Eigenfunction Hartree-Fock

Hartree-Fock wavefunction, is an eigenfunction of and the corresponding oth-order energy Eg° is equal to the sum or orbital energies for the occupied molecular... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

The Hartree-Fock determinant and all of the substituted determinants are eigenfunctions of Hg these are the solutions to the part of the divided Hamiltonian for which we have a solution. Thus ... 

The most famous case concerns the symmetry breaking in the Hartree-Fock approximation. The phenomenon appeared on elementary problems, such as H2, when the so-called unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism, using different orbitals for a and p electrons, was first proposed by G. Berthier [5] in 1954 (and immediately after J.A. Pople [6] ) for problems where the number of a andp electrons were different. This formulation takes the freedom to deviate from the constraints of being an eigenfunction. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

In the cases other than [case A] and [case B), so called "open-shell SCF methods are employed. The orbital concept becomes not quite certain. The methods are divided into classes which are "restricted 18> and "unrestricted 19> Hartree-Fock procedures. In the latter case the wave function obtained is no longer a spin eigenfunction. 

In a molecular-orbital-type (Hartree-Fock or Kohn-Sham density-functional) treatment of a three-dimensional atomic system, the field-free eigenfunctions ir e can be rigorously separated into radial (r) and angular (9) components, governed by respective quantum numbers n and l. In accordance with Sturm-Liouville theory, each increase of n (for... 

Here a is an adjustable parameter, usually determined by comparing Hartree-Fock and Xa atomic calculations. In the spin-unrestricted version, the spin-up and spin-down orbitals are distinct, so that in general the resulting wavefunction is not a spin eigenfunction. 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the F >v matrix elements depend on the Cv,i LCAO-MO coefficients which are, in turn, solutions of the so-called Roothaan matrix Hartree-Fock equations- Zv F >v Cv,i = Zv S v Cvj. One should also note that, just as F (f>j = j (f>j possesses a complete set of eigenfunctions, the matrix Fp,v, whose dimension M is equal to the number of atomic basis orbitals used in the LCAO-MO expansion, has M eigenvalues j and M eigenvectors whose elements are the Cv>i- Thus, there are occupied and virtual molecular orbitals (mos) each of which is described in the LCAO-MO form with CV)i coefficients obtained via solution of... 

We now consider the theoretical calculation of excited-state wave functions. This is more difficult than ground-state calculations because we are dealing with open-shell configurations. The Hartree-Fock equations for a state of an open-shell configuration have a more complicated form than for closed shells, and there exist close to a dozen different approaches to excited-state Hartree-Fock calculations. As noted earlier, the Hartree-Fock wave function for a closed-shell state is a single determinant, but for open-shell states, we may have to take a linear combination of a few Slater determinants to get a Hartree-Fock function that is an eigenfunction of S and Sz and has the correct spatial symmetry. 

The spin orbitals can be separated into a spatial part, if/ (orbital or molecular orbital) and a spin eigenfunction, a (or for the opposite spin). In restricted Hartee-Fock theory (RHF), the spatial part is independent on the spin state, in contrast to unrestricted Hartree-Fock (UHF) where it is spin dependent. Consequently, the RHF spin orbitals can be written as %i= whereas in UHF the corresponding relation is Xi= even number of electrons (closed shell system), but can easily be extended to treat also UHF1. 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

In a molecule, the one-electron eigenfunctions (Mulliken and Hund s molecular orbitals, M. 0.) are determined by a core field U(x,y,z) and have well-defined symmetry type yn. However, the actual calculation of such M. 0. is very difficult in polyatomic molecules, whereas the Hartree-Fock method can be applied to monatomic entities when large electronic computers are available (13,14). I have written a book about several of the principal problems regarding the concept of one-electron functions... 

Hartree-Fock M.O. previous models, eigenfunction of the with Watson s correlation effects1 -25 already core-field (Ux,y,z). found in gaseous monatomic ions. ... 

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