Fock’s equation

These equations reduce the problem to that of determining the eigenfunctions and eigenvalues of a three-dimensional linear Hermitean operator F which itself involves the eigenfunctions which are to be determined. In principle, therefore, the equations can be solved by an iterative procedure in which one postulates an initial set of eigenfunctions, calculates a new set from Fock s equations, and rep>eats this process until convergence is obtained. This, however, is an extremely arduous task, even for atoms, and for molecules it is absolutely necessary to imp>ose further restrictions on the molecular orbitals y>, before it becomes even remotely practicable to determine the form of these orbitals. The most important of these further restrictions is the assumption embodied... 

It should be mentioned at this point that, although the above wave functions, particularly those obtained by the solution of Fock s equations, are the best possible one-electron wave functions, they give values for the energies of atoms which are incorrect by approximately 0.5 volt per electron. For this reason any results obtained from calculations based upon one-electron wave functions can be only quali-... 

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... 

Although for brevity s sake the 3-MCSE has not been considered here, it may be convenient to mention it in these final comments. This equation, which depends on the 1-CSE, does not have a unique solution. Indeed, this equation is satisfied not only by the FCI 3-RDM but also by the Hartree-Fock one. Alcoba [48] performed a series of calculations with the 3-MCSE for the beryllium iso-electronic series. Alcoba took as initial data a set of RDMs that corresponded to a state that had already some correlation and whose energy was below the Hartree-Fock s one. The results of these calculations showed that there was a smooth although very slow convergence toward the exact solution. For larger systems the situation will probably be similar to the 4-MCSE one and a strict... 

The first few 4-dimensional hyperspherical harmonics K i, ,m(u) are shown in Table 5. Shibuya and Wulfman [19] extended Fock s momentum-space method to the many-center one-particle Schrodinger equation, and from their work it follows that the solutions can be found by solving the secular equation (63). If Fock s relationship, equation (67), is substituted into (65), we obtain ... 

Exact solutions of the Schrddinger equation are, of course, impossible for atoms containing 90 electrons and more. The most common approximation used for solving Schrddinger s equation for heavy atoms is a Hartree-Fock or central field approximation. In this approximation, the individual electrostatic repulsion between the electron i and the N-1 others is replaced by a mean central field giving rise to a spherically symmetric potential... 

Fock s argument rests on the theory of the Fourier transform. In particular, he uses the momentum-space version of the Schrodinger equation. We let f denote the Fourier transform of / e... 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

Thus four of the seven lowest H20 MOs are linear combinations of the four a, symmetry orbitals listed above, and are a, MOs similarly, the two lowest b2 MOs are linear combinations of 02p and H,1j — H21.s, and the lowest bx MO is (in this minimal-basis calculation) identical with 02px. The coefficients in the linear combinations and the orbital energies are found by iterative solution of the Hartree-Fock-Roothaan equations. One finds the ground-state electronic configuration of H20 to be... 

Now we have FC = SCe (5.57), the matrix form of the Roothaan-Hall equations. These equations are sometimes called the Hartree-Fock-Roothaan equations, and, often, the Roothaan equations, as Roothaan s exposition was the more detailed and addresses itself more clearly to a general treatment of molecules. Before showing how they are used to do ab initio calculations, a brief review of how we got these equations is in order. 

Roothaan s equations. In 1951 using Fock s method the American physicist C. C. Roothaan worked out a system ol nonlinear algebraic equations providing the AO coefficients of Eq. (1) ... 

There exist several SCF codes for the solution of radial equations the Hartree-Fock [16] equations are only one example, and the case described above is that of the single configuration approximation, in which each electron has well-defined values of n and l. There exist several other possibilities as stressed above, in Hartree s original method, the exchange term was left out in the Hartree-Slater method [17], an approximate expression is used for the form of the exchange term. The Cowan code [20] is a pseudorelativistic SCF method, which avoids the complete four-component wavefunctions by simulating relativistic effects. 

All three single-particle equations, Hartree, Hartree-Fock and Kohn-Sham can also be interpreted as approximations to Dyson s equation (38), which can be rewritten as [48]... 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

This equation was solved for the hydrogen atom by Vladimir Fock [2,3]. The solution in p space revealed the four-dimensional symmetry responsible for the degeneracy of states with the same n but different I quantum numbers in the hydrogen atom. This is a fine example where the momentum-space perspective led to fresh and deep insight. Fock s work spawned much further research on dynamical groups and spectrum-generating algebras. 

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