Hartree-Fock equations/theory application

The description of electron motion and electronic states that is at the heart of all of chemistry is included in wave function theory, which is also referred to as self-consistent-field (SCF) or, by honouring its originators, Hartree-Fock (HF) theory [7]. In principle, this theory also includes density functional theory (DFT) approaches if one uses densities derived from SCF densities, which is common but not a precondition [2] therefore, we treat density functional theory in a separate section. Many approaches based on wave function theory date back to when desktop supercomputers were not available and scientists had to reduce the computational effort by approximating the underlying equations with data from experiment. This approach and its application to the elucidation of reaction mechanisms are outlined in Section 7.2.3. 

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. 

The pioneering work on the application of the many-body perturbation theory to atomic and molecular systems was performed by Kelly.5-17-21 He applied the method to atoms using numerical solutions of the Hartree-Fock equations. Many other calculations on atomic systems were subsequently... 

P. Fischer and M. Defranceschi, Representation of the Atomic Hartree-Fock Equations in a Wavelet Basis by Means of the BCR Algorithm, in Wavelets Theory, Algorithms, and Applications, (C.K. Chui. L. Montefusco and L. Puccio Eds), Academic Press, New York, (1994), pp. 495-506. 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

Integration of the system of equations (9) yields trajectories of classical nuclei dressed with END. This approach can be characterized as being direct, and non-adiabatic or as fully non-linear time-dependent Hartree-Fock (TDHF) theory of quantum electrons and classical nuclei. This simultaneous dynamics of electrons and nuclei driven by their mutual instantaneous forces requires a different approach to the choice of basis sets than that commonly encountered in electronic structure calculations with fixed nuclei. This aspect will be further discussed in connection with applications of END. 

As we have already observed above, within the hardness (interaction) representation (see Tables 1, 3) the FF indices provide important weighting factors in combination formulas which express the global CS and potentials in terms of the local properties, relevant for the resolution in question. The FF expressions from the EE equations are invalid in the MO resolution, since no equalization of the orbital potentials can take place, due to obvious constraints on the MO occupations in the Hartree-Fock (HF) theory [61]. Moreover, standard chain-rule transformations of derivatives are not applicable in the MO resolution since some of the derivatives involved are not properly defined. Various approaches to the local FF, f(f), have been proposed e.g., those expressing f(f), in terms of the frontier orbital densities [11, 25], or the spin densities [38]. Also the finite difference estimates of the chemical potential (electronegativity) and hardness have been proposed in the MO and Kohn-Sham theories for various electron configurations [10, 11, 19, 52, 61b, 62, 63]. 

The 1960s saw the applications of the many-body perturbation theory developed during the 1950s by Brueckner [13], Goldstone [38] and others to the atomic structure problem by Kelly [63-72]." These applications used the numerical solutions to the Hartree-Fock equations which are available for atoms because of the special coordinate system. Kelly also reported applications to some simple hydrides in which the hydrogen atom nucleus is treated as an additional perturbation. 

The many-body perturbation theory was applied to atoms and some simple molecular systems by Kelly using a reference function obtained from finite difference solutions of the Hartree-Fock equations. The introduction of the algebraic approximation using finite basis sets during the 1970s opened up applications to arbitrary polyatomic molecular systems. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

I have dealt at length with the Hartree and the Hartree-Fock models. The father of this field, Sir William Hartree, was concerned with the atomic problem where it is routinely possible to integrate numerically the HF integro-differential equations in order to produce (numerical) wavefunctions that correspond to the Hartree-Fock limit. For molecular applications the LCAO variant of HF theory assumes a dominant role because of the reduced symmetry of the problem. 

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