Fock matrix diagonalization

W. PoUard and R. Eriesner (1993) Efficient Fock matrix diagonalization by a Krylov-space method. J. Chem. Phys. 99, p. 6742... 

Figure 4 Typical timing behavior of the quadratic Fock matrix formation versus the cubically scaling diagonalization step (small prefactor) in SCF energy calculations. The timings for a conventional Fock matrix formation, the linear-scaling CFMM/LinK schemes (as explained later in this review), and Fock matrix diagonalization for a series of DNA molecules (A-T), = 1 — 16 are depicted. Integral threshold is 10 , basis set 6-31G. ...

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

In ub initio calculations all elements of the Fock matrix are calculated using Equation (2.226), ii re peifive of whether the basis functions ip, cp, formally bonded. To discuss the semi-empirical melh ids it is useful to consider the Fock matrix elements in three groups (the diagonal... 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

Using the same nomenclature as for the INDO approximation, the elements of the MINDO/3 UHF Fock matrix are described below. When (]) j and (jty are on different centers the off-diagonal elements... 

The developers of ZINDO found that the parameters required to reproduce orbital energy orderings and UV spectra are different from those required to reproduce accurate structures by geometry optimization. They introduced anew pair of parameters, called the overlap weighting factors, to account for this. These parameters are provided in HyperChem in the Semi-empirical Options dialog box. Their effect is to modify the resonance integrals for the off-diagonal elements of the Fock matrix. 

Diagonalize the Fock matrix (see Chapter 13 for details). The eigenvectors contain the new MO coefficients. 

The gradient of the energy is an off-diagonal element of the molecular Fock matrix, which is easily calculated from the atomic Fock matrix. The second derivative, however, involves two-electron integrals which require an AO to MO transformation (see Section 4.2.1), and is therefore computationally expensive. 

The Huckel methods perform the parameterization on the Fock matrix elements (eqs. (3.50) and (3.51)), and not at the integral level as do NDDO/INDO/CNDO. This means that Huckel methods are non-iterative, they only require a single diagonalization of the Fock (Huckel) matrix. The Extended Huckel Theory (EHT) or Method (EHM), developed primarily by Hoffmann again only considers the valence electrons. It makes use of Koopmans theorem (eq. (3.46)) and assigns the diagonal elements in the F... 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

Just as the variational condition for an HF wave function can be formulated either as a matrix equation or in terms of orbital rotations (Sections 3.5 and 3.6), the CPFIF may also be viewed as a rotation of the molecular orbitals. In the absence of a perturbation the molecular orbitals make the energy stationary, i.e. the derivatives of the energy with respect to a change in the MOs are zero. This is equivalent to the statement that the off-diagonal elements of the Fock matrix between the occupied and virtual MOs are zero. 

An equation for the elements can be obtained from the condition that the Fock matrix is diagonal, and expanding all involved quantities to first-order. 

Without detailing the derivation, the diagonal Fock matrix elements take the following form, in the CNDO model, for d> on atom A and [Pg.21]

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