Electron affinities of boron, aluminum, gallium, indium, and thallium

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1 PHYSICAL REVIEW A VOLUME 56, NUMBER 6 DECEMBER 1997 Electron affinities of boron, aluminum, gallium, indium, and thallium Ephraim Eliav School of Chemistry, Tel Aviv University, Tel Aviv, Israel Yasuyuki Ishikawa Department of Chemistry, University of Puerto Rico, P.O. Box 23346, San Juan, Puerto Rico Pekka Pyykkö Department of Chemistry, University of Helsinki, P.O. Box 55 (A. I. Virtasen aukio 1), FIN-00014, Helsinki, Finland Uzi Kaldor School of Chemistry, Tel Aviv University, Tel Aviv, Israel Received 3 June 1997 The relativistic coupled-cluster method in a large four-component Gaussian-spinor basis is applied to the ionization potentials and electron affinities of group-13 elements B, Al, Ga, In, and Tl. Many shells are correlated up to 35 electrons for Tl to account for core effects. Calculated ionization potentials are in excellent agreement within ev with experiment. The electron affinities are ev for B experimental , ev for Al ), ev for Ga , ev for In , and ev for Tl experimental The first four are in close agreement with recent multiconfiguration Dirac-Fock MCDF values; it is shown that the smaller affinity predicted by MCDF for Tl ev is due to the neglect of dynamic correlation, in particular that of the 5d electrons, which has a substantial contribution in this case. S X PACS numbers: Jv, v, Dv, w I. INTRODUCTION While the electron affinities EA of the two lightest group-13 elements, boron and aluminum, are experimentally well known, this is not the case for the heavier elements, where errors in recommended values 1,2 range from 50% for Ga to 100% for Tl. Very few calculations for the latter elements appeared in the literature; in fact, the only computations known to us are by Arnau et al. 3 using pseudopotentials and multireference configuration interaction MRCI, our relativistic coupled-cluster RCC work on Tl 4, and the multiconfiguration Dirac-Fock MCDF computation of Wijesundera 5, which appeared while the present work was in progress. Comparison with experimental values are meaningful only for boron and aluminum. The MRCI values for Al 0.45 ev and the MCDF results for B 0.26 ev and Al 0.43 ev are in good agreement with the Hotop and Lineberger values 0.28 and 0.44 ev, respectively. The MRCI and MCDF EAs for the other atoms agree with each other 0.29 and 0.30 ev for Ga, 0.38 and 0.39 ev for In, 0.27 and 0.29 ev for Tl. The RCC EA of Tl is much higher at ev. A major difference between the RCC and the other two methods lies in the number of electrons correlated. While 3 and 5 correlate valence electrons only, three for the neutral atom and four for the anion, we correlated 35 electrons in Tl and 36 in Tl. The goal of the present work is twofold: to present RCC electron affinities for all five elements of group 13, and to study the effect of inner-shell correlation and the virtual space used in the calculation on the EAs, particularly for Tl. II. METHOD The relativistic coupled-cluster method has been described in our previous publications 6,7, and only a brief review is given here. We start from the projected Dirac- Coulomb or Dirac-Coulomb-Breit Hamiltonian 8,9, where in atomic units H H 0 V, 1 H 0 i i h D i i, 2 h D ic i p i c 2 i 1V nuc iui, 3 V i j i j V eff ij j i i i Ui i. 4 Here h D is the one-electron Dirac Hamiltonian. An arbitrary potential U is included in the unperturbed Hamiltonian H 0 and subtracted from the perturbation V. This potential is chosen to approximate the effect of the electron-electron interaction; in particular, it may be the Dirac-Fock selfconsistent-field potential. The nuclear potential V nuc includes the effect of finite nuclear size. i are projection operators onto the positive energy states of the Dirac Hamiltonian h D. Because of their presence, the Hamiltonian H has normalizable, bound-state solutions. This approximation is known as the no-virtual-pair approximation, since virtual electronpositron pairs are not allowed in intermediate states. The form of the effective potential V eff depends on the gauge /97/566/45325/$ The American Physical Society

