Overlap matrices

Solving the previous matrix equation tor the coefficients C describing the LCAO expan sion of th e orbitals and orbital energies n requires a matrix dia.i>onaliz(ition. If the overlap matrix were a... 

Because of the presence of the overlap matrix, however, yon m n st first diagonalize the overlap matrix ... 

In a /no computation where the overlap matrix is assumed to be til c un It m atrix, th e diagori al elemcri ts of the den sity matrix... 

If the functions Oj are orthonormal, then the overlap matrix S reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form ... 

After forming the overlap matrix, the new orthonormal funetions are defined as follows ... 

For first- and seeond-row atoms, the Is or (2s, 2p) or (3s,3p, 3d) valenee-state ionization energies (aj s), the number of valenee eleetrons ( Elee.) as well as the orbital exponents (es, ep and ej) of Slater-type orbitals used to ealeulate the overlap matrix elements Sp y eorresponding are given below. 

Finally, the matrix S from Equation 31 is the overlap matrix, indicating the overlap between orbitals. 

The integrals involving atomic orbitals are often collected together into a matrix called the overlap matrix S... 

This means that the overlap matrix is a unit matrix, in this case... 

Think of ethene, where we use tt basis functions and Xa- if identify these as ordinary atomic 2p orbitals, then we can calculate the overlap matrix using the methods described earlier. I will write it as... 

The only remaining question is the nature of the error function. Pulay suggested the difference FDS — SDF (S is the overlap matrix), which is related to the gradient of the SCF energy with respect to the MO coefficients. This has been found to work well in practice. 

The sum of the product of MO coefficients and the occupation numbers is the density matrix defined in Section 3.5 (eq. (3.51)). The sum over the product of the density and overlap matrix elements is the number of electrons. 

Finally, since the MOs are orthonormal, the derivatives of the coefficients may be replaced by derivatives of the overlap matrix. 

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). 

In order to proceed further, it is now necessary to study the linear dependence of the projected functions 0V 02, 03>. . . which is done by investigating their overlap matrix... 

---