Extrapolations from Principal Expansions

Although the convergence of the FCI principal expansion is slow, it is systematic [12], In fact, for a sufficiently large basis, it has been found that each STO in the FCI principal expansion of He contributes an amount of energy that, to a good approximation, is given by the expression [47, 48] 

Note that the energy contribution depends only on the principal quantum number n. Therefore, each of the n2 orbitals that constitute the shell n contributes the same amount of energy, justifying the use of the principal expansion. Summing the energy contributions from all orbitals, we obtain 

By summing the contributions from all neglected shells, it is now easy to estimate the error that arises when the principal expansion is truncated after n = X  

This empirical result is consistent with the theoretical analysis of the partial-wave expansion (where the truncation of the FCI expansion is based on the angular-momentum quantum number rather than on the principal quantum number n), for which it has been proved that the truncation error is proportional to L-3 when all STOs up to l = L are included in the FCI wavefunction [49, 50], 

Guided by Eq. (5.12), we assume that the calculated He energy is well represented by the expression 

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For example, if we carry out calculations with X = 3 and Y = 4 using optimized numerical orbitals (i.e., no longer simple STOs), we obtain errors in the energy of 4.9 and 2.1 kJ/mol, respectively. The error in the energy extrapolated from these two results using Eq. (5.14) is less than 0.1 kJ/mol, which would require a FCI principal expansion with X = 10 or more. 

Calculation of the state-dependent nonlogarithmic contribution of order a(Za) is a difficult task, and has not been done for an arbitrary principal quantum number n. The first estimate of this contribution was made in [63]. Next the problem was attacked from a different angle [64, 65]. Instead of calculating corrections of order a(Za) an exact numerical calculation of all contributions with one radiative photon, without expansion over Za, was performed for comparatively large values of Z (n = 2), and then the result was extrapolated to Z = 1. In this way an estimate of the sum of the contribution of order a(Za) and higher order contributions a(Za) was obtained (for n = 2 and Z = 1). We will postpone discussion of the results obtained in this... 

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