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Electric dipole polarizabilities of negative ions of the coinage metal atoms
10May 1996
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 253 (1996) 383-389
Electric dipole polarizabilities of negative ions of the coinage metal atoms Vladimir Kell6 a, Miroslav Urban a, Andrzej J. Sadlej b a Department of Physical Chemistry, Faculty of Sciences, Comenius University, Mlynska dolina, SK-842 15 Bratislaua, Slovakia b Department of Theoretical Chemistry, Chemical Centre, University ofLund, P.O.B. 124, S-221 O0 Lurid, Sweden Received 14 February 1996
Abstract
The dipole polarizability of the coinage metal anions is calculated in different approximations for both the electron correlation and relativistic effects. The large polarizabilities of these ions give a good illustration of the performance of different computational methods. The spin-averaged Douglas-Kroll no-pair scheme combined with the CCSD(T) treatment of the electron correlation predict the following values of polarizabilities: 243 au for Cu-, 213 au for Ag- and only about 93 au for Au-. The non-relativistic CCSD(T) results are 278, 299 and 256 au, respectively. Both the non-relativistic and relativistic results agree with the QCISD(T) pseudopotential data, though serious discrepancies occur at the level of the QCISD approximation. The use of the scalar Pauli approximation for the estimation of the relativistic contribution to the dipole polarizabilities of the coinage metal ions is limited to Cu- and Ag-. For Au- this approximation strongly overestimates the magnitude of the relativistic effect.
1. Introduction
Most atoms with incompletely filled valence shells can bind an additional electron [1]. The energies released in such processes, i.e. the electron affinities (EA), are usually of the order of 1-2 eV [1] and the added electron is only loosely bound. In theoretical studies the interest in negative ions and their physical properties arises mostly because of the diffuse character of the electron density distribution in the outer region. This makes the electron correlation problem difficult to handle and simultaneously challenging [2,3]. In the case of heavy negative ions one can also expect significant contributions to their properties arising from relativistic effects [4]. Owing to the magnitude of these contributions the negative ions of heavy elements provide a convenient play-
ground for the investigation of the performance of different relativistic methods of quantum chemistry. In this Letter we report investigations of the dipole polarizability of the coinage metal anions (Cu-, A g - , A u - ). All of the Group Ib elements are known to form negative ions [ 1]; their experimental electron affinites are equal to 1.228 eV for Cu, 1.302 eV for Ag and 2.309 eV for Au [1]. Schwerdtfeger and Bowmaker [5] have recently carried out a series of calculations of the dipole polarizability of the negative ions of these elements. The electron correlation effects have been studied at the level of the many-body perturbation theory (MBPT) of second, third and fourth order [6] and by using the so-called quadratic configuration interaction (QCI) [7] approach with single and double excitations (QCISD) and corrected for triple exci-
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V. Kell5 et a l . / Chemical Physics Letters 253 (1996) 383-389
tations (QCISD(T)) [7]. The electron correlation contributions to anionic polarizabilities calculated by Schwerdtfeger and Bowmaker [5] exhibit a strong dependence on the level of approximation. This indicates a poor, if any, convergence of the electron correlation perturbation series and is typical of the polarizabilities of negative ions [3]. The relativistic effect upon dipole polarizabilities has been estimated by Schwerdtfeger and Bowmaker [5] from parallel calculations with non-relativistic and j-averaged relativistic pseudopotentials. The relativistic contribution to the dipole polarizability has been found to be large for Au and quite sizeable for Ag [5]. From the data presented by Schwerdtfeger and Bowmaker [5] one can also conclude that, at least for Au-, the mutual interplay of the relativistic and electron correlation effects [8] is important. In the present study the electron correlation effects will be evaluated by using the coupled cluster (CC) [6,9,10] approaches rather than the MBPT and QCI methods employed by Schwerdtfeger and Bowmaker [5]. The MBPT series, as illustrated by the results of these authors, does not seem to converge. The poor performance of the QCI methods for the polarizabilities of highly ionic systems is known as well [11] while the CCSD [6] and CCSD(T) schemes [9,10] seem to perform much better. Particular attention will be given to the evaluation of the relativistic contribution to the dipole polarizabilities of the coinage metal anions. The relativistic effects will be studied by using the scalar Pauli (MVD, mass-velocity + Darwin) approximation [12,13] and the method based on the Douglas-Kroll spin-averaged (no-pair) Hamiltonian [ 14-16]. The latter, as long as the spin-orbit coupling effects can be neglected, appears to be the best one-component quasi-relativistic scheme available at the moment [ 17-19]. The Douglas-Kroll no-pair (DK np) results will be considered as a measure of the performance of the other methods. They will be used to assess the quality of the pseudopotential data and to follow the failure of the MVD approximation for the Au- ion.
