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Basis set limit extrapolation of ACPF and CCSD(T) results for the third and fourth lanthanide ionization potentials
7 December 2001
Chemical Physics Letters 349 (2001) 489±495 www.elsevier.com/locate/cplett
Basis set limit extrapolation of ACPF and CCSD(T) results for the third and fourth lanthanide ionization potentials Xiaoyan Cao a
a,b
, Michael Dolg
a,*
Institut f ur Physikalische und Theoretische Chemie, Universitat Bonn, Wegelerstr. 12, D-53115 Bonn, Germany b Department of Biochemistry, Zhongshan University, 510275 Guangzhou, People's Republic of China Received 31 August 2001; in ®nal form 8 October 2001
Abstract Theoretical values for the third and fourth ionization potentials of the lanthanide elements from La to Lu are presented. The results are based on large-scale multi-reference averaged coupled-pair functional calculations starting from a complete active space reference wavefunction and single-reference coupled-cluster calculations. Relativistic eects were accounted for by small-core energy-consistent ab initio pseudopotentials. The results obtained for optimized uncontracted valence basis sets including up to i functions were extrapolated to the basis set limit. The importance of higher angular momentum basis functions in ab initio calculations for energy dierences between con®gurations with dierent 4f occupation numbers is discussed. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The chemistry of lanthanides and actinides received much attention among theoretical chemists in recent years [1±6]. Several lanthanide compounds possess low-lying electronic states arising from lanthanide metals with dierent oxidation states and correspondingly dierent 4f occupation numbers. Experimentally and theoretically studied examples are the competition between Ce 4f 1 and 4f 0 in Ce C8 H8 2 [7±9] or between Yb 4f 13 and 4f 14 in YbO [10±12]. Since for most molecules energy dierences between electronic states arising from dierent 4f occupation numbers have not been
*
Corresponding author. Fax: +49-228-73-9066. E-mail address: [email protected] (M. Dolg).
measured accurately, the spectroscopic data available for lanthanide atoms and ions may serve as suitable reference for calibration purposes. Of considerable interest are the third and fourth ionization potentials (IP3 and IP4 ), where, with a few exceptions, the 4f occupation changes by one electron. Unfortunately the experimentally determined values for some cases bear large error bars, i.e., up to several tenths of an electron volt [13]. Much of the data result from a combination of actually measured energy dierences and extra- or intrapolated data as well as estimated values [14,15]. The accurate calculation of these IPs is a considerable challenge, since both relativistic eects as well as electron correlation eects are large (1 eV and more). Previous results for lanthanides [16±18] and actinides [19] were obtained both with energy-
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 2 1 1 - 8
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X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
consistent relativistic ab initio pseudopotentials (PP) and a wavefunction-based correlated treatment using standard basis sets also suitable for molecular calculations as well as with relativistic all-electron gradient-corrected density functional methods. Here we present an investigation of differential electron correlation eects in ab initio PP calculations for IP3 and IP4 of the lanthanides using large uncontracted basis sets and basis set extrapolation techniques to approach the experimental values as close as possible. 2. Method The method of relativistic energy-consistent ab initio PPs applied in the present study is described in detail elsewhere [20,21]. Small-core PPs in semilocal form were used for La to Lu [17,22], i.e., the 1s±3d shells were included in the PP core, while all shells with main quantum number larger than 3 were treated explicitly. The free parameters of the valence model Hamiltonian were adjusted to reproduce the valence total energies of a multitude of low-lying electronic states of the neutral atom and its ions. The reference data for the scalar-relativistic terms have been taken from relativistic all-electron calculations using the socalled Wood±Boring Hartree±Fock approach, whereas the multi-con®guration Dirac±Hartree± Fock approach based on the Dirac±Coulomb Hamiltonian was used for the spin±orbit part. The valence orbitals were expanded in optimized (14s13p10d8f6g) Gaussian basis sets [17], to which a (2s2p2d2f2g) diuse set was added. The resulting (16s15p12d10f8g) basis set was augmented by (8h) and (8h8i), respectively, using the same exponents for h and i as for g. The M O L P R O [23±27] program system was applied for all scalar-relativistic calculations using D2h symmetry. The averaged coupled-pair functional (ACPF) approach [28] based on a state-averaged complete active space multi-con®guration self-consistent ®eld (CASSCF) wavefunction was used to account for electron correlation eects. All orbitals belonging to partly occupied shells were active. The state average was taken over all components of the lowest energy LS state of each
