Segmented Contracted Error-Consistent Basis Sets of Double- and

Article pubs.acs.org/JCTC

Segmented Contracted Error-Consistent Basis Sets of Double- and Triple‑ζ Valence Quality for One- and Two-Component Relativistic All-Electron Calculations Patrik Pollak† and Florian Weigend*,†,‡ †

Institut für Physikalische Chemie, Abteilung für Theoretische Chemie, Karlsruher Institut für Technologie, Kaiserstraße 12, 76131 Karlsruhe, Germany ‡ Institut für Nanotechnologie, Karlsruher Institut für Technologie, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany S Supporting Information *

ABSTRACT: Segmented contracted Gaussian basis sets optimized at the one-electron exact two-component (X2C) level − including a finite size model for the nucleus − are presented for elements up to Rn. These basis sets are counterparts for relativistic all-electron calculations to the Karlsruhe “def2” basis sets for nonrelativistic (H−Kr) or effective core potential based (Rb−Rn) treatments. For maximum consistency, the bases presented here were obtained from the latter by modification and reoptimization. Additionally we present extensions for self-consistent two-component calculations, required for the splitting of inner shells by spin−orbit coupling, and auxiliary basis sets for fitting the Coulomb part of the Fock matrix. Emphasis was put both on the accuracy of energies of atomic orbitals and on the accuracy of molecular properties. A large set of more than 300 molecules representing (nearly) all elements in their common oxidation states was used to assess the quality of the bases all across the periodic table. Dyall,7−13 by Balabanov and Peterson,14 by Nakajima and Hirao,15,16 or by Watanabe et al.17 For many computer programs in quantum chemistry it is much more economic to use so-called segmented contracted sets instead, for which atomic shells are not fitted as a whole, but by segments. Here, for instance the 1s orbital is described by the first cGTF, which also describes the inner part of the 2s orbital (from the nucleus to the first radial node). The part beyond the first radial node is described by a second cGTF and so on. This multiuse of cGTFs requires simultaneous optimization of exponents and contraction coefficients and care in the choice of the start values for these parameters. Further, in particular for heavier elements, it is not easy to maintain this scheme in a strict way without losing accuracy in both atomic and molecular properties. On the other hand, as strictly following such a scheme concerning both core/valence and polarization functions is difficult or even impossible anyways, one might rather focus on the pragmatic aspect of reliability of results, which was the basic idea of the system of Karlsruhe basis sets (“def2-bases”)18 for nonrelativistic all-electron treatments up to Kr and effective core potential (ECP) based treatments19−21 for Rb to Rn, in its first version without lanthanides. That time, a large set of molecules representing (nearly) each element in most of its typical

1. INTRODUCTION Application of quantum chemical methods to molecules of chemical interest requires expansion of (one-particle) wave functions by basis functions. In this way, tasks are reduced mainly to integral evaluations and matrix operations and thus tailored to the abilities of contemporary computer architectures, which allows for calculations of systems consisting of several hundred atoms. Concerning economy, a proven way is the employment of “contracted” Gaussian-type functions (cGTFs),1 combinations of primitive Gaussian-type functions (pGTFs) with fixed expansion coefficients, as integral evaluation is much easier for Gaussian than for Slater-type functions, albeit the latter may be physically more reasonable. Usually one cGTF is used for each core shell and X functions for each valence shell, resulting in ‘X-tuple-ζ valence’ bases, which have to be extended by GTFs of higher angular quantum number for the description of polarization effects (at any theoretical level) as well as for correlation effects in post-HF methods. For the latter, there are well-known schemes for the consistent extension,2 derived from the contributions of functions to the total correlation energy in post-HF methods. Often the same functions are also used for describing polarization effects. The conceptually easiest way for construction of cGTFs is the usage of the same set of pGTFs for all cGTFs. Such generally contracted sets were developed also for relativistic calculations, for instance by the Lund group,3−6 by © 2017 American Chemical Society

Received: June 8, 2017 Published: July 5, 2017 3696

DOI: 10.1021/acs.jctc.7b00593 J. Chem. Theory Comput. 2017, 13, 3696−3705

Article

Journal of Chemical Theory and Computation

derivatives of the (finite-nucleus-one-component) X2C-HF energy with respect to these parameters, followed by a relaxation procedure until reaching convergence. The whole procedure was carried out in the absence of polarization functions, which were readded in the end. For consistency with def2-bases, the s-, p-, and d-elements were optimized at the restricted open shell Hartree−Fock level and for the lanthanides at the unrestricted Hartree−Fock level; the chosen states are the same as previously and are listed in the Supporting Information, Table S1. For the elements up to Kr optimizations were started from the “def2-“ type bases, except for the SV bases for the 3d elements, for which a diffuse d-function was added (this turned out to be necessary for covalent and weakly ionic bonds). For Rb−Xe the nonrelativistic bases by May and Ahlrichs,46 “SVPall” and “TZVPall”, served as a starting point, as respective def2-bases were designed for the usage in connection with effective core potentials. Those sets were not tested for the above-mentioned molecular test set that time and in fact exhibit several smaller deficits concerning error balance, e.g. due to the lack of a diffuse p-function and also a diffuse d-function for the s- and the d-elements. Further, the p-functions are only loosely contracted. Thus, at first improved nonrelativistic basis sets were designed and optimized (at the nonrelativistic level with analytical gradients), which concerning the core and valence space differ from the SVPall/TZVPall sets mainly by the p-set and the additional diffuse d-function. For the s-functions of the TZV bases the looser contraction scheme of the previous TZVPall bases was kept. Additional polarization functions were taken from the def2-sets. After adding one steep s-function to the first cGTF to account for the changed shape in the region of the nucleus, the optimization at the X2C level was carried out in the way described above. For Cs−Rn, for which no segmented contracted nonrelativistic all-electron basis sets are available in our basis set system, developments were started by designing an SV and a TZV set for Au similar to that of the lighter homologue, Ag. The contraction length of the first cGTF of each angular quantum number was chosen in a way that all functions contribute with the same phase; the subsequent cGTFs often consist of three pGTFs. In such a cGTF, two pGTFs model the respective segment; the third corrects the deviations in the region of the radial node. Its exponent typically is between the two smallest exponents of the previous cGTF; the sign of the contraction coefficient is opposite to that of the two others. This finally yielded a {8433311/7433/53211/51} set for the double-ζ basis and a {(10)5111111111/854311/631111/61} for the triple-ζ basis. It is evident that both bases differ from the strict DZV/TZV scheme for the reasons mentioned in the Introduction. For instance, the double-ζ basis contains an additional d-function, which turned out to be necessary to prevent huge errors in the case of oxidation state 0. For the TZV basis the use of a comparably large number of primitive s-functions turned out be the best way to account for the slightly different positions of radial nodes (for instance, for the 5s orbital the fourth radial node is located at 0.419 Bohr, and for the 6s orbital the fourth radial node is located at 0.403 Bohr). The double-use of a respective segment leads to errors at the atom which may be tolerable for double-ζ but not for triple-ζ bases. The p-space of this basis is somewhat larger because of a certain flexibility needed also in the 5p shell, which at least for the lighter elements in this period may be regarded as “semi-core” shell. The d space was extended for the same reasons as in the case of

