Hartree-Fock calculations INDEX

Suppose states D > and A > are one-determinant many-electron functions, which are written in terms of (real) molecular orbitals and where a is the spin index, a a, p. These are the optimized canonical orbitals obtained from Hartree-Fock calculations of states D and A. Using the standard rules of matrix element evaluations[18], one can obtain an appropriate expression for Eq. (1) in terms of MO s of the system. 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

If the Kohn-Sham orbitals [52] of density functional theory (DFT) [53] are used instead of Hartree-Fock orbitals in the reference state [54], the RI can become essential for the realization of electron propagator calculations. Modern implementations of Kohn-Sham DFT [55] use the variational approximation of the Coulomb potential [45,46] (which is mathematically equivalent to the RI as presented above), and four-index integrals are not used at all. A very interesting example of this combination is the use of the GW approximation [56] for molecular systems [54],... 

M. Towler, An introductory guide to Gaussian basis sets in solid state electronic structure calculations, http //www.orystal.unito.it/tutojan2004/tutorials/index.html A.D. McLean, R.S. McLean, Roothaan-Hartree-Fock atomic wave functions (Slater basis-set expansions for Z=55-92),... 

The preceding step to both MP2 and coupled-cluster calculations is to solve the Hartree-Fock equations. The standard approach is, of course, to solve the equations in a basis set expansion (Roothaan-Hall method), using atom-centered basis functions. This set of basis functions is used to expand the molecular orbitals and we will call it orbital basis set (OBS). It spans the computational (finite) orbital space. Occupied spin orbitals will be denoted (pi and virtual (unoccupied) spin orbitals pa- In order to address the terms that miss in a finite OBS expansion, the set of virtual spin orbitals in a formally complete space is introduced, pa- If we exclude from this space all those orbitals which can be represented by the OBS, we obtain the complementary space, with orbitals denoted cp i. The subdivision of the orbital space and the index conventions are summarized in the left part of Fig. 2. 

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