Fock matrix perturbed

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. 

TABLE 10. Vicinal oHX —Fock matrix by second-order perturbation theory (energies in kcal mol-1)... 

NBO analysis can be used to quantify this phenomenon. Since tire NBOs do not diagonalize the Fock operator (or tire Kohn-Sham operator, if the analysis is carried out for DFT instead of HF), when the Fock matrix is formed in the NBO basis, off-diagonal elements will in general be non-zero. Second-order perturbation tlieory indicates that these off-diagonal elements between filled and empty NBOs can be interpreted as the stabilization energies... 

Mpller-Plesset perturbation theory (MPPT) uses the orbitals and orbital energies obtained from a closed-shell Hartree-Fock-Roothaan (HFR) calculation. The HFR (or canonical) orbitals correspond to the eigenvectors of the inactive Fock matrix... 

In the first expression the integrals are in the covariant AO representation (in which they are calculated), and the one-index transformed density elements are in the contravariant representation (obtained from the MO basis in usual one- and two-electron transformations). The second expression is useful whenever the transformation matrix is calculated directly in the covariant AO representation and requires the transformation of the Fock matrix to the contravariant representation. The last expression is convenient when the number of perturbations is large, since it avoids the transformation of the covariant AO Fock matrix to the MO or contravariant AO representations. 

This initial guess may then be inserted on the right-hand sides of the equations and subsequently used to obtain new amplitudes. The process is continued until self-consistency is reached. For the special case in which canonical Hartree-Fock molecular orbitals are used, the Fock matrix is diagonal and the T2 amplitude approximation above is exactly the same as the first-order perturbed wave-function parameters derived from Moller-Plesset theory (cf. Eq. [212]). In that case, the Df and arrays contain the usual molecular orbital energies, and the initial guess for the T1 amplitudes vanishes. 

If the density matrix in Eq. (64) is obtained by performing the self-consistent field (SCF) calculation using the appropriately perturbed one-electron Fock matrix elements, the electrostatic and polarization contributions are contained in the first two terms of Eq. (68). 

As noted above, adoption of the CIS method necessitates use of the FIXDM keyword and rules out -dependent NBO options such as second-order perturbative analysis or Fock matrix deletions. However, numerous NBO descriptors remain useful for analyzing the CIS wave function, including NBO Lewis structure, orbital shapes and occupancies, and NRT weightings and bond orders. In the following, we illustrate the use of these descriptors for the three lowest CIS excited states. Our emphasis is not on the absolute accuracy of the CIS wave functions themselves but on how the NBO/NRT descriptors provide a chemist s picture of the wave functions, useful for predicting structural rearrangements... 

Rydberg orbitals. The contribution to the total density matrix from unoccupied orbitals is generally small however, antibonds play an important role since they represent unused valence-shell capacity. The energy of a molecule can be decomposed into two components associated with covalent and noncovalent structures the former corresponds to the ideal Lewis contributions, and the latter is associated with non-Lewis contributions. The nondiagonal elements of the Fock matrix in the NBO basis are interpreted as the stabilizing interaction between occupied orbitals of the formal Lewis structure and unoccupied ones. Since the corrections to the energy of the Lewis-type picture are usually small, they can be approximated by second-order perturbation theory, Eq. (78),... 

Once again, the derivative of the density matrix P can be obtained as solution of first-order coupled-perturbed Hartree-Fock equation with derivar tive Fock matrix given by eq. (1.70), exactly as for the nuclear shielding. 

The perturbation due to a static homogeneous electric field, E, is H = -eE r. For the Fock matrix in AO basis, the perturbation term is added to the one-electron part of the unperturbed Fock matrix. 

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