Energy-consistent pseudopotentials and

Energy-consistent pseudopotentials and correlation consistent basis sets for the 5d elements Hf–Pt Detlev Figgen, Kirk A. Peterson, Michael Dolg, and Hermann Stoll Citation: J. Chem. Phys. 130, 164108 (2009); doi: 10.1063/1.3119665 View online: http://dx.doi.org/10.1063/1.3119665 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v130/i16 Published by the AIP Publishing LLC.

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THE JOURNAL OF CHEMICAL PHYSICS 130, 164108 共2009兲

Energy-consistent pseudopotentials and correlation consistent basis sets for the 5d elements Hf–Pt Detlev Figgen,1 Kirk A. Peterson,2,a兲 Michael Dolg,3 and Hermann Stoll4 1

Centre for Theoretical Chemistry and Physics, New Zealand Institute for Advanced Study, Massey University Albany, Auckland 0745, New Zealand 2 Department of Chemistry, Washington State University, Pullman, Washington 99164, USA 3 Institut für Theoretische Chemie, Universität zu Köln, D-50939 Köln, Germany 4 Institut für Theoretische Chemie, Universität Stuttgart, D-70550 Stuttgart, Germany

共Received 18 February 2009; accepted 26 March 2009; published online 23 April 2009兲 New relativistic energy-consistent pseudopotentials have been generated for the 5d transition metals Hf–Pt. The adjustment was done in numerical two-component multiconfiguration Hartree–Fock calculations, using atomic valence-energy spectra from four-component multiconfiguration Dirac– Hartree–Fock calculations as reference data. The resulting two-component pseudopotentials replace the 关Kr兴4d104f 14 cores of the 5d transition metals and can easily be split into a scalar-relativistic and a spin-orbit part. They reproduce the all-electron reference energy data with deviations of ⬃0.01 eV for configurational averages and ⬃0.05 eV for individual relativistic states. Full series of correlation consistent basis sets from double to quintuple-zeta have also been developed in this work for use with the new pseudopotentials. In addition, all-electron triple-zeta quality correlation consistent basis sets are also reported in order to provide calibration for the pseudopotential treatment. The accuracy of both the pseudopotentials and basis sets are assessed in extensive coupled cluster benchmark calculations of atomic ionization potentials, electron affinities, and selected excitation energies of all the 5d metal atoms, including the effects of spin-orbit coupling. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3119665兴 I. INTRODUCTION

Relativistic pseudopotentials 共PPs兲 simulate chemically invariant frozen cores of heavy atoms and, at the same time, implicitly account for relativistic effects 共both direct and indirect ones兲 in formally nonrelativistic valence-only calculations.1,2 Their use involves a trade-off between computational simplicity and accuracy. Very often, the accuracy of a specific PP is not so much determined by the potential itself, but rather by the valence basis set published together with the potential. With our new generation of energyconsistent PPs,3 we adjust the potentials in basis-set-free numerical atomic calculations, and endow them with series of correlation consistent polarized valence n-zeta basis sets4 that are analogous to the all-electron case. This enables the user to adapt the accuracy of the calculations to his specific needs, and eventually also to extrapolate to the complete basis set 共CBS兲 limit in order to determine the intrinsic accuracy of the chosen electronic structure method. Recently we have also optimized triple-zeta quality, all-electron correlation consistent basis sets for use in a Douglas–Kroll–Hess 共DKH兲 scalar-relativistic scheme in order to determine the intrinsic errors of the PPs themselves. Our current set of energy-consistent PPs is adjusted in two-component multiconfiguration Hartree–Fock 共MCHF兲 calculations to valence-energy spectra derived from fully relativistic four-component multiconfiguration Dirac– Hartree–Fock 共MCDHF兲 calculations. Thus, intrinsic errors a兲

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of the PPs may be related either to errors of the above fit, but also to transferability errors that arise when applying the atomically fitted potentials in molecular calculations and/or when including dynamical correlation effects within the valence shell. In two recent papers,5,6 we showed for the 4d elements Y–Pd that our newly generated MCDHF-adjusted PPs simulate the 关Ar兴3d10 core quite efficiently and accurately. Fitting errors were of the order of 0.01 eV for configurational averages, and of 0.06 eV for individual relativistic states. After inclusion of valence correlation, we find PP errors of 0.4 kcal/mol for atomic ionization potentials 共IP兲 and of 1.3 kcal/mol for s → d excitation energies 共absolute mean兲. Typical accuracies for diatomic molecules, in comparison to all-electron calculations, were 0.001–0.004 Å for bond lengths, and 1 – 2 cm−1 for harmonic wavenumbers. In the present paper, we offer corresponding PPs and basis sets for the 5d elements Hf–Pt. The new MCDHFadjusted PPs are meant to supersede our older MWB-type PPs,7 which 2 decades ago were adjusted to less accurate Wood-Boring 共WB兲 reference data for a multitude of manyelectron states and were essentially of one-component type. 关Spin-orbit 共SO兲 potentials were adjusted in a separate step, to reference data for highly charged single-valence-electron ions.兴 The present two-component adjustment, which yields scalar-relativistic PPs and SO potentials simultaneously and on the same footing, seems to be definitely more appropriate for the 5d elements where relativistic effects are large. In addition since the present work aims to provide systematic sequences of basis sets that allow for accurate extrapolations to the CBS limit, it is important to remove any uncertainties

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due to the more limited WB reference data and base the PP fit on a significantly broader basis with respect to the number of configurations. A detailed comparison of the MWB and MCDHF PPs can be found in Ref. 3. The following section contains the new PP parameters, together with information on details of the adjustment procedure and on the accuracy of the fit. The accompanying basis sets are characterized and discussed in Sec. III. Previously reported basis sets for these elements of triple-zeta or higher quality include the triple- and quadruple-zeta basis sets of Weigend and Ahlrichs,8 which are based on the older Andrae et al.7 quasirelativistic PPs. All-electron, relativistic basis sets based on the DKH Hamiltonian include the recent atomic natural orbital 共ANO兲 sets of Roos et al.,9 the sets of Koga et al.,10and those of Tsuchiya et al.11 Dyall12 also reported correlation consistent basis sets of double- through quadruple-zeta quality, but for use in four-component relativistic calculations. The PPs and basis sets of the present work are analogous to those previously developed for the 4d transition metal elements,6 as well as of course Au/Hg,13,14 and offer the largest range 共DZ through 5Z兲 for both frozen-core and outer-core correlations, making them appropriate for the calculation of atomic and molecular properties from qualitative accuracy to the CBS limit. A careful investigation of the accuracy of the new PPs and basis sets in high-level ab initio calculations for atomic IPs, electron affinities 共EAs兲, and s2dm−2 → s1dm−1 excitation energies follows in Sec. IV, including a comparison to allelectron calculations and experimental data, as well as a discussion of core contributions 共e.g., from the 4f shell兲, correlation contributions beyond the widely used CCSD共T兲 level, and SO coupling effects. Finally, conclusions are drawn in Sec. V. II. PSEUDOPOTENTIAL ADJUSTMENT

The adjustment of our energy-consistent PPs has been described for the 4d elements in Ref. 6. Thus, only a short X for a summary is given here for the 5d elements. The PP VPP given atom X has a one-center form and constitutes the oneelectron potential of a formally nonrelativistic atomic valence Hamiltonian HX, nval

HX = 兺 hX共i兲 + i

n

1 val 1 兺 , 2 i⫽j rij

X hX共i兲 = − 21 ⌬i + VPP 共i兲,

where i and j run over the number nval of valence electrons X , only. In the semilocal ansatz for VPP X =− VPP

