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Hartree-Fock equations solving

An approximate or semiempirical Hartree-Fock molecular-orbital method in which electron repulsion integrals and some Hamiltonian matrix elements are approximated and the approximate Hartree-Fock equations solved and integrated to self-consistency... [Pg.455]

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. [Pg.76]

SCF (self-consistent field) procedure for solving the Hartree-Fock equations SCI-PCM (self-consistent isosurface-polarized continuum method) an ah initio solvation method... [Pg.368]

The first two kinds of terms are called derivative integrals, they are the derivatives of integrals that are well known in molecular structure theory, and they are easy to evaluate. Terms of the third kind pose a problem, and we have to solve a set of equations called the coupled Hartree-Fock equations in order to find them. The coupled Hartree-Fock method is far from new one of the earliest papers is that of Gerratt and Mills. [Pg.240]

The coupled Hartree-Fock equations are then solved (Figure 17.5). [Pg.291]

The Hartree-Fock equations have to be solved by the coupled Hartree-Fock method. The following article affords a typical example. [Pg.300]

When deriving the Hartree-Fock equations it was only required that the variation of the energy with respect to an orbital variation should be zero. This is equivalent to the first derivatives of the energy with respect to the MO expansion coefficients being equal to zero. The Hartree-Fock equations can be solved by an iterative SCF method, and... [Pg.117]

The last term in Eq. 11.47 gives apparently the "average one-electron potential we were asking for in Eq. 11.40. The Hartree-Fock equations (Eq. 11.46) are mathematically complicated nonlinear integro-differential equations which are solved by Hartree s iterative self-consistent field (SCF) procedure. [Pg.226]

This means that one has to be extremely careful in making physical interpretations of the results of the unrestricted Hartree-Fock scheme, even if one has selected the pure spin component desired. In many cases, it is probably safer to carry out an additional variation of the orbitals for the specific spin component under consideration, i.e., to go over to the extended Hartree-Fock scheme. In the unrestricted scheme, one has obtained mathematical simplicity at the price of some physical confusion—in the extended scheme, the physical simplicity is restored, but the corresponding Hartree-Fock equations are now more complicated to solve. We probably have to accept these mathematical complications, since it is ultimately the physics of the system we are interested in. [Pg.315]

Thus, once we know the various contributions in equation (5-15) we have a grip on the potential Vs which we need to insert into the one-particle equations, which in turn determine the orbitals and hence the ground state density and the ground state energy by employing the energy expression (5-13). It should be noted that Veff already depends on the density (and thus on the orbitals) through the Coulomb term as shown in equation (5-13). Therefore, just like the Hartree-Fock equations (1-24), the Kohn-Sham one-electron equations (5-14) also have to be solved iteratively. [Pg.62]

In the literature we may find the procedure for creating localized Hartree-Fock orbitals via an energy minimization based on a Cl procedure employing monoexcitations (see for instance Reference [24]). The scheme starts from a set of given (guess) orbitals and solves iteratively the Hartree-Fock equations via the steps ... [Pg.141]

The presence of the nonlocal exchange potentials in the Hartree-Fock equations greatly complicates their solution and necessitates further approximations. Several of these are discussed in the following subsection. In the evaluation of any calculations, it is important to recognize their common (and imperfect) origin, as well as the seriousness of the particular approximations made in solving the equations. [Pg.531]

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. [Pg.365]

Note that the exchange term is of the form / y(r,r ) h(r )dr instead of the y (r) (r) type. Equation (1.12), known as the Hartree-Fock equation, is intractable except for the free-electron gas case. Hence the interest in sticking to the conceptually simple free-electron case as the basis for solving the more realistic case of electrons in periodic potentials. The question is how far can this approximation be driven. Landau s approach, known as the Fermi liquid theory, establishes that the electron-electron interactions do not appear to invalidate the one-electron picture, even when such interactions are strong, provided that the levels involved are located within kBT of Ep. For metals, electrons are distributed close to Ep according to the Fermi function f E) ... [Pg.59]

