Models for Calculation of Dissociation Energies of Homonuclear Diatomic Molecules BY LEOBREWERAND JOHN S. WINN Materials and Molecular Research Division Lawrence Berkeley Laboratory and Department of Chemistry University of California Berkeley California 94720 U.S.A. Received 29th August 1979 The variation of known dissociation energies of the transition metal diatomics across the Periodic Table is rather irregular in a manner similar to the irregular variation of the enthalpies of sublimation of the bulk metals. This has suggested that the valence-bond model used for bulk metallic systems might be applicable to the gaseous diatomic molecules as well as to the various clusters intermediate between the bulk and the diatomic molecules. The available dissociation energies were converted to valence-state bonding energies considering various degrees of promotion to optimize the bonding.It was found that the model used for the bulk metals was applicable to the diatomic molecules. The degree of promotion of electrons to increase the number of bonding electrons is smaller than for the bulk but the trends in bonding energy parallel the behaviour found for the bulk metals. Thus using the established trends in bonding energies for the bulk elements it was possible to calculate all un- known dissociation energies to provide a complete table of dissociation energies for all M2molecules from H2to Lr2. The details of the calculations and final values are presented. For solids such as Mg Al Si and most of the transition metals large promotion energies are offset by strong bonding between the valence state atoms.The main question is whether bonding in the diatomics is adequate to sustain extensive promo- tion. The most extreme example for which a considerable difference would be expected between the bulk and the diatomics would be that of the Group IIA and IIB metals. The first section of this paper which deals with the alkaline earths Mg and Ca will demonstrate a significant influence of the excited valence state even for these elements. The next section will then expand the treatment to transition metals. THE ALKALINE EARTHS While most of the diatomic metals have at least one unpaired electron per atom to contribute towards bonding the Group IIA and IIB metals do not With ground state configurations ns2 and (n -l)d1'ns2 respectively the diatomics of these metals should be van der Waals molecules analogous to the rare gas diatomics with very small dissociation energies.However the first excited states of the rare gases involve excitation to a shell of the next higher principal quantum number but the Group IIA and IIB atoms have nsnp and ns(n -l)d excited configurations available at considerably lower energy. These low-lying configurations are certainly of importance in the bulk metal bonding. In this section we show how these low-lying states influence even the weakly bound diatomics and how spectroscopic data on weakly bound species may be treated to yield accurate estimates of the dissociation energy. We restrict the spectroscopic analysis to Mg and Ca, the only two diatomics of these groups for which detailed spectral constants of the ground electronic state have L.BREWER AND J. S. WINN been measured. The approach is to invert spectroscopic constants (ae,LO,^, Be a, etc.) to the parameters of a potential function expansion. We use the expansion where ;1= 1 -(R,/R)P. This potential function has been applied to several weakly bound diatomics with very good The parameters are e (with units of energy) the correction coeffi- cients en the equilibrium bond length Re and the parameter p which need not be integral. Note however if p = 6 and en = 0 n = 1 2 3 . . . then eqn (1) is the familiar Lennard-Jones (6-12) potential. The expressions relating these parameters to spectroscopic constants have appeared in the literat~re.