On the relationship between one-electron potential and densities of Fisher information and Shannon entropy

It is shown that there is an analytical relationship between the one-electron potential (OEP) and the densities of Shannon entropy and the two forms of the Fisher information. Moreover, following the relationship between OEP and the quantum potentials in many electron systems we found that the local quantum potentials can also be related to the information theoretic measures.     

Optimized effective potential method for individual low-lying excited states

This paper presents an optimized effective potential (OEP) approach based on density functional theory (DFT) for individual excited states that implements a simple method of taking the necessary orthogonality constraints into account. The amended Kohn-Sham (KS) equations for orbitals of excited states having the same symmetry as the ground one are proposed. Using a variational principle with some orthogonality constraints, the OEP equations determining a local exchange potential for excited states are derived. Specifically, local potentials are derived whose KS determinants minimize the total energies and are simultaneously orthogonal to the determinants for states of lower energies. The parametrized form of an effective DFT potential expressed as a direct mapping of the external potential is used to simplify the OEP integral equations. A performance of the presented method is examined by exchange-only calculations of excited state energies for simple atoms and molecules.     

On finite basis set implementation of the exchange-only optimized effective potential method

In this paper, we analyze a structure of the basis set optimized effective potential (OEP) equations from the Fredholm alternative point of view and present one of possible numerical schemes to solve the OEP equation in a stable manner. The solution is constructed as a sum of a unique solution on the subspace of eigenfunctions of the response matrix with non-zero eigenvalues and a non-unique solution on a counterpart subspace with singular eigenvalues. Non-uniqueness of a solution is exploited to obtain a local effective potential that satisfies the condition for the highest occupied molecular orbital (HOMO) without restricting the variational freedom of the optimization procedure. Unlike the existing methods we implement the HOMO condition using the functions of the null-subspace. Numerical results for the total and orbital energies based on the proposed scheme are close to the corresponding literature data.     

On orthogonality constrained multiple core‐hole states and optimized effective potential method

An attempt to construct a multiple core‐hole state within the optimized effective potential (OEP) methodology is presented. In contrast to the conventional Δ‐self‐consistent field method for hole states, the effects of removing an electron is achieved using some orthogonality constraints imposed on the orbitals so that a Slater determinant describing a hole state is constrained to be orthogonal to that of a neutral system. It is shown that single, double, and multiple core‐hole states can be treated within a unified framework and can be easily implemented for atoms and molecules. For this purpose, a constrained OEP method proposed earlier for excited states (Glushkov and Levy, J. Chem. Phys. 2007, 126, 174106) is further developed to calculate single and double core ionization energies using a local effective potential expressed as a direct mapping of the external potential. The corresponding equations, determining core‐hole orbitals from a one‐particle Schrödinger equation with a local potential as well as correlation corrections derived from the second‐order many‐body perturbation theory are given. One of the advantages of the present direct mapping formulation is that the effective potential, which plays the role of the Kohn–Sham potential, has the symmetry of the external potential. Single and double core ionization potentials computed with the presented scheme were found to be in agreement with data available from experiment and other calculations. We also discuss core‐hole state local potentials for the systems studied. © 2012 Wiley Periodicals, Inc.     

Study on the electric double layer of a cylindrical reverse micelle with functional theoretical approach

Some surfactants, such as AOT (bis-(2-ethylhexyl sodium sulfosuccinate), have such a special structure with a smaller hydrophilic head group but a bigger hydrophobic tail. Some mixtures of surfactants (or surfactant/co-surfactant) also take the same special structure[1―3]. If their concentrations are much higher than their critical micelle concentrations (cmc) in oil/water system, these surfactants or mixtures usually assemble as W/O cylindrical (or wormlike) micelles with their lengths bei…     

Effective local potentials for orbital-dependent density functionals

Practicality of the Kohn-Sham density functional scheme for orbital-dependent functionals hinges on the availability of an efficient procedure for constructing local exchange-correlation potentials in finite basis sets. We have shown recently that the optimized effective potential (OEP) method, commonly used for this purpose, is not free from difficulties. Here we propose a robust alternative to OEPs, termed effective local potentials (ELPs), based on minimizing the variance of the difference between a given nonlocal potential and its desired local counterpart. The ELP method is applied to the exact-exchange-only problem and shown to be promising for overcoming troubles with OEPs.     

