The electron correlation cusp

The Kais function is an exact solution of the Schrödinger equation for a pair of electrons trapped in a parabolic potential well with r ₁₂ ^?1 electron-electron interaction. Partial wave analysis (PWA) of the Kais function yields E _L = E + C₁(L + \-C ^?1 ₂)^?3 + O(L ^?5) where E is the exact energy and E _L the energy of a renormalized finite sum of partial waves omitting all waves with angular momentum ? > L. Slight rearrangement of an earlier result by Hill shows that the corresponding full CI energy differs from E _L only by terms of order O(L ^?5) with FCI values of C ₁ and \-C ^?1 ₂ identical to PWA values. The dimensionless \-C ₂ parameter is weakly dependent upon the size of the physical system. Its value is 0.788 for the Kais function, and 0.893 for the less diffuse helium atom, and approaches \-C ₂→ 1 in the limit of an infinitely compact charge distribution. The ?th energy increment satisfies an approximate virial theorem which becomes exact in the high ? limit. This analysis, formulated to facilitate use of the Maple system for symbolic computing, lays the mathematical ground work for subsequent studies of the electron correlation cusp problem. The direction of future papers in this series is outlined.     

Localizability of dynamic electron correlation

The convergence of the intrapair correlation energy for a localized internal orbital is investigated as the virtual subspace is enlarged. At variance with previous investigations of this kind, the virtual subspace is represented in atomic orbitals. This allows to define spatial relations between the orbitals involved. Typically, over 98% of the pair correlation energy is recovered by a small local basis set, consisting of the valence orbitals of the atoms with which the electron pair is associated. This opens the possibility of an efficient Cl procedure based on localized pairs.     

Wavefunction-based electron correlation methods for solids

In this article we provide an overview of the most common ways of treating electron correlation effects in 3D-periodic systems with some emphasize on wavefunction-based correlation methods such as the method of increments and the local MP2 method implemented in the Cryscor program. We discuss strengths and weaknesses of the different approaches and give examples for their application. Additionally, for the method of increments we discuss recent developments for its application to open shell systems and problems related to the treatment of graphene sheets.     

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.     

On electron correlation in NaCl2

This article reports the intrapair and interpair electron correlation energies of the radical NaCl₂. The total interpair correlation energy dominates. Hence, the interpair electron correlation energy must be considered in building models for correcting computed correlation energies. The 6-311+G* basis set recovers only 32% of the total estimated correlation energy and 44% of this amount came from the core electrons. © 1995 John Wiley & Sons, Inc.     

Response function theory of electron correlation

The full perturbation expansion for the response (or density—density correlation) function is examined in order to provide a useful general theory of excitation energies, oscillator strengths, dynamic polarizabilities, etc., that is more accurate than the random phase approximation. It is first shown how the formal partition of the diagrammatic version of the perturbation expansion into reducible and irreducible diagrams is generally useless as the latter category contains all the difficult terms which have heretofore resisted analysis in all but a haphazard form. It is then shown how the diagram for the response function can be partitioned into “correlated” and “uncorrelated” subsets. Restricting attention to the particle—hole blocks of the full response function, the “uncorrelated” diagrams desecribe the propagation of a particle—hole pair in an N-electron system where the particle and hole are each interacting with the remaining electrons but they are not interacting with each other. The “correlated” diagrams are those containing the hole—particle interactions, and, by defining a new class of reducible and irreducible diagrams, these are all summed to provide a perturbation expansion of the effective two-body hole—particle interaction that appears in the inverse of the response function. The “uncorrelated” diagrams are further partitioned into two sets, one of which is summed to all orders, while the other set is inverted in an order by order fashion. The final result presents a perturbation expansion for the inverse of the response function that is analogous to the Dyson equation for one-electron Green functions. Maintaining the perturbation expansion through first order for the inverse of the response function yields the eigenvalue equation of the familiar random phase approximation, while truncation at second order provides the most advanced theories that have been generated by the equations-of-motion method.     

Assessment of transition operator reference states in electron propagator calculations

The transition operator method combined with second-order, self-energy corrections to the electron propagator (TOEP2) may be used to calculate valence and core-electron binding energies. This method is tested on a set of molecules to assess its predictive quality. For valence ionization energies, well known methods that include third-order terms achieve somewhat higher accuracy, but only with much higher demands for memory and arithmetic operations. Therefore, we propose the use of the TOEP2 method for the calculation of valence electron binding energies in large molecules where third-order methods are infeasible. For core-electron binding energies, TOEP2 results exhibit superior accuracy and efficiency and are relatively insensitive to the fractional occupation numbers that are assigned to the transition orbital.     

