Nuclear electric quadrupole moment of potassium ... - APS Link Manager

PHYSICAL REVIEW A 91, 032516 (2015)

Nuclear electric quadrupole moment of potassium from the molecular method Tiago Quevedo Teodoro,1,2 Roberto Luiz Andrade Haiduke,1,* and Lucas Visscher2,† 1

Departamento de Qu´ımica e F´ısica Molecular, Instituto de Qu´ımica de São Carlos, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400–CP 780, 13560-970 São Carlos, SP, Brazil 2 Amsterdam Center for Multiscale Modeling, VU University Amsterdam, de Boelelaan 1083, 1081 HV Amsterdam, Netherlands (Received 25 February 2015; published 26 March 2015) The current standard nuclear quadrupole moments (NQMs) of the 39 K, 40 K, and 41 K isotopes have recently been contested by Singh and co-workers on the basis of their atomic computational data [Singh et al., Phys. Rev. A 86, 032509 (2012)]. Thus we performed relativistic calculations of electric field gradients at the potassium nuclei in three diatomic molecules (KF, KCl, and KBr) and combined these values with accurate experimental nuclear quadrupole coupling constants to provide an independent assessment of these NQMs. Our most accurate results, obtained by treating electron correlation with coupled cluster theory, employing a four-component Hamiltonian that includes the Gaunt two-electron correction, and with an incremented relativistic basis set of quadruple-ζ quality, yield Q(39 K) = 60.3(6), Q(40 K) = –75.0(8), and Q(41 K) = 73.4(7) mb. These values are in better agreement with the results obtained by Singh et al. and indicate that the standard NQMs should be revised. DOI: 10.1103/PhysRevA.91.032516

PACS number(s): 31.15.aj, 21.10.Ft

I. INTRODUCTION

The International Union of Pure and Applied Chemistry (IUPAC) defines the nuclear electric quadrupole moment (NQM), Q, as “a parameter which describes the effective shape of the equivalent ellipsoid of the nuclear charge distribution, Q > 0 for prolate; Q < 0 for oblate nuclei” [1]. This simple quantity is fundamental to understanding the structure of atomic nuclei with spin quantum number I ࣙ 1 [2]. While a direct measurement of an NQM is difficult, one may combine either atomic or molecular experimental spectroscopic data with electronic structure calculations to determine its value. In some cases, however, these “atomic” and “molecular” methods to obtain NQMs yield different results. For example, according to the latest compilation by Pyykkö [3], a standard NQM of 58.5(6) mb is recommended for the 39 K nucleus on the basis of the molecular result proposed by Kellö and Sadlej in 1998 [4]. This value was challenged in 2012 by Singh et al. [5], who recommended a revision to Q = 61.4(6) mb on the basis of atomic data. With the proposed error bars of 1% for both NQMs, there is indeed a clear disagreement between both values. Given the fact that the molecular calculations can nowadays be done to higher accuracy, we decided to carry out a new investigation with the molecular method, as this remains one of the most reliable ways to determine NQMs [3]. The combination of experimental nuclear quadrupole coupling constants (NQCCs) with theoretical calculations of electric field gradients (EFGs) yields the NQM through the following relation: νQ (X) Q (X) = , (1) 234.9647q (X) where Q(X), νQ (X), and q(X) are, respectively, the NQM in barns (b), the NQCC in megahertz (MHz), and the EFG in

* †

[email protected] [email protected]

1050-2947/2015/91(3)/032516(4)

atomic units (a.u.) at the nucleus X in a linear molecule. Thus, by means of high-level relativistic calculations, including the coupled cluster methodology with single and double excitations (CCSD) and different variants of perturbative triple corrections [CCSD+T, CCSD(T), and CCSD-T], we believe we are able to obtain a highly accurate NQM of 39 K. The NQMs of 40 K and 41 K are furthermore straightforwardly obtained via the experimental Q ratios between 39 K and these other isotopes. II. GENERAL CONSIDERATIONS

Accurate equilibrium NQCCs of the 39 K nucleus in diatomic halides are available in the compilation presented by Lovas and Tiemann [6]: −7.982 18(160), −5.695 03(65), and −5.0343(28) MHz in KF, KCl, and KBr, respectively. Since the equilibrium distances are also known to great accuracy [7], it is sufficient to determine the EFGs at the potassium nuclei in these molecules only at such geometries. The employed bond ˚ for KF, lengths were thus 2.171 457, 2.666 65, and 2.820 78 A KCl, and KBr, respectively. The calculations were carried out with the relativistic four-component Dirac-Coulomb (DC) and Dirac-Coulomb-Gaunt (DG) Hamiltonians as implemented within the DIRAC14 package [8]. Default code parameters have been used: a Gaussian nuclear model, the speed of light set to 137.035 999 8 a.u. and a neglecting of (SS|SS) two-electron integrals [9]. In addition we used the DC Hamiltonian in conjunction with the B3LYP functional [10–12] in density functional theory (DFT) calculations. All basis sets were used in their uncontracted form. III. RESULTS AND DISCUSSION A. Basis set increment study

