5.2.3.6 Basis Functions and the Roothaan–Hall Equations
5.2.3.6.1 Deriving the Roothaan–Hall Equations
As they stand, the Hartree–Fock equations5.44,5.46or 5.47 are not very useful for
molecular calculations, mainly because (1) they do not prescribe a mathematically
viable procedure getting the initial guesses for the MO wavefunctionsci, which we
need to initiate the iterative process (Section 5.2.3.5), and (2) the wavefunctions
may be so complicated that they contribute nothing to a qualitative understanding
of the electron distribution.
For calculations onatoms, which obviously have much simpler orbitals than
molecules, we could use for thec’s atomic orbital wavefunctions based on the
solution of the Schr€odinger equation for the hydrogen atom (taking into account the
increase of atomic number and the screening effect of inner electrons on outer ones
[ 25 ]). This yields the atomic wavefunctions as tables ofcat various distances from
the nucleus. This is not a suitable approach for molecules because among molecules
there is no prototype species occupying a place analogous to that of the hydrogen
atom in the hierarchy of atoms, and as indicated above it does not readily lend itself
to an interpretation of how molecular properties arise from the nature of the
constituent atoms.
In 1951 Roothaan and Hall independently pointed out [ 26 ] that these problems
can be solved by representing MO’s as linear combinations of basis functions (just
as in the simple H€uckel method, inChapter 4, thepMO’s are constructed from
atomicporbitals). Roothaan’s paper was more general and more detailed than
Hall’s, which was oriented to semiempirical calculations and alkanes, and the
method is sometimes called the Roothaan method. For a basis-function expansion
of MO’s we write
c 1 ¼c 11 f 1 þc 21 f 2 þc 31 f 3 þ(((þcm 1 fm
c 2 ¼c 12 f 1 þc 22 f 2 þc 32 f 3 þ(((þcm 2 fm
c 3 ¼c 13 f 1 þc 23 f 2 þc 33 f 3 þ(((þcm 3 fm
...cm¼c 1 mf 1 þc 2 mf 2 þc 3 mf 3 þ(((þcmmfm$ð 5 : 51 ÞIn devising a more compact notation for this set of equations it is very helpful to
use different subscripts to denote the MO’scand the basis functionsf. Conven-
tionally, Roman letters have been used for thec’s and Greek letters for thef’s, or
i,j,k,l,...for thec’s andr,s,t,u,...for thef’s. The latter convention will be
adopted here, and we can write the equations (5.51) as
5.2 The Basic Principles of the ab initio Method 197