Electron pair density information measures in atomic systems

Shannon entropies of the pair density, conditional entropies, and mutual information are studied in position and in momentum space for ground state neutral atoms and selected excited states at the Hartree‐Fock level. We show that the mutual information, a measure of correlation, is larger in position space than in momentum space. This result also holds for a mutual information defined in terms of the exchange density; however, these quantities display much more structure than the corresponding ones based on the pair densities. The interpretation of this behavior is that exchange effects are smaller in momentum space than in position space in these systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Statistical mechanics of worm-like polymers from a new generating function

We present a mathematical approach to the worm-like chain model of semiflexible polymers. Our method is built on a novel generating function from which all the properties of the model can be derived. Moreover, this approach satisfies the local inextensibility constraint exactly. In this paper, we focus on the lowest order contribution to the generating function and derive explicit analytical expressions for the characteristic function, polymer propagator, single chain structure factor, and mean square end-to-end distance. These analytical expressions are valid for polymers with any degree of stiffness and contour length. We find that our calculations are able to capture the fully flexible and infinitely stiff limits of the aforementioned quantities exactly while providing a smooth and approximate crossover behavior for intermediate values of the stiffness of the polymer backbone. In addition, our results are in very good quantitative agreement with the exact and approximate results of five other treatments of semiflexible polymers.     

On the electron-electron counterbalance hole

When a many-electron system has spatial inversion symmetry, the electron-electron counterbalance hole implies that two electrons with parallel spins cannot be at opposite positions with respect to the inversion center, and its presence was pointed out in the literature [T. Koga, J. Chem. Phys. 108, 2515 (1998)] for any pairs of Hartree-Fock orbitals with the same inversion parity. We report here a generalized result that in all two-electron systems with spatial inversion symmetry, the electron-electron counterbalance hole always exists for any approximate and exact wave functions with even inversion parity. The same is also true in momentum space. An extension of the hole to systems with three or more electrons is discussed.     

Fundamental importance of the Coulomb hole sum rule to the understanding of the Colle-Salvetti wave function functional

In this paper we consider the general form of the correlated-determinantal wave function functional of Colle and Salvetti (CS) for the He atom. The specific form employed by CS is the basis for the widely used CS correlation energy formula and the Lee-Yang-Parr correlation energy density functional of Kohn-Sham density functional theory. We show the following: (i) The key assumption of CS for the determination of this wave function functional, viz., that the resulting single-particle density matrix and the Hartree-Fock theory Dirac density matrix are the same, is equivalent to the satisfaction of the Coulomb hole sum rule for each electron position. The specific wave function functional derived by CS does not satisfy this sum rule for any electron position. (ii) Application of the theorem on the one-to-one correspondence between the Coulomb hole sum rule for each electron position and the constraint of normalization for approximate wave functions then proves that the wave function derived by CS violates charge conservation. (iii) Finally, employing the general form of the CS wave function functional, the exact satisfaction of the Coulomb hole sum rule at each electron position then leads to a wave function that is normalized. The structure of the resulting approximate Coulomb holes is reasonably accurate, reproducing both the short- and the long-range behavior of the hole for this atom. Thus, the satisfaction of the Coulomb hole sum rule by an approximate wave function is a necessary condition for constructing wave functions in which electron-electron repulsion is represented reasonably accurately.     

An innovative computational comparison of exact and centrifugal sudden quantum properties of the N + N2 reaction

The development of an innovative computational strategy suited to provide an accurate quantum evaluation of the detailed properties of the N + N(2) exchange reaction has been undertaken by carrying out an extended theoretical study of such reaction. To this end exact and approximate quantum calculations (based on both time-independent and time-dependent techniques) of state-specific and state-to-state probabilities of the title reaction have been performed by considering values of the total angular momentum quantum number up to 20, values of total energy up to 2.3 eV and by making a combined use of both high throughput and high performance computing platforms. The comparison of the results obtained from calculations performed by taking into account the full Coriolis coupling of the allowed helicity states with those obtained when neglecting the Coriolis coupling or even a model energy shift treatment has allowed us to find out when a workflow managing the distribution of the jobs can replace exact treatments with approximate ones and for what type of properties this is possible.     

