Fock eigenfunctions

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 two sets of coefficients result in two sets of Fock matrices (and their associated density matrices), and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper dissociation to separate atoms, correct delocalized orbitals for resonant systems, and other attributes characteristic of open shell systems. However, the eigenfunctions are not pure spin states, but contain some amount of spin contamination from higher states (for example, doublets are contaminated to some degree by functions corresponding to quartets and higher states). 

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 ... 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

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 ... 

An important consequence of the only approximate treatment of the electron-electron repulsion is that the true wave function of a many electron system is never a single Slater determinant We may ask now if SD is not the exact wave function of N interacting electrons, is there any other (necessarily artificial model) system of which it is the correct wave function The answer is Yes it can easily be shown that a Slater determinant is indeed an eigenfunction of a Hamilton operator defined as the sum of the Fock operators of equation (1-25)... 

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. 

How do CMOs and LMOs differ The CMOs are symmetry-adapted eigenfunctions of the Fock (or Kohn-Sham) operator F, necessarily reflecting all the molecular point-group symmetries of F itself,26 whereas the LMOs often lack... 

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... 

Since the J and K operators depend on the occupied orbitals, the pseudoeigenvalue problem must be solved iteratively until consistency is achieved between orbitals that determine J and K and those that emerge as eigenfunctions of Heff, which in this approximation is known as the Fock operator F. 

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. 

The Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

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... 

Recall that the minimum requirement for a many-electron wave function is that it be written as a suitably antisymmetrized sum of products of one-electron wave functions, that is, as a Slater determinant of MOs [see equation (A.68)] In Chapter 2 and Appendix A, we find that the condition that this be the best possible wave function of this form is that the MOs be eigenfunctions of a one-electron operator, the Fock operator [recall equation (A.42)], from which one can choose the appropriate number of the lowest energy. The Fock operator in restricted form, F( 1) [RHF, the UHF form was given in equation (A.41)], is given by... 

Therefore, the condition that the orbitals yield a stationary point (hopefully a minimum) on the energy hypersurface with respect to variations is that the orbitals are eigenfunctions of the Fock operator, with associated orbital energy, e,... 

In summary, to obtain a many-electron wave function of the single determinantal form [equation (A.12)] which will give the lowest electronic energy [equation (A.14) or (A.27)], one must use one-electron wave functions (orbitals) which are eigenfunctions of the one-electron Fock operator according to equation (A.42). There are many, possibly an infinite number of, solutions to equation (A.42). We need the lowest Ne of them, one for each electron, for equation (A. 12) [or (A.27)]. When the Ne MOs of lowest energy satisfy equation (A.42), then Eq=Ehf [equation (A.27)] and o= hf [equation (A.12)]. 

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