Hartree Fock

HF theory allows a set of molecular orbitals, ) , to be constructed which are solutions to the Fock equations  

In practical problems, the molecular orbitals themselves are constructed using the idea of a linear combination of atomic orbitals or, more generally, a linear combination of basis functions, xf- 

Slater showed that the Hartree equations can be obtained if the variation principle is applied to a product of spin orbitals. The Russian theoretical physicist Vladimir A. Lock pointed out that certain symmetry conditions are not obeyed in the Hartree method, of which the most important one is the antisymmetric property of the total wave function. The variation principle was now applied to an antisymmetrized product of spin orbitals, that is, a Slater determinant. This is a fnndamen-tal method in electronic structure calculations and is referred to as the Hartree-Fock method or simply Hartree-Eock.  

In atomic units and assuming a fixed nucleus, we may write the Hamiltonian for 

The derivation of Equation 2.27 is mathematically simple and straightforward, but lengthy, so it is left out here. Notice that capitals are used for many-electron wave functions, while lowercase symbols are used for spin orbitals. 

The Coulomb operator Jj corresponds to the Coulomb repulsion at i from the electron in spin orbital x fj and is defined by 

The integral operator Kj is almost negligible, except when it acts on its own spin orbital xpj. Since K /j(x) = JjM j(x), JjX tj(x) and Kj tj(x) cancel in Equation 2.27. Physically, this means that the electron in spin orbital ij does not repel itself. There 

It was decided to improve these calculations by using better electronic wavefunctions 0. Single configuration molecular orbital wavefunctions were still used. However, the molecular orbitals were expressed in terms of a so-called extended basis set of gaussian atomic orbitals (for details see reference (3)). The Hartree-Fock-self-consistent-field (HFSCF) procedure was carried out with the digital computer program POLYATOM, The quality of the wavefunctions is not quite what would be called Hartree-Fock limit wavefunctions. Calculations were carried out at several intemuclear distances and C was calculated with the inclusion of the factor A correctly calculated. The calculations were extended to include the ground states of several ions and also of HCl. 

Cade and Huo (13) have calculated near Hartree-Fock limit wavefunctions for LiH, BH, HF and HCl. The molecular orbital coefficients c j are available in Hie literature again at only the equilibrium Intemuclear distance. Thus again Cl values cannot be completely calculated. In Table I, the C values at the experimental equilibrium intemuclear distances calculated for LiH, BH, HF, and HCl with (1) minimum basis set wavefunctions, with (2) our extended basis set wavefunctions and with (3) the near Hartree-Fock limit wavefunctions are compared. In order to assess the quality of the various wavefunctions, the respective electronic energies are compared with those of the corresponding near Hartree-Fock limit wavefunctions. For the minimum basis set and the near Hartree-Fock limit calculations, the correct CL values of the extended basis set calculations were employed to calculate C. It is seen that both the C values and the A C values for the tended basis set calculations approach closely those of the near Hartree-Fock limit calculations. For H2, we carried out calculations not only for an extended basis set but also for a large extended basis set which 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

---

Hartree-Fock orbital Relatively accurately calculated orbital shapes. 

Even Hartree-Fock calculations are diflTicult and expensive to apply to large molecules. As a result, fiirther simplifications are often made. Parts of the Fock operator are ignored or replaced by parameters chosen by some sort of statistical procedure to account, in an average way, for the known properties of selected... 

Although it is now somewhat dated, this book provides one of the best treatments of the Hartree-Fock approximation and the basic ideas involved in evaluating the correlation energy. An especially valuable feature of this book is that much attention is given to how these methods are actually implemented. 

A highly readable account of early efforts to apply the independent-particle approximation to problems of organic chemistry. Although more accurate computational methods have since been developed for treating all of the problems discussed in the text, its discussion of approximate Hartree-Fock (semiempirical) methods and their accuracy is still useful. Moreover, the view supplied about what was understood and what was not understood in physical organic chemistry three decades ago is... 

If one uses a Slater detemiinant to evaluate the total electronic energy and maintains the orbital nomialization, then the orbitals can be obtained from the following Hartree-Fock equations ... 

This expression is not orbitally dependent. As such, a solution of the Hartree-Fock equation (equation (Al.3.18) is much easier to implement. Although Slater exchange was not rigorously justified for non-unifonn electron gases, it was quite successfiil in replicating the essential features of atomic and molecular systems as detennined by Hartree-Fock calculations. 

Douketis C, Socles G, Marchetti S, Zen M and Thakkar A J 1982 Intermolecular forces via hybrid Hartree-Fock SCF plus damped dispersion (HFD) energy calculations. An improved spherical model J. Chem. Phys. 76 3057... 

Schmidt M W and Ruedenberg K 1979 Effective convergence to compiete orbitai bases and to the atomic Hartree-Fock iimit through systematic sequences of Gaussian primitives J. Chem. Phys. 71 3951-62... 

Ziegler T, Rauk A and Baerends E J 1977 On the calculation of multiplet energies by the Hartree-Fock-Slater method Theor. Chim. Acta 43 261-71... 

Becke A D 1983 Numerical Hartree-Fock-Slater calculations on diatomic molecules J. Chem. Phys. 76 6037 5 Case D A 1982 Electronic structure calculation using the Xa method Ann. 

PIsanI C, DovesI R and RoettI C 988 Hartree-Fock Ab initio Treatment of Crystaiiine Systems (Lecture Notes in Chemistry, voi 48) (Berlin Springer)... 

Ravenek W and Geurts EMM 1986 Hartree-Fock-Slater-LCAO implementation of the moderately large-embedded-cluster approach to chemisorption. Calculations for hydrogen on lithium (100) J. Chem. Phys. 84 1613-23... 

Pisani C, Doves R and Nada R 1990 Ab initio Hartree-Fock perturbed-cluster treatment of local defects in crystals J. Chem. Phys. 92 7448... 

Bacskay G B 1981 A quadratically convergent Hartree-Fock (QC-SCF) method. Applications to the closed-shell case Chem. Phys. 61 385... 

Head-Gordon M and Pople J A 1988 Optimization of wavefunotion and geometry in the finite basis Hartree-Fock method J. Phys. Chem. 92 3063... 

Field M J 1991 Constrained optimization of ab initio and semiempirical Hartree-Fock wavefunctions using... 

Hartke B and Carter E A 1992 Spin eigenstate-dependent Hartree-Fock molecular dynamics Chem. Phys. Lett. 189 358... 

Kurnikov I V and Beratan D N 1996 Ab initio based effective Hamiltonians for long-range electron transfer Hartree-Fock analysis J. Chem. Phys. 105 9561-73... 

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. 

Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree-Fock [68] wave functions to heat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the elech onic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69-71] restricted themselves to adiabatic problems, the method can incorporate non-adiabatic effects directly in the electionic wave function. 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

A simple example would be in a study of a diatomic molecule that in a Hartree-Fock calculation has a bonded cr orbital as the highest occupied MO (HOMO) and a a lowest unoccupied MO (LUMO). A CASSCF calculation would then use the two a electrons and set up four CSFs with single and double excitations from the HOMO into the a orbital. This allows the bond dissociation to be described correctly, with different amounts of the neutral atoms, ion pair, and bonded pair controlled by the Cl coefficients, with the optimal shapes of the orbitals also being found. For more complicated systems... 

K. Goeke and P.-G. Reinhard, Time-Dependent Hartree-Fock and Beyond, Lecture Notes in Physics, Vol. 171 (Springer-Verlag, Berlin, 1982). 

---