Functional derivative of the kinetic energy functional for spherically symmetric systems

Ensemble non-interacting kinetic energy functional is constructed for spherically symmetric systems. The differential virial theorem is derived for the ensemble. A first-order differential equation for the functional derivative of the ensemble non-interacting kinetic energy functional and the ensemble Pauli potential is presented. This equation can be solved and a special case of the solution provides the original non-interacting kinetic energy of the density functional theory.     

Information-theoretic approach to kinetic-energy functionals: the nearly uniform electron gas

We strengthen the connection between information theory and quantum mechanical systems using a recently developed dequantization procedure which results in a decomposition of the kinetic energy as the sum of a classical term and a purely quantum term. For the nearly uniform electron gas, we thereby approximate the noninteracting kinetic energy as the sum of the Thomas-Fermi term, which is exact for the uniform electron gas, and the Weizsäcker term, which is proportional to the Fisher information. Electron correlation is included via a nonlocal analytical expression which is a functional of the (N-1)-conditional probability density. This expression is evaluated via a statistically rigorous Monte-Carlo procedure to obtain the correlation energy as a functional of the electron density. We show that this functional is well aproximated by a term which is proportional to the Shannon entropy. Thus the kinetic energy is expressed as the standard Thomas-Fermi term plus terms which are proportional to two of the cornerstones of information theory: the Fisher information, which is a measure of localization, and the Shannon entropy, which is a measure of delocalization.     

Random activation energy model and disordered kinetics, from static to dynamic disorder

We suggest a unified path integral approach for random rate processes with random energy barriers, which includes systems with static and dynamic disorder as particular cases. We assume that the random component of the activation energy barrier can be described by a generalized Zubarev-McLennan nonequilibrum statistical ensemble that can be derived from the maximum information entropy approach by assuming that the time history of the fluctuations of the random components of the energy barrier are known. We show that the average survival function, which is an experimental observable in disorderd kinetics, can be computed exactly in terms of the characteristic functional of this generalized Zubarev-McLennan nonequilibrium statistical ensemble. We investigate different types of disorder described by our approach, ranging from static disorder with infinite memory to random processes with long or short memory, and finally to rapidly fluctuating independent random processes with no memory. We derive expressions of the average survival function for all these types of disorder and discuss their implications in the evaluation of kinetic parameters from experimental data. We illustrate our approach by studying a simple model of dynamic disorder of the renewal type. Finally we discuss briefly the implications of our approach in molecular biology and genetics.     

Information theoretical analysis of the hydrogen atom

This is an analysis of the statistical nature of the time-independent Schrödinger equation through the use of the information entropy concept. We first study the Schrödinger equation in a general way and then by actually computing entropies of various states of the hydrogen atom for a re-examination of the problem. It is found that there exists a variational procedure involving maximizing entropy for obtaining all solutions once one solution is known. Based on certain observations of the particular single system, some general conclusions can be deduced. First of all, we can safely say that the Schrödinger equation, among many other interpretations, is but the consequence of a principle of minimum potential energy expectation with certain proper constraints imposed. In addition, the ensemble concept in statistical thermodynamics is also useful in understanding microscopic quantum systems and many quantum mechanical concepts such as energy quantization and wave nodal properties can be discussed in the light of information theory and statistics in general.     

密度泛函活性理论中的Rényi熵, Tsallis熵和Onicescu信息能

根据密度泛函理论,分子的电子密度确定了该体系基态下的所有性质,其中包括结构和反应活性.如何运用电子密度泛函有效地预测分子反应活性仍然是一个有待解决的难题.密度泛函活性理论(DFRT)倾力打造这样一个理论和概念架构,使得运用电子密度以及相关变量准确地预测分子的反应特性成为可能.信息理论方法的香农熵和费舍尔信息就是这样的密度泛函,研究表明,它们均可作为反应活性的有效描述符.本文将在DFRT框架中介绍和引进三个密切相关的描述符, Rényi熵、Tsallis熵和Onicescu信息能.我们准确地计算了它们在一些中性原子和分子中的数值并讨论了它们随电子数量和电子总能量的变化规律.此外,以第二阶Onicescu信息能为例,在分子和分子中的原子两个层面上,系统地考察了其随乙烷二面角旋转的变化模式.这些新慨念的引入将为我们深入洞察和预测分子的结构和反应活性提供额外的描述工具.     

Shannon entropy as a predictor of avoided crossing in confined atoms

Avoided crossing is one of the unique spectroscopic features of a confined atomic system. Shannon information entropy of the ground state and some of the excited states of confined H atom as a predictor of avoided crossing is studied in this work. This is accomplished by varying the strength of the confinement and examining structure properties like ionization energy and Shannon information entropy. Along with the energy level repulsion at the avoided crossing, Shannon information entropy is also exchanged between the involved states. This work also addresses a question: In addition to that regarding localization, what other property of the system can be extracted from Shannon entropy? Insightful connection is discovered between Shannon entropy and the average value of confinement potential, Coulomb potential, and kinetic energy.     