Q + 兺 Vlj共r兲Plj , r lj

X the first term is the long-range part of VPP , where Q = Z − ncore is equal to the atomic number Z of atom X reduced by the number of core electrons ncore that are replaced by the PP. For reasons given in Ref. 6, we chose a “small-core” definition, with ncore = 60, for the 5d elements, i.e., we treat

the 5sp orbitals together with the 5d and 6sp ones explicitly in the valence space. The 4f shell, on the other hand, is fully X replaced by VPP , which seems to be justified in view of the good spatial separation of 4f and 5spd orbitals. Note, however, that the 4f orbital energy is higher than the 5p one for Hf, Ta, and W, and a good energetic separation is only achieved for Os, Ir, and Pt. We shall discuss this point further in Sec. IV. Together with the long-range term, the shortrange lj-dependent potentials Vlj account for all effects of the replaced core electrons 共Coulomb and exchange potentials, Pauli-repulsion, relativistic effects兲 as well as for direct relativistic effects on the valence electrons with quantum numbers l and j = l ⫾ 1 / 2; Plj is a projector on the corresponding lj-spinors. Vlj is expanded as Vlj共r兲 = 兺 rnljkBljk exp共− ␤ljkr2兲, k

and the parameters nljk, Bljk, and ␤ljk are fitted in such a way that the spectrum of HX agrees as closely as possible to the valence spectrum of the full atomic Hamiltonian. The reference configurations for the 共simultaneous兲 “multielectron” fit of the spd parts of Vlj共r兲 closely correspond to those of our previous work.6 For element X of the mth group of the periodic table, we chose • X0 : 5dm−26s2 , 5dm−26s16p1 , 5dm−16s1 , 5dm−16p1 , 5dm , 5dm−36s26p1 , 5dm−36s27s1 , 5dm−36s27p1, • X+ : 5dm−26s1 , 5dm−26p1 , 5dm−27s1 , 5dm−1 , 5dm−36s2 , 5dm−36s16p1, • X2+ : 5dm−2 , 5dm−36s1 , 5dm−36p1, • X共m−1兲+ : 5s25p6共6 , 7 , 8兲s1 , 5s25p6共6 , 7 , 8兲p1 , 5s25p6 共5 , 6 , 7兲d1, • Xm+ : 5s25p6, • X共m+1兲+ : 5s25p5 , 5s15p6. These 29 configurations lead to between 474 共hafnium兲 and 2228 共rhenium兲 relativistic levels in the valence spectrum. We calculated all these relativistic levels in state-averaged four-component MCDHF calculations using the program 15 GRASP. The Dirac–Coulomb Hamiltonian was used, and the frequency-dependent Breit correction to the two-electron terms was included perturbatively, with the frequency of the exchanged photon being calculated from the orbital energy differences.16 To represent the finite nucleus, we used the two-parameter Fermi charge distribution as implemented in 17 GRASP. The fit to the reference relativistic levels was done in corresponding valence-only MCHF calculations with the formally nonrelativistic Hamiltonian HX, using a modified version of GRASP.18 We did not optimize total valence energies but rather energy differences ⌬Eij between individual relativistic states within the spectrum. The PP valence energies for the 5d elements are 共in contrast to those for 4d兲 somewhat lower than the corresponding all-electron values; the differences amount to values between −0.59Eh 共Hf兲 and −0.12Eh 共Pt兲 and remain below 1.5% of the total valence energies for all atoms.

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Pseudopotentials and basis sets for Hf–Pt

TABLE I. Parameters 共in atomic units兲 of the PPs for Hf through W. l

j

nlj1

Blj1

␤lj1

nlj2

Blj2

␤lj2

nlj3

Blj3

␤lj3

Hf

s p p d d f f g g

1/2 1/2 3/2 3/2 5/2 5/2 7/2 7/2 9/2

0 0 0 0 0 0 0 0 0

478.237 404 0 303.140 454 9 318.296 641 4 107.788 667 2 109.891 220 5 11.227 535 9 11.232 503 7 ⫺10.432 316 1 ⫺10.381 119 1

10.248 855 5 8.683 462 2 7.916 920 0 5.189 556 9 5.253 228 6 1.912 607 9 1.880 740 0 2.724 180 1 2.708 952 9

2 2 2 2 2

0.388 023 9 0.397 937 4 0.679 661 8 0.064 779 2 0.056 235 2

10.729 127 3 9.379 242 7 8.244 208 8 5.937 871 5 5.712 906 3

0 0 0 0 0

⫺1.071 231 6 ⫺1.223 741 3 ⫺0.842 725 4 ⫺0.701 447 2 ⫺0.141 415 4

1.964 856 4 1.596 686 4 1.710 280 5 1.014 855 8 0.473 711 6

Ta

s p p d d f f g g

1/2 1/2 3/2 3/2 5/2 5/2 7/2 7/2 9/2

0 0 0 0 0 0 0 0 0

454.600 649 5 290.732 349 5 293.775 647 9 114.924 938 9 116.064 952 7 13.434 802 8 13.436 791 8 ⫺12.789 147 4 ⫺12.712 162 6

10.318 069 2 8.743 342 5 7.916 222 7 5.447 313 5 5.212 545 5 2.161 275 4 2.125 939 4 3.145 920 1 3.127 941 8

2 2 2 2 2

2.837 974 7 14.437 571 0 9.507 768 6 2.007 332 3 0.715 992 3

10.540 267 1 9.275 736 0 8.101 675 3 5.884 357 9 5.649 578 5

0 0 0 0 0

⫺0.814 735 7 ⫺1.377 518 1 ⫺0.966 879 5 ⫺0.768 068 6 ⫺0.769 266 3

2.574 725 7 2.077 127 1 2.750 372 3 1.388 180 4 1.294 398 3

W

s p p d d f f g g

1/2 1/2 3/2 3/2 5/2 5/2 7/2 7/2 9/2

0 0 0 0 0 0 0 0 0

419.227 598 6 322.044 331 2 322.049 352 7 147.203 196 4 164.472 593 6 14.542 435 7 14.544 853 0 ⫺15.306 624 1 ⫺15.197 818 0

11.063 794 9 9.338 187 9 8.430 448 1 6.205 433 1 6.122 156 6 2.307 953 4 2.270 608 7 3.583 490 8 3.562 514 9

0 2 2 2 2

41.191 306 7 0.076 325 9 0.077 843 0 0.048 841 7 0.036 593 8

8.217 641 5 9.490 020 0 9.489 947 4 6.274 556 1 6.226 375 0

0 0 0 0

⫺0.351 553 5 0.445 032 8 ⫺0.221 441 4 ⫺0.349 543 2

1.882 997 2 1.906 971 6 1.963 875 2 1.888 286 6

The f- and g-components of Vlj共r兲, in turn, were adjusted in single-electron fits to the following orbital configurations: • X共nval−1兲+ : 关Kr兴4d104f 14共5 , 6 , 7 , 8兲f 1 • X共nval−1兲+ : 关Kr兴4d104f 14共5 , 6 , 7 , 8兲g1. For Hf and Ta, only three f configurations were used since 关Kr兴4d104f 145f 1 could not be converged. The resulting PP parameters are listed in Tables I and II. The quality of the fit 共see Table III兲 can be characterized by an average root mean square error of 0.01 eV for energy differences between configurational averages, and of ⬃0.05 eV for energy differences between individual relativistic states. The maximum absolute deviations of 0.08–0.35 eV usually affect relativistic levels that are very high in energy and reduce by approximately 50% when only low-lying relativistic levels for each configuration are considered, i.e., levels less than 3 eV above the ground state level of an orbital configuration. In spite of the larger scalar-relativistic effects and SO splittings for the 5d elements, as compared to the 4d ones, the fitting errors are roughly comparable in the two cases. Interestingly, the errors for configurational averages actually decrease from the middle of the 5d row toward the end, in contrast to the 4d case, leading to significantly lower errors at the end of the row in the former case, although the number of PP parameters is the same 共except for an additional g projector for the 5d elements兲. Table IV lists