Virial exchange energies from model exact-exchange potentials

It is shown by the example of Slater's averaged exchange potential that a poor approximation to the optimized effective potential (OEP) can yield a deceptively accurate energy via the conventional Kohn-Sham energy functional. For a trial exchange potential to be correct, its Kohn-Sham energy must coincide with the value obtained by the Levy-Perdew virial relation. Significant discrepancies between Kohn-Sham and the virial exchange energies are found for self-consistent Slater, Becke-Johnson, and effective local potentials (ELPs); their relative magnitudes are used to argue that, as approximations to the exact-exchange OEP, ELPs are the most accurate. Virial energy discrepancies vanish for Yang-Wu OEPs when the orbital and auxiliary basis sets are balanced, and remain surprisingly small for oscillatory OEPs obtained with unbalanced basis sets.     

Numerically stable optimized effective potential method with balanced Gaussian basis sets

A solution to the long-standing problem of developing numerically stable optimized effective potential (OEP) methods based on Gaussian basis sets is presented by introducing an approach consisting of an exact exchange OEP method with an accompanying construction and balancing scheme for the involved auxiliary and orbital Gaussian basis sets that is numerically stable and that properly represents an exact exchange Kohn-Sham method. The method is a purely analytical method that does not require any numerical grid, scales like Hartree-Fock or B3LYP procedures, is straightforward to implement, and is easily generalized to take into account orbital-dependent density functionals other than the exact exchange considered in this work. Thus, the presented OEP approach opens the way to the development and application of novel orbital-dependent exchange-correlation functionals. It is shown that adequately taking into account the continuum part of the Kohn-Sham orbital spectrum is crucial for numerically stable Gaussian basis set OEP methods. Moreover, it is mandatory to employ orbital basis sets that are converged with respect to the used auxiliary basis representing the exchange potential. OEP calculations in the past often did not meet the latter requirement and therefore may have led to erroneously low total energies.     

Ab initio density functional theory: the best of both worlds?

Density functional theory (DFT), in its current local, gradient corrected, and hybrid implementations and their extensions, is approaching an impasse. To continue to progress toward the quality of results demanded by today's ab initio quantum chemistry encourages a new direction. We believe ab initio DFT is a promising route to pursue. Whereas conventional DFT cannot describe weak interactions, photoelectron spectra, or many potential energy surfaces, ab initio DFT, even in its initial, optimized effective potential, second-order many-body perturbation theory form [OEP (2)-semi canonical], is shown to do all well. In fact, we obtain accuracy that frequently exceeds MP2, being competitive with coupled-cluster theory in some cases. Furthermore, this is accomplished within a relatively fast computational procedure that scales like iterative second order. We illustrate our results with several molecular examples including Ne2,Be2,F2, and benzene.     

Kohn–Sham potentials by an inverse Kohn–Sham equation and accuracy assessment by virial theorem

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one-dimensional model of a He atom and an H₂ molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange-correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground-state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem.     

Optimized effective potentials from electron densities in finite basis sets

The Wu-Yang method for determining the optimized effective potential (OEP) and implicit density functionals from a given electron density is revisited to account for its ill-posed nature, as recently done for the direct minimization method for OEP's from a given orbital functional [T. Heaton-Burgess, F. A. Bulat, and W. Yang, Phys. Rev. Lett. 98, 256401 (2007)]. To address the issues on the general validity and practical applicability of methods that determine the Kohn-Sham (local) multiplicative potential in a finite basis expansion, a new functional is introduced as a regularized version of the original work of Wu and Yang. It is shown that the unphysical, highly oscillatory potentials that can be obtained when unbalanced basis sets are used are the controllable manifestation of the ill-posed nature of the problem. The new method ensures that well behaved potentials are obtained for arbitrary basis sets.     

A simple and effective technique to variationally interpret the structure of SUSY partners of mirror image potentials

Precise supersymmetric partner potentials can be generated for exactly solvable problems of the stationary Schrödinger equation. Construction of isospectral potential is not always possible for exactly solvable systems. This is a restriction, as most problems are not exactly solvable. Employment of mirror-image property can help to construct an exact isospectral partner of that potential. These potentials have chemical relevance to enantiomers. In this paper, we present a formulation as modelling to explore the form of SUSY pair of these potentials. Through polynomial fit, we correlate all possible basic SUSY partners and optimise it to best fit polynomial to present a typical energy value of N = 50.     

