Exact correlation energy

Eqs. (20), (24), and (25), known as the Kohn-Sham equations, are formally exact and contain only one unknown term, EUp]- It is Exc that is approximated in Kohn-Sham DPT, not the conventional exchange-correlation energy Exact treatment of the kinetic energy as an orbital-dependent functional is cmcial to the practicality of this scheme because TTp] and Tsip] are notoriously difficult to approximate as explicit... 

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

The main difference between the G1/G2 and CBS methods is the way in which they try to extrapolate the correlation energy, as described below. Both tire G1/G2 and CBS methods come in different flavours, depending on the exact combinations of metliods used for obtaining the above four contributions. 

The variation principle tells us that EHF is always larger than the exact energy, and this implies that the total correlation energy is always a negative quantity. From Eq. 11.70 it follows then that 7 corr is always positive, whereas VC0TT is negative according to the formulas ... 

According to Eq. 11.67, the correlation energy is simply defined as the difference between the exact energy and the energy of the Hartree-Fock approximation. Let us repeat this definition in a more precise form ... 

The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. 

In molecular applications the calculation of the HF energy is a still more difficult problem. It should be observed that, in the SCF-MO-LCAO now commonly in use, one does not determine the exact HF functions but only the best approximation to these functions obtainable within the framework given by the ordinarily occupied AO s. Since the set of these atomic orbitals is usually very far from being complete, the approximation may come out rather poor, and the correlation energy estimated from such a calculation may then turn out to be much too large in absolute order of magnitude. The best calculation so far is perhaps Coulson s treatment of... 

It should be observed that the subscript exact here refers to the lowest eigenvalue of the unrelativistic Hamiltonian the energy is here expressed in the unit Aci 00(l+m/Mz) 1 and Z is the atomic number. If the HE energies are taken from Green et al.,8 we get the correlation energies listed in the first column of Table I expressed in electron volts. The slow variation of this quantity is noticeable and may only partly be understood by means of perturbation theory. 

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. 

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