mean unsigned percent errors for atomic properties not used in the PP fitting procedure like orbital energies ␧ and r expectation values for orbital densities. The errors are between 0.1% and 1% for ␧ and between 0.2% and 2% for 具r典 and 具r2典, with an increase in the errors when going from valence to outer-core orbitals. The explanation for the latter observation comes from the fact that the pseudo-orbital transformation 共elimination of radial nodes兲 is more severe for the outer-core 5sp orbitals than for the 6s orbital, and from the fact that the reference data contain more information about 5d excitations than 5sp ones. Note that there is no consistent difference in the errors between the first and the second half of the 5d row; only for the outer-core 5sp orbitals, orbital energies are less accurate within the first half due to the high-lying 4f shell. However, there is a definite increase of the deviations with respect to the 4d row, by about a factor of 2–3 for both ⌬␧ / ␧ and ⌬具rn典 / 具rn典. Apparently, the larger size of the cores to be simulated by the PPs and the increase in relativistic effects implicitly included in the PPs becomes effective here. III. BASIS SET OPTIMIZATION

The form and construction details of the correlation consistent basis sets for Hf–Pt closely follow those reported previously for Y–Pd,6 and of course are identical to those of Au and Hg.14 Thus only a brief account will be given here. In

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Figgen et al. TABLE II. Parameters 共in atomic units兲 of the PPs for Re through Pt. l

j

nlj1

Blj1

␤lj1

nlj2

Blj2

␤lj2

Re

s p p d d f f g g

1/2 1/2 3/2 3/2 5/2 5/2 7/2 7/2 9/2

0 0 0 0 0 0 0 0 0

421.970 299 7 265.445 729 8 265.180 829 5 107.906 076 6 107.946 265 5 16.903 932 6 16.903 716 1 ⫺17.943 616 1 ⫺17.788 925 4

12.163 814 4 9.684 597 1 9.476 214 1 6.509 888 4 6.091 215 8 2.562 658 4 2.521 548 7 4.034 598 5 4.009 627 6

0 0 0 0 0 0 0 0 0

50.134 438 7 31.303 015 0 30.688 114 5 13.350 752 8 13.738 887 2

7.107 594 6 7.668 065 8 5.055 156 2 4.164 006 0 4.407 378 6

Os

s p p d d f f g g

1/2 1/2 3/2 3/2 5/2 5/2 7/2 7/2 9/2

0 0 0 0 0 0 0 0 0

424.390 376 2 264.837 008 2 264.729 095 5 111.944 270 9 111.922 746 0 19.432 775 8 19.430 429 2 ⫺20.658 021 7 ⫺20.450 886 9

13.002 001 5 10.498 940 9 9.992 523 0 7.083 440 8 6.721 179 4 2.826 275 7 2.781 198 2 4.496 980 9 4.467 738 2

0 0 0 0 0 0 0 0 0

57.122 503 0 35.145 036 0 34.372 509 5 13.875 968 2 13.700 172 7

6.962 763 5 6.588 247 7 5.037 167 5 3.794 626 2 3.748 125 0

Ir

s p p d d f f g g

1/2 1/2 3/2 3/2 5/2 5/2 7/2 7/2 9/2

0 0 0 0 0 0 0 0 0

426.958 149 0 264.226 326 8 264.187 155 4 116.083 146 1 116.063 456 1 22.128 423 1 22.124 798 7 ⫺23.424 067 2 ⫺23.153 752 8

13.831 473 9 11.106 433 1 10.456 552 1 7.548 754 3 7.265 307 9 3.098 705 8 3.049 461 3 4.969 789 7 4.935 613 9

0 0 0 0 0 0 0 0 0

65.338 812 6 38.046 779 4 36.953 924 0 15.375 619 6 15.772 600 4

7.047 913 9 6.379 614 0 5.064 726 0 3.800 968 8 3.718 280 1

Pt

s p p d d f f g g

1/2 1/2 3/2 3/2 5/2 5/2 7/2 7/2 9/2

0 0 0 0 0 0 0 0 0

429.646 087 0 264.068 751 8 263.997 294 0 108.894 629 6 108.948 512 7 24.990 512 3 24.986 719 3 ⫺26.216 434 8 ⫺25.875 944 7

14.604 500 3 11.577 161 8 10.883 843 0 7.699 609 6 7.550 807 6 3.379 869 0 3.326 255 4 5.452 019 6 5.412 584 9

0 0 0 0 0 0 0 0 0

73.156 884 4 41.046 823 4 41.121 976 2 17.546 489 8 18.986 220 9

7.218 287 2 6.424 402 9 5.224 198 1 3.961 163 5 3.872 777 5

conjunction with the new PPs discussed above, basis sets from double-共cc-pVDZ-PP兲 to quintuple-zeta 共cc-pV5Z-PP兲 quality were optimized. The underlying Hartree–Fock 共HF兲 primitive sets ranged in quality: 共8s7p6d兲 for DZ, 共10s9p8d兲 for TZ, 共14s11p10d兲 for QZ, and 共16s13p11d兲 for 5Z. In

each case these Gaussian exponents were fully optimized with only one constraint, that the ratio of two exponents of the same angular momentum must be greater than 1.6, in order to avoid coalescence and linear dependency problems. Given the plethora of low-lying electronic states in the tran-

TABLE III. Accuracy of the PP fit: maximum absolute deviations of the PP results from the all-electron valence-energy differences, with root mean square deviations given in parentheses 共all in eV兲. Relativistic states Hf Ta W Re Os Ir Pt

0.08 0.16 0.23 0.29 0.28 0.35 0.20

共0.018兲 共0.035兲 共0.050兲 共0.061兲 共0.070兲 共0.070兲 共0.057兲

Configurational averages 0.01 0.02 0.04 0.04 0.04 0.03 0.02

共0.003兲 共0.006兲 共0.010兲 共0.010兲 共0.009兲 共0.008兲 共0.008兲

Relativistic states ⱕ3 eV 0.04 0.07 0.11 0.16 0.21 0.16 0.10

a

共0.012兲 共0.023兲 共0.042兲 共0.061兲 共0.072兲 共0.062兲 共0.041兲

Relativistic states with energies ⱕ3 eV above the ground state level of each orbital configuration.

a

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Pseudopotentials and basis sets for Hf–Pt

TABLE IV. Accuracy of the PP fit: mean absolute errors of orbital energies, ⑀, and radial orbital expectation values, 具r典 and 具r2典 共in percent兲. Hf–Pt

Outer core orbitals 5d orbitals 6s orbitals All orbitals

Hf–W only

Re–Pt only

⑀

具r典

具r 典

⑀

具r典

具r 典

⑀

具r典

具r2典

0.87 0.47 0.12 0.64

2.06 0.97 0.18 1.45

1.65 0.89 0.26 1.22

1.13 0.40 0.08 0.75

1.92 0.90 0.17 1.36

1.48 0.93 0.24 1.14

0.68 0.52 0.15 0.56

2.17 1.02 0.19 1.53

1.78 0.86 0.28 1.28

2

sition metals, a key concern in developing basis sets for these elements is that the underlying HF functions should not be strongly biased toward any one state. This is known to be most important for the d-type functions and in the present work these exponents have been optimized for the average HF energy of three states with differing 5d occupations: 5dm−26s2, 5dm−16s1, and 5dm6s0, where m is the number of valence electrons in the neutral atom. The s-type functions were optimized only for the 5dm−26s2 states, while the p-type functions were obtained from calculations on the excited 5dm−36s26p1 states. In each case these primitive sets were then minimally contracted to 关2s2p1d兴 using a general contraction scheme and atomic orbital coefficients from stateaveraged HF calculations involving either two 共5dm−26s2, 5dm−16s1兲 or three 共the previous plus the 5dm6s0兲 electronic states 共two for Hf–Os and three for Ir and Pt兲, as well as the 5dm−36s26p1 state for the valence 6p contraction. The term symbols for the states used in the basis set optimizations are given in Table V, which also includes the ground states of the atomic cations and anions. With regard to the functions added to these contracted basis sets for polarization and electron correlation, the functions of s, p, and d symmetries were treated very differently than those of higher angular momentum. In the former case, ANO contractions utilizing the HF primitives were employed where the ANO coefficients were obtained from frozen-core 共5d6s兲 singles and doubles configuration interaction 共CISD兲 density matrices averaged over the same states used in the HF d exponent optimizations described above. For the DZ set a 1s1p1d set of ANO contractions was utilized, a 2s2p2d set was added in the TZ case, etc. In order to provide more flexibility, the most diffuse function of each angular momentum was also uncontracted. Shells of higher angular momen-

