been studied in terms of the variation of probability due to small departures from the most probable configuration. The results obtained enable the determination of the peculiarities of the molecular ordering, as well as the construction of models of the structures of FLUORO1 and FLUORO2 in different modes of interaction. The mesophase nature has been correlated with the parameters introduced in this paper.     

Semiclassical statistical mechanics of two-dimensional hard-body fluids

The problem of calculating the thermodynamic properties of two-dimensional semiclassical hard-body fluids is studied. Explicit expressions are given for the first-order quantum corrections to the free energy, equation of state, and virial coefficients. The numerical results are calculated for the planar hard dumbbell fluid. Significant features are the increase in quantum corrections with increasing eta and increasing L*=L/sigma(0).     

Thermodynamics and statistical mechanics of multilayer adsorption

The multilayer adsorption models of Brunauer-Emmett-Teller and Guggenheim-Anderson-de Boer are reconsidered. The relationship between the fitting parameters and the physical parameters of the equation is discussed. The preexponential factors of the parameters are shown to be in general far different from unity, contrary to a widespread use. A thermodynamical derivation illuminates the hypothesis on which the multilayer sorption equation is dependent and frees it from too restrictive hypothesis usually taken as necessary for its validity. Equations are derived for the number fraction of sorption sites occupied by different numbers of molecules. The Guggenheim-Anderson-de Boer equation is shown to imply incomplete occupation (jamming) of the first sorption layer at saturation.     

Semiclassical statistical mechanics of hard-body fluid mixtures

The thermodynamic properties of semiclassical hard-body fluid mixtures are studied. Explicit expressions are given for the free-energy, equation of state and virial coefficients of the classical hard convex-body fluid mixtures. The numerical results are discussed under different conditions. The agreement with the exact data is good in all cases. The first-order quantum corrections are also studied. The quantum effects depend on the condition, shape parameters L11* and L22*, and concentrations x1 and x2 in general and increase with an increase of packing fraction eta, in particular.     

A formulation of statistical mechanics of ordered systems

Expressions are derived for the thermodynamic functions (Gibbs free energy, Helmholtz free energy, etc.) of an ordered system in terms of the single-particle distribution function,p(x), and correlation functions. The thermodynamic functions are treated as functionals of the single-particle distribution function. By minimizing the Helmholtz free energy with respect top(x) under constraints of constantT, V andN, an integral equation is obtained from whichp(x) can be determined. The correlation function of the ordered state in the region near the coexistence surface between ordered and disordered state is expanded about the correlation function of the disordered state, and the series is truncated. Methods for calculating the thermodynamic functions and the single-particle distribution function are presented, and our result is discussed in relation to other treatments of phase coexistence in the literature.     

Theoretical study on aquation reaction of cis-platin complex: RISM-SCF-SEDD, a hybrid approach of accurate quantum chemical method and statistical mechanics

The ligand exchange process of cis-platin in aqueous solution was studied using RISM-SCF-SEDD (reference interaction site model-self-consistent field with spatial electron density distribution) method, a hybrid approach of quantum chemistry and statistical mechanics. The analytical nature of RISM theory enables us to compute accurate reaction free energy in aqueous solution based on CCSD(T), together with the microscopic solvation structure around the complex. We found that the solvation effect is indispensable to promote the dissociation of the chloride anion from the complex.     

Benchmark ab initio proton affinity and gas-phase basicity of α-alanine based on coupled-cluster theory and statistical mechanics

