Detlev Figgen, Erich Goll, and Hermann Stoll b) Institut für Theoretische Chemie, Universität Stuttgart, D Stuttgart, Germany

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1 JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 21 1 DECEMBER 2003 Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group elements Kirk A. Peterson a) Department of Chemistry, Washington State University, Pullman, Washington and Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington Detlev Figgen, Erich Goll, and Hermann Stoll b) Institut für Theoretische Chemie, Universität Stuttgart, D Stuttgart, Germany Michael Dolg c) Institut für Theoretische Chemie, Universität zuköln, D Köln, Germany Received 30 June 2003; accepted 9 September 2003 A series of correlation consistent basis sets have been developed for the post-d group elements in conjunction with small-core relativistic pseudopotentials of the energy-consistent variety. The latter were adjusted to multiconfiguration Dirac Hartree Fock data based on the Dirac Coulomb Breit Hamiltonian. The outer-core (n 1)spd shells are explicitly treated together with the nsp valence shell with these PPs. The accompanying cc-pvnz-pp and aug-cc-pvnz-pp basis sets range in size from DZ to 5Z quality and yield systematic convergence of both Hartree Fock and correlated total energies. In addition to the calculation of atomic electron affinities and dipole polarizabilities of the rare gas atoms, numerous molecular benchmark calculations HBr, HI, HAt, Br 2,I 2,At 2, SiSe, SiTe, SiPo, KrH, XeH, and RnH ) are also reported at the coupled cluster level of theory. For the purposes of comparison, all-electron calculations using the Douglas Kroll Hess Hamiltonian have also been carried out for the halogen-containing molecules using basis sets of 5Z quality American Institute of Physics. DOI: / I. INTRODUCTION In the preceding paper by one of the present authors, 1 the development of a series of correlation consistent-type basis sets from double- to quintuple- were reported for all of the post-d group elements using existing, accurate, smallcore relativistic pseudopotentials PPs. The goal of this work was not just to optimize large Gaussian basis sets for use with relativistic PPs, but to generate a family of basis sets for heavy elements that systematically converge to the complete basis set limit in correlated atomic and molecular calculations much like the correlation consistent basis sets of Dunning and co-workers for light atoms. 2,3 Armed with such sets, the errors due to basis set truncation can be removed from the calculations and the contributions due to effects such as incomplete electron correlation recovery, corevalence correlation, and spin orbit coupling can be much more unambiguously determined. The present paper not only reports the analogous basis sets for the post-d group elements, but also new small-core pseudopotentials adjusted in the same manner as the ones used in the previous study. In order to thoroughly test the new PPs and accompanying basis sets, coupled cluster benchmark calculations have been carried out on the atomic electron affinities, rare gas atom dipole polarizabilities, and the spectroscopic constants a Electronic mail: kipeters@wsu.edu b Electronic mail: stoll@theochem.uni-stuttgart.de c Electronic mail: m.dolg@uni-koeln.de of a variety of diatomic molecules. The new basis sets, denoted cc-pvnz-pp n D, T, Q, 5, yield systematic convergence of both Hartree Fock and correlation energies in atomic and molecular calculations, and this is reflected in the regular convergence of the calculated atomic and molecular properties. As in the previous group study, a number of all-electron calculations were also carried out using large correlation consistent basis sets in order to assess the impact of the pseudopotential approximation on the calculated properties. The errors due to the pseudopotential approximation are found to be nearly negligible and much smaller than the effects expected due to core-valence correlation and spin orbit coupling. Section II briefly describes the details of both the pseudopotential adjustment and basis set construction, while Sec. III discusses the results