Microcanonical analysis of the kinetic method. the meaning of the "apparent entropy"

A rigorous analysis of the kinetic method is carried out using Rice-Ramsperger-Kassel-Marcus (RRKM) theory of microcanonical statistical unimolecular dissociation rates. The model employs a kinetics treatment appropriate for metastable ion dissociation. Proton-bound alkoxide dimer anions are used as model systems, with realistic vibrational and rotational parameters calculated by ab initio methods for the cluster ion and transition states leading to the competitive dissociation channels. The numerical simulations show that the kinetic method plots of ln(I2/I1) versus AAH are nearly linear but can exhibit significant curvature. The apparent entropy obtained in the extended kinetic method is not approximately equal to the thermodynamic entropy difference for dissociation, AAS(T), or for activation, deltadeltaS++(T), either at the effective temperature or at any fixed equilibrium temperature. Instead, the apparent entropy term can be related to the ratio of the microcanonical sum of states of the dissociation transition states for the kinetically selected internal energy of the dissociating ions.     

Validation of Markov state models using Shannon's entropy

Markov state models are kinetic models built from the dynamics of molecular simulation trajectories by grouping similar configurations into states and examining the transition probabilities between states. Here we present a procedure for validating the underlying Markov assumption in Markov state models based on information theory using Shannon's entropy. This entropy method is applied to a simple system and is compared with the previous eigenvalue method. The entropy method also provides a way to identify states that are least Markovian, which can then be divided into finer states to improve the model.     

Ensemble Regression Coefficient Analysis for Application to Near-Infrared Spectroscopy

A new variable selection method called ensemble regression coefficient analysis is reported on the basis of model population analysis. In order to construct ensemble regression coefficients, many subsets of variables are randomly selected to calibrate corresponding partial least square models. Based on ensemble theory, the mean of regression coefficients of the models is set as the ensemble regression coefficient. Subsequently, the absolute value of the ensemble regression coefficient can be applied as an informative vector for variable selection. The performance of ensemble regression coefficient analysis was assessed by four near infrared datasets: two simulated datasets, one wheat dataset, and one tobacco dataset. The results showed that this approach can select important variables to obtain fewer errors compared with regression coefficient analysis and Monte Carlo uninformative variable elimination.     

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.     

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.     

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.     

Transport of matter and energy in a mesoscopic thermo-hydrodynamic approach

We derive a thermo-hydrodynamic theory for particles and energy flow, based on a nonequilibrium grand-canonical ensemble formalism. The time-dependent kinetic coefficients are explicitly given in terms of microscopic mechanical quantities. The time evolution equations describing the coupled flow of energy and particles are derived. The second-rank tensorial fluxes of current of energy and particles present in the nonequilibrium ensemble are nondiagonal. We obtain a generalized Fick's Law, which presents the effect of the energy flow on the particle diffusion equation.     

Orbital interpretation of kinetic energy density and a direct space comparison of chemical bonding in tetrahedral network solids

We present how the kinetic energy density (KED) can be interpreted on the basis of the orbital interactions within the Kohn-Sham theory and propose how to utilize a direct space function in chemical bonding analysis, the relative entropy density (RED), which is constructed from the KED, the Thomas-Fermi KED (TF-KED), and the electron density. From the detailed analysis of the KED of wave functions and the TF-KED from the free electron model, it is shown that the RED can reveal the nodal properties of individual wave functions and provide a variationally meaningful way of accumulating chemical bonding information from the wave functions, hence allowing quantitative bonding analysis in direct space. To substantiate the proposal, the RED function has been tested on the tetrahedral network solids, including the group 14 elements and the III-V binary compounds with the zinc blende structure. The direct space maps of the RED quantitatively reflect the trend in metallicity and the polarity of their two-center, two-electron bonds in terms of the absolute values of the RED, the location of the minimum values, and the behavior of the deformation from the spherical symmetry of the atomic RED.     

Mean field kinetic theory for a lattice gas model of fluids confined in porous materials

We consider the mean field kinetic equations describing the relaxation dynamics of a lattice model of a fluid confined in a porous material. The dynamical theory embodied in these equations can be viewed as a mean field approximation to a Kawasaki dynamics Monte Carlo simulation of the system, as a theory of diffusion, or as a dynamical density functional theory. The solutions of the kinetic equations for long times coincide with the solutions of the static mean field equations for the inhomogeneous lattice gas. The approach is applied to a lattice gas model of a fluid confined in a finite length slit pore open at both ends and is in contact with the bulk fluid at a temperature where capillary condensation and hysteresis occur. The states emerging dynamically during irreversible changes in the chemical potential are compared with those obtained from the static mean field equations for states associated with a quasistatic progression up and down the adsorption/desorption isotherm. In the capillary transition region, the dynamics involves the appearance of undulates (adsorption) and liquid bridges (adsorption and desorption) which are unstable in the static mean field theory in the grand ensemble for the open pore but which are stable in the static mean field theory in the canonical ensemble for an infinite pore.     

Electrical resistivity correlations in rigid and plasticized polymers

Empirical correlations in the literature between room-temperature electrical resistivity and activation energy in various polymers are shown to be equivalent to the compensation law. Assuming that electrical conduction is an ionic process, following the Eyring theory, the compensation law can be expressed as a direct proportionality between activation energy and activation entropy. It is found that plasticizers do not change the ion number density or the ionic jump distance but only the temperature at which energy and entropy effects balance. An approximation of the ion number density is consistent with the idea that the mobile ions result from the dissociation of absorbed water.