HandSare square matrices andcand 0 are column matrices (Eq.4.51), andEis
a scalar (an ordinary number). We have been developing these equations for a
system of two basis functions, so there should be two MOs, each with its own
energy and its own pair ofc’s (Fig.4.11). We need two energy values and fourc’s:
we want to be able to calculatec 11 andc 21 ofc 1 (MO 1 , energy level 1) andc 12 and
c 22 ofc 2 (MO 2 , 0 energy level; in keeping with common practice the energies of the
MOs are designatede 1 ande 2. Equation4.53can be extended (our simple derivation
shortchanges us here) [ 36 ] to encompass the fourc’s and twoe’s; the result is
HC¼SC« '(4:54)
We now have only square matrices; in Eq.4.53cwas a column matrix andEwas
not a matrix, but rather a scalar – an ordinary number. The matrices are (four
equations Eqs (4.55)):
H¼
H 11 H 12
H 21 H 22
C¼
c 11 c 12
c 21 c 22S¼
S 11 S 12
S 21 S 22
e¼e 1 0
0 e 2'(4:55)
TheHmatrix is an energy-elements matrix, theFock^27 matrix, whose elements
are integralsHij(Eqs.4.44). Fock actually pointed out the need to take electron spin
into account in more elaborate calculations than the simple H€uckel method; we will
meet “real” Fock matrices inChapter 5. For now, we just note that in the simple
(and extended) H€uckel methods as an ad hoc prescription at most two electrons,
paired, are allowed in each MO. EachHijrepresents some kind of energy term,
sinceHˆis an energy operator (Section 4.3.3). The meaning of theHij’s is discussed
later in this section.
TheCmatrix is thecoefficient matrix, whose elements are the weighting factors
cijthat determine to what extent each basis functionf(roughly, each atomic orbital
on an atom) contributes to each MOc. Thusc 11 is the coefficient off 1 inc 1 ,c 21 the
coefficient off 2 inc 1 , etc., with the first subscript indicating the basis function and
the second subscript the MO (Fig.4.11). In each column ofCthec’s belong to the
same MO.
(^27) Vladimer Fock, born St. Petersburg, 1898. Ph.D. Petrograd University, 1934. Professor Lenin-
grad University, also worked at various institutes in Moscow. Worked on quantum mechanics and
relativity, e.g. the Klein–Fock equation for particles with spin in an electromagnetic field. Died
Leningrad, 1974.
124 4 Introduction to Quantum Mechanics in Computational Chemistry