Correlation energy magnitude

The magnitude of the core correlation can be evaluated by including the oxygen Is-electrons and using the cc-pCVXZ basis sets the results are shown in Table 11.9. The extrapolated CCSD(T) correlation energy is —0.370 a.u. Assuming that the CCSD(T) method provides 99.7% of the full Cl value, as indicated by Table 11.7, the extrapolated correlation energy becomes —0.371 a.u., well within the error limits on the estimated experimental value. The core (and core-valence) electron correlation is thus 0.063 a.u.. 

The effect of core-electron correlation is small, as shown in Table 11.16. It should be noted that the valence and core correlation energy per electron pair is of the same magnitude, however, the core correlation is almost constant over the whole energy surface and consequently contributes very little to properties depending on relative energies, like vibrational frequencies. It should be noted that relativistic corrections for the frequencies are expected to be of the order of 1 cm" or less. ... 

Let us now try to get an idea of the order of magnitude of the unrelativistic correlation energy for some actual systems. In considering the difference... 

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... 

In order to get an estimate of the order of magnitude of the correlation energy, Froman makes it plausible that the correlation energy should be roughly proportional to the total Coulomb repulsion energy of all the electrons within the system, and he suggests the formula... 

The higher-order contributions to the correlation energy [such as CCSD(T)-MP2] are more than an order of magnitude smaller than their second-order counterparts. However, the basis set convergence to the CCSD(T)-R12 limit does not follow the simple linear behavior found for the second-order correlation energy. This is a consequence of the interference effect described in Eq. (2.2). The full Cl or CCSD(T) basis set truncation error is attenuated by the interference factor (Fig. 4.9). The CBS correction to the higher-order components of the correlation energy is thus the difference between the left-hand sides of Eqs. (2.2) and... 

We now turn to the problem of simplifying the recovery of the dynamic correlation energy. We consider the simplest situation, viz., where the zeroth-order wavefunction can be chosen as the SCF approximation. A challenging disparity exists between the energetic smallness of these refinements and the complexity and magnitude of the computational efforts required for their variational determination. In order to reduce this disproportion, various semiempirical approaches have been proposed (56-61), notably in particular the introduction of semiempirical elements into MP2 theory which has led to the successful Gn methods (62-64). 

It follows from the relation (79) that both contributions to the HF correlation energy AT + AE = E " are of comparable magnitude and partially cancel out, leading to... 

We see that the HF correlation energy Eg " is a very small quantity indeed - four orders of magnitude smaller than the exchange energy. When compared with the known results for the KS correlation energy Eg s E or the QM one [Eq. (27)], it happens to be two orders of magnitude smaller. 

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