Combining two-body density correlation functionals with multiconfigurational wave functions using natural orbitals and occupation numbers

We study the performance of different approaches that combine multiconfigurational wavefunctions with correlation functionals for the calculation of magnetic coupling constants of several materials and molecules. The systems under study include four antiferromagnetic materials: NiO, KNiF(3), K(2)NiF(4) and La(2)CuO(4); two biradicals: alpha-4-Dehydrotoluene and 1,1',5,5'-Tetramethyl-6,6'-dioxo-3,3'-biverdazyl; two molecular complexes: [Cu(2)Cl(6)](-2) and Copper(II) acetate monohidrate; and the prototypical H-He-H system. On average, the best results are obtained with a recently proposed method [Phys. Rev. A 75, 012503 (2007)] that estimates the correlation energy of density functionals from a pair of alternative spin densities built from the natural orbitals and occupation numbers of the multiconfigurational wavefunction.

Accurate quantum chemical methods for the prediction of spin-state energy gaps for strongly correlated systems are computationally expensive and scale poorly with the size of the system. This makes calculations for many experimentally interesting molecules impractical even with abundant computational resources. In previous work, we have shown that the localized active space (LAS) self-consistent field (SCF) method is an efficient way to obtain multi-configuration SCF wave functions of comparable quality to the corresponding complete active space (CAS) ones. To obtain quantitative results, a post-SCF method is needed to estimate the complete correlation energy. One such method is multiconfiguration pair-density functional theory (PDFT), which calculates the energy based on the density and on-top pair density obtained from a multiconfiguration wave function. In this work we introduce localized-active-space pair-density functional theory, which uses a LAS wave function for subsequent PDFT calculations. The method is tested for computing spin-state energy gaps in conjugated organic molecules and bimetallic compounds and is shown to give results within 0.05 eV of the corresponding CAS-PDFT results at a significantly lower cost.

We have developed a method for analyzing the (hyper)polarizabilities of open-shell molecular systems. This method employs the (hyper)polarizability densities based on the natural orbitals and occupation numbers, which enables us to analyze the contributions of odd electrons having various open-shell (diradical) characters. Within broken-symmetry, i.e., spin-unrestricted, single-determinant molecular orbital and density functional theory approaches, we can also remove the spin contamination effects on these quantities through spin projection. To do that, an approximate spin projected method has been elaborated and applied to the analysis of the (hyper)polarizability of multi-radical systems. As examples, typical open-shell singlet systems, 1,3-dipoles and rectangular graphene nanoflakes, are examined.

Ab initio calculations including electron correlation are reported for the water and methylene molecules as a function of geometry. A large contracted gaussian basis set is used and the multiconfiguration wave functions, optimized by the iterative natural orbital procedure, include 277 and 617 configurations for H2O and CH2 respectively. The method of selecting configurations, yielding “first-order” wave functions, is discussed in some detail. For H2O, the SCF geometry is r=0,942 A, θ=105,8°, the correlated result is r=0,968 A, θ=103,2°, and the experimental r=0,957 A, θ=104,5°. The water stretching force constants, in millidynes/A, are 8,72 (SCF), 8,75 (CI), and 8,4 (experiment). Bending force constants are 0,88 (SCF), 0,83 (CI), and 0,76 (experiment). For methylene the SCF geometry is r=1,072 A, θ=129,5°, while the result from first-order wave functions is r=1,088 A, θ=134°. The predicted CH2 force constants are 6,16 (SCF) and 6,13 (CI) for stretching and 0,44 (SCF) and 0,33 (CI) for bending.

The natural orbitals and their corresponding occupation numbers are constructed for several interesting problems to demonstrate that the existence of negative natural orbital occupation numbers for single reference correlation methods provides a simple diagnostic for the need for a multiconfigurational description of the wave function.

Two theorems on the eigenvalues of differences of idempotent matrices determine the natural occupation numbers and orbitals of electronic detachment, attachment, or excitation that pertain to transitions between wavefunctions that each consist of a single Slater determinant. They are also applicable to spin density matrices associated with Slater determinants. When the ranks of the matrices differ, unit eigenvalues occur. In addition, there are ±w pairs of eigenvalues where |w| ≤ 1, whose values are related to overlaps, t, between the corresponding orbitals of Amos and Hall, and Löwdin by the formula w=±1-t2 12. Generalized overlap amplitudes, including Dyson orbitals and their probability factors, may be inferred from these eigenvalues, which provide numerical criteria for: classifying transitions according to the number of holes and particles in final states with respect to initial states, identifying the most important effects of orbital relaxation produced by self-consistent fields, and the analysis of Fukui functions. Two similar theorems that apply to sums of idempotent matrices regenerate formulae for the natural orbitals and occupation numbers of an unrestricted Slater determinant that were published first by Amos and Hall.