con®guration in D2h symmetry in order to avoid symmetry breaking. The considered LS states and active open shells are listed in Tables 1 and 2. In addition we report coupled-cluster results with single and double excitations as well as perturbative triple excitations (CCSD(T)) for La, Ce, Eu, Gd, Yb and Lu. For each state the reference wavefunction was a single-determinant symmetrybroken Hartree±Fock solution. Excitations were allowed from all occupied orbitals in the ACPF and CCSD(T) calculations. Spin±orbit contributions were derived from complete con®guration interaction calculations within all open-shell orbitals (COSCI) with and without spin±orbit potential and then added to the scalar-relativistic results. All CASSCF, ACPF and CCSD(T) values reported here include spin±orbit corrections. Modi®ed versions of the ®nite-dierence programs MCHF [29] and GRASP [30] were applied exploiting the spherical symmetry. CASSCF/ACPF and CCSD(T) results are reported for the uncontracted (16s15p12d10f8g8h8i) basis sets truncated after f (l 3), g (l 4), h (l 5) and i (l 6) functions, respectively. Empirically the total valence energies (or valence correlation energies) obtained with truncation at l > 3 were found to be linear in 1=l3 with high accuracy and extrapolated to l ! 1 in order to estimate the basis set limit. The correlation coef®cient of the linear regression was in all cases larger than 0.9999, i.e., the worst correlation was 0.99991 for the ACPF energy of La2 , whereas the deviation from 1 is even one or two orders of magnitude smaller for the heavier elements of the series. Other extrapolation formulas were tried, but all were found to yield lower accuracy. 3. Results and discussion The current results for IP3 and IP4 are compared to experimental data in Tables 1 and 2, respectively. The mean absolute errors (m.a.e.'s) with respect to the available experimental values [13] are reported for the ®rst (La±Gd) and second (Tb±Lu) halves of the lanthanide series in order to distinguish between cases where the 4f shell is up to half-®lled and more than half-®lled, respec-
X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
491
Table 1 IP3 of the lanthanides (in eV) Ln
State
CAS
Initial La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
1
d f2 f3 f4 f5 f6 f7 f 7 d1 f9 f 10 f 11 f 12 f 13 f 14 f 14 s1
Final 1
2
D3=2 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 9 D2 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1 S0 2 S1=2 3
f1 f2 f3 f4 f5 f6 f7 f8 f9 f 10 f 11 f 12 f 13 f 14
S0 F5=2 3 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 7 F6 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1 S0 2
m.a.e. La±Gd m.a.e. Tb±Lu
ACPF
Expt.
f
g
h
i
Extr.
18.19 18.34 19.80 19.89 19.87 21.10 22.47 19.93 17.98 19.06 18.46 17.95 18.92 20.37 19.97
18.56 18.49 20.06 20.58 20.97 22.18 23.42 20.36 19.96 21.18 21.17 21.09 22.12 23.43 20.71
18.69 19.48 20.99 21.46 21.80 23.02 24.13 20.50 20.85 21.99 21.91 21.77 22.72 23.99 20.79
18.76 19.76 21.26 21.71 22.05 23.25 24.34 20.58 21.13 22.25 22.15 22.01 22.94 24.16 20.81
18.78 19.88 21.38 21.83 22.15 23.36 24.45 20.60 21.26 22.37 22.27 22.13 23.05 24.27 20.81
18.82 20.05 21.54 21.98 22.30 23.44 24.58 20.65 21.42 22.53 22.41 22.28 23.18 24.37 20.82
1.85 3.90
1.22 1.47
0.54 0.85
0.33 0.65
0.24 0.55
0.14 0.42
19.18 20.20 21.62 22.1 0.3 22.3 0.4 23.4 0.3 24.92 0.10 20.63 0.10 21.91 0.10 22.8 0.3 22.84 0.10 22.74 0.10 23.680.10 25.05 0.03 20.96 0:15 0:10
Results from CASSCF and multi-reference ACPF calculations in comparison to experimental data (expt. [13]). The results were obtained with relativistic energy-consistent ab initio pseudopotentials and a (16s15p12d10f8g8h8i) valence basis set truncated after the angular quantum number l denoted by the labels f, g, h and i. The results of an extrapolation to the basis set limit using 1=l3 (extr.) are also given. All results were corrected for spin±orbit interaction contributions. The m.a.e. in eV is given for the ®rst (La±Gd) and second (Tb±Lu) half of the lanthanide series. The averaged error bars of the experimental values are also given if larger than 0.01 eV.
