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Accurate core binding energies of ions from Dirac-Fock calculations combined with experimental atomic binding energies
and Related Phenomena, 26 (1982) 269-266 Elsevier Scientific Publishmg Company, Amsterdam - Printed in The Netherlands
Journal of Electron Spectroscopy
ACCURATE CORE BINDING ENERGIES OF IONS FROM DIRAC-FOCK CALCULATIONS COMBINED WITH EXPERIMENTAL ATOMIC BINDING ENERGIES
M. S. BANNA, R. J. KEY and C. S. EWIG
Department
of Chemistry,
Vanderbilt Unwersity, Nashville, TN 37235 (U.S.A.)
(First received 28 December 1981; in final form 9 February 1982)
ABSTRACT Core binding energies are reported for the singly-charged alkali ions obtained by removing the outer s electron and for the singly-charged halide ions obtained by adding a pJIp electron. The levels studied are Li Is, Na Is, K 2p, Rb 3p, Cs 3d, F ls, Cl 2p, Br 3p and 13d. The binding energies were obtained by combining Dirac-Fock ASCF bindingenergy shifts between the ions and the corresponding isoelectronic rare gases with experimental rare-gas binding energies, and also by combining the calculated atom-ion shifts with experimental atomic binding energies where available. It is shown that the former approach corrects accurately for the correlation energy, which is not included in singleconfiguration calculations.
INTRODUCTION
Although it is not yet possible to measure core binding energies of ions by X-ray photoelectron spectroscopy, there is nevertheless considerable current interest in these quantities. Extensive Hartree-Fock atom-ion bindingenergy shifts for levels frequently studied by XPS have recently been published [l]. A more limited study of atom-ion shifts has been reported by Key et al. [ 23, who used a numerical Dirac-Fock program [ 31 and obtained better absolute binding energies than is possible using single-configuration nonrelativistic calculations. Highly ionic solids such as the alkali halides have been the subject of numerous XPS studies [4], and recently at least two groups have begun a study of the alkali halides in the gas phase. Mathews et al. [ 51 reported 3d binding energies for gaseous CsI and Aksela et al. [ 6] performed a number of Auger measurements on the gaseous alkali halides. Accurate free-ion binding energies would be useful in interpreting both Auger and XPS spectra.
0368-2048/82/0000-0000/$02.75
o 1982 Elsevier Scientific Publishing Company
260 METHOD
In this study we have used the numerical Dirac-Fock program of Desclaux [3] to obtain single-configuration “ASCF” core binding energies of one (possibly spin-orbit split) level in each of the alkali cations Li+, Na+, K+, Rb+ and Cs+ and of the same level in each of the corresponding halide anions F-, Cl-, Br- and I: The binding energies thus obtained should be in good agreement with experiment, judging from the agreement obtained for atoms [2,7]. Discrepancies are expected to remain, however, due primarily to the neglect of correlation energy. It is possible to obtain excellent agreement with experiment when approximate correlation effects are included explicitly for atoms for which such a calculation is feasible [ 81. However, we present here a simple semiempirical scheme that should be capable of yielding accurate core binding energies for the levels under study. The method relies on the availability of accurate experimental core binding energies for the rare gases, which are isoelectronic with the alkali and halide ions. The experimental binding energies are combined with the computed rare-gas-ion shifts, which are considerably more accurate than the computed absolute binding energies, to obtain good semiempirical values. A related but somewhat less accurate method is to combine the calculated atom-ion shifts with experimental atomic binding energies. For the systems we have chosen, this is possible for the alkali ions since atomic core binding energies are available [g-11]. The values thus obtained are compared with those obtained from the shifts between isoelectronic pairs. The reason for the accuracy of values extrapolated from isoelectronic rare gases as opposed to those from neutral atoms of the same atomic number 2 lies largely in the approximate invariance of correlation energy with changes in 2. For example, the striking constancy of the correlation energy in