The CCSD Valence Correlation Component of TAE

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. 

For all intents and purposes then, we are concerned here with the CCSD (coupled cluster with all single and double substitutions [30]) correlation energy. Its convergence is excruciatingly slow Schwartz [31] showed as early as 1963 that the increments of successive angular momenta l to the second-order correlation energy of helium-like atoms converge as 

His conclusions were generalized to other methods and general pair correlation energies by Hill [32] and by Kutzelnigg and Morgan [33], 

The denominator shift of 1/2 was chosen as a compromise between the situation for hydrogen and helium (where n = 1 + 1 for the cc-pVnZ basis set) and main-group elements (where n = 1). As is immediately obvious upon series expansion, there is considerable coupling between the denominator shift and the exponent. As a result, the three-point extrapolation generally leads to exponents well in excess of three [34], 

For the smaller basis sets used in W1 theory, the regime where the leading Eoo + A/L3 term dominates convergence behavior has not yet been reached, and using the formula in its unmodified form leads to overestimated (in absolute value) CCSD limits. One unelegant solution would be the use of three-term extrapolations like Eoo + A/L3 + B/L4, but in light of the poor quality of the VDZ basis set this is a most unsatisfactory alternative. Another alternative is the use of a two-point extrapolation Eoo + A/LQ, in which a is a fixed empirical parameter. By minimizing the deviation from the W2 CCSD limit for the so-called W2-1 set of 28 molecules (vide infra), we determined a = 3.22, which is the value used in W1 theory and its variants. 

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Extrapolation of the CCSD valence correlation component of TAE from the medium and large basis sets (Wl) or from the large and extra-large basis sets (W2) employing the two-point formula E(n) = Eoo + A/Ba, where a = 3.22 (Wl) or 3 exactly (W2). 

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