2 56 ELECTRON AFFINITIES OF BORON, ALUMINUM, used. In Coulomb gauge it becomes in atomic units, correct to second order in the fine-structure constant ) 10 V eff 1 r 12 B 12 O 3, where the frequency-independent Breit interaction is 5 B r r 12 2 r 12 /r The Dirac-Coulomb-Breit Hamiltonian H may be rewritten in terms of normal-ordered products of secondquantized spinor operators, r s and r s ut 8,11, HH 0H 0 rs f rs r s 1 rstur s ut, 4 rstu where f rs and rstu are, respectively, elements of oneelectron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac fourcomponent spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly brackets in the equation above, which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive energy state, and the negative energy states are ignored. The no-pair approximation leads to a natural and straightforward extension of the nonrelativistic open-shell CC theory. The multireference valence-universal Fock-space coupled-cluster approach is employed here, which defines and calculates an effective Hamiltonian in a low-dimensional model or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. According to Lindgren s formulation of the open-shell CC method 12, the effective Hamiltonian has the form H eff PHP, where is the normal-ordered wave operator, exps. The excitation operator S is defined in the Fock-space coupled-cluster approach with respect to a closed-shell reference determinant. In addition to the traditional decomposition into terms with different total (l) number of excited electrons, S is partitioned according to the number of valence holes (m) and valence particles (n) to be excited with respect to the reference determinant, S m0 n0 lmn S l m,n. 10 The upper indices in the excitation amplitudes reflect the partitioning of the Fock space into sectors, which correspond to the different numbers of electrons in the physical system. This partitioning allows for partial decoupling of the openshell CC equations, since the equations in each sector do not involve excitation amplitudes from higher sectors. The eigenvalues of the effective Hamiltonian 8 in a sector give directly the correlated energies in that sector with respect to the correlated 0,0 reference state. These transition energies may be ionization potentials IP, electron affinities, or excitation energies, according to the presence of valence holes and/or valence particles. In the present application we use the 0,0, 0,1, 0,2, 1,0, and 2,0 sectors. The lower index l in Eq. 10 is truncated at l2. The resulting coupled cluster with single and double excitations CCSD scheme involves the fully self-consistent, iterative calculation of all one- and two-body virtual excitation amplitudes, and sums all diagrams with these excitations to infinite order. Negative energy states are excluded from the Q space, and the diagrammatic summations in the CC equations are carried out only within the subspace of the positive energy branch of the Dirac-Fock spectrum. III. CALCULATIONS The Fock-space relativistic coupled-cluster method was applied to the group-13 elements B, Al, Ga, In, and Tl. The reference configuration was the closed-shell ns 2 M. Two electrons were added in the np shell, one at a time, to obtain M, M 0,0 M0,1 M 0,2. 11 The two heaviest elements, In and Tl, are sufficiently relativistic to make the ns 2 2 np 1/2 configuration closed shell, providing an alternative route to the electron affinity and ionization potential, with two electrons ionized from the np 1/2 shell: M 0,0 M1,0 M 2,0. 12 The Dirac-Fock 11 and RCC 6,7 programs are both written for spherical symmetry, utilizing the angular decomposition of the wave function and CC equations in a central field. The energy integrals and CC amplitudes which appear in the Goldstone-type diagrams defining the CC equations are decomposed in terms of vector-coupling coefficients, expressed by angular-momentum diagrams, and reduced Coulomb-Breit or S matrix elements, respectively. The re- TABLE I. Even-tempered basis sets 15 for group-13 elements. Members of the s-basis series used in the various l sectors are given. Basis s p d f g h i B In 35s26p20d14f 9g6h4i Tl 35s27p21d15f 9g6h4i