twoelectron systems from H- to C 4+ has been noted by Lowdin [12]. This was later justified rigorously by Stanton [13], who demonstrated the following: any parameter in the total Hamiltonian operator that affects only the oneelectron systems from H- to C4+ has been noted by Lowdin [12]. This was in comparison with those that affect two-electron terms. Thus a change in 2, which affects only the (one-electron) nuclear attraction operator, has only a small effect on the correlation energy. Although Stanton’s proof was in reference to nonrelativistic HartreeFock calculations, exactly the same principle applies to the DiraFFock method as well. The DirapFock Hamiltonian may be written as
where the two summations represent one- and two-electron operators respec-
261
tively, Hn (i) is the single-particle Dirac Hamiltonian, pi is an electron charge, and rir the interelectronic distance. H,,(i) may be written, with conventional notation, as H,(l)
= (Y,
l
P,C
+ &c”
-
ppZ/ri
in which all terms except the last are matrices representing the relativistic kinetic energy, and rr is the distance from the nucleus. Formally, the summa. . tions m HDF may be considered as carried out over an arbitrarily large number of electrons, pz being set to zero for all except the bound electrons. If 2 is considered as the variable parameter, and pr is held fixed as in going from a neutral atom to the isoelectronic ion, only H,(i) is affected and thus the change in correlation energy is small. However, if 2 is held fixed and pI is varied (for example, a change by unity corresponds to going from a neutral atom to its singly-charged ion), both one- and two-electron terms are affected. Thus the corresponding change in correlation energy is expected to be relatively large in this case. So if neglect of correlation energy is assumed to be the predominant source of error in the Dirac-Fock ASCF approach, this principle means that reference to an isoelectronic system should always be the most accurate approximation.
RESULTS AND DISCUSSION
The experimental atomic binding energies used in calculating the binding energies of free ions by the two methods outlined above are given in Table 1. Our results are presented in Table 2. The binding energies in columns 3-6 represent an attempt to correct for the deficiencies in the DF and HF calculations. This is done in two ways. In columns 3 and 4 we combine the shift between the ASCF binding energies of the isoelectronic ion and the corresponding rare gas, calculated relativistically (column 3) or nonrelativistically [14] (column 4), with the accurately known experimental binding energies of the rare gas. It is expected that the correlation energies of isoelectronic species would be almost equal, which would result in an accurate calculated rare-gas-isoelectronic-ion shift and hence accurate ionic binding energies. In columns 5 and 6, the ionic binding energies are obtained from calculated atom-ion shifts combined with the known experimental alkali-atom binding energies. The shifts in column 5 were calculated using the program of Desclaux [3] while the non-relativistic shifts used in column 6 were obtained from the work of Broughton and Bagus [ 11. The former cannot be used for systems having two open subshells of the same symmetry; therefore relativistic Li 1s and Na 1s atomic binding energies were obtained from the Diracr Slater work of Huang et al. [ 151. Since the atom and the corresponding ion differ by one electron, the correlation energies are not expected to cancel to
262 TABLE 1 EXPERIMENTAL CORE BINDING ENERGIES USED IN THIS WORK FOR THE RARE GASES AND THE ALKALIMETAL ATOMS Atom
Level
Binding energy (eV)
He Ne Ar Ar Kr Kr Xe Xe Li Na K K Rb Rb cs cs
IS 15 2PllZ 2PY2 3P1/2 3Ps12 3&z 3&/z 1s 1s 2Plf2 2PY2 3Plf2 3~~2 3&/z 3ds/2
24.5ga 870.21b 250.78’ 248.63’ 221.Bd 214.2d 689.3tid 676.70d 64.8ge 1078.9f 303.6g 300.7g 254.3h 245.4h 745.6f 731.6f
a From ref. 10. b From H. &ren, J. Nordgren, L. Selander, C. Nordling and K. Siegbahn, J. Electron Spectroac. Relat. Phenom., 14 (1978) 27. c From J. Nordgren, H. &ren, C. Nordling and K. Siegbahn, Phys. Ser., 19 (1979) 5. d From ref. 16. e From ref. 10, averaged over multipleta. f From ref. 9, reduced by 0.2 eV because of the new value of Ne 1s reported in the work quoted in footnote b above. a From ref. 11. h From ref. 9.