Iterative computational method for the rapid analysis of iron and plutonium by controlled potential coulometry and some interesting observations on the coulogram of the Pu(III) /Pu(IV/ couple

An iterative computational method for the determination of metal ions in aqueous solutions which form reversible couples such as Fe(II)/Fe(III), Pu(III)/Pu(IV) etc. by controlled potential coulometry has been developed. The method involves carrying out the electrolysis to about 95–97% and calculating the total amount present in the sample by an iterative computational method. The method utilizes the direct application of the Nernst equation. The important criterion to be met is that the coulogram of the couple should strictly obey the Nernst equation. The validity of the method has been checked by analyzing about 50 samples of a standard iron solution. Results of analysis of mixtures of Pu and Fe by the iterative technique show that the interference of Fe can almost entirely be eliminated. However, analysis of Pu samples by this procedure gives results about 2–3% lower than the expected value. A careful examination of the experimental coulograms of Pu in lM HClO₄ indicates a slight deviation from the theoretical coulogram, where as those of Fe match exactly.     

Exchange methods in Kohn-Sham theory

Differences between exchange methods in exchange-only Kohn-Sham theory are highlighted by calculations of diatomic molecule total energies, uncoupled isotropic NMR shieldings, and HOMO-LUMO eigenvalue differences. Optimised effective potential (OEP) and Wu-Yang (WY) results are very similar. Localised Hartree-Fock (LHF) and Krieger-Li-Iafrate (KLI) results are close to one another, but are different to OEP and WY. Becke 1988 exchange (B88X) is different again. Shieldings reduce from OEP/WY to LHF/KLI to B88X, which is consistent with an observed reduction in HOMO-LUMO gaps. LHF, KLI, and B88X shieldings and HOMO-LUMO gaps are closer to near-exact, correlated values, than are the OEP values. These variations arise entirely due to differences in the one-electron exchange potentials, which is clearly evident in potential difference plots, relative to OEP, for the N2 molecule. Density difference plots are also presented, which exhibit a spatial correlation with the potential differences. HOMO and LUMO probability density difference plots show a contraction of the LUMO relative to OEP, which is consistent with the NMR and HOMO-LUMO findings. Plots are also presented for near-exact, correlated Kohn-Sham calculations. The features are qualitatively similar to those observed in the LHF, KLI, and B88X plots, highlighting correlated character in these approximate exchange-only calculations.     

The effective local potential method: implementation for molecules and relation to approximate optimized effective potential techniques

We have recently formulated a new approach, named the effective local potential (ELP) method, for calculating local exchange-correlation potentials for orbital-dependent functionals based on minimizing the variance of the difference between a given nonlocal potential and its desired local counterpart [V. N. Staroverov et al., J. Chem. Phys. 125, 081104 (2006)]. Here we show that under a mildly simplifying assumption of frozen molecular orbitals, the equation defining the ELP has a unique analytic solution which is identical with the expression arising in the localized Hartree-Fock (LHF) and common energy denominator approximations (CEDA) to the optimized effective potential. The ELP procedure differs from the CEDA and LHF in that it yields the target potential as an expansion in auxiliary basis functions. We report extensive calculations of atomic and molecular properties using the frozen-orbital ELP method and its iterative generalization to prove that ELP results agree with the corresponding LHF and CEDA values, as they should. Finally, we make the case for extending the iterative frozen-orbital ELP method to full orbital relaxation.     

An analysis of the siegert eigenvalue problem for autoionizing states

We show that the divergent integrals which appear in a direct matrix solution to the Siegert problem for autoionizing (or electron scattering) state energies and widths can be cancelled exactly. When this is done the Siegert problem becomes essentially a bound state problem. We also show that the resulting non-hermitian secular equation which requires several non-hermitian diagonalizations in the iterative solution for the complex energy can be exactly reduced by a partitioning technique to a single hermitian diagonalization (for a single open channel) with the subsequent iterative solution of a simple algebraic equation.     

