Computational Chemistry

function, each composed of two Gaussians, and an outer 2s,2px,2pyand 2pz(2s^00 ,
2 px^00 ,2py^00 ,2pz^00 ) function, each composed of one Gaussian, making nine basis
functions. The terms inner and outer derive from the fact that the Gaussian of the
outer shell has a smallerathan the Gaussians of the inner shell, and so the former
function falls off more slowly, i.e. it is more diffuse and effectively spreads out
further, into the outer regions of the molecule. The purpose of splitting the valence
shell is to give the SCF algorithm more flexibility in adjusting the contributions of
the basis functions to the molecular orbitals, thus achieving a more realistic
simulated electron distribution. Consider carbene, CH 2 (Fig.5.15). We can denote
the basis functionsf 1 #f 13 :

C1s:f 1

C2s^0 ,2px^0 ,2py^0 ,2pz^0 :f 2 ,f 3 ,f 4 ,f 5 (inner valence shell)

C2s^00 ,2px^00 ,2py^00 ,2pz^00 :f 6 ,f 7 ,f 8 ,f 9 (outer valence shell)

H 11 s^0 :f 10 (inner shell)

H 11 s^00 :f 11 (outer shell)

H 21 s^0 :f 12 (inner shell)

H 21 s^00 :f 13 (outer shell)

Thirteen basis functions (“atomic orbitals”) give thirteen LCAO MO’s:

c 1 ¼c 11 f 1 þc 21 f 2 þ(((þc 13 ; 1 f 13

c 2 ¼c 12 f 1 þc 22 f 2 þ(((þc 13 ; 2 f 13

...

c 13 ¼c 1 ; 13 f 1 þc 2 ; 13 f 2 þ(((þc 13 ; 13 f 13

Note that since there are 13 MO’s but only eight electrons to be accommodated,
only the first four MO’s (c 1 – c 4 ) are occupied (recall that we are talking about
closed-shell molecules in the ground electronic state). The nine empty MO’s are
calledunoccupiedorvirtual molecular orbitals. We shall see that virtual MO’s are
important in certain kinds of calculations. Now, in the course of the SCF process the
coefficients of the various inner-shell and outer-shell basis functions can be varied
independently to find the best wavefunctionsc(those corresponding to the lowest
energy). As the iterations proceed some outer-shell functions, say, could be given
greater (or lesser) emphasis, independently of any inner-shell functions, allowing a

H H

13 basis functions

8 electrons

C, 9 basis functions

H, 2 basis functions H, 2 basis functions

C

Fig. 5.15 Carbene, with
3–21G basis functions

244 5 Ab initio Calculations