the same degree as for isoelectronic species. This, coupled with the fact that alkali-atom binding energies are known less accurately than those of the rare gases, means that this method is likely to give less reliable ionic binding energies than the isoelectronic-species method. The Dirac-Fock “ASCF” energies used are given in column 7 with the relaxation energies in parentheses. The DF ASCF results for the rare gases as well as for the alkali atoms have been given elsewhere [Z] and are not included here, except the He 1s value of 23.45 eV. Finally, we show in column 8 the nonrelativistic ASCF results recently published by Broughton and Bagus [l] , together with the corresponding relaxation energies. The Hartree-Fock results of Broughton and Bagus [l] do not suffer from any basis-set deficiency since they were obtained numerically. However,
18 1s
Li+ Na+ K+ K+ Rb+ Rb+ C3+ CS+ FCiCiBrBrII-
75.608 1088.71 310.73 307.94 260.5 251.7 752.02 738.11 679.79 198.54 196.91 187.4 180.9 630.77 619.30 197.47
680.11
1088.26
75.5oh 1088.7h 310.7 307.8 260.8 251.9 751.4 737.4 75.62 1088.3 310.7 307.8 260.7 251.8 751.3 737.3
74.46 1087.97 310.22 307.47 263.75 254.64 751.70 737.88 679.05 198.03 196.43 190.58 183.70 630.45 619.07
( 1.53) (22.99) (11.37) (11.15) ( 8.97) ( 8.56) (18.65) (17.93) (24.62) (12.61) (12.45) (10.52) (10.16) (19.87) (19.23)
633.90
(18.97)
( 9.93)
(12.35)
197.07 184.37
(17.61) (24.34)
( 8.29)
(10.99)
( 1.52) (22.46)
755.32 678.54
255.36
308.43
74.46 1086.70
HF*
s Obtained by combining DF ion-isoelectronic rare-gas ASCF shift with experimental rare-gas binding energies from Table 1. b Same as a, with shifts calculated nonrelativisticaily by Bagus [ 141. Weighted means of the experimental rare-gas spin-orbit components’ binding energies were used. c Obtained by combining DF atom-ion ASCF shifts with experimental atomic binding energies for the alkali atoms from Table 1. d Same as c, with HF atom-ion shifts from ref. 1. e Dirac-Fock ASCF results using the program described in ref. 3. Relaxation energies are given in parentheses. * Hartree-Fock ASCF results from ref. 1. Relaxation energies are given in parentheses. s The ASCF DF He 1s value used is 23.45 eV. h Obtained using Dirac-Slater ASCF atomic binding energies from ref. 15.
3dY2
3dY2
3~~2
Qpl/2
2PY2
2Pll2
1s
34l2
wz
~PYZ
3p~2
2P3/2
SPY2
Level
Ion
CORE BINDING ENERGIES OF THE ALKALI AND HALIDE IONS (eV)
TABLE 2
E
264
neither relativistic effects nor correlation-energy differences between the ground and core-hole states are included. The former are expected to amount from a fraction of an eV to perhaps 1 eV or more for these systems as found from a comparison of relativistic and nonrelativistic binding energies of atoms [2]. For example, the relativistic 1s binding energy of Na+ is 1.3eV higher than the nonrelativistic binding energy (columns 7 and 8 in Table 2). When the level is spin-orbit split, we have argued previously [2] that for p levels the nonrelativistic result is always closer to the binding energy of the pY2 level, while for d levels it is closer to that of ds2. This is clearly seen in Table 2. Both the relativistic and the nonrelativistic relaxation energies (the difference between the negative of the orbital energy and the ASCF energy) are given in Table 2. We note once again [2;5] that the spin-orbit components have different relaxation energies, with the 1 - 4 level having greater relaxation energy than the 1 + f level. Furthermore, the relaxation energies obtamed in the DF calculations are invariably higher than the corresponding nonrelativistic ones. It is interesting to note that binding energies generally increase on going from the ASCF value to the semiempirical value. This is because the inclusion of correlation energy, which is being done semiempirically in this case, tends to increase the binding energy by lowering the energy of the ground state more than the energy of the hole state. A dramatic departure from this trend, however, is exhibited by Br- and