oxidation states by at least one compound was used to identify deficits of existing22−25 or newly developed bases and to correct them where necessary. This resulted in a system of “error consistent” bases with typical similar errors all across the periodic table for polarized X-tuple-ζ valence bases; for instance at the HF or DFT level for X = 2(3) typical errors in atomization energies amount to 20(5) kJ/mol per atom, and typical errors in dipole moments amount to 0.5(0.1) D. Subsequent further developments done for this system were the addition of diffuse functions for excited state calculations,26 extensions for calculations accounting for spin−orbit splitting,27,28 and the extension to lanthanides.29 The goal of the present work is the adaption of this system of proven sets for relativistic all-electron treatments, more precisely for calculations within the exact two-component decoupling (for the one-electron part), X2C,30,31 employing a finite-nucleus model32 basing on a Gaussian charge distribution with the parameters given in ref 32. A reason for using relativistic all-electron techniques instead of ECPs is the accessibility of inner shells, which are not at all described in the ECP formalism and inaccurately in the nonrelativistic formalism. Thus, for respective bases special care has to be taken for these shells, regarding energy and spatial shape as well as respective splitting when accounting for spin−orbit coupling in two-component self-consistent field procedures. In the past decade, a system of − comparably large − segmented contracted basis sets optimized for scalar-relativistic all-electron treatments within the Douglas-Kroll-Hess (DKH) technique33,34 was developed by the Sapporo group35−37 with great care, another one in the group of Jorge,38−40 and further a loosely (or almost uncontracted) system of bases by Neese and co-workers.41−43 For the lanthanides, a significantly improved variant was provided by Dolg.44 In Section 2 we first describe the design and optimization for double- and triple-ζ valence bases for elements up to Rn for one-component relativistic all-electron treatments, that is, for covering scalar relativistic effects only, and next specify extensions for two-component treatments. Additionally we present fitting basis sets for the Coulomb part of the Fock operator. In Section 3 the quality of the resulting bases is assessed, at first concerning accuracy of atomic orbitals and next concerning molecular properties (bond energies, dipole moments, energies of molecular orbitals). The section also contains a comparison to the above-mentioned basis sets developed by Jorge and co-workers as well as to that by the Sapporo group.

2. DESIGN, OPTIMIZATION, AND SPECIFICATION Orbital Basis Sets for One-Component Treatments. All calculations were done with the X2C implementation31 in TURBOMOLE.45 In all cases the optimization of X2C bases started from an existing or previously optimized nonrelativistic basis. First, improved contraction coefficients were obtained by resolving contractions and taking the MO coefficients resulting from an X2C-HF calculation as new contraction coefficients. This was done successively, that is, one started by resolving the first contracted s-function and took the respective MO coefficients of the 1s level as new contraction coefficients; next, the second contracted s-function was resolved, and the respective MO coefficients of the 2s level were taken as contraction coefficients, etc. From this improved starting point the simultaneous optimization of exponents and contraction coefficients was carried out by repeated calculation of numerical 3697


Article

Journal of Chemical Theory and Computation Table 1. Contraction Schemes for Bases x2c-SV(P)all, x2c-SVPall, x2c-TZVPall, and x2c-TZVPPall

Li Be B−Ne Na Mg Al−Ar K Ca Sc−Cu Zn Ga,Ge As−Kr Rb,Sr Y−Ag Cd In−Xe Cs, Ba La Ce−Yb Lu Hf−Au Hg Tl−Rn

SV

(P)

P

TZV

P

PP

s/p/d/f 511 511 511/31 533/5 533/5 533/511 63311/53 63311/53 63311/53/41 63311/53/41 63311/5311/41 63311/5311/41 733311/633/5211 733311/6331/5211 733311/6331/5211 733311/63311/53 8433311/7433/5211/4 8433311/7433/53211/4 8433311/74321/53211/511 8433311/74321/53211/51 8433311/7433/53211/51 8433311/74331/53211/51 8433311/7432111/533/51

p/d/f 21 31 -/1 1/1 11/1 -/1 1/11 1/31 1/1 1/1 -/1 -/1 1/1 1/1 11/1 -/1 1/1 1/1 1/1 1/1 1/1 1/1 -/1

p/d/f 21 31 -/1 1/1 11/1 -/1 1/11 1/31 1/1/1 1/1/1 -/1 -/1 1/1 1/1/1 11/1/1 -/1 1/1 1/1 1/1/-/1 1/1/-/1 1/1 1/1 -/1

s/p/d/f 62111 62111 62111/411 73211/51 73211/51 73211/51111 842111/631 842111/631 842111/631/411 842111/631/411 842111/63111/51 842111/64111/51 94211111/9531/62111 94211111/95311/62111 94211111/9531/62111 94211111/953111/631 (10)5111111111/854311/63111/4 (10)5111111111/854311/631111/41 (10)5111111111/854311/631111/5111 (10)5111111111/854311/631111/61 (10)5111111111/854311/631111/61 (10)5111111111/854311/631111/61 (10)5111111111/8543111/64111/61

p/d/f 111 121 -/11/1 11/111 11/111 -/21/1 1/111 11/211 11/1/1 11/1/11/1 -/11/1 -/11/1 11/11/1 1/1/1 11/1/1 -/1/11 1/1 1/1 1/1/-/1 1/1/-/1 1/1 1/1 -/1/1

p/d/f/g 111/1 121/1 -/11/1 11/111 11/111 -/111/1 1/111 11/211 11/1/11/1 11/1/11/1 -/11/1 -/11/1 11/11/11/1 1/1/11/1 11/1/11/1 -/1/11 1/1/1/1 1/1/1/1 1/1/-/21 1/1/-/21 1/1/1/1 1/1/1/1 -/1/1

