Principal expansion

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

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 ... 

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. 

Figure 13 Thermal expansion of TEA(TCNQ)2. The largest principal expansion ax is along the chain axis at any temperature. Above 200 K a2 and a3 are roughly parallel and perpendicular to the N—H N interchain interactions below 200 K, a2 and a3 are directed roughly along the transverse and elongation axes of the average bonded molecular species. (From Ref. 23.)...

A similar conclusion holds for the principal expansion (the orbital space is expanded by successively including all orbitals with the same principal quantum number ri), which is the rationale behind the correlation-consistent basis sets of Dunning, Peterson, Woon and co-workers. As shown by Carroll et al and Kutzelnigg , each shell n contributes a correlation energy increment azn, which again leads to a decay of the residual error ccn. This relation was also observed empirically for molecules, when correlation-consistent basis sets, e.g. the cc-pVXZ series, are employed. 

We shall refer to this expansion of the helium wave function as the principal expansion. The first plotted point corresponds to the expectation value of the Is Hartree-Fock state... 

The numerical optimization of the orbitals reduces the error in the energy (by a factor of 3-4 at each level) but does not improve substantially on the intrinsically slow convergence of the Cl expansion. A more detailed discussion of the convergence of the principal expansion (7.3.9) is given in Section 7.5. In routine molecular calculations, we employ AOs more flexible than the single-zeta STOs in (7.3.6) but less flexible than the numerical orbitals in (7.3.9) - in practice, the results are close to those of the numerical Cl expansion in Figure 7.6, see Section 8.4.2. 

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

The convergence rate of the partial-wave expansion discussed in Section 7.4 is theoretically interesting but difficult to realize in practice since, for each partial wave, an infinite expansion in orbitals is required. In the Cl framework, a more useful approach is to consider a series of calculations where a (reasonably) fast convergence of the correlation energy is obtained in a small orbital space. This is achieved by means of the principal expansion discussed in the present section. 

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...

In Table 7.3, the total energies of the principal expansion are listed, together with the energy contributions from each new shell ... 

In Figure 7.6, the curve marked as numerical O corresponds to the energies obtained by the principal expansion in this table. Comparing the energies in Table 7.3 with those in Table 7.1, we find that, at each level iV = L -I-1, the energy is lower for the partial-wave expansion than for the principal expansion. This difference is expected since the partial-wave expansion L +1 contains - in addition to all the determinants included in the principal expansion N - all other determinants with / single-determinant Hartree-Fock wave function, whoeas at the first level of the partial-wave expansion we employ a Cl wave function in the complete set of s orbitals. 

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.

We have seen that the partial-wave and principal expansions give energy increments that are inversely proportional to the fourth power of L and N, respectively - see (7.4.13) and (7.5.2). This behaviour may be understood by considering the energy contributions from the individual orbitals [12]. 

Thus, in this approximation, the contribution to the energy is independent of the angular momentum and inversely proportional to the sixth power of the principal quantum number. We shall now use this expression to predict the energy increments for a given L in the partial-wave expansion and for a given N in the principal expansion, summing the contributions from all orbitals added at each level as illustrated in Figure 7.9. 

Next, recalling that there are orbitals of a given principal quantum number N, we obtain for the energy increments of the principal expansion... 

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. 

In Table 8.17 the correlation energies obtained with the cc-pVXZ and n-cc-pVXZ sets are listed for X < 5. For these cardinal numbers, the numerical sets recover 86-99% of the correlation energy, whereas the analytical sets recover 77-99%. Thus, at the double-zeta level, the numerical n-cc-pVDZ set recovers about 8.5% more of the energy than does the corresponding analytical set for higher cardinal numbers, the differences are 2.6%, 0.7% and 0.2%. The cc-pVXZ sets therefore recover nearly all the correlation energy available in the principal expansion beyond the triple-zeta level. 

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 ... 

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