Rb+. It is well known that the singleparticle description breaks down for Kr 3p, owing to the presence of doublyionized states close in energy to the 3p hole state [16]. The DF ASCF binding energy for Xr 3p is 3 eV higher than the experimental value [2] and the correlation-energy contribution is negative [8]. Similar behavior is expected for the isoelectronic species Rb+ and Br: The largest discrepancy between BE3 and BE4 on the one hand and BE1 on the other is observed for Cs+ 3d. In this case the correlation energy would lower the atomic DF binding energy by -0.4 eV. This can be seen by comparing the ASCF DF binding energy for the atom with experiment [ 21. Inclusion of correlation brings the DF Cs 3d binding energies into excellent agreement with experiment [17,18]. On the other hand, the DF ASCF value for Cs+ 3d is probably too low, judging from the results for Xe 3d, for which the DF value is 0.3 eV lower than experiment [2]. Thus the calculated atomion shift for Cs 3d is probably too small by -0.7 eV. Comparison of the ionic binding energies with experiment is not possible except for Li’, the 1s binding energy of which is 75.641 eV [lo] *. The 1s binding energy in F-and Na+ can be compared with the accurate theoretical values of Beck and Nicolaides 1191, who obtained 679.8 and 1088.6 eV, respectively. For all three levels the agreement is excellent. The ionic binding energies obtained for Cs 3dsj2 and I 3dsj2 enable us to * We used 1 eV = 8065.479 cm-’
in calculating energies from the data in Ref. 10.
265
determine the accuracy of another semiempirical but less reliable estimate of these binding energies reported by Mathews et al. [5]. These authors obtained 736.6 and 619.3eV for the 3ds12 levels of Cs+ and I-, respectively. We note that the latter is in excellent agreement with our result, while the Cs3ds12 value is -1.5 eV too small. The values of Mathews et al. were obtained from the experimental binding energies of gaseous CsI corrected by the l/R contribution due to the presence of the counter-ion. As the authors pointed out, this simple point-charge model neglects relaxation as well as the incomplete transfer of the electron from the cesium to the iodine atom. It therefore appears that, since these two contributions have opposite sign for I 3ds1* in CsI but the same sign for Cs 3dsj2, cancellation of errors gave a good 13dsf2 value while their additivity gave a relatively poor Cs 3dsi2 value in that earlier work. In summary, the method outlined here appears to give accurate and reliable estimates of core binding energies of free ions. Thus it affords an improvement over the limitations in the DF ASCF method applied to these systems, which are difficult to study experimentally. As such it complements semiempirical schemes developed for neutral atoms, such as those proposed by Broughton and Perry [ZO] and Johanssen and M&rtensson [213.
ACKNOWLEDGMENTS
The authors thank D. R. Beck and R. D. Mathews for useful discussions and suggestions, and J. Q. Broughton for helpful comments. This work was supported in part by the National Science Foundation (Grant No. CHE7918390). Acknowledgment is also made to the Donors of the Petroleum Reseamh Fund, administered by the American Chemical Society, for partial support.
REFERENCES J. Q. Broughton and P. S. Bagus, J. Electron Spectrosc. Re&t, Phenom., 20 (1980) 127. R. J. Key, M. S. Banna and C. S. Ewig, J. Electron Spectrosc. Relat. Phenom., 24 (1981) 173. J, P. Desclaux, Comput. Phys. Commun., 9 (1975) 31. See for example, S. P. Kowalczyk, F. R. McFeely, L. Ley, R. A. Pollak and D. A. Shirley, Phys. Rev, B, 9 (1974) 3573. R. D. Mathews, R. J. Key, A. Sur, C. S. Ewig and M. S. Banna, J. Chem. Phys., ‘74 (1981) 5407. S. Aksela, 1981, private communication. W. Mehlhorn, B. Breuekmann and D. Hausamann, Phya. Ser., 16 (1977) 177. D.R. Beck and C.A. Nicolaides, Phye. Rev. A, in press. M. S. Banna, B. Wallbank, D. C. Frost, C. A. McDowell and J. S. H. Q. Perera, J. Chem. Phys., 68 (1978) 5459.