“valence-1” shell in ECP-based basis sets showed that the main problem is the description of the more compact part of the subshells (p1/2, d3/2).27 That time, one additional function was added for each nonvalence p- and d-shell that is not covered by the ECP. The analogous route was pursued this time. In detail, for elements up to Xe, a single pGTF was added to each nonvalence p- and d-shell and optimized by minimization of the total energy at the two-component X2C level. An exception is the 2p shell, where this turned out to be not sufficient. Here a cGTF was added with exponents taken from the first p-type cGTF of the one-component set. Contraction coefficients were obtained by taking the MO coefficients of the 2p1/2 spinor as start values and optimizing them by minimizing the total X2C energy (without changing the exponents). For Cs to Rn one more (steep) d-pGTF turned out to be necessary, and further the cGTF for the 2p shell had to be extended by one (steep) pfunction. The extensions were optimized for the triple-ζ valence bases but may also be used for the double-ζ valence bases. The contraction schemes (p/d) of the extensions thus are {5} for Na to Ar, {61} for K to Kr, {811/1} for Rb to Xe, and {9111/ 111} for Cs to Rn. For these bases the suffix “-2c” was added. Coulomb-Fitting Basis Sets. Density-fitting (or resolution of the identity, RI) for the Coulomb part (RI-J) meanwhile is standard in most quantum-chemical codes. It significantly accelerates the execution, in particular in calculations with nonhybrid density functionals, and it is a comparably robust and accurate approximation. For instance, with previously optimized RI-J bases47 one typically gets errors of less than 20 μEh (1 Eh = 2625.5 kJ/mol) for an isolated atom and less than 100 μEh per atom for the total energy in a compound, which is small compared to the errors obtained with double-, triple-, and usually also quadruple-ζ bases. In the case of RI-J, where only the total density needs to be approximated, the design of reasonable auxiliary basis sets is not very critical and may even be done automatically.48 Nevertheless, we decided to provide

the double-ζ basis. The finally resulting basis for Au was used as a starting point for the neighbor atoms. In detail, at first exponents were scaled with factors determined by energy minimization; next, start values for contraction coefficients were obtained by step-by-step decontraction, HF calculation, and recontraction with the respective MO coefficients. Starting from this point, the full optimization was done. By this procedure for each element the same type of minimum is reached. For the other elements of the period rather small changes were necessary. For the lanthanides higher flexibility in the f space is required (double-ζ: {511}, triple-ζ {5111}), and for the s-elements a single cGTF ({4}) is sufficient; for the pelements the p-space had to be improved toward higher flexibility, whereas the d-space could be slightly reduced. In the end, the polarization functions from the respective def2-bases were added. For consistency with previous developments the resulting basis sets were termed x2c-SV(P)all, x2c-SVPall, x2cTZVPall, and x2c-TZVPPall. The resulting contraction schemes are displayed in Table 1. For the calculation of properties − in particular for molecular response calculations − the augmentation by diffuse functions might be advisible. For the present bases, this is not a straightforward procedure, as the bases in part already contain moderately diffuse functions, for example in order to keep errors in dipole moments below the limits. Thoroughly designed respective extensions for the def2-bases were proposed by Rappoport and Furche,26 which may also be used for the bases presented here. Extensions for Two-Component Treatments. Spin− orbit coupling leads not only to energetic but also spatial splitting of inner shells yielding a more compact subshell for j = l − s and a more diffuse subshell for j = l + s. This splitting has to be covered by the basis set, otherwise errors are huge; the change in total energy caused by spin−orbit coupling may even have the wrong sign. More detailed analysis carried out for the 3698


Article


core and valence space even-tempered sets were employed with a factor of 4 10 between the exponents. The number of functions per angular quantum number was determined by the requirement that any further steep or diffuse function changes the atomic energy by less than 10 μEh. For polarization the respective functions of the dhf-QZVPP basis sets were taken. The schemes of the resulting reference bases are shown in Table S2. In the following, the differences between the x2cbases and the reference bases are termed “errors”. Total and Orbital Energies of Atoms. In Table S3 the errors in total one- and two-component Hartree−Fock energies are listed. As expected, errors increase with the atomic number, for SV(P) basis sets from ca. 20 mEh for Be to 466 mEh for As and further to 1500 mEh for Bi at the one-component level; respective numbers for TZVP bases are 0.5 mEh, 78 mEh, and 300 mEh. Errors at the two-component level usually are slightly larger, in particular for the heavier elements, but still similar to those of def2-bases for nonrelativistic treatments (for instance the errors of def2-bases for As amount to 322 mEh with def2SVP and 44 mEh with def2-TZVP). Of higher relevance are energies of orbitals, in particular that of inner shells, as their accessibility is one of the advantages of all-electron methods compared to ECPs. For the innermost shells (1s, 2s, 2p), for which energies and also absolute errors are largest, the average deviations are listed separately for each period in Table 3. For the two-component basis sets additionally the errors in the spin−orbit splitting are given. For triple-ζ bases at the HF level mean errors are below 1 eV throughout, with double-ζ bases typically around 1 eV is achieved for the 2s2p shell and somewhat larger numbers for the 1s shell. The spin−orbit splitting shows errors of 0.3 eV or less. Standard deviations (not listed) are smaller than mean values by a factor of 4 to 10 in all cases. The overall maximum error for double-ζ bases is observed for the 1s-shell of Tc, 6.7 eV, for triple-ζ bases for the 1s shell of Cd, 2.9 eV. Generally, within a period the errors increase from the left to the right. The maximum errors of the sixth period may serve as the most striking example. In the case of the triple-ζ bases, for Cs the maximum error amounts to 0.37 eV; for Po, which is the worst case within this period, 2.57 eV are found; similar is true for the double-ζ bases: 1.67 eV for Cs to 6.52 eV for the worst case, At. For DFT treatments, for which basis sets are not optimized, errors for the 2s and 2p shells tend to be slightly higher amounting to up to 2 eV. Surprisingly, the errors for the 1s shell are in the same range but not larger. These numbers have to be related both to the orbital energies themselves (e.g., ca. −90000 eV for the 1s orbital of Bi and a spin−orbit splitting of ca. 2400 eV for the 2p shell) and further to the differences caused by changing the quantum chemical method; for instance the energy of the Bi(1s) orbital obtained with DFT(BP86) is higher by ca. 735 eV than with HF. Further, the energy of this orbital significantly depends on the model employed for the nucleus; when for instance employing a point charge model instead of the finite-size model it changes by ca. 50 eV. The error of the RI approximation for energies of (occupied) orbitals is very small, typically 10−3 eV; the largest error was observed for the 2p orbital of Te at the one-component SVP level, 0.03 eV. Characteristic Molecular Quantities for a Representative Set of Molecules. The reliability of the x2c basis sets for practical use was ensured by determining the errors in atomization energies, dipole moments, and HOMO energies for a large set of molecules representing (nearly) each element

optimized RI-J auxiliary basis sets for the relativistic all-electron bases, which were derived from those optimized for the def2bases previously in the following way. The s-type functions, which for totally symmetric systems yield the only contribution, were replaced by n (primitive) functions resulting from a welltempered fit of the form ⎞ ⎛ i2 ; ηi + 1 = βηi⎜1 + γ 2⎟ (n + 1) ⎠ ⎝

