Canonical Hartree-Fock

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 ... 

These conditions determine a unique set of molecular orbitals, the canonical molecular orbitals, (CMO s), . Inserting the conditions (25) in the SCF Eqs. (17), one sees that the CMO s are solutions of the canonical Hartree-Fock equations 10)... 

At this point it should be noted that, in addition to the discussed previously, the canonical Hartree-Fock equations (26) have additional solutions with higher eigenvalues e . These are called virtual orbitals, because they are unoccupied in the 2iV-electron ground state SCF wavefunction 0. They are orthogonal to the iV-dimensional orbital space associated with this wavefunction. 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant,... 

For many ionization energies and electron affinities, diagonal selfenergy approximations are inappropriate. Methods with nondiagonal self-energies allow Dyson orbitals to be written as linear combinations of reference-state orbitals. In most of these approximations, combinations of canonical, Hartree-Fock orbitals are used for this purpose, i.e. 

Contrarily to conventional MP2 theory, the original formulation of MP2-R12 theory (3,4) did not provide the same results when canonical or localized molecular orbitals were used. Indeed, for calculations on extended molecular systems, unphysical results were obtained when the canonical Hartree-Fock orbitals were rather delocalized (5). In order to circumvent this problem, an orbital-invariant MP2-R12 formulation was introduced in 1991, which is the preferred method since then (6),... 

As an example of the application of matrix methods to MO theory, consider the transformation between the set of delocalized canonical Hartree-Fock MOs energy-localized MOs linear combinations of the canonical MOs ... 

Neglecting off-diagonal elements of the self-energy matrix in the canonical Hartree-Fock basis in (1.15) constitutes the quasiparticle approximation. With this approximation, the calculation of EADEs is simplified, for each KT result may be improved with many-body corrections that reside in a diagonal element of the self-energy matrix. 

Instead of the standard Hartree-Fock reference calculation, a grand-canonical Hartree-Fock calculation [35] is used with the occupation number of a single spin-orbital (i.e., the transition spin-orbital) set to 0.5. Upon convergence, appreciable corrections to the relaxation energy are included in the transition spin-orbital s energy [23, 24], Usually a very close agreement with the ASCF method [36] is obtained [26], The second order electron propagator is applied to the ensemble... 

The theory is usually expressed in terms of canonical Hartree-Fock equations... 

Equivalently, time-dependent canonical Hartree-Fock equations are assumed to take the same form as the time-dependent Schrodinger equation. 

Chemically speaking there is little to say. Canonical Hartree-Fock molecular orbitals leave no place for classical chemical concepts such as bonds between atoms or groups, lone pairs, resonance hybrids, etc. However, chemists still utilize these concepts because they are extremely useful in correlating and understanding chemical facts. Even when one manages to localize the canonical molecular orbitals (which is not always straightforward) in regions such that they could be associated with lone pairs or individual chemical bonds, it is important to bear in mind that the orbitals represent localized one-electron states, and not a two-electron chemical bond between atoms or a lone pair of electrons, as will be discussed further. 

Let us first evaluate the canonical Hartree-Fock functions = Vj, (r2,. .. a tn which are solutions to the equations... 

If canonical Hartree-Fock orbitals are chosen, the first term is zero by Bril-louin s theorem. 

This partitioning, when applied in conjunction with the set of canonical Hartree-Fock orbitals (in which is diagonal), corresponds to the Moller-Plesset variant of many-body perturbation theory. A Hartree-Fock determinant, which is an eigenfunction of Pjq, is therefore the natural choice for the zeroth-order wavefunctiond... 

Again assuming canonical Hartree-Fock orbitals, the terms containing Fock matrix elements are reduced to include the diagonal elements only ... 

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. 

Quantum chemistry, however, seems to contradict to the above well known facts, as both the AT-electron wavefunction V(X],. ..,xx) and the canonical Hartree-Fock one-particle orbitals [Pg.45]

Canonical Hartree-Fock orbitals have been the usual choice among quantum chemists. Effects of electron correlation and orbital relaxation in final states are described by the self-energy operator, X(E). The latter operator can be written as a sum of energy-independent and energy-dependent parts according to... 

With the 6-311+ + G basis, the total number of MOs is 568. There are 84 occupied MOs. Table 2 presents the VDEs of this anion calculated with a variable number of virtual orbitals retained according to the QVOS scheme [30]. The difference between the current QVOS code and the one described previously [30] is that in the latter all MOs had to be included, whereas in the current variant any orbital window can be chosen at the second-order step. Columns 1 and 2 of Table 2 list VDEs calculated either (1) with no reduction of the virtual space or (2) by simply omitting core and high-energy virtual MOs. In the latter case, the original set of canonical, Hartree-Fock MOs is employed. 

If the Hartree-Fock determinant dominates the wavefunction, some of the occupation numbers will be close to 2. The corresponding MOs are closely related to the canonical Hartree-Fock orbitals. The remaining natural orbitals have small occupation numbers. They can be analysed in terms of different types of correlation effects in the molecule . A relation between the first-order density matrix and correlation effects is not immediately justified, however. Correlation effects are determined from the properties of the second-order reduced density matrix. The most important terms in the second-order matrix can, however, be approximately defined from the occupation numbers of the natural orbitals. Electron correlation can be qualitatively understood using an independent electron-pair model . In such a model the correlation effects are treated for one pair of electrons at a time, and the problem is reduced to a set of two-electron systems. As has been shown by Lowdin and Shull the two-electron wavefunction is determined from the occupation numbers of the natural orbitals. Also the second-order density matrix can then be specified by means of the natural orbitals and their occupation numbers. Consider as an example the following simple two-configurational wavefunction for a two-electron system ... 

Figure 7. Canonical Hartree-Fock valence orbitals of NO constructed in a minimal cartesian gausslan basis set (28). Values shown refer to a 6 by 16 plane, with N on the left 0 on the right.

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