~*~ For Mg, we have used the constants obtained by Vidal and Scheingraber6 in their analysis of the spectrum reported by Balfour and Douglas7 For Ca, we used the constants by Balfour and Whitlock.8 The parameters one obtains for the potential functions of the ’C,+ ground states are given in table 1.Dissociation energies are TABLE 1 .-PARAMETERS OF EQN (1) FOR X’Zc,+GROUND STATES OF Mg2 AND Ca2 eo = 785.94 K eo = 2570.8 K p = 3.59 p = 3.57 ela = 0 ela = 0 e2 = 0.058 99 e2 = -0.2317 e3 = 0.079 66 e3 = -0.1200 e4 = -0.122 9 e4 = 0.059 7 e5 = -0.147 6?6 = 0.108 Re = 3.890A Re = 4.277 4 A a The constant el is identically zero by our choice for determiningp as discussed in ref.(1)-(5). obtained by setting A = 1 in eqn (1). The predictions are dissociation energies of 768 K for Mg and 1820 K for Ca,. These values are 20 and 15 % respectively Iarger than spectroscopic estimates6*s of the dissociation energy and are in all likeli- hood truly in error by these amounts. The source of this error can be traced to the very informative parameter p. Note from table 1 that p x 3.6 for Mg and Ca, which means eqn (1) approaches the separated atom limit at large R like R-3-6. In contrast one knows that the proper large R behaviour should be R-6,in accordance with dispersion theory. Thus eqn (1) rises toward the dissociation plateau too slowly and thereby overestimates the dissocia- tion limit. The parameter p (as well as the others) is evaluated from equilibrium properties of the diatomic and perhaps should not be expected to give the proper long-range behaviour to the full potential.Yet in many case~l~~*~ as diverse as Ar, NaAr and BeAr+ the value ofp is large enough to give the proper long-range behaviour. (Actually theoretical arguments predict3 that p will be closer to the value n -1 than DISSOCIATION ENERGY MODEL to n where n is the expected long-range exponent. This prediction is observed in the previously reported molecules.) Therefore the small value of p for Mg and Ca is informative. For most chemically bound diatomics p is in the range 0.4-2.5 [and parenthetically eqn (1) does not converge at all well for these molecules]. Thus the alkaline earth diatomics have potential functions with a shape near Re which is inter- mediate between that of truly non-bonded diatomics such as Ar and NaAr and that of ordinary chemically bound diatomics.Perturbation theory expressions for p indicate the role of excited state mixing in determining the value ofp. It is clear that one is observing the effects of this mixing in the alkaline earth ground states even though the bonding remains very weak. TRANSITION METAL DIATOMICS The alkaline earth example illustrates that promotion from the ground atomic state plays a small but definite role in the bonding of even the weakly bound Group I1 element diatomics. Most atoms have a filled valence s orbital in the ground state; promotion of an s electron to provide two bonding electrons is important for the bulk metals.In addition promotion of inner shell d or felectrons can play an important role. The lanthanide elements provide a clear illustration of the role of promotion of 4f electrons in the homonuclear diatomic gases. Kant and Lin noted that the dis- sociation energies of the diatomic lanthanides decreased steadily from cerium to europium with a large increase for gadolinium with again a steady decrease to ytter- bium. They pointed out that the trends were parallel to those for the enthalpies of sublimation of the bulk metals and that the trends were due to the increasing difficulty of promotion of 4f electrons with increasing nuclear charge. Examination of the experimental values given in table 2 indicates similar parallel trends for the 3d transi-TABLE 2.-vALENCE STATE BONDING ENTHALPIES OF DIATOMICS element (AHi/R)/kK" reference valence state valence bonding in kKper electron d SP H 51.967 * 0.001 1s 52.0 He 0.Li 12.16 f0.1 2s J 2. Be B (<0.3) 35. f3 33. C 72.0 &1 36. N 113.25 f0.1 38. 0 59.36 & 0.02 29.7 F 18.59 f.0.07 18.6 Ne 0.025 f0.004 Na 8.36 Z!I 0.1 3s 8. Mg 0.5814 f.0.002 A1 20.0 rt 2 3P 20.0 Si 37.3 f.1 3P2 19. P 58.41 410.03 3P3 19.5 S 50.704 * 0.03 3P4 25. CI 28.774 0.001 3P5 29. Ar 0.122 30.002 L. BREWER AND J. S. WINN 129 TABLE2.-continued ~~ valence bonding element (AH,"/R)/kK" reference valence state in kK per electron d SP K 6.0 -4 0.1 4s 6.0 Ca 1.5 f 0.2 sc 19.1 & 3.0 3d24s 17.18. Ti 16. f3 3d '54s4p * 8. 19. V 28.6 rt 2.0 3d3''4s4po* 7.2 20. Cr 18. f 3.0 3d4'54s4po*5 2.5 21. Mn 5. f 3.0 '* Fe 14.6 f 2.5 3d6*'4.~4~ 5.5 22.0 co 20. f 3.0 3d 54s4po* 10.0 22.5 Ni 26. f 2.5 3d8* 54s4p0. 19.5 23.0 cu 23.5 12 4s 23.5 Zn 2.0 f 0.5 Ga 16.6 &fl 4P 16.6 Ge 32.6 i. 1.5 4P2 16.3 As 45.95 f 0.01 4P3 15.3 Se 39.58 f 0.03 4P4 19.8 Br 22.873 f 0.001 4P5 22.9 Kr 0.182 f 0.002 Rb 5.7 f 0.5 5s 5.7 Sr (1.7) Y 18.8 -4:3 4d25s 8.7 13 Zr (40) 4d35s 3.3 14 Nb 56 35 4d45s 0.3 15 Mo 44. 35 4d55s 5.6 16 Tc (34) 4d65s 6.1 17 Ru '(37) 4d75s 6.3 18 Rh 32.8 k3 4d85s 7.2 18.5 Pd 12.6 f 2.5 4d95s 2.5 19 Ag 19.3 f 0.8 5s 19.3 Cd 1.1 i. 0.2 In 12.0 It1 5P 12 Sn 23.O k2 5p2 12.5 Sb 35.9 f.0.5 5P3 12 Te 31.07 f 0.1 5P4 15.5 I 17.899 k 0.001 5P5 18 Xe 0.266 f 0.003 cs 4.57 f 0.1 6s 4.6 Ba (3) f2 5d6s 14 15 La 29. f3 Sd26s 10 17 Ce 29. f3 4f5d26s 9 18 Pr 18 & 3.5 4f25d26s 8.5 19 Nd 10 f 3.5 4f35d26s 8.5 20 Pm (8.5) 4f45d26s 8 21 Sm 7 f3 4f65do*56s6p0'5 19 22 Eu 4 &2 4f75do'56s6po'5 17 23 Gd 20.5 +4 4j'5d26s 7 24 Tb 15 f3 4J"5d26s 7 25 DY 8 f4 4f96s6p 26 DISSOCIATION ENERGY MODEL TABLE2.-continued valence bonding element (AH,"/R)/kK" reference valence state in kK per electron d SP Ho f3 4f O6s6p 27 Er +3 4fI26s6p 27 Tm f2 4f l36s6p 27 Yb f2 4f 146s6p 26 Lu 5d6s6p 18 26 Hf f6 5d26s6p 14 26 Ta f6 5d46s 11 26 W 58 5d56s 8 26 Re + 10 5d56s6p 8 26 0s +6 5d76s 11 26 Ir &6 5d86s 12 26 Pt &5 5d96s 13.5 26.5 Au stl 6s 26.7 Hg 0.15 T1 f3 7 Pb f3 5 Bi fl 8 Po f3 9 At 10 Rn Fr f1 7s 4 Ra fl Ac f7 6d 27s 11 16 Th f.4 6d37s 11 17 Pa 5f6d37s 10.5 17.5 U +6 5j36d27s 10 18 NP &7 5f46d27s 10 19 Pu 13 Am Cm f7 5f 76d27s 10 22 Bk It2 Cf Es Fm Md No Lr f10 7s27p 20 tion metals.l09l1 However the quantitative analysis of the data to be illustrated below shows that there are substantial differences between the bonding in the M2 gas and in the bulk solid for many elements.The second column of table 2 presents values of AHi/R for M,(g) = 2M(g). Calculated or estimated values are given in parentheses. Uncertainties are listed for all experimental values based on a critical evaluation of the literature.When a review paper adequately covers the literature and arrives at a value considered ac- ceptable only a reference to the review paper is given. Otherwise references are L. BREWER AND I. S. WINN given to the original papers. As the experimental values were used to calibrate the variation of bonding with nuclear charge across the Periodic Table the calculated values obtained by interpolation of bonding values have uncertainties close to those of adjoining elements but generally larger by "1 kK. Thus uncertainties are not indicated as they can be obtained from the uncertainties given for neighbouring experimental values. However where extrapolations are necessary or if there is reason to suspect the accuracy of the bonding trends uncertainties are also indicated for the calculated values.The fourth column of table 2 gives the electronic con- figuration of the atomic valence state selected as illustrated below. No configuration is shown for van der Waals molecules. The trends in bonding are shown in the last column where AHi/R for dissociation of M,(g) in its ground state to the atoms in the indicated valence state has been divided by the number of bonding electrons per atom with a separation into bonding per d-or per sp-electron for the transition elements. The method of determining the effective electronic configuration in the valence state is