i = 0, 1, ...n − 2

n was chosen to be 12 for Li to Ne, 14 for Na to Ar, 19 for K to Kr, 25 for Rb to Cd, 26 for In to Xe, and 34 for Cs to Rn. The parameters β and γ were optimized by maximization of the Coulomb energy of the atoms for orbitals obtained at the onecomponent X2C-UHF level in C1 symmetry with the x2cTZVPall basis set. Next, all contractions of functions of higher angular quantum number were resolved. For several atoms with partially filled shells this was not sufficient to keep errors for the atom below the above-mentioned threshold, as these shells cause deviations from the total symmetry of the electron density of the atom, which needs to be described by the RI-J fitting functions; for instance, for the boron atom in C1 symmetry one gets a singly occupied pz (or px or py) orbital, which needs to be represented by a respective d-function in the fitting basis. This might be considered to be an artifact of the symmetry-free treatment, but as for instance atomization energies often are calculated exactly in this manner, it was decided to account for this by modifying the d-functions for the p-elements, the g-functions for the d-elements, and the ifunctions for the f-elements when necessary, either by adding a steep function or − if errors are still larger than noted above − by a well-tempered set. The schemes of the finally resulting sets are listed in Table 2 together with average errors for groups of Table 2. RI-J Auxiliary Basis Setsa s/p/d/f/g/h/i Li−Ne Na−Ar K−Ca, Ga−Kr Sc−V, Fe−Ni Cr−Mn, Cu−Zn Rb−Sr In−Xe Y−Nb Mo−Cd Cs−Ba, Tl−Rn La, Hf−Hg Ce−Lu

12/5/4/2/1 14/5/5/2/1 19/5/5/3/1 19/5/5/3/4 19/5/5/3/3 25/5/5/3/1 26/5/5/3/1 25/5/6/3/5 25/5/5/3/5 34/5/5/3/1 34/5/5/3/5 34/7/6/6/6/3/5

one-component 1.98 4.19 4.85 6.93

± ± ± ±

2.03 2.97 3.69 4.94

two-component 1.28 3.35 3.36 6.10

± ± ± ±

1.19 2.12 2.47 3.95

3.86 ± 3.56

1.53 ± 1.00

4.17 ± 4.53

4.11 ± 6.52

5.89 ± 4.99 11.54 ± 8.39 10.07 ± 7.19

3.09 ± 1.83 8.86 ± 8.33 7.81 ± 8.24

a

Column “s/p/d/f/g/h/i” lists the number of (primitive) Gaussians per angular quantum number, the subsequent columns average error (±standard deviation) at the atom in μEh, when used in connection with x2c-TZVPall bases at the one-component HF level and with the x2c-TZVPall-2c bases at the two-component HF level.

elements with the same scheme. For both one- and twocomponent treatments errors amount to a few μEh and thus are in the range of previously developed RI-J bases for nonrelativistic and ECP-based sets.

3. DOCUMENTATION OF ACCURACY Reference Basis Sets. For the evaluation of the accuracy large reference basis sets were constructed as follows. For the 3699


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Journal of Chemical Theory and Computation Table 3. Errors in Atomic Orbital/Spinor Energies in eV for the 1s and 2s2p Shellsa Hartree−Fock

DFT(BP86)

SV(P)

TZVP

SV(P)-2c

TZVP-2c

SV(P)

TZVP

SV(P)-2c

TZVP-2c

0.21 1.11 0.31 0.20

0.02 0.12 0.03 0.03

0.11 0.27 0.09 0.08

0.15 0.06 0.06

0.65 0.79 1.13

1.48 0.88 0.97

2.25 1.01 1.25

0.68 0.34 0.22

1.93 1.29 2.00

1.61 1.10 1.32

3.78 1.30 0.98

0.59 0.40 0.38

0.02 0.11 0.04 0.04 0.01 0.14 0.03 0.04 0.09 0.62 0.23 0.11 0.12 0.59 0.25 0.38 0.32

0.26 0.66 0.30 0.18

1.51 0.39 0.32

0.21 1.61 0.43 0.32 0.02 3.11 0.62 0.34 0.09 4.38 1.80 0.88 0.17 4.82 1.55 1.26 0.28

2.49 1.23 1.58

2.04 1.33 1.44

0.26 1.79 0.65 0.59 0.02 2.40 0.32 0.51 0.06 4.95 2.01 1.08 0.41 5.60 1.34 1.11 0.40

0.11 0.17 0.08 0.07 0.03 0.54 0.57 0.62 0.06 0.77 0.27 0.16 0.05 0.74 0.27 0.31 0.31

Li−Kr,1s Na−Ar,1s 2s 2p 2p3/2-2p1/2 K−Kr,1s 2s 2p 2p3/2-2p1/2 Rb−Xe,1s 2s 2p 2p3/2-2p1/2 Cs−Rn,1s 2s 2p 2p3/2-2p1/2

a Listed are absolute deviations averaged over the respective period. For the two-component cases the line “2p” refers to the 2p1/2 spinor, here additionally the errors in the spin-orbit splitting, “2p3/2-2p1/2”, are given.

Table 4. Errors in Atomization Energies per Atom in kJ/mol at the HF Level (Left) and the BP86 Level (Right)a Hartree−Fock N 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p a

