Hartree-Fock-Roothaan based

HARTREE-FOCK-ROOTHAAN BASED SEMIEMPIRICAL METHODS... 

Now we are ready to start the derivation of the intermediate scheme bridging quantum and classical descriptions of molecular PES. The basic idea underlying the whole derivation is that the experimental fact that the numerous MM models of molecular PES and the VSEPR model of stereochemistry are that successful, as reported in the literature, must have a theoretical explanation [21], The only way to obtain such an explanation is to perform a derivation departing from a certain form of the trial wave function of electrons in a molecule. QM methods employing the trial wave function of the self consistent field (or equivalently Hartree-Fock-Roothaan) approximation can hardly be used to base such a derivation upon, as these methods result in an inherently delocalized and therefore nontransferable description of the molecular electronic structure in terms of canonical MOs. Subsequent a posteriori localization... 

Semi-empirical calculations for the simple vinyl cation C2H3+ have been reported by Hoffmann (1964) and by Yonezawa et ad., (1968). More rigorous calculations by Sustmann et ad. (1969) are based on a semi-empirical method based on the neglect of diatomic differential overlap (NDDO) calibrated to results of ab initio Hartree-Fock-Roothaan SCF calculations. Recent work by Hopkinson et al. (1971) is entirely based on a non-empirical LCAO-MO-SCF method. 

Hartree-Fock-Roothaan methods have often been quite successful in the calculation of properties despite the fact that the variational principle upon which they are based ensures only the best total energy. In particular, other energetic properties such as force constants and charge-distribution properties such as electron-density distributions and electric-field gradients are well reproduced. 

Table 5.11. Electronic structure of the COj cluster molecular-orbital compositions in terms of atomic-orbital components based on ab initio Hartree-Fock-Roothaan SCF calculations"...

A term used to refer to a Hartree-Fock-Roothaan calculation, often applied to calculations with small bases... 

Each of these methods is based on the AFDF approach. Within the framework of the conventional Hartree-Fock-Roothaan-Hall self-consistent field linear combination of atomic orbitals (LCAO) ab initio representation of molecular wave functions built from molecular orbitals (MOs), the AFDF principle can be formulated using fragment density matrices. For a complete molecule M of some nuclear configuration K, using an atomic orbital (AO) basis of a set of n AOs density matrix P can be determined using the coefficients of AOs in the occupied MOs. The electronic density p(r) of the molecule M, a function of the three-dimensional position variable r, can be written as... 

F. Fenske. We demonstrate for transition metal complexes that the non-empirical Fenske-Hall (FH) approach provides qualitative results that are quite similar to the more rigorous treatment given by density functional theory (DFT) and are quite different from Hartree-Fock-Roothaan (HFR) calculations which have no electron correlation. For example, the highest occupied molecular orbital of ferrocene is metal based for both DFT and FH while it is ligand (cyclopentadienyl) based for HFR. In the doublet (S = 1/2) cluster, Cp2Ni2(pi-S)2(MnCO)3, the unpaired electron is delocalized over the complex in agreement with the DFT and FH results, but localized on Mn in the HFR calculation. A brief description of the theory of FH calculations is used to rationalize the origin of its similarity to DFT. 

While this endeavor was made in the efforts to discredit the MO approach and the orbital concept in general, we believe that atomic orbitals and their linear combination provide the set of elementary properties of mater on which base the whole chemistry can be rationalized based on a single (i.e., the eigen-value problem) principle, either in Schrodinger, Hartree-Fock/ Roothaan or Kohn-Sham/Density Frmctional Theory (see below) approaches. 

Numerical discretization methods pose an interesting consequence for fully numerical Dirac-Hartree-Fock calculations. These grid-based methods are designed to directly calculate only those radial functions on a given set of mesh points that occupy the Slater determinant. It is, however, not possible to directly obtain any excess radial functions that are needed to generate new CSFs as excitations from the Dirac-Hartree-Fock Slater determinant. Hence, one cannot directly start to improve the Dirac-Hartree-Fock results by methods which capture electron correlation effects based on excitations that start from a single Slater determinant as reference function. This is very different from basis-set expansion techniques to be discussed for molecules in the next chapter. The introduction of a one-particle basis set provides so-called virtual spinors automatically in a Dirac-Hartree-Fock-Roothaan calculation, which are not produced by the direct and fully numerical grid-based approaches. 

Our general procedure is to represent the atoms in a molecule using the Hartree-Fock orbitals of the individual atoms occurring in the molecule. (We will also consider the interaction of molecular fragments where the Hartree-Fock orbitals of the fragments are used.) These are obtained with the above bases in the conventional way using Roothaan s RHF or ROHF procedure[45], extended where necessary. 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

Carsky, R, Hubac, I., and Staenamler, V., Correlation energies in open shell systems. Comparison ofCEPA, PNO—Cl, and perturbation treatments based on the restricted Roothaan-Hartree-Fock formalism, Theor. Chim. Acta 60, 445—450 (1982). 

As will be seen from this account, Pople s simplification of Roothaan s self-consistent theory was aimed at clarifying the description of the ground state, particularly of alternant hydrocarbons.65 It was not concerned, in the first instance, with the characteristic problems which arise when one is discussing excited electronic states. However, Pople and Hush63 developed from it a theory of the ionization potentials and electron affinities of aromatic hydrocarbons, based on a consideration of the Hartree-Fock eigenvalues Ef. According to Koopmann s theorem, the ionization potential of a closed shell should approximate to the... 

Determinantal MO s may be obtained by a large number of computational methods based on Roothaan s self-consistent field formalism 94> for solving the Hartree-Fock equation for molecules which differ in degree of sophistication as regards the completeness and kind of the set of starting atomic wave functions, as well as the completeness of the Hamiltonian used 9S>. So a chain of various kinds of approximations is available for calculations starting from different ways of non-empirical ab initio" calculations 96>, viasemiempiricalmethods for all-valence electrons with inclusion of electronic interaction 95-97>98)... 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

Ab initio calculations are based on iterative procedures and provide the basis for self-consistent field-molecular orbital SCF-MO) methods. Electron-electron repulsion is specifically taken into account. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved in the Roothaan method by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved. 

---