2

tum functions were then added in the usual correlation consistent pattern for transition metals, i.e., 1f for DZ, 2f1g for TZ, and up to 4f3g2h1i for 5Z, but in contrast to the spd cases, these were always completely uncontracted. The exponents of these functions were optimized for the average frozen-core CISD energy of the same two to three states outlined above and were constrained to follow an even tempered sequence, i.e., ␣k = ␣0␤k. The resulting basis sets are denoted cc-pVnZ-PP 共n = D, T, Q, 5兲 with contractions 关4s4p3d1f兴, 关5s5p4d2f1g兴, 关6s6p5d3f2g1h兴, and 关7s7p6d4f3g2h1i兴, respectively. These are appropriate for valence electron 共5d6s兲 correlation. Separate basis sets, denoted by cc-pwCVnZ-PP 共n = D, T, Q, 5兲, were developed to simultaneously describe both valence 共5d6s兲 and outer-core 共5s5p兲 electron correlation effects. The additional functions were optimized at the CISD level of theory on the 5dm−26s2 states employing the usual weighted core-valence 共CV兲 prescription, i.e., the optimization was strongly weighted to minimize the intershell CV correlation energy rather than simply the sum of the corecore and CV correlation energies. Starting with the spd parts of the cc-pVnZ-PP basis sets, a set of uncontracted 1s1p1d functions were optimized in this manner and added to the cc-pVDZ-PP set, while 2s2p2d sets of functions were added to each of the higher members of the series 共cc-pVTZ-PP, etc.兲. Since the higher angular momentum functions optimized for core correlation exhibited significant overlap with the existing cc-pVnZ-PP fghi valence functions, the new core correlating functions were optimized simultaneously with all the higher angular momentum valence functions except for the most diffuse valence function of each angular momentum. The optimization criterion was chosen to be the

TABLE V. Neutral atom states 共cation ground states in parentheses and anion ground states in square brackets兲 used in the basis set optimizations and calculations of the present work with their corresponding electronic configurations. Note that m refers to the total number of valence electrons 共6s and 5d兲. m 共neutral兲 Hf Ta W Re Os Ir Pt Au Hg

4 5 6 7 8 9 10 11 12

5dm−26s2 3

F,a 共 2D兲, 关 4F兴 4 a 5 F, 关 D兴 5 a 6 D, 关 S兴 6 a 5 S, 关 D兴 5 a 4 D, 关 F兴 4 a 3 F, 关 F兴 3 F, 关 2D兴 2 D, 关 1S兴 1 a S

5dm−16s1 5 F D, 共 5F兲 7 S, 共 6D兲 6 D, 共 7S兲 5 F, 共 6D兲 4 F, 共 5F兲 3 Da 2 a S 共 2S兲 6

5dm6s0 5

D S 5 D 4 F 3 F 2 D 1 S, 共 2D兲 共 1S兲 ¯ 6

5dm−36s26p1 3

F G 5 G 6 F 7 P 6 D 5 F 4 D 3 P 4

Neutral atom ground state 共in the absence of SO coupling兲.

a

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sum of the CISD weighted CV correlation energy and the frozen-core CISD total energy 共the latter being an average of two to three states兲. Thus the cc-pwCVnZ-PP basis sets are not simply the union of cc-pVnZ-PP basis sets with some additional tight functions, i.e., all of the f and higher angular momentum functions were reoptimized with the additional tight functions. For the description of negative ions, polarizabilities, and weak intermolecular interactions, diffuse-augmented basis sets were also developed, denoted aug-cc-pVnZ-PP 共and aug-cc-pwCVnZ-PP兲. These sets share a common set of diffuse functions and were constructed by simply adding an additional function in each existing angular momentum symmetry. The exponents of these functions were obtained from an even tempered extension of the cc-pwCVnZ-PP basis sets. In order to accurately assess the accuracy of the PPs of the present work, correlation consistent, all-electron basis sets of triple-zeta quality have also been developed with use of the scalar-relativistic Douglas–Kroll–Hess 共DKH兲 Hamiltonian:19 cc-pVTZ-DK, cc-pwCVTZ-DK, and aug-ccpVTZ-DK. The HF primitive sets 共32s28p19d11f兲 were optimized as in the PP case, but with use of the second-order DKH Hamiltonian 共DKH2兲. Most of the exponents were also not completely optimized, but utilized a six-term Legendre expansion as described by Petersson et al.20 In the cases of the s, p, and d functions, however, the outer five exponents of each series were optimized without this constraint 共simultaneously with the other exponents兲. The subsequent optimization of the correlating functions followed the same procedure as in the PP basis sets, including the ANO contractions of the correlating spd functions. The resulting basis sets were very similar in quality to the TZ-PP sets. Two versions of these contracted DK basis sets were actually produced that used identical primitive Gaussian functions. One used contraction coefficients 共both HF and CISD ANO兲 obtained using DKH2 while the other 共denoted cc-pVTZ-DK3, etc.兲 employed the third-order DKH Hamiltonian 共DKH3兲.21 A final concern that was addressed in the present work was the impact of not explicitly treating the 4f electrons but including them in the PPs, especially for the earlier members of this period, i.e., Hf, Ta, etc. For this purpose the cc-pwCVTZ-DK all-electron basis sets were augmented with an additional set of tight 2f2g1h functions optimized for 4f electron correlation 共final sets denoted cc-pwCVTZ-DK+ 4f兲. All basis set optimizations of this work utilized the MOL22 PRO suite of ab initio programs. IV. ATOMIC BENCHMARK CALCULATIONS

Stringent tests of the accuracy and convergence characteristics of the new PPs and basis sets involved a variety of atomic properties—IPs, EAs, and the 5dm−26s2 → 5dm−16s1 excitation energies. Molecular benchmarks are the subject of a subsequent paper.23 As in our earlier 4d transition metal work,5,6 the present benchmarks are based on restricted open-shell HF coupled cluster treatments, the majority of which were carried out at the singles and doubles coupled cluster level of theory with a perturbative estimate of con-

nected triples, R/UCCSD共T兲.24 Calculations were also carried out at the CCSDT and CCSDTQ levels of theory25 using the MRCC program of Kállay26 interfaced to MOLPRO. In each case the HF orbitals were fully symmetry equivalenced, but the coupled cluster calculations were carried out on only one of the 共usually兲 degenerate components of a given term. One of the principle advantages of using a sequence of correlation consistent basis sets for a particular atomic or molecular property is the ability to extrapolate the results to the CBS limit, thereby removing the error due to an incomplete basis set. This of course also allows one to elucidate the true intrinsic error of the chosen electronic structure method for the property of interest. In the present work we have −3 formula,27 assumed that the usual ᐉmax corr Ecorr n = ECBS +

