Fully optimized contracted Gaussian basis sets for atoms Li to Kr

Fully optimized contracted Gaussian basis sets for atoms Li to KrAnsgar Schkifer, Hans Horn, and Reinhart AhlrichsInstitut ftir Physikalische Chemie und Elektrochemie, Lehrstuhl ftir Theoretische Chemie, UniversitiitKarlsruhe, Kaiserstr. 12, 7500 Karlsruhe, Germany(Received 18 March 1992; accepted 6 May 1992)Various contracted Gaussian basis sets for atoms up to Kr are presented which have beendetermined by optimizing atomic self-consistent field ground state energies withrespect to all basis set parameters, i.e., orbital exponents and contraction coefficients.INTRODUCTIONChemical bonding usually does not affect inner shellsof atoms. Atomic contributions from the valence shell,however, may contract or expand and be polarized dependingon the actual bonding situation. The smallest (atomcentered) basis set which can be expected to reliably accountfor these effects, e.g., in self-consistent field (SCF)treatments, is of SVP (split valence polarization) type,which means that inner shell atomic orbitals (AOs) aredescribed by a single basis function, two basis functions areprovided for each valence shell AO, augmented by a set ofpolarization functions.CGTO (contracted Gaussian-type orbital) basis setscan, of course, be obtained by segmented or generalizedcontraction of atom optimized GTO basis sets. Since wewill not deal with this popular and successful procedure,we refer the reader to recent reviews. l-5 It should be noted,however, that the basis sets obtained by segmented contractionare clearly not fully optimized, i.e., the basis setparameters do not minimize an appropriate measure foraccuracy such as the atomic ground state energy. Atomoptimized basis sets in a generalized contraction are clearlyfully optimized.6 Since we consider it advantageous to usefully optimized segmented CGTO sets, we will present variousbasis sets of this kind, especially SV basis sets.The most widely and successfully used atom optimizedbasis sets of SV type have been determined by Pople et al.,the 4-31 G (Ref. 7) and 6-3 1 G (Ref. 8) basis sets. Themain and probably only drawback of 4-31 G bases, to givean example, concerns the relatively poor energy especiallyfor second row elements: the 4-3 1 G atomic energies for Alto Cl are even higher than those of Huzinaga’s (9~5~)basis.g The SV type 4-31 G basis does not appear to be anefficient contraction of a (12&p) primitive basis.2SV contractions can be conveniently obtained fromatom optimized SZ (single zeta) basis sets by decontractingvalence AOs. SZ basis sets are available for variousnumbers of GTOs per A0 for atoms up to Rn from thework of Huzinaga and co-workers.” These SZ sets haveproved invaluable to us since their study provided importanthints for the design of fully atom optimized CGTOsets presented in this article.METHOD OF COMPUTATION, ACCURACY, ANDNOMENCLATUREThe methodology of GTO basis set optimization hasbeen developed and described in detail by Faegri and Alm-lbf.” These authors consider energy optimization of GTObasis sets by means of analytical first and second orderderivatives. We refer the reader to this paper, which containsvaluable remarks, and sketch technical aspects onlybriefly.The basis sets presented in this article were optimizedwith the program system TURBOMOLE,'~ which allows forthe calculation of analytic gradients of the SCF energywith respect to orbital exponents and contraction coefficients(for atoms and molecules). The general expressionfor the SCF energy gradient is as follows’3 (closed shellcase assumed for simplicity) :withE’ = 2 D,d;v - c. W&v+; p;L D,.Jh 1.W I d 1PP-&.+‘~)I’ (1)Dp,=2 c c,&, (density),a.zoccw/&v=2 c ~&lCvm (energy weighted density ) .(IEOCC(3)The indices ,u, Y, K, il run over all basis functions(CGTOs in general). The density matrices D and W areobtained from the preceding SCF calculation and the maintask of a gradient step is to calcul$e differentiated integrals’1,14~15 of the general form ( 0 denotes an arbitraryone-electron operator)(~lo^lY)~=(~‘l~IY)+(yl~‘IY)+~~IOhIY~)r (4)(~u~I~/z)‘=(~‘~I~il)+(~~‘l~il)+(~~l~’il)+(~~lKil’).A CGTO GM centered at nucleus A at r, = (x~,~A,zA) isdefined asGM= .i? &a, (6)r=lwithgL4=Nz(X-XA)k(Y-YA)‘(z-z~)~e-~i(r-r~)z, (7)Nz= (22) l”( 1x77y2w+1/2),(2k-1)!!(21-1)!!(2m-l)!!) r’s(2)(5)(8)J. Chem. Phys. 97 (4), 15 August 1992 0021-9606/92/l 62571-07$006.00 @ I992 American Institute of Physics 2571