2. Methods and computational details. Basis sets
The methods which are either used or discussed in this Letter in the context of the evaluation of the
electron correlation contribution to polarizabilities belong to the standard tools of modern quantum chemistry and their detailed description can be found elsewhere [6,7,9,10]. The same applies to methods for the estimation of the relativistic effect on polarizabilities [8,12-16]. One should mention that all the polarizability data obtained in this study follow from the so-called finite perturbation calculations, i.e. they are the numerical second-order derivatives of the electric field dependent energies [20]. A similar numerical technique is also used in calculations of the mixed relativistic correlation contribution in the MVD approximation [8]. The values of the external electric field strength (F) and the numerical parameter (W) used in finite perturbation calculations within the MVD scheme [8] depend on the magnitude of the calculated polarizability and relativistic effects. The values of F used in our calculations were in the range 1 0 - 4 - - 1 0 -3 au. For the lower values of F a parabolic approximation for the F-dependence of energies has been found to be sufficiently accurate while for the higher ones a quartic polynomial fit was used. For the MVD perturbation strength the corresponding parameter W was in the range 0.0010.01 au with the higher value used in MVD calculations for Cu-. It should also be mentioned that some convergence problems occur when using the MVD finite perturbation scheme [8] for uncontracted basis sets with high-exponent GTOs; the p4 (mass-velocity) contribution to Fock matrix elements [13] may give rise to large matrix elements of the Fock operator and lead to divergences at the level of the SCF HF approximation. This problem does not occur when (core) contracted basis sets are employed. Then the large integrals are quenched by small contraction coefficients. Most of the calculations reported in this Letter have been carried out with GTO/CGTO basis sets obtained by extending the so-called polarized sets developed in our earlier studies [21,22]. One should remark that the parent polarized basis sets (PolMe for non-relativistic + MVD [21] and NpPolMe for DK np calculations [22]) are designed for molecular calculations under the assumption that no extreme ionicity is present in the given molecule. While using these sets for negative ions one needs to extend them to take into account the diffuseness of the charge
V. Kell5 et al. / Chemical Physics Letters 253 (1996) 383-389
distribution. Moreover, the appropriate polarization and correlating orbitals need to be added. Because of the difference in one-electron Hamiltonians in the non-relativistic and DK np approaches the parent basis sets for non-relativistic + MVD calculations (PolMe [21]) differ from those used in the context of the DK np scheme (NpPolMe basis sets [22]) by contraction coefficients. However, their size and flexibility in the outer region of the electron density distribution is essentially the same. For this reason the same primitive s, p, d and f functions were added to the parent sets in both cases. One should add that the parent PolMe and NpPolMe sets are essentially uncontracted in the valence and nextto-valence shells A comprehensive study of the basis set dependence of the calculated ionic polarizabilities has been carried out and some of the results are presented in Table 1. They show the effect of different extensions of the parent G T O / C G T O PolMe [21] sets on the ionic polarizability in non-relativistic and quasi-relativistic (MVD) SCF HF and CCSD(T) approximations. Several features of the quasi-relativistic MVD approximation which can be seen from this tabula-
385
tion and the apparent failure of the MVD scheme for the Au- ion will be discussed in the next section. Extensions of the parent basis sets have been generated by assuming for the pertinent GTO exponents a geometric progression based on the two lowest exponents of the given subset in PolMe sets. For the above-discussed reasons both the non-relativistic and MVD SCF HF results calculated with the PolMe sets are much too low. However, adding a (3s3pld) set of diffuse functions is quite sufficient for stabilizing the SCF HF values. The same applies essentially to the CCSD(T) results, though there is some effect of extending the p-subset by one more diffuse orbital. Since the extended sets are also used in other calculations for the negative ions of the coinage metals, we found it of some advantage to use the (3s3p2dlf) extensions of the PolMe sets rather than to further increase the number of diffuse p-type GTOs. The NpPolMe sets extended in the same way will be used in DK np calculations of ionic polarizabilities. The software used in the present study is the Molcas-3 system of quantum chemistry programs [23] linked to the Titan package for CCSD(T) calcu-
Table 1 Dipole polarizabilities of the coinage metal anions. The basis set dependence of the non-relativistic and quasi-relativistic (MVD) data. All values in au Ion
Method a
Basis set PolMe b
PolMe + 3s3pld
PolMe + 3s4pld
PolMe + 3s3p3d
PolMe + 3s3p3d2f
PolMe + 3s3pldl f
378 209 518 262 543 242
580 278 702 298 630 254
580 277 702 289 631 240
580 279 702 298 630 254
580 280 702 300 -
579 278 702 299 630 256
Quasi-relativistic (MVD) results c CuSCF 356 CCSD(T) 190 AgSCF 425 CCSD(T) 196 AuSCF 204 CCSD(T) 48
523 244 520 205 100 23
524 243 520 197 100 17
524 245 520 205 100 24
524 246 518 205 -
523 245 519 206 101 23
non-relativistic results CuSCF CCSD(T) AgSCF CCSD(T) Au SCF CCSD(T)
a All CCSD(T) results correspond to correlating 20 electrons of the valence and next-to-valence shells. b The PolMe (Me = Cu, Ag, Au) are the polarized G T O / C G T O basis sets generated in Ref. [21] for non-relativistic and quasi-relativistic (MVD) calculations of electric properties. c The MVD correction is included at both the SCF HF and CCSD(T) levels of approximations and takes into account the mixed relativistic correlation effects.
1/. Kell5 et aL / Chemical Physics Letters 253 (1996) 383-389
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lations [24]. The DK np-pair calculations have been carried out by using the same software and a code for computing the core Hamiltonian matrix in the DK np approximation written by Hess [25].