19 40 12 31 10 47 24 8 34 20 8 58 33 17

SV(P) 4.1 6.2 1.8 15.3 5.6 13.0 7.8 8.0 18.4 13.7 12.9 4.0 13.6 10.5

± ± ± ± ± ± ± ± ± ± ± ± ± ±

4.2 6.3 1.8 9.2 4.0 8.7 5.5 4.4 16.3 10.5 7.3 3.2 9.4 9.3

SVP 2.9 4.0 1.4 13.1 5.4 9.0 6.4 4.0 6.1 12.0 12.6 4.5 13.1 9.6

± ± ± ± ± ± ± ± ± ± ± ± ± ±

3.2 2.7 1.6 9.3 4.0 6.6 6.3 5.0 5.6 11.9 7.6 3.2 9.6 10.0

DFT(BP86) TZVP 3.3 1.4 1.5 1.7 3.4 3.3 1.4 3.3 6.0 2.6 3.8 1.6 4.7 2.8

± ± ± ± ± ± ± ± ± ± ± ± ± ±

2.1 0.8 1.6 1.1 1.8 2.7 1.0 1.3 3.8 2.3 1.8 1.8 2.8 2.2

TZVPP 2.4 0.7 1.4 1.4 3.2 1.9 1.2 1.3 2.1 2.5 1.1 1.1 0.9 2.8

± ± ± ± ± ± ± ± ± ± ± ± ± ±

2.2 0.4 1.6 1.1 1.9 1.9 1.0 0.7 1.7 2.4 0.8 1.0 0.9 2.2

SV(P) 3.7 6.8 2.5 10.5 7.8 4.9 3.6 5.2 6.0 8.0 10.5 7.2 4.3 6.3

± ± ± ± ± ± ± ± ± ± ± ± ± ±

3.7 7.2 2.5 6.8 6.8 3.3 2.9 5.3 4.7 6.1 5.7 4.9 2.9 5.5

SVP 2.3 4.1 2.3 8.8 7.7 5.5 2.6 6.0 5.7 7.1 10.4 7.9 4.0 6.1

± ± ± ± ± ± ± ± ± ± ± ± ± ±

TZVP 3.1 3.9 2.6 6.1 6.9 3.6 2.6 4.8 3.3 6.7 5.8 5.1 2.9 5.8

3.0 1.3 1.3 1.3 3.4 1.9 1.1 2.5 3.5 1.5 1.8 1.4 2.5 1.9

± ± ± ± ± ± ± ± ± ± ± ± ± ±

1.9 1.1 1.4 0.8 2.0 1.6 0.8 1.2 1.8 1.3 1.3 1.3 1.2 1.3

TZVPP 2.1 0.5 1.3 1.0 3.3 1.2 1.1 1.1 0.8 1.6 0.7 1.3 0.8 2.0

± ± ± ± ± ± ± ± ± ± ± ± ± ±

1.9 0.5 1.5 0.7 2.1 1.1 0.8 0.7 0.7 1.3 0.6 1.4 0.5 1.2

Listed are average values and standard deviations. N denotes the number of compounds.

and b) one overall does not observe systematic dependence of errors on the element group (2s, 2p, 3s,...). We now compile the most important features of errors for the characteristic molecular properties, starting with atomization energies (per atom). For this property, the anionic species in the test set were omitted, as for them atomization energies are problematic. If one calculates the decomposition to n−1 atoms and one ion, the error becomes unusually large but due to the atomic ion, not due to the ionic molecule; on the other hand, neglecting the charge for the products (atoms) as done previously violates the charge balance and thus is not of practical relevance. For the higher polarized variant of the double-ζ sets, x2c-SVPall, at the HF level one typically gets average errors of 1−13 kJ/mol and standard deviations of 2−12 kJ/mol, which in the following is abbreviated as (1···13) ± (2··· 12) kJ/mol. For the 2s and the 3s elements they are somewhat smaller, (1···3) ± (2···3) kJ/mol, than for the others, (4···13) ± (3···12) kJ/mol. This is very similar to the errors of the def2bases (see Figure 2 in ref 18), indicating ca. 6 ± 9 kJ/mol for

in (nearly) all its common oxidation states by at least one compound, just like for the previously developed errorbalanced Karlsruhe sets in their original version (“def2”) for s-, p-, and d-elements18 and lanthanides29 as well as in their improved version adapted for Dirac-Hartree−Fock ECPs (“dhf”).28 The set used here combines the previous sets; Cartesian coordinates are listed in Table S4. These extensive tests were done for the basis sets for one-component treatments only, as two-component extensions affect only the inner shells and do not change the quality of the valence/ polarization space. Statistical evaluation resolved by compounds of groups of elements (2s, 2p, 3s, etc.) is shown in Table 4 (atomization energies at levels HF/SV(P)/P, HF/TZVP/PP, BP86/SV(P)/P, BP86/TZVP/PP), Table 5 (dipole moments), and Table 6 (HOMO energies). Overall, results are similar to that obtained for nonrelativistic or ECP-based sets, that is, a) errors significantly and systematically are smaller for triple-ζ than for double-ζ bases, 3700


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Table 5. Errors in Dipole Moments in 10−2 D at the HF Level (Left) and the BP86 Level (Right) for the Compounds of the Test Set with Nonzero Dipole Momenta Hartree−Fock N 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p a

7 23 6 15 7 19 14 8 19 15 8 46 15 11

SV(P) 21 17 27 30 38 30 16 31 39 16 23 22 26 22

± ± ± ± ± ± ± ± ± ± ± ± ± ±

10 11 18 19 31 32 10 18 32 12 13 19 17 18

SVP 18 13 26 28 39 27 16 27 21 17 22 20 25 22

± ± ± ± ± ± ± ± ± ± ± ± ± ±

DFT(BP86)

TZVP

12 8 19 16 30 27 10 19 12 11 12 17 16 19

11 5 12 7 13 7 4 10 15 5 13 8 8 7

± ± ± ± ± ± ± ± ± ± ± ± ± ±

7 4 12 4 11 5 3 7 11 3 13 6 5 6

TZVPP 8 2 12 5 13 5 3 6 7 5 15 8 4 7

± ± ± ± ± ± ± ± ± ± ± ± ± ±

8 2 12 4 11 4 2 8 6 4 12 7 3 6

SV(P) 36 17 42 29 68 12 24 52 7 26 58 15 10 33

± ± ± ± ± ± ± ± ± ± ± ± ± ±

24 12 32 17 40 10 20 42 7 18 34 15 9 20

SVP 34 11 42 25 69 11 24 53 7 27 58 16 9 33

± ± ± ± ± ± ± ± ± ± ± ± ± ±

27 8 32 14 39 9 20 41 9 18 34 11 9 20

TZVP 10 7 11 7 17 4 5 10 9 5 25 6 4 6

± ± ± ± ± ± ± ± ± ± ± ± ± ±

8 6 17 5 12 3 4 9 6 4 19 5 2 6

TZVPP 8 3 11 5 17 4 5 9 5 5 23 6 3 6

± ± ± ± ± ± ± ± ± ± ± ± ± ±

8 2 7 3 12 4 4 9 4 4 16 6 3 6

Listed are average values and standard deviations. N is the number of compounds in the respective group.

Table 6. Errors in Energies of the Highest Occupied Molecular Orbital in 10−2 eV at the HF Level (Left) and the BP86 Level (Right)a Hartree−Fock N 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p a

19 40 12 31 10 48 26 8 34 22 8 58 34 18

SV(P) 14 9 17 11 34 16 11 23 13 13 29 9 15 17

± ± ± ± ± ± ± ± ± ± ± ± ± ±

15 9 18 10 23 14 9 29 12 12 28 9 13 12

SVP 13 11 17 12 34 15 12 22 10 13 29 9 15 17

± ± ± ± ± ± ± ± ± ± ± ± ± ±

16 11 19 10 23 13 9 27 8 12 28 9 13 12

DFT(BP86) TZVP 4 2 3 3 8 3 4 5 4 4 5 3 4 4

± ± ± ± ± ± ± ± ± ± ± ± ± ±

4 2 3 2 5 3 3 5 4 3 5 4 4 4

TZVPP 3 2 3 3 8 2 4 4 3 4 3 2 2 4

± ± ± ± ± ± ± ± ± ± ± ± ± ±

3 2 3 2 5 3 3 3 2 3 3 3 2 4

SV(P) 18 37 15 29 36 18 28 18 10 31 20 13 11 28

± ± ± ± ± ± ± ± ± ± ± ± ± ±

20 30 19 17 29 14 21 20 9 23 25 14 6 19

SVP 18 38 15 29 37 17 28 17 9 31 20 14 11 28

± ± ± ± ± ± ± ± ± ± ± ± ± ±

20 29 19 17 29 14 20 19 9 23 26 14 6 19

TZVP 6 7 3 7 10 5 7 5 3 7 5 6 3 6

± ± ± ± ± ± ± ± ± ± ± ± ± ±

5 6 5 3 7 12 5 6 3 5 4 10 2 5

TZVPP 4 6 3 7 10 4 7 4 2 7 3 5 2 6

± ± ± ± ± ± ± ± ± ± ± ± ± ±

4 5 5 3 7 12 5 5 2 5 3 10 1 5

Listed are average values and standard deviations. N denotes the number of compounds.