of the various benchmark calculations. Conclusions are presented in Sec. IV. II. METHODOLOGY A. Pseudopotential adjustment Small-core energy-consistent relativistic pseudopotentials have been derived for the post-d main-group elements Se Kr, Te Xe, and Po Rn. The (n 1)spd outer-core shell is treated explicitly in the valence space, together with the genuine nsp valence orbitals, and only the Ne, Ar 3d 10, and Kr 4d 10 4 f 14 cores, respectively, are replaced by pseudopotentials. Both one-component potentials describing the Pauli repulsion of the cores, their Coulomb and /2003/119(21)/11113/11/$ American Institute of Physics

2 11114 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Peterson et al. TABLE I. Parameters in a.u. for the MCDHF-adjusted energy-consistent small-core pseudopotentials of Se, Te, and Po. Se Te Po l j B B B s 1/ p 1/ p 3/ d 3/ d 5/ f 5/ f 7/ g 7/ g 9/ exchange effects on the valence space, and scalar-relativistic corrections as well as their two-component extensions describing outer-core and valence spin orbit interaction are provided. The adjustment of the pseudopotentials PP has been done using reference atomic valence energies derived from all-electron four-component multiconfiguration Dirac Hartree Fock MCDHF calculations for a multitude of valence- and outer-core excited relativistic states. The present pseudopotentials are meant to supplement similar, previously published PPs for the post-d elements of groups ,4 Thus, only a short description is given here on the details of the adjustment procedure, with special emphasis on problems encountered and properties characteristic for group elements. The functional form used for the full two-component scalar-relativistic spin orbit pseudopotentials is V PP r Q r ljk B k lj exp k lj r 2 P lj, where Q is the inner-core charge (Q 24, 25, 26, for group 16, 17, and 18 elements, respectively ; the sum is over a 1 TABLE II. Parameters in a.u. for the MCDHF-adjusted energy-consistent small-core pseudopotentials of Br, I, and At. Br I At l j B B B s 1/ p 1/ p 3/ d 3/ d 5/ f 5/ f 7/ g 7/ g 9/

3 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Systematic convergent basis sets. II TABLE III. Parameters in a.u. for the MCDHF-adjusted energy-consistent small-core pseudopotentials of Kr, Xe, and Rn. Kr Xe Rn l j B B B s 1/ p 1/ p 3/ d 3/ d 5/ f 5/ f 7/ g 7/ g 9/ Gaussian expansion index k of semilocal short-range radial potentials which are different for different orbital angularmomentum quantum numbers l and, for a given l, for the two total one-electron angular-momentum quantum numbers j l 1/2 (P lj is the projector onto the complete space of functions with angular symmetry l, j around the core under study. The B k lj and k lj are adjusted, in a least-squares sense, so that V PP in two-component valence-only atomic calculations reproduces as closely as possible a set of all-electron reference data. The nonlinear fit of the parameters B k lj and k lj is done to valence energies total energies minus energies of the X Q cores of all relativistic states accessible in numerical allelectron MCDHF average-level AL calculations with a perturbative treatment of the Breit interaction 5 for a set of orbital configurations. This set includes 15 low-lying (n 1)s 2 (n 1)p 6 (n 1)d a ns b np c nd d (n 1)s e (n 1)p f (a 9, 10; e, f 0, 1 configurations of the neutral, singly and doubly charged atoms giving rise to relativistic states, supplemented by 4 high-charge (n 1)s 2 (n 1)p 6 n f 1 (n 4) configurations and, for the heaviest elements Po,At,Rn, (n 1)s 2 (n 1)p 6 n g 1 (n 5) configurations. Adjustment to these reference data was done by means TABLE IV. Atomic ground-state SCF results in a.u. with the MCDHF-adjusted small-core group 16 pseudopotentials excitation energy E, ionization potential IP, electron affinity EA, orbital energies, and r n orbital expectation values. Results in parentheses are differences to corresponding all-electron values. Se Te Po E( 3 P 1 3 P 2 ) E( 3 P 0 3 P 2 ) E( 1 D 2 3 P 2 ) E( 1 S 0 3 P 2 ) IP a EA a (ns) (np 1/2 ) (np 3/2 ) r (ns) r (np 1/2 ) r (np 3/2 ) r 2 (ns) r 2 (np 1/2 ) r 2 (np 3/2 ) a Energy difference between configurational averages.