3. Results and discussion According to the data and discussion of different basis set extensions presented in Section 2 the major part of our calculations has been carried out with the PolMe (NpPolMe) basis set extended by three s-, three p-, one d- and one f-type GTOs of low exponents. A comparison of our non-relativistic, MVD and DK np results for the three anions reveals several interesting features of the methods used here to account for relativistic contributions to their dipole polarizabilities. The corresponding results are displayed in Table 2. It can be seen that for both Cu- and Ag- the MVD approximation leads to results which are close to those calculated by using the DK np method. This observation applies to both the SCF HF and correlated levels of approximation. However, for Auquite a large difference between the two approaches occurs already for the SCF HF results. Thus, the first-order MVD correction is not capable of correctly accounting for the relativistic contraction of the 6s orbital in Au- and strongly overestimates this
effect, suggesting that higher-order approximations are necessary. Using the MVD approximation beyond first order in the perturbation treatment of relativistic contributions would raise several objections [26] and would be incompatible with the origins of the method [13]. One should stress that the Au- ion is so far one of a few examples of the major failure of the MVD scheme and, to some extent, defines the limits of its use. On the other hand, the same approximation performs in a quite satisfactory way for the Hg atom which is isoelectronic with Au-. This shows that the nuclear charge of the heavy atoms is not the only parameter which determines the quality of the MVD scheme for the calculation of relativistic corrections to atomic and molecular properties. The diffuseness of the valence ns orbital and, in particular, the relativistic shift of its nodes cannot be represented by using the first-order MVD correction to the SCF HF energy. In this context one should also remark that the same approximation is satisfactory for singly positive ions of the coinage metals [22], whose dipole polarizability is primarily determined by that of the next-to-valence (n - 1)d t° shell. Our results presented in Table 2 also give some illustration of the role of the electron correlation in both non-relativistic and relativistic calculations of the dipole polarizability of the coinage metal anions. Already in the non-relativistic approximation the
Table 2 Dipole polarizabilities of the coinage metal anions. Results of non-relativistic, MVD and DK no-pair calculations in different approximations for the electron correlation effects a. All values in au Method
SCF MBPT2
CCSD
CCSD(T)
N b
12 18 20 12 18 20 12 18 20
C-
Ag-
Au-
nr
MVD
DK np
nr
MVD
DK np
nr
MVD
DK np
579 127 76 88 298 291 297 266 270 278
523 82 32 44 264 259 265 233 238 245
526 84 34 46 266 261 267 236 240 247
702 129 60 73 361 345 350 304 290 299
519 43 -20 - 9 263 249 254 211 200 206
523 52 - 8 2 266 253 257 218 207 213
630 62 - 4 6 318 303 307 263 249 256
101 - 106 -136 - 138 48 44 45 27 21 23
193 13 -12 109 105 97 92 -
a All results of this table correspond to calculations with PolMe (NpPolMe) sets extended by the (3s3pldlf) diffuse set. See Section 2 for details. b N is the number of correlated electrons. For A u - in the DK np approximation the largest number of correlated electrons is 18 since the 5s level falls below the 4f levels.
V. Kell5 et a l . / Chemical PhySics Letters 253 (1996) 383-389
second-order MBPT2 correction strongly overestimates the electron correlation contribution, leading in some cases to disastrous (negative) results. Moreover, its value depends considerably upon the number of correlated electrons. The infinite order CC methods show that the latter effect is an artifact of the MBPT2 approximation; the CC dipole polarizabilities calculated with either 12 or 20 correlated electrons do not differ significantly. When the MBPT2 method is combined with the MVD approach to the evaluation of mixed relativistic-correlation contributions this leads to unphysical results already for Ag-. One should mention, however, that the DK np scheme suffers in the same way. The second-order treatment of the electron correlation in negative ions of the coinage metal atoms leads to an overestimation of both pure correlation and relativistic correlation corrections to dipole polarizabilities. A little disturbing is the difference between CCSD(T) and CCSD results which amounts to about
Table 3 Dipole polarizabilities of the coinage metal anions. Results of the DK np calculations with fully uncontracted ( 3 s 3 p l d l f ) extended NpPolMe basis sets and their comparison with the pseudopotential data of Schwerdtfeger and Bowman. All values in an Method this work SCF MBPT2
CCSD
CCSD(T)
reference results [5] b SCF MBPT2 QCISD QCISD(T)
N a
CU-
Ag-
Au-
12 18 20 12 18 20 12 18 20
528 80 31 45 269 264 269 234 236 243
523 49 - 10 - l 267 254 258 217 206 213
195 15 - 10 112 108 98 93 -
20 20 20
537 71 216 ¢ 317 c
543 15 327 213
205 3 118 96
a See footnote b to Table 2. b In Ref. [5] the polarizability data are given in ,~3. The conversion factor used here is: 1 au of the dipole polarizability -~ 0.14818 ¢ These results are likely to be incorrect because of some convergence problems which are mentioned by Schwerdtfeger and Bow-
man [5].
387
30 au for Cu- and increases to about 50 au for Ag-. This may indicate that including iterative contributions due to T 3 operators and perhaps also those from the higher order cluster operators may be of some importance. This effect is, however, much less pronounced in the case of Au-. The corresponding difference amounts to only about 10 au and makes the present CCSD(T) reliable. It has already been pointed out that the DK np approach to scalar relativistic effects in atoms and molecules appears to be currently the most successful tool of relativistic quantum chemistry [19]. The present data show its superiority to the MVD scheme in the case of heavy ions with diffuse electron density distribution. Though computationally advantageous, the DK np approach does not offer an immediate answer to the question of the importance of relativistic effects. This can be achieved only by comparing the DK np results with the non-relativistic ones; the latter should be obtained with fully compatible basis sets. On the other hand, the MVD corrections are calculated on top of the usual non-relativistic calculations. They offer a perturbative insight into the role of relativistic effects. At some point the MVD approach must fail in the same way as any first-order perturbation scheme does for high enough values of the perturbation strength. So far our major interest was in comparative studies of different methods. In order to obtain the best values of the dipole polarizability of the coinage metal anion we have carried out DK np calculations with fully uncontracted ( 3 s 3 p l d l f ) extended NpPolMe basis sets. The results are presented in Table 3 and compared with the data calculated by Schwerdtfeger and Bowmaker [5]. Because of some convergence problems in the QCI calculations for Cu- [5] the corresponding results of these authors are uncertain and most likely incorrect for they do not follow the pattern which one would expect on the basis of our CCSD(T) calculations. Hence our comparisons will be limited to the data for Ag- and Au-. With allowance made for differences in both methods and other details of calculations the QCISD(T) results of Schwerdtfeger and Bowmaker are surprisingly close to our CCSD(T) data. Also the pattern of the MBPT2 polarizabilities for all ions appears to be similar. Thus, one would conclude that
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V. Kell5 et al./ Chemical Physics Letters 253 (1996) 383-389
in this case the QCI and CC methods lead to compatible results. However, for Ag- the CCSD polarizability differs significantly from the QCISD result though for Au- the corresponding difference is within the limits of accuracy of both calculations. Some additional insight into the differences between the QCI and CC techniques can be gained by comparing our non-relativistic results of Table 2 with the non-relativistic pseudopotential data of Schwerdtfeger and Bowmaker [5]. For Ag- our CCSD(T) value (299 au) is nearly the same as that of the other authors (300 au). However, at the level of the CCSD and QCISD approximations the present calculation gives 350 au whereas Schwerdtfeger and Bowmaker obtain 438 au. One should mention that these values follow the pattern of relativistic polarizabilities of Ag-. However, surprisingly enough, this pattern is not followed by the QCISD data for Au-. The relativistic pseudopotential calculations show that the contribution of triple excitations as accounted for by the QCISD(T) method is small (see Table 3) while their non-relativistic results predict that this contribution should be of the order of 140 au [5]. In the present case the difference between the CCSD(T) and CCSD nonrelativistic data for Auamounts to about 50 au and is reduced to about 15 au in the DK np approximation. It is difficult to guess to what extent the behaviour of the QCI values of polarizabilities [5] follows from the faults in the method [11,27] and to what extent it is related to the use of a relativistic pseudopotential. One should mention that both non-relativistic and relativistic pseudopotentials used by Schwerdtfeger and Bowmaker [5] are the atomic energy-adjusted 19-electron pseudopotentials [28] whose performance for ions may not be as good as for neutral atoms • [21,22]. According to the present data one needs to be careful when assessing the quality of the QCI results obtained with a restricted excitation manifold. This conclusion complies with the earlier findings conceming the discrepancies between QCI and CC results [11].
