Roothan-Hartree-Fock

E. Clement and C. Roetti, Roothan-Hartree-Fock atomic wavefunctions,... 

The ground state 9, (0 of the neutral target atom is represented by a Roothan-Hartree-Fock wavefunction that may be expanded in terms of Slater orbitals [45]. Denoting the triply differential cross section by... 

The coefficients Z> and b in the above expressions come from the description of the Roothan-Hartree-Fock wavefunctions of the particular atom which may range from hydrogen to argon inclusively [45], The quantities A, Q.%, and x are common to both the p shell and the s shell in all atoms, yet they have specific values for the two theories. For the CDW model we have... 

Next turning to the 2p and 3p Roothan-Hartree-Fock orbitals, the post form of the square of the CDW/CDW-EIS scattering amplitudes may be given as... 

All the above methods are somehow based on an orbital hypothesis. In fact, in the multipolar model, the core is typically frozen to the isolated atom orbital expansion, taken from Roothan Hartree Fock calculations (or similar [80]). Although the higher multipoles are not constrained to an orbital model, the radial functions are typically taken from best single C exponents used to describe the valence orbitals of a given atom [81]. Even tighter is the link to the orbital approach in XRCW, XAO, or VOM as described above. Obviously, an orbital assumption is not at all mandatory and other methods have been developed, for example those based on the Maximum Entropy Method (MEM) [82-86] where the constraints/ restraints come from statistical considerations. 

E. dementi and C. Roetti, Roothan-Hartree-Fock atomic wave functions. Basic functions and their coefficients for ground and certain excited states and ionized atoms, Z < 54. Atomic Data, Nuclear Data Tables 14, 177-478 (1974). 

Since the exact solution of the Hartree-Fock equation for molecules also proved to be impossible, numerical methods approximating the solution of the Schrodinger s equation at the HF limit have been developed. For example, in the Roothan-Hall SCF method, each SCF orbital is expressed in terms of a linear combination of fixed orbitals or basis sets ((Pi). These orbitals are fixed in the sense that they are not allowed to vary as the SCF calculation proceeds. From n basis functions, new SCF orbitals are generated by... 

The Hamiltonian, Hea, which is called the Hartree-Fock-Roothan operator is a 1-electron operator whose application yields the energy of an electron moving in the average field of the other electrons and nuclei. In principle an SCF theory approach will lead to a well-defined expression for Hett for closed and open shell systems (188, 189), and with the aid of modern computers Hm integrals can be evaluated numerically even for transition metal complexes. This type of ab initio calculation has been reported for a reasonable number of organometallic complexes of first-row transition elements by Hillier, Veillard, and their co-workers (48, 49, 102, 103, 111-115 58, 68, 70, 187, 228, 229). 

The many-electron theory (e.g., Hartree-Fock-Roothan) of molecular systems within the LCAO approximation involves the electron repulsion integrals of the type which represent... 

Sum Over States (SOS). This method computes molecular orbitals, from which values for transition fi equencies may be calculated. First the electronic ground state of the molecular system is determined, after which one may apply either the Hartree-Fock-Roothan method or the LCAO (hnear combination of atomic orbitals) approximation. Then one accoimts for correlations by configuration interaction calculations to form the lowest-energy excited states and transition dipole moments of the molecule. Finally transition frequencies and dipole moments are employed along with the formulas for the hyperpolarizabilities. The SOS method needs as input, energies and transition moments for excited states. It yields Pico) directly (eq. 1) identification of contributing excited states is important. 

Cohen, H. D. and Roothan, C. C. J. 1965. Electric dipole polarizahility of atoms by the Hartree-Fock method. I. Theory for closed-shell systems. J. Chem. Phys. 43 S34-S39. 

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