H^0 ¼C^0 eC(^0) " 1
(4.104)
In other words, usingS"1/2we transform the original Fock matrixH, which is
not directly diagonalizable to eigenvector and eigenvalue matricesCand«, into a
related matrixH^0 whichisdiagonalizable to eigenvector and eigenvalue matricesC^0
and«. The matrixC^0 is then transformed to the desiredCby multiplying byS"1/2
(Eq.4.100). So without using the drasticS¼ 1 approximation we can use matrix
diagonalization to get the coefficients and energy levels from the Fock matrix.
The orthogonalizing matrixS"1/2is calculated fromS: the integralsSare
calculated and assembled intoS, which is then diagonalized:
S¼PDP"^1 (4.105)
Now it can be shown that any function of a matrixAcan be obtained by taking
the same function of its corresponding diagonal alter ego and pre- and postmulti-
plying by the diagonalizing matrixPand its inverseP"^1 :
fðAÞ¼PfðDÞP"^1 (4.106)and diagonal matrices have the nice property thatf(D) is the diagonal matrix whose
diagonal elementi,,j¼f(elementi,jofD). So the inverse square root ofDis the
matrix whose elements are the inverse square roots of the corresponding elements
ofD. Therefore
S"^1 =^2 ¼PD"^1 =^2 P"^1 (4.107)
and to findD"1/2we (or rather the computer) simply take the inverse square root of
the diagonal (i.e. the nonzero) elements ofD. To summarize:Sis diagonalized to
giveP,P"^1 andD, Dis used to calculateD"1/2, then the orthogonalizing matrix
S"1/2is calculated (Eq.4.107) fromP,D"1/2andP"^1. The orthogonalizing matrix
is then used to convertHtoH^0 (Eq.4.102), which can be diagonalized to give the
eigenvalues and the eigenvectors (Section 4.4.2).
4.4.1.3 Review of the EHM Procedure
The EHM procedure for calculating eigenvectors and eigenvalues, i.e. coefficients
(or in effect molecular orbitals – thec’s along with the basis functions comprise the
MOs) and energy levels, bears several important resemblances to that used in more
advanced methods (Chapters 5 and 6 ) and so is worth reviewing.
- An input structure (a molecular geometry) must be specified and submitted to
calculation. The geometry can be specified in Cartesian coordinates (probably
the usual way nowadays) or as bond lengths, angles and dihedrals (internal
158 4 Introduction to Quantum Mechanics in Computational Chemistry