Independent particle theory with electron correlation

We formulate an effective independent particle model where the effective Hamiltonian is composed of the Fock operator and a correlation potential. Within the model the kinetic energy and the exchange energy can be expressed exactly leaving the correlation energy functional as the remaining unknown. Our efforts concentrate on finding a correlation potential such that exact ionization potentials and electron affinities can be reproduced as orbital energies. The equation-of-motion coupled-cluster approach enables us to define an effective Hamiltonian from which a correlation potential can be extracted. We also make the connection to electron propagator theory. The disadvantage of the latter is the inherit energy dependence of the potential resulting in a different Hamiltonian for each orbital. Alternatively, the Fock space coupled-cluster approach employs an effective Hamiltonian which is energy independent and universal for all orbitals. A correlation potential is extracted which yields the exact ionization potentials and electron affinities and a set of associated molecular orbitals. We also describe the close relationship to Brueckner theory.     

Quantum statistics and classical mechanics: real time correlation functions from ring polymer molecular dynamics

We propose an approximate method for calculating Kubo-transformed real-time correlation functions involving position-dependent operators, based on path integral (Parrinello-Rahman) molecular dynamics. The method gives the exact quantum mechanical correlation function at time zero, exactly satisfies the quantum mechanical detailed balance condition, and for correlation functions of the form C(Ax)(t) and C(xB)(t) it gives the exact result for a harmonic potential. It also works reasonably well at short times for more general potentials and correlation functions, as we illustrate with some example calculations. The method provides a consistent improvement over purely classical molecular dynamics that is most apparent in the low-temperature regime.     

Analytic correlated wave functions in momentum space and related properties of helium like ions

We obtain analytic correlated wave functions in momentum space as the Fourier transform of correlated wave functions which are able to incorporate almost all of the correlation energy for the ground-state of two-electron atoms. Then we study the atomic momentum-density, the Compton profile and the elastic and inelastic scattering factors for this kind of wave functions. The scattering factors are also compared with the ones provided by a more accurate correlated wave function. All the calculations can be analytically performed, provided the correlated wave function in position space has been determined.     

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.     

Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation

It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.     

Transformed Dirac equation for the hydrogen atom,comparison with previous approaches in momentum space,and the anomalous Zeeman effect in momentum representation

The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum‐space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Lévy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum‐space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth‐order solutions of the new equation are recovered in the limit c → ∞ where c is the speed of light. These are manifestly separable into positive‐ and negative‐energy forms. For positive energy, these solutions have nonvanishing upper components that are two‐component spinors. The latter exactly correspond to the single‐component, nonrelativistic, momentum‐space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v²/c² but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum‐space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well‐known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Search for a small hole in a cavity wall by intermittent bulk and surface diffusion

We study the search of a small round hole in the wall of a spherical cavity by a diffusing particle, which can reversibly bind to the cavity wall and diffuse on the surface being in the bound state. There are two channels for the particle first passage to the hole, through the bulk, and through the surface. We propose a coarse-grained model of the search process and use it to derive simple approximate formulas for the mean time required for the particle to reach the hole for the first time and for the probability of the first passage to the hole through the bulk channel. This is done for two distributions of the particle starting point: (1) Uniform distribution over the cavity volume and (2) uniform distribution over the cavity wall. We check the accuracy of the approximate formulas by comparing their predictions with the corresponding quantities found by solving the mixed bulk-surface diffusion problem numerically by the finite difference method. The comparison shows excellent agreement between the analytical and numerical results.     

Path integral based calculations of symmetrized time correlation functions. II

Schofield's form of quantum time correlation functions is used as the starting point to derive a computable expression for these quantities. The time composition property of the propagators in complex time is exploited to approximate Schofield's function in terms of a sequence of short time classical propagations interspersed with path integrals that, combined, represent the thermal density of the system. The approximation amounts to linearization of the real time propagators and it becomes exact with increasing number of propagation legs. Within this scheme, the correlation function is interpreted as an expectation value over a probability density defined on the thermal and real path space and calculated by a Monte Carlo algorithm. The performance of the algorithm is tested on a set of benchmark problems. Although the numerical effort required is considerable, we show that the algorithm converges systematically to the exact answer with increasing number of iterations and that it is stable for times longer than those accessible via a brute force, path integral based, calculation of the correlation function. Scaling of the algorithm with dimensionality is also examined and, when the method is combined with commonly used filtering schemes, found to be comparable to that of alternative semiclassical methods.     