4 11116 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Peterson et al. TABLE V. Atomic ground-state SCF results in a.u. with the MCDHF-adjusted small-core group 17 pseudopotentials excitation energy E, ionization potential IP, electron affinity EA, orbital energies, and r n orbital expectation values. Results in parentheses are differences to corresponding all-electron values. Br I At E( 2 P 1/2 2 P 3/2 ) IP a EA a (ns) (np 1/2 ) (np 3/2 ) r (ns) r (np 1/2 ) r (np 3/2 ) r 2 (ns) r 2 (np 1/2 ) r 2 (np 3/2 ) a Energy difference between configurational averages. of a weighted least-squares fit in two-component valence average-level AL calculations with a formally nonrelativistic Hamiltonian including the PP Ref. 6 for the same states. PP parameters for the s, p, d projectors were optimized simultaneously, while the ( f,g) excitation/ionization data were fitted for each l separately, keeping the spd part of the potential fixed. For simplicity, we attributed equal weights, within each least-squares fit, to all relativistic states of a given orbital configuration, and we attributed equal total weights to every individual orbital configuration. Our target accuracy for reproducing individual reference energies is 10 2 ev. In order to reach this accuracy, a twoterm Gaussian expansion of the radial potentials k max 2in Eq. 1 is usually sufficient, for each (l, j) combination. For the highest l value (l max 3 for the post-3d and -4d elements, l max 4 for the post-5d ones, often even a single term does the job. For the d projector, we need three functions for the post-3d elements where the potential is attractive and also for some of the post-4d elements I,Xe. Finally, we need a three-term expansion also for the s projector of groups 17 and 18; here, the effective valence charge Q is so high that specially prepared Gaussians are necessary in order to compensate the attractive potential for very small r values so as to avoid humps in the inner-core region of the (n 1)s orbitals. An alternative would be the use, in the expansion, of repulsive terms with a singularity at r 0, i.e., functions of the form exp( k lj r 2 )/r. 7 The final parameters are collected in Tables I III. A transcription of the resulting lj-dependent PPs to scalar-relativistic and SO potentials is easily possible cf., e.g., Ref. 8 and will be made available on the webpage of the Stuttgart-Köln pseudopotentials As an indication of the quality of the fit, we present in Tables IV VI both PP and all-electron results for total valence energies, orbital energies, and orbital r expectation values for the ground state configurations of the atoms under study. Note that no information on orbital properties has been used in the fit of the PPs. The accuracy of the valence orbital energies essentially parallels that of the total valence energies: the absolute deviations from the all-electron values are ev at most with the exception of 6p1/2 of Po, At, Rn, where deviations of up to 0.06 ev occur. When comparing pseudopotential and all-electron r and r 2 expectation values, it should be kept in mind that full agreement cannot be expected, anyway, because of the differences in the core region. Nevertheless, only small differences are obtained which are of the order of Å at most except TABLE VI. Atomic ground-state SCF results in a.u. with the MCDHF-adjusted small-core group 18 pseudopotentials ionization potential IP, excitation energy E of the ion, orbital energies, and r n orbital expectation values. Results in parentheses are differences to corresponding all-electron values. Kr Xe Rn IP a E( 2 P 1/2 2 P 3/2 ) (ns) (np 1/2 ) (np 3/2 ) r (ns) r (np 1/2 ) r (np 3/2 ) r 2 (ns) r 2 (np 1/2 ) r 2 (np 3/2 ) a Energy difference between configurational averages.

