Canonical Hartree-Fock equations

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. 

When this is done, the resulting spin orbitals are termed the Hartree-Fock canonical orbitals. Read section 3.3 in Szabo and Ostlund for various fun things to do with the Hartree-Fock equations. 

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. 

In the canonical case, we are hence looking for solution to the Hartree-Fock equations of the form ... 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

The correlated methods discussed up to this point provide a delocalized description of the electronic system. The delocalized nature of these methods arises from their use of canonical orbitals (i.e., the eigenvectors of the Hartree-Fock equations) of Eq. (33). To treat large systems, it is better to express the theory in terms of orbitals that are localized in space, extending over only a few atoms. The virtual excitations then occur predominantly locally in the molecule (among localized occupied and virtual orbitals). As a result, the number of excitation amplitudes increases only linearly with system size. 

The IGLO method starts with the MO-LCAO-SCF calculation which produces the conventional (canonical) molecular orbitals [fa). These satisfy Hartree-Fock equations written in the form... 

The starting point is an ab initio SCF calculation of a saturated hydrocarbon C H. We recall that—because of equations 12 and 13—the underlying Hartree-Fock equations take the canonical form... 

Section 3.2 constitutes a derivation of the results of the previous section. The order of presentation of these two sections is such that the derivations of Section 3.2 can be skipped if necessary. For a fuller appreciation of Hartree-Fock theory, however, it is recommended that the derivations be followed. We first present the elements of functional variation and then use this technique to minimize the energy of a single Slater determinant. A unitary transformation of the spin orbitals then leads to the canonical Hartree-Fock equations. 

The unique set of spin orbitals obtained from a solution of this eigenvalue equation is called the set of canonical spin orbitals. We henceforth drop the primes and write the Hartree-Fock equations as... 

Solving the canonical Hartree-Fock equations for a particular molecule directly yields the canonical spinors of that molecule. How the equations can be solved is discussed from a technical point of view in the next section, while their explicit solution procedures are presented in chapters 9 and 10. The terms "Dirac-Hartree-Fock", "Dirac-Fock", and "four-component Hartree-Fock" are used synonymously in research papers. 

Reformulate the coupled Hartree-Fock equations (11.9.11) to admit non-canonical spin-orbitals, which do not diagonalize the zeroth-order Fock operator. What advantages might there be in using such orbitals [Hint With a large basis there may be a large number of very diffuse virtual orbitals.]... 

A further difficulty arises from the fact that neither the WF s nor the wave functions obtained from a localization inside a molecule are solutions of the canonical Hartree-Fock equations. Therefore, any method treating correlation (perturbation theory, coupled cluster expansion, etc.) has to be reformulated for this case. This work has been done in the case of perturbation theory for the localization inside a molecule /27/ and it is in progress for the coupled cluster method /28/. 

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

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