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.     

Fermi and Coulomb holes of molecules in excited states

Correlation holes of electrons with the same (Fermi hole) and different (Coulomb hole) spins in the ground (X¹Σ⁺), first (A¹Σ⁺) and second (B¹II) excited states of LiH were constructed from full configuration interaction (CI ) wave functions. It was found that the shapes of both the Fermi and Coulomb holes in these states are dependent on the location of the reference electron. When the reference electron is chosen to be close to the Li nucleus, the Fermi correlation results in a large negative hole for all three states. However, the A¹Σ⁺ excited state is further characterized by displaying a second hole around the H nucleus, and in the B¹II state, the hole is elongated along the molecular axis. Coulomb correlation shows up strongly in the A¹Σ⁺ state and, in addition, there is clearly correlation of electrons at the two nuclei. These features of the correlation holes were compared with those from a two-Slater-determinant model wave function. The Hartree, Fermi, and Coulomb screening potentials in these states were also studied in the light of possible modeling of the correlation functionals for the excited states. © 1995 John Wiley & Sons, Inc.     

Effect of modified virtual orbitals on local Coulomb correlation holes in FCN

Local Coulomb correlation hole distribution functions may be used to assess the extent to which electron correlation effects are present in large scale SCF + CI wave functions. From a set of modified virtual orbitals, ordered according to their interaction with the SCF configuration, we have constructed a limited SCF + CI wave function with improved convergence characteristics with respect to that formed from the canonical virtual orbital set. These wave functions, of the same size yet with different energies, have been used to examine the range and depth of local Coulomb correlation holes in FCN. In all cases, the depth of the local Coulomb hole is no more than 10% or so of that of the corresponding Fermi hole, and the range Fermi correlation is generally less than that of Fermi correlation. This is particularly marked in the high density regions around the nuclei. The significance of our results is discussed in relation to a recent proposal for the incorporation of Coulomb correlation into the local exchange method.     

The Colle–Salvetti Wavefunction Revisited: a Comparison Between Three Approaches for Obtaining the Correlation Energy

The correlation factor of Colle and Salvetti is studied by comparing the behavior of three different correlation functionals. The normalization, sum rule, Coulomb hole, correlation energy integrand, and the Wigner exclusion hole have all been analyzed by applying the three approaches. The results indicate that the correlation factor proposed by Colle–Salvetti is a very good choice for modeling electron correlation in atoms. The flaws appearing in the development of the Colle–Salvetti equations seem mainly due to an inadequate use of the first mean value theorem of integral calculus. The Gaussian summation used for the two-body density matrix seems to be a good approximation to obtain the correlation factor equations.     

Explicit and implicit multi‐center integrations

We present truncated expansions of multicenter one‐electron nuclear attraction and two‐electron repulsion integrals over localized basis functions in terms of one‐ and two‐center integrals of “Coulomb,” “exchange,” and “hybrid” type. Two variants are discussed: the “Explicit Multi‐center Integrations” and the “Implicit Multi‐Center Integrations” (abbreviated as “EMCI” and “IMCI”, respectively). While EMCI also deals with individual integrals, the IMCI option is the more appealing one: it enables us to evaluate the entire matrix elements of “Restricted Hartree–Fock”‐type in a very effective and chemically meaningful way. Due to the diatomic nature of our expansions, integrations over “Slater‐Type Orbitals” become well‐feasible, too. © 2012 Wiley Periodicals, Inc.     

Unpaired electrons at the second‐order reduced density matrix level: Covalent bonding,and coulomb and fermi correlations in closed shell systems

The usual one‐electron populations in atomic orbitals of closed shell systems are split into unpaired and paired at the (spin‐dependent) second‐order reduced density matrix level. The unpaired electron in an orbital is defined as the “simultaneous occurrence of an electron and an electron hole of opposite spins in the same spatial orbital,” which for simplicity is called “electropon.” The electropon population in a given orbital reveals whether and to what degree the Coulomb correlations, and hence, the chemical bonding between this orbital and the remaining orbitals of the system are globally favorable or unfavorable. The interaction of two electropons in two target orbitals reveals the quality (favorable or unfavorable) and the strength of the covalent bonding between these orbitals; this establish a bridge between the notion of “unpaired electrons” and the traditional covalent structure of valence‐bond (VB) theory. Favorable/unfavorable bonding between two orbitals is characterized by the positive/negative (Coulomb) correlation of two electropons of opposite spins, or alternatively, by the negative/positive (Fermi) correlation of two parallel spin electropons. A spin‐free index is defined, and the relationship between the electropon viewpoint for chemical bonding and the well‐known two‐electron Coulomb and Fermi correlations is established. Benchmark calculations are achieved for ethylene, hexatriene, benzene, pyrrole, methylamine, and ammonia molecules on the basis of physically meaningful natural orbitals. The results, obtained in the framework of both orthogonal and nonorthogonal population analysis methods, provide the same conceptual pictures, which are in very good agreement with elementary chemical knowledge and VB theory. © 2013 Wiley Periodicals, Inc.     