266 10 11 12 13 14 16 16 17 18 19 20 21
S. Bashkin and J. 0. Stoner, Atomic Energy Levels and Grotrian Diagrams, Vol. 1, North-Holland, Amsterdam, 1975. S. Aksela, M. Kellokumpu, H. Aksela and J. Vayrynen, Phys. Rev. A, 23 (1981) 2374. P. -0. Lowdin, Advances in Chemical Physics, Vol. II, Interscience, New York, 1959. R. E. Stanton, J. Chem. Phys., 36 (1962) 1298. P. S. Bagus, Phys. Rev., 139 (1965) A619. K.-N. Huang, M. Aoyagi, M. H. Chen, B. Crasemann and H. Mark, At. Data Nucl. Data Tables, 18 (1976) 243. S. Svensson, N. MELrtensson, E. Basiher, P. A. MaImquist, U. Gelius and K. Siegbahn, Phys Ser., 14 (1976) 141. D. R. Beck and C. A. Nicolaides, in C. A. Nicolaides and D. R. Beck (eds.), Excited States in Quantum Chemistry, Reidel, Boston, 1978, p. 329. D. R. Beck and C. A. Nicolaides, Int. J. Quantum Chem., Quantum Chem. Symp., 14 (1980) 323. D. R. Beck and C. A. Nicolaides, J. Electron Spectrosc. Relat. Phenom., 8 (1976) 249. J. Q. Broughton and D. L. Perry, J. Electron Spectrosc. Relat. Phenom., 16 (1979) 45. B. Johanssen and N. Mglrtensson, Phys. Rev. B, 21(1980) 4427.
Journal of Electron Spectroscopy
ACCURATE CORE BINDING ENERGIES OF IONS FROM DIRAC-FOCK CALCULATIONS COMBINED WITH EXPERIMENTAL ATOMIC BINDING ENERGIES
M. S. BANNA, R. J. KEY and C. S. EWIG
Department
of Chemistry,
Vanderbilt Unwersity, Nashville, TN 37235 (U.S.A.)
(First received 28 December 1981; in final form 9 February 1982)
ABSTRACT Core binding energies are reported for the singly-charged alkali ions obtained by removing the outer s electron and for the singly-charged halide ions obtained by adding a pJIp electron. The levels studied are Li Is, Na Is, K 2p, Rb 3p, Cs 3d, F ls, Cl 2p, Br 3p and 13d. The binding energies were obtained by combining Dirac-Fock ASCF bindingenergy shifts between the ions and the corresponding isoelectronic rare gases with experimental rare-gas binding energies, and also by combining the calculated atom-ion shifts with experimental atomic binding energies where available. It is shown that the former approach corrects accurately for the correlation energy, which is not included in singleconfiguration calculations.
INTRODUCTION
Although it is not yet possible to measure core binding energies of ions by X-ray photoelectron spectroscopy, there is nevertheless considerable current interest in these quantities. Extensive Hartree-Fock atom-ion bindingenergy shifts for levels frequently studied by XPS have recently been published [l]. A more limited study of atom-ion shifts has been reported by Key et al. [ 23, who used a numerical Dirac-Fock program [ 31 and obtained better absolute binding energies than is possible using single-configuration nonrelativistic calculations. Highly ionic solids such as the alkali halides have been the subject of numerous XPS studies [4], and recently at least two groups have begun a study of the alkali halides in the gas phase. Mathews et al. [ 51 reported 3d binding energies for gaseous CsI and Aksela et al. [ 6] performed a number of Auger measurements on the gaseous alkali halides. Accurate free-ion binding energies would be useful in interpreting both Auger and XPS spectra.
