Essentials of Computational Chemistry

7.4.3 Multireference
7.4 PERTURBATION THEORY 223
The generalization of MPn theory to the multireference case involves the obvious choice
of using an MCSCF wave function for (0) instead of a single-determinant RHF or UHF
one. However, it is much less obvious what should be chosen for H (0) ,astheMCSCFMOs
do not diagonalize any particular set of one-electron operators. Several different choices
have been made by different authors, and each defines a unique ‘flavor’ of multireference
perturbation theory (see, for instance, Andersson 1995; Davidson 1995; Finley and Freed
1995). One of the more popular choices is the so-called CASPT2N method of Roos and
co-workers (Andersson, Malmqvist, and Roos 1992). Often this method is simply called
CASPT2 – while this ignores the fact that different methods having other acronym endings
besides N have been defined by these same authors (e.g., CASTP2D and CASPT2g1), the
other methods are sufficiently inferior to CASPT2N that they are typically used only by
specialists and confusion is minimized.
Most multireference methods described to date have been limited to second order in perturbation
theory. As analytic gradients are not yet available, geometry optimization requires
recourse to more tedious numerical approaches (see, for instance, Page and Olivucci 2003).
While some third order results have begun to appear, much like the single-reference case,
they do not seem to offer much improvement over second order.
An appealing feature of multireference perturbation theory is that it can correct for some
deficiencies associated with an incomplete active space. For instance, the relative energies
for various electronic states of TMM (Figure 7.1) were found to vary widely depending
on whether a (2,2), (4,4), or (10,10) active space was used; however, the relative energies
from corresponding CASPT2 calculations agreed well with one another. Thus, while the
motivation for multireference perturbation theory is to address dynamical correlation after
a separate treatment of non-dynamical correlation, it seems capable of handling a certain
amount of the latter as well.
7.4.4 First-order Perturbation Theory for Some Relativistic Effects
In Møller–Plesset theory, first-order perturbation theory does not improve on the HF energy
because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order
perturbation theory can be useful for estimating energetic effects associated with operators
that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity
and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly
difficult to self-consistently optimize wavefunctions for systems where these terms are explicitly
included in the Hamiltonian, but an estimate of their energetic contributions may be
had from simple first-order perturbation theory, since that energy is computed simply by
taking the expectation values of the operators over the much more easily obtained HF
wave functions.
The mass-velocity correction is evaluated as
1
Emv = HF −
∇
4
i
HF
(7.49)
8c 2
i