3 4534 ELIAV, ISHIKAWA, PYYKKÖ, AND KALDOR 56 TABLE II. Ionization potential IP and electron affinity EA of B ev. Methods are relativistic CC with singles and doubles RCCSD, nonrelativistic CC with singles and doubles NRCCSD or with singles, doubles, and triples NRCCSDT, and multiconfiguration Hartree-Fock MCHF. RCCSD big basis NRCCSD smaller basis NRCCSDT smaller basis Triples RCCSD corrected for triples MCHF Expt. 1, IP EA TABLE IV. Dependence of Tl EA on amount of correlation and coupled-cluster scheme ev. Basis Correlated Tl electrons Scheme 12 Scheme 11 l6 4f 14 5s 2 5p 6 5d 10 6s 2 6p f 14 5p 6 5d 10 6s 2 6p p 6 5d 10 6s 2 6p d 10 6s 2 6p s 2 6p l5 6s 2 6p l4 6s 2 6p l3 6s 2 6p MCDF 5 l3 6s 2 6p duced equations for single and double excitation amplitudes are derived using the Jucys-Levinson-Vanagas theorem 12 and solved iteratively. This technique makes possible the use of large basis sets. To avoid variational collapse 13, the Gaussian spinors in the basis are made to satisfy kinetic balance 14. They also satisfy relativistic boundary conditions associated with a finite nucleus, described here as a sphere of uniform proton charge 11. The atomic masses are for B, for Al, for Ga, for In, and for Tl. The speed of light c is atomic units. The uncontracted universal basis set of Malli, Da Silva, and Ishikawa 15 is used. It consists of Gaussian-type orbitals, with exponents given by the geometric series n n1, , Orbitals with l values from 0 to 6 are included; details are given in Table I. Atomic orbitals with the same l but different k number e.g., p 1/2 and p 3/2 ) are expanded in the same basis functions. Smaller sets are used to estimate the contribution of triple excitations to the electron affinities of B and Al. These are the atomic natural orbital bases of Widmark et al. 16, with diffuse functions added to describe the anions, giving a (15s10p5d4 f )/8s8p5d4f set. To account for possible core effects, many core electrons were correlated. The number of correlated electrons is 5 for B, 11 for Al (K shell excluded, 27 for Ga not including K and 2s electrons, 21 for In freezing K, L, and M electrons, and 35 for Tl (4 f 5s5p5d6s6p). Virtual orbitals with high orbital TABLE III. Ionization potential IP and electron affinities EA of Al ev. IP EA 3 P 0 EA 1 D 2 RCCSD big basis NRCCSD big basis NRCCSD smaller basis NRCCSDT smaller basis Triples RCCSD corrected for triples Expt. 1,2, energies have been found to contribute very little to correlation effects on ionization potentials and electron affinities; orbitals higher than 100 a.u. are therefore eliminated from the calculation, effecting considerable savings in computational effort. IV. RESULTS AND DISCUSSION A. Boron and aluminum: Contribution of triple excitations The ionization potential and electron affinity of boron are shown in Table II. The CCSD approximation yields an excellent IP 17 but a rather poor EA 1. It has been observed before 18 that three-body terms have significant effect on transition energies in first-row atoms. To check this point we calculated three-body excitations in a smaller basis 16. A nonrelativistic program was used, but this hardly matters, since relativistic effects on the EA of B are minute. The smaller basis gives IP and EA rather close to the big basis values Table II; it is therefore reasonable to assume that it provides a good estimate of the contribution of triples. This contribution is substantial; when added to the CCSD values of the big basis, agreement within the experimental error bounds of ev is obtained. The present EA of boron is also in excellent agreement with that of Froese Fischer et al. 19, having estimated error bounds of ev. The IP and EA of Al were calculated using the same procedure as for B Table III. Here the CCSD values are in rather good agreement with experiment. The three-body effects on the EA are much smaller than for B, in line with previous observations showing rather small effects for TABLE V. Dependence of In EA on amount of correlation and coupled-cluster scheme ev. Basis Correlated In electrons Scheme 12 Scheme 11 l6 4s 2 4p 6 4d 10 5s 2 5p p 6 4d 10 5s 2 5p d 10 5s 2 5p s 2 5p l5 5s 2 5p l4 5s 2 5p l3 5s 2 5p MCDF 5 l3 5s 2 5p 0.393

4 56 ELECTRON AFFINITIES OF BORON, ALUMINUM, TABLE VI. Group-13 electron affinities ev. Atom Expt. 1,2 RCCSD MCDF 5 B a Al ( 48 ) a Ga In Tl TABLE VII. Ionization potentials of group-13 elements ev. Atom Expt. 17 RCCSD B Al Ga In Tl a Including triple excitations. second-row atoms 20. Values corrected for triple excitations show very good agreement with experiment Table III. B. Thallium: Effect of dynamic correlation No accurate experimental determination of the EA of Tl exists 1. We have previously calculated this value as ev 4. More recently, Wijesundera 5 reported a multiconfiguration Dirac-Fock EA of ev, in agreement with an earlier MRCI calculation 3. The RCC values for the EAs of the other group-13 elements are close to corresponding MCDF numbers; we