Energy Hartree-Fock

The sum of the zeroth-order and first-order energies thus corresponds to the Hartree-Fock energy (compare with Equation (2.110), which gives the equivalent result for a closed-shell system) ... 

To obtain an improvement on the Hartree-Fock energy it is therefore necessary to use Moller-Plesset perturbation theory to at least second order. This level of theory is referred to as MP2 and involves the integral J dr. The higher-order wavefunction g is... 

In a higher level energy calculation, values for the energy computed using the more accurate procedure appear shortly after the Hartree-Fock energy. Here is the output from a formaldehyde calculation done at the MP2 level (RMP2 replaces RHF in the route section ... 

Adding E and E yields the Hartree-Fock energy (since Hg+V is the full Hamiltonian) ... 

Thus, the value of E the first perturbation to the Hartree-Fock energy, will always be negative. Lowering the energy is what the exact correction should do, although the Moller-Plesset perturbation theory correction is capable of overcorrecting it, since it is not variational (and higher order corrections may be positive). 

The extrapolation to the complete basis set energy limit is based upon the MoUer-Plesset expansion E= + E + E + E + E +. .. as described earlier in this appendix. Recall that E + E is the Hartree-Fock energy. We will denote E and all higher terms as E , resulting in this expression for E ... 

According to Eqs. II. 11 and 11.44, the Hartree-Fock energy of our molecular system is now given by the formula... 

In considering the Hartree-Fock energy Z HF given by Eq. 11.45, we observe that, even for atoms, there are actually only a few such energies published in the literature and that, even in simple cases, there may be discrepancies between the results of different authors. The point is that the atomic Hartree-Fock functions are usually tabulated with only three decimal figures, and this numerical accuracy is often not sufficient for obtaining the accuracy desired... 

The first orbital X is very closely related to the ordinary SCF function, which is also shown by comparing the total energy of the wave function (xx)2 with the Hartree-Fock energy ... 

The only nonvanishing matrix elements HKL associated with the SCF determinant will thus be the diagonal element, which is identical with the Hartree-Fock energy, and the interaction elements with the doubly excited determinants... 

The results show that it is possible to improve the Hartree-Fock energy —2.86167 at.u. considerably by means of a simple correlation factor, but also that it is essential to scale the total function W properly to fulfil the virial theorem. The parameters in the best function u of the form of Eq. III. 121 are further given below ... 

The CHF formulae (75)-(77) for the response tensors are established by expanding the Hartree-Fock energy [8],... 

Table 2 Restricted Hartree-Fock energies (Hartrees) for Ceo and C70 and their muon adducts. AE is the difference in energy between the carbon allotrope and its adduct. In all cases, except where indicated by f, only the six carbon atoms in the immediate vicinity of the muon have had there positions optimised, f means that a full geometry optimisation has been carried out. The type specifies the defect and for C70 is identified in Table 1. is the spin density at the muon in atomic units (and the hyperfine coupling constant in MHz). JMuon constrained to lie in equatorial plane. indicates geometry not fully optimized.

Table 3 Unrestricted Hartree-Fock energies (Hartrees) for Ceo and C70 and their muon adducts, muon in atomic units (and MHz) after quartet spin state annihilation. The value of the total spin operator, < >, is also given after quartet spin state...

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

With sufficiently large basis set, the Hartree-Fock (HF) method is able to account for 99% of the total energy of the chemical systems. However, the remaining 1% is often very important for describing chemical reaction. The electron correlation energy is responsible for the same. It is defined as the difference between the exact nonrelativistic energy of the system ( 0) and Hartree-Fock energy (E0) obtained in the limit that the basis set approaches completeness [36] ... 

In the second row of Table 1.4, we have listed the corrections to the Hartree-Fock energies that are obtained from CCSD calculations. Clearly, we now have a better description of the atomization process, the error in the calculated AE being only -19.6 kJ/mol (2 %). Still, we are far away from the prescribed target accuracy of 1 kJ/mol. 

Hartree-Fock convergence does not present an insurmountable difficulty there are clear indications that the molecular Hartree-Fock energy converges as exp (—aX) and thus rather rapidly [51-53],... 

Hartree-Fock energy 227 Hartree-Fock molecular orbitals 224 Hartree-Fock theory 229 helical domains 94 heroin 81... 

The corresponding Hartree-Fock energies hf are shown from Ref. [61], All quantities are given in atomic units. 

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