Basis-Set Completeness Profiles in Two Dimensions ALEXANDER A. AUER,1 TRYGVE HELGAKER,2,∗ WIM KLOPPER3 1 Institute
of Physical Chemistry, University of Mainz, D-55099 Mainz, Germany 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
2 Department
Received 23 October 2000; Accepted 9 November 2000
Abstract: A two-electron basis-set completeness profile is proposed by analogy withthe one-electron profile
introduced by D. P. Chong (Can J Chem 1995, 73, 79). It is defined as Y (α, β) = m n Gα (1)Gβ (2)|(1/r12 )| m (1) n (2) m (1) n (2)|r12 |Gα (1)Gβ (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
Two-Dimensional Completeness Profiles
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) =
Gα (1) m (1) m (1)Gα (1)
(1)
m
against the exponent α of the scanning Gaussian-type atomic orbital Gα . { 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 different regions and with respect to different molecular properties was gained by inspecting and comparing the corresponding oneelectron 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 ability 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 discussed, and examples for a few carbon–atom basis sets will be presented.
In the framework of the explicitly correlated R12 methods, covariant two-electron basis functions of the type |χ = r12 Gα (1)Gβ (2)
(2)
are used.2 The overlap of this function with the contravariant twoelectron function 1 χ˜ | = Gα (1)Gβ (2) r12
(3)
provides a two-electron analogue of Chong’s one-electron profile, namely: 1 (4) Gα (1)Gβ (2) r12 Gα (1)Gβ (2) = 1. r12 ∗ Permanent address: Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway
Correspondence to: W. Klopper; e-mail:
[email protected] 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.
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Software News and Updates
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Vol. 23, No. 0
The resolution of the identity (RI) in terms of an orthonormal two-electron basis set reads m (1) n (2) m (1) n (2) 1. (5) m
n
Insertion of the RI into eq. (4) yields the two-electron analogue of the basis-set completeness profile: 1 Gα (1)Gβ (2) m (1) n (2) Y (α, β) = r12 m n
× m (1) n (2)r12 Gα (1)Gβ (2) , (6)
with Y (α, β) = 1 in the limit of a complete basis set { m }. The two-electron equivalent of the completeness profile is a two-dimensional function—that is, a surface. Furthermore, Y (α, β) is not necessarily smaller than or equal 1. A major difference from the one-electron completeness profiles is that the scanning functions Gα and Gβ can be functions of different angular momentum or be placed on different centers. In the following, ss-profiles and pp-profiles refer to profiles in which both Gα and Gβ are chosen to be s- or p-type functions, respectively. In the sp-profiles, Gα is an s-type function and Gβ a p-type function. In Figures 1 to 6, two-electron completeness profiles of the cc-pVXZ (X = D, T, Q),3 aug-cc-pVTZ,4 cc-pCVTZ,5 and SVP-auxiliary basis sets6 are shown for carbon. These surfaces were obtained by plotting Y (α, β) against log (α) and log (β) for scanning functions Gα and Gβ with exponents α and β in the range of 10−4 to 104 . All calculations were performed using a local version of the DALTON
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Journal of Computational Chemistry
program package,7 which has been modified to make use of auxiliary basis sets. Our two-electron profiles are motivated by the expression for the basis-set truncation error found in explicitly correlated secondorder Møller–Plesset perturbation theory.8 In this theory, the basisset truncation correction Vij2
(7)
Vij − Uij
is added to the conventional pair energies eij for the pair of occupied orbitals φi and φj , leading to the following expression for the explicitly correlated pair energies fij = eij +
Vij2 Vij − Uij
,
(8)
with Vij
1 1 1 = − [φi φj ] [φp φq ] [φp φq ]r12 [φi φj ] , 2 2 p