5 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Systematic convergent basis sets. II TABLE VII. Compositions of the cc-pvnz-pp basis sets of this work and Ref. 1. post-3d (Ga Kr) post-4d (In Xe) post-5d (Tl Rn) cc-pvdz-pp (8s7p7d)/ 4s3p2d (8s6p6d)/ 4s3p2d (8s6p6d)/ 4s3p2d cc-pvtz-pp (10s11p9d1 f )/ 5s4p3d1f (12s11p9d1 f )/ 5s4p3d1f (12s11p8d1 f )/ 5s4p3d1f cc-pvqz-pp (14s11p12d2 f 1g)/ 6s5p4d2f1g (14s11p12d2 f 1g)/ 6s6p4d2f1g (14s11p11d2 f 1g)/ 6s6p4d2f1g cc-pv5z-pp (16s13p13d3 f 2g1h)/ 7s6p5d3f2g1h (16s13p13d3 f 2g1h)/ 7s7p5d3f2g1h (16s13p12d3 f 2g1h)/ 7s7p5d3f2g1h for the sixth-row elements Po and At again, where the deviations rise to up to Å. B. Construction of cc-pvnz-pp basis sets The details of the basis set development in this work are identical to those described previously for the post-d group elements 1 and hence will only be briefly described here. The MOLPRO program suite 9 was used for all the calculations, only the pure spherical harmonic components of the dfgh polarization functions were used, and the Hartree Fock HF orbitals were fully symmetry equivalenced in all cases. All of the correlated calculations in this work employed the frozen core approximation, e.g., only the 4s4p (Se Kr), 5s5p (Te Xe), and 6s6p (Po Rn) electrons were correlated. The exponent optimizations were carried out within a scaled parameter space using double-sided numerical derivatives. The gradients were typically converged to better than 10 6 a.u. In addition, all of the correlating functions were optimized at the singles and doubles configuration interaction CISD level of theory. The size of the HF primitive sets chosen for a particular cc-pvnz-pp basis set was determined primarily on its HF error relative to the estimated HF limit relative to the error in the correlation energy near the complete basis set CBS limit. Both the sizes of the HF primitive basis sets associated with a given cc-pvnz-pp basis set, as well as the number and type of correlating functions, were identical to those determined previously for the group elements. 1 These choices were, however, tested in the cases of Br and I and were found to also be appropriate for the group elements. The composition of the final cc-pvnz-pp basis sets (n D, T, Q, 5 are shown in Table VII. For the description of negative ions and weak interactions, such as hydrogen bonding and van der Waals molecules, the cc-pvnz-pp basis sets were augmented with additional diffuse functions, resulting in aug-cc-pvnz-pp basis sets (n D, T, Q, 5. A single diffuse function in each angular symmetry present in the cc-pvnz-pp basis set was optimized for the total energy of the atomic anion. The s and p exponents were obtained at the HF level of theory, while the higher angular momentum functions were obtained with the CCSD T method. 10 Of course, for very weakly bound systems, doubly augmented basis sets are more appropriate, and these can be easily generated 11 as even tempered extensions of the aug-cc-pvnz-pp sets. As in the group work, a small number of allelectron Douglas Kroll Hess 12 DK calculations were also carried out in order to test the impact of the pseudopotential approximation in atomic and molecular calculations. The basis sets for these calculations were derived from the standard cc-pvnz and aug-cc-pvnz basis sets for the post-3d elements 3 by simply recontracting the spd functions in atomic HF-DK calculations 13 subsequently denoted cc-pvnz-dk). New cc-pv5z-dk basis sets for both iodine and astatine were developed that were identical in composition to those described previously for Sb and Bi, respectively. 1 Namely a (28s23p17d) HF primitive set was optimized for I, while a (32s26p19d12f ) set was optimized for At. Both of these were constructed using the extended even-tempered ExtET framework of Ref. 1 in HF-DK calculations. After uncontracting the 4 most diffuse s and p primitives and the 3 most diffuse d primitives in each case, CISD-DK-optimized, even-tempered (1d3 f 2g1h) correlating functions were added to form the final cc-pv5z-dk basis sets for these elements. In addition, aug-cc-pv5z-dk basis sets were also determined for I and At using HF-DK s and p exponents and CCSD T -DK (dfgh exponents calculations on the atomic anions. III. ATOMIC RESULTS Atomic electron affinities calculated at the CCSD T level of theory are given in Table VIII and are depicted in Fig. 1 as a function of the aug-cc-pvnz-pp basis sets. Smooth convergence towards the apparent CBS limits are observed in each case and nearly identical basis set convergence rates are observed among the group 16 and group 17 elements. The results for the electron affinities of Se and Br obtained in all-electron CCSD T -DK calculations are very similar to the PP-based ones, with the former yielding electron affinities larger by just kcal/mol with the augcc-pv5z-dk basis set compared to the aug-cc-pv5z-pp val- FIG. 1. Calculated CCSD T electron affinities in kcal/mol as a function of the aug-cc-pvnz-pp basis sets.