2576 Schafer, Horn, and Ahlrichs: Optimized basis sets for atoms Li to KrAEscF=EscF(molecule) - C Es&atoms).The results demonstrate the quality of SVP basis sets inSCF treatments, e.g., in comparison to DZP or TZP, andconfirms the quantitative accuracy of the SCF approximationas far as structure constants are concerned. The SVPbasis also shows no shortcomings with respect to SCFbinding energies for P4 or AlCls. Only for SF6 which showsa rather delicate balance of ionic (,!@+I?-) and hypervalent(3d on S) contributions to bonding, a larger basis isrequired for a quantitative account of energetics of bonding.Finally we report results for TiCl, as a transition metalcompound which can be treated on the SCF level. Theequilibrium geometry was determined with SVP and DZPbasis sets [~~=0.065 (Ti), qd=0.34 (Cl)] with followingresults listed in the order SVP/DZP/exp:*’TiC&( r,)RJpm=217.3/219.5/219,AEs,&kJ mol-‘= - 1081/- 1106.Again the SVP basis shows relatively good accuracy withrespect to molecular structure and bonding energy as comparedto DZP.RECOMMENDATIONSCALCULATIONSFOR WlOLECULARThe use of atom optimized basis sets in molecular calculationsrequires some modifications and extensionswhich have been extensively discussed in the literature.‘”We only make a few comments in connection with thebasis sets presented in this article.Polarization functions should be added to SV or largerbasis sets to obtain a reliable account of bonding effects inSCF calculations. An SVP basis appears to yield equilibriumgeometries sufficiently close to the HF limit. Polarizationfunctions for H are here of lesser importance,5 andhydrocarbons, benzene or related compounds, andfullerenes can even be treated on the SV level.18Alkaline earth metals. Since it is absolutely necessaryto have valence p functions added to the basis sets of alkalineearth elements in molecular calculations, we have optimizedthe corresponding SV and DZ basis sets for theexcited state 3P (i.e., ns’np’). Alternatively one can addtwo p sets to the SV basis obtained for the ground statewith exponents similar to those of the valence s functions.For boron we recommend to scale thep exponents by afactor of 1.3, since thep contribution to molecular orbitals(MOs) is always markedly contracted as compared to the(relatively weakly bound) 2p A0 of the free atom.SVP versus larger basis sets. If more extended basis setsthan SVP are to be used, we recommend to go directly toTZP [or TZDP] for atoms up to Ar and the DZP basis setsfor K to Kr. The present SV basis may be decontracted forthe shell below the valence shell (by taking out the mostdiffuse function), but we have not encountered cases so farwhere this led to marked improvements. We recommend inany case that an SVP treatment for a structure determina-tion is followed by a single point calculation with a largerbasis to obtain a more reliable energy-unless sufiicientexperience about the performance of SVP bases is alreadyavailable.Transition metal compounds show marked variationsof d occupation. To be on the safe side it is therefore recommendedto contract the (5d) GTO set {3113 in a TZtype fashion. For the electron rich atoms Ni or Cu a (4113contraction of the (6d) basis may be necessary since thisgives a better balanced description of 3dn4? vs 3d”+‘4s’(see Table II). Transition metal complexes with formaloxidation of +2 or larger for the metal atom show onlyweak 4s and 4p occupations. In these cases one can actuallydrop the more diffuse of the two GTOs describing the 4sA0 and add a 4p set with the same exponent as the steepers GTO. In complexes with formal charges L@ or M+l, werecommend to add two p GTO sets with exponents as forthe 4s subshell to account for polarization effects or to usethe 4p functions determined by Wachters for excitedstates.i7Relativistic e&c& are already noticeable for post 3delements, so that it might be better in some cases to useeffective core potentials (ECPs) with relativistic correctionsinstead of all electron basis sets.