Computational Chemistry

overlap integrals are also unity, since the basis functions are normalized (cf.
Section 4.4.1.2).
In the Mulliken scheme each electron inciis taken to contribute a “fraction of an
electron”c 1 ic 1 iS 11 ¼c^21 ito basis functionf 1 , and a fraction of an electron 2c 1 ic 2 iS 12
(see Eq. (5.210)) to thef 1 /f 2 overlap region, and in general to contribute a fraction
of an electronc^2 rito the basis function (loosely, the atomic orbital)fron an atom,
and a fraction of an electron 2cricsiSrsto thefr/fsinteratomic overlap space; see
Fig.5.39a. This seems reasonable since (1), the terms sum to one (the “fractions” of
the electron must add to one), and (2) it seems reasonable to partition out the
contribution of electrons to basis functions and to overlap regions from the

fs 1 fs 2

b

nr nr/s (^1) nr/s 2
(population due solely to fr)
MO yi, formed from basis functions f 1 and f 2
a

• 
  

  –

  
  


• 
  


• 
  


• 
  


• 
  


• – ––


• 
  

fr fr

fr fr fs

Fig. 5.39 The Mulliken scheme for partitioning electron density

346 5 Ab initio Calculations