quite straightforward for most elements. The enthalpy of dissociation of M,(g) to two M(g) in their ground state is given by AH,"/R= (n -l)Ed/R+ E,/R -2P for a transition metal with ground state P2s2 and a valence state d"-'s.The promotion of a ground state atom to the valence state requires P kK for one atom or 2P kK for two atoms. Ed is the bonding energy (more strictly enthalpy but at 0 K they are essentially identical) per d electron and E is the bonding energy per s electron. The promotion energies to levels of each electronic configuration for elements other than the lanthanides and actinides are tabulated by Moore.35 Due to lack of data for the lanthanides and actinides a model for prediction of promotion energies had previously been de~eloped.~~ The recent review3' of values for the lanthanides has confirmed the reliability of the model and where experimental data are still lacking the predictions of the model can be confidently used.As noted earlier," the energy corresponding to the lowest state of each configuration can be accurately used in place of a weighted mean of all the levels of a configuration if the valence state bonding energies are obtained from experimental data using the same basis for the promotion energies. For the transition metals there are often two configurations e.g.,4dnW15sor 4dn-25s5p that might contribute significantly. One can differentiate the energy eqn (10) to obtain the optimum mix but the data are not accurate enough to specify more closely than one-half electron as in 3d2*54s4p0.s for Ti. Table 3 gives the TABLE 3 .-PROMOTION ENERGIES TO VALENCE STATES element ground state promotion energy /kK dfl-1 dn-2sp sc 3d4s2 16.575 22.550 Ti 3d24s2 9.434 22.844 V 3d 24s2 3.039 23.541 Cr 3d54s 0.0 35.929 Fe 3d64s2 9.968 27.842 co 3d74s2 5.011 33.973 Ni 3ds4s2 0.295 37.054 Y 4d5s2 15.737 21 SO9 Zr 4d25s2 7.008 21.270 Nb 4d45s 0.0 23.988 Mo 4d55s 0.0 40.094 132 DISSOCIATION ENERGY MODEL TABLE3.-continued element ground state promotion energy /kK Tc 4d55s2 d“-ls 3.702 d“-’sp 23.638 Ru 4d75s 0.0 36.278 Rh Pd 4d85s 4d’O 0.0 9.444 (48.0) <73.0 Ba 6s’ 12.998 17.648 f’-3d2s f”-3d5p La 5d6s2 3.867 19.078 Ce 4f5d6s2 3.409 19.444 Pr 4f 36s2 9.660 26.080 Nd 4f46s2 12.661 29.167 Pm 4f56s2 (14.4) (31) f”-2ds j”-2sp Sm 4f66s2 15.540 19.850 ELI 4f76s2 18.595 20.241 f n-3d2~ f” -3dsp Gd 4f75d6s2 9.177 20.195 Tb 4f 96s2 1 1.784 21.6 J’”-’ds f”-2sp DY 4f lo6s2 25.201 22.398 Ho 4f 116s2 27.146 22.812 Er 4f”6s’ 27.858 23.483 Tm 4f 136s2 29.362 24.088 Yb 4f146s2 35.235 24.875 d”-ls d”-’sp Lu 5d6s2 27.123 25.074 Hf 5d26s2 20.276 20.169 Ta 5d36s2 14.041 25.01 3 W 5d46s2 4.246 27.897 Re 5d56s2 16.912 27.265 0s 5d66s2 7.401 33.758 Ir 5d76s2 4.079 37.851 Pt 5d96s 0.0 43.390 Ac 6d7s2 13.261 19.730 Th 6d27s2 8.004 20.812 Pa U NP Pu Am Cm 5f 26d7s2 5f36d7s2 5f46d 7s2 5f67s2 5f 77s2 5f 76d7s2 8.991 (10.8) 21.455 (21.O) 14.597 (10.0) [f3d2s] [f4d2s] [f5d2s] [f 7ds] [f 7d2~] [fd3s] (23.0) 21.070 (21-0) 22.300 22.457 21.945 [fd2SPl [f 3dsp] Cf4dSP1 [f6sp] [f 7sp] [f 7dsp] promotion energies for those transition-metals lanthanides and actinides where one might have to consider the contribution of two configurations.With the various promotion energies available the procedure for calculation of L. BREWER AND J. S. WINN unknown dissociation energies involves the combination of the promotion energy for a given valence state with the interpolated bonding energies. For some elements with no unpaired electrons in the ground atomic state one calculates that no reasonable bonding energies could offset the promotion to even the lowest excited state and the cohesion of the atoms must be due primarily to van der Waals interactions. The noble gases the Group I1 elements Zn to Hg Be to Sr and Ra and the actinides Bk to No and probably Pu and Am fall into the van der Waals class.Most of these actinides have unpaired 5f electrons but the 5f electrons are so localized