Fock matrix canonical orthonormalization

Figure 4.9 Application of the canonical orthonormalization procedure of Section 3.6 to the calculation of the 1 s and 2s eigenfunctions and eigenvalues approximations for the Is and 2s orbitals in hydrogen over Slater functions. Note the exact fit of the Is Slater, which is an eigenfunction of the Fock matrix for the hydrogen atom and the relatively close agreement of the ls/2s linear combinations based on simple canonical orthogonalization and also direct orthonormalization using the matrix procedure of Section 3.7.

Retransform the coefficients for the orthonormal linear combinations of the canonically orthogonalized Slater linear combinations, which diagonalize the Fock matrix, into linear coefficients over the original Slater functions, with... 

Secondly, the canonical orthonormalization procedure to diagonalize the overlap matrix and then the application of the Jacobi transformation to diagonalize the Fock matrix in the eigenfunctions of the overlap matrix, returns two eigenvalues, the values —0.50000 and —0.12352 Hartrees, in canonical B 18 and B 19. This is the important elementary point that we can make two linear combinations of two functions and so there are two possible eigenvalues to be calculated. These eigenvalues, of course, are present in the calculation set out in the other worksheet, based on the Schmidt procedure. The Is... 

Figure 6.12a Canonical orthonormalization of the Fock matrix for the HeH+ calculation using the sto-lg ls) basis proposed by Whitten (48). Without polarization, C = C2 = 0.0 [cells G 23 and H 23] the electronic energy is found to be —3.87700473 Hartree, while the total energy is —2.51013759 Hartree. A slightly better result is returned using Stewart s sto-lg) basis (32).

This is the easiest initial guess to obtain, but may be a poor one in more complicated situations. The next step is to transform the Fock matrix to the canonically orthonormalized basis set. 

The Hartree-Fock method can be applied to truly large molecules containing several hundred atoms. For such systems, it becomes impossible to construct a set of orthonormal orbitals (5.1.5), much less a set of canonical orbitals. However, as we shall see in Chapter 10, all information about the Hartree-Fock wave function is contained in the one-electron density matrix, which may be expressed directly in the basis of AOs. For large molecules, the density-matrix elements can be optimized by an algorithm whose complexity scales linearly with the size of the system. 

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