6 11118 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Peterson et al. TABLE VIII. CCSD T total energies and electron affinities for the post-d group 16 and 17 atoms. The total energies (E CCSD T )) are in hartrees and the electron affinities EA are in kcal/mol. Only the valence ns and np electrons have been correlated. E CCSD T) Atom Basis set Anion Neutral EA a Se aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp Experiment b Br aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp Experiment a Te aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp Experiment b I aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp Experiment a Po aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp Experiment b 44 7 At aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp Experiment b 65 5 a Experimental values from Ref. 14 have been approximately corrected for spin orbit effects by averaging the atomic multiplets Ref. 19. b Due to the lack of appropriate experimental data, these values are not corrected for spin orbit effects. ues. Analogous calculations for I and At exhibit differences of just 0.00 and 0.15 kcal/mol, respectively. As shown in Table VIII, the CCSD T method near the CBS limit yields accurate electron affinities compared to experiment, 14 as long as spin orbit effects are accounted for. Dipole polarizabilities calculated for the rare gas atoms at the SCF and CCSD T levels of theory are shown in Table IX. These calculations employed the finite field method with TABLE IX. Atomic dipole polarizabilities in a.u. calculated for Kr, Xe, and Rn compared to experiment. Basis Kr Xe Rn SCF CCSD T SCF CCSD T SCF CCSD T aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp aug-cc-pvqz-dk aug-cc-pv5z-dk Expt. a a Reference 15. field strengths of a.u. The rate of convergence with respect to basis set is relatively rapid for each element and similar at both the SCF and CCSD T levels of theory. Compared to the all-electron polarizabilities calculated for Kr, the aug-cc-pvnz-pp values are slightly larger by about 0.3%. Compared to experiment, 15 the CCSD T /aug-cc-pv5z-pp polarizabilities are larger by just 0.14 and 0.28 a.u. for Kr and Xe, respectively. The present values for Kr and Xe can also be compared to the all-electron CCSD T /ANO results of Rice et al. 16 after including their SCF corrections for scalar relativity. These are just slightly larger 0.15 a.u. and smaller 0.04 a.u., respectively, than the 5Z results shown in Table IX. Our results are also in very good agreement with the previous relativistic PP results SCF and CCSD T of Nicklass et al. 17 IV. MOLECULAR RESULTS Near-equilibrium potential energy functions for a variety of diatomic molecules were calculated at the CCSD T level of theory by sampling 9 bond lengths over the range 0.4 a o r r e 0.7 a o. After accurately fitting to polynomials in simple internal displacement coordinates, spectro-

7 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Systematic convergent basis sets. II TABLE X. CCSD T total energies and spectroscopic constants calculated for Br 2,I 2, and At 2 compared to experiment. a Basis E min E r e Å e e x e D e kcal/mol Br 2 cc-pvdz-pp cc-pvtz-pp cc-pvqz-pp cc-pv5z-pp cc-pvqz cc-pv5z cc-pvqz-dk cc-pv5z-dk Expt. b I 2 cc-pvdz-pp cc-pvtz-pp cc-pvqz-pp cc-pv5z-pp cc-pv5z-dk Expt. c At 2 cc-pvdz-pp cc-pvtz-pp cc-pvqz-pp cc-pv5z-pp cc-pv5z-dk a Spin orbit effects have been approximately removed from the experimental dissociation energies using the atomic splittings of Moore Ref. 19. b References 30 and 31. c Reference 31. scopic constants were evaluated via the usual Dunham analysis. 