Basis-Set Completeness Proﬁles in Two Dimensions ALEXANDER A. AUER, 1 TRYGVE HELGAKER, 2,* WIM KLOPPER 3 1 Institute of Physical Chemistry, University of Mainz, D-55099 Mainz, Germany 2 Department of Chemistry, University of Cambridge, Lensﬁeld Road, Cambridge CB2 1EW, UK 3 Theoretical Chemistry Group, Debye Institute, Utrecht University, P.O. Box 80052, NL-3508 TB Utrecht, The Netherlands Received 23 October 2000; Accepted 9 November 2000 Abstract: A two-electron basis-set completeness proﬁle is proposed by analogy with the one-electron proﬁle introduced by D. P. Chong (Can J Chem 1995, 73, 79). It is deﬁned as Y(α,β) = ∑ m ∑ n 〈G α (1)G β (2)|(1/r 12 )|  m (1) n (2)〉〈 m (1) n (2)|r 12 |G α (1)G β (2)〉 and motivated by the expression for the basis-set truncation correction that occurs in the framework of explicitly correlated methods. The two-electron basis-set proﬁles provide a visual assessment of the suitability of basis sets to describe electron-correlation effects. Furthermore, they provide the opportunity to assess the quality of the basis set as a whole—not only of the individual angular momentum subspaces, as is the case for the one-electron basis-set proﬁles. The two-electron completeness proﬁles of the cc-pVXZ(X = D, T, Q), aug-cc-pVTZ, cc-pCVTZ, and SVP-auxiliary basis sets for the carbon atom are presented as illustrative examples. © 2002 John Wiley & Sons, Inc. J Comput Chem 23: 1–6, 2002 Key words: basis sets; electon correlation; completeness; visualization Introduction In 1995, Chong introduced a method to give a visual assessment of the quality of one-electron basis sets. 1 His basis-set completeness proﬁle was obtained by plotting the function Y(α) =  m  G α (1)    m (1)   m (1)   G α (1)  (1) against the exponent α of the scanning Gaussian-type atomic or- bital G α . { m } is the (orthonormalized) one-electron basis set under study. This proﬁle can take values between 0 ≤ Y(α) ≤ 1 and approaches 1 in the limit of a complete basis. Various plots for the s, p, and d subspaces of the basis of interest were constructed, and insight into the quality of several one-electron basis sets in dif- ferent regions and with respect to different molecular properties was gained by inspecting and comparing the corresponding one- electron basis-set completeness proﬁles. It is possible to deﬁne a two-electron analogue of Chong’s one-electron basis-set proﬁle, which is suited to assess the abil- ity of basis sets to describe electron-correlation effects. In the present short communication, the construction and opportunities of such a two-electron basis-set completeness proﬁle will be dis- cussed, and examples for a few carbon–atom basis sets will be presented. Two-Dimensional Completeness Proﬁles In the framework of the explicitly correlated R12 methods, covari- ant two-electron basis functions of the type |χ 〉= r 12   G α (1)G β (2)  (2) are used. 2 The overlap of this function with the contravariant two- electron function 〈˜ χ |=  G α (1)G β (2)   1 r 12 (3) provides a two-electron analogue of Chong’s one-electron proﬁle, namely:  G α (1)G β (2)     1 r 12 r 12     G α (1)G β (2)  = 1. (4) ∗ Permanent address: Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway Correspondence to: W. Klopper; e-mail: w.m.klopper@chem.uu.nl Contract/grant sponsor: The Royal Netherlands Academy of Arts and Sciences Contract/grant sponsor: Norwegian Research Council; contract/grant number: NN118K © 2002 John Wiley & Sons, Inc.