Rigorous integral screening for electron correlation methods

We derive rigorous multipole-based integral estimates (MBIE) in order to account for the distance dependence occurring in atomic-orbital (AO) formulations of electron correlation theory, where our focus is on AO-MP2 theory within a Laplace scheme. We find for the exact transformed integral products an extremely early onset of a linear-scaling behavior and a very small number of significant products. To preselect the significant integral products we adapt our MBIE method as rigorous upper bound. In this way it is possible to exploit the favorable scaling behavior observed and to reduce the scaling of estimated products asymptotically to linear, without sacrificing accuracy or reliability. By separating Coulomb- and exchange-type contractions only half-transformed integrals need to be computed. Furthermore, our scheme of rigorously preselecting transformed integral products via MBIE seems to offer particularly interesting perspectives for a direct formation of half- or fully transformed integrals by using multipole expansions and auxiliary basis sets.     

Role of electron correlation in hyperfine interaction

The spin density near the nucleus of a magnetic atom is examined in the first order with respect to the nondiagonal matrix element for the electron interaction with the corresponding excitation level. A system containing one electron outside the closed shells is discussed.     

Intershell nl4f electron correlation effects

The pair correlation energies for some nl4f pairs of the ground state of the Yb atom are calculated for the first time. The partial wave (PW ) increments to the second-order pair energies are generated using numerical first-order radial pair functions obtained as the solution of two-dimensional differential equations. The analysis of the PW s contributions shows the dominant role of the df, fg, and gh PW s for the 4d4f pair, of the pf and dg PW s for the 4p4f and 5p4f pairs, and of the sf and pg PW s for the 4s4f, 5s4f, and 6s4f pairs. A discussion of the similarities and differences of the structure of the correlation energy found in this paper with those calculated earlier for smaller atoms is given.     

Large electron correlation effect leading to BeBe bond

The bonding of the beryllium diatomic molecule (Be₂) in the ground state is exclusively made from the electron correlation effect. Unlike the ordinary van der Waals bond, where the electron correlation of the dispersion type makes weak bond energy (D_e) at large bond distance (R_e), the BeBe bond is surprisingly strong with D_e = 830 cm^?1 and R_e = 245 pm. This paper presents in an analytical way the different electron correlation effects with the corresponding spectroscopic data.     

Mutual information and electron correlation in momentum space

Mutual information and information entropies in momentum space are proposed as measures of the nonlocal aspects of information. Singlet and triplet state members of the helium isoelectronic series are employed to examine Coulomb and Fermi correlations, and their manifestations, in both the position and momentum space mutual information measures. The triplet state measures exemplify that the magnitude of the spatial correlations relative to the momentum correlations depends on and may be controlled by the strength of the electronic correlation. The examination of one- and two-electron Shannon entropies in the triplet state series yields a crossover point, which is characterized by a localized momentum density. The mutual information density in momentum space illustrates that this localization is accompanied by strong correlation at small values of p.     

Local-MP2 electron correlation method for nonconducting crystals

Rigorous methods for the post-HF (HF-Hartree-Fock) determination of correlation corrections for crystalline solids are currently being developed following different strategies. The CRYSTAL program developed in Torino and Daresbury provides accurate HF solutions for periodic systems in a basis set of Gaussian type functions; for insulators, the occupied HF manifold can be represented as an antisymmetrized product of well localized Wannier functions. This makes possible the extension to nonconducting crystals of local correlation linear scaling On techniques as successfully and efficiently implemented in Stuttgart's MOLPRO program. These methods exploit the fact that dynamic electron correlation effects between remote parts of a molecule (manifesting as dispersive interactions in intermolecular perturbation theory) decay as an inverse sixth power of the distance R between these fragments, that is, much more quickly than the Coulomb interactions that are treated already at the HF level. Translational symmetry then permits the crystalline problem to be reduced to one concerning a cluster around the reference zero cell. A periodic local correlation program (CRYSCOR) has been prepared along these lines, limited for the moment to the solution of second-order Moller-Plesset equations. Exploitation of point group symmetry is shown to be more important and useful than in the molecular case. The computational strategy adopted and preliminary results concerning five semiconductors with tetrahedral structure (C, Si, SiC, BN, and BeS) are presented and discussed.     

A finite-difference approach to the electron correlation problem

A variant of the transcorrelated method of Boys and Handy employing finite differences is presented. It is based upon the following two properties of the transcorrelated Hamiltonian operator C^?1HC: (1) C^?1HC possesses an energy eigenvalue spectrum which is identical to that associated with H itself; and (2) if \documentclass{article}\pagestyle{empty}\begin{document}$$ C \equiv \begin{array}{*{20}c} \pi & {e^{r_{ij} /2} } \\ {i > j} & {} \\ \end{array} $$\end{document} then C^?1HC is free of the singularities of H at the points where the interelectron separation r_ij is zero. A bivariational principle for approximating the eigenvalues and the left and right eigenfunctions of C^?1HC is introduced and the resulting set of coupled integro-differential equations are solved in finite-difference form by means of a coupled self-consistent field, Newton Raphson algorithm. As a preliminary test of the method, a calculation of the ground-state energy of the helium atom is presented.