Hartree Fock average potential

We determined the coefficients in eqs. (17) and (18) by fitting to general databases of experimental AHvap and AHsub, using Hartree-Fock electrostatic potentials [13,14] however Rice et al reparametrized these equations at the B3LYP/6-31G level in terms of data pertaining specifically to energetic compounds [85]. Their average absolute deviations for AHvap and AHsub were 1.2 and 2.7 kcal/mole, respectively. 

The SCF-Xa-SW method makes the following major assumptions. First one replaces the nonlocal Hartree-Fock exchange potential with the Xa local exchange potential that corresponds to the average exchange potential of a free electron gas. The exchange potential is related to the local electronic density by... 

The perturbation H is the difference between the true interelectronic repulsions and the Hartree-Fock interelectronic potential (which is an average potential). 

The last term in Eq. 11.47 gives apparently the "average one-electron potential we were asking for in Eq. 11.40. The Hartree-Fock equations (Eq. 11.46) are mathematically complicated nonlinear integro-differential equations which are solved by Hartree s iterative self-consistent field (SCF) procedure. 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

It is also reasonable that using the average value of the interelectronic potential should correct the energy only to the Hartree-Fock level. There is no correlation between the two electrons in T°, and Equation 11.18 introduces none. However, the instantaneous value of (Vr12) does introduce correlation. Note that the correlation is in p. and not in T0, which is unchanged. 

The penetration contribution to the electrostatic potential at R, is evaluated by application of the general expression of Eq. (8.49) for per for the spherical density (lt = ml = 0). The point-charge term, proportional to 1/Rfj-, must subsequently be subtracted. Due to the rapid decrease of the penetration terms with increasing R j, convergence is quickly achieved. For spherically averaged Hartree-Fock atom densities, inclusion of penetration terms for atoms within 10 A of the point under consideration is more than adequate. 

Now consider the normalized Hartree-Fock spatial orbitals, namely, 4>is, 4>is, , with energies ei, 25,..., respectively, occupied by v, (=0, 1, or 2) electrons. The potential energy of an electron with energy e, includes, on average, the repulsion between this electron and all the other electrons. The sum of the orbital energies Vi i thus counts each interelectronic repulsion twice. The Hartree-Fock energy... 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

Jj( 1) is the potential energy of interaction between the point charge of electron 1 and electron 2 considered to be smeared out into a hypothetical charge cloud of charge density (charge per unit volume) - e)Hartree-Fock method considers average interelectronic interactions, rather than instantaneous inter-... 

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