*lSmall basis sets for methyl, -CHs, and tertiary butyl,-C(CHs),. Methyl (Me) and tertiary butyl (‘Bu) groupsare often used in preparative chemistry for steric protection.A proper description of steric requirements can beachieved by relatively small basis sets of basically SZ quality.A simultaneous optimization of basis sets and structureconstants has been performed for methane and neopentaneas parent compounds for Me and ‘Bu. The basis sets chosenfor this purpose are H: {33, C: {42/3) or {5 1 l/3 !3, wherethe larger SV basis is used only for the central carbon atomwhich then connects these groups to the rest of the molecule.The following geometries were obtained [values inparentheses: SV basis at all atoms ({3 13 is used as SV andDZ basis at H)/DZP basis [rl,=O.46 (H); qd=0.46 (C)/experiment]CH,:C-H= 108.3( 109.3/108.8(108.522) pm,C(CH3),:C-C= 154.3( 154.2/153.3/153.4*0.323) pm,C-H= 108.7( 109.5/109.1/l 11.4hO.8) pm,The present fully optimized minimal basis sets clearlygive a reliable description of the steric demand of thesebulky groups.CONCLUSIONFull optimization of all basis set parameters (exponentsand contraction coefficients) by means of gradienttechniques leads to basis sets of relatively small size withleast possible loss in accuracy. These basis sets are expectedto be superior to those created by just contracting atomJ. Chem. Phys., Vol. 97, No. 4, 15 August 1992

Schafer, Horn, and Ahlrichs: Optimized basis sets for atoms Li to Kr 2577optimized GTO sets. The present fully optimized SV basissets are considered to be the best compromise of expenseand accuracy.AVAILABILITYThe basis sets presented in this article are available viaFTP (file transfer protocol) at internet address“tchibm3.chemie.uni-karlsruhe.de” (internet number129.13.108.8) with login-ID “anonymous” in the directory“/pub/basis.” Besides those mentioned in Tables I and II,the files include also basis sets of type (7,4), (8,4), (9,5),(9,5>/[5,3]/{51111/311}, (10,6), (11,7), (11,7)/[6,4]/{611111/4111), (13,9) for first row, (10,7), (11,7),(12,8), (12,9), (13,10), (13,10>/[8,6]/{61111111/5111113, (15,12), (17,12) for second row, (14,9,5) forthird row transition metals, and (14,11,5), (14,11,5)/[8,6,2]/{6211111 l/61 111 l/413 for third row main groupelements.ACKNOWLEDGMENTSThis work was supported by the Volkswagen-Stiftungand by the Fonds der Chemischen Industrie.‘T. H. Dunning and P. J. Hay, in Modern Theoretical Chemistry: Methodsof Electronic Structure Theory, edited by H. F. Schaefer III (Plenum,New York, 1977), Vol. 3.‘R. Ahlrichs and P. R. Taylor, J. Chim. Phys. 78, 315 (1981).‘S. Huzinaga, Comp. Phys. Rep. 2, 279 (1985).4E. R. Davidson and D. Feller, Chem. Rev. 86, 681 (1986).‘W. J. Hehre, L. Random, P. v. R. Schleyer, and J. A. Pople, Ab initioMolecular Orbital Theory (Wiley, New York, 1986).6R. C. Raffenetti, J. Chem. Phys. 58, 4452 (1973).‘R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys. 54, 724(1971); W. J. Hehre and W. A. Lathan, ibid. 56, 5255 (1972).s W. J. Hehre, R. Ditchfield, and J. A. Pople, J. Chem. Phys. 56, 2257(1972); M. M. Francl, W. J. Pietro, W. J. Hehre, J. S. Binkley, M. S.Gordon, D. J. DeFrees, and J. A. Pople, ibid. 77, 3654 (1982).‘S. Huzinaga, Approximate Atomic Functions I, II (Division of TheoreticalChemistry, Department of Chemistry, University of Alberta, 1971).lo Gaussian Basis Sets for Molecular Calculations, edited by S. Huzinaga(Physical Sciences Data, Elsevier, Amsterdam, 1984), Vol. 16.“K. Faegri, Jr. and J. AlmlBf, J. Comput. Chem. 7, 396 (1986).“R. Ahlrichs, M. Bar, M. Hiiser, H. Horn, and C. Kolmel, Chem. Phys.Lett. 162, 165 (1989); Recent extensions required for the optimizationof CGTOs are described in A. Schafer, diploma thesis, Karlsruhe, 1991.I3 P. Pulay, Mol. Phys. 17, 197 (1969).i4J. Almliif and P. R. Taylor, Int. J. Quantum Chem. 27, 743 (1985).15G Tonachini and H. B. Schlegel, J. Chem. Phys. 87, 514 (1987).i6F.’ B. van Duijneveldt, IBM Technical Research Report No. RJ-945,(1971).“A. J. H. Wachters, J. Chem. Phys. 52, 1033 (1970).“S Richard (private communication).I’M HIser, J. Almlof, and G. E. Scuseria, Chem. Phys. Lett. 181, 497(1991).*‘A. F. Wells, Structural Inorganic Chemistry, 5th ed. (Oxford University,Oxford, 1984).“P. A. Christiansen, W. C. Ermler, and K. S. Pitzer, Annu. Rev. Phys.Chem. 36, 407 (1985).**L. S. Bartell, K. Kuchitsu, and R. J. deNeui, J. Chem. Phys. 33, 1254(1960).“L. S. Bartell and W. F. Bradford, J. Mol. Struct. 37, 113 (1977).24C. Froese Fischer, The Hartree-Fock Method for Atoms. A NumericalApproach (Wiley, New York, 1977).“E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, 177 (1974).J. Chem. Phys., Vol. 97, No. 4, 15 August 1992