The first task in EFG evaluations is the definition of a basis set that is able to accurately represent the wave function not only in the valence region but also in points close to the nucleus under study. We used the quadruple-ζ relativistic prolapse-free family of basis sets, RPF-4Z [13], which has been shown

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TEODORO, HAIDUKE, AND VISSCHER


to be well suited for the application in EFG calculations (see Ref. [14] and references therein). We first assessed the stability of the EFG values obtained upon extension of the basis, carrying out Hartree-Fock (HF) and B3LYP calculations in which the effect of additional tight and diffuse functions of s, p, d, f , and g symmetries, as well as h polarization functions, was determined. The exponents of these functions were obtained via the polynomial version of the generator coordinate Dirac-Fock (p-GCDF) method [15] through the application of the parameters given by Haiduke and da Silva [16] and following the strategy outlined in Ref. [13]. The DC Hamiltonian was used in this part of the study. The extra functions, displayed in Tables S1–S3 in the Supplemental Material (SM) [17], were independently added to each symmetry block (s, p, d, f , g, and h) of the RPF-4Z set of potassium in the KF molecule (with the RPF-4Z set for fluorine). Functions that resulted in variations larger than 0.1% of the total EFG with any of the two methods (HF or B3LYP) were included in the augmented set. This procedure yielded a final set, denoted mod-RPF4Z, consisting of the RPF-4Z basis set augmented by four extra tight d (exponents: 9.010 100 407, 40.587 473 49, 197.038 514 5, and 1132.863 241) and four extra tight f (exponents: 0.774 730 968 8, 3.485 408 888, 14.857 338 27, and 74.704 464 60) functions (21s15p11d7f 1g functions in total). As expected, the EFG was hardly affected by adding diffuse functions because we already started from a polarized basis set. Usually, increments in the basis set centered at the nuclei other than that under study are not very important to such EFG determinations (see, e.g., Ref. [18]), but as the halogens exhibit a significant anionic character, we decided to use the aug-RPF-4Z sets for the F (15s10p4d3f 2g), Cl (19s14p4d3f 2g), and Br (22s18p12d3f 2g) atoms. This was done to allow for an even better description of these ionic systems. This augmented version includes one extra diffuse function for each symmetry block of the original set [13]. Finally, as already noted in the previous section, both the mod-RPF-4Z (for K) and aug-RPF-4Z (for the halogens) sets were kept in their uncontracted form to ensure maximum basis set flexibility in all the following calculations. B. Electric field gradients

In order to obtain accurate values, we first performed an analytical evaluation of the EFGs at the potassium nuclei in the KX (X = F, Cl, and Br) molecules using the HF wave function. The electron correlation contribution was then evaluated through a finite-difference technique in a two-point form, corr ∂E (λ) E corr (+λ) − E corr (−λ) , (2) ≈ ∂λ 2λ λ→0 in which λ, set here as 1 × 10−7 a.u., is the scale factor attached to the EFG interaction term that is added to the Hamiltonian. This two-step approach, in which only the electron correlation contribution is obtained via finite differentiation, is numerically more stable than differentiating the total energy [19]. In the correlation calculations we considered both second-order Møller-Plesset (MP2) and coupled cluster [CCSD, CCSD+T, CCSD(T), and CCSD-T] theories correlating all electrons and

TABLE I. EFGs (in a.u.) at the K nuclei given by different levels of calculation.a Method DC-HF DG-HF DC-B3LYP DC+G-MP2 DC+G-CCSD DC+G-CCSD+T DC+G-CCSD(T) DC+G-CCSD-T DC+G-CCSDTb Ref. [4]c