Table 2 IP4 of the lanthanides (in eV) Ln
State
CAS
Initial La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
6
p f1 f2 f3 f4 f5 f6 f7 f8 f9 f 10 f 11 f 12 f 13 f 14
Final 1
S0 F5=2 3 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 7 F6 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1 S0 2
m.a.e. La±Gd m.a.e. Tb±Lu For explanations cf. Table 1.
5
p p6 f1 f2 f3 f4 f5 f6 f7 f8 f9 f 10 f 11 f 12 f 13
2
P3=2 S0 2 F5=2 3 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 7 F6 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1
ACPF
Expt.
f
g
h
i
Extr.
49.05 35.23 37.58 39.24 39.44 39.59 40.93 43.00 35.98 38.09 39.37 38.96 38.49 39.58 41.25
49.62 34.55 37.31 39.05 39.68 40.16 41.69 43.54 37.44 39.50 41.00 41.06 41.05 42.19 43.83
49.85 35.60 38.40 40.08 40.64 41.12 42.55 44.31 38.44 40.44 41.86 41.83 41.74 42.79 44.40
49.93 35.86 38.71 40.38 40.93 41.40 42.83 44.56 38.79 40.75 42.15 42.11 42.02 43.04 44.59
49.96 35.98 38.85 40.52 41.06 41.53 42.95 44.68 38.94 40.90 42.29 42.25 42.15 43.17 44.72
50.01 36.15 39.04 40.70 41.23 41.64 43.12 44.83 39.15 41.08 42.46 42.42 42.32 43.33 44.83
1.40 3.68
1.21 1.63
0.42 0.85
0.26 0.58
0.27 0.44
0.33 0.27
49.95 0.06 36.76 0.01 38.98 0.02 40.4 0.4 41.1 0.6 41.4 0.7 42.7 0.6 44.0 0.7 39.37 0.10 41.4 0.4 42.5 0.6 42.7 0.4 42.7 0.4 43.56 0.10 45.25 0.03 0:39 0:29
492
X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
tively. The m.a.e.'s given under the entry CAS also indicate the importance of dynamical electron correlation eects. It is rather obvious that electron correlation is less important for the ®rst half of the series (CAS m.a.e. IP3 1.85 eV, IP4 1.40 eV) than for the second half (3.90 eV, 3.68 eV). However, due to the more diuse character of the 4f shell at the beginning of the series and the higher number of empty 4f orbitals accepting virtual excitations, electron correlation contributions are more dicult to account for the light lanthanide elements. Using standard generalized [17] or segmented [18] contracted basis set including up to g functions at the spin±orbit corrected CASSCF/ACPF level for the evaluation of IP3 and IP4 leads to disappointingly large underestimations of the experimental values by up to 5% and 2%, respectively. Notably only IP4 of Gd is overestimated by about 1.5%. We want to demonstrate here by an extrapolation to the basis set limit that these errors are mostly due to the limitations in the electron correlation treatment, and to a much lesser extent in the applied PPs. Our extrapolated results for IP3 have m.a.e. of 0.13 eV (La±Gd) and 0.42 eV (Tb±Lu). These results are considerably better than those reported previously for calculations using augmented standard basis sets including up to h functions (La±Gd 0.39 eV, Tb±Lu 0.69 eV [17]). The m.a.e. for La±Gd even lies within the mean experimental error bar of 0.15 eV, whereas for Tb±Lu the mean experimental error bar of 0.10 eV is signi®cantly below our m.a.e. For IP4 the present m.a.e.'s of 0.32 eV (La±Gd) and 0.27 eV (Tb±Lu) are both below the mean experimental
