chem.Rev.1988.86.661-696 881 Basis Set Selection for Molecular Calculations ERNEST R. DAVIDSON' and DAVID FELLER IMnn ~~, Blwmhgton. Indlane 47405 Recehred January 10. 1986 (Rsvlsed Msnuscript Received April 5. 1988) IntroducNon The quantum chemistry literature containa references to a plethora of basis seta, currently numbering almost 100. While professional quantum chemists might be- come familiar with several dozen of these in a lifetime of calculations, the OCeasionaJ user of ab initio programs probably wishes to ignore all but the two or three sets which, through habitual use, have become personal fa- vorites. Unfortunately, this attitude has its drawbacks. Intelligent reading of the literature requires at least a cursory knowledge of the limitations of other basis sets. Information conceming the likely aceuracy of a specific basis for a particular property is essential in order to judge the adequacy of the computational method and, hence, the soundness of the results. Occasionally, for I. reasons of economy or computational feasibility, a basii Ernest R. Davfdson was born in Terre Haute. IN. in 1938. He set is selected for which the computed results are nearly reghnrd a B.Sc. wee in chermcal engindng from RoseHman without significance. InsUtute in 1958 and his Ph.D. in 1961 from Indiana University In In light of the large number of publications reporting chemistry. He was an NSF postdoctoral fellow at the University of Wismnsh in 1961. From 1962 until 1984 he was on the facuity new basis sets or detailing the performance of existing at the university of Washington. In 1984 he returned to Indiana sets the task of remaining informed has become very University where he is presently a distinguished professor and diffteult for experts and nonexperts alike. The existence director of the Guantum Chemlshy Program Exchange and of the of such a vast multitude of basis sets is attributable, at Chemical Physics program. He was elected to the International Academy of Molecular Quantum Science in 1981. He has over least in part, to the difficulty of finding a single set of 200 publications in the field of computational quantum chemistry functions which is flexible enough to produce 'good" including work on denslty matrices, charge. spin. and momentum results over a wide range of molecular geometries and distributions. electronic Structure of excited states. the Structure is still small enough to leave the problem computa- of organic radicals and diradicals, and ab initio computational tionally tractible and economically within reason. methods. The driving force behind much of the research effort in small basis sets is the fact that the computer time required for some parts of an ab initio calculation is very strongly dependent on the number of basis func- tions. For example, the integral evaluation goes as the fourth power of the number of Gaussian primitives. Fortunately this is the only step which explicitly de- pends on the number of primitives. All subsequent steps depend on the number of contracted functions formed from the primitives. The concept of primitive and contracted functions will be discussed later. Consider a collection of K identical atoms, each with n doubly occupied orbitals and N unoccupied (or vir- tual) orbitals. The SCF step increases as (n + N'K', Daw FwM was ban h Dlhqtie, IA, in 1950. He rwceivwd a B.S. while the full transformation of the integrals over the wee from Loras COILS@ in 1972 and a PhD. in physical chem original basis functions to integrals over molecular or- lsby from Iowa State University under the direction of Professor bitals goes as (n + N5P.Methods to account for Kbus Ruedenberg In 1979. He then ioined the research groups 01 Professors Wes Wden and Ernest Davidson at the University correlation effects vary greatly. Only a few of the of Washington as a postdoctoral associate. In 1982 he was popular ones will be considered here. Second order pamtedto Um poWm of Research AssocktelFanmy. Cunently Moller-Plesset (MP2) perturbation theory goes as he holds an Associate Scientist posmn at Indiana University. His n2NX4but still requires an nN4Fintegral transfor- research interests include the application of very extended basis mation. MP3 goes as n2N4P,while a Hartree-Fock set ab initio techniques for determining molecular properties. singles and doubles CI will have n2WK4configurations, (n2NW')' hamiltonian matrix elements of which n2N4P replacing the full (n+ N5K5integral transformation will be nonzero. Pople and co-workers' have proposed with a n2N4Ppartial transformation every CI iteration. 