4. Conclusions In the present study of dipole polarizabilities of the coinage metal anions we have carried out exten-
sive non-relativistic, MVD and DK np calculations including the electron correlation effects at the level of the CC approximation. The DK np results obtained with fully uncontracted and extended GTO basis set (see Table 3) at the level of the CCSD(T) method predict the dipole polarizabilities to be 243 au for Cu-, 213 au for Ag- and only about 93 au for Au-. The non-relativistic CCSD(T) results of this Letter (see Table 2) are 278, 299 and 256 au, respectively. The Z-dependence of the non-relativistic results predict that all these ions should have their dipole polarizabilities about the same magnitude. The relativistic effects lead to their decrease with the increasing value of the nuclear charge. In consequence, the polarizability of the Au- ion is reduced to about 100 au and Cu- turns out to be the most polarizable anion in the series. The polarizability values for the coinage metal anions strongly depend on the electron correlation effects. The significance of relativistic and mixed relativistic correlation contributions are already well seen for Ag- and their proper treatment becomes indispensable for Au-. The relativistic pseudopotentials seem to perform quite well at the level of the QCISD(T) approximation, though the QCISD results appear to suffer from truncations of the true CC expansion. The MVD approximation for relativistic contributions works well for Cu- and Ag-. However, this approximation becomes completely unsuitable for Au- leading to its dipole polarizability being lower than that of the neutral atom.
Acknowledgement The authors wish to express their thanks to Professor B.A. Hess for permission to use his code which generates the matrix elements of the DK no-pair one-electron Hamiltonian. The financial support of the European Science Foundation extended to the authors (VK and MU) through the REHE Programme is gratefully acknowledged.
References [1] H. Hotop and W.C. Lineberger, J. Phys. Chem. Ref. Data 14 (! 975) 731.
V. Kellfi et al./ Chemical Physics Letters 253 (1996) 383-389
[2] A.J. Sadlej, J. Phys. Chem. 83 (1979) 1653. [3] S. Wilson and A.J. Sadlej, Theoret. Chim. Acta 60 (1981) 19; C. Nelin, B.O. Roos, A.J. Sadlej and P.E.M. Siegbahn, J. Chem. Phys. 77 (1982) 3607; B.O. Roos, A.J. Sadlej and P.E.M. Siegbahn, Phys. Rev. A 26 (1982); G.H.F. Diercksen and A.J. Sadlej, Mol. Phys. 47 (1982) 33; G.H.F. Diercksen and A.J. Sadlej, Theoret. Chim. Acta 61 (1982) 485; S.A. Kucharski, Y.S. Lee, G.D. Purvis Ill and R.J. Bartlett, Phys. Rev. A 29 (1984) 1619. [4] P. PyykkiS, Chem. Rev. 88 (1988) 563; in: The effects of relativity in atoms, molecules, and the solid state, eds. S. Wilson, I.P. Grant and B.L. Gyorffy (Plenum Press, New York, 1991)p. 1. [5] P. Schwerdtfeger and G.A. Bowmaker, J. Chem. Phys. 100 (1994) 4487. [6] M. Urban, I. (~ernu~ik, V. Kell~ and J. Noga, in: Methods in computational chemistry, Vol. 1, ed. S. Wilson (Plenum Press, New York, 1987) p. 117, and references therein. [7] J.A. Pople, M. Head-Gordon and K. Raghavachari, J. Chem. Phys. 87 (1987) 5968. [8] A.J. Sadlej and M. Urban, Chem. Phys. Letters 176 (1991) 293. [9] K. Raghavachari, G.W. Trucks J.A. Pople and M. HeadGordon, Chem. Phys. Letters 157 (1989) 479. [10] R.J. Bartlett, J. Phys. Chem. 93 (190) 1697; T.J. Lee, A.P. Rendell and P.R. Taylor, J. Phys. Chem. 94 (1990) 5463. [11] J. Watts, M. Urban and R.J. Bartlett, Theoret. Chim Acta 90 (1995) 341. [12] R.D. Cowan and D.C. Griffin, J. Opt. Soc. Am. 66 (1976) 1010. [13] V. Kellb and A.J. Sadlej, J. Chem. Phys. 93 (1990) 8112. [14] M. Douglas and N.M. Kroll, Ann. Phys. 82 (1974) 89; J. Sucher, Phys. Rev. A 22 (1980) 348.