therefore set out to elucidate the source of the difference. Both MRCI 3 and MCDF calculations 5 correlated only the 6s6p electrons of Tl, while RCC correlated the external 35 electrons. A careful study of the contribution of the different core shells to the electron affinity was therefore undertaken. Results Table IV show that correlation of the 4 f,5s, and 5p shells has small but not insignificant effect ev; the 5d shell, on the other hand, has a substantial contribution 0.08 ev, accounting for most of the difference between RCC and MCDF values. Additional contributions to this difference come from the truncation of the MCDF space at l3; the RCC value with this truncation and correlating only the 6s and 6p electrons is very close to the MCDF results, confirming the source of the difference. Neglect of dynamic correlation in MCDF calculations has affected energies in other cases: the RCC values for excitation energies in Pr 3 21 have one-fourth the error of corresponding MCDF values 22; and dynamic correlation changes the order of the two low levels of rutherfordium element One problem we did not resolve involves the significant difference between EAs calculated by schemes 11 and 12. While the contributions of the dynamic correlation of inner shells are similar in the two schemes, there is a difference of 0.1 ev in the basic valence-electron EA. It should be noted that both schemes give values larger than 5. Scheme 12 should be more reliable, as it starts from the Tl orbitals. This difference leads to assigning a rather high uncertainty to our final EA, which is ev. Errors are much smaller for the other atoms. C. Other group-13 elements CCSD EAs for the other group-13 elements are not far from MCDF values. A study similar to the one just described for Tl was carried out for In Table V. Dynamic correlation effects were one-third those of Tl (0.04 ev, explaining the good agreement between MCDF and CCSD. No significant difference is found between EAs calculated by schemes 11 and 12 in this case. The electron affinities of all group-13 elements are shown in Table VI. Only B and Al have reliable experimental electron affinities. In order to map the periodic trends, modern measurements for Ga, In, and Tl would be most welcome. The calculated ionization potentials Table VII are in excellent agreement with experiment, with errors of ev. Finally, we compare our results with some recent calculations for the two heavier group-13 elements. For In, Guo and Whitehead 24 used a quasirelativistic local-spindensity-functional theory with the correlation corrections of Vosko et al. 25 and obtained an electron affinity of ev, depending on the exact form of the correlation, which is within 0.05 ev of our value. Their ionization potential is ev, too low by 0.3 ev. We are not aware of EA calculations for Tl, except the two applications discussed in the Introduction. Several recent papers describe calculations of the Tl IP. They start from the Dirac-Fock function and correct it perturbatively. Results are ev 26, ev 27, or ev 28; a somewhat different scheme, using configuration interaction, gave ev 29. All these values are somewhat further from experiment ev than the RCC result reported here ev. V. SUMMARY AND CONCLUSION Ionization potentials and electron affinities of group-13 elements are calculated by the relativistic coupled-cluster method. Large sets of four-component Gaussian spinors with l up to 6 are used, and many shells are correlated. Results are in excellent agreement with experimental values a few hundredths of an ev when the latter are known all IPs, EAs of B and Al, and may provide the best values when experimental errors are large EAs of Ga, In, and Tl. The EAs for the lighter four elements agree well with MCDF results 5, but the MCDF EA for Tl is lower by 0.12 ev than the corresponding CCSD value. The difference is ascribed to the neglect of dynamic correlation by MCDF. In particular, the correlation of the 5d shell contributes 0.08 ev to the EA; the other shells and the smaller virtual space open to the MCDF function account for the rest. ACKNOWLEDGMENTS The research reported above was supported at TAU by the U.S.-Israel Binational Science Foundation and the Israeli Ministry of Science. Computer time on the Israeli HPCU IBM SP2 machine is acknowledged. Y.I. was supported by the National Science Foundation through Grant No. PHY P.P. receives financial support from The Academy of Finland.