The analytical calculation of geometrical derivatives in molecular electronic structure theory is reviewed. The simplifications arising from the use of variational wave functions (Wigner’s 2n+l rule) are described and it is shown how energies for non-variational wave functions may be written in a variational form to take advantage of these simplifications. After a discussion on Hamiltonian derivatives, general expressions are given for the molecular gradients and Hessians for the following ab initio electronic structure techniques: self-consistent field (SCF) and multiconfigurational (MC) SCF wave functions, configuration interaction (CI) and coupled cluster (CC) wave functions, and Moller-Plesset (MP) perturbation theory. Finally, use of translational and rotational symmetries is discussed.

The occupation numbers and natural orbitals of several model π-electron molecules are presented and analyzed, which are directly calculated by employing a theory developed earlier by one of the authors. Computational details are discussed briefly, and the errors which arise in our calculations are estimated through a comparison with the density matrix of a full configuration interaction wave function for the pyridine molecule.

Natural orbitals (NOs) are central constituents for evaluating correlation energies through efficient approximations. Here, we report the closed-form expression of the NOs of two-electron quantum rings, which are prototypical finite-extension systems and new starting points for the development of exchange-correlation functionals in density functional theory. We also show that the natural occupation numbers for these two-electron paradigms are in general non-vanishing and follow the same power law decay as atomic and molecular two-electron systems.

Accurate quantum chemical methods for the prediction of spin-state energy gaps for strongly correlated systems are computationally expensive and scale poorly with the size of the system. This makes calculations for many experimentally interesting molecules impractical even with abundant computational resources. Previous work has shown that the localized active space (LAS) self-consistent field (SCF) method can be an efficient way to obtain multiconfiguration SCF wave functions of comparable quality to the corresponding complete active space (CAS) ones. To obtain quantitative results, a post-SCF method is needed to estimate the complete correlation energy. One such method is multiconfiguration pair-density functional theory (PDFT), which calculates the energy based on the density and on-top pair density obtained from a multiconfiguration wave function. In this work, we introduce localized-active-space PDFT, which uses a LAS wave function for subsequent PDFT calculations. The method is tested by computing spin-state energies and gaps in conjugated organic molecules and a bimetallic compound and comparing to the corresponding CAS-PDFT values.

(Hyper)polarizability density analysis for open-shell molecular systems based on natural orbitals and occupation numbers

Approximate wave functions for the hydrogen molecule ground state previously available in the literature are analyzed quantitatively into approximate natural spin orbital functions with particular attention to the corresponding occupation numbers. The analysis demonstrates the very great similarity of all such trial wave functions, and especially the largely molecular orbital nature of the wave function. In addition it shows a close relationship between the molecular orbital and valence bond functions, and the importance for allowing for angular correlation of the electrons by including terms dependent upon the azimuthal coordinate. The analysis particularly demonstrates that approximate natural spin orbital occupation numbers are nearly invariant under a wide variety of choices of basis functions, and therefore are particularly suitable for comparison of different approximate functions and for discussion of their respective properties.

By examining the natural orbital occupation numbers, it was discovered that unrestricted Møller–Plesset perturbation theory (MP2) can violate N-representability by having natural occupation numbers greater than 2 or less than 0, even though the energy appears well behaved. Analytically, this problem stems from the fact that the MP2 effective one-particle density matrix is dependent on the inverse of the orbital hessian matrix (A′). When a molecular system goes through the point where spin-restricted orbitals become unstable to unrestriction, one of the eigenvalues of the A′ matrix will become 0, making it a singular matrix, thereby causing problems in the relaxed one-particle density matrix and discontinuities in the first order properties such as nuclear forces and dipole moments. Since the new ‘double-hybrid’ density functionals also involve a second-order perturbation expression for correlation using Kohn–Sham orbitals, these functionals also exhibit similar issues near the unrestriction point. The unphysical orbital occupations can be eliminated and the first derivative discontinuities removed by optimizing orbitals with inclusion of the second order correlation energy. Connections to the second derivative discontinuities in Hartree–Fock and Kohn–Sham density functional theory are also discussed.

Whereas a density functional that incorporates dispersion interaction has remained elusive to date, we demonstrate that in principle the dispersion energy can be obtained from a density matrix functional. In density matrix functional theory one tries to find suitable approximations to the two-particle reduced density matrix (2RDM) in terms of natural orbitals (NOs) and natural orbital occupation numbers (ONs). The total energy is then given as a function(al) of the NOs and ONs, i.e., as an implicit functional of the 1RDM. The left-right correlation in a (dissociating) bond, as well as various types of dynamical correlation, can be described accurately with a NO functional employing only J and K integrals (JK-only functional). We give a detailed analysis of the full CI wavefunction of the He(2) dimer, from which the dispersion part of the two-particle density matrix is obtained. It emerges that the entirely different physics embodied in the dispersion interaction leads to an essentially different type of exchange-correlation orbital functional for the dispersion energy (non-JK). The distinct NO functionals for the different types of correlation imply that they can be used in conjunction without problems of double counting. Requirements on the (primitive) basis set for Van der Waals bonding appear to be more modest than for other types of correlation.

CASSCF study of the relation between the Fe charge and the Mössbauer isomer shift