0368-2048/82/0000-0000/$02.75
o 1982 Elsevier Scientific Publishing Company
260 METHOD
In this study we have used the numerical Dirac-Fock program of Desclaux [3] to obtain single-configuration “ASCF” core binding energies of one (possibly spin-orbit split) level in each of the alkali cations Li+, Na+, K+, Rb+ and Cs+ and of the same level in each of the corresponding halide anions F-, Cl-, Br- and I: The binding energies thus obtained should be in good agreement with experiment, judging from the agreement obtained for atoms [2,7]. Discrepancies are expected to remain, however, due primarily to the neglect of correlation energy. It is possible to obtain excellent agreement with experiment when approximate correlation effects are included explicitly for atoms for which such a calculation is feasible [ 81. However, we present here a simple semiempirical scheme that should be capable of yielding accurate core binding energies for the levels under study. The method relies on the availability of accurate experimental core binding energies for the rare gases, which are isoelectronic with the alkali and halide ions. The experimental binding energies are combined with the computed rare-gas-ion shifts, which are considerably more accurate than the computed absolute binding energies, to obtain good semiempirical values. A related but somewhat less accurate method is to combine the calculated atom-ion shifts with experimental atomic binding energies. For the systems we have chosen, this is possible for the alkali ions since atomic core binding energies are available [g-11]. The values thus obtained are compared with those obtained from the shifts between isoelectronic pairs. The reason for the accuracy of values extrapolated from isoelectronic rare gases as opposed to those from neutral atoms of the same atomic number 2 lies largely in the approximate invariance of correlation energy with changes in 2. For example, the striking constancy of the correlation energy in twoelectron systems from H- to C 4+ has been noted by Lowdin [12]. This was later justified rigorously by Stanton [13], who demonstrated the following: any parameter in the total Hamiltonian operator that affects only the oneelectron systems from H- to C4+ has been noted by Lowdin [12]. This was in comparison with those that affect two-electron terms. Thus a change in 2, which affects only the (one-electron) nuclear attraction operator, has only a small effect on the correlation energy. Although Stanton’s proof was in reference to nonrelativistic HartreeFock calculations, exactly the same principle applies to the DiraFFock method as well. The DirapFock Hamiltonian may be written as
where the two summations represent one- and two-electron operators respec-
261
tively, Hn (i) is the single-particle Dirac Hamiltonian, pi is an electron charge, and rir the interelectronic distance. H,,(i) may be written, with conventional notation, as H,(l)
= (Y,
l
P,C
+ &c”
-
ppZ/ri
in which all terms except the last are matrices representing the relativistic kinetic energy, and rr is the distance from the nucleus. Formally, the summa. . tions m HDF may be considered as carried out over an arbitrarily large number of electrons, pz being set to zero for all except the bound electrons. If 2 is considered as the variable parameter, and pr is held fixed as in going from a neutral atom to the isoelectronic ion, only H,(i) is affected and thus the change in correlation energy is small. However, if 2 is held fixed and pI is varied (for example, a change by unity corresponds to going from a neutral atom to its singly-charged ion), both one- and two-electron terms are affected. Thus the corresponding change in correlation energy is expected to be relatively large in this case. So if neglect of correlation energy is assumed to be the predominant source of error in the Dirac-Fock ASCF approach, this principle means that reference to an isoelectronic system should always be the most accurate approximation.
RESULTS AND DISCUSSION
The experimental atomic binding energies used in calculating the binding energies of free ions by the two methods outlined above are given in Table 1. Our results are presented in Table 2. The binding energies in columns 3-6 represent an attempt to correct for the deficiencies in the DF and HF calculations. This is done in two ways. In columns 3 and 4 we combine the shift between the ASCF binding energies of the isoelectronic ion and the corresponding rare gas, calculated relativistically (column 3) or nonrelativistically [14] (column 4), with the accurately known experimental binding energies of the rare gas. It is expected that the correlation energies of isoelectronic species would be almost equal, which would result in an accurate calculated rare-gas-isoelectronic-ion shift and hence accurate ionic binding energies. In columns 5 and 6, the ionic binding energies are obtained from calculated atom-ion shifts combined with the known experimental alkali-atom binding energies. The shifts in column 5 were calculated using the program of Desclaux [3] while the non-relativistic shifts used in column 6 were obtained from the work of Broughton and Bagus [ 11. The former cannot be used for systems having two open subshells of the same symmetry; therefore relativistic Li 1s and Na 1s atomic binding energies were obtained from the Diracr Slater work of Huang et al. [ 151. Since the atom and the corresponding ion differ by one electron, the correlation energies are not expected to cancel to
262 TABLE 1 EXPERIMENTAL CORE BINDING ENERGIES USED IN THIS WORK FOR THE RARE GASES AND THE ALKALIMETAL ATOMS Atom
Level
Binding energy (eV)
He Ne Ar Ar Kr Kr Xe Xe Li Na K K Rb Rb cs cs
IS 15 2PllZ 2PY2 3P1/2 3Ps12 3&z 3&/z 1s 1s 2Plf2 2PY2 3Plf2 3~~2 3&/z 3ds/2
24.5ga 870.21b 250.78’ 248.63’ 221.Bd 214.2d 689.3tid 676.70d 64.8ge 1078.9f 303.6g 300.7g 254.3h 245.4h 745.6f 731.6f
a From ref. 10. b From H. &ren, J. Nordgren, L. Selander, C. Nordling and K. Siegbahn, J. Electron Spectroac. Relat. Phenom., 14 (1978) 27. c From J. Nordgren, H. &ren, C. Nordling and K. Siegbahn, Phys. Ser., 19 (1979) 5. d From ref. 16. e From ref. 10, averaged over multipleta. f From ref. 9, reduced by 0.2 eV because of the new value of Ne 1s reported in the work quoted in footnote b above. a From ref. 11. h From ref. 9.
