Recent Advances in Multireference Methods

The following sections are included:

• FOREWORD

• AUTHOR LIST

• CONTENTS

The rationale for carrying out multireference single- and double-excitation configuration interaction (CI) calculations to obtain highly correlated electronic wave-functions for atoms and molecules is discussed. A detailed description of computational algorithms for implementing a configuration-driven approach for this type of theoretical treatment is given, with emphasis on the use of CI tables to enable an efficient evaluation of Hamiltonian matrix elements between Slater determinants spanning configurations with many open shells. These techniques have recently been incorporated in a Table-Direct CI program version (TD-CI) which minimizes I/O operations and thereby greatly expands the range of applicability of the configuration-driven approach. With the use of improved methods for recognition of excitation relationships between configurations it has become possible in the current implementation to solve secular equations of order exceeding one million randomly selected linear combinations of Slater determinants which are eigenfunctions of the S² operator. Finally, timing results are presented for a number of test examples and a survey of possible future extensions of these methods is undertaken.

Various approaches to multi-reference perturbation theory are discussed. In this context, multi-reference is taken to imply the diagonalization of an effective Hamiltonian after perturbation corrections are applied. Available methods include Møller-Plesset formulations with orbital energy denominators, Epstein-Nesbet formulations with diagonal elements of H as denominators, and shifted denominators that are between these. A few sample results from the literature are included.

We present in this paper linear response theories based on our recently formulated state-specific multi-reference coupled cluster (SS-MRCC) formalism with a complete active space (CAS). The coupled cluster Ansatz for the wave operator excites each of the reference determinants to all the virtual functions via its own cluster operator. The linear dependence of the cluster operators implicit in such a choice is eliminated by positing suitable sufficiency conditions while ensuring size-extensivity. The use of a CAS then also guarantees size-consistency. After a resume of the SS-MRCC method, which serves both to highlight the essentials and to introduce the notations, we discuss two distinct response formulations. The first is a straightforward numerically oriented approach exploiting our SS-MRCC to compute static second order properties via perturbed orbitals in a finite field. The other, a genuinely analytical linear response theory using excitation operators, generalizes the single reference CC-based linear response theory to a multi-reference function for obtaining excitation energies. Both these formulations are illustrated with example applications.

This contribution primarily discusses coupled electron pair (CEPA) type variants of the multi-reference single and double excitation configuration interaction (MRSDCI) method. This type of theory takes advantage of simplicity and flexibility of configuration interaction method. However they incorporate more or less the knowledge from cluster expansion method. In the early stage of this note, we review configuration interaction and cluster expansion methods briefly. As the state-selective (state-specific) approach of the CEPA type variants of MRSDCI is extensively reviewed by Szalay, this report will mainly focus our attention on the state-universal approach of the CEPA type variants of MRSDCI and will be compared with the other approach from a theoretical point of view. Reliability of computational results will be discussed.

An analytic energy gradient method for second-order quasidegenerate perturbation theory with multiconfigurational self-consistent field reference functions (MC-QDPT) is presented. An expression for the energy gradients is derived using the Lagrange multiplier method The gradients are calculated without solving coupled perturbed equations Instead, it is necessary to solve eight sets of linear equations for the multipliers. Six of the eight equations reduces to simple partial differential forms which directly give the multipliers, and only the remaining two are large scale linear equations that need iterative procedure. The gradients are given as the product of the first derivative integrals and the effective densities that depends on the obtained multipliers and the parameters. The expression for the conventional quasidegenerate perturbation theory and numerical results for the methylene molecule are also presented.

Applications of the Complete Active Space (CAS) SCF method in conjunction with multiconfigurational second-order perturbation theory (CASPT2) in electronic spectroscopy of organic molecules are reviewed. Since the first applications in spectroscopy were performed at the beginning of the present decade, the CASSCF/CASPT2 method has been used to study electronic spectra of a large number of compounds. The experience gained from this global investigation is illustrated in the present contribution through several examples. In most cases, the CASSCF reference function does characterize with sufficient accuracy the states of interest, which supports the use of a single reference perturbation theory in spectroscopic studies of organic systems. The CASSCF/CASPT2 method is capable of yielding accurate results for relative energies and other properties of excited states, provided that flexible one-electron basis sets are used and an appropriate active space can be chosen. The overall accuracy of the approach is high. The excitation energies are usually found to be within ±0.2 eV of the available experimental energies for correctly assigned transitions. The review covers some of the most recent applications in the spectroscopy of organic compounds: absorption spectrum of free base porphin employing an extended treatment; vertical, nonvertical, and emission energies of long polyenals; spectra of trans- and cis-stilbene together with an analysis of certain static aspects of the photo-induced isomerization process; absorption spectra of purine DNA base monomers and related compounds, and analysis of the spectroscopic features of polypeptides based on intra- and interpeptide charge transfer transitions. In addition, the electronic spectra of organic compounds with interacting double bonds are rationalized. These studies either confirm existing experimental assignments or lead to new predictions and a novel understanding of the electronic spectra of the corresponding organic molecules.

The electronic structure of many molecular systems in their ground and excited states show characteristic features due to electrons occupying weakly overlapping atomic-like orbitals and to complex spin couplings. Such systems can be conveniently represented with CAS multiconfiguration wavefunctions which are able to capture these essential non-dynamical electron correlation effects and to yield semi-quantitative structures, spectra, and other properties for such systems. Multireference perturbation theoretical treatments of the dynamical electron correlation effects can then be applied to obtain quantitative accuracy. In this chapter we discuss selected examples of such molecular systems, including the design of CAS wavefunctions and we analyze the calculated structural and spectroscopic data. We end with a comparison of the present CAS approach with the broken symmetry UHF methodology. The resuls indicate that the UHF approach reproduces the CAS results well.