We determine the proton affinity (PA) and gas-phase basicity (GB) of amino acid α-alanine at a chemically accurate level by performing explicitly-correlated CCSD(T)-F12b/aug-cc-pVDZ geometry optimizations and normal mode vibrational frequency calculations as well as CCSD(T)-F12b/aug-cc-pVTZ energy computations at the possible neutral and protonated geometries. Temperature effects at 298.15 K considering translational, rotational, and vibrational enthalpy and entropy corrections are obtained via standard statistical mechanics utilizing the molecular geometries and the harmonic vibrational energy levels. Both the amino nitrogen (N) and the carbonyl oxygen (O) atoms are proven to be potential protonation sites and a systematic conformational search reveals 3 N- and 9 O-protonated conformers in the 0.00–7.88 and 25.43–30.43 kcal/mol energy ranges at 0 K, respectively. The final computed PA and GB values at (0)298.15 K in case of N-protonation are (214.47)216.80 and 207.07 kcal/mol, respectively, whereas the corresponding values for O-protonation are (189.04)190.63 and 182.31 kcal/mol. The results of the benchmark high-level coupled-cluster computations are utilized to assess the accuracy of several lower-level cost-effective methods such as MP2 and density functional theory with various functionals (SOGGA11-X, M06-2X, PBE0, B3LYP, M06, TPSS).     

Size-distribution of droplets in emulsions by statistical mechanics calculation

This report is concerned with theoretical demonstration of the spontaneous emulsification which has been observed in a soft contact of nitrobenzene with water without surfactant [K. Aoki, M. Li, J. Chen, T. Nishiumi, Electrochem. Commun. 11 (2009) 239]. The demonstration is based on the model of spherical oil droplets with any size in equilibrium. The droplets are composed of the smallest droplets, the total number of which is given. An assembly of small droplets has larger surface energy than that of large ones because the surface energy is proportional to the surface area. The former has larger configurational entropy than the latter because the number of small droplets is bigger than that of the large ones. Since the free energy is determined by the competition between the surface energy and the entropy, it is not clear which assembly has lower free energy. This question was solved numerically here by statistical mechanics calculation of the size distributions, which contained only a parameter of the surface energy. The results of the computation at small number of droplets were used for deriving approximate equations for extremely large number of droplets. The size distribution was localized both to the smallest and the largest droplets. The diameter of the largest droplet was estimated from the dynamics in which coalescence by diffusion of droplets is disturbed by gravitational convection. The size then predicted was of the order of micrometer, being close to experimental values.     

Nonlinear coupling in the quantum statistical theory of proton transfer dynamics

The possible role of the nonlinear coupling on the character of the dynamics of particle transfer process is investigated. The analysis and solutions of the kinetic equation indicate that nonlinear coupling causes symmetry breaking of particle transfer potential and determines possible equilibrium structure of the system. Dissipative coupling characterizes the rate of the system to reach thermodynamic equilibrium and along with nonlinear coupling and parameters of the system determines in a unique way the resulting equilibrium structure of the system.     

Cytochrome P450CAM enzymatic catalysis cycle: a quantum mechanics/molecular mechanics study

The catalytic pathway of cytochrome P450cam is studied by means of a hybrid quantum mechanics/molecular mechanics method. Our results reveal an active role of the enzyme in the different catalytic steps. The protein initially controls the energy gap between the high- and low-spin states in the substrate binding process, allowing thermodynamic reduction by putidaredoxin reductase and molecular oxygen addition. A second electron reduction activates the delivery of protons to the active site through a selective interaction of Thr252 and the distal oxygen causing the O--O cleavage. Finally, the protein environment catalyzes the substrate hydrogen atom abstraction step with a remarkably low free energy barrier ( approximately 8 kcal/mol). Our results are consistent with the effect of mutations on the enzymatic efficacy and provide a satisfactory explanation for the experimental failure to trap the proposed catalytically competent species, a ferryl Fe(IV) heme.     

