can of course be augmented (Section 5.3.3) with diffuse functions, and the 6–31G*
with H polarization functions (6–31G). MP2 calculations increase rapidly in
complexity with the number of electrons and orbitals, involving as they do asum
of terms(rather than just one term as in HHe+), each representing the promotion of
an electron pair from an occupied to a virtual orbital; thus an MP2 calculation on
CH 2 with the 6–31G* basis involves eight electrons and 19 MOs (four occupied and
15 virtual MOs).
In MP2 calculations doubly excited states (doubly excited configurations) inter-
act with the ground state (the integral in Eq.5.162involvesc 1 with electrons 1 and
2, andc 2 with electrons 1 and 2). In MP3 calculations doubly excited states interact
with one another (there are integrals involving two virtual orbitals). In MP4
calculations singly, doubly, triply and quadruply excited states are involved. MP5
and higher expressions have been developed, but MP2 and MP4 are by far the most
popular Møller–Plesset levels (also called MBPT(2) and MBPT(4) – many-body
perturbation theory). MP2 calculations, which are much slower than Hartree–Fock,
can be speeded up somewhat by specifying MP2(fc), MP2 frozen-core, in contrast
to MP2(full); frozen-core means that the core (non-valence electrons) are “frozen”,
i.e. not promoted into virtual orbitals, in contrast to full MP2 which takes all the
electrons into account in summing the contributions of excited states to the lower-
ing of energy. Most programs, e.g. Gaussian, Spartan) perform MP2(fc) by default
when MP2 is specified, and “MP2” usually means frozen-core. When seen in this
book referring to a specific calculation rather than a general method, it may be taken
as shorthand for MP2(fc). MP4 calculations are sometimes done omitting the triply
excited terms (MP4SDQ) but the most accurate (and slowest) implementation is
MP4SDTQ (singles, doubles, triples, quadruples).
Calculated properties like geometries and relative energies tend to be better (to be
closer to the true ones) when done with correlated methods (Sections 5.5.1–5.5.4).
To save time, energies are often calculated with a correlated method on a Hartree–
Fock geometry, rather than carrying out the geometry optimization at the correlated
level. This is called asingle-pointcalculation (it is performed at a single point on the
HF potential energy surface, without changing the geometry). A single-point MP2
(fc) calculation using the 6–311G* basis, on a structure that was optimized with the
Hartree–Fock method and the 6–31G basis, is designated as MP2(fc)/6–311G//
HF/6–31G. A HF/6–31G (say) geometry optimization, without a subsequent
single-point calculation, is sometimes designated HF/6–31G//HF/6–31G, and
an MP2 optimization MP2/6–31G//MP2/6–31G. The correlation treatment (HF,
MP2, MP4,...) is often called themethod, and the basis set (STO-3G, 3–21G(),
6–31G,...) thelevel, but we will often find it convenient to letleveldenote the
combined procedure of method and basis set, referring, say, to an MP2/6–31G
calculation as being at a higher level than an HF/6–31G one.
Figure5.20shows the rationale behind the use of single-point calculations for
obtaining relative energies. In the diagram a single-point MP2 calculation on a
stationary point at the HF geometry gives the same energy as would be obtained by
optimizing the species at the MP2 level, which is often approximately true (it would
be exactly true if the MP2 and HF geometries were identical). For example, the
5.4 Post-Hartree–Fock Calculations: Electron Correlation 265