error bar of 0.39 eV and 0.29 eV, respectively, as well as signi®cantly smaller than the theoretical m.a.e. obtained previously (La±Gd 0.39 eV, Tb±Lu 0.76 eV [17]). Recently it has been demonstrated that the IP4 of Ce can be calculated at the CCSD(T) level with an error of less than 0.06 eV (0.3%) with respect to experimental data (36.82 eV, expt. 36:76 0:01 eV), using a (slightly dierent) basis set extrapolation technique and the PP of the present work [17]. Similarly, an error of )0.02 eV ()0.1%) was obtained for IP3 of La (19.16 eV, expt. 19.18 eV) [17]. The atomic states of the lanthanide ions required to calculate IP3 and IP4 mostly need a multi-reference treatment, however, besides La and Ce at the beginning of the series, single-reference CCSD(T) calculations are also applicable for some IPs of a few heavier lanthanide ions, i.e., for Eu, Gd, Yb and Lu. Our corresponding results for IP3 and IP4 are listed in Table 3. The estimated CCSD(T) basis set limits for Lu, i.e., IP3 20.92 eV (expt. 20.96 eV; error )0.2%), and IP4 45.33 eV (expt. 45:25 0:03 eV; error 0.2%), have a similar accuracy as the results for La and Ce reported previously. The extrapolated CCSD(T) result for IP3 of Eu (25.02 eV, expt. 24:92 0:10 eV; error 0.4%) is also signi®cantly improved with respect to the corresponding ACPF value. The same is true for IP3 of Yb (25.00 eV, expt. 25:05 0:03 eV; error )0.2%). Disregarding Gd for the moment, the m.a.e. of the extrapolated ACPF results (Tables 1 and 2) for the systems reported in Table 3 is 0.37 eV. At the CCSD and CCSD(T) levels the m.a.e.'s are 0.21 and 0.06 eV, respectively. The signi®cant improvement upon the perturbative
Table 3 Selected CCSD(T) results for IP3 and IP4 of the lanthanides (in eV) Ln La La Ce Eu Gd Yb Lu Lu
IP3 IP4 IP4 IP3 IP4 IP3 IP3 IP4
f
g
h
i
Extr.
Expt.
18.81 49.68 34.93 23.79 43.55 23.95 20.78 44.26
18.98 49.91 36.11 24.55 44.34 24.58 20.87 44.87
19.06 49.99 36.42 24.78 44.59 24.76 20.89 45.07
19.09 50.02 36.53 24.89 44.71 24.88 20.90 45.21
19.14 50.07 36.74 25.02 44.86 25.00 20.92 45.33
19.18 49.950.06 36.760.01 24.920.10 44.00.7 25.050.03 20.96 45.250.03
For further explanations and involved electronic states cf. Tables 1 and 2.
X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
inclusion of triples emphasizes the importance of such excitations. Most likely the lack of higher excitations is also the main remaining source of error in the CASSCF/ACPF results reported here. In principle such higher excitations can be included in CASSCF/ACPF by enlarging the active orbital space, but from the present ACPF wavefunctions it is not obvious to us as to how to do this in a systematic way and still keep the calculation size feasible for our current hardware and software. It is noteworthy, however, that the extrapolated CCSD(T) result for the IP4 of Gd (44.86 eV, expt. 44:0 0:7 eV; error 1.9%) shows a signi®cant disagreement with experimental data. Since already the ACPF IP4 , which is typically below the corresponding CCSD(T) value, is by almost 1 eV larger than the experimental value, we suppose that the experimental result is too low by roughly this amount. Besides the comparison with experiment the in¯uence of higher angular momentum functions on the dynamical correlation contributions is interesting. We use the CASSCF (0%) and extrapolated CASSCF/ACPF (100%) values as reference points and list correlation contributions obtained with uncontracted basis sets including functions up to a given angular momentum in Table 4. Due
493
to the special electronic con®gurations involved, the results for La have to be considered separately, whereas the values are changing monotonously with increasing nuclear charge for the heavier elements of the series. With an spdf (spdfg) basis set 59.1% (79.6%) and 59.3% (83.4%) of the correlation contributions to IP3 and IP4 , respectively, are recovered for La. An even faster convergence is observed for Lu, where an spdf (spdfg) basis set yields 86.6% (95.8%) and 71.9% (87.7%). These results already demonstrate the need for g basis