- Orthogonalizing matrix
As explained above, we (a) diagonalizeS, (b) calculateD"1/2, then (c) calculate
the orthogonalizing matrixS"1/2:
(a) DiagonalizeS
S¼
10 : 435
0 :435 1
¼
0 :707 0: 707
0 : 707 " 0 : 707
1 :435 0
00 : 565
0 :707 0: 707
0 : 707 " 0 : 707
PDP"^1
(4.112)
(b) CalculateD"1/2D"^1 =^2 ¼^1 :^435
" 1 = (^20)
00 : 565 "^1 =^2
¼
0 :835 0
01 : 330
(4.113)
(c) Calculate the orthogonalizing matrixS"1/2S"^1 =^2 ¼
0 :707 0: 707
0 : 707 " 0 : 707
0 :835 0
01 : 330
0 :707 0: 707
0 : 707 " 0 : 707
¼
1 : 083 " 0 : 248
" 0 :248 1: 083
PD"^1 =^2 P"^1
(4.114)
- Transformation of the original Fock matrixHtoH^0
Using Eq.4.102:
H^0 ¼
1 : 083 " 0 : 248
" 0 :248 1: 083
" 13 : 6 " 14 : 5
" 14 : 5 " 24 : 6
1 : 083 " 0 : 248
" 0 :248 1: 083
¼
" 9 : 67 " 7 : 65
" 7 : 68 " 21 : 74
S"^1 =^2 HS"^1 =^2
(4.115)
- Diagonalization ofH^0
From Eq.4.104(H^0 ¼C^0 «C^0 "^1 ), diagonalization ofH^0 gives an eigenvector
matrixC^0 and the eigenvalue matrix«; the columns ofC^0 are the coefficients of
the transformed, orthonormal basis functions:
H^0 ¼
" 9 : 67 " 7 : 65
" 7 : 68 " 21 : 74
¼
0 :436 0: 899
0 : 900 " 0 : 437
" 25 : 50
0 " 5 : 95
0 :436 0: 900
0 : 899 " 0 : 437
C^0 e C^0 "^1
(4.116)162 4 Introduction to Quantum Mechanics in Computational Chemistry