Computational Chemistry

F 11 c 11 þF 12 c 21 þF 13 c 31 þ(((¼eðS 11 c 11 þS 12 c 21 þS 13 c 31 þ(((Þ

i.e.

Xm

s¼ 1

csiFrs¼e

Xm

s¼ 1

csiSrs ð 5 : 60 Þ

But this is the first equation of the set (5.53-1). Continuing in this way we see
that matching each element of (the multiplied-out) matrixFC(5.58) with the
corresponding element of (the multiplied-out) matrixSC«gives one of the equa-
tions of the set5.54-1to (5.54-m), i.e. of the set (5.56). This can be so only ifFC¼
SCe, so this matrix equation is indeed equivalent to the set of equations (5.56).
Now we haveFC¼SCe(5.57), the matrix form of the Roothaan–Hall equa-
tions. These equations are sometimes called the Hartree–Fock–Roothaan equations,
and, often, the Roothaan equations, as Roothaan’s exposition was the more detailed
and addresses itself more clearly to a general treatment of molecules. Before
showing how they are used to do ab initio calculations, a brief review of how we
got these equations is in order.
Summary of the derivation of the Roothaan–Hall equations.

1. The total wavefunctionCof an atom or molecule was expressed as a Slater
   determinant of spin MO’sc(spatial)aandc(spatial)b, Eq.5.12.


2. From the Schr€odinger equation we got an expression for the electronic energy of
   the atom or molecule,E¼ CjH^jC
   , Eq.5.14.


3. Substituting the Slater determinant for the total molecular wavefunctionCand
   inserting the explicit form of the Hamiltonian operatorH^into (5.14) gave the
   energy in terms of the spatial MO’sc, (Eq.5.17):

E¼ 2

Xn

i¼ 1

Hiiþ

Xn

i¼ 1

Xn

j¼ 1

ð 2 Jij#KijÞ:

4. MinimizingEin Eq.5.17with respect to thec’s (to find the bestc’s) gave the
   Hartree–Fock equationsF^c¼ec(5.44).


5. Substituting into the Hartree–Fock equationsF^c¼ec(5.44) the Roothaan–Hall
   linear combination of basis functions (LCAO) expansionsci¼

P

csifs(5.52)

for the MO’scgave the Roothaan–Hall equations (Eqs.5.56), which can be

written compactly asFC¼SCe(Eqs.5.57).

5.2.3.6.2 Using the Roothaan–Hall Equations to do ab initio Calculations – the
SCF Procedure

The Roothaan–Hall equationsFC¼SCe(Eqs.5.57)(F,C,Sandeare defined in
connection with Eqs.5.58and5.59; the matrix elementsFandSare defined by
Eqs.5.54and5.55) are of the same matrix form as Eq. 4.54,HC¼SCe, in the

5.2 The Basic Principles of the ab initio Method 203