INDEX Hartree-Fock

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. 

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. 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant,... 

We recognize this as the normal Fock matrix in closed shell Hartree-Fock theory, but only for those matrix elements where the second index, i, corresponds to an occupied orbital. The first index is arbitrary. The condition for an optimum wave function is then Fpi = Fip. This is trivially fulfilled when also p corresponds to an occupied orbital, since the wave function is invariant towards rotations among these orbitals. On the other hand, when p is a virtual orbital we have Fip = 0 (all density matrix elements in (4 42) are then zero). 

A theoretical evaluation of the aromaticity of the pyrones pyromeconic acid, maltol, and ethylmaltol along with their anions and cations was carried out at several levels (Hartree-Fock, SVWN, B3LYP, and B1LYP) using the 6-311++G(d,p) basis set <2005JP0250>. The relative aromaticity of these compounds was evaluated by harmonic oscillator model of aromaticity (HOMA), nucleus-independent chemical shifts (NICSs), and /6 indexes and decreases in the order cation > neutral molecule > anion. 

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],... 

Hartree-Fock Scheme.- For the sake of simplicity, we will consider the eigenvlue relations (2.3) for a non-degenerate eigenvalue X, for which 0. If Ca and Da are two approximate solutions - characterized by the index "a"- the best approximations are obtained by using the bi-variational principle (B.3.6). In the Hartree-Fock scheme for N fermions, the approximate solutions are represented by two Slater determinants ... 

Let us now consider the special case when = T = T, i.e. when the original operator T is self-adjoint and real. In the conventional Hartree-Fock scheme - in the following indicated by an index c (=conventional) - which is based on relation (3.48), one has pc = pc and Teff c = Teff c, and it is obvious that the classical canonical form tc must be diagonal with only real eigenvalues. 

Here is the exact total energy of the system, are solutions of the Hartree-Fock problem, and e is the sum of Hartree-Fock orbital energies over occupied spinorbitals. Then the eigenvalue in eqn. (4.66), E, becomes directly the correlation energy in the i th electronic state. Since our concern is focused on the ground state, i.e. i B 0, the index i in eqn, (4.70) may be dropped and the respective contributions to the correlation energy can be expressed as... 

I. Flamant et al. successfully applied the FGSO basis set in Fourier Space Restricted Hartree Fock (FS-RHF) in a study of identification of conformational signatures in valence band of polyethylene. In 1998, they used a distributed basis set of s-type Gaussian function (DSGF) in FS-RHF. The method briefly is to use RHF-Bloch states (p (k,r), which are doubly occupied up to the Fermi energy Ep and orthonormalized. k and n the wave number and the band index, respectively. 

An important feature of the 1-CSE is that it is exactly satisfied by at least two sets of RDM s [6,7,13,19], the set corresponding to the Hartree-Fock solution and the set corresponding to the FCI solution. In what follows, the set of HF matrices, as well as the corresponding energy will be distinguished from those corresponding to the exact FCI solution by an upper index ( ). Thus the HF form of Eq.(14) is written... 

For Eq. (11) S is the Bragg vector S = 2ttH, IT is the row vector (htk,l) and the scalar S - S = 4ir sin 0/A. The index / covers the N atoms in the unit cell. The atomic scattering factor f (S) is the Fourier-Bessel transform of the electronic, radial density function of the isolated atom. This density function is usually derived from a spin-restricted Hartree-Fock wave function for the atom in its ground state. The structure fac-... 

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. 

For a free electron gas, it is possible to evaluate the Hartree-Fock exchange energy directly [3,16], The Slater determinant is constmcted using free electron orbitals. Each orbital is labelled by a k and a spin index. The Coulomb... 

The main computational attraction of the SCF method is that it reduces to a one-electron matrix problem the manipulations are those of matrices of the size of the number of basis functions. This means, for example, that any orbital transformations which may have to be done involve transformations of matrix representations of one-electron operators no time-consuming transformations of the four-index electron-repulsion integrals have to be done explicitly. The electron-repulsion terms are all contained in the Hartree-Fock matrix via G or,... 

To define Kab and more general multicenter indices at a post Hartree-Fock level, one must invoke more sophisticated techniques [64-67]. We follow our work [36] in which the full Cl and some approximated models were considered in detail. The conventional definition of the generalized bond index is based on identification of Kab with a charge density fluctuation measured via the second-order joint statistical moment [64] (generally, the joint cumulant) ... 

We can obtain a matrix equation for the C i by substituting the linear expansion (3.133) into the Hartree-Fock equation (3.132). Using the index V, gives... 

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