Fock matrix corrections

Without the minus sign in the definition the name is not quite adequate. 

The expression for C%(N) has a clear physical interpretation. The first term represents the interaction of the charge distribution (of electron 1, 

In the cfpq(N) correction, in the summation over I, we have neglected the exchange term -jYliZh-N rs srHpr %)- The reason for this was that we have been convinced, that P] vanishes very fast, when cell / separates from cell h. Subsequent reasoning would then be ea the most important term (/ = h) would be Jlrs )- it Contains the differential overlap a (1) a (1), which decays 

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Fock Matrix Corrections Total Energy Corrections Multipole Expansion Applied to the Fock Matrix Multipole Expansion Applied to the Total Energy... 

The parentheses [] mean the corresponding Cartesian multipole mcxnent. When computing the Fock matrix correction, the first multipole moment [] stands for the multipole mcxnent rd" the charge distribution XpX - tt secraid. of the unit cell. For example. for the correctira Cp (fV) is equal to... 

The preceding discussion means that the Matrix equations already described are correct, except that the Fock matrix, F, replaces the effective one-electron Hamiltonian matrix, and that Fdepends on the solution C ... 

In step 3, a criterion of convergence may be introduced to terminate the iterations. Two other points should be mentioned instead of taking the correct Singles-CI matrix, we may resort to a simpler one, omitting single bi-electronic integrals and using only Fock-matrix elements as ... 

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

This is a common form, but by no means the only one. Often, the corrected x is instead given by some nonlinear procedure. As an example, an SCF procedure can be implemented by applying corrections to the Fock matrix, which is diagonalized to provide new occupied orbitals. The variable x is best represented (in SCF) by a density matrix, which then depends in a highly non-linear way upon the corrections to the Fock matrix. 

An equation for the llO) elements can be obtained from the condition that the Fock matrix is diagonal, and expanding all involved quantities to first-order. and solving the CPHF equations is usually not the bottleneck in these cases. Without the Lagrange technique for non-variational wave functions (Cl, MP and CC), the nth-order CPHF is needed for the nth-derivative. Consider for example the MP2 energy correction, eq. (4.45). ... 

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... 

In evaluating the first-order correction to two-electron integrals (entering the first-order Fock matrix) we can again use the expansion of the gauge factor... 

From a computational point of view, - H could be evaluated analytically ab initio) and then be added to the semiempirical core Hamiltonian matrix. This procedure, however, introduces an imbalance between the one- and two-electron parts of the Fock matrix as long as the two-electron integrals are not subjected to the same exact transformation (J)), which would sacrifice the computational efficiency of semiempirical methods and is therefore not feasible. Hence the orthogonalization corrections to the one-electron integrals must instead be represented by suitable parametric functions. Their essential features can be recognized from the analytic expressions for the matrix elements of in the simple case of a homonuclear diatomic molecule with two orbitals at atom A, at atom B) ... 

The effective Hartree-Fock matrix equation for a many-shell system has been derived in Chapter 14 and used in several applications open shells and some MCSCF models. So far, it has been seen simply as the formally correct equation to generate SCF orbitals for these many-sheU structures without any interpretation. In particular, the fact that the effective Hartree-Fock matrix (the McWeenyan ) contains many arbitrary parameters has not been addressed, nor has the practical problem of the actual grounds for the choice of values for these parameters been systematised. In looking at this problem we must bear two points in mind ... 

We can see, therefore, that in a certain sense it is good luck that the UHF or closed-shell SCF process can be solved uniquely without the involvement of the additional off-diagonal terms in the Fock matrix. If there is only one occupied shell then the empty shell is uniquely determined. Expressing the stationary condition in terms of the occupied-empty projection of the Fock matrix is formally correct but not useful. 

Again writing the expansion of the first-order correction in terms of the eigenvectors of the unperturbed Hartree-Fock matrix ... 

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),... 

With the goal of minimizing CPU time that is required to assemble the Fock matrix, VanAlsenoy proposed the multiplicative integral approxima-tion" o in which products of contracted Gaussians are approximated by linear combinations of 13 auxiliary functions. The resulting three-index integrals are used in fast construction of the Fock matrix, which is later corrected with the ERIs for which the error of approximation exceeds some predetermined threshold. 

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