Eigenvalue equations Hamiltonian diagonalization

The eigenvalue equations of the two diagonal blocks of the effective Hamiltonian matrix is characterized by the equations... 

The use of atomic symmetry block-diagonalizes the matrix representation of the DHFB Hamiltonian of (95) giving the generalized matrix eigenvalue equations... 

The one-electron matrix Schrodinger equation itself is a quite practical approach and so we outline its derivation. We need two sets of one-electron matrix elements. The first set consists of those of the Hamiltonian Hjk = f droverlap matrix S Sjk = / dr( (r) fc(r). Often one assumes, not quite correctly, that the overlap matrix is diagonal and so is the identity matrix. The unknown coefficients in the expression for the wave function V i(j ) are the solution of the matrix eigenvalue equation... 

The Fock matrix is by design an effective one-electron Hamiltonian whose diagonalization yields a set of MOs, the canonical orbitals, from which a variationally r timized Hartree-Fock wave function may be constructed. However, since the Fock matrix (10.3.24) contains contributions from each occupied canonical orbital, we cannot solve the Hartree-Fock pseudo-eigenvalue equations (10.3.3) in a single step but must instead resort to some iterative scheme. 

The next step might be to perform a configuration interaction calculation, in order to get a more accurate representation of the excited states. We touched on this for dihydrogen in an earlier chapter. To do this, we take linear combinations of the 10 states given above, and solve a 10 x 10 matrix eigenvalue problem to find the expansion coefficients. The diagonal elements of the Hamiltonian matrix are given above (equation 8.7), and it turns out that there is a simplification. 

These equations show that if we ignore AH e) and diagonalize Hin the subspace 5 we are in effect diagonalizing the projected Hamiltonian P HP thus the complementary subspace 5 is treated implicitly through the energy-dependent operator AH s). A useful way of thinking about the relationship between Eqs. (2-2) and (2-3) follows from the observation that ifis completely determined by the requirement that it should reproduce the exact eigenvalues E ,... 

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