A

3 , ᐉmax

共1兲

is also applicable to the correlation energy of the 5d transition metal elements and was utilized with QZ and 5Z basis sets. Of course in the case of transition metal cc basis sets, ᐉmax = n + 1, where n is the cardinal number of a correlation consistent basis set 共n = 4 for QZ, n = 5 for 5Z兲, and this prescription was utilized throughout the present work. The HF contributions were not extrapolated since the 5Z results were already well converged at this level of theory for the present atomic properties. Two different schemes were used in the present work to account for the effects of SO coupling on the chosen atomic properties. For the IPs and EAs, frozen-core SO multireference singles and doubles CI 共SO-MRCISD兲 calculations28 were carried out with the COLUMBUS program29 using the full PPs and aug-cc-pVTZ-PP basis sets. In these calculations the lowering of the energy due to SO coupling was calculated for the ground state of each atom 共neutral and charged兲 from the difference between two separate calculations, one without SO 共MRCISD兲 and one with 共SO-MRCISD兲. In each case the orbitals 共identical for both calculations兲 were taken from full valence 共5d and 6s兲 complete active space self-consistent field 共CASSCF兲 calculations with full symmetry equivalencing. In the case of the 5dm−26s2 → 5dm−16s1 excitation energies this procedure proved to be impractical in several cases since it was not possible to converge the SO-MRCISD calculations to the desired excited electronic states because of the very large number of intervening SO levels. So in these cases the SO effects were obtained from two-component MCDHF calculations as the difference between results from using the averaged and full PPs with the GRASP program.15 In these calculations both orbital occupations were treated in one simultaneous calculation, to describe not only the SO interaction between relativistic levels having the same orbital occupation, but also between levels with different orbital occupations. The disadvantage of this approach, however, is the lack of dynamical electron correlation effects on the SO splittings and the use of orbitals that are averaged over both orbital occupations. As shown below, this leads in some cases to an underestimation of SO effects by up to 30%. One last correction that was included in the present work was the effect due to the Breit interaction. This is included 共perturbatively兲 in the adjustment of the PPs, but one unfor-

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Pseudopotentials and basis sets for Hf–Pt

TABLE VI. Calculated atomic IPs compared to their experimental values. All values are in kcal/mol. Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

149.17 149.93 150.29 150.44 150.7 150.6

166.23 168.22 168.96 169.29 169.7 169.7

170.85 173.56 174.74 175.21 175.9 175.7

170.28 173.79 175.38 175.99 176.8 176.7

185.29 188.45 190.02 190.64 191.6 191.4

197.76 200.60 202.52 203.19 204.3 204.0

196.28 201.78 203.76 204.60 205.8 205.4

200.23 206.05 208.08 209.08 210.5 210.1

231.33 234.78 236.52 237.33 238.8 238.4

1.55 3.43 4.53 4.88 5.4

2.19 2.23 2.14 2.12 2.1

1.81 1.89 1.85 1.83 1.8

1.82 1.89 1.90 1.88 1.8

1.85 1.86 1.86 1.82 1.8

1.95 1.91 1.91 1.85 1.8

3.30 3.05 3.02 2.93 2.8

2.92 2.66 2.63 2.53 2.4

2.11 2.03 2.06 1.99 1.9

⌬DK 共DKH3—PP兲c Val: aug-cc-pVTZ Val+ CV aug-cc-pwCVTZ

⫺0.77 ⫺1.01

0.19 0.09

0.34 0.19

0.38 0.08

0.24 ⫺0.06

0.19 ⫺0.16

⫺0.07 ⫺0.59

0.18 ⫺0.39

0.26 ⫺0.26

⌬4f d ⌬CCSDT e ⌬CCSDTQ f Final without SOg Expt. 共J-averaged兲 ⌬SO h ⌬Breit i Final Expt. 共no J-averaging兲j

1.22 0.10 0.11 156.4 154.78 2.66 0.16 159.2 157.39

0.98 0.94 0.24 174.1 174.63 ⫺0.25 ⫺0.15 173.7 174.10

0.73 0.89 0.29 179.6 180.45 1.00 ⫺0.27 180.3 181.35

0.70 0.14 0.19 179.6 180.65 0.70 ⫺0.21 180.1 180.65

0.68 ⫺0.05 0.18 194.0 195.40 0.17 ⫺0.25 193.9 194.59

0.67 ⫺0.20 0.18 206.3 205.85 1.03 ⫺0.27 207.1 206.78

0.90 ⫺1.03 ⫺0.02 207.5 204.17 ⫺0.29 ⫺0.28 206.9 206.59

0.76 ⫺0.41 0.10 212.6 212.75 0.04 ⫺0.29 212.4 212.75

0.63 ⫺0.52 0.21 240.4 240.69 ⫺0.05 ⫺0.28 240.1 240.69

Valence corr., CCSD共T兲 cc-pVDZ-PP cc-pVTZ-PP cc-pVQZ-PP cc-pV5Z-PP CBS共Q5兲a CBS共aug Q5兲a ⌬CV b cc-pwCVDZ-PP cc-pwCVTZ-PP cc-pwCVQZ-PP cc-pwCV5Z-PP CBS共Q5兲a

Using Eq. 共1兲 with QZ and 5Z 共ᐉmax = 5 and 6兲. Effect of 5s5p CV correlation, CCSD共T兲共5d6s + 5s5p兲 / cc-pwCVnZ-PP− CCSD共T兲共5d6s兲 / cc-pwCVnZ-PP. c Difference between DKH3 and PP results, with both valence-only correlation 共val兲 and valence+ 5s5p 共val+ CV兲 correlation. d Effect of correlating the 4f electrons, DKH2-CCSD共T兲共5d6s + 4f5s5p兲 / aug-cc-pwCVTZ-DK4f − DKH2-CCSD共T兲共5d6s + 5s5p兲 / aug-cc-pwCVTZ-DK4f. e Difference between frozen-core 共5d6s兲 CCSDT and CCSD共T兲 calculations with the aug-cc-pVTZ-PP basis set. f Difference between frozen-core CCSDTQ and CCSDT calculations with the aug-cc-pVDZ-PP basis set. g Final= valence corr., CCSD共T兲 / CBS共aug Q5兲 + ⌬CV/ CBS共Q5兲 + ⌬DK共val+ CV兲 + ⌬4f + ⌬CCSDT+ ⌬CCSDTQ. h Effects of SO coupling calculated via SO-MRCISD/aug-cc-pVTZ-PP. See the text. i Effects due to the Breit interaction obtained perturbatively from MCDHF calculations. j References 36 and 37. a

b

tunate side effect of calibrating the PP results with DKH3 calculations is that any effects due to the Breit interaction 共which is not included in DKH3兲 is effectively removed. For the present atomic properties the contributions due the Breit term have been calculated perturbatively using the GRASP program at the MCDHF level of theory. Our present values for the Au atom 共see Tables VI–IX兲 can be compared to the previous coupled cluster results of Eliav et al.30 In this case the MCDHF-based values are within 0.1 kcal/mol of their more accurate treatment. A. Ionization potentials

Calculated results for the IPs of Hf through Hg are shown in Table VI where they are also compared to their experimental values. For these results, as well as for the EAs and excitation energies presented below, the values are organized in six sections: frozen-core CCSD共T兲 results with cc-pVnZ-PP basis sets, CCSD共T兲 effects of outer-core electron 共5sp兲 correlation with cc-pwCVnZ-PP basis sets, and then corrections due to the difference between all-electron

