Hartree-Fock-Roothaan Orbitals

Following Roothaan s proposal, the Hartree-Fock orbitals are usually represented as linear combinations of a set of known basis functions xl - 

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. 

Commonly, one uses normalized Slater-type orbitals 

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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... 

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

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... 

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)) ... 

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... 

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, ... 

Mpller-Plesset perturbation theory (MPPT) uses the orbitals and orbital energies obtained from a closed-shell Hartree-Fock-Roothaan (HFR) calculation. The HFR (or canonical) orbitals correspond to the eigenvectors of the inactive Fock matrix... 

Hartree-Fock-Roothaan SCF theory, using molecular orbitals expanded as linear combinations of atomic orbital basis functions (LCAO-MO theory)... 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

At each point r, the electronic density p(r,K) of a molecule of nuclear conformation K can be computed by the Hartree-Fock-Roothaan-Hall SCF LCAO ab initio method. Using a basis set cp(K) of atomic orbitals (pj(r,K)... 

This is closely analogous to the Hartree equations (Eq. (1.7)). The Kohn-Sham orbitals are separable by definition (the electrons they describe are noninteracting) analogous to the HF MOs. Eq. (1.50) can, therefore, be solved using a similar set of steps as was done in the Hartree-Fock-Roothaan method. 

This is a secular equation whose roots give the orbital energies. These Hartree-Fock-Roothaan (HER) equations must be solved by an intera-tive process, since the Fjk integrals depend on the orbitals <]) which in turn depend on the unknown coefficients c, . 

Table 4.1. Comparison of orbital ionization potentials (in eV) for SiO observed experimentally from uv photoelectron spectroscopy with those calculated by the MS-SCF-Ya method, ab initio Hartree-Fock-Roothaan method, incorporating the ASCF approach (HFR ASCF), and with perturbation theory (HFR + PT)...

Additional information on orbital type and composition is available from (e,2e) or electron momentum spectroscopy (Moore et al., 1982 see Appendix B) performed on Sip4 by Fantoni et al. (1986). Electron momentum distributions measured at various binding energies have been compared with those from ah initio Hartree-Fock-Roothaan SCF calculations using a double- wave function with a single Si 3of polarization... 

Table 4.10. The Si-O bond lengths (R(Si-O)] and vibrational frequencies calculated" using Hartree-Fock-Roothaan (SCF) molecular-orbital methods with different basis sets and compared with experimental properties of Si(OH)4 in Dj, symmetry...

Table 4.11. Hartree-Fock-Roothaan SCF molecular-orbital calculations of Si-O-Si angles (in degrees) for (SiH l O and (OH)3SiOSi(OH)3...

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"...

There has been considerable effort directed towards the assessment of the covalency or ionicity of various solids. The output of standard ab initio (SCF) Hartree-Fock-Roothaan calculations contains Mulliken (1955) charge distribution analysis parameters such as the atomic-orbital populations, net atomic charges, and bond overlap populations deseribed earlier (Chapter 3), which are often used to discuss the relative covalency or ionicity of materials. Considerable caution is required in using such parameters, however, since net atomic charges and other such quantities are not quantum-mechanical observables that is, they cannot even in principle be measured, and are highly basis-set dependent, as noted by Hehre et al. (1986 pp. 336-41). This is illustrated for molecules more relevant to mineralogy in Table 7.1, in which a number of properties of CO2 and SiOj are shown calculated at various basis-set levels. It is clear... 

The set of mathematical functions used to expand the molecular orbitals in a Hartree-Fock-Roothaan calculation... 

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