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[15] B.A. Hess, Phys. Rev. A 33 (1986) 3742. [16] B.A. Hess and P. Chandra, Physica Scripta 36 (1987) 412. [17] M.M. Gleichmann and B.A. Hess, Chem. Phys. Letters 227 (1994) 1. [18] U. Kaldor and B.A. Hess, Chem. Phys. Letters 230 (1994) 229. [19] C.L. Collins, K.D. Dyall and H.F. Schaeffer Iii, J. Chem. Phys. 102 (1995) 2024. [20] A.D. McLean and M. Yoshimine, J. Chem. Phys. 46 (1967) 3682; A.D. McLean and M. Yoshimine, J. Chem. Phys. 47 (1967) 3256. [21] P. Neogrady, V. KellY5, M. Urban and A.J. Sadlej, Theoret. Chim. Acta, in press. [22] V. Kellb and A.J. Sadlej, Theoret. Chim. Acta, submitted for publication. [23] K. Andersson, M.R.A. Blomberg, M. F'tilscher, V. Kell5, R. Lindh, P.-,~. Malmqvist, J. Noga, J. Olsen, B.O. Roos, A.J. Sadlej, P.E.M. Siegbahn, M. Urban and P.-O. Widmark, MOLCAS System of Quantum Chemistry Programs, Release 3. Theoretical Chemistry, University of Lund, Lund, Sweden and IBM Sweden (1994). [24] T.J. Lee, A.P. Rendell and J.E. Rice, TITAN, a set of electronic structure programs included in the MOLCAS System; T.J. Lee, A.P. Rendell and P.R. Taylor, J. Phys. Chem. 94 (1990) 5463. [25] B.A. Hess, private communication. [26] W. Kutzelnigg, Z. Physik D 11 (1989) 15; 15 (1990) 27. [27] J. Paldus, J. Ci~ek and B. Jeziorski, J. Chem. Phys. 90 (1989) 4356; 93 (1990) 1485. [28] M. Dolg, H. Wedig, H. Stoll and H. Preuss, J. Chem. Phys. 88 (1987) 866; P. Schwerdtfeger, M. Dolg, W.H.E. Schwarz, G.A, Bowmaker and P.W.D. Boyd, J. Chem. Phys. 91 (1989) 1762; D. Andrae, U. H~iusermann, M. Dolg, H. Stoll and H. Preuss, Theoret. Chim. Acta 77 (1990) 123.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 253 (1996) 383-389
Electric dipole polarizabilities of negative ions of the coinage metal atoms Vladimir Kell6 a, Miroslav Urban a, Andrzej J. Sadlej b a Department of Physical Chemistry, Faculty of Sciences, Comenius University, Mlynska dolina, SK-842 15 Bratislaua, Slovakia b Department of Theoretical Chemistry, Chemical Centre, University ofLund, P.O.B. 124, S-221 O0 Lurid, Sweden Received 14 February 1996
Abstract
The dipole polarizability of the coinage metal anions is calculated in different approximations for both the electron correlation and relativistic effects. The large polarizabilities of these ions give a good illustration of the performance of different computational methods. The spin-averaged Douglas-Kroll no-pair scheme combined with the CCSD(T) treatment of the electron correlation predict the following values of polarizabilities: 243 au for Cu-, 213 au for Ag- and only about 93 au for Au-. The non-relativistic CCSD(T) results are 278, 299 and 256 au, respectively. Both the non-relativistic and relativistic results agree with the QCISD(T) pseudopotential data, though serious discrepancies occur at the level of the QCISD approximation. The use of the scalar Pauli approximation for the estimation of the relativistic contribution to the dipole polarizabilities of the coinage metal ions is limited to Cu- and Ag-. For Au- this approximation strongly overestimates the magnitude of the relativistic effect.
1. Introduction
Most atoms with incompletely filled valence shells can bind an additional electron [1]. The energies released in such processes, i.e. the electron affinities (EA), are usually of the order of 1-2 eV [1] and the added electron is only loosely bound. In theoretical studies the interest in negative ions and their physical properties arises mostly because of the diffuse character of the electron density distribution in the outer region. This makes the electron correlation problem difficult to handle and simultaneously challenging [2,3]. In the case of heavy negative ions one can also expect significant contributions to their properties arising from relativistic effects [4]. Owing to the magnitude of these contributions the negative ions of heavy elements provide a convenient play-
ground for the investigation of the performance of different relativistic methods of quantum chemistry. In this Letter we report investigations of the dipole polarizability of the coinage metal anions (Cu-, A g - , A u - ). All of the Group Ib elements are known to form negative ions [ 1]; their experimental electron affinites are equal to 1.228 eV for Cu, 1.302 eV for Ag and 2.309 eV for Au [1]. Schwerdtfeger and Bowmaker [5] have recently carried out a series of calculations of the dipole polarizability of the negative ions of these elements. The electron correlation effects have been studied at the level of the many-body perturbation theory (MBPT) of second, third and fourth order [6] and by using the so-called quadratic configuration interaction (QCI) [7] approach with single and double excitations (QCISD) and corrected for triple exci-
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V. Kell5 et a l . / Chemical Physics Letters 253 (1996) 383-389
tations (QCISD(T)) [7]. The electron correlation contributions to anionic polarizabilities calculated by Schwerdtfeger and Bowmaker [5] exhibit a strong dependence on the level of approximation. This indicates a poor, if any, convergence of the electron correlation perturbation series and is typical of the polarizabilities of negative ions [3]. The relativistic effect upon dipole polarizabilities has been estimated by Schwerdtfeger and Bowmaker [5] from parallel calculations with non-relativistic and j-averaged relativistic pseudopotentials. The relativistic contribution to the dipole polarizability has been found to be large for Au and quite sizeable for Ag [5]. From the data presented by Schwerdtfeger and Bowmaker [5] one can also conclude that, at least for Au-, the mutual interplay of the relativistic and electron correlation effects [8] is important. In the present study the electron correlation effects will be evaluated by using the coupled cluster (CC) [6,9,10] approaches rather than the MBPT and QCI methods employed by Schwerdtfeger and Bowmaker [5]. The MBPT series, as illustrated by the results of these authors, does not seem to converge. The poor performance of the QCI methods for the polarizabilities of highly ionic systems is known as well [11] while the CCSD [6] and CCSD(T) schemes [9,10] seem to perform much better. Particular attention will be given to the evaluation of the relativistic contribution to the dipole polarizabilities of the coinage metal anions. The relativistic effects will be studied by using the scalar Pauli (MVD, mass-velocity + Darwin) approximation [12,13] and the method based on the Douglas-Kroll spin-averaged (no-pair) Hamiltonian [ 14-16]. The latter, as long as the spin-orbit coupling effects can be neglected, appears to be the best one-component quasi-relativistic scheme available at the moment [ 17-19]. The Douglas-Kroll no-pair (DK np) results will be considered as a measure of the performance of the other methods. They will be used to assess the quality of the pseudopotential data and to follow the failure of the MVD approximation for the Au- ion.