5 4536 ELIAV, ISHIKAWA, PYYKKÖ, AND KALDOR 56 1 H. Hotop and W. C. Lineberger, J. Phys. Chem. Ref. Data 4, ; 14, D. Calabrese, A. M. Covington, J. S. Thompson, R. W. Marawar, and J. W. Farley, Phys. Rev. A 54, F. Arnau, F. Mota, and J. J. Novoa, Chem. Phys. 166, E. Eliav, U. Kaldor, Y. Ishikawa, M. Seth, and P. Pyykkö, Phys. Rev. A 53, W. P. Wijesundera, Phys. Rev. A 55, E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 49, E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 50, J. Sucher, Phys. Rev. A 22, ; Phys. Scr. 36, W. Buchmüller and K. Dietz, Z. Phys. C 5, I. Lindgren, in Many-Body Methods in Quantum Chemistry, edited by U. Kaldor, Lecture Notes in Chemistry Vol. 52 Springer-Verlag, Heidelberg, 1989, p Y. Ishikawa and H. M. Quiney, Phys. Rev. A 47, ; Y. Ishikawa, ibid. 42, ; Y. Ishikawa, R. Baretty, and R. C. Binning, Chem. Phys. Lett. 121, I. Lindgren and J. Morrison, Atomic Many-Body Theory, 2nd ed. Springer-Verlag, Berlin, W. Kutzelnigg, Int. J. Quantum Chem. 25, R. E. Stanton and S. Havriliak, J. Chem. Phys. 81, G. L. Malli, A. B. F. Da Silva, and Y. Ishikawa, Phys. Rev. A 47, P.-O. Widmark, P. Å. Malmquist, and B. O. Roos, Theor. Chim. Acta 77, ; P.-O. Widmark, B. J. Persson, and B. O. Roos, ibid. 79, C. E. Moore, Atomic Energy Levels, Natl. Bur. Stand. U.S. Circ. No. 467 U.S. GPO, Washington, DC, 1947, Vol. I; ibid. U.S. GPO, Washington, DC, 1952, Vol. II; ibid. U.S. GPO, Washington, DC, 1958, Vol. III. 18 U. Kaldor and A. Haque, Chem. Phys. Lett. 128, C. Froese Fischer, A. Ynnerman, and G. Gaigalas, Phys. Rev. A 51, U. Kaldor, Int. J. Quantum Chem. Symp. 20, E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 51, Z. Cai, V. Meiser Umar, and C. Froese Fischer, Phys. Rev. Lett. 68, E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. Lett. 74, Y. Guo and M. A. Whitehead, Phys. Rev. A 38, S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and O. P. Sushkov, J. Phys. B 20, V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and O. P. Sushkov, in Relativistic, Quantum Electrodynamic, and Weak Interaction Effects in Atoms, edited by W. Johnson, P. Mohr, and J. Sucher, AIP Conf. Proc. No. 189 AIP, New York, 1989, p S. A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Rev. A 42, V. A. Dzuba, V. V. Flambaum, and M. G. Kozlov, Phys. Rev. A 54,

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