Computational Chemistry

The total one-electron energy matrix,Hcore, is

Hcore¼TþVðHÞþVðHeÞ¼

# 1 : 6606 # 1 : 3160

# 1 : 3160 # 2 : 3030



ð 5 : 120 Þ

This matrix represents the one-electron energy (the energy the electron would
have if interelectronic repulsion did not exist) of an electron in H#Heþ, at the
specified geometry, for this STO-1G basis set. The (1,1), (2,2) and (1,2) terms
represent, ignoring electron–electron repulsion, the energy of an electron inf 1 ,f 2 ,
and thef 1 #f 2 overlap region, respectively; the values are the net result of the
various kinetic energy and potential energy terms discussed above.
(b) The two-electron matrix
The two-electron matrixG, the electron repulsion matrix (Eq. 5.104), is
calculated from the two-electron integrals (Eqs.5.110) and the density matrix
elements (Eq.5.81). This is intuitively plausible since each two-electron integral
describes one interelectronic repulsion in terms of basis functions (Fig.5.10) while
each density matrix element represents the electron densityon(the diagonal
elements ofPin Eq.5.80) orbetween(the off-diagonal elements ofP) basis
functions. To calculate the matrix elementsGrs(Eqs.5.106–5.108) we need the
appropriate integrals (Eqs.5.110) and density matrix elements. These latter are
calculated from

Ptu¼ 2

Xn

j¼ 1

c$tjcuj t¼ 1 ; 2 ;...;m and u¼ 1 ; 2 ;...;m ð 5 : 121 ¼ 5 : 81 Þ

EachPrsinvolves the sum over the occupied MO’s (j¼1–n; we are dealing with a
closed-shell ground-state molecule with 2nelectrons) of the products of the coeffi-
cients of the basis functionsfrandfs. As pointed out in Section 5.2.3.6.2 the
Hartree–Fock procedure is usually started with an “initial guess” at the coefficients.
We can use as our guess the extended H€uckel coefficients we obtained for HeH+, with
this same geometry (Section 4.4.1.2); we need thec’s only for theoccupiedMO’s:

c 11 ¼ 0 : 249 ; c 21 ¼ 0 : 867 ð 5 : 122 Þ

(Usually we need morec’s than the small basis set of an extended H€uckel or other
semiempirical calculation supplies; aprojectedsemiempirical wavefunction is then
used, with the missingc’s extrapolated from the available ones). Using thesec’s and
Eq.5.121¼5.81we calculate the initial-guessP’s for Eqs.5.106–5.108; since there
is only one occupied MO (n¼1 in Eq. 5. 121) the summation has only one term:

P 11 ¼ 2 c 11 c 11 ¼ 2 ð 0 : 249 Þ 0 : 249 ¼ 0 : 1240

P 12 ¼ 2 c 11 c 21 ¼ 2 ð 0 : 249 Þ 0 : 867 ¼ 0 : 4318

P 22 ¼ 2 c 21 c 21 ¼ 2 ð 0 : 867 Þ 0 : 867 ¼ 1 : 5034

ð 5 : 123 Þ

222 5 Ab initio Calculations