2. Methods and computational details. Basis sets
The methods which are either used or discussed in this Letter in the context of the evaluation of the
electron correlation contribution to polarizabilities belong to the standard tools of modern quantum chemistry and their detailed description can be found elsewhere [6,7,9,10]. The same applies to methods for the estimation of the relativistic effect on polarizabilities [8,12-16]. One should mention that all the polarizability data obtained in this study follow from the so-called finite perturbation calculations, i.e. they are the numerical second-order derivatives of the electric field dependent energies [20]. A similar numerical technique is also used in calculations of the mixed relativistic correlation contribution in the MVD approximation [8]. The values of the external electric field strength (F) and the numerical parameter (W) used in finite perturbation calculations within the MVD scheme [8] depend on the magnitude of the calculated polarizability and relativistic effects. The values of F used in our calculations were in the range 1 0 - 4 - - 1 0 -3 au. For the lower values of F a parabolic approximation for the F-dependence of energies has been found to be sufficiently accurate while for the higher ones a quartic polynomial fit was used. For the MVD perturbation strength the corresponding parameter W was in the range 0.0010.01 au with the higher value used in MVD calculations for Cu-. It should also be mentioned that some convergence problems occur when using the MVD finite perturbation scheme [8] for uncontracted basis sets with high-exponent GTOs; the p4 (mass-velocity) contribution to Fock matrix elements [13] may give rise to large matrix elements of the Fock operator and lead to divergences at the level of the SCF HF approximation. This problem does not occur when (core) contracted basis sets are employed. Then the large integrals are quenched by small contraction coefficients. Most of the calculations reported in this Letter have been carried out with GTO/CGTO basis sets obtained by extending the so-called polarized sets developed in our earlier studies [21,22]. One should remark that the parent polarized basis sets (PolMe for non-relativistic + MVD [21] and NpPolMe for DK np calculations [22]) are designed for molecular calculations under the assumption that no extreme ionicity is present in the given molecule. While using these sets for negative ions one needs to extend them to take into account the diffuseness of the charge
V. Kell5 et al. / Chemical Physics Letters 253 (1996) 383-389
distribution. Moreover, the appropriate polarization and correlating orbitals need to be added. Because of the difference in one-electron Hamiltonians in the non-relativistic and DK np approaches the parent basis sets for non-relativistic + MVD calculations (PolMe [21]) differ from those used in the context of the DK np scheme (NpPolMe basis sets [22]) by contraction coefficients. However, their size and flexibility in the outer region of the electron density distribution is essentially the same. For this reason the same primitive s, p, d and f functions were added to the parent sets in both cases. One should add that the parent PolMe and NpPolMe sets are essentially uncontracted in the valence and nextto-valence shells A comprehensive study of the basis set dependence of the calculated ionic polarizabilities has been carried out and some of the results are presented in Table 1. They show the effect of different extensions of the parent G T O / C G T O PolMe [21] sets on the ionic polarizability in non-relativistic and quasi-relativistic (MVD) SCF HF and CCSD(T) approximations. Several features of the quasi-relativistic MVD approximation which can be seen from this tabula-
385
tion and the apparent failure of the MVD scheme for the Au- ion will be discussed in the next section. Extensions of the parent basis sets have been generated by assuming for the pertinent GTO exponents a geometric progression based on the two lowest exponents of the given subset in PolMe sets. For the above-discussed reasons both the non-relativistic and MVD SCF HF results calculated with the PolMe sets are much too low. However, adding a (3s3pld) set of diffuse functions is quite sufficient for stabilizing the SCF HF values. The same applies essentially to the CCSD(T) results, though there is some effect of extending the p-subset by one more diffuse orbital. Since the extended sets are also used in other calculations for the negative ions of the coinage metals, we found it of some advantage to use the (3s3p2dlf) extensions of the PolMe sets rather than to further increase the number of diffuse p-type GTOs. The NpPolMe sets extended in the same way will be used in DK np calculations of ionic polarizabilities. The software used in the present study is the Molcas-3 system of quantum chemistry programs [23] linked to the Titan package for CCSD(T) calcu-
Table 1 Dipole polarizabilities of the coinage metal anions. The basis set dependence of the non-relativistic and quasi-relativistic (MVD) data. All values in au Ion
Method a
Basis set PolMe b
PolMe + 3s3pld
PolMe + 3s4pld
PolMe + 3s3p3d
PolMe + 3s3p3d2f
PolMe + 3s3pldl f
378 209 518 262 543 242
580 278 702 298 630 254
580 277 702 289 631 240
580 279 702 298 630 254
580 280 702 300 -
579 278 702 299 630 256
Quasi-relativistic (MVD) results c CuSCF 356 CCSD(T) 190 AgSCF 425 CCSD(T) 196 AuSCF 204 CCSD(T) 48
523 244 520 205 100 23
524 243 520 197 100 17
524 245 520 205 100 24
524 246 518 205 -
523 245 519 206 101 23
non-relativistic results CuSCF CCSD(T) AgSCF CCSD(T) Au SCF CCSD(T)
a All CCSD(T) results correspond to correlating 20 electrons of the valence and next-to-valence shells. b The PolMe (Me = Cu, Ag, Au) are the polarized G T O / C G T O basis sets generated in Ref. [21] for non-relativistic and quasi-relativistic (MVD) calculations of electric properties. c The MVD correction is included at both the SCF HF and CCSD(T) levels of approximations and takes into account the mixed relativistic correlation effects.
