Christian Kollmar

A size extensive energy functional derived from a double configuration interaction approach: The role of N representability conditions

Starting from a configuration interaction (CI) approach including only doubly excited configurations, the corresponding energy functional is modified by introduction of a topological factor in the normalization condition of the CI coefficients in such a way that it gets inherently size extensive. Constraints imposed by N representability conditions on the modified functional are discussed and lead to a specific choice of the topological factor. The basic variable in the modified energy functional is the second-order reduced density matrix parametrized in terms of CI-like coefficients. Test calculations for a variety of small molecules show that the numerical results obtained with the new functional are in very good agreement with those obtained from coupled cluster singles doubles calculations.

The structure of the second-order reduced density matrix in density matrix functional theory and its construction from formal criteria

Some formal requirements for the second-order reduced density matrix are discussed in the context of density matrix functional theory. They serve as a basis for the ad hoc construction of the second-order reduced density matrix in terms of the first-order reduced density matrix and lead to implicit functionals where the occupation numbers of the natural orbitals are obtained as diagonal elements of an idempotent matrix the elements of which represent the variational parameters to be optimized. The numerical results obtained from a first realization of such an implicit density matrix functional give excellent agreement with the results of full configuration interaction calculations for four-electron systems like LiH and Be. Results for H2O taken as an example for a somewhat larger molecule are numerically less satisfactory but still give reasonable occupation numbers of the natural orbitals and indicate the capability of density matrix functional theory to cope with static electron correlation.

A new approach to density matrix functional theory

Starting from a pair-excitation multiconfiguration self-consistent field approach considering pairwise excitations of two electrons of opposite spin from a single occupied molecular orbital to a single virtual molecular orbital, we present a natural orbital functional for the electronic energy containing the natural orbitals and the pair-excitation coefficients as variational parameters to be optimized. The occupation numbers of the natural orbitals can be determined from the pair-excitation coefficients in this implicit functional. Test calculations for the water molecule give occupation numbers of the natural orbitals in reasonable agreement with the results of full configuration interaction calculations.

The coupled electron pair approximation: variational formulation and spin adaptation

It is demonstrated that the coupled electron pair approximation (CEPA) equations are very close to the variational condition for an energy functional. Thus, fully relaxed density matrices can be obtained without solving Z vector equations for the amplitudes. It is also shown that CEPA/1 is the only one among the various CEPA approaches which can be spin-adapted in a consistent fashion.

An orbital-invariant and strictly size extensive post-Hartree-Fock correlation functional.

A strictly size extensive post-Hartree-Fock correlation functional being invariant with respect to orbital transformations within the occupied and virtual subspaces is presented. While avoiding the necessity to solve additional Z vector equations for the calculation of properties and energy gradients, this functional reproduces almost exactly the results of coupled-cluster singles doubles (CCSD) calculations. In particular, it is demonstrated that the method is rigorous in the sense that it can be systematically improved by the perturbative inclusion of triple excitations in the same way as CCSD. As to the computational cost, the presented approach is somewhat more expensive than the CCSD if the energy is variationally optimized with respect to both the orbitals and the excitation amplitudes. Replacement of orbital optimization by the Brueckner condition reduces the computational cost by a factor of two, thus making the method less expensive than CCSD.

The role of energy denominators in self‐consistent field (SCF) calculations for open shell systems

The transformation of the molecular orbitals (MO’s) of open shell systems during a self‐consistent field iteration step is compared for two different approaches: the usual procedure of iterative diagonalization of a Fock matrix and a simplified direct approach using the gradient and the one‐electron Hessian matrix in a space of orbital rotation parameters. It is shown that the frequent failure of the former is due to inappropriate energy denominators given by the difference of MO energies. In contrast to closed shell systems, the elements of the one‐electron Hessian matrix cannot be represented consistently by such differences in the case of open shell systems. The results of numerical calculations support this finding.

The LEDO expansion for diatomic overlap densities: application to the density functional theory of molecules

The limited expansion of differential overlap (LEDO) approach for the expansion of diatomic overlap densities in terms of mono-centre densities is discussed in the context of density functional theory (DFT). It is shown that it leads to a particularly simple construction scheme for major parts of the secular matrix, i.e. the electron-electron interaction and the exchange-correlation potential: using the LEDO expansion coefficients, matrix elements between atomic orbitals located on different centres can be expressed in terms of the corresponding mono-centre elements, thus allowing the reduction of three-centre and four-centre integrals to two-centre integrals. This results in the first DFT method with formal N 2 scaling for the construction of the secular matrix, with N being the dimension of the atomic orbital (AO) basis set. Test calculations show that numerical agreement with the results of conventional DFT calculations is excellent.

A partially restricted Hartree–Fock approach

We present a Hartree–Fock method which gives the molecular orbitals (MOs) of a single determinant wave function consisting of a restricted part (one orbital assigned to each pair of spin‐up and spin‐down electrons) and an unrestricted part (different orbitals for different spins). The MOs of the restricted part transform as irreducible representations of the point group of the molecule, whereas those of the unrestricted part may be symmetry broken. A modified coupling operator formalism is used for the determination of the MOs which result from different Fock operators for the closed shell orbitals (restricted part) and for the spin‐up and spin‐down orbitals of the unrestricted part. The method is illustrated by considering the symmetry breaking in the conjugated π system of short‐chain radical intermediates formed in the course of the solid state polymerization reaction of diacteylene crystals.

The static response function in Kohn-Sham theory: An appropriate basis for its matrix representation in case of finite AO basis sets

The role of the static Kohn-Sham (KS) response function describing the response of the electron density to a change of the local KS potential is discussed in both the theory of the optimized effective potential (OEP) and the so-called inverse Kohn-Sham problem involving the task to find the local KS potential for a given electron density. In a general discussion of the integral equation to be solved in both cases, it is argued that a unique solution of this equation can be found even in case of finite atomic orbital basis sets. It is shown how a matrix representation of the response function can be obtained if the exchange-correlation potential is expanded in terms of a Schmidt-orthogonalized basis comprising orbitals products of occupied and virtual orbitals. The viability of this approach in both OEP theory and the inverse KS problem is illustrated by numerical examples.

The relationship between double excitation amplitudes and Z vector components in some post-Hartree-Fock correlation methods

The relationship between Z vector components and excitation amplitudes is analyzed for several post-Hartree-Fock correlation methods limited to double excitation amplitudes. An analytical formula approximating the Z vector for the coupled cluster doubles method is presented and shown to be quite accurate. This approximation is also used to determine the prefactor of the norm of doubly excited states in averaged coupled pair functional-type energy functionals self-consistently leading to better agreement with coupled cluster results.