Thermodynamics and statistical mechanics of the ideally polarisable electrode: Gibbs isotherm and surface tension

A definition of the ideally polarisable electrode which allows a general statistical mechanical treatment is given, comparisons with the usual thermodynamic approach being made. It is shown that, with this definition, the system behaves as a pure capacitance to an imposed alternating potential of sufficiently low frequency. The equilibrium states of the system are described by the grand canonical distribution. Thermodynamic functions, given in terms of the grand canonical partition function, can be computed in molecular terms. In particular, general expressions for the surface tension and the pressure are obtained, and compared to those proposed by various authors. The separation of the pressure into electrical and non-electrical parts is analyzed. Expressions are given for the electrochemical potentials of charged and of dipolar species. With these microscopic definitions, the Gibbs adsorption isotherms and the Lippmann equation are derived. For several simple models, including that of Gouy and Chapman for an ionic solution and a metal electrode, these relations are explicity considered. The inadmissability, in a coherent model, of reducing the solvent to a medium of fixed dielectric constant, is emphasized.     

Quantum statistical mechanics with Gaussians: equilibrium properties of van der Waals clusters

The variational Gaussian wave-packet method for computation of equilibrium density matrices of quantum many-body systems is further developed. The density matrix is expressed in terms of Gaussian resolution, in which each Gaussian is propagated independently in imaginary time beta=(k(B)T)(-1) starting at the classical limit beta=0. For an N-particle system a Gaussian exp[(r-q)(T)G(r-q)+gamma] is represented by its center qinR(3N), the width matrix GinR(3Nx3N), and the scale gammainR, all treated as dynamical variables. Evaluation of observables is done by Monte Carlo sampling of the initial Gaussian positions. As demonstrated previously at not-very-low temperatures the method is surprisingly accurate for a range of model systems including the case of double-well potential. Ideally, a single Gaussian propagation requires numerical effort comparable to the propagation of a single classical trajectory for a system with 9(N(2)+N)/2 degrees of freedom. Furthermore, an approximation based on a direct product of single-particle Gaussians, rather than a fully coupled Gaussian, reduces the number of dynamical variables to 9N. The success of the methodology depends on whether various Gaussian integrals needed for calculation of, e.g., the potential matrix elements or pair correlation functions could be evaluated efficiently. We present techniques to accomplish these goals and apply the method to compute the heat capacity and radial pair correlation function of Ne(13) Lennard-Jones cluster. Our results agree very well with the available path-integral Monte Carlo calculations.     

Intra-annular cyclophane diamines as proton sponges: a computational study

Gas-phase proton affinities of cyclophanes containing intra-annular amino groups were calculated using density functional theory (DFT) at the B3LYP/6-31+G^∗∗//B3LYP/6-31G^∗ level. They are higher in magnitude as those for proton sponges such as 1,8-bisaminonaphthalene, however, they are slightly weaker bases than 1,8-bis(dimethylamino)naphthalene. The high basicity of the cyclophane diamines is attributed mainly to their structural flexibility, which allows them to maximize the hydrogen bond strength in the cations by achieving N-H?N linearity, while strain relief upon protonation is less important. Another contributing factor is the stabilizing interaction of the added proton with adjacent phenyl π systems of the cyclophanes. Barriers for proton transfer between the nitrogen atoms of the diamine cations are also reported.     

Quantum statistical mechanics of closed‐ring molecular chains

A model for a closed‐ring unhindered three‐dimensional macromolecular chain, based on Quantum Mechanics, is presented. Upon starting from an exact non‐relativistic Hamiltonian operator, we integrate out all electronic degrees of freedom, in the Born‐Oppenheimer framework, giving rise to an effective vibro‐rotational Hamiltonian for the chain. Then, assuming a harmonic oscillator‐like vibrational potential between nearest‐neighbour atoms, the integration of the atomic radial degrees of freedom is carried in the limit of high frequencies. Thus, all bond lengths become fixed, including the one which makes the chain to become a closed ring. This formulation leads to a specific Hamiltonian for the remaining angular variables of the closed‐ring chain, and constitutes an alternative in comparison with standard Gaussian models, which do not. Use is made of a variational inequality by Peierls to find an approximate quantum partition function for the angular variables of the system. We then proceed to obtain approximately another representation for the angular partition function in the classical limit. Several features of the classical partition function are discussed.