00082665/8610788-0881$~.50/0 8 1986 Amsrican Chemical Society 682 Chemical Reviews, 1986, Vol. 86, No. 4 Davidson and Feller For certain ranges of n, N, and K this may represent considered the ratio is reduced even more. A detailed an improvement. The MP3 step by itself only requires comparison of the relative performance of STO and one iteration. Gaussian d functions using a wide range of position and As is evident from the above discussion, even a small momentum space properties found that a single STO reduction in the size of the basis set can result in an was worth no more than two Gaussians in the copper enormous overall speedup. Since the amount of com- atom.5 In a comparisions of STO and Gaussian basis puter time in question can often be measured in days sets in molecular property calculations Carsky and there is great interest in finding smaller basis sets that Urban6 and Rosenberg and Shavitt7 find similar values produce results equivalent to much larger ones. with similar size basis sets. Since the choice of basis set must be guided by con- To some extent the disadvantage associated with the siderations of desired accuracy in the results and com- larger number of Gaussians compared to STO’s was putational costs, it becomes imperative that sufficient ameliorated through the work of Clementis and Whit- information be available so that an intelligent choice teng who introduced the concept of “contracted” is possible. Some of the new basis sets appearing in the Gaussians as approximations to atomic orbitals. In this literature are used for a single project and then disap- scheme, the basis functions for molecules became, not pear into oblivion. Others are proposed, without ex- individual Gaussians, but rather fixed linear combina- tensive calibration, as serious contenders to replace tions. Early work dealt with Gaussian lobe functions basis sets in common use. in which, for example, a p orbital was approximated by In practice, most popular computer programs for ab differences of s orbitals slightly displaced from each initio calculations contain internally defined basis sets other. For higher angular momentum, the number of from which the user must select an appropriate one. In terms required, and the loss of accuracy from differ- principle this should facilitate switching from one set encing in computing the integrals, made the method to another as it becomes warranted by the course of the unwieldy. investigation. For example, preliminary determinations Clementi and Davisxo extended the use of of all relevant equilibrium and transition state geome- “contracted” functions to include Cartesian Gaussians. tries on a given potential energy surface may be made Although individual integrals with Cartesian Gaussians with a small basis set. Subsequent refinement at a are somewhat more complicated than with Gaussian chosen subset of these points can then be made with lobes, the ease in extending the basis set to higher an- a larger set. gular momentum has made this procedure the method In general the evolution of “black box”-type quantum of choice. Contrary to the misapprehensions about the use of Gaussians as basis functions, fewer contracted chemistry programs (such as Gaussian-82 by Pople and co-workers), coupled with the increasing performance Gaussians than uncontracted Slater-type orbitals are of computer hardware, has resulted in a tremendously required, in fact, for a given level of accuracy. One of expanded role for ab initio techniques in academic and the first widely distributed program packages to permit commercial applications. It may be possible in the contracted Gaussians was IBMOL. The 1966 user’s future to have programs make an informed decision guide” to version 2 boasts the impressive ability to about the choice of basis set based on the results of handle 700 Gaussians of s, p, d, or f symmetry on an thousands of previous calculations which are accessible IBM 7094 with 35,000 32-bit words of memory. in a data base, but that time has not yet come. The responsibility still rests with the program user. Definitions In this article we will seek to review basis sets of the type known as “contracted Cartesian Gaussians” from Slater-type orbitals are defined in spherical polar the point of view of the nonspecialist who is interested coordinates as in a practical guide to what is generally available today and what are some of the pitfalls to avoid. Other recent reviews on this subject from slightly different points of view have been written by Huzinaga2 and Wil~on.~ where I and m are angular momentum quantum num- Boys4 first advocated the use of Gaussian-type basis bers, N is a normalization constant and {is the orbital functions on the practical ground that all of the inte- exponent. The main claims to superiority of these or- grals required for a molecular calculation could be easily bitals are their short- and long-range behavior. Actu- and efficiently evaluated. Because a Gaussian has the ally, of course, a linear combination of such functions wrong behavior both near the nucleus and far from the will behave correctly only if the smallest orbital expo- nucleus, it was clear that many more Gaussians would nent is exactly (U)’i2,where I is the ionization potential be required to describe an atomic orbital than if Sla- in atomic units.