s=1mm basis functions sth basis functionc of the sth basis function of ith MOyi = ∑csi fs (^) i = 1, 2, 3, ..., m (m MOs)
ith MO
ð 5 : 52 Þ
We are expanding each MOcin terms ofmbasis functions. The basis functions
are usually (but not necessarily) located on atoms, i.e. for the functionf(x,y,z),
wherex,y,zare the coordinates of the electron being treated by this one-electron
function, the distance of the electron from the nucleus is:
r¼½ðx#x 0 Þ^2 þðy#y 0 Þ^2 þðz#z 0 Þ^2 ^1 =^2 ð 5 : 53 Þ
wherex 0 ,y 0 ,z 0 are the coordinates of the atomic nucleus in the coordinate system
used to define the geometry of the molecule. Because each basis function may
usually be regarded (at least vaguely) as some kind of atomic orbital, this linear
combination of basis functions approach is commonly called a linear combination
of atomic orbitals (LCAO) representation of the MO’s, as in the simple and
extended H€uckel methods (Sections 4.3.4. and 4.4.1). The set of basis functions
used for a particular calculation is called thebasis set.
We need at least enough spatial MO’scto accommodate all the electrons in the
molecule, i.e. we need at leastnc’s for the 2nelectrons (recall that we are dealing
with closed-shell molecules). This is ensured because even the smallest basis
sets used in ab initio calculations have for each atom at least one basis function
corresponding to each orbital conventionally used to describe the chemistry of the
atom, and the number of basis functionsfis equal to the number of (spatial) MOsc
(Section 4.3.4). An example will make this clear: for an ab initio calculation on
CH 4 , the smallest basis set would specify for C:
fðC; 1 sÞ;fðC; 2 sÞ;fðC; 2 pxÞ;fðC; 2 pyÞ;fðC; 2 pzÞ
and for each H:
fðH; 1 sÞ
These nine basis functionsf(5 on C and 4' 1 ¼4 on H) create nine spatial
MO’sc, which could hold 18 electrons; for the ten electrons of CH 4 we need only
five spatial MO’s. There is noupperlimit to the size of a basis set: there are
commonly many more basis functions, and hence MO’s, than are needed to hold all
the electrons, so that there are usually many unoccupied MO’s. In other words, the
number of basis functionsmin the expansions (5.52) can be much bigger than the
198 5 Ab initio Calculations