Hartree Fock equation limit

Since the exact solution of the Hartree-Fock equation for molecules also proved to be impossible, numerical methods approximating the solution of the Schrodinger s equation at the HF limit have been developed. For example, in the Roothan-Hall SCF method, each SCF orbital is expressed in terms of a linear combination of fixed orbitals or basis sets ((Pi). These orbitals are fixed in the sense that they are not allowed to vary as the SCF calculation proceeds. From n basis functions, new SCF orbitals are generated by... 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

The HF equations are approximate mainly because they treat electron-electron repulsion approximately (other approximations are mentioned in the answer suggested for Chapter 5, Harder Question 1). This repulsion is approximated as resulting from interaction between two charge clouds rather than correctly, as the force between each pair of point-charge electrons. The equations become more exact as one increases the number of determinants representing the wavefunctions (as well as the size of the basis set), but this takes us into post-Hartree-Fock equations. Solutions to the HF equations are exact because the mathematics of the solution method is rigorous successive iterations (the SCF method) approach an exact solution (within the limits of the finite basis set) to the equations, i.e. an exact value of the (approximate ) wavefunction l m.. 

The complete neglect of differential overlap (CNDO) procedure was developed by Pople and co-workers (2) and has been widely used. It suffers from some of the same limitations as EH but makes different approximations. It is a simplification of the exact Hartree-Fock equations for a molecule. In this procedure mathematical approximations leading to neglect of small terms are employed rather than the intuitive approximations employed in EH. In addition, electrons having different spin are treated in this procedure. 

When the basis set contains many terms (effectively infinite), one obtains the best possible result from solution of the Hartree-Fock equation the Hartree-Fock limit. Improvements beyond this limit are most usually achieved by allowing the molecular wavefunction I7 to be a linear combination of antisymmetrized products of orbitals i,... 

If a -electron wave function is limited to a Slater determinant of n spin orbitals, one stays within the frame of the independent-particle model, and the best model of that sort (for a discussion, see 22>) for a given problem is that in which the orbitals used to construct the wave function are solutions of the Hartree-Fock equations. This model is only an approximation of the correct wave function. As mentioned in Sect. 3.1, the wave function should be written as a linear combination of Slater determinants, as in Eq. (3.4). To illustrate this, let us consider a two-electron system where the spin can be separated off, so that it is sufficient to consider a function ip (1,2) depending only on the space coordinates of the two particles 1 and 2. For a singlet state ip (1,2) is symmetric with respect to space coordinates ... 

There have been continuous attempts to develop schemes to circumvent the limitations imposed by the finite basis sets, e.g. the counterpoise method of Boys and Bernardi and its variants (28,29) or the complete basis set approach (30). It seems that the availibility of exact solutions of the Hartree-Fock equations for diatomic molecules could be of help in devising such methods by allowing the dependence of the basis set truncation and superposition errors on internuclear separation to be monitored (16). 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

So, although these type of limited-basis expansions do formally solve the Hartree-Fock equations, it is better, perhaps, to refer to them as merely satisfying the SCF equations rather than solving them to try and distinguish between actively solving the equations and the rather passive fact of satisfying the equa tions by dint of some accidental fact of symmetry. The Hiickel and SCF orbitals of pyridine, for example, are different even in a minimal basis since pyridine has a lower symmetry than benzene. 

A third class of methods is the sparse matrix methods (60), which solve the Hartree-Fock equations for large systems. Disadvantages include that it is limited to the ground state, formulated (much less implemented ) only for semi-enq)irical methods, and intrinsically incapable of treating electron transport and optical properties such as two-photon absorption. 

Actually Lowdin goes quite a long way to develop the MCSCF formalism. He defines a full Cl on a limited one-electron basis set and points out that since the expansion is no longer complete, it is now important not only to determine the Cl coefficients but also the MOs in order to obtain a solution which is as accurate as possible. When the number of spin orbitals, M equals the number of electrons, A, this becomes the ordinary Hartree-Fock equations, but when M > N v/q obtain what he calls the extended Hartree-Fock equations. Twenty-five years later it became known as the CASSCF method. He writes the condition for optimum orbitals in the form... 

Early work on the finite basis set problem in relativistic calculations has been reviewed by Kutzelnigg. Spurious unphysical solutions of the Dirac equation or the Dirac-Hartree-Fock equations are observed with too small a kinetic energy, leading to an overestimation of the binding energy. Furthermore, these solutions are found neither to tend to the solutions of the Schrodinger equation in the limit c- co nor to vary systematically with increasing size of basis set. 

Ab-initio methods based on solution of the Hartree-Fock equations are well established (Ref 14), and are used in routine molecular and electronic calculations on small organic and inorganic molecules. For such molecules, extremely accurate predictions of many spectroscopic properties can be made using methods such as CAS-CI, MCSCF, coupled-cluster theory, and multireference methods. Recent advances in supercomputer technology coupled with improved algorithms have made it possible to perform full Cl calculations for small systems. Due to their size and complexity, such calculations have been limited mostly to diatomic molecules. However, where cost is not a problem, it is quite feasible to perform full Cl calculations on quite large systems, and such calculations have been carried out on small energetic molecules by Haskins and Cook at RARDE (Ref 16). 

The failure to properly reproduce the particle-particle coalescence asymptotics bears upon the rates of convergence of the computed energies and other observables to their complete-basis-set (CBS) limits [1]. Whereas in practice this convergence is sufficiently rapid for the solutions of the Hartree-Fock equations [2, 3], obtaining accurate approximations to correlated electronic wave-functions is much more difficult [4, 5]. In order to alleviate this problem, two distinct strategies have been developed, namely inclusion of a correlation factor in the trial function [6-8] and extrapolation to the CBS limit [9, 10]. Successful implementations of the latter approach hinge upon understanding how the approximate wavefunction approaches its exact counterpart as the size of the basis set increases. 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

Solution of the numerical HF equations to full accuracy is routine in the case of atoms. We say that such calculations are at the Hartree-Fock limit. These represent the best solution possible within the orbital model. For large molecules, solutions at the HF limit are not possible, which brings me to my next topic. 

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

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