KF

KCl

KBr

−0.6240 −0.6231 −0.5593 −0.5525 −0.5765 −0.5568 −0.5610 −0.5617 −0.5603 −0.5771

−0.4360 −0.4355 −0.3900 −0.3982 −0.4108 −0.4021 −0.4025 −0.4025 −0.4023 −0.4174

−0.3839 −0.3834 −0.3374 −0.3532 −0.3639 −0.3565 −0.3569 −0.3570 −0.3567 –

a

MP2 and coupled cluster results are obtained by summing the analytic EFG from the DG-HF calculation and the respective electron correlation contribution obtained through Eq. (2). b Given by the sum of the CCSD-T values with the T correction from calculations with a smaller active space (see Table S4 [17]). c Spin-free Douglas-Kroll-Hess-CCSD(T) calculation.

allowing excitations to virtual spinors with energies lower than 100 a.u. These data are displayed in Table I. The values in Table I show that the contribution of the Gaunt two-electron correction is almost negligible in all molecules considered here (less than 0.15%). We therefore expect that the inclusion of still smaller two-electron interaction corrections, such as the gauge part of the full Breit term, will have no significance for these EFG results. The computationally efficient B3LYP method yields EFG values that are in good agreement with the highest-level coupled cluster results for this set of molecules. Electron correlation effects are important, with MP2 increasing the EFG relative to HF by up to 11.5%. Although MP2 typically overestimates correlation effects, its values are quite close to the CCSD-T ones. As expected, the largest contribution from electron correlation effects comes from single and double excitations, but the contribution of triple substitutions calculated perturbatively by means of the CCSD-T method leads to EFG values about 2% less negative than the respective CCSD ones. In order to assess the effect of still higher orders of electron correlation, we used the MRCC code [20,21], as interfaced to DIRAC, to calculate the effect of the explicit treatment of triple excitations with CCSDT. However, such calculations were only feasible with a smaller active space including spinors with energy between −1.0 and 3.0 a.u. (12 electrons in the KF and 14 electrons in the KCl and KBr molecules). Results are shown in Table S4 [17]. The values are slightly different from those in Table I due to the smaller active spaces considered, but the relative difference between CCSD and CCSD-T EFGs remains the same, approximately 2%. We therefore used these results to deduce a T correction (difference between perturbative and full treatment of triples) that could be added to the CCSD-T EFGs in Table I. We were not able to assess the importance of electron correlation beyond triple excitations, but the close agreement between CCSD-T and CCSDT gives us confidence in the convergence with respect to the treatment of electron correlation. An interesting observation is that the CCSD(T) values are closer to the CCSDT results than the

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CCSD-T ones, especially for KCl and KBr, although the differences between the two ways of calculating perturbative triple corrections are small. The difference is larger for KF but remains consistent, so the composite result reached in Table I (labeled DC+G-CCSDT) is not affected by the choice of the perturbation treatment. The EFGs calculated by Kellö and Sadlej [4] (which were combined with the same experimental data used here) to produce the current standard NQM value are also shown in Table I. The differences of 3%–4% between those values and the CCSD(T) results obtained in the present study are outside the error bars that were assumed at that time. By comparing details of both calculations we may uncover the factors that give rise to this discrepancy. The first noticeable difference between our calculations and theirs is related to the basis set completeness. Kellö and Sadlej choose incremented versions of the sets from Huzinaga and Klobukowski [22] with a 21s14p6d2f final size for potassium. The tighter exponents in the d and f blocks of this set (9.141 988 and 0.819 777 50, respectively) are very similar to the first extra d and f tight functions added in our convergence study. However, as can be deduced from Table S1 [17], the other 3d and 3f tighter functions selected to compose the mod-RPF-4Z set increased the EFG of potassium in KF at the DC-HF level in about 2% of the total value. To verify this possible cause of discrepancy, we also performed calculations with the same sets used in Ref. [4] (but in their uncontracted form) while keeping our other computational parameters constant (using the DC Hamiltonian and correlating all electrons). These DC-CCSD(T) results are −0.5776 and −0.4116 a.u. in KF and KCl, respectively, which are very consistent, especially in the KF case, with the ones reported in Ref. [4]. Thus, this extra flexibility provided by the RPF-4Z versions (mod- and aug-) used in the present paper appears to be the main cause for the discrepancy with the EFGs given in Ref. [4]. Another difference can be found in the active space used in both electron correlation calculations. While we correlated all electrons and included a large virtual space, Kellö and Sadlej used only the valence and subvalence shells of the KF and KCl molecules: 16 and 24 electrons, respectively. From our DC-HF calculations, we noticed that the individual contributions of the orbitals in these spaces to the EFGs correspond to 86% (KF) and 89% (KCl) of the total electronic contribution. Hence, it is possible that correlation of the low-lying orbitals provides a small correction to the EFG. In that manner we carried out calculations with an active space similar to that used in Ref. [4], thus including only spinors with energy between −10.5 and 20.0 a.u. (corresponding to 16, 24, and 34 electrons in the KF, KCl, and KBr molecules, respectively). These results are plotted in Table S5 in the Supplemental Material [17]. An increase of about 0.9% with respect to (CCSD-T) values in Table I is found for the three molecules, which is about the same order as the error bars suggested for the current NQMs, making this factor indeed significant. We conclude that in order to achieve the desired precision of 1% in the NQM of potassium, deep core correlation effects must be considered. Finally, we evaluated the influence of the relativistic Hamiltonian employed on the calculations. Kellö and Sadlej treated the relativistic effects through the spin-free Douglas-Kroll-Hess second-order approximation (DKH2), while we use a more