18 In each case the experimental dissociation energies that are used for comparison have been adjusted for the experimental spin orbit splitting in the atoms, 19 since the present calculations do not include spin orbit coupling. While the systematic convergence of the total energies of this work should make possible their accurate extrapolation to the complete basis set limit as in previous all-electron studies cf., Ref. 20, this has not been carried out in the present work. Total energies and spectroscopic constants calculated for the dihalogens are shown in Table X, those for the hydrogen halides are given in Table XI, while those for SiSe, SiTe, and SiPo are shown in Table XII. Calculated spectroscopic constants for the protonated rare gas atoms, KrH, XeH, and RnH, are displayed in Table XIII. In each case the available experimental results are also shown for comparison. Focusing first on the results for the dihalides shown in Table X, the calculated spectroscopic constants are observed to smoothly converge with increasing basis set size. Since these results are indicative of those for the other diatomics, the basis set convergence of both r e and D e are also depicted in Fig. 2 for Br 2,I 2, and At 2. While the rate of convergence for D e is observed to be nearly the same for all three molecules, the basis set convergence towards the CBS limit for the equilibrium bond length clearly becomes slower from Br 2 to At 2. On comparison to the all-electron CCSD T /cc-pv5z-dk results also shown in Table X, the CCSD T /cc-pv5z-pp bond lengths are smaller by just Å, Å, and Å for Br 2,I 2, and At 2, respectively. The corresponding differences in D e are just 0.01, 0.24, and 0.12 kcal/mol, respectively. Differences with respect to experiment are as expected given the neglect of core-valence correlation and spin orbit coupling, as well as residual basis set incompleteness, i.e., bond lengths are too long with the cc-pv5z-pp basis set by about 0.01 Å and dissociation energies too small by 2 4 kcal/mol. For I 2, shortening of r e due to correlation of the 4d electrons has been previously determined 21 to be 0.02 Å, while r e is lengthened by about the same amount when spin orbit coupling is included. 22,23 As demonstrated by Feller et al., 24 both molecular core-valence correlation and molecular spin orbit coupling also have strong effects on the D e of I 2, increasing it by a total of 4 kcal/mol. The sum of these effects was much more modest in Br 2, 0.8 kcal/mol, as reported by the same authors. From the all-electron calculations on Br 2 both with cc-pv5z-dk basis set and without cc-pv5z basis set relativistic effects using the DK Hamiltonian, the effects of scalar relativity on the spectroscopic constants is determined to be Å on r e, 1.2 cm 1 on e, and 0.54 kcal/mol on D e. Apparently the inclusion of spin orbit coupling results in a considerable lengthening of the bond in At 2, since our results for r e are shorter by about 0.16 Å compared to the 4-component relativistic CCSD T results of Visscher and Dyall. 23 Large-core PP CCSD T calculations including core-valence correlation effects by means of a core-polarization potential CPP yielded a spin orbit

8 11120 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Peterson et al. TABLE XI. CCSD T total energies and spectroscopic constants calculated for HBr, HI, and HAt compared to experiment. a Basis E min E r e Å e e x e D e kcal/mol HBr aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp aug-cc-pv5z-dk Expt. b HI aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp aug-cc-pv5z-dk Expt. b HAt aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp aug-cc-pv5z-dk a Spin orbit effects have been approximately removed from the experimental dissociation energies using the atomic splittings of Moore Ref. 19. b References 31 and 32. induced lengthening of the At 2 bond length by 0.13 Å. 25 Adding the spin orbit and CPP contributions from that work to our present PP CCSD T /cc-pv5z-pp results for Br 2 /I 2, we obtain R e 2.281/2.657 Å, e 327.1/212.4 cm 1, and D e 45.5/37.0 kcal/mol in quite good agreement with the experimental data of R e 2.281/2.666 Å, e 325.2/214.5 cm 1, and D e 45.9/36.0 kcal/mol. Our prediction for the spectroscopic constants of At 2 derived in this way, i.e., R e Å, cm 1 and D e 20.6 kcal/mol, is in quite good agreement with the estimates made by Viss- TABLE XII. CCSD T total energies and spectroscopic constants calculated for SiSe, SiTe, and SiPo compared to experiment. a Basis E min E r e Å e e x e D e kcal/mol SiSe cc-pvdz-pp cc-pvtz-pp cc-pvqz-pp cc-pv5z-pp cc-pv5z-dk Expt. b SiTe cc-pvdz-pp cc-pvtz-pp cc-pvqz-pp cc-pv5z-pp Expt. b SiPo cc-pvdz-pp cc-pvtz-pp cc-pvqz-pp cc-pv5z-pp a Spin orbit effects have been approximately removed from the experimental dissociation energies using the atomic splittings of Moore Ref. 19. b Reference 31.