Iterative Solution Method for the Linearized Poisson-Boltzmann Equation: Indirect Boundary Integral Equation Approach

An iterative solution scheme is proposed for solving the electrical double-layer interactions governed by the linearized Poisson-Boltzmann equation. The method is based on the indirect integral equation formulation with the double-layer potential kernel of the linearized Poisson-Boltzmann equation. In contrast to the conventional direct integral equation approach that yields Fredholm integral equations of the first kind, the indirect integral equation approach yields well-posed Fredholm integral equations of the second kind. The eigenvalue analysis reveals that the spectral radius of the double-layer integral operator is always less than one. Thus, iterative solution schemes can be successfully implemented for solving the electrical double-layer interactions for very large and complex systems. The utility of the iterative indirect method is demonstrated for several examples which include spherical and spheroidal particles. Copyright 2001 Academic Press.     

A simple effective potential for exchange

The optimized effective potential (OEP) for exchange was introduced some time ago by Sharp and Horton and by Talman and Shadwick. The integral equation for the OEP is difficult to solve, however, and a variety of approximations have therefore been proposed. These are explicitly orbital dependent and require the same two-electron integrals as Hartree-Fock theory. We have found a remarkably simple approximate effective potential that closely resembles the Talman-Shadwick potential in atoms. It depends only on total densities and requires no two-electron integrals.     

Calculating particle pair potentials from fluid‐state pair correlations: Iterative ornstein–zernike inversion

An iterative Monte Carlo inversion method for the calculation of particle pair potentials from given particle pair correlations is proposed in this article. The new method, which is best referred to as Iterative Ornstein–Zernike Inversion, represents a generalization and an improvement of the established Iterative Boltzmann Inversion technique (Reith, Pütz and Müller‐Plathe, J. Comput. Chem. 2003, 24, 1624). Our modification of Iterative Boltzmann Inversion consists of replacing the potential of mean force as an approximant for the pair potential with another, generally more accurate approximant that is based on a trial bridge function in the Ornstein–Zernike integral equation formalism. As an input, the new method requires the particle pair correlations both in real space and in the Fourier conjugate wavenumber space. An accelerated iteration method is included in the discussion, by which the required number of iterations can be greatly reduced below that of the simple Picard iteration that underlies most common implementations of Iterative Boltzmann Inversion. Comprehensive tests with various pair potentials show that the new method generally surpasses the Iterative Boltzmann Inversion method in terms of reliability of the numerical solution for the particle pair potential. © 2018 Wiley Periodicals, Inc.     

Crystal environments probed by EPR spectroscopy. Variations in the EPR spectra of Co(II)(octaethylporphyrin) doped in crystalline diamagnetic hosts and a reassessment of the electronic structure of four-coordinate cobalt(II)

The powder and single-crystal EPR spectra of Co(II)(OEP) (OEP is the dianion of octaethylporphyrin) doped into a range of diamagnetic crystals including simple four-coordinate hosts, H(2)(OEP), the triclinic B form of Ni(II)(OEP), the tetragonal form of Ni(II)(OEP) and Zn(II)(OEP); five-coordinate hosts, micro-dioxane)[Zn(II)(OEP)](2) and (py)Zn(II)(OEP); six-coordinate hosts, (py)(2)Zn(II)(OEP) and (py)(2)Mg(II)(OEP); and hosts containing fullerenes, C(60).2Zn(II)(OEP).CHCl(3), C(70).Ni(II)(OEP).C(6)H(6).CHCl(3), and C(60).Ni(II)(OEP).2C(6)H(6) have been obtained and analyzed. Spectra were simulated using a program that employed the exact diagonalization of the 16 x 16 complex spin Hamiltonian matrix. The EPR spectra of these doped samples are very sensitive to the environment within each crystal with the crystallographic site symmetry determining whether axial or rhombic resonance patterns are observed. For Co(II)(OEP) doped into tetragonal Ni(II)(OEP) (which displays a very large g( perpendicular ) of 3.405 and a very small g( parallel ) of 1.544) and several other crystals containing four-coordinate metal sites, the g components could not be fit using existing theory with the assumption of the usual z(2) ground state. However, reasonable agreement of the observed EPR parameters could be obtained by assuming that the unpaired electron resides in an xy orbital in the four-coordinate complexes.