complete four-component Hamiltonian. To assess the effect of this feature, calculations were done with the nonrelativistic (NR), the spin-free exact two-component (SF-X2C), and the DKH2 Hamiltonians. These EFGs are displayed in Tables S6–S8 in the Supplemental Material [17]. The nonrelativistic EFGs for potassium can deviate up to 1% from the respective DC+G ones. This difference is largely captured by the DKH2 approach. Including only scalar relativistic effects at the DKH2 level leads to differences no higher than 0.3% with respect to the four-component Hamiltonian selected here (from the CCSD-T results in all cases). C. Nuclear quadrupole moments

Having been able to explain the differences between the molecular computational data, we can now recompute the NQMs by applying Eq. (1) with the newly calculated EFGs and the equilibrium NQCCs from Ref. [6] (the latter values are also presented in Sec. II.). This yields the 39 K NQMs displayed in Table II. These results are also an additional test on the reliability of the calculations as NQMs should be independent of the molecule. Thus, at the highest level of calculation, DC+G-CCSDT, the NQMs from the three molecules indeed result in a small mean absolute deviation (MAD) of 0.2 mb or, approximately, 0.3% of the average value. Since the errors in the experimental NQCCs are almost negligible, the error bar in this final NQM of 60.3 mb should reflect remaining deficiencies in the EFG calculations. As discussed in Secs. III A and III B, errors from the basis set incompleteness are minor as are deviations from neglecting further relativistic corrections to the two-electron interaction. Given the systematic spread of approximately 2% between the CCSD and CCSD-T results, higher orders of electron correlation are probably the largest residual source of error. Although we observed that explicit inclusion of triple excitations (CCSDT) yields virtually the same results as CCSD-T, one still needs to be cautious with specifying the error bar. Hence, we indicate a conservative TABLE II. NQMs (in mb) of the 39 K nucleus given by different levels of calculation.a Method

KF

KCl

KBr

Average

MADb

DC-HF DG-HF DC-B3LYP DC+G-MP2 DC+G-CCSD DC+G-CCSD+T DC+G-CCSD(T) DC+G-CCSD-T DC+G-CCSDTc Ref. [4] Ref. [5]d

54.4 54.5 60.7 61.5 58.9 61.0 60.6 60.5 60.6 58.9 –

55.6 55.7 62.1 60.9 59.0 60.3 60.2 60.2 60.2 58.1 –

55.8 55.9 63.5 60.7 58.9 60.1 60.0 60.0 60.1 – –

55.3 55.4 62.1 61.0 58.9 60.5 60.3 60.2 60.3 58.5(6) 61.4(6)

0.6 0.6 0.9 0.3 0.0 0.4 0.2 0.2 0.2 0.4 –

a

NQMs obtained through Eq. (1) by the combination of theoretical EFGs (shown in Table I) and experimental NQCCs (Sec. II). b Mean absolute deviation. c These values are given by adding the contribution of the full triple excitations from EFG calculations that used a smaller active space than that employed in the lower levels of correlation. d Result from atomic calculations.

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error bar of 1.0% or 0.6 mb, bringing our results into better agreement with the reexamination done in Ref. [5] through atomic calculations, 61.4(6) mb. This final value of 60.3(6) mb is still out of the range of the current standard 39 K NQM, 58.5(6) mb, which we consider to be due to the too optimistic error bar proposed at the time. Additionally, the NQMs for the 40 K and 41 K nuclei are determined by means of experimental ratios [23], Q(40 K)/ Q(39 K) = –1.244 ± 0.002 and Q(41 K)/Q(39 K) = 1.2175 ± 0.001, giving rise to NQMs of −75.0(8) and 73.4(7) mb for 40 K and 41 K, respectively.

in combination with accurate experimental NQCCs, yield the following NQMs: Q(39 K) = 60.3(6), Q(40 K) = –75.0(8), and Q(41 K) = 73.4(7) mb. Hence, considering that these values are closer to the ones recently obtained by Singh et al. [5], it is clear that the current standard NQMs of these nuclei [3,4] should be revised. Finally, by taking into account the accuracy of the calculations carried out in this study and the consistency of their correspondent data, we recommend that the above-mentioned results be adopted as the new standard NQMs of the referred potassium isotopes.

IV. CONCLUSIONS

ACKNOWLEDGMENTS

The EFGs at the potassium nuclei in three diatomic molecules calculated at the DC+G-CCSDT level of theory,

T.Q.T. thanks FAPESP for support provided by a doctoral fellowship (Grant No. 2014/02939-5).

[1] IUPAC. Compendium of Chemical Terminology, 2nd ed. (the “Gold Book”), edited by A. D. McNaught and A. Wilkinson (Blackwell Scientific Publications, Oxford, 1997). XML online corrected version available at http://goldbook.iupac.org (2014), created by M. Nic, J. Jirat, and B. Kosata; updates compiled by A. Jenkins. [2] G. Neyens, Rep. Prog. Phys. 66, 633 (2003). [3] P. Pyykkö, Mol. Phys. 106, 1965 (2008). [4] V. Kellö and A. J. Sadlej, Chem. Phys. Lett. 292, 403 (1998). [5] Y. Singh, D. K. Nandy, and B. K. Sahoo, Phys. Rev. A 86, 032509 (2012). [6] F. J. Lovas and E. Tiemann, J. Phys. Chem. Ref. Data 3, 609 (1974). [7] NIST Chemistry WebBook, edited by P. J. Linstrom and W. G. Mallard, NIST Standard Reference Database No. 69 (National Institute of Standards and Technology, Gaithersburg, MD, 2011). Available at http://webbook.nist.gov (accessed October 01, 2014). [8] DIRAC, a relativistic ab initio electronic structure program, Release DIRAC14 (2014), written by T. Saue, L. Visscher, H. J. Aa. Jensen, and R. Bast, with contributions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, A. S. P. Gomes, T. Helgaker, J. Henriksson, M. Iliaˇs, Ch. R. Jacob, S. Knecht, S. Komorovský, O. Kullie, C. V. Larsen, J. K. Lærdahl, Y. S. Lee, H. S. Nataraj, P. Norman, G. Olejniczak, J. Olsen, Y. C. Park, J. K. Pedersen, M. Pernpointner, R. di Remigio, K. Ruud, P. Sałek, B. Schimmelpfennig, J. Sikkema, A. J. Thorvaldsen, J. Thyssen, J. van Stralen, S. Villaume, O. Visser, T. Winther,

and S. Yamamoto; see http://www.diracprogram.org (accessed November 01, 2014). L. Visscher, Theor. Chem. Acc. 98, 68 (1997). A. D. Becke, J. Chem. Phys. 98, 5648 (1993). C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J. Phys. Chem. 98, 11623 (1994). T. Q. Teodoro, A. B. F. da Silva, and R. L. A. Haiduke, J. Chem. Theory Comput. 10, 3800 (2014). R. T. Santiago, T. Q. Teodoro, and R. L. A. Haiduke, Phys. Chem. Chem. Phys. 16, 11590 (2014). R. L. A. Haiduke, L. G. M. de Macedo, R. C. Barbosa, and A. B. F. da Silva, J. Comput. Chem. 25, 1904 (2004). R. L. A. Haiduke and A. B. F. da Silva, J. Comput. Chem. 27, 61 (2006). See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevA.91.032516 for further data about the alternative EFG calculations addressed in the main text. R. L. A. Haiduke, Chem. Phys. Lett. 544, 13 (2012). M. Pernpointner and L. Visscher, J. Chem. Phys. 114, 10389 (2001). MRCC, a quantum chemical program suite written by M. Kállay, Z. Rolik, I. Ladjánszki, L. Szegedy, B. Ladóczki, J. Csontos, and B. Kornis. See also Z. Rolik, L. Szegedy, I. Ladjánszki, B. Ladóczki, and M. Kállay, J. Chem. Phys. 139, 094105 (2013), as well as http://www.mrcc.hu (accessed January 01, 2015). M. Kállay and P. R. Surján, J. Chem. Phys. 115, 2945 (2001). S. Huzinaga and M. Klobukowski, J. Mol. Struct.: THEOCHEM 167, 1 (1988). E. P. Jones and S. R. Hartmann, Phys. Rev. B 6, 757 (1972).

[9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20]

[21] [22] [23]

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