9 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Systematic convergent basis sets. II TABLE XIII. CCSD T total energies and spectroscopic constants calculated for KrH,XeH, and RnH compared to experiment. Basis E min E r e Å e e x e D e kcal/mol KrH aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp aug-cc-pv5z-dk Expt. a XeH b aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp Expt. c RnH b aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp a References 33 and 34. b The dissociation energy is cited relative to Rg H and was derived using the experimental ionization potentials of Rg and H H: ev, Xe: ev, Rn: ev and the calculated proton affinity. See and references therein. c References 34 and 35. cher and Dyall on the basis of their 4-component all-electron CCSD T results, i.e., R e Å, e cm 1, and D e kcal/mol. 23 We note that the above procedure to estimate spectroscopic constants is not entirely correct, since the CPP besides core-valence correlation effects also accounts for the static polarizability of the X 7 core, which is partly taken into account in the small-core PP calculations already. It is justified, however, since the dynamic core-polarization effects are larger than the static ones. The present CCSD T results for the hydrogen halides are shown in Table XI and exhibit similar trends as the dihalides, although the spectroscopic constants of the hydrogen halides converge much more rapidly with basis set. The errors due to the pseudopotential approximation as judged by comparison to CCSD T /aug-cc-pv5z-dk calculations are again observed to be relatively small, amounting to just , , and Å in r e for HBr, HI, and HAt, respectively, and 0.02, 0.19, and 0.02 kcal/mol, respectively, for D e. The errors with respect to experiment are due primarily to the neglect of both core-valence correlation and spin orbit coupling. On comparison of our augcc-pvtz-pp results to the 4-component relativistic calculations of Visscher et al. 26 with a similar valence basis set, spin orbit coupling is expected to lengthen the r e values by just Å for HI, but about 0.03 Å for HAt. The large-core PP/CPP calculations of Dolg yielded spin orbit induced bond length increases of and Å for HI and HAt, respectively, whereas no change was observed for HBr. 25 Adding again the spin orbit and CPP contributions from this work to our PP CCSD T /cc-pv5z-pp results for HBr/HI, we obtain R e 1.413/1.607 Å, e / cm 1, and D e 91.4/75.6 kcal/mol in satisfactory agreement with the experimental values of R e 1.414/1.609 Å, e / cm 1, and D e 90.4/73.8 kcal/mol. For AtH our corresponding estimate for the binding energy D e 58.2 kcal/mol coincides with the measured D kcal/mol, 27 whereas our bond length R e Å and vibrational frequency e cm 1 are predictions. A detailed study of the effects of spin orbit coupling with the new PPs and cc-pvnz-pp basis sets, which will remove the uncertainties of the present estimates, is currently planned. From the above discussion it is apparent that the differences between PP and DK results are somewhat larger for the iodine cases compared to Br and At. To a certain extent, similar comparisons can also be found in the As/Sb/Bi results in the preceding paper. 1 While at first one might expect a smooth variation from Br to I to At, iodine and other elements of this row are not just intermediate to Br and At. Specifically, for the post-4d elements the unoccupied 4 f shell is relatively compact due to missing orthogonality with occupied inner shells of the same symmetry. This is in contrast to the post-3d elements where the unoccupied 4d shell is diffuse due to orthogonality constraints with the occupied 3d shell, and the post-5d elements where the 6d and 5 f shells are diffuse due to orthogonality to the occupied 5d and 4 f shells. Thus one might not expect a smooth variation from Br through I to At. At this point it is not clear if the current, small differences are due to defects in the PP adjustment, PP basis sets, or the all-electron DK basis sets, and this subject is currently under investigation. Nevertheless it should be stressed that there is still very good agreement in this work between the PP and DK results for the post-4d elements.