particularly for the second half of the series that they contribute insignificantly to the bonding. Ba is an exception among the Group I1 elements in that the 5d6s configuration is close enough to the ground 6s2 configuration to allow substantial contribution al- though the net contribution to AH,"/Ris still only 3 & 2 kK. For Eu and Yb like- wise the calculations indicate that they are not van der Waals molecules. For transition-metals of Groups 111-VI the valence state configurations are essen- tially the same for the diatomic and the solids in consisting of a mixing of the a"-ls and d"-2sp configurations with less p contribution for diatomic Zr Ta W and Group I11 and more p contribution for diatomic Hf than for the solid.A much more dramatic difference is found for Fe to Cu which can promote to a dn-2*5sp1.5 valence state in the solid but can only achieve dfl-1*5sp0.5 for Fe Co and Ni diatomics and no substantial promotion for Cu which uses the ground dlOs configuration. For the 4d and 5d Groups VII-XI all use the d'*-ls valence state for the diatomic with the exception of Re which is able to promote to d5sp. Examination of the bonding energies given in table 2 shows that the irregular behaviour of the dissociation energies of the diatomic is due to three contributions that change in different ways with variation of position in the Periodic Table. There is first the contribution from promotion energies which are known quite accurately for most elements.Secondly there is the increase of the s p bonding with increasing nuclear charge for a given period with a reduction in bonding per electron for multiple bonding and a reduction in p bonding when the core includes the closed s subshell of the outer shell. Thirdly there is the reduction in d bonding with nuclear charge for a given period up to the d5 configuration and an increase in bonding per d electron beyond the d5 configuration as the most localized orbitals are used by non-bonding electrons and the most extended orbitals are used by bonding electrons. The contri- bution of d bonding is greatly increased from 3d to 4d to 5d due to the contraction of the ns2np6subshell with increasing nuclear charge relative to the nd orbital.These same trends are found for the bulk metals and the simple smooth trends found for each of these factors makes the prediction of bonding energies and therefore dissociation energies quite straightforward and reasonably accurate. We thank J. H. Goble for his help in the analysis of the Mg and Ca potential functions. J. S. Winn acknowledges partial support from an Alfred P. Sloan Research Fellowship. This work was supported by the Division of Materials Sciences Office of Basic Energy Sciences U.S. Department of Energy under contract No. W-7405-Eng-48. J. H. Goble D. C. Hartman and J. S. Winn J. Chem. Phys. 1977 67 4206. J. H. Goble and J. S. Winn J. Chem. Phys. 1979 70,2051.J. H. Goble and J. S. Winn J. Chem. Phys. 1979 70 2058. J. H. Goble S. M. Walsh and J. S. Winn to be published. A. J. Thakkar J. Chem. Phys. 1975 62,1693. C. R. Vidal and H. Scheingraber J. Mol. Spectr. 1977 65 46. DISSOCIATION ENERGY MODEL 'W. J. Balfour and A. E. Douglas Canad. J. Phys. 1970,48,901; W. C. Stwalley Chem. Phys. Letters 1970 7,600. W. J. Balfour and R. F. Whitlock Canad. J. Phys. 1975 53,472. A. Kant and S. S. Lin Monatsh. 