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Advanced Review

Atomic orbital basis setsFrank Jensen

Electronic structure methods for molecular systems rely heavily on using basis

sets composed of Gaussian functions for representing the molecular orbitals. Anumber of hierarchical basis sets have been proposed over the last two decades,and they have enabled systematic approaches to assessing and controlling theerrors due to incomplete basis sets. We outline some of the principles for con-structing basis sets, and compare the compositions of eight families of basis setsthat are available in several different qualities and for a reasonable number ofelements in the periodic table. C2012 John Wiley & Sons, Ltd.

How to cite this article:

WIREs Comput Mol Sci2012. doi: 10.1002/wcms.1123

INTRODUCTION

T he use of basis sets for expanding the molec-ular orbitals in wave function or KohnShamdensity functional methods is an essential compo-nent of contemporary methods for describing theelectronic structure of molecular and extended sys-tems. Nuclear-centered Slater-1 or Gaussian-type2

functions have dominated for molecular systems, withGaussian functions being preferred due to the bettercomputational efficiency. Plane-wave basis functionsare often used for extended systems, as they are natu-

rally suited to periodic boundary conditions; but forany reasonable number of plane waves, this necessi-tates the use of a pseudo-potential for representing theatomic core electrons/potential. Nuclear-centered ba-sis functions can also be used for periodic systems, andthis treats all electrons on an equal footing. Recent de-velopments have investigated the use of finite-elementmethods where piecewise polynomials are used forrepresenting the orbitals.35

The goal of a basis set is to provide the bestrepresentation of the unknown molecular orbitals (orelectron density), with as small a computational costas possible. Because different theoretical methods and

molecular properties have different basis set demands,different computer architectures and algorithms havedifferent efficiency requirements, and the desired ac-curacy varies with the application, it is not possibleto design one optimum basis set. Indeed, the large

Correspondence to: frj@chem.au.dk

Department of Chemistry, Aarhus University, Aarhus, Denmark

DOI: 10.1002/wcms.1123

number of different basis sets proposed over the yearsis a testament to these conflicting demands.

In this review, we will summarize some of theprinciples for constructing and classifying basis sets,with focus on modern hierarchical basis sets. Only ba-sis sets composed of nuclear-centered Gaussian-typefunctions will be discussed, but many of the principlesand conclusions hold for Slater-type functions as well.We will concentrate on basis sets for the first 36 atomsin the periodic table [hydrogen (H) to krypton (Kr)]as many of the popular basis sets are only availablefor these elements, but the trends and principles carry

over to elements in the remaining part of the periodictable as well.

All-electron calculations for systems containingatoms from the lower part of the periodic table mustinclude relativistic effects for accurate results, andthis leads to some differences in the basis set require-ments compared to nonrelativistic methods. We willnot discuss basis sets for relativistic methods in thisreview. A significant computational saving can be ob-tained for systems with many-electron atoms by re-placing the core electrons by a pseudo-potential ormodel potential, and this can also to some extent

account for relativistic effects. The use of pseudo-potentials has been the subject of recent reviews byDolg and Cao, and these can be consulted for furtherinformation.6,7 A discussion of plane wave or finite-element methods, and the topic of auxiliary basis setsfor density fitting, is beyond the scope of the presentreview.

We will focus on eight families of basis sets thatare available in several quality levels and defined for areasonable number of elements in the periodic table:

Vo l u m e 0 0 , Ja n u a ry / F e b ru a ry 2 0 1 2 1c 2 0 1 2 Jo h n Wi l e y & S o n s , L td .

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Advanced Review wires.wiley.com/wcms

1. The Pople-stylek-lmnG basis sets.820

2. The Ahlrichs SVP, TZP, QZP basis sets intheir Def2 version.21

3. The XZP basis sets developed by Jorge andcoworkers.2226

4. The Sapporo basis sets developed by Koga

and coorkers.2730

5. The ANO basis sets developed by Roos andcoworkers.3135

6. The cc-pVXZ basis sets developed byDunning36 and Peterson and coworkers.3739

7. ThenZaP basis sets developed by Peterssonand coworkers.4042

8. The pc-nbasis sets developed by Jensen andcoworkers.4348

PRIMITIVE AND CONTRACTED BASISFUNCTIONS

An unknown one-electron function, such as a molec-ular orbital , can be expanded in a set of knownfunctions , the basis set49,50:

=M

i=1cii . (1)

The coefficients ci are in HartreeFock (HF)and KohnSham density function theory (DFT) deter-mined by minimizing the total energy, which by tra-

ditional methods lead to a matrix eigenvalue problemthat is solved iteratively to provide a self-consistentfield (SCF) solution. The matrix elements containmultidimensional integrals over basis functions, ofwhich those involving the electronelectron interac-tion (two-electron integrals) completely dominate thecomputational effort. We will, in the present context,not be concerned with determination of the molecularexpansion coefficients, but focus on the basis func-tions, which will be taken as Cartesian Gaussian-typefunctions:

i

=N(x

X)k (y

Y)l (z

Z)m ei (rR)

2

. (2)

The center of such a primitive function is R(X,Y,Z), typically a nuclear position, the sum of k,l, andmdefines the angular momentum (e.g.,k + l+m= 1 is a p-function), i is the exponent providingthe radial extent of the function, and Nis a normal-ization constant. The accuracy of the expansion inEq. (1) is determined by the number of functions M,their distribution in terms of angular momentum, andthe values of the exponents i [Eq. (2)].

The primitive basis set is for computational rea-sons usually contracted, by forming K fixed linearcombinationsj fromM primitive functions i (K