Helium atom principal expansion

The principal expansion of the ground-state helium atom... 

Table 73 The convergence of the principal expansion of the ground-state energy of the helium atom. is the energy (in Eh) obtained in a basis containing all AOs of principal quantum number N and lower...

Fig. 7.10. The energy increments logio(—fi/v) of the principal expansion of the ground-state helium atom plotted against logic (A — 1 /2). The thick grey line represents the function C4 (Af — 1 /2) where the constant C4 has been determined by a fit to the calculated numbers. Atomic units are used.

Fig. 7.11. The helium ground-state wave function with one electron fixed at a position 0.5oo from the nucleus for different principal expansions 2

In Section 7.5, we investigated the principal expansion of the correlation energy in the helium atom. The numerical orbitals generated by this expansion have the same composition and nodal structure as the correlation-consistent basis sets cc-pVXZ. In the present subsection, we shall refer to the numerical orbitals of the principal expansion as the numerical correlation-consistent basis sets n-cc-pVXZ, by analogy with the standard analytical correlation-consistent basis sets cc-pVXZ. By comparing the analytical and numerical basis sets, we should obtain some impression of the performance of the standard cc-pVXZ sets and in particular of their deficiencies vis-d-vis a fully optimized set of correlating orbitals. 

To carry out the extrapolations, some simple analytical model for the convergence of the enei y must be assumed. In our discussion of the principal expansion of the helium atom in Section 7.5, we found that the error in the FCI energy obtained by neglecting all orbitals of principal quantum number n greater than N is given by the expression... 

For the helium atom, the correlation-consistent sets cc-pVYZ have the same composition as the orbitals of the principal expansion with N = X. Identifying the cardinal number X with N in (8.4.1), we then obtain the following simple expression for the error in the cc-pVYZ basis ... 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

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