Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation.

In Paper I, the performances of pre-screening (PS), extended PS (EPS), and cumulant (CU) approximations to the fourth-order density matrix were examined in the context of second-order N-electron valence state perturbation theory (NEVPT2). It has been found that the CU, PS, and even EPS approximations with loose thresholds may introduce intruder states. In the present work, the origin of these "false intruder" states introduced by approximated density matrices is discussed. Canonical NEVPT2 implementations employ a rank reduction trick. By analyzing its residual error, we find that the omission of the rank reduction leads to a more stable multireference perturbation theory for incomplete active space reference wave functions. Such a full rank (FR)-NEVPT2 formulation is equivalent to the conventional NEVPT2 method for the complete active space self-consistent field/complete active space configuration interaction reference wave function. A major drawback of the FR-NEVPT2 formulation is the necessity of the fifth-order density matrix. To avoid the construction of the high-order density matrices, the combination of the FR-NEVPT2 with the CU approximation is studied. However, we find that the CU approximation remains problematic as it still introduces intruder states. The question of how to robustly and efficiently perform internally contracted multireference perturbation theories with approximate densities remains a challenging field of investigation.

An efficient implementation of the NEVPT2 and CASPT2 methods avoiding higher-order density matrices.

A factorization of the matrix elements of the Dyall Hamiltonian in N-electron valence state perturbation theory allowing their evaluation with a computational effort comparable to the one needed for the construction of the third-order reduced density matrix at the most is presented. Thus, the computational bottleneck arising from explicit evaluation of the fourth-order density matrix is avoided. It is also shown that the residual terms arising in the case of an approximate complete active space configuration interaction solution and containing even the fifth-order density matrix for two excitation classes can be evaluated with little additional effort by choosing again a favorable factorization of the corresponding matrix elements. An analogous argument is also provided for avoiding the fourth-order density matrix in complete active space second-order perturbation theory. Practical calculations indicate that such an approach leads to a considerable gain in computational efficiency without any compromise in numerical accuracy or stability.

An alternative choice of the zeroth-order Hamiltonian in CASPT2 theory.

A zeroth-order Hamiltonian based on Koopmans matrices for complete active space second-order perturbation theory (CASPT2) is presented. This Hamiltonian involves three types of Fock matrices. The original CASPT2 Fock matrix is retained for excitation classes where the excitation does not change the number of electrons in the complete active space (CAS). For excitation classes involving a change in the number of electrons in the CAS, two alternative Fock matrices corresponding to either positive or negative ionization of the CAS are introduced. These are constructed such that they exactly reproduce the Koopmans matrices for a singly ionized CAS. Test calculations indicate that the method gives better excitation energies than CASPT2 without using empirical parameters, for example, the ionization potential-electron affinity shift, which is also designed to improve excitation energies. The method is also less prone to intruder states than conventional CASPT2. Moreover, the dissociation curve for the chromium dimer looks much more reasonable than the one obtained with conventional CASPT2.

A perturbation-based super-CI approach for the orbital optimization of a CASSCF wave function.

A perturbation theory-based algorithm for the iterative orbital update in complete active space self-consistent-field (CASSCF) calculations is presented. Following Angeli et al. (J. Chem. Phys. 2002, 117, 10525), the first-order contribution of singly excited configurations to the CASSCF wave function is evaluated using the Dyall Hamiltonian for the determination of a zeroth-order Hamiltonian. These authors employ an iterative diagonalization of the first-order density matrix including the first-order correction arising from single excitations, whereas the present approach uses the single-excitation amplitudes directly for the construction of the exponential of an anti-Hermitian matrix resulting in a unitary matrix which can be used for the orbital update. At convergence, the single-excitation amplitudes vanish as a consequence of the generalized Brillouin's theorem. It is shown that this approach in combination with direct inversion of the iterative subspace (DIIS) leads to very rapid convergence of the CASSCF iteration procedure. © 2019 Wiley Periodicals, Inc.

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.

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 orbital products in the optimized effective potential method.

The orbital products of occupied and virtual orbitals are employed as an expansion basis for the charge density generating the local potential in the optimized effective potential method thus avoiding the use of auxiliary basis sets. The high computational cost arising from the quadratic increase of the dimension of this product basis with system size can be greatly reduced by elimination of the linearly dependent products according to a procedure suggested by Beebe and Linderberg [Int. J. Quantum Chem. 12, 683 (1977)]. Numerical results from this approach show a very good agreement with those obtained from balancing the auxiliary basis for the expansion of the local potential with the orbital basis set.

Optimized effective potential method: is it possible to obtain an accurate representation of the response function for finite orbital basis sets?

The optimized effective potential (OEP) equations are solved in a matrix representation using the orbital products of occupied and virtual orbitals for the representation of both the local potential and the response function. This results in a direct relationship between the matrix elements of local and nonlocal operators for the exchange-correlation potential. The effect of the truncation of the number of such products in the case of finite orbital basis sets on the OEP orbital and total energies and on the spectrum of eigenvalues of the response function is examined. Test calculations for Ar and Ne show that rather large AO basis sets are needed to obtain an accurate representation of the response function.

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.

Optimization of auxiliary basis sets for the LEDO expansion and a projection technique for LEDO-DFT.

We present a systematic procedure for the optimization of the expansion basis for the limited expansion of diatomic overlap density functional theory (LEDO-DFT) and report on optimized auxiliary orbitals for the Ahlrichs split valence plus polarization basis set (SVP) for the elements H, Li--F, and Na--Cl. A new method to deal with near-linear dependences in the LEDO expansion basis is introduced, which greatly reduces the computational effort of LEDO-DFT calculations. Numerical results for a test set of small molecules demonstrate the accuracy of electronic energies, structural parameters, dipole moments, and harmonic frequencies. For larger molecular systems the numerical errors introduced by the LEDO approximation can lead to an uncontrollable behavior of the self-consistent field (SCF) process. A projection technique suggested by Löwdin is presented in the framework of LEDO-DFT, which guarantees for SCF convergence. Numerical results on some critical test molecules suggest the general applicability of the auxiliary orbitals presented in combination with this projection technique. Timing results indicate that LEDO-DFT is competitive with conventional density fitting methods.

The "JK-only" approximation in density matrix functional and wave function theory.

Various energy functionals applying the "JK-only" approximation which leads to two-index two-electron integrals instead of four-index two-electron integrals in the electron-electron interaction term of the electronic energy are presented. Numerical results of multiconfiguration self-consistent field calculations for the best possible "JK-only" wave function are compared to those obtained from the pair excitation multiconfiguration self-consistent (PEMCSCF) method and two versions of density matrix functional theory. One of these is derived making explicit use of some necessary conditions for N representability of the second-order density matrix. It is shown that this method models the energy functional based on the best possible "JK-only" wave function with good accuracy. The calculations also indicate that only a minor fraction of the total correlation energy is incorporated by "JK-only" approaches for larger molecules.

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.