Illustrating correlation methods for the beryllium atom

Since beryllium only has four electrons, CISDTQ is a full Cl treatment and completely equivalent to a CCSDTQ calculation. The multi-reference character displays itself as a relatively slow convergence of the perturbation series, with millihartree accuracy being attained at the MP6 level and inclusion of terms up to MP20 is required in order to converge the energy to within 10 au of the exact answer. Note also that the correlation energy is overestimated at order seven, i.e. the perturbation series oscillates at higher orders. The contribution from triply excited states is minute, as expected 

Excitation Type CISDTQ tcCSDTQ fcCSDTQ tcCSDTQ 

The coefficients in the CISDTQ and CCSDTQ wave functions (using intermediate normalization) for the dominating excitations are given in Table 4.6. 

There is little difference between the full Cl and CCSD coefficients for the three most important doubly excited states. The quadruply excited states enter the Cl wave function with non-negligible weights, contributing 4% of the correlation energy (Table 4.5), but as shown in the last column of Table 4.6, these contributions are estimated very well by the product terms in the CC wave function. The CCSD energy in Table 4.5 and the u amplitude in Table 4.6 show that the quadruply excited states in the Cl wave function are mainly of the product type, and not a true quadruply excited state. It is this feature that makes CC superior to Cl-based methods. 

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