Pair natural orbitals

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. 

For aP electron pairs, the coefficient matrix C may be diagonalized, yielding the pair natural orbital (PNO) expansion of the pair energies ... 

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. 

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],... 

Figure 4-1 Convergence of the correlation energy for the helium ground state with the number N of pair natural orbitals.

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... 

A Complete Basis Set Model Chemistry. IV. An Improved Atomic Pair Natural Orbital Method. 

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

The CEPA computer programs, developed by two German groups78,104 are based on the use of pair natural orbitals and, therefore, it is appropriate to refer to their approach as the CEPA-PNO method. The construction of H-matrix elements over the PNO s is the same as with the PNO-CI (this was noted in the preceding section). 

Even though computers were an essential tool in quantum chemical calculations, the main challenge was the further development of methods and concepts to describe even more facets of chemistry and with higher accuracy. Methods that account for electron correlation were extended to be able to describe energy surfaces more reliably. Several variants of the CEPA Ansatz (CEPA-1, CEPA-2) were developed as well as the method of self-consistent electron pairs (SCEP). Formulations using canonical or localized orbitals (e.g., pair natural orbitals, PNO, as a kind of optimized virtual orbitals) were put forth. These methods were extensively used for two decades, primarily in Germany, until coupled cluster formulations became more popular. ... 

PNO-CI (Pair Natural Orbital Configuration Interaction) and CEPA-PNO (Coupled... 

Electron Pair Approximation with Pair Natural Orbitals) Calculations of Molecular Systems. I. Outline of the Method for Closed-Shell States. 

As in lEPA-PNO (see Section 4.E.) one calculates the pair natural orbitals for each pair of occupied spinorbitals and constructs from them the (lowest) doubly excited states. But in contrast to lEPA-PNO, the PNO-CI wave functions contain the ground state and doubly excited configurations which correspond to excitations from all pairs. This, of course, brings about difficulties in constructing the Hamiltonian matrix elements, H.. , because each pair generates its own set of PNO s,... 

The CEPA computer programs, developed by two German groups are based on the use of pair natural orbitals and, therefore, it... 

Montgomery, J.A., Ochterski, J.W., Petersson, G.A. A complete basis-set model chemistry. 4. An improved atomic pair natural orbital method. J. Chem. Phys. 1994,101(7), 5900-09. 

MO calculations have been reported for the BH and BH3 molecules, using the pair natural orbital configuration interaction (PNO—Cl) and coupled electron pair approximation with natural orbitals (CEPA—PNO) methods." The force constant and equilibrium distance of BH agreed very well with experimental values. 

Aluminium Hydrides.—PNO-CI (pair natural orbital/configuration interaction) and CEPA-PNO (coupled electron pair approximation with pair natural orbitals) MO calculations on AIH3 give a binding energy (referred to A1 + 3H) of ca. 205 kcal mor. °... 

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. 

R. Ahhichs and E. Driessler, Determination of pair natural orbitals. A new method to solve the multiconfiguration Hartree-Eock problem for two-electron wave functions, Theor. Chim. Acta, 36 (1975) 275. W. Meyer, Theory of self-consistent electron pairs. An iterative method for correlated many-electron... 

Two ab initio methods, which were well known and much discussed in the 1970s and 1980s, were the pair natural orbital Cl (PNO-CI) method and the coupled electron pair approximation (CEPA) method. They were proposed by Meyer [67] in 1973 and 2 years later improved by Ahlrichs et al. [68]. In 1983, Burton and Senff [69] applied the method of Ahlrichs et al. to an analysis of the anisotropy of (H2)2 interaction near the minimum in the van der Waals interaction energy. 

We normalize Pci such that the coefficient of is equal to 1. The spin orbitals (pa, natural spin orbitals of the respective lEPA pair correction function My 22) jhis will be the case for all of the methods considered and we shall not stress this again. One ought to add the abbreviation PNO (for Pair-Natural-Orbitals, sometimes interpreted as Pseudo-Natural-Orbitals) to the terms lEPA and CEPA as well, i.e. to speak of lEPA-PNO and CEPA-PNO rather than just of lEPA and CEPA. Using the PNO s one gets mutually orthogonal 0if though the PNO s of different pairs are nonorthogonal. Also the explicit expressions for Hab are not too complicated. 

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