Why the generalized gradient approximation works and how to go beyond it

The local spin density (LSD) approximation, while of only moderate accuracy, has proven extremely reliable over three decades of use. We argue that any gradient-corrected functional should preserve the correct features of LSD even if the system under study contains no regions of small density gradient. The Perdew-Wang 1991 (PW91) functional respects this condition, while, e.g., the Lee-Yang-Parr (LYP) correlation functional violates it. We extend this idea to the next generation of density functionals, those which incorporate exact exchange via the optimized effective potential (OEP), with a model in which the correlation hole is constructed from the exact exchange hole. The resulting exchange-correlation hole is deeper and less diffuse than the exact exchange hole. We denote such a functional as “locally correlated Hartree-Fock” and list a variety of conditions such a functional should satisfy. We demonstrate the promise of this approach with a crude simple model. © 1997 John Wiley & Sons, Inc.     

The nature of electron correlation in a dissociating bond

We have constructed the unrestricted Hartree-Fock (UHF), restricted Hartree-Fock (RHF), and full configuration interaction (FCI) position and momentum intracules and holes for H···H at bond lengths R from 1 to 10 bohrs. We trace the recently discovered inversion of the UHF position hole at intermediate R to over-localization of the spin-orbitals, and support this by a correlation energy component analysis. The RHF and UHF momentum holes are found to be more complicated; however their features are explained through decomposition of electron correlation effects. The UHF momentum hole is also found to invert and exhibits interesting behavior at large R. The RHF (but not UHF) and FCI momentum intracules exhibit Young-type interference patterns related to recent double photoionization experiments. Our analyses yield the most comprehensive picture to date of the behavior of the electrons during homolytic bond fission.     

Scaled particle theory for hard sphere pairs. I. Mathematical structure

We develop an extension of the original Reiss-Frisch-Lebowitz scaled particle theory that can serve as a predictive method for the hard sphere pair correlation function g(r). The reversible cavity creation work is analyzed both for a single spherical cavity of arbitrary size, as well as for a pair of identical such spherical cavities with variable center-to-center separation. These quantities lead directly to a prediction of g(r). Smooth connection conditions have been identified between the small-cavity situation where the work can be exactly and completely expressed in terms of g(r), and the large-cavity regime where macroscopic properties become relevant. Closure conditions emerge which produce a nonlinear integral equation that must be satisfied by the pair correlation function. This integral equation has a structure which straightforwardly generates a solution that is a power series in density. The results of this series replicate the exact second and third virial coefficients for the hard sphere system via the contact value of the pair correlation function. The predicted fourth virial coefficient is approximately 0.6% lower than the known exact value. Detailed numerical analysis of the nonlinear integral equation has been deferred to the subsequent paper.     

Hardness of intermolecular forces and thermal diffusion

An approximate momentum transfer theorem leads to a direct relation between the thermal diffusion ratio and the logarithmic derivative of the potential. The correlation with exact results is excellent.     

A universal state-selective approach to multireference coupled-cluster non-iterative corrections

A new form of the asymmetric energy functional for multireference coupled cluster (MRCC) theories is discussed from the point of view of an energy expansion in a quasidegenerate situation. The resulting expansion for the exact electronic energy can be used to define the non-iterative corrections to approximate MRCC approaches. In particular, we show that in the proposed framework the essential part of dynamic correlation can be encapsulated in the so-called correlation Hamiltonian, which in analogy to the effective Hamiltonian, is defined in the model space (M(0)). The proper parametrization of the exact/trial wavefunctions leads to the cancellation of the overlap-type numerators and to a connected form of the correlation Hamiltonian and size-extensive energies. Within this parametrization, when the trial wavefunctions are determined without invoking a specific form of the MRCC sufficiency conditions, the ensuing correction can be universally applied to any type of the approximate MRCC method. The analogies with other MRCC triples corrections to MRCC theories with singles and doubles (MRCCSD) are outlined. In particular, we discuss the approach, which in analogy to the Λ-Mk-MRCCSD(T) method [F. A. Evangelista, E. Prochnow, J. Gauss, H. F. Schaefer III, J. Chem. Phys. 132, 074107 (2010)], introduces an approximate form of the triply-excited clusters into the effective and correlation Hamiltonians. Since the discussed corrections can be calculated as a sum of independent reference-related contributions, possible parallel algorithms are also outlined.