the s-, 8 ± 12 for the p-, and 12 ± 10 kJ/mol for the d-element compounds; note the numbers in the previous work meant the difference to QZVPP bases, whereas the term “errors” in the present work means the deviation to a reference basis much larger than QZVPP. As previously, neglecting the polarizing ffunctions for the 3d and 4d elements (basis x2c-SV(P)all) yields significantly larger errors for the respective compounds, in particular those for the 4d element compounds (18 ± 16 kJ/ mol) are larger than desired. Further − smaller −differences are due to the presence/absence of a p-function at H. At the DFT level errors are somewhat smaller, and the sensitivity on polarization functions is less pronounced, just as observed previously. Here the x2c-SV(P)all set is sufficient to obtain average errors between 4 and 11 kJ/mol and standard deviations between 3 and 7 kJ/mol. Also for triple-ζ bases results are very similar to that obtained with def2-bases. For the higher polarized variant, x2c-TZVPPall, errors amount to (0.7··· 3.2) ± (0.4···2.4) kJ/mol at the HF level and to very similar numbers at the DFT level. For the latter, no large changes are observed for the lower polarized set, x2c-TZVPall, but at the

HF level deficits for the d-elements are evident, e.g. 6.0 ± 3.8 kJ/mol for 4d. Results for dipole moments are displayed in Table 5. For the 213 systems with nonvanishing dipole moments we checked that for any molecule the vector of the dipole moment pointed to the same direction for all bases. Thus, only the absolute values are considered. For dipole moments with double-ζ bases errors for s-element compounds are larger than for the others, for instance at level DFT/x2c-SV(P)all one obtains (0.36··· 0.68) ± (0.24···0.42) D for the s-elements and (0.07···0.33) ± (0.07···0.20) D for the others. This trend holds independent from the method and the polarization set and was also seen before: for the def2-SV(P) bases 0.42 ± 0.30 D were obtained for the s-elements, but 0.09 ± 0.23 D were obtained for the p-, and 0.00 ± 0.19 D were obtained for the d-elements, see Figure 4 in ref 18. The reason for this is rather the basis of the bond partner than that of the s-element, which for this special purpose is not flexible enough (electronic overload caused by highly ionic bond situation). Like in the case of the def2-bases, this imbalance is much less pronounced for triple-ζ bases. For the lower polarized variant at 3701


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Table 7. Contraction Schemes (s/p/d/f) and Total Number of Spherical Harmonic Functions for Bases DKH-DZP, x2cSV(P)all, Sapporo-DZP, x2c-TZVPall, and Sapporo-TZP for Elements O, Cl, Au, and Pb DKH-DZP O Cl Au Pb

4/2/1 7/4/1 8/7/4/2 8/7/4/2

x2c-SV(P)all 15 24 63 63

3/2/1 4/3/1 7/5/6/2 7/7/4/2

Sapporo-DZP 14 18 66 62

5/4/2 6/5/3 9/7/6/3 9/8/7/3

x2c-TZVPall 27 36 81 89

5/3/2/1 5/5/2/1 11/7/7/2 11/7/6/3

Sapporo-TZP 31 37 81 83

7/5/3/2 8/7/5/3 11/9/7/4/2 11/10/8/5/2

51 75 119 134

Table 8. Electron Density ρ = ρα + ρβ, Spin Density σ = ρα − ρβ, and Electrostatic Potential V Arising from the Electrons at the Nucleus (All in au) and Energies of the Inner Orbitals in eV at the HF and DFT(BP86) Levelsa reference size

DKH-DZP 3.941

0.529

ρ·106 σ V ε(1s) ε(2s) ε(2p) ε(3s) ε(3p) ε(3d) ε(4s) ε(4p) avg

6.9554 395.69 −629.95 −81164.91 −144753 −125677 −3483.68 −2918.11 −2286.31 −792.61 −603.15

−1.8555 −105.70 6.23 −155.72 −95.48 −114.16 −40.60 4.59 −61.17 −11.39 5.86 48.66

ρ·106 σ V ε(1s) ε(2s) ε(2p) ε(3s) ε(3p) ε(3d) ε(4s) ε(4p) avg

7.0637 439.16 −6307 −80479.10 −14197.20 −123334 −3351.78 −2807.03 −2199.15 −727.52 −547.75

−1.9587 −138.79 6.34 −158.54 −99.61 −117.15 −43.25 0.35 −62.50 −10.76 6.64 49.32

x2c-SV(P)all

Sapporo-DZP

0.554 Hartree−Fock −1.4088 −84.14 0.12 5.73 1.36 0.44 0.46 0.03 0.23 0.07 −0.07 0.27 DFT(BP86) −1.5170 −88.89 0.39 2.84 −1.19 −2.26 −0.97 −1.51 −1.44 −0.29 −0.42 1.31

x2c-TZVPall

Sapporo-TZP

0.681

0.681

1.00

−1.1708 −79.38 −2.30 21.05 2.14 0.01 0.22 −0.22 −0.48 −0.06 −0.20 0.36

−0.3896 −30.24 0.02 0.68 0.35 0.34 0.20 0.01 0.02 0.09 −0.03 0.11

−1.1800 −81.24 −2.25 21.23 2.59 0.57 0.61 0.24 −0.01 0.14 0.01 0.34

−1.2789 −81.30 −2.02 19.79 −0.47 −2.92 −1.29 −1.89 −2.30 −0.24 −0.36 1.71

−0.4827 −36.34 0.16 −2.06 −1.38 −1.47 −0.10 −0.35 −0.49 0.59 0.45 0.66

−1.2860 −86.50 −1.94 18.85 −0.83 −3.26 −1.36 −1.94 −2.35 −0.24 −0.34 1.81

a

The column “reference” contains the absolute numbers, and the subsequent columns contain the differences to this column for the respective basis set. The line “size” indicates the ratio of the number of basis functions for a given basis with respect to that of the Sapporo-TZP basis. The line “avg” displays the average absolute deviations for the energies of the inner orbitals apart from the 1s orbital and weighted by the shell degeneracy.

