Fock matrix block-diagonality

The presence of /,/ and components requires an iterative solution of this equation—an approach that necessitates storage of the T3 amplitudes in each iteration This scheme is unreasonable because the number of such amplitudes would rapidly become the computational bottleneck as the size of the molecular system increased. This problem may be circumvented, however, by utilizing the so-called semicanonical molecular orbital basis in which the occupied-occupied and virtual-virtual blocks of the Fock matrix are diagonal. In this basis, the two final terms in the T3 equation above vanish, and the conventional noniterative computational procedure described earlier in the chapter may be employed. 

In calculating the canonical Hartree-Fock orbitals, we carry out a full diagonalization of the Fock matrix. Complete diagonalization is not necessary, however, to ensure that the elements of the electronic gradient vanish. Indeed, any set of orbitals that brings the Fock matrix into block-diagonal form with vanishing occupied-virtual elements... 

For k = 0, the Fock matrix and its derivatives with respect to the displacements of the nuclei are always block diagonal. Then one can directly apply the analytical derivative methods developed for finite systems to extended systems [69,86,87,88]. But when the displacements break the translational symmetry, the Fock matrix and its derivatives are no longer block diagonal. To solve the CPHF equations, one needs to use the symmetrized (normal mode) coordinates instead of the Cartesian coordinates of the nuclei. Efficient analytical methods have been developed to calculate the energy derivatives for k / 0 with both plane wave [89-90] and general basis functions [85]. The latter can be functions of nuclear coordinates and have linear dependence. These methods reduce the computational cost required to calculate the phonon spectrum with k 7 0 to the same as that needed for the spectrum at k = 0. 

Similarly, a specific choice of an adequate effective Fock matrix is the closed shell Fock matrix with a correction such that the off-diagonal blocks associated with the open-shell orbitals are adjusted to be proportional to the orbital gradient... 

Using the block-diagonal nature of the density matrix, besides reducing the requirement in CPU time, also allows one to label the DMEV by the appropriate particle number(ne) and the z— component of the total spin of the block [Mg,a)- The Fock space of the individual sites that are added at each iteration are eigenstates of the site spin and number operators. This allows targeting a definite particle number Nf.) and a definite projected spin (M ) state of the total system. 

As a second step, symmetry type 2 can be applied to the set of the s k points, which allows one to further reduce the Fock matrix into a block-diagonal form. By transforming the basis set into an equivalent set of symmetry adapted basis functions, every block of the transformed matrix in Figure 4, which corresponds to one particular point ky, reveals, in turn, a block-diagonal structure, for example, of the kind depicted in Figure 20. 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

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The unitary transformation required for the block diagonalization of the relativistic Fock operator can be obtained in one step if a matrix representation of the Fock operator is available this is achieved by the so-called eXact-2-Component (X2C) approach [725-728,731-734]. An important characteristic of the X2C approach is its noniterative construction of the key operator X of Eq. (11.2). In this noniterative construction scheme, the matrix operator X is obtained from the electronic eigenvectors of the relativistic (modified) Roothaan Eq. (14.13),... 

Bagus et al. have chosen to treat the overlap in a different way. They chose a specific order in which they (Schmidt) orthogonalize the orbitals on all the previous ones, e.g., (occ A) -h (occ. B) - - (virt. B) -h (virt. A). In the Fock matrix in this basis, in which the overlap S matrix is diagonal, blocks are successively allowed to be non-zero. After the step based on (occ. A) H- (occ. B) only, the set (virt. B) is allowed to mix... 

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