Angular aspects of exchange correlation and the fermi hole

The complete (nonreduced) αα probability density functions evaluated from the Hartree–Fock and simple Hartree product wavefunctions have been used to elucidate the angular features of spin correlation and the Fermi hole in the 2³S state of helium and the ground state of beryllium. This approach shows that the local Fermi holes in these two cases are very similar and that the Fermi hole is essentially spherically symmetric when the reference electron is close to the nucleus. As the reference electron is removed to larger radial distances, appreciable polarization of the Fermi hole is observed. The polarization is greater in the direction of the nucleus than away from the nucleus, contrary to the situation in the Coulomb hole of the helium ground state where the polarization is greater away from the nucleus than toward the nucleus. Several other differences between the He 2³S Fermi hole and the He 1¹S Coulomb hole are noted.     

Electron correlation: the many-body problem at the heart of chemistry

The physical interactions among electrons and nuclei, responsible for the chemistry of atoms and molecules, is well described by quantum mechanics and chemistry is therefore fully described by the solutions of the Schr?dinger equation. In all but the simplest systems we must be content with approximate solutions, the principal difficulty being the treatment of the correlation between the motions of the many electrons, arising from their mutual repulsion. This article aims to provide a clear understanding of the physical concept of electron correlation and the modern methods used for its approximation. Using helium as a simple case study and beginning with an uncorrelated orbital picture of electronic motion, we first introduce Fermi correlation, arising from the symmetry requirements of the exact wave function, and then consider the Coulomb correlation arising from the mutual Coulomb repulsion between the electrons. Finally, we briefly discuss the general treatment of electron correlation in modern electronic-structure theory, focussing on the Hartree-Fock and coupled-cluster methods and addressing static and dynamical Coulomb correlation.     

Effect of Coulomb interaction between the electrons on two-electron redox-mediated tunneling

Two-electron non-adiabatic redox-mediated tunneling through a symmetric electrochemical contact with a bridge molecule having one electron energy level participating in tunneling is considered under ambient conditions. It is shown that the current/overpotential dependence in this system can disclose two distinct or overlapping clear-cut maxima depending on the value of the effective Coulomb repulsion energy. This new effect is due to the opening of the channel for tunneling of second electron with the variation of the electrode potential. The system manifests also a rectification effect in the current/bias voltage curve which depends on the value of the effective Coulomb repulsion energy.     

The Coulomb Hole of the Ne Atom

We analyze the Coulomb hole of Ne from highly-accurate CISD wave functions obtained from optimized even-tempered basis sets. Using a two-fold extrapolation procedure we obtain highly accurate results that recover 97 % of the correlation energy. We confirm the existence of a shoulder in the short-range region of the Coulomb hole of the Ne atom, which is due to an internal reorganization of the K-shell caused by electron correlation of the core electrons. The feature is very sensitive to the quality of the basis set in the core region and it is not exclusive to Ne, being also present in most of second-row atoms, thus confirming that it is due to K-shell correlation effects.     

Electron correlation effects in SCF+CI wave functions—Fermi and coulomb correlation holes in HCN

The Coulomb correlation hole distribution function has been computed with respect to various reference centers in the HCN molecule, using standard SCF +CI type wave functions. The extent to which statistical correlation between unlike-spin electrons is introduced into an SCF wave function through the inclusion of configuration interaction has been assessed by an examination of the range and depth of such holes, and compared with the behavior of analogous Fermi distribution functions. Our results show that the range of Fermi correlation is consistently longer than that of the corresponding Coulomb correlation.     

