### Abstract

Within the framework of many‐body perturbation theory, the total correlation energy can be partitioned into: intraorbital pair energies, *e* _{ i i }; interorbital pair energies, ^{αβ} *e* _{ i j } and ^{αα} *e* _{ i j }; double‐excitation pair‐coupling terms *e* _{ i j,k l }(*D*); and higher‐excitation pair‐coupling terms, *e* _{ i j,k l }(*S*,*T*,*Q*,...). The asymptotic convergence of pair natural orbital expansions for each of these terms is determined for the model problem of *n* infinitely separated helium‐like ions with infinite nuclear charge. For example, the asymptotic form of the basis set truncation error in an αβ‐interorbital pair energy is Limit_{ N i j →∞}Δ^{αβ} *e* _{ i j } =^{αβ} *f* _{ i j } (J_{μ=1} ^{ N i j } *C* _{μ i j })^{2} (^{(−225}/ _{4608})) (*N* _{ i j }+δ_{ i j })^{−} ^{1} , where *N* _{ i j } is the number of pair natural orbitals and *C* _{μ i j } is the coefficient of pair natural orbital configuration μ_{ i j }. Numerical studies of the neon atom verify that this model behavior applies to real many‐electron systems. The pair‐coupling terms beyond third‐order contribute less than 1% of the total correlation energy in a variety of atoms and molecules and can therefore be neglected. As a practical test of the use of the asymptotic forms to extrapolate the remaining terms, a double zeta plus polarization set of pair natural orbitals was used. Extrapolation of each of the neon pair energies to the value for a complete basis set yields an independent electron pair approximation equal to −0.4233 hartree, which is 108.6% of the experimental correlation energy (−0.3896±0.001 hartree). Including the third‐order MP‐MBPT pair‐coupling terms and extrapolating to a complete basis set gives a total correlation energy equal to −0.3904 hartree, which is 100.2±0.2% of the experimental value. A similar

calculation on H_{2}O gave equally good results (calc. −0.3706; expt. −0.370±0.003 hartree) indicating that this DZ+P CBS method is applicable to polyatomic potential energy surfaces.

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