variety of mathematical tricks. Among these are: the use of symmetry to avoid
duplicate calculation of identical integrals; testing two-electron integrals quickly to
see if they are small enough to be neglected (as is the case for functions on distant
nuclei; this decreases the time of a calculation from ann^4 dependence on the number
of basis function to about ann2.3dependence); recalculating integrals to avoid the
bottleneck of hard-drive access (direct SCF,Section 5.3.2); representing the MOs as a
set of gridpoints in space (in addition to a basis set expansion), which eliminates the
need to explicitly calculate two-electron integrals. Thispseudospectralmethod
speeds up ab initio calculations by a factor of perhaps 3 or 4. Methods of speeding
up calculations are explained, with references to the literature, by Levine [ 33 ].
The method of calculating wavefunctions and energies that has been described in
this chapter applies toclosed-shell,ground-statemolecules. The Slater determinant
we started with (Eq.5.12) applies to molecules in which the electrons are fed
pairwise into the MO’s, starting with the lowest-energy MO; this is in contrast to
free radicals, which have one or more unpaired electrons, or to electronically
excited molecules, in which an electron has been promoted to a higher-level MO
(e.g. Fig.5.9, neutral triplet). The Hartree–Fock method outlined here is based on
closed-shell Slater determinants and is called therestricted Hartree–Fockmethod
or RHF method; “restricted” means that the electrons ofaspin are forced to occupy
(restricted to) the same spatial orbitals as those ofbspin: inspection of Eq.5.12
shows that we do not have a set ofaspatialorbitals and a set ofbspatialorbitals. If
unqualified, a Hartree–Fock (i.e. an SCF) calculation means an RHF calculation.
The common way to treat free radicals is with theunrestricted Hartree–Fock
method or UHF method. In this method, we employ separate spatial orbitals for the
aand thebelectrons, giving two sets of MO’s, one foraand one forbelectrons.
Less commonly, free radicals are treated by therestricted open-shell Hartree–Fock
or ROHF method, in which electrons occupy MO’s in pairs as in the RHF method,
except for the unpaired electron(s). The theoretical treatment of open-shell species
is discussed in various places in references [ 1 ] and in [ 12 ].
Excited states, and those unusual molecules with electrons of opposite spin singly
occupying different spatial MO’s (open-shell singlets) cannot be properly treated
with a single-determinant wavefunction. They must be handled with approaches
beyond the Hartree–Fock level, such as configuration interaction (Section 5.4).
5.3 Basis Sets................................................................
5.3.1 Introduction.......................................................
We encountered basis sets in Sections 4.3.4, 4.4.1.2, and 5.2.3.6.1. A basis set is a
set of mathematical functions (basis functions), linear combinations of which yield
molecular orbitals, as shown in Eqs.5.51and5.52. The functions are usually, but
not invariably, centered on atomic nuclei (Fig.5.7). Approximating molecular
orbitals as linear combinations of basis functions is usually called the LCAO or
linear combination of atomic orbitals approach, although the functions are not
232 5 Ab initio Calculations