CCSD共T兲 DKH3 and PP calculations with TZ basis sets, the effects of 4f correlation, the impact of correlation beyond the CCSD共T兲 level of theory, and finally SO coupling effects. With regard to the basis set dependence of the frozencore CCSD共T兲 IPs, the convergence is qualitatively very similar to that reported previously for the 4d metals: An increase in the basis set always leads to an increase in the IP and the very early metals 共Hf and Ta兲 exhibit only a very small dependence on basis set. Even through the Ir atom, the IPs only increase by about 6 kcal/mol between cc-pVDZ-PP and the CBS limit. The dependence on basis set increases somewhat in the cases of Pt and Au where the difference between cc-pVDZ-PP and the CBS limit is 9.1 and 9.9 kcal/ mol, respectively. There is also little difference in the CBS limits obtained by extrapolating the cc-pVnZ-PP or aug-cc-pVnZ-PP correlation energies, with Pt–Hg showing the largest differences of just 0.4 kcal/mol. At the CBS limit the effect of correlating the outer-core 5s and 5p electrons is on the average of only about 2 kcal/ mol and are generally smaller than the analogous values re-

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TABLE VII. Calculated atomic EAs compared to their experimental values 共see footnotes to Table VI兲. All values are in kcal/mol. Ta

W

Re

Os

Ir

Pt

Au

5.10 6.58 7.09 7.22 7.3

30.79 31.92 32.48 32.59 32.7

⫺13.80 ⫺9.52 ⫺7.66 ⫺6.92 ⫺6.0

16.67 20.00 21.92 22.70 23.7

34.43 38.30 40.15 40.85 41.6

44.52 46.27 47.08 47.40 47.7

48.15 49.98 50.93 51.21 51.5

⌬CV aug-cc-pwCVDZ-PP aug-cc-pwCVTZ-PP aug-cc-pwCVQZ-PP aug-cc-pwCV5Z-PP CBS共Q5兲

⫺0.42 0.38 0.77 0.95 1.2

⫺1.90 ⫺0.72 ⫺0.16 0.10 0.5

⫺0.44 0.09 0.41 0.66 1.0

⫺1.83 ⫺1.11 ⫺0.70 ⫺0.39 0.0

⫺2.57 ⫺1.81 ⫺1.45 ⫺1.15 ⫺0.7

2.08 1.80 1.70 1.41 1.0

2.18 1.85 1.73 1.60 1.4

⌬DK 共DKH3—PP兲 Val: aug-cc-pVTZ Val+ CV: aug-cc-pwCVTZ

⫺0.67 ⫺0.34

⫺1.08 ⫺0.40

0.87 1.03

0.46 0.90

0.14 0.87

0.13 ⫺0.23

⫺0.04 ⫺0.48

⌬4f ⌬CCSDT ⌬CCSDTQ Final without SO ⌬SO ⌬Breit Final Expt. 共no J-averaging兲a

⫺0.33 0.00 0.49 8.3 ⫺1.48 0.26 7.1 7.4

⫺0.24 ⫺0.53 0.20 32.2 ⫺12.90 0.73 20.0 18.8

⫺0.61 ⫺0.05 0.37 ⫺4.3 5.21 0.10 1.0 3.4

⫺0.51 ⫺0.20 0.27 24.2 1.40 0.30 25.9 24.85

⫺0.40 ⫺0.47 0.24 41.1 ⫺4.99 0.54 36.7 36.07

0.26 ⫺0.51 0.31 48.5 ⫺0.35 0.12 48.3 49.01

0.28 ⫺0.58 0.26 52.4 0.04 ⫺0.12 52.3 53.24

Valence corr., CCSD共T兲 aug-cc-pVDZ-PP aug-cc-pVTZ-PP aug-cc-pVQZ-PP aug-cc-pV5Z-PP CBS共Q5兲

a

Reference 38.

ported previously for the 4d metals. The Hf atom is the clear exception with a CV effect of 5.4 kcal/mol and a pronounced basis set dependence. In this regard it is much more similar to Y than Zr, and like the former it is the only species in its row where ionization involves removing a d electron rather than from the valence s orbital 共see Table V兲. For the other elements the CV effect is within about 0.2 kcal/mol of the CBS limit with just the cc-pwCVTZ-PP basis set. The impact of the PP approximation, as well as fitting errors arising in the PP adjustment procedure, is estimated in this work by comparing the PP results with all-electron DKH3 CCSD共T兲 calculations. These are shown in Table VI and are denoted by ⌬DK. Two values are shown, the frozencore result with aug-cc-pVTZ basis sets 共-DK3 and -PP兲 and also this correction with valence and outer-core electrons correlated 共5d6s + 5s5p兲 using aug-cc-pwCVTZ basis sets. In the latter case it is important to note that with the DKH calculations it was required in the cases of Hf–W to move some of the 4f orbitals below the 5p orbitals before the correlation calculation in order to correlate the same electrons as in the PP calculations. This would require careful consideration in molecular calculations. Overall as shown in Table VI there are very little differences between the PP and DKH3 results for the IPs, with an unsigned average deviation of just 0.3 kcal/mol. The largest differences are observed for Hf 共⫺1.01 kcal/mol兲, Pt 共⫺0.59 kcal/mol兲, and Au 共⫺0.39 kcal/ mol兲. Presumably this is related to the d occupation changing during the ionization in Hf, while in Pt and Au these are the

only cases where the cations have 6s0 ground states. The effects of 4f electron correlation on the IPs are calculated at the DKH2-CCSD共T兲 level of theory to be nonnegligible but nearly constant across the row, ranging from 0.6 to 1.2 kcal/mol. Obviously for high accuracy work this cannot be ignored, but this must be carried out in either all-electron calculations or through use of core polarization potentials. In some cases the difference between CCSDT and CCSD共T兲 is as large as the 4f correlation effect, particularly Ta, W, and Pt, where it approaches 1 kcal/mol. As shown in Table VI, however, the contributions from connected quadruple excitations are generally very small, reaching a maximum of 0.29 kcal/mol for the W atom. The final values given in Table VI without SO coupling can in principle be compared to the experimental values that have been J-averaged over their fine structure states. At this level of comparison, most of the calculated IPs are well within 1 kcal/mol accuracy. Notable exceptions are Hf 共+1.6 kcal/ mol兲, Re 共⫺1.1 kcal/mol兲, Os 共⫺1.4 kcal/ mol兲, and Pt 共+3.3 kcal/ mol兲. Explicit inclusion of SO coupling effects, however, remove much of the remaining errors in the latter three species and, except for Hf, all calculated IPs are now within 1 kcal/mol of experiment. Configurational mixing in the Pt atom was especially strong and required a proper SO treatment in order to accurately compare to experiment 共or analogously, simple J-averaging over the experimental fine structures of the Pt 3D and Pt+ 2D terms

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Pseudopotentials and basis sets for Hf–Pt

TABLE VIII. Calculated atomic 5dm−26s2 → 5dm−16s1 excitation energies compared to their experimental values 共see footnote to Table VI兲. All values are in kcal/mol.