the DFT level we get (0.10···0.25) ± (0.08···0.19) D for the selements and (0.04···0.09) ± (0.02···0.06) D for the others. Results for HF are similar; the larger polarization set does not lead to significant improvements. Errors of energies of the highest occupied orbitals were calculated for the entire test set, including the anionic species and shown in Table 6. At the double-ζ level they amount to (0.1···0.4) ± (0.1···0.3) eV for both polarization sets and methods, and maximum errors are observed for anionic compounds and compounds containing O or F, probably for similar reasons as discussed for the dipole moments. Triple-ζ bases exhibit much smaller errors, (0.02··· 0.10) ± (0.02···0.12) eV, again largely independent from the polarization set and the method. These errors are by far smaller than the differences between HF and DFT orbital energies. If one considers the sum of average error and standard deviation as a short-hand measure for the “typical” error, we may briefly summarize as follows. For atomization energies typical errors amount to ca. 20 kJ/mol with double-ζ and to ca. 5 kJ/mol with triple-ζ sets, if the higher polarized variant is

used for HF and the lower for DFT. For dipole moments and HOMO energies results do not strongly depend on the employed polarization set. For the former with double-ζ sets one overall gets typical errors about 0.5 D but somewhat larger numbers for the s-element compounds; as for these the description of the electronic overload at the bond partner is critical with small bases. With triple-ζ bases typical errors of 0.1 D are achieved; HOMO energy errors amount to 0.5 eV for double-ζ and 0.1 eV for triple-ζ bases. In contrast to our previous studies we did not perform extensive tests at correlated levels like MP2, as the bases in this work differ from previous sets mainly in the description of the inner shells, which are usually frozen in MP2 treatments anyways. Thus, no large differences to previous sets are expected. For clarity we note that the basis sets presented here are suited for MP2 within the frozen-core approximation only, as no functions for the description of core correlation are present. 3702


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Journal of Chemical Theory and Computation Table 9. Atomization Energies, Dipole Moments, and HOMO Energies of Four Selected Compounds at the HF and DFT(BP86) Levelsa reference size

3.456

AuCl Au2 PbO PbO2 avg

−82.82 −39.24 −43.42 4.45


−142.96 −110.15 −265.83 −251.91

AuCl PbO

5.48 5.94

AuCl PbO

3.16 4.31


−10.87 −7.87 −10.04 −11.89


−6.92 −6.47 −6.36 −7.03

DKH-DZP

x2c-SV(P)all

0.448

Sapporo-DZP

0.448 0.631 HF Atomization Energy per Atom/kJ/mol −28.74 6.90 0.33 −68.28 3.01 −0.51 5.97 16.83 2.02 7.36 18.11 2.09 27.59 11.20 1.13 DFT(BP86) Atomization Energy per Atom/kJ/mol −44.06 5.71 1.60 −98.60 4.58 −0.08 −3.22 7.36 1.53 −2.29 9.28 2.34 36.79 6.73 1.39 HF Dipole Moment/Debye 0.001 −0.036 0.049 −0.193 −0.416 −0.038 DFT(BP86) Dipole Moment/Debye −0.027 −0.278 0.028 −0.435 −0.634 −0.064 HF Energy of the Highest Occupied Orbital/eV 0.361 0.176 0.005 −0.030 0.020 −0.017 0.197 0.125 −0.027 0.218 0.302 −0.063 0.202 0.156 0.028 DFT(BP86) Energy of the Highest Occupied Orbital/eV 0.699 0.132 −0.006 1.347 −0.137 −0.043 0.274 0.203 −0.034 0.349 0.343 0.001 0.667 0.204 0.021

x2c-TZVPall 0.631

Sapporo-TZP 1.00

3.22 1.92 4.30 3.88 3.20

−1.59 −2.96 −1.45 −1.60 1.90

3.04 3.43 2.08 2.16 2.68

−1.21 −2.31 −1.08 −1.05 1.41

0.050 −0.084

−0.003 0.030

−0.050 −0.125

−0.002 0.031

0.039 −0.001 0.034 0.046 0.030

−0.008 −0.013 0.001 −0.007 0.007

0.017 −0.059 0.080 0.078 0.059

−0.010 −0.016 0.001 −0.003 0.008

a

The column “reference” contains the absolute numbers, and the subsequent columns contain the differences to this column for the respective basis set. The line “size” indicates the ratio of the sum of basis functions over the four compounds for a given basis with respect to that of the SapporoTZP basis. “avg” denotes the average of the absolute deviations.

Comparison to Other All-Electron Bases. So far, we have demonstrated that the newly developed bases are the consistent modification/extension of the proven def2- or dhfseries for all-electron relativistic treatments. We now compare the x2c-SV(P)all and the x2c-TZVPall sets to three other segmented contracted basis sets, DKH-DZP, Sapporo-DZP, and Sapporo-TZP for the gold atom and for four compounds containing sixth row elements, Au2, AuCl, PbO, and PbO2. The Sapporo basis sets were optimized at the DKH3/HF level within a finite size model for the nucleus, and the DKH-DZP bases were optimized at the DKH2/HF level with point-shaped nuclei. The contraction schemes of these five sets and the total number of spherical harmonic basis functions are listed in Table 7. By far the largest is the Sapporo-TZP set; despite the name it is rather a quadruple-ζ basis set or even larger, as evident e.g. for oxygen: the valence p space is described by five p-functions, polarized by a 3d2f set; the balanced and correlation consistent sets by most other groups contain three p-functions and a 2d1f set for polarization. Smallest are, with very similar sizes, the DKH-DZP and x2c-SV(P)all, which consist of about half as many functions as Sapporo-TZP. Sapporo-DZP and x2cTZVPall are in-between, both with ca. 2/3 of the number of basis functions of the Sapporo-TZP series. Again, the Sapporo set is larger than one would expect from the name.