Bare Coulomb field and atomic reciprocal form factor

The reciprocal form factor of N‐electron closed shells systems in a bare Coulomb field is shown to be a spherically symmetric, positive, and decreasing function of the radial distance. Nonmonotonicities of the reciprocal form factor appear when studying bare Coulomb field open‐shell systems. Analysis of the weight of the interelectronic repulsion term is carried out for some isoelectronic series as well as neutral atoms with N = 1–103. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

A simultaneous probability density for the intracule and extracule coordinates

We introduce the intex density X(R,u), which combines both the intracular and extracular coordinates to yield a simultaneous probability density for the position of the center-of-mass radius (R) and relative separation (u) of electron pairs. One of the principle applications of the intex density is to investigate the origin of the recently observed secondary Coulomb hole. The Hartree-Fock (HF) intex densities for the helium atom and heliumlike ions are symmetric functions that may be used to prove the isomorphism 2I(2R)=E(R), where I(u) is the intracule density and E(R) is the extracule density. This is not true of the densities that we have constructed from explicitly correlated wave functions. The difference between these asymmetric functions and their symmetric HF counterparts produces a topologically rich intex correlation hole. From the intex hole distributions (X(exact)(R,u)-X(HF)(R,u)), we conclude that the probability of observing an electron pair with a very large interelectronic separation increases with the inclusion of correlation only when their center-of-mass radius is close to half of their separation.     

A new look at correlations in atomic and molecular systems. I. Application of fermion monte carlo variational method

We are engaged in research directed toward the development of compact and accurate correlation functions for many-electron systems. Our computational tool is the variational method in which the many-electron integrals are calculated by Monte Carlo using the fermion Metropolis sampling algorithm. That is, a many-fermion system is simulated by sampling the square of a correlated antisymmetric wave function. The principal advantage of the method is that interelectronic distance r_ij may be included directly in the wave function without adding significant computational complexity. In addition, other quantities of physical and theoretical interest such as electron correlation functions and representations of Coulomb and Fermi “holes” are very easily obtained. Preliminary results are reported for He, H₂, and Li₂.     

A nonlocal correlation energy density functional from a Coulomb hole model

A nonlocal correlation energy density functional based on the approximation of a model Coulomb hole is presented. The functional is constructed to describe both the homogeneous electron gas and nonuniform systems. In the nonuniform case, the functional satisfies all uniform, as well as most nonuniform, coordinate-scaling constraints. The numerical results for the homogeneous electron gas and for atoms He through Ar compare favorably with those of other correlation functionals. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 603–616, 1997     

Gaussian and finite-element Coulomb method for the fast evaluation of Coulomb integrals

The authors propose a new linear-scaling method for the fast evaluation of Coulomb integrals with Gaussian basis functions called the Gaussian and finite-element Coulomb (GFC) method. In this method, the Coulomb potential is expanded in a basis of mixed Gaussian and finite-element auxiliary functions that express the core and smooth Coulomb potentials, respectively. Coulomb integrals can be evaluated by three-center one-electron overlap integrals among two Gaussian basis functions and one mixed auxiliary function. Thus, the computational cost and scaling for large molecules are drastically reduced. Several applications to molecular systems show that the GFC method is more efficient than the analytical integration approach that requires four-center two-electron repulsion integrals. The GFC method realizes a near linear scaling for both one-dimensional alanine alpha-helix chains and three-dimensional diamond pieces.     

Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a (nonlinear, approximate) density functional of three spatial dimension one‐electron density for an N‐electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas‐Fermi kinetic energy (～∫ρ^5/3d r ₁) and Parr electron–electron repulsion energy (～∫ρ^4/3d r ₁) main‐term functionals are accepted, and they should, the later described, compact one‐electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg–Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground‐state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one‐electron density models. © 2008 Wiley Periodicals, Inc. J Comput Chem 2009     

On the proof by reductio ad absurdum of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states of Coulomb systems

It is demonstrated that the original reductio ad absurdum proof of the generalization of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states for isolated many‐electron Coulomb systems with Coulomb‐type external potentials by Gross and colleagues is self‐contradictory, since the to‐be‐refuted assumption (negation) regarding the ensemble one‐electron densities and the assumption regarding the external potentials are logically incompatible to each other due to the Kato electron‐nuclear cusp theorem. It is proved, however, that the Kato theorem itself provides a satisfactory proof of this theorem. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006