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
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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.
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