1/. Kell5 et aL / Chemical Physics Letters 253 (1996) 383-389
386
lations [24]. The DK np-pair calculations have been carried out by using the same software and a code for computing the core Hamiltonian matrix in the DK np approximation written by Hess [25].
3. Results and discussion According to the data and discussion of different basis set extensions presented in Section 2 the major part of our calculations has been carried out with the PolMe (NpPolMe) basis set extended by three s-, three p-, one d- and one f-type GTOs of low exponents. A comparison of our non-relativistic, MVD and DK np results for the three anions reveals several interesting features of the methods used here to account for relativistic contributions to their dipole polarizabilities. The corresponding results are displayed in Table 2. It can be seen that for both Cu- and Ag- the MVD approximation leads to results which are close to those calculated by using the DK np method. This observation applies to both the SCF HF and correlated levels of approximation. However, for Auquite a large difference between the two approaches occurs already for the SCF HF results. Thus, the first-order MVD correction is not capable of correctly accounting for the relativistic contraction of the 6s orbital in Au- and strongly overestimates this
effect, suggesting that higher-order approximations are necessary. Using the MVD approximation beyond first order in the perturbation treatment of relativistic contributions would raise several objections [26] and would be incompatible with the origins of the method [13]. One should stress that the Au- ion is so far one of a few examples of the major failure of the MVD scheme and, to some extent, defines the limits of its use. On the other hand, the same approximation performs in a quite satisfactory way for the Hg atom which is isoelectronic with Au-. This shows that the nuclear charge of the heavy atoms is not the only parameter which determines the quality of the MVD scheme for the calculation of relativistic corrections to atomic and molecular properties. The diffuseness of the valence ns orbital and, in particular, the relativistic shift of its nodes cannot be represented by using the first-order MVD correction to the SCF HF energy. In this context one should also remark that the same approximation is satisfactory for singly positive ions of the coinage metals [22], whose dipole polarizability is primarily determined by that of the next-to-valence (n - 1)d t° shell. Our results presented in Table 2 also give some illustration of the role of the electron correlation in both non-relativistic and relativistic calculations of the dipole polarizability of the coinage metal anions. Already in the non-relativistic approximation the
Table 2 Dipole polarizabilities of the coinage metal anions. Results of non-relativistic, MVD and DK no-pair calculations in different approximations for the electron correlation effects a. All values in au Method
SCF MBPT2
CCSD
CCSD(T)
N b
12 18 20 12 18 20 12 18 20
C-
Ag-
Au-
nr
MVD
DK np
nr
MVD
DK np
nr
MVD
DK np
579 127 76 88 298 291 297 266 270 278
523 82 32 44 264 259 265 233 238 245
526 84 34 46 266 261 267 236 240 247
702 129 60 73 361 345 350 304 290 299
519 43 -20 - 9 263 249 254 211 200 206
523 52 - 8 2 266 253 257 218 207 213
630 62 - 4 6 318 303 307 263 249 256
101 - 106 -136 - 138 48 44 45 27 21 23
193 13 -12 109 105 97 92 -
a All results of this table correspond to calculations with PolMe (NpPolMe) sets extended by the (3s3pldlf) diffuse set. See Section 2 for details. b N is the number of correlated electrons. For A u - in the DK np approximation the largest number of correlated electrons is 18 since the 5s level falls below the 4f levels.
V. Kell5 et a l . / Chemical PhySics Letters 253 (1996) 383-389
second-order MBPT2 correction strongly overestimates the electron correlation contribution, leading in some cases to disastrous (negative) results. Moreover, its value depends considerably upon the number of correlated electrons. The infinite order CC methods show that the latter effect is an artifact of the MBPT2 approximation; the CC dipole polarizabilities calculated with either 12 or 20 correlated electrons do not differ significantly. When the MBPT2 method is combined with the MVD approach to the evaluation of mixed relativistic-correlation contributions this leads to unphysical results already for Ag-. One should mention, however, that the DK np scheme suffers in the same way. The second-order treatment of the electron correlation in negative ions of the coinage metal atoms leads to an overestimation of both pure correlation and relativistic correlation corrections to dipole polarizabilities. A little disturbing is the difference between CCSD(T) and CCSD results which amounts to about
Table 3 Dipole polarizabilities of the coinage metal anions. Results of the DK np calculations with fully uncontracted ( 3 s 3 p l d l f ) extended NpPolMe basis sets and their comparison with the pseudopotential data of Schwerdtfeger and Bowman. All values in an Method this work SCF MBPT2
CCSD
CCSD(T)
reference results [5] b SCF MBPT2 QCISD QCISD(T)
N a
CU-
Ag-
Au-
12 18 20 12 18 20 12 18 20
528 80 31 45 269 264 269 234 236 243
523 49 - 10 - l 267 254 258 217 206 213
195 15 - 10 112 108 98 93 -
20 20 20
537 71 216 ¢ 317 c
543 15 327 213
205 3 118 96
a See footnote b to Table 2. b In Ref. [5] the polarizability data are given in ,~3. The conversion factor used here is: 1 au of the dipole polarizability -~ 0.14818 ¢ These results are likely to be incorrect because of some convergence problems which are mentioned by Schwerdtfeger and Bow-
man [5].
