Beyond Hartree-Fock

The motion of each electron in the Hartree-Fock approximation is solved for in the presence of the average potential of all the remaining electrons in the system. Because of this, the Hartree-Fock approximation, as discussed earlier, does not provide an adequate description of the repulsion between pairs of electrons. If the electrons have parallel spin, they are effectively kept apart in the Hartree-Fock method by the antisymmetric nature of the wavefunction, producing what is commonly known as the Fermi hole. Electrons of opposite spin, on the other hand, should also avoid each other, but this is not adequately allowed for in the Hartree-Fock method. The avoidance in this latter case is called the Coulomb hole. 

if a proper account of electronic interactions is to be made, the motion of all the electrons must be correlated so that the position of one depends on the instantaneous position of the other, rather than on the average position of the other 

Dreyfus, B. Maigret, and A. Pullman, Theor. Chim. Acta, 1970, 17, 109. 

Another way that additional configurations can be added to the the ground-state wave function is by the use of Moller-Plesset perturbation theory (MPPT). As it happens, a Hamiltonian operator constructed from a sum of Fock operators has as its set of solutions the HF single determinantal wave function and all other determinantal wave 

If carried out with a good basis set [6-31G(d) or better], the benefits of MPPT, carried out to second order (MP2), include moderate improvements in structures and relative energies and often significant improvement in the values of secondary properties such as dipole moments, vibrational frequencies, infrared and Raman absorption intensities, and NMR chemical shifts. Modem quantum chemistry codes such as the GAUSSIAN package incorporate analytical calculation of MP2 forces and force constants. Although it adds substantially to the time required to carry out the calculations, the results of MPPT usually make the extra effort worthwhile. 

Another class of methods uses more than one Slater determinant as the reference wave function. The methods used to describe electron correlation within these calculations are similar in some ways to the methods listed above. These methods include multiconfigurational self-consistent field (MCSCF), multireference single and double configuration interaction (MRDCI), and /V-clcctron valence state perturbation theory (NEVPT) methods.5 

All of the levels of theory listed in Table 1.1 predict the C-H bond length with accuracy within 1 %. One piece of cheering information from Table 1.1 is that the DFT method predicts this bond length as accurately as the much more computationally expensive CCSD approach. The error in the ionization energy predicted by HF is substantial, but all three of the other methods give better predictions. The higher levels of theory (MP2 and CCSD) give considerably more accurate results for this quantity than DFT. 

Now we look at the properties of CH4 predicted by a set of calculations in which the level of theory is fixed and the size of the basis set is varied. 

TABLE 1.1 Computed Properties of CH4 Molecule for Four Levels of Theory Using pVTZ Basis Set  

Level of Theory C H (A) Percent Error Ionization (eV) Percent Error Relative Time 

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In the RISM-SCF theory, the statistical solvent distribution around the solute is determined by the electronic structure of the solute, whereas the electronic strucmre of the solute is influenced by the surrounding solvent distribution. Therefore, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that SCF (self-consistent field) applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. The MO part of the method can be readily extended to the more sophisticated levels beyond Hartree-Fock (HF), such as configuration interaction (Cl) and coupled cluster (CC). 

With such calculations one can approach Hartree-Fock accuracy for a particular cluster of atoms. These calculations yield total energies, and so atomic positions can be varied and equilibrium positions determined for both ground and excited states. There are, however, drawbacks. First, Hartree-Fock accuracy may be insufficient, as correlation effects beyond Hartree-Fock may be of physical importance. Second, the cluster of atoms used in the calculation may be too small to yield an accurate representation of the defect. And third, the exact evaluation of exchange integrals is so demanding on computer resources that it is not practical to carry out such calculations for very large clusters or to extensively vary the atomic positions from calculation to calculation. Typically the clusters are too small for a supercell approach to be used. 

P. Hobza et al., Significant structural deformation of nucleic acid bases in stacked base pairs an ab initio study beyond Hartree-Fock. Chem. Phys. Lett. 288, 7-14 (1998)... 

The bulk of previous work has concentrated on the X1S+ ground state, but there is now an increasing number of studies on the excited states, particularly with wave-functions beyond Hartree-Fock, and therefore including electron correlation. For comparison with this work, the near-HF calculation of Cade and Huo104 remains a useful comparison point. 

Hobza P, Kabelac M, Sponer J, Mejzlik P, Vondrasek J (1997) Performance of empirical potentials (AMBER, CFF95, CVFF, CHARMM, OPS, POLTEV), semiemprical quantum chemical methods (AMI, MNDO/M, PM3) and ab initio Hartree-Fock method for interaction of DNA bases comparison of nonempirical beyond Hartree-Fock results, J Comp Chem, 18 1136-1150... 

P. R. Taylor, Int.J. Quantum Chem., 31,521 (1987). Integral Processing in Beyond-Hartree-Fock Calculations. 

Kratochvil, M., Engkvist, O., Sponer, J., Jungwirth, P., and Hobza, P. (1998) Uracil dimer potential energy and free energy surfaces. Ab initio beyond Hartree-Fock and empirical potential surfaces, J. Phys. Chem. A 102, 6921-6926. 

Often dispersion energy is described as the interaction between mutually induced dipoles, one on each atom. One can see this as a correlation between two dipoles. It is not obvious how this correlation is related to Lowdin s beyond-Hartree-Fock-correlation [23]. In this appendix it is shown how the latter correlation and dispersion are interrelated. Earlier this connection was shown [63] in a somewhat different manner. 

There are two broad categories of methods that go beyond Hartree-Fock in constructing wavefunctions configuration interaction (Cl), and many-body perturbation theory. In Cl one begins by noting that the exact... 

The Hiickel and extended Hiickel approximations are one-electron approximations. They are quite useful but they do not take us beyond Hartree-Fock. To do so we need either to use the weak version of the CNDO approximation, as given in Eq. (19) above or to make a further simplification, the strong CNDO approximation, in which we allow only electrons (of opposite spins) which are on the same site to repel. In other words, in the two-center Coulombic repulsion integrals [ijlij], we allow only i = j and... 

Contents Experimental Basis of Quantum Theory. -Vector Spaces and Linear Transformations. - Matrix Theory. -- Postulates of Quantum Mechanics and Initial Considerations. - One-Dimensional Model Problems. - Angular Momentum. - The Hydrogen Atom, Rigid, Rotor, and the H2 Molecule. - The Molecular Hamiltonian. - Approximation Methods for Stationary States. - General Considerations for Many-Electron Systems. - Calculational Techniques for Many-Electron Systems Using Single Configurations. - Beyond Hartree-Fock Theory. 

Beyond Hartree-Fock, the energies (and wavefunctions) may be improved at several levels various orders of the perturbation method at the Moller-Plesset level (1934) (MP2, MP3, MP4) for open-shell systems we could use either unrestricted MP2 (Pople et al. 1976) or one of several variants of the perturbation method based on the ROHF wavefunction the Z-averaged perturbation theory (ZAPT) (Lee and Jayatilaka 1993 Lee et al. 1994) and RMP (Knowles et al. 1980 Lauderdale et al. 1991) configuration interaction (Cl) method (Brooks and Schaefer 1979 Ivanic and Ruedenberg 2001). 

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