Pair Natural Orbital Extrapolations

Twenty years ago, we extended asymptotic extrapolations to polyatomic molecules by transformation of the Schwartz formulae to a symmetry-independent form based on the total number N of pair natural orbitals (PNOs) [8-10, 14], 

The algorithm employed for these PNO extrapolations selects the value of N giving the largest (i.e. most negative) value for the CBS pair energy with the constraint of N Nmin. Convergence to the exact CBS 

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Something called a pair natural orbital extrapolation to estimate the error due to using a finite basis set... 

The CBS models use the known asymptotic convergence of pair natural orbital expansions to extrapolate from calculations using a finite basis set to the estimated complete basis set limit. See Appendix A for more details on this technique. 

A MP2/6-311- -G(2df,2p) calculation is carried out, which automaticaUy yields the corresponding HF energy. The MP2 result is extrapolated to the basis set limit by the pair natural orbital method. 

We have recently employed the Dunning correlation-consistent basis sets for our pair natural orbital CBS extrapolation algorithm, Eqs. (2.1) and (2.2) [50]. The results produced a substantial improvement over the raw second-order energies, but were inferior to the (lmax + ) 3 extrapolations listed in Table 4.4. The residual underestimation of... 

There are three basic CBS methods CBS-4 (for fourth-order extrapolation), CBS-Q (for quadratic Cl) and CBS-APNO (or CBS-QCI/APNO, for asymptotic pair natural orbitals), in order of increasing accuracy (and increasing computer time). These methods are available with keywords in the Gaussian 94 and 98 programs. 

The CBS family employs the asymptotic convergence of pair natural orbital expansions to extrapolate to the MP2 limit and the higher order contributions for the various components of the electronic energy are evaluated with a smaller basis set selected to balance the errors and CPU times [224],... 

Complete Basis Set Methods Petersson et al.61-63 developed a series of methods, referred to as complete basis set (CBS) methods, for the evaluation of accurate energies of molecular systems. The central idea in the CBS methods is an extrapolation procedure to determine the projected second-order (MP2) energy in the limit of a complete basis set. This extrapolation is performed pair by pair for all the valence electrons and is based on the asymptotic convergence properties of pair correlation energies for two-electron systems in a natural orbital expansion. As in G2 theory, the higher order correlation contributions are evaluated by a sequence of calculations with a variety of basis sets. 

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