The Hartree-Fock Procedure

An outline of the Hartree-Fock procedtire. The Fock matrix F is a sum of a one-electron part, and a density-dependent two-electron part, G(D), where D is the electronic 

The major components of the Hartree-Fock procedure, listed roughly in order of decreasing computational requirements, are  

In a parallel Hartree-Fock program, these steps can be performed with either replicated or distributed data. When distributed data is used, the matrices are blocked in such a way that all the elements corresponding to the basis functions within a given pair of shells are always found on the same processor. This grouping of the elements by shells is used because integral libraries, for computational efficiency, compute full shell blocks of integrals including all the functions within the involved shells. For the parallel computation of 

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The full explanation of why the 4s 3d configuration is adopted in scandium, even though the 3d level has a lower energy, emerges from the peculiarities of the way in which orbital energies are defined in the Hartree-Fock procedure. The details are tedious but have been worked out and I refer anyone who is interested in pursuing this aspect to the literature (Melrose, Scerri, 1996).6,7... 

In a recent paper Ostrovsky has criticized my claiming that electrons cannot strictly have quantum numbers assigned to them in a many-electron system (Ostrovsky, 2001). His point is that the Hartree-Fock procedure assigns all the quantum numbers to all the electrons because of the permutation procedure. However this procedure still fails to overcome the basic fact that quantum numbers for individual electrons such as t in a many-electron system fail to commute with the Hamiltonian of the system. As aresult the assignment is approximate. In reality only the atom as a whole can be said to have associated quantum numbers, whereas individual electrons cannot. 

In order to find a good approximate wave function, one uses the Hartree-Fock procedure. Indeed, the main reason the Schrodinger equation is not solvable analytically is the presence of interelectronic repulsion of the form e2/r. — r.. In the absence of this term, the equation for an atom with n electrons could be separated into n hydrogen-like equations. The Hartree-Fock method, also called the Self-Consistent-Field method, regards all electrons except one (called, for instance, electron 1), as forming a cloud of electric charge... 

The most obvious defect of the Thomas-Fermi model is the neglect of interaction between electrons, but even in the most advanced modern methods this interaction still presents the most difficult problem. The most useful practical procedure to calculate the electronic structure of complex atoms is by means of the Hartree-Fock procedure, which is not by solution of the atomic wave equation, but by iterative numerical procedures, based on the hydrogen model. In this method the exact Hamiltonian is replaced by... 

If Slater determinants obtained from the Hartree-Fock procedure are used in equations (4) and (5), we obtain the uncoupled Hartree-Fock (UCHF) scheme because the field effects upon the electron-electron interactions are not taken into account [14-15]. To go beyond this crude approximation, the wavefimctions are built as linear combina-... 

The set of molecular orbitals leading to the lowest energy are obtained by a process referred to as a self-consistent-field or SCF procedure. The archetypal SCF procedure is the Hartree-Fock procedure, but SCF methods also include density functional procedures. All SCF procedures lead to equations of the form. 

Unfortunately, the Slater-type orbitals become increasingly less reliable for the heavier elements, including to some extent the first transition series these limitations are described in a recent review by Craig and Nyholm (5 ). The most accurate wave functions to use in these calculations would be the SCF functions obtained by the Hartree-Fock procedure outlined above, but this method leads to purely numerical radial functions. However, Craig and Nyholm (5S) have drawn attention to relatively good fits obtained by Richardson (59) to SCF 3d functions by means of two-parameter orbitals of the type... 

The Cl procedure just described uses a fixed set of orbitals in the functions An alternative approach is to vary the forms of the MOs in each determinantal function O, in (1.300), in addition to varying the coefficients c,. One uses an iterative process (which resembles the Hartree-Fock procedure) to find the optimum orbitals in the Cl determinants. This form of Cl is called the multiconfiguration SCF (MCSCF) method. Because the orbitals are optimized, the MCSCF method requires far fewer configurations than ordinary Cl to get an accurate wave function. A particular form of the MCSCF approach developed for calculations on diatomic molecules is the optimized valence configuration (OVC) method. 

The solution of the secular equation Fy —F5y = 0 requires the evaluation of the constituent matrix terms Fy. The Fy s are, however, themselves functions of the coefficients of the atomic orbitals amt through Pjel and therefore can only be evaluated by solving the secular equation. The Hartree-Fock procedure thus requires that a preliminary guess be made as to the values of the molecular population distribution terms Pici these values are then used to calculate the matrix elements Fy and thence solve the secular determinant. This, in turn, provides a better approximation to the wave function and an. .improved set of values of Pm. The above procedure is repeated with this first improved set and a second improved set evaluated. The process is repeated until no difference is found between successive improved wave functions. Finally, it may be shown that when such a calculation has been iterated to self-consistency the total electronic energy E of a closed shell molecule is given by... 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

Each Prs involves the sum over the occupied MO s (j = 1 -n we are dealing with a closed-shell ground-state molecule with 2n electrons) of the products of the coefficients of the basis functions 4>r and cf)s. As pointed out in Section 5.2.3.6.2 the Hartree-Fock procedure is usually started with an initial guess at the coefficients. We can use as our guess the extended Hiickel coefficients we obtained for HeH+, with this same geometry (Section 4.4.1.2) we need the c s only for the occupied MO s ... 

In the Hartree-Fock procedure, the wave function of the system is taken as a Slater determinant... 

Unfortunately, even with an incomplete one-electron basis, a full Cl is computationally intractable for any but the smallest systems, due to the vast number of. V-electron basis functions required (the size of the Cl space is discussed in section 2.4.1). The Cl space must be reduced, hopefully in such a way that the approximate Cl wavefunction and energy are as close as possible to the exact values. By far the most common approximation is to begin with the Hartree-Fock procedure, which determines the best single-configuration approximation to the wavefunction that can be formed from a given basis set of one-electron orbitals (usually atom centered and hence called atomic orbitals, or AOs). This yields a set of molecular orbitals (MOs) which are linear combinations of the AOs ... 

The conventional structure factor formalism utilized in standard structure determinations invokes the concept of the promolecule the superposition of isolated (spherical) atomic densities derived, for example, via the Hartree-Fock procedure [45], While this model mimics the dominant topological features of the ED (local maxima at the nuclear positions) reasonably well, it completely neglects density deformations due to bonding. Unfortunately, this omission leads to biases in estimates of the structural [46,47] and thermal parameters [48]. 

The Hartree-Fock procedure adds to the methods of elementary quantum mechanics a series of approximations which turn the problem of the TV-electron atom into an equivalent set of coupled one-electron problems to be solved self-consistently. Despite its simplicity and success, this procedure is far from being obvious, as can be appreciated if we move from quantum to classical mechanics. The possible occurrence of chaos... 

Both the 4-component and the DKH or NESC methods allow also for more advanced treatments of spin-orbit effects. It is possible to ignore spin-orbit coupling effects in the Hartree-Fock procedure but include them afterwards in... 

As in the simpler jellium model, we retain the simple description of independent electrons, each moving in a confining potential U(r). Here however, that potential is not an arbitrary, made-up model potential chosen to fit data or to make a calculation convenient this potential includes such effects as the exchange interaction with the other electrons. In this, the present approach is quite reminiscent of the Hartree-Fock self-consistent procedure, which will be described next. There is one essential difference. Unlike the Hartree-Fock procedure, here the exchange term is approximated as a local function, depending only on the one-electron density. This approximation yields fast convergence to a self-consistent density. As in the Hartree-Fock... 

In the past few years, we have proposed a model for correcting the wrong description of the electron density near the nucleus of the semiclassical approaches [4], This provides values of average atomic properties at the accuracy of the Hartree-Fock procedure with a very simple approach [5], allows relativistic extensions [6], and can describe negative ions [7],... 

The second part contains detailed discussions and performance analyses of parallel algorithms for a number of important and widely used quantum chemistry procedures and methods. The book presents schemes for the parallel computation of two-electron integrals, details the Hartree-Fock procedure, considers the parallel computation of second-order Moller-Plesset energies, and examines the difficulties of parallelizing local correlation methods. 

For hydrogen the exact wave function is known. For helium and lithium, very accurate wave functions have been calculated by including interelectronic distances in the variation functions. For atoms of higher atomic number, the best approach to finding a good wave function lies in first calculating an approximate wave function using the Hartree-Fock procedure, which we shall outline in this section. The Hartree-Fock method is the basis for the use of atomic and molecular orbiteils in many-electron systems. 

Here, we end the illustration of the Hartree-Fock procedure by the two coupled harmonic oscillators. 

The Hartree-Fock procedure represents a variational method. The variational function takes the form of a single... 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

The periodic potential V(r) determines the properties of the Bloch function. It includes the interaction between all electrons and ions. One applies the Hartree-Fock procedure in self-consistent approximation in order to find the potential V(r). Consequently, the many-partide problem is reduced to the one-electron problem by means of the averaged field V(r). 

The Hartree-Fock procedure produces a set xj of 2K spin orbitals. The Hartree-Fock ground state. 

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