The Hartree-Fock approximation leads to a set of coupled differential equations (the Hartree-Fock equations), each involving the coordinates of a single electron. While they may be solved numerically, it is advantageous to introduce an additional approximation in order to transform the Hartree-Fock equations into a set of algebraic equations. [Pg.25]

The main difficulty in solving the Hartree-Fock equation is caused by the non-local character of the potential in which an electron is orbiting. This causes, in turn, a complicated dependence of the potential, particularly of its exchange part, on the wave functions of electronic shells. There have been a number of attempts to replace it by a local potential, often having an analytical expression (e.g. universal Gaspar potential, Slater approximation for its exchange part, etc.). These forms of potential are usually employed to find wave functions when the requirements for their accuracy are not high or when they serve as the initial functions. [Pg.336]

M is an positive integer number. Optimal c,- and a, values are found by solving the Hartree-Fock equations in an analytical form, i.e., varying the energy functional with respect to these parameters. Determining non-linear parameters a,- is rather difficult. Therefore quite often they are chosen to be the same for all shells with the given l values. An efficient method of... [Pg.340]

The pecularities of solving the various versions of the Hartree-Fock equations are described in more detail in monographs [16, 45], There are a number of widely used universal computer programs to solve the non-relativistic [16, 45] and relativistic [57, 182] versions of the Hartree-Fock equations, used separately or as a part of the more general complex, e.g. calculating energy spectra, etc. [183]. [Pg.341]

Expressions (29.8) and (29.9) are the Hartree-Fock equations in a multi-configurational approximation [222], or the Hartree-Fock-Jucys equations. They must be solved together with the equations... [Pg.349]

In this approach, for the main configuration (i = 1), the single-configuration Hartree-Fock equations, whereas for the admixed ones (i 1), the two-configurational equations dealing only with the functions of admixed configuration, must be solved. From the practical point of view, the process of solution of these equations is much easier than that of equations (29.8) and (29.9), without losing too much accuracy in the results. Formulas (29.12) and (29.13) are called the simplified Hartree-Fock-Jucys equations. Pecularities of their solution are discussed in [226]. [Pg.350]

In spite of these gross approximations, the method proved to be extremely useful and was extensively used to correlate the chemical properties of conjugated systems. Several attempts were subsequently made to introduce the repulsions between the n electrons in the calculations. These include the work of Goeppert-Mayer and Sklar 4> on benzene and that of Wheland and Mann 5> and of Streitwieser 6> with the a> technique. But the first general methods of wide application were developed only in 1953 by Pariser and Parr 7> (interaction of configuration) and by Pople 8> (SCF) following the publication by Roothaan of his self-consistent field formalism for solving the Hartree-Fock equation for... [Pg.5]

In the framework of the A-potential model, combined with the frozen-cage approximation, the problem is solved simply. Namely, HF wavefunctions and energies of the encaged atom, solutions of the extended to encaged atoms Hartree-Fock equations (2), must be substituted into corresponding formulae for the photoionization of an nl subshell of the free atom, Equations (18)-(26), thereby turning them into formulae for the encaged atom (to be marked with superscript " A") rrni(o>) —> a A(co), Pni(fi>) Yni o>) - and 8ni((o) - 8 A(co). This accounts... [Pg.25]

We now furn to the problem of how the Hartree-Fock equations are solved in practice. The standard way to proceed is to expand the wavefunction in a basis. The orbital is expanded in terms of a linear combination of basis functions ... [Pg.14]

The Hartree-Fock equations are the coupled differential equations of the SCF procedure. The LCAO approximation transforms these differential equations into an ensemble of algebraic equations, which are substantially easier to solve. [Pg.253]


See other pages where Hartree-Fock equations solving is mentioned: [Pg.72]    [Pg.74]    [Pg.74]    [Pg.131]    [Pg.368]    [Pg.213]    [Pg.8]    [Pg.9]    [Pg.80]    [Pg.81]    [Pg.531]    [Pg.77]    [Pg.130]    [Pg.130]    [Pg.274]    [Pg.334]    [Pg.348]    [Pg.446]    [Pg.39]    [Pg.516]    [Pg.14]    [Pg.113]   
See also in sourсe #XX -- [ Pg.215 ]




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