1972 103 757. loL. Brewer Viewpoints of Stability of Metallic Structures in Phase Stability in Metals and Alloys ed. P. Rudman J. Stringer and R. L. Jaffee (McGraw-Hill New York 1967) pp. 39-61 241-9 344-6 560-8. L. Brewer Science 1968 161 115. l2 The Committee on Data for Science and Technology (CODATA) of the International Council of Scientific Unions has critically evaluated the Doand AH of formation values for atomic and diatomic states of H N P 0 S C1 Br and I.The Brz and l2values were slightly revised to correspond to the recent values reported by Barrow et al. (R. F. Barrow D. F. Broyd L. B. Pederson and K. K. Yee Chem. Phys. Letters 1973 18 357). The S2value was changed in acknowledgement of the objection raised by Huber and Herzbergl3 to the use of a Do value that does not relate to the actual lowest rotational level of S2. The Do values in cm-' given in CODATA Report Part I Bulletin 5 (Dec. 1971) and Part V (Sept. 1975) were multiplied by hc/k = 1.4388 cm K to obtain the values in K. Brz and Clz differ from the others in not having a predominant isotope thus resulting in a small difference between Doand AH5 of dissociation of the dimer.The AH values reported by CODATA in J. Chem. Thermodynamics 1976 8,603 were converted to AH and divided by R = 8.314 33 J K-' but the uncertainties are those of the original D0" values. l3 K. P. Huber and G. Herzberg Molecular Spectra and Molecular Structure. IV Constants of Diatomic Molecules (Van Nostrand Reinhold New York 1979). l4 D. D. Konowalow and M. L. Olson J. Chem. Phys. 1979,71,450. l5 K. A. Gingerich J. Cryst. Growth 1971 9 31. l6 C. A. Stearns and F. J. Kohl High Temp. Sci. 1973 5,113. l7 C. Chatillon A. Michel and A. Pattoret Compt. rend. C 1975 280 1505. W. J. Balfour J. Chem. Educ. 1979 56 452. l9 A. Kant J. Chem. Phys. 1964 41 1872 and 1968 49 5144.E. Rutner and G. L. Haury J. Chem. Eng. Data 1974 19 19 obtained a different value upon repeating Kant's third-law calculations but they used the atomic weight of nickel rather than twice the atomic weight of nickel for the molecular weight of Ni2. 2o A Neckel and G. Sodeck Monatsch. 1972,103,367. 21 K. D. Carlson and K. R. Kushnir J. Phys. Chem. 1964,68 1566. 22 J. Drowart and S. Smoes J.C.S. Faraday 11 1977 73 1755. 23 S. K. Gupta and K. A. Gingerich Inorg. Chem. 1978 17 321 and J. Chem. Phys. 1979 70 5350 report values of D(Nbz) and D(Mo,) from third-law calculations which include no elec- tronic contributions in the calculation of -(Go -H&)/RT. Brewer and LamoreauxZ4 have pointed out that even with a IC ground state low-lying electronic states of higher multiplicity are expected to be populated at the temperature range of measurements and electronic terms must be included.The value given for Nbz has been corrected in a manner similar to the correction for M02.24 24 L. Brewer and R. H. Lamoreaux Atomic Energy Reuiew Molybdenum Part I Physicochemical Properties of Its Compounds and Alloys (International Atomic Energy Agency Vienna 1980). 25 D. L. Cocke and K. A. Gingerich J. Chem. Phys. 1974 60 1958. 26 V.Piacente G. Balducci and G. Bari J. Less-Common Metals 1974 37 123. 27 J. Kordis K. A. Gingerich and R. J. Seyse J. Chem. Phys. 1974 61 51 14. 28 D. L. Cocke and K. A. Gingerich J. Phys. Chem. 1971,75 3264. 29 M. Guido and G. Balducci J. Chem. Phys. 1972 57 561 1. 3o G.D. Blue R. S. Carbonara and C. A. Alexander Proc. 18th Annual ConJ on Mass Spectro- metry and Allied Topics San Francisco June 1970 and personal communiation reporting upper limit of 90 kcal mol-I for D(Pt2). 31 J. Drowart and R. E. Honig J. Phys. Chem. 1957 61,980. 3z D. R. Stull and G. C. Sinke Thermodynamic Properties of the Elements Adv. Chem. Ser. 1956 No. 18. 33 K. A. Gingerich High Temp. Sci. 1969 1 258. 34 R. Stern and N. Lang Lawrence Eivermore Laboratory San Francisco Bay Area Conference on High-Temperature Science and Technology 8th March 1979 report an upper limit of 15 kK for D(U2)compared to 20 41 5 kK reported by K. A. Gingerich and G. D. Blue J. Chem. Phys. 1967,47,5447 and 26 & 3 kK reported by L. N. Gorokhov A. M. Emel'yanov and Yu.S. Khodeev High Temperature 1974 12 1 156. L. BREWER AND J. S. WINN 35 C. E. Moore Atomic Energy Levels (U. S. Government Printing Office Washington D.C. 1949 1952 1958) VOI. 1-3. 36 L. Brewer J. Opt. SOC. Amer. 1971 61 1101. 37 W. C. Martin R. Zalubas and L. Hagan Atomic Energy Levels-The Rare-Earth Elements NSRDS-NBS 60 (U.S. Government Printing Office Washington D.C. 1978).