Valence corr., CCSD共T兲 cc-pVDZ-PP cc-pVTZ-PP cc-pVQZ-PP cc-pV5Z-PP CBS共Q5兲 CBS共aug Q5兲 ⌬CV cc-pwCVDZ-PP cc-pwCVTZ-PP cc-pwCVQZ-PP cc-pwCV5Z-PP CBS共Q5兲

Hf

Ta

W

Re

Os

39.86 39.79 39.76 39.82 39.9 39.9

22.88 22.90 22.97 23.08 23.2 23.2

⫺9.00 ⫺8.11 ⫺7.69 ⫺7.49 ⫺7.2 ⫺7.2

46.58 42.93 41.52 40.87 40.0 40.1

20.12 17.67 16.28 15.62 14.7 14.8

0.36 ⫺0.83 ⫺1.55 ⫺1.81 ⫺2.2

0.82 ⫺0.40 ⫺1.08 ⫺1.38 ⫺1.8

2.56 1.11 0.35 ⫺0.01 ⫺0.5

1.54 0.56 ⫺0.01 ⫺0.40 ⫺0.9

0.86 0.70

1.02 0.57

1.46 0.55

0.50 0.50 0.11 39.5 39.01 2.61 ⫺0.32 41.8 40.29

0.40 0.65 0.22 23.2 23.94 4.51 ⫺0.43 27.3 27.90

0.30 0.74 0.26 ⫺5.9 ⫺4.30 15.81b ⫺0.86 9.1 8.44

⌬DK 共DKH3—PP兲 Val: aug-cc-pVTZ Val+ CV: aug-cc-pwCVTZ ⌬4f ⌬CCSDT ⌬CCSDTQ Final without SO Expt. 共J-averaged兲 ⌬SO a ⌬Breit Final with SO Expt. 共no J-averaging兲c

Ir

Pt

Au

6.94 4.21 2.66 2.11 1.3 1.4

⫺7.19 ⫺10.58 ⫺12.28 ⫺12.77 ⫺13.4 ⫺13.3

⫺40.75 ⫺42.13 ⫺43.70 ⫺44.14 ⫺44.7 ⫺44.6

3.39 2.24 1.57 1.12 0.5

4.40 3.26 2.65 2.20 1.6

5.46 4.48 3.94 3.50 2.9

7.75 6.64 6.06 5.57 4.9

⫺0.76 ⫺1.15

⫺0.33 ⫺1.06

⫺0.14 ⫺1.09

0.17 ⫺1.04

0.76 ⫺0.82

0.77 ⫺0.07 0.08 38.8 40.57 ⫺3.38 ⫺0.25 35.2 33.61

0.73 ⫺0.07 0.09 15.0 18.39 0.96 ⫺0.45 15.5 14.71

0.65 ⫺0.01 0.09 2.6 3.18 6.03 ⫺0.67 8.0 8.11

0.56 0.05 0.09 ⫺10.7 ⫺9.15 8.34b ⫺0.75 ⫺3.1 ⫺2.36

0.77 0.03 0.11 ⫺39.6 ⫺40.23 13.99b ⫺0.90 ⫺26.5 ⫺26.19

a

Effects of SO coupling calculated via 2-component numerical MCDHF PP calculations. See the text. SO-MRCISD results as in Tables VI and VII. The corresponding MCDHF values are 10.72 共W兲, 8.53 共Pt兲, and 12.90 共Au兲 kcal/mol. c References 36 and 37. b

does not yield an accurate SO-averaged IP兲. Overall the effect of the Breit interaction is to decrease the IPs by just 0.2–0.3 kcal/mol. B. Electron affinities

Perhaps not surprisingly, the basis set dependence of the calculated atomic EAs shown in Table VII is somewhat stronger than that of the IPs. This is particularly true for the Re atom where the anion is clearly unstable without the inclusion of SO coupling even at the FCI/CBS level of theory. This has also been noted previously by Roos et al.9 in CASPT2 calculations. In any event the basis set convergence toward the CBS limits as the cc basis sets are increased is

well-behaved in both frozen-core 共aug-cc-pVnZ-PP兲 and CV correlation treatments 共aug-cc-pwCVnZ-PP basis sets兲. In the latter case the final CBS limit ⌬CV corrections are all on the order of 1 kcal/mol or less, and except for Ir, increase the EA. Unfortunately, however, the basis set dependence for ⌬CV is relatively strong and basis sets at least as large as aug-cc-pwCVTZ-PP must be used to approach within 1 kcal/ mol of the CBS limits. The differences between DKH3 and the PP treatment for the EA are generally below 1 kcal/mol, but as for the IPs there are large differences between the frozen-core and val + CV values. Hence these corrections, if applied, should not be carried out just at the frozen-core level. In contrast to the results for the IPs, the effects of including the 4f electrons in

TABLE IX. Differences between second- and third-order DKH IPs, EAs, and excitation energies 共⌬Es兲 of this work calculated at the CCSD共T兲/aug-cc-pwCVTZ-DK level of theory 共calculated as DKH3—DKH2 with correlation of the 5s5p5d6s electrons兲. All values are in kcal/mol.

IP EA ⌬E共5dm−26s2 → 5dm−16s1兲 ⌬E共5dm−26s2 → 5dm6s0兲

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

⫺0.19

0.19 ⫺0.18 0.31

0.21 ⫺0.23 0.34

0.23 ⫺0.22 0.39

0.30 ⫺0.27 0.48

0.36 ⫺0.31 0.57 0.97

0.50 0.30 0.66 1.22

0.58 0.09 0.69

0.59

0.27

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the correlation treatment are relatively small for the EAs, ranging from just 0.2 to 0.6 kcal/mol. Also shown in Table VII are the contributions due to full iterative connected triple and quadruple excitations in the coupled cluster treatment. Overall while these are non-negligible, they are not on average as large as previously calculated for the EAs of the 4d metals where the sum of the CCSDT and CCSDTQ corrections reached 1.2 kcal/mol for both Zr and Nb. In the case of the 5d metals the largest net post-CCSD共T兲 contribution is obtained for the Ta atom, 0.49 kcal/mol. The inclusion of SO effects is particularly important for the calculation of the EAs of these atoms since the fine structure levels of the anions are typically not known from experiment. As seen in Table VII the contributions of SO coupling to the EAs can be very large, e.g., ⫺12.9 kcal/mol for W, 5.2 kcal/mol for Re, and ⫺5.0 kcal/mol for Ir. In particular as mentioned above, the EA of Re only becomes positive with the inclusion of SO coupling. Contributions from the Breit interaction are generally small, except in the cases of W and Ir where the EAs are calculated to increase by 0.7 and 0.5 kcal/mol, respectively. While the present value for Re is the most accurate ab initio value reported to date, this is also the atom with the largest final error with respect to experiment 共⫺2.4 kcal/mol兲. Presumably a more accurate result would require a more sophisticated, fully two-component treatment of SO effects together with large basis sets and dynamic electron correlation. In all other cases, however, the final EA values are well within ⬃1 kcal/ mol of experiment using the present composite approach. All of the calculations of the present work were also carried out on Hf, including SO-MRCISD, yet the EA of Hf was still negative by 5 kcal/ mol. Therefore consistent with experiment, the Hf atom does not bind an electron. The calculated SO effects on the EAs of W and Re can be compared to those previously calculated in a variation-perturbation approach by Roos et al.9 at the CASSCF/CASPT2 level of theory. In their work SO effects of ⫺16.1 and +10.8 kcal/ mol were reported for the EAs of W and Re, respectively. In the case of Re this is more than a factor of two larger than the present SO-MRCISD result shown in Table VII, which is another indication of the difficulty in accurately calculating the EA of the Re atom. C. Electronic excitation energies

Calculated excitation energies for the atomic 5dm−26s2 → 5dm−16s1 transitions are shown in Table VIII. Except for the three early metals and particularly for the Hf atom where the frozen-core CCSD共T兲 ⌬E is essentially converged with the cc-pVDZ-PP basis set, the frozen-core excitation energies converge with basis set from above, i.e., the state with the larger number of d electrons 共5dm−16s1兲 has the strongest basis set dependence. The overall basis set effect between cc-pVDZ-PP and the CBS limits range from just ⫺0.04 共Hf兲 to +6.1 共Pt兲 kcal/mol. No more than a 0.1 kcal/ mol difference is observed between the CBS limits obtained from the cc-pVnZ-PP and aug-cc-pVnZ-PP series. As in the EAs, the ⌬CV corrections smoothly converge to their CBS limits but cover a relatively large range of values from ccpwCVDZ-PP. Even the cc-pwCVTZ-PP results differ in