Summarized in brief, x2c-SV(P)all yields much smaller errors than DKH-DZP, x2c-TZVPall and Sapporo-DZP show errors similar to each other and significantly smaller than x2cSV(P)all, and the large Sapporo-TZP basis does not lead to improvement over Sapporo-DZP in the present cases. In detail: For the gold atom we calculated the total electron density, the spin density, and the electrostatic potential arising from the electrons at the position of the nucleus as well as the energies of the inner orbitals (1s-4p) at levels HF and DFT(BP86), Table 8. Errors are largest for the DKH-DZP basis; in particular the typical errors in orbital energies of 50 eV are very high compared to the other bases, for which they amount to ca. 1 eV. Among them, errors are about the same for Sapporo-DZP, Sapporo-TZP, and x2c-SV(P)all; for x2c-TZVPall they are smaller than for the others by a factor of about three. The 1s orbital obviously is very sensitive to the computational level for which the basis is designed. Here the errors for the Sapporo sets are comparably large, ca. 20 eV. Similar holds for the electrostatic potential, for which the error is about 6 atomic units (au) for DKH-DZP, ca. 2 au for the Sapporo sets, and ca. 0.4 au for x2c-SV(P)all at the DFT level. For x2c-TZVPall at the DFT level and for both bases at the HF level about 0.1 au are achieved. For total and spin densities largest errors are observed for DKH-DZP, smallest errors are observed for x2c3703



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TZVPall (smaller by a factor of more than three), and the other three bases are in-between with errors similar to each other. Atomization energies, dipole moments, and HOMO energies (HF and DFT(BP86)) are shown in Table 9. Errors in atomization are largest for DKH-DZP, about 30 kJ/mol per atom; with x2c-SV(P)all − which is of the same size − they are smaller on average by a factor of about three (HF) or about six (DFT). For the other bases errors amount to ca. 3 kJ/mol per atom (x2c-TZVPall) and 1−2 kJ/mol for the Sapporo bases. It has to be noted that the reason for the impressively small errors for the Sapporo bases in particular for the TZP set partly is a comparably large basis set superposition error, which in most cases even leads to an overestimation of the bond energy. The basis set superposition error per atom (calculated with a standard counterpoise procedure) at DFT(BP86) for Au2 amounts to 0.99 kJ/mol for x2c-TZVPall, to 2.07 kJ/mol for Sapporo-DZP, and to 2.64 kJ/mol for Sapporo-TZP (respective numbers for x2c-SV(P)all and DKH-DZP are 3.35 and 74.42 kJ/mol). For HOMO energies a similar picture is revealed. Worst errors are observed for DKH-DZP, somewhat smaller errors are observed for x2c-SV(P)all, and significantly (about 1 order of magnitude) smaller errors are observed for the three others. Here the Sapporo-TZP set outperforms the two others. For dipole moments, one again may distinguish between DKHDZP and x2c-SV(P)all on the one side and the larger bases on the other, but here the x2c-SV(P)all bases are slightly worse than the DKH-DZP bases and x2c-TZVPall are slightly worse than the Sapporo bases. The reason lies in the comparably small sets for O and Cl (see above), which we kept for reasons of consistency with def2- and dhf-bases. When replacing the x2c-SV(P)all bases at Cl and O with the x2c-TZVPall bases, the comparably large errors at the DFT level are reduced to −0.01 D for AuCl and to −0.09 D for PbO. Nevertheless, such unbalanced choices may cause other problems (like basis set superposition errors), and one should rather employ the x2cTZVPall bases throughout for such cases.

Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.7b00593. Tables of the electronic states chosen for the optimizations, the contraction schemes of the reference bases, the differences in total energies of the optimized bases to the reference bases, and the Cartesian coordinates of all compounds of the molecular test set (PDF) x2c-SV(P)all basis sets in TURBOMOLE format (TXT) x2c-SVPall basis sets in TURBOMOLE format (TXT) x2c-TZVPall basis sets in TURBOMOLE format (TXT) x2c-TZVPPall basis sets in TURBOMOLE format (TXT) Basis set extensions for two-component calculations in TURBOMOLE format (TXT) RI-J auxiliary basis sets in TURBOMOLE format (TXT)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: fl[email protected]. ORCID

Florian Weigend: 0000-0001-5060-1689 Funding

P.P. acknowledges financial support by the TURBOMOLE GmbH. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Hehre, W. J.; Stewart, R. F.; Pople, J. A. Self-consistent molecularorbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J. Chem. Phys. 1969, 51, 2657−2664. (2) Ahlrichs, R.; Scharf, P.; Jankowski, K. The impact of higher polarization basis functions on molecular ab-initio results. 2. The ground-states of CO, N2, NO, O2, and F2. Chem. Phys. 1985, 98, 381−386. (3) Roos, B. O.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. New relativistic ANO basis sets for transition metal atoms. J. Phys. Chem. A 2005, 109, 6575−6579. (4) Roos, B. O.; Veryazov, V.; Widmark, P.-O. Relativistic atomic natural orbital type basis sets for the alkaline and alkaline-earth atoms applied to the ground-state potentials for the corresponding dimers. Theor. Chem. Acc. 2004, 111, 345−351. (5) Roos, B. O.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. Main group atoms and dimers studied with a new relativistic ANO basis set. J. Phys. Chem. A 2004, 108, 2851−2858. (6) Roos, B. O.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. New relativistic ANO basis sets for actinide atoms. Chem. Phys. Lett. 2005, 409, 295−299. (7) Dyall, K. G. Relativistic and nonrelativistic finite nucleus optimized triple-zeta basis sets for the 4p, 5p and 6p elements. Theor. Chem. Acc. 2002, 108, 335−340. (8) Dyall, K. G. Relativistic double-zeta, triple-zeta, and quadruplezeta basis sets for the 5d elements Hf-Hg. Theor. Chem. Acc. 2004, 112, 403−409. (9) Dyall, K. G. Relativistic quadruple-zeta and revised triple-zeta and double-zeta basis sets for the 4p, 5p, and 6p elements. Theor. Chem. Acc. 2006, 115, 441−447. (10) Dyall, K. G. Relativistic double-zeta, triple-zeta, and quadruplezeta basis sets for the 4d elements Y-Cd. Theor. Chem. Acc. 2007, 117, 483−489.

4. SUMMARY We presented segmented contracted Gaussian all-electron basis sets of double-ζ and triple-ζ valence quality optimized at the X2C level including a finite size model for the nucleus for elements up to Rn. Accuracy of molecular properties is very similar to that of their nonrelativistic or ECP-based counterparts, the Karlsruhe “def2” bases. Quality was assessed for a large set of more than 300 molecules representing (nearly) all elements in their common oxidation states. Typical errors for atomization energies amount to 20(5) kJ/mol per atom at the double(triple)-ζ level, errors for dipole moments amount to 0.5(0.1) D, and errors for HOMO energies amount to 0.5(0.1) eV. For inner shells errors are in the range of ca. 1 eV, for the 1s orbital of heavy elements up to 5 eV, which is still not much if compared to the energies themselves and to method-induced changes. Comparison to other bases for scalar relativistic allelectron treatments reveals the following picture. The x2cSV(P)all bases presented here yield much smaller errors than DKH-DZP bases, which are of similar size; x2c-TZVPall and Sapporo-DZP are similar to each other concerning both size and errors. We additionally provided extensions for selfconsistent two-component treatments, which are inevitable for this purpose, as well as Coulomb-fitting basis sets suited for all the presented bases. 3704


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Segmented Contracted Error-Consistent Basis Sets of Double- and

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