387
30 au for Cu- and increases to about 50 au for Ag-. This may indicate that including iterative contributions due to T 3 operators and perhaps also those from the higher order cluster operators may be of some importance. This effect is, however, much less pronounced in the case of Au-. The corresponding difference amounts to only about 10 au and makes the present CCSD(T) reliable. It has already been pointed out that the DK np approach to scalar relativistic effects in atoms and molecules appears to be currently the most successful tool of relativistic quantum chemistry [19]. The present data show its superiority to the MVD scheme in the case of heavy ions with diffuse electron density distribution. Though computationally advantageous, the DK np approach does not offer an immediate answer to the question of the importance of relativistic effects. This can be achieved only by comparing the DK np results with the non-relativistic ones; the latter should be obtained with fully compatible basis sets. On the other hand, the MVD corrections are calculated on top of the usual non-relativistic calculations. They offer a perturbative insight into the role of relativistic effects. At some point the MVD approach must fail in the same way as any first-order perturbation scheme does for high enough values of the perturbation strength. So far our major interest was in comparative studies of different methods. In order to obtain the best values of the dipole polarizability of the coinage metal anion we have carried out DK np calculations with fully uncontracted ( 3 s 3 p l d l f ) extended NpPolMe basis sets. The results are presented in Table 3 and compared with the data calculated by Schwerdtfeger and Bowmaker [5]. Because of some convergence problems in the QCI calculations for Cu- [5] the corresponding results of these authors are uncertain and most likely incorrect for they do not follow the pattern which one would expect on the basis of our CCSD(T) calculations. Hence our comparisons will be limited to the data for Ag- and Au-. With allowance made for differences in both methods and other details of calculations the QCISD(T) results of Schwerdtfeger and Bowmaker are surprisingly close to our CCSD(T) data. Also the pattern of the MBPT2 polarizabilities for all ions appears to be similar. Thus, one would conclude that
388
V. Kell5 et al./ Chemical Physics Letters 253 (1996) 383-389
in this case the QCI and CC methods lead to compatible results. However, for Ag- the CCSD polarizability differs significantly from the QCISD result though for Au- the corresponding difference is within the limits of accuracy of both calculations. Some additional insight into the differences between the QCI and CC techniques can be gained by comparing our non-relativistic results of Table 2 with the non-relativistic pseudopotential data of Schwerdtfeger and Bowmaker [5]. For Ag- our CCSD(T) value (299 au) is nearly the same as that of the other authors (300 au). However, at the level of the CCSD and QCISD approximations the present calculation gives 350 au whereas Schwerdtfeger and Bowmaker obtain 438 au. One should mention that these values follow the pattern of relativistic polarizabilities of Ag-. However, surprisingly enough, this pattern is not followed by the QCISD data for Au-. The relativistic pseudopotential calculations show that the contribution of triple excitations as accounted for by the QCISD(T) method is small (see Table 3) while their non-relativistic results predict that this contribution should be of the order of 140 au [5]. In the present case the difference between the CCSD(T) and CCSD nonrelativistic data for Auamounts to about 50 au and is reduced to about 15 au in the DK np approximation. It is difficult to guess to what extent the behaviour of the QCI values of polarizabilities [5] follows from the faults in the method [11,27] and to what extent it is related to the use of a relativistic pseudopotential. One should mention that both non-relativistic and relativistic pseudopotentials used by Schwerdtfeger and Bowmaker [5] are the atomic energy-adjusted 19-electron pseudopotentials [28] whose performance for ions may not be as good as for neutral atoms • [21,22]. According to the present data one needs to be careful when assessing the quality of the QCI results obtained with a restricted excitation manifold. This conclusion complies with the earlier findings conceming the discrepancies between QCI and CC results [11].
4. Conclusions In the present study of dipole polarizabilities of the coinage metal anions we have carried out exten-
sive non-relativistic, MVD and DK np calculations including the electron correlation effects at the level of the CC approximation. The DK np results obtained with fully uncontracted and extended GTO basis set (see Table 3) at the level of the CCSD(T) method predict the dipole polarizabilities to be 243 au for Cu-, 213 au for Ag- and only about 93 au for Au-. The non-relativistic CCSD(T) results of this Letter (see Table 2) are 278, 299 and 256 au, respectively. The Z-dependence of the non-relativistic results predict that all these ions should have their dipole polarizabilities about the same magnitude. The relativistic effects lead to their decrease with the increasing value of the nuclear charge. In consequence, the polarizability of the Au- ion is reduced to about 100 au and Cu- turns out to be the most polarizable anion in the series. The polarizability values for the coinage metal anions strongly depend on the electron correlation effects. The significance of relativistic and mixed relativistic correlation contributions are already well seen for Ag- and their proper treatment becomes indispensable for Au-. The relativistic pseudopotentials seem to perform quite well at the level of the QCISD(T) approximation, though the QCISD results appear to suffer from truncations of the true CC expansion. The MVD approximation for relativistic contributions works well for Cu- and Ag-. However, this approximation becomes completely unsuitable for Au- leading to its dipole polarizability being lower than that of the neutral atom.
Acknowledgement The authors wish to express their thanks to Professor B.A. Hess for permission to use his code which generates the matrix elements of the DK no-pair one-electron Hamiltonian. The financial support of the European Science Foundation extended to the authors (VK and MU) through the REHE Programme is gratefully acknowledged.
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