J. Chem. Phys. 130, 164108 共2009兲

FIG. 1. The convergence of the excitation energy ⌬E for the atomic 5dm−26s2 → 5dm−16s1 excitation from CCSD共T兲 calculations using the series of aug-cc-pwCVnZ-PP basis sets with the 5s5p5d6s electrons correlated. In addition, the differences between all-electron 共DKH3兲 and PP results at the aug-cc-pwCVTZ level are shown.

some cases from the CBS limit by more than 1.5 kcal/mol. The overall convergence, however, of the val+ CV excitation energies with basis set is very regular, as shown in Fig. 1, and with the valence and outer-core electrons correlated, all of the CCSD共T兲 excitation energies converge from above. Obviously the late transition metals exhibit a much slower convergence to the CBS limit, and this was also observed previously for both the 3d and 4d transition metals.6,31 Also shown in Fig. 1 is the difference between all-electron DKH3 and the PP values with the aug-cc-pwCVTZ basis sets 共val + CV兲. For Re–Au these are of the same magnitude and sign of the residual basis set error in the 5Z basis set, about 1 kcal/mol. The ⌬DK values for Hf–W are smaller but with opposite sign. As shown in Table VIII, correlation of the 4f electrons results in a preferential stabilization of the 6s2 states in all cases by 0.3 共W兲 to 0.8 共Re and Au兲 kcal/mol. Somewhat surprisingly, most of the larger values occur in the late metals rather than in Hf—W. On the other hand, in the case of higher level correlation contributions, the differences between CCSDT and CCSD共T兲 is largest in the early metal atoms where they are generally larger than 0.5 kcal/mol. Likewise the contribution of connected quadruples via CCSDTQ calculations are also non-negligible for these elements, ranging from 0.1 to 0.3 kcal/mol. Both of these corrections work to stabilize the 6s2 states and their magnitudes are reminiscent of those calculated previously for the early 4d metals. However neither the CCSDT nor the CCSDTQ corrections are significant for Re through Au, which deviates somewhat from the behavior of Tc–Ag. The calculated values to this point can be compared to the experimental excitation energies provided the latter were first averaged over their SO states. These J-averaged experimental values are also shown in Table VIII. For several of the elements this comparison is very satisfactory, i.e., differences are less than 1 kcal/mol, but there are a few notable exceptions. The ⌬E without SO for W differs from the J-averaged experimental value by ⫺1.6 kcal/mol, while Os

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Pseudopotentials and basis sets for Hf–Pt

and Pt differ by ⫺3.4 and ⫺1.6 kcal/mol, respectively. Some of these differences arise from inaccuracies of the J-averaging procedure due to configurational mixing between SO coupled electronic states. When SO and Breit effects are included in the ab initio values, the Os and Pt excitation energies then agree with experiment to within 0.8 and 0.7 kcal/mol, respectively. In the cases of W and Au there are considerable differences between the MCDHF and SO-MRCISD SO effects, 10.72 versus 15.81 kcal/mol and 12.90 versus 13.99 kcal/mol, respectively. The SO-MRCISD values, which include dynamic electron correlation effects, should be more accurate and their application yields final excitation energies for W共 5D0 → 7S3兲 and Au共 2S1/2 → 2D5/2兲 that differ from experiment by just +0.7 and ⫺0.3 kcal/mol, respectively. In both cases the agreement with experiment was greatly improved by addition of Breit corrections of ⫺0.9 kcal/mol in each case. Unlike our previous results for ⌬Breit in the 4d metal atoms,6 its contribution to the present excitation energies increases from Hf to W, drops, and then steadily increases again from Re to Au. D. Third-order versus second-order DKH

While the difference in total energies between DKH2 and DKH3 is substantial, there is evidence in literature that DKH2 calculations already very accurately describe scalarrelativistic effects in the case of energy differences.32 In the present work both DKH2 and DKH3 calculations have been carried out for all of the atomic properties of the present work. The results at the CCSD共T兲 level of theory with all 5spd and 6s electrons correlated are shown in Table IX. The resulting differences between the DKH2 and DKH3 values are actually not insignificant, ranging from 0.2 to 0.7 kcal/ mol for the IPs and ⌬E’s, but more modest for the EAs, 0.1 to 0.3 kcal/mol. Also shown in this table are the 5dm−26s2 → 5dm6s0 excitation energies for both Ir and Pt, where the differences between DKH2 and DKH3 nearly double compared to the 5dm−26s2 → 5dm−16s1 transition. This is certainly in the range of concern for accurate thermochemistry treatments and the use of at least DKH3 for the 5d transition metals is recommended, either in all-electron calculations or for PP calibration as in the present work. V. CONCLUSIONS

New relativistic, small-core, energy-consistent PPs with accompanying sequences of correlation consistent basis sets from double to quintuple zeta have been generated for the 5d transition metals Hf–Pt. The two-component PPs reproduce all-electron MCDHF atomic reference data to within 0.05 eV for individual relativistic states 共within 0.01 eV for configurational averages兲. The sets of correlation consistent basis sets, both for valence-only and valence plus outer-core correlations, exhibit systematic convergence toward the CBS limit as is commonly observed 共and exploited兲 for main group elements. In order to calibrate the PP treatment, allelectron triple-zeta basis sets have also been developed within the DKH framework for all the 5d transition metals. The accuracy of the new PPs and basis sets were assessed by carrying out coupled cluster benchmark calcula-

tions on atomic IPs, EAs, and electronic excitation energies. In each case all major sources of error were systematically addressed: basis set incompleteness within both the frozencore and core correlated levels with the CCSD共T兲 method, correlation of the 4f electrons in all-electron DKH CCSD共T兲 calculations, contributions due to higher levels of electron correlation in frozen-core calculations at the CCSDT and CCSDTQ levels of theory, and finally the effects of SO coupling using either large-scale SO-MRCISD or MCDHF calculations. Basis set incompleteness in the frozen-core calculations is observed to be the most critical factor to address. For example the use of only DZ or TZ basis sets would typically contribute at least 3–7 kcal/mol to the overall error in these atomic properties. Neglect of correlation of the outer-core electrons, i.e., the 5s and 5p electrons, is observed to contribute a modest 1–3 kcal/mol at the basis set limit, although in a couple of cases 共IP of Hf and 6s2 → 6s1 ⌬E in Au兲 this reaches about 5 kcal/mol. Obtaining accuracies of 1 kcal/mol require correlation of the 4f electrons as well as consideration of at least replacing the perturbative triples of CCSD共T兲 with full iterative triples, CCSDT. The effects of the PP approximation was also investigated by comparison to all-electron DKH3 results. At least for the atomic properties studied in this work, the differences were generally below 1 kcal/mol and hence much smaller than previously observed for the 4d transition metals. Comparison of DKH3 to DKH2 indicated that the former is required for accurate work on the 5d elements, which is also in contrast to the 4d transition metals. Finally, not surprisingly for these heavy 5d elements, comparison of the theoretical results to simple J-averaged experimental values is not very accurate in several cases due to relatively strong configurational mixing among SO states, and a proper treatment of SO effects as in the present work is then required. Basis set and PP data will be available for download on the web pages of the authors,33 the Pacific Northwest National Laboratory basis set exchange website,34 and the 22 MOLPRO program website. The basis set parameters are also available as supplemental material.35 ACKNOWLEDGMENTS

K.A.P would like to thank the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, U.S. Department of Energy 共DOE兲 under Grant No. DE-FG02-03ER15481 共catalysis center program兲, as well as the National Science Foundation 共Grant No. CHE-0723997兲, for partial support of this work. Financial support to D.F. by the Deutsche Forschungsgemeinschaft and the Land Baden-Württemberg through Grant No. SFB 706 is kindly acknowledged. 1

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