Hartree-Fock-Roothaan approximation

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... 

If RCI expansions are used or orbitals are subdivided into inactive and active groups, or both, then variation of the orbitals themselves may lead to an essential energy decrease (in contrast to the FCI method where it does not happen). Such combined methods that require both optimization of Cl coefficients and LCAO coefficients in MOs are called MCSCF methods. Compared with the Cl method, the calculation of the various expansion coefficients is significantly more complicated, and, as for the Hartree-Fock-Roothaan approximation, one has to obtain these using an iterative approach, i.e. the solution has to be self-consistent (this gives the label SCF). 

The fundamental tool for the generation of an approximately transferable fuzzy electron density fragment is the additive fragment density matrix, denoted by Pf for an AFDF of serial index k. Within the framework of the usual SCF LCAO ab initio Hartree-Fock-Roothaan-Hall approach, this matrix P can be derived from a complete molecular density matrix P as follows. 

Considering the different calculated values for an individual complex in Table 11, it seems appropriate to comment on the accuracy achievable within the Hartree-Fock approximation, with respect to both the limitations inherent in the theory itself and also to the expense one is willing to invest into basis sets. Clearly the Hartree-Fock-Roothaan expectation values have a uniquely defined meaning only as long as a complete set of basis functions is used. In practice, however, one is forced to truncate the expansion of the wave function at a point demanded by the computing facilities available. Some sources of error introduced thereby, namely ghost effects and the inaccurate description of ligand properties, have already been discussed in Chapter II. Here we concentrate on the... 

The Hartree-Fock-Roothaan (HFR) scheme [24] consists in approximating the one-particle orbitals linear combinations of suitable basis functions... 

Within the framework of the 77-electron approximation Ea is assumed to be simply a constant and the expression for Ew is used to find the optimum 77-electron LCAO MOs, that is, the Hartree-Fock-Roothaan... 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

If the basis set used is finite and incomplete, solution of the secular equation yields approximate, rather than exact, eigenvalues. An example is the linear variation method note that (2.78) and (1.190) have the same form, except that (1.190) uses an incomplete basis set. An important application of the linear variation method is the Hartree-Fock-Roothaan secular equation (1.298) here, basis AOs centered on different nuclei are nonorthogonal. Ab initio and semiempirical SCF methods use matrix-diagonalization procedures to solve the Roothaan equations. 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

The idea to employ a finite basis set of AOs to represent the MOs as linear combinations of the former apparently belongs to Lennard-Jones [68] and had been employed by Hiickel [37] and had been systematically explored by Roothaan [38]. That is why the combination of the Hartree-Fock approximation with the LCAO representation of MOs is called the Hartree-Fock-Roothaan method. 

Application of ab initio MO theory usually begins at the monoconfigurational level, with the Hartree-Fock-Roothaan or LCAO-SCF methodology [4,5]. In this scheme the wave function for a closed-shell molecule containing N electrons is approximated as an antisymmetrized product (determinant) of spin-orbitals, ... 

Calculated values Hartree-Fock-Roothaan (using Koopmans Approximation) Basis Set STO-3G 14.7 15.6... 

Table 3.7. Calculated equilibrium structural properties of H,0 and NHj [bond lengths, / (0-H) and 7 (N-H) in A, bond angles

It has been suggested that virtually all samples of MgO (and probably silicates as well) contain small amounts of HjO and CO2 (Freund, 1981). These impurities lead to formation of O species. The species CO4 " has also been postulated to occur on the surface of MgO exposed to COj. This species has recently been studied by ab initio 8CF Hartree-Fock-Roothaan MO calculations, both in its anion form and as the protonated cluster C(0H)4 (Gupta et al., 1981). Its calculated equilibrium bond distance is intermediate between those observed for B emd N in tetrahedral coordination with oxygen, and there seems to be no intrinsic source of instability. Thus, such a species seems stable. However, the calculations indicate a charge on C in 04" " very similar to that in COj, arguing for the formulation O 04 as first approximation to the electronic structure, rather than the C°(04 ) formulation suggested by Freund (1981). Perhaps such a species is better formulated as a chemisorption complex of a anion (a bent, 18-valence-electron system) and an O. . . 0 ... 

The method is an approximate self-consistent-field (SCF) ab initio method, as it contains no empirical parameters. All of the SCF matrix elements depend entirely on the geometry and basis set, which must be orthonormal atomic orbitals. Originally, the impetus for its development was to mimic Hartree-Fock-Roothaan [5] (HFR) calculations especially for large transition metal complexes where full HFR calculations were still impossible (40 years ago). However, as we will show here, the method may be better described as an approximate Kohn-Sham (KS) density functional theory (DFT)... 

Whereas the concepts and method described in this contribution are equally applicable to various approximate and more advanced quantum-chemical representations, the basic concepts will be discussed and illustrated within the framework of the conventional Hartree-Fock-Roothaan-Hall SCF LCAO ab initio representation of molecular wave functions and electronic densities, as can be computed, for example, using the Gaussian family of computer programs of Pople and co-workers. The essence of the shape analysis methods will be discussed with respect to some fixed nuclear arrangement K note, however, that the generalizations will involve changes in the nuclear arrangement K. 

Abstract The momentum representation of the electron wave functions is obtained for the nonrelativistic hydrogenic, the Hartree-Fock-Roothaan, the relativistic hy-drogenic, and the relativistic Hartree-Fock-Roothaan models by means of Fourier transformation. All the momentum wave functions are expressed in terms of Gauss-type hypergeometric functions. The electron momentum distributions are calculated by the use of these expressions, and the relativistic effect is demonstrated. The results are applied for calculations of inner-shell ionization cross sections by charged-particle impact in the binary-encounter approximation. The reiativistic effect and the wave-function effect on the ionization cross sections are discussed. 

When the Roothaan equations (13.157) [or (13.179)] are solved exactly, the canonical MOs and the calculated values of molecular properties do not change if one changes the orientation of the coordinate axes the calculated values are said to be rota-tionally invariant Likewise, the results do not change if each basis AO on a particular atom is replaced by a linear combination of the basis AOs on that atom, and the results are hybridizntioruilly invariant When approximations are made in solving the Hartree-Fock-Roothaan equations, rotational and hybridizational invariance may not hold. 

All-electron (AE) calculations are certainly the most rigorous way to treat atoms and molecules, however, the computational requirements are sometimes prohibitive, especially for molecular Dirac-Hartree-Fock-Roothaan (DHFR) and subsequent configuration interaction (Cl) calculations. Nevertheless, very accurate all-electron calculations on small systems yield important reference data for the calibration of more approximate computational schemes, e.g. valence-electron (VE) methods, which may be applied to larger systems. 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

A variation function with variable orbital exponents in the 1 atomic orbitals making up the LCAOMOs ofEq. (20.3-5) gives Oe = 3.49 eV and re = 73.2 pm with an apparent nuclear charge equal to 1.197 protons (a larger charge than the aetual nuclear charge). A careful Hartree-Fock-Roothaan calculation gives De = 3.64 eV and = 74 pm." If this result approximates the best Hartree-Fock result, the correlation error is approximately 1.11 eV. 

In the second expression for the 2a orbital the values of the coefficients were chosen to maintain approximately the same relative weights of the atomic orbitals as in the Hartree-Fock-Roothaan orbital. Figure 20.16 shows the orbital region of the 2a LCAOMO and shows that it is a bonding orbital with overlap between the nuclei. It... 

This method was mentioned in Chapter 19, and has become a common method in quantum chemical research. Detailed discussion of it is beyond the scope of this book. It has been found that the approximation schemes that have been developed work at least as well as Hartree-Fock-Roothaan methods with configuration interaction for most molecular properties such as bond lengths and energies of molecular ground states. 

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