the same degree as for isoelectronic species. This, coupled with the fact that alkali-atom binding energies are known less accurately than those of the rare gases, means that this method is likely to give less reliable ionic binding energies than the isoelectronic-species method. The Dirac-Fock “ASCF” energies used are given in column 7 with the relaxation energies in parentheses. The DF ASCF results for the rare gases as well as for the alkali atoms have been given elsewhere [Z] and are not included here, except the He 1s value of 23.45 eV. Finally, we show in column 8 the nonrelativistic ASCF results recently published by Broughton and Bagus [l] , together with the corresponding relaxation energies. The Hartree-Fock results of Broughton and Bagus [l] do not suffer from any basis-set deficiency since they were obtained numerically. However,
18 1s
Li+ Na+ K+ K+ Rb+ Rb+ C3+ CS+ FCiCiBrBrII-
75.608 1088.71 310.73 307.94 260.5 251.7 752.02 738.11 679.79 198.54 196.91 187.4 180.9 630.77 619.30 197.47
680.11
1088.26
75.5oh 1088.7h 310.7 307.8 260.8 251.9 751.4 737.4 75.62 1088.3 310.7 307.8 260.7 251.8 751.3 737.3
74.46 1087.97 310.22 307.47 263.75 254.64 751.70 737.88 679.05 198.03 196.43 190.58 183.70 630.45 619.07
( 1.53) (22.99) (11.37) (11.15) ( 8.97) ( 8.56) (18.65) (17.93) (24.62) (12.61) (12.45) (10.52) (10.16) (19.87) (19.23)
633.90
(18.97)
( 9.93)
(12.35)
197.07 184.37
(17.61) (24.34)
( 8.29)
(10.99)
( 1.52) (22.46)
755.32 678.54
255.36
308.43
74.46 1086.70
HF*
s Obtained by combining DF ion-isoelectronic rare-gas ASCF shift with experimental rare-gas binding energies from Table 1. b Same as a, with shifts calculated nonrelativisticaily by Bagus [ 141. Weighted means of the experimental rare-gas spin-orbit components’ binding energies were used. c Obtained by combining DF atom-ion ASCF shifts with experimental atomic binding energies for the alkali atoms from Table 1. d Same as c, with HF atom-ion shifts from ref. 1. e Dirac-Fock ASCF results using the program described in ref. 3. Relaxation energies are given in parentheses. * Hartree-Fock ASCF results from ref. 1. Relaxation energies are given in parentheses. s The ASCF DF He 1s value used is 23.45 eV. h Obtained using Dirac-Slater ASCF atomic binding energies from ref. 15.