10 11122 J. Chem. Phys., Vol. 119, No. 21, 1 December 2003 Peterson et al. are given in Table XIII. Near the basis set limit, the CCSD T spectroscopic constants are in very good agreement with the accurate experimental results, with the remaining small discrepancies being attributed mainly to the neglect of core-valence correlation. V. CONCLUSIONS In the present work, small-core valence-electron PPs and correlation consistent-type basis sets, cc-pvnz-pp and aug-cc-pvnz-pp, (n D, T, Q, 5 for the post-d group elements have been generated, which together with previous work 1,4 completes the entire block of post-d elements. These basis sets and PPs have been tested in CCSD T calculations of atomic electron affinities, rare gas polarizabilities, and the spectroscopic constants of several closed-shell diatomic molecules. Comparisons were also made to all-electron calculations using cc-pv5z basis sets, where relativistic effects were recovered via the DK Hamiltonian. New cc-pv5z-dk basis sets were generated for both I and At for this purpose. In all cases the errors due to the pseudopotential approximation were calculated to be nearly negligible, e.g., Å in r e and 0.3 kcal/mol in D e. The new cc-pvnz-pp and aug-cc-pvnz-pp) basis sets are shown to exhibit the systematic convergence and accuracy characteristic of their all-electron counterparts developed for lighter elements. Taken together, the MCDHF small-core PPs and cc-pvnz-pp basis sets should greatly facilitate accurate, systematic studies of the molecular properties of the heavy post-d elements using correlated methods. The basis sets and pseudopotentials of this work can be obtained from the EMSL Gaussian Basis Set Library. 29 They are also available from the Stuttgart/Köln pseudopotential reference file at pseudopotentials. ACKNOWLEDGMENTS FIG. 2. Convergence of the calculated CCSD T equilibrium dissociation energies (D e ) and bond lengths (r e ) as a function of cc-pvnz-pp basis set for a Br 2, b I 2, and c At 2. Lastly, benchmark CCSD T calculations for SiSe and its post-4d and -5d analogs are shown in Table XII. These results are also observed to be consistent with those discussed previously. Overall, the basis set dependence of r e and D e are observed to be very similar between SiSe, SiTe, and SiPo. The results for SiSe shown in Table XII are nearly identical to the CCSD T /SDB-cc-pVnZ results (n T,Q) of Martin and Sundermann, 28 where large-core PPs were employed. This is in contrast to molecules containing group elements, where it had been observed 1 that their use of large-core PPs yielded bond lengths and dissociation energies somewhat too small and too large, respectively. Calculated CCSD T spectroscopic constants using the aug-cc-pvnz-pp basis sets for the protonated rare gas atoms The financial support of the Deutsche Forschungsgemeinschaft DFG for D.F. is gratefully acknowledged. K.A.P. acknowledges the support of the National Science Foundation CHE , as well as the Division of Chemical Sciences in the Office of Basis Energy Sciences of the U.S. DOE. Some of this research was performed in the William R. Wiley Environmental Molecular Sciences Laboratory EMSL at the Pacific Northwest National Laboratory PNNL. Operation of the EMSL is funded by the Office of Biological and Environmental Research in the U.S. Department of Energy DOE. PNNL is operated by Battelle Memorial Institute for the U.S. DOE under Contract No. DE- AC06-76RLO K. A. Peterson, J. Chem. Phys. 119, , preceding paper. 2 T. H. Dunning, Jr., J. Chem. Phys. 90, ; D.E.WoonandT.H. Dunning, Jr., ibid. 98, A. K. Wilson, K. A. Peterson, D. E. Woon, and T. H. Dunning, Jr., J. Chem. Phys. 110, B. Metz, M. Schweizer, H. Stoll, M. Dolg, and W. Liu, Theor. Chem. Acc. 104, ; B. Metz, H. Stoll, and M. Dolg, J. Chem. Phys. 113, GRASP: atomic numerical MCDHF program package; K. G. Dyall, I. P.

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