Hartree-Fock problem

We describe in this Subsection the application of local-scaling transformations to the calculation of the energy for the lithium and beryllium atoms at the Hartree-Fock level [113]. (For other reformulations of the Hartree-Fock problem see [114] and referenres therein.) The procedure described here involves three parts. The first part is orbital transformation already discussed in Sect. 2.5. The second is intra-orbit optimization described in Sect. 4.3 and the third is inter-orbit optimization discussed in Sect. 4.6. 

The static Hartree-Fock problem assumes T-reversal invariance and T-even single-particle density matrix. In this case, Skyrme forces can be limited by only T-even densities psif) Tsif) In the case of dy-... 

The set of initial atomic functions %i is called the basis set. Although the complete solution of the Hartree-Fock problem requires an infinite basis set, good approximations can be achieved with a limited number of atomic orbitals. The minimum number of such functions corresponding approximately to the number of electrons involved in the molecule is the minimal" basis set. The coefficients amt which measure the importance of each atomic orbital in the respective molecular orbitals are parameters determined by a variational procedure, i.e. chosen so as to minimize the expression... 

Among the numerous approximations which could be used to simplify the Hartree-Fock problem, the all-valence electrons, N.D.D.O. method is particularly appropriate, due to the simplicity and adequacy of its approximations. These are ... 

The requirements of space invariance restrict further simplification of the Hartree-Fock problem to one of two distinct routes. 

With these notations the Hartree-Fock problem acquires the form of an eigen-value/eigenvector problem ... 

The projection operator formulation of the Hartree-Fock problem can be used for constructing a perturbation procedure for determining the electronic structure in terms of the latter.20 The simplest formulation departs from the Hartree-Fock equation for the projection operator to the occupied MOs eq. (1.152). Let us assume that the bare perturbation (see below) concerns only the one-electron part of the Fock operator so that ... 

The one-electron density matrix corresponding to the solution of the Hartree-Fock problem in the CLS is, like any Hartree-Fock density matrix, an operator (matrix) P... 

THE ROOTHAAN-HALL MATRIX FORMULATION OF THE HARTREE-FOCK PROBLEM... 

To simplify the Hartree-Fock problem, Pople introduced CNDO/1 (1965), then CNDO/2 (1967), and then INDO (1967) to yield computer programs that mimic ab initio programs with a minimum of fuss. Jaffe66 improved CNDO to fit spectroscopic absorptions (with a minimum of Cl) this was CNDO/S (1968). Later, Dewar67 introduced MINDO/3 (1975), then MNDO (1977), AMI (1985), and PM3 (1989). For transition metals, Zerner68 introduced ZINDO (1984) these were progressive improvements on INDO, but parameterized to fit thermochemical data, dipole moments, absorption spectra, and so on, to the fitful extent that they are available from experiment. 

Xi Solution of the Hartree Fock problem yields the single-determinant wave function, written in abbreviated form,... 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

Equation (4.67) represents actually the Hartree-Fock problem and we assume that its complete solution is known. Our goal is to find the solution of eqn, (4.66) under the assumption that 4 changes into if the perturbation W is switched on. RSPT gives us... 

Here is the exact total energy of the system, are solutions of the Hartree-Fock problem, and e is the sum of Hartree-Fock orbital energies over occupied spinorbitals. Then the eigenvalue in eqn. (4.66), E, becomes directly the correlation energy in the i th electronic state. Since our concern is focused on the ground state, i.e. i B 0, the index i in eqn, (4.70) may be dropped and the respective contributions to the correlation energy can be expressed as... 

The Hartree—Fock problem in its simplest form (Hartree, 1927 Fock, 1930) consists in finding the best orbitals a) so that the configuration p) approximates as closely as possible the lowest-energy eigenstate of H in the symmetry manifold /j. This is done using the variation theorem. 

Equn. (5.17) is a formal statement of the Hartree—Fock problem in the simplest, single determinant, form. We must find orbitals that satisfy the equation. 

The Hartree—Fock problem with the Dirac Hamiltonian (3.153) is called Dirac—Fock. The coordinate—spin representation of the orbital rj) is... 

The extension of the matrix solution of section 4.3 for one-electron bound states to the Hartree—Fock problem has many advantages. It results in radial orbitals specified as linear combinations of analytic functions, usually normalised Slater-type orbitals (4.38). This is a very convenient form for the computation of potential matrix elements in reaction theory. The method has been described by Roothaan (1960) for a closed-shell or single-open-shell structure. 

We illustrate the method by applying it to the simplest closed-shell form (5.29) of the Hartree—Fock problem. The radial orbitals u (r) are expressed as a linear combination of basis radial orbitals /i/(r). 

So that, at self-consistency, there is a set of unmixed shell Hartree-Fock problems to be solved whose physical interpretation is an obvious generalisation of the simpler cases. This point is taken up in detail in Chapter 23. 

Finite basis set expansions are ubiquitous in ab initio molecular elecuonic sUucture studies and are widely recognized as one of the major sources of error in contemporary calculations [1-5]. Since the pioneering work by Hartree and his co-workers in the 1930s, finite difference methods have been used in atomic Hartree-Fock calculations. It is only in the past fifteen years or so that finite difference techniques [6-10] (and more recently, finite element methods [11-14]) have been applied to the molecular Hartree-Fock problem. By exploiting spheroidal co-ordinates, two-dimensional Haitree-Fock calculations for diatomic molecules have become possible. These calculations have provided benchmarks which, in turn, have enabled the finite basis set approach to be refined to the point where matrix Hartree-Fock calculations for diatomic molecules can yield energies which approach the p-Hartree level of accuracy [15-18]. Furthermore, these basis sets can then be employed in calculations for polyatomic molecules [19,20] which are not, at present, amenable to finite difference or finite element techniques. 

In (6.106) we gave a single-determinantal wave fimction for a system of n electrons if there are N functions, we can define 2N so-called spin-orbital basis functions of the type ifix(a) and V (j6) which may be linearly combined into 2N spin-orbitals Xi. Solution of tile Hartree Fock problem yields the single-determinant wave function, written in abbreviated form,... 

The atomic Hartree-Fock problem is treated in considerable detml in J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Co., Vol. 1 and Vol. 2, 1960). 

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