Calculations on correlation effects in molecules : convergence and size-consistency of multi-reference methods

Abstract

This thesis is about methods for electronic structure calculations on molecular systems. The ultimate goal is to construct methods that yield potential energy surfaces of sufficient accuracy to allow a qualitatively correct description of the chemistry of these systems; i.e. heat of formation, isomerisation barriers, equilibrium geometries, and vibrational spectra. In order to properly calculate the potential energy surfaces for all these properties a multi-configurational starting point is essential. This means that all methods that will be discussed are based on a multi-reference wavefunction where the reference function is optimised using the multi-configurational Hartree-Fock (MCHF) method. Beyond the MCHF method there are various methods to account for the correlation energy of a molecule. Not all of these methods are equally suited to calculating potential energy surfaces. In chapter 1 the basic notions are introduced and a list of required qualities for a method is compiled. This list includes size consistency, independence of the orbital representation of the reference wavefunction, efficiency, and others. The methods discussed in this thesis will be checked against these requirements. The main part of this thesis treats a perturbation method that was first formulated by Møller and Plesset (MP) in 1934. In its original formulation the method is applied starting from a single closed shell determinant. It was known that in this case the method is strictly size consistent. It was known also that the method diverges in cases when two states are close in energy. The basic idea at the start of the work described here was to avoid these divergences by including all nearly degenerate states in the reference function thus generalising the method to the multi-reference case. If this would be possible while retaining the size consistency a very efficient and highly accurate multi-reference Møller-Plesset method (MRMP) would be obtained. In chapter 2 the implementation of this method for a general reference wavefunction is described. Although the test applications yielded encouraging results, a few results suggested divergences may still show up. In chapter 3 a method to detect divergences is proposed and it was applied to suspicious systems. It is found that the multi-reference perturbation theory may be more strongly divergent than the single-reference approach. Also, the multi-reference results were not exactly size consistent. A detailed study of this problem is?118 given in chapter 4 were it is concluded from theory that the method should be exactly size consistent. In chapter 5 the practical aspects involved in a size consistent multi-reference perturbation theory are described. The results show that a size consistent approach can be obtained. The crucial aspects are that the projection operators to construct the zeroth-order Hamiltonian should each project onto a subspace of a single excitation level, the orthogonalisation method to generate the orthonormal excited states should be highly accurate, and in open-shell calculations applying the unitary group generators twice is not enough to generate all required spin states. Perturbation theory is not the only method that yields size consistent results. Already in the sixties it was known that some electron pair approximations yield exactly size consistent correlation energies also. In the single reference case it was shown that the coupled electron-pair approximation (CEPA) could be used to approximate coupled cluster in the singles-doubles configuration space. Ruttink et al. have generalised this approach to the multi-reference case (MRCEPA(0)). Although this approach is not exactly size consistent it is the best alternative to MRMP we have available. For this reason the MRMP results in this thesis are often compared to results obtained with MRCEPA(0). At the heart of the MRCEPA(0) is the Davidson diagonalisation method that is used to iteratively solve the eigenvalue equations. The efficiency of the MRCEPA depends primarily on the rapid convergence of the Davidson method. Essentially, the Davidson method calculates the best approximation to the wavefunction from a given set of vectors. Through extending this set by one vector (the update vector) in every iteration convergence is guaranteed. The speed of convergence depends on how appropriate the update vectors are. However, Sleijpen and van der Vorst realised that if the method was applied exactly as suggested by Davidson it would never converge. A detailed analysis led to improvements enhancing the speed of convergence. The application of these improvements in quantum chemistry is discussed in chapter 6. In chapter 7 the results from the main chapters are checked against the requirements list compiled in chapter 1. The conclusion is that although some requirements could be met, none of the methods satisfies all requirements. Because the alternatives employing a determinantal basis are nearly exhausted it is suggested that future developments should go in other directions, e.g. explicitly correlated wavefunctions.