Symmetry in Hartree-Fock Theory

Basis Sets Correlation Consistent Sets Configuration Interaction Coupled-cluster Theory Density Functional Applications Density Functional Theory Applications to Transition Metal Problems G2 Theory Integrals of Electron Repulsion Integrals Overlap Linear Scaling Methods for Electronic Structure Calculations Localized MO SCF Methods Mpller-Plesset Perturbation Theory Monte Carlo Quantum Methods for Electronic Structure Numerical Hartree-Fock Methods for Molecules Pseudospectral Methods in Ab Initio Quantum Chemistry Self-consistent Reaction Field Methods Symmetry in Hartree-Fock Theory. 

The contribution of the ERIs to the Fock matrix, including near and far range contributions to the coulombic part of the Fock matrix, are discussed by P. Gill in Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field. The topic of symmetry utilization and the computation of ERIs over symmetry-adapted basis functions will also not be covered in this chapter. The reader is referred to the article by R. Pitzer on Symmetry in Hartree-Fock Theory. In addition, the topic of computation of integral derivatives will not be covered. The extension of the formalism to derivative integrals is straightforward and does not introduce any new problems in comparison with those encountered in integral evaluation. 

Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Gradient Theory Symmetry in Chemistry Symmetry in Hartree-Fock Theory. 

See Symmetric Group and Symmetry and Chirality Continuous Measures Symmetry in Chemistry and Symmetry in Hartree-Fock Theory. 

Russeii M. Pitzer Ohio State University, Columbus, OH, USA Symmetry in Hartree-Fock Theory 4 2929... 

Since the many-electron Hamiltonian, the effective one-electron Hamiltonian obtained in Hartree-Fock theory (the Fock operator ), and the one- and two-electron operators that comprise the Hamiltonian are all totally symmetric, this selection rule is extremely powerful and useful. It can be generalized by noting that any operator can be written in terms of symmetry-adapted operators ... 

It is important to emphasize that the bond-antibond interactions depicted in equation (17) or equation (21) are the only physically significant delocalizations in Hartree-Fock theory. The remaining symmetry adaptation that converts LMOs into MOs (i.e., that converts equation 20b into equation 20a) is only decorative, with no physical effect on the HF wave-function or other measurable property. 

Note that the frequency calculation produces many more frequencies than those listed here. We ve matched calculated frequenices to experimental frequencies using symmetry types and analyzing the normal mode displacements. The agreement with experiment is generally good, and follows what might be expected of Hartree-Fock theory in the ground state. ... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Efficient use of symmetry can greatly speed up localized-orbital density-functional-exchange-and-correlation calculations. The local potential of density functional theory makes this process simpler than it is in Hartree-Fock-based methods. The greatest efficiency can be achieved by using non-Abelian point-group symmetry. Such groups have multidimensional irreducible representations. Only one member of each such representation need be used in the calculation. However efficient localized-orbital evaluation of the chosen matrix element requires the sum of the magnitude squared of the components of all the members on one of the symmetry inequivalent atoms, based on Eq. 13. 

Linear transformations of the integrals previously listed, in order to prepare for the determination of the effective Hamiltonian and take advantage of the symmetry properties of the molecule. The computing time needed for this increases at least as w even if the molecule has no symmetry at all, the presence of exchange terms in the Hartree-Fock theory makes a part of the transformation still necessary, namely the construction of the integral list. 

Section 3.5 contains a detailed illustration of the closed-shell ab initio SCF procedure using two simple systems the minimal basis set descriptions of the homonuclear (H2) and heteronuclear (HeH" ) two-electron molecules. We first describe the STO-3G minimal basis set used in calculations on these two molecules. We then describe the application of closed-shell Hartree-Fock theory to H2. This is a very simple model system, which allows one to examine the results of calculations in explicit analytical form. Finally, we apply the Roothaan SCF procedure to HeH. Unlike H2, the final SCF wave function for minimal basis HeH is not symmetry determined and the HeH example provides the simplest possible illustration of the iterative SCF procedure. The description of the ab initio HeH calculation given in the text is based on a simple FORTRAN program and the output of a HeH calculation found in Appendix B. By following the details of this simple but, nevertheless, real calculation, the formalism of closed-shell ab initio SCF calculations is made concrete. 

So far we have not taken time-reversal symmetry into account. From the preceding chapters, we expect that incorporating time-reversal symmetry in a Kramers-restricted Dirac-Hartree-Fock theory will result in a reduction of the work, and possibly also a reduction in the rank of the Fock matrix. The basis set we will use is a basis set of Kramers pairs. We develop the theory for a closed-shell reference, for which all Kramers pairs are doubly occupied. ... 

The purpose of the present chapter is to discuss the structure and construction of restricted Hartree-Fock wave functions. We cover not only the traditional methods of optimization, based on the diagonalization of the Fock matrix, but also second-order methods of optimization, based on an expansion of the Hartree-Fock eneigy in nonredundant orbital rotations, as well as density-based methods, required for the efficient application of Hartree-Fock theory to large molecular systems. In addition, some important aspects of the Hartree-Fock model are analysed, such as the size-extensivity of the energy, symmetry constraints and symmetry-broken solutions, and the interpretation of orbital energies in the canonical representation. 

As discussed in the present section and also in Sections 10.5 and 10.6, canonical Hartree-Fock theory offers many advantages. However, for open-shell RHF states, this approach is not always possible because of the constraints imposed by the spin- and space-symmetry adaptation of the wave fiinction. In such cases, the wave function must be calculated using the more general methods of Section 10.8 in the present section, we restrict our attention to closed-shell systems. In addition, the application of canonical Hartree-Fock theory is restricted to small and moderately large systems since it requires an amount of work that scales at least cubically with the size of the system for large systems, the methods of Section 10.7 must be used instead. 

It should be emphasized that both the RHF solution and the UHF solution in Figure 10.6 are approximations to the same wave function. The 2 RHF wave function transforms in the same manner as the true wave function but is higher in energy than the symmetry-broken UHF solution. The difference in energy is very small, however, compared with the tme excitation energies of the system. Neither wave function may be classified as best and, for a satisfactory resolution of this symmetry dilemma of Hartree-Fock theory, a more advanced wave function must be used, as described in Section 12.8. 

---