3dY2
3dY2
3~~2
Qpl/2
2PY2
2Pll2
1s
34l2
wz
~PYZ
3p~2
2P3/2
SPY2
Level
Ion
CORE BINDING ENERGIES OF THE ALKALI AND HALIDE IONS (eV)
TABLE 2
E
264
neither relativistic effects nor correlation-energy differences between the ground and core-hole states are included. The former are expected to amount from a fraction of an eV to perhaps 1 eV or more for these systems as found from a comparison of relativistic and nonrelativistic binding energies of atoms [2]. For example, the relativistic 1s binding energy of Na+ is 1.3eV higher than the nonrelativistic binding energy (columns 7 and 8 in Table 2). When the level is spin-orbit split, we have argued previously [2] that for p levels the nonrelativistic result is always closer to the binding energy of the pY2 level, while for d levels it is closer to that of ds2. This is clearly seen in Table 2. Both the relativistic and the nonrelativistic relaxation energies (the difference between the negative of the orbital energy and the ASCF energy) are given in Table 2. We note once again [2;5] that the spin-orbit components have different relaxation energies, with the 1 - 4 level having greater relaxation energy than the 1 + f level. Furthermore, the relaxation energies obtamed in the DF calculations are invariably higher than the corresponding nonrelativistic ones. It is interesting to note that binding energies generally increase on going from the ASCF value to the semiempirical value. This is because the inclusion of correlation energy, which is being done semiempirically in this case, tends to increase the binding energy by lowering the energy of the ground state more than the energy of the hole state. A dramatic departure from this trend, however, is exhibited by Br- and Rb+. It is well known that the singleparticle description breaks down for Kr 3p, owing to the presence of doublyionized states close in energy to the 3p hole state [16]. The DF ASCF binding energy for Xr 3p is 3 eV higher than the experimental value [2] and the correlation-energy contribution is negative [8]. Similar behavior is expected for the isoelectronic species Rb+ and Br: The largest discrepancy between BE3 and BE4 on the one hand and BE1 on the other is observed for Cs+ 3d. In this case the correlation energy would lower the atomic DF binding energy by -0.4 eV. This can be seen by comparing the ASCF DF binding energy for the atom with experiment [ 21. Inclusion of correlation brings the DF Cs 3d binding energies into excellent agreement with experiment [17,18]. On the other hand, the DF ASCF value for Cs+ 3d is probably too low, judging from the results for Xe 3d, for which the DF value is 0.3 eV lower than experiment [2]. Thus the calculated atomion shift for Cs 3d is probably too small by -0.7 eV. Comparison of the ionic binding energies with experiment is not possible except for Li’, the 1s binding energy of which is 75.641 eV [lo] *. The 1s binding energy in F-and Na+ can be compared with the accurate theoretical values of Beck and Nicolaides 1191, who obtained 679.8 and 1088.6 eV, respectively. For all three levels the agreement is excellent. The ionic binding energies obtained for Cs 3dsj2 and I 3dsj2 enable us to * We used 1 eV = 8065.479 cm-’
in calculating energies from the data in Ref. 10.
265
determine the accuracy of another semiempirical but less reliable estimate of these binding energies reported by Mathews et al. [5]. These authors obtained 736.6 and 619.3eV for the 3ds12 levels of Cs+ and I-, respectively. We note that the latter is in excellent agreement with our result, while the Cs3ds12 value is -1.5 eV too small. The values of Mathews et al. were obtained from the experimental binding energies of gaseous CsI corrected by the l/R contribution due to the presence of the counter-ion. As the authors pointed out, this simple point-charge model neglects relaxation as well as the incomplete transfer of the electron from the cesium to the iodine atom. It therefore appears that, since these two contributions have opposite sign for I 3ds1* in CsI but the same sign for Cs 3dsj2, cancellation of errors gave a good 13dsf2 value while their additivity gave a relatively poor Cs 3dsi2 value in that earlier work. In summary, the method outlined here appears to give accurate and reliable estimates of core binding energies of free ions. Thus it affords an improvement over the limitations in the DF ASCF method applied to these systems, which are difficult to study experimentally. As such it complements semiempirical schemes developed for neutral atoms, such as those proposed by Broughton and Perry [ZO] and Johanssen and M&rtensson [213.
ACKNOWLEDGMENTS
The authors thank D. R. Beck and R. D. Mathews for useful discussions and suggestions, and J. Q. Broughton for helpful comments. This work was supported in part by the National Science Foundation (Grant No. CHE7918390). Acknowledgment is also made to the Donors of the Petroleum Reseamh Fund, administered by the American Chemical Society, for partial support.
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