functions for accurate investigations. The situation is signi®cantly worse for the lighter lanthanide elements with a partially occupied 4f shell. The most dicult case appears to be Ce: an spdf (spdfg) basis set recovers 8.4% (66.3%) of the correlation contributions to IP3 , whereas for IP4 )74.4% (40.5%) are accounted for, i.e., the correlation contributions calculated with an spdf basis set lead to signi®cantly worse results than the CASSCF treatment. The inclusion of higher angular momentum functions, i.e., at least h functions, appears to be mandatory when ab initio evaluations of energy dierences between Ce(III) and Ce(IV) compounds are attempted. Finally, a discussion of possible errors in the results of this work applies. First of all, the PPs for
Table 4 Percentage (%) of dierential electron correlation contributions to IP3 and IP4 of the lanthanides Ln La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
IP3
IP4
spdf
+g
+h
+i
>i
spdf
+g
+h
+i
>i
59.1 8.4 14.7 33.0 45.1 46.0 45.3 60.9 57.6 61.0 68.2 72.5 75.1 76.5 86.6
20.5 57.9 53.3 42.2 34.4 36.0 33.7 18.6 25.8 23.4 19.1 15.6 14.1 14.1 9.2
10.7 16.3 15.6 12.0 10.0 10.0 10.0 11.0 8.0 7.5 6.0 5.7 5.1 4.1 2.4
3.2 7.4 6.9 5.5 4.4 4.6 4.9 2.9 3.7 3.5 3.1 2.7 2.6 2.8 0.3
6.4 9.9 9.5 7.3 6.1 3.3 6.1 6.6 4.9 4.6 3.7 3.5 3.1 2.5 1.5
59.3 )74.4 )18.6 )13.0 13.0 27.9 35.0 29.3 46.1 47.1 52.9 60.7 66.9 69.7 71.9
24.1 114.9 74.7 70.0 54.2 46.9 39.0 42.2 31.5 31.6 27.6 22.5 17.8 16.0 15.8
8.4 28.8 21.4 21.1 15.9 13.6 12.6 13.5 11.0 10.3 9.3 8.1 7.4 6.7 5.3
3.0 13.2 9.5 9.2 7.3 6.1 5.8 6.7 4.6 4.9 4.5 3.9 3.4 3.5 3.6
5.1 17.5 13.0 12.8 9.6 5.5 7.6 8.3 6.7 6.2 5.7 4.9 4.5 4.1 3.3
The percentages are recovered with basis sets containing up to f functions (spdf), as well as by addition of g (+g), h (+h), i (+i) and higher (> i) angular momentum functions. 0% and 100% refer to the entries CAS and ACPF, extr., respectively, in Tables 1 and 2.
494
X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
Ce to Yb have not been adjusted to ions with charges larger than +1 and errors of a few tenths of an electron volt might result for IP3 and IP4 . Second, the Breit interaction, which increases most of the IP3 and IP4 by one- or two-tenths of an electron volt, was not considered in the reference data for the PP adjustment. Third, the COSCI calculations used here to evaluate spin±orbit contributions do not account for a coupling with dynamic electron correlation eects. Finally, the empirical extrapolation scheme used to determine the basis set limit might be a source of small errors. Based on the good agreement of the extrapolated ACPF and especially CCSD(T) results with the experimental values we estimate the sum of these errors in IP3 and IP4 to amount to at most 1%. Within this accuracy the present values are the best available set of theoretical data for IP3 and IP4 of the lanthanides. For many lanthanides more reliable experimental values are desirable and would allow a more conclusive testing of the applied PPs, basis sets and correlation methods. More re®ned extrapolation techniques, or an explicit inclusion of the interelectronic distance in the wavefunction [31] in connection with an all-electron treatment based on the best relativistic Hamiltonian at hand, i.e., the Dirac±Coulomb±Breit Hamiltonian [32], might yield improved theoretical results which may both guide further experimental investigations and also help in the calibration of approximate computational schemes such as eective core potentials. However, to our knowledge such studies have not been attempