Roothaan Hartree-Fock

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... 

At the other end of the spectrum are the ab initio quantum molecular models, which are rigorous within the Hartree-Fock/ Roothaan—Hall (HF/RH) formalisms. Electronic structure is calculated, and dependent properties are derivable. In theory, full reaction profiles can be modeled. In practice, however, their speed makes it impractical to apply the more accurate... 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

A Consistent Calculation of Atomic Energy Shell Corrections Strutinsky s Method in the Hartree-Fock-Roothaan Scheme... 

Formulation of Strutinsky s method in the Hartree-Fock-Roothaan scheme... 

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

This representation permits analytic calculations, as opposed to fiiUy numerical solutions [47,48] of the Hartree-Fock equation. Variational SCF methods using finite expansions [Eq. (2.14)] yield optimal analytic Hartree-Fock-Roothaan orbitals, and their corresponding eigenvalues, within the subspace spanned by the finite set of basis functions. 

Theoretical quantum-chemical study of pyridine adsorption at Hg electrode (including its charged surface) has been described by Man ko et al. [137,138]. An ab initio Hartree-Fock-Roothaan method has been employed. The electrode was modeled as a planar seven-atomic Hg-7 cluster. The deepest minimum of the total energy of the adsorption system was found for positive charge density and Py interacting with the metal through the lone electron... 

These O, are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5)), a simple set of equations (the Hartree-Fock-Roothaan equations) is obtained which can be used to determine the optimum coefficients Cti. For those systems where the space part of each MO is doubly occupied, i.e. there are two electrons in each 0, with spin a and spin respectively so that the complete MOs including spin are different, the total wavefunction is... 

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)) ... 

We now consider the PPP, CNDO, INDO, and MINDO two-electron semiempirical methods. These are all SCF methods which iteratively solve the Hartree-Fock-Roothaan equations (1.296) and (1.298) until self-consistent MOs are obtained. However, instead of the true Hartree-Fock operator (1.291), they use a Hartree-Fock operator in which the sum in (1.291) goes over only the valence MOs. Thus, besides the terms in (1.292), f/corc(l) m these methods also includes the potential energy of interaction of valence electron 1 with the field of the inner-shell electrons rather than attempting a direct calculation of this interaction, the integrals of //corc(/) are given by various semiempirical schemes that make use of experimental data furthermore, many of the electron repulsion integrals are neglected, so as to simplify the calculation. 

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. 

Thus four of the seven lowest H20 MOs are linear combinations of the four a, symmetry orbitals listed above, and are a, MOs similarly, the two lowest b2 MOs are linear combinations of 02p and H,1j — H21.s, and the lowest bx MO is (in this minimal-basis calculation) identical with 02px. The coefficients in the linear combinations and the orbital energies are found by iterative solution of the Hartree-Fock-Roothaan equations. One finds the ground-state electronic configuration of H20 to be... 

Calculations of the electronic structure of the phosphorus molecule were carried out by ab initio MO LCAO Hartree-Fock-Roothaan SCF method in the restricted (RHF), restricted-open (ROHF) or unrestricted Hartree-Fock (UHF)... 

Now we have FC = SCe (5.57), the matrix form of the Roothaan-Hall equations. These equations are sometimes called the Hartree-Fock-Roothaan equations, and, often, the Roothaan equations, as Roothaan s exposition was the more detailed and addresses itself more clearly to a general treatment of molecules. Before showing how they are used to do ab initio calculations, a brief review of how we got these equations is in order. 

Fig. 5.6 Summary of the steps in the Hartree-Fock-Roothaan-Hall SCF procedure...

Lee, C.M. (1974). Spectroscopy and collision theory. III. Atomic eigenchannel calculation by a Hartree-Fock-Roothaan method, Phys. Rev. A 10, 584-600. 

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. 

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... 

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