The differential Hartree-Fock equation

The Hartree-Fock equation of the previous chapter is a non-separable partial differential equation in three dimensions. In this chapter we derive an equation which is satisfied by an approximation to the differential Hartree-Fock equation. This matrix equation provides the techniques and concepts for the vast majority of quantum chemistry. 

A more theoretical investigation what formal properties do these virtual orbitals have and how do they relate to any corresponding properties of the (differential) Hartree-Fock equation. 

That is, we have to rely on physical and chemical intuition in our choice of parametrisation to try to ensure that our solution of the SCF (finite-basis approximated) Hartree-Fock equations are, indeed, realistic and acceptable approximations to the solutions of the (differential) true Hartree-Fock equations. Our parameters are the expansion coefficients, so that these matters all hinge on choice of basis function by which the parameters are multiplied we must ensure that there are sufficient basis functions available to locate a minimum in the energy function which reflects both the existence and form of the putative solution of the differential Hartree-Fock equations that we seek. ... 

The similarity between the Kohn-Sham (KS) and the (differential) Hartree-Fock equation is so great that is too tempting not to try to use the same linear expansion methods for its approximate (parametric) solution. We already have the mathematics and the software technology to evaluate all the terms in the matrix form of the Kohn-Sham equation. 

It is, however, natural to ask if these orbitals have any meaning do they have any physical interpretation for example, are they approximations to any orbitals associated with the system under investigation or are they just an artifact of the use of the LCAO technique In the case of the orbitals of the single determinant — the occupied orbitals — the LCAO expansion is a more or less good approximation to the solutions of the differential Hartree-Fock equation depending on the quality and length of the AO expansion. As the quality and size of the basis is improved, the occupied MOs presumably become better and better approximations to these orbitals. What about the unoccupied molecular orbitals ... 

The case of virtual orbitals has been discussed earlier in a different context since these orbitals are not even formally available from the differential Hartree-Fock equation. 

These are the glorious Hartree-Fock equations derived in general in the spin orbital basis. But wait - there s a problem. These are coupled integro-differential equations, and while they are not strictly unsolvable, they re a pain. It would be nice to at least uncouple them, so let s do that. 

The widespread application of MO theory to systems containing a bonds was sparked in large part by the development of extended Hiickel (EH) theory by Hoffmann (I) in 1963. At that time, 7r MO theory was practiced widely by chemists, but only a few treatments of a bonding had been undertaken. Hoffmann s theory changed this because of its conceptual simplicity and ease of applicability to almost any system. It has been criticized on various theoretical grounds but remains in widespread use today. A second approximate MO theory with which we are concerned was developed by Pople and co-workers (2) in 1965 who simplified the exact Hartree-Fock equations for a molecule. It has a variety of names, such as complete neglect of differential overlap (CNDO) or intermediate neglect of differential overlap (INDO). This theory is also widely used today. 

In an analogous fashion to the atomic Hartree-Fock equations, the angular variables can be separated and integrated out using the Wigner-Eckart theorem in the Dirac equation to yield a set of coupled differential equations depending on r (29). 

Fundamental to almost all applications of quantum mechanics to molecules is the use of a finite basis set. Such an approach leads to computational problems which are well suited to vectoris-ation. For example, by using a basis set the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients - a set of matrix equations. The absolute accuracy of molecular electronic structure calculations is ultimately determined by the quality of the basis set employed. No amount of configuration interaction will compensate for a poor choice of basis set. 

Under the conditions of our derivation, i.e. S-state atoms A and B with vanishing differential overlap, we can show that the localized orbitals Ip) and Icr) are identical (apart from mixing possibly degenerate orbitals) to the orbitals obtained by solving the monomer Hartree-Fock equations (in the dimer basis)... 

Originally, Hartree-Fock atomic calculations were done by using numerical methods to solve the Hartree-Fock differential equations (11.12), and the resulting orbitals were given as tables of the radial functions for various values of r. [The Numerov method (Sections 4.4 and 6.9) can be used to solve the radial Hartree-Fock equations for the radial factors in the Hartree-Fock orbiteds the angular factors are spherical harmonics. See D. R. Heu tree, The Calculation of Atomic Structures, Wiley, 1957 C. Froese Fischer, The Hartree-Fock Method for Atoms, Wiley, 1977.]... 

To solve the unrestricted Hartree-Fock equations (3.312) and (3.313), wc need to introduce a basis set and convert these integro differential equations to matrix equations,just as we did when deriving Roothaan s equations. We thus introduce our set of basis functions = 1, 2,..., X and... 

Variation of the Hartree-Fock total electronic energy expectation value with respect to the orbitals yields the radial Hartree-Fock equations [479,480] that can be written as homogeneous differential equations... 

In 1951, Hall [6] and, independently, Roothaan [7] put the Hartree-Fock equations - the ubiquitous independent particle model - in their matrix form. The Hartree-Fock equations describe the motion of each electron in the mean field of all the electrons in the system. Hall and Roothaan invoked the algebraic approximation in which, by expanding molecular orbitals in a finite analytic basis set, the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients which are well-suited to computer implementation. 

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. 

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. 

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. 

Hartree-Fock Equations. The set of differential equations resulting from application of the Born-Oppenheimer and Hartree-Fock Approximations to the many-electron Schrodinger Equation. 

In most atomic programs (5) is actually solved self-consistently either in a local potential or by the relativistic Hartree-Fock method. There is, however, an important time-saving device that is often used in energy band calculations for actinides where the same radial Eq. (5) must be solved If (5.a) is substituted into (5.b) a single second order differential equation for the major component is obtained... 

The KS equations are obtained by differentiating the energy with respect to the KS molecular orbitals, analogously to the derivation of the Hartree-Fock equations, where differentiation is with respect to wavefunction molecular orbitals (Section 5.2.3.4). We use the fact that the electron density distribution of the reference system, which is by decree exactly the same as that of the ground state of our real system (see the definition at the beginning of the discussion of the Kohn-Sham energy), is given by (reference [9])... 

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. 

It has been customary to classify methods by the nature of the approximations made. In this sense CNDO, INDO (or MINDO), and NDDO (Neglect of Diatomic Differential Overlap) form a natural progression in which the neglect of differential overlap is applied less and less fully. It is now clearer that there is a deeper division between methods, related to their objectives. On the one hand are approximate methods which set out to mimic the ab initio molecular orbital results. The objective here is simply to find a more economical method. On the other hand, some workers, recognizing the defects of the MO scheme, aim to produce more accurate results by the extensive use of parameters obtained from experimental data. This latter approach appears to be theoretically unsound since the formalism of the single-determinant wavefunction and the Hartree-Fock equations is retained. It can be argued that the use of the single-determinant wavefunction prevents the consistent achievement of predictions better than those obtained by the ab initio scheme where no further... 

It is obvious that the Hartree-Fock equations are rather complicated integro-differential equations of a non-linear nature with bifurcations etc., and it was hence of fundamental importance when Ueb and Simon [21] in 1977 could show the mathematical existence of solutions to these equations. There are still some mathematical problems associated with the Hartree-Fock scheme, particularly the connection between the starting point of the c culatlons and the final result, which is usually associated with a "local minimum" of the energy . We note further that the concept of "self-consistency is related to some form of "numerical convergence" in a specified number... 

By minimizing the energy of d>, in Eq. (3.12), we obtain a set of coupled integro-differential equations, the Hartree-Fock equations, which may be expressed in the following form for closed-shell systems (for open-shell cases see Szabo and Ostlund, 1989) ... 

Iron), and Eq. (3.12) is a one-electron differential equation. This has been indicated by writing F and T>, as functions of the coordinates of electron 1 of course, the coordinates of any electron could have been used. The operator F is peculiar in that it depends on its own eigenfunctions, which are not known initially. Hence the Hartree-Fock equations must be solved by an iterative process. One obtains approximate solutions for the ( ), and from these constructs the first approximation to F. Equation (3.13) is then solved to obtain a new set of 4>, (which are generally occupied according to their order of e,) and a new F is constructed. Such a process is called a self-consistent-field approach, and the process is terminated when the orbitals output from one step are virtually identical to those that are input from the preceding step (in practice, the energy E is usually monitored). 

These very complicated inhomogeneous coupled differential equations can again be simplified by using Slater s approximation. This method is therefore called the relativistic Hartree-Fock-Slater or Dirac-Fock-Slater (DFS) 52—53) calculations, and they have also been done by several authors for the superheavy elements 54-56). 

Differentiating the Hartree-Fock equation is more involved than differentiating the Schrodinger equation because the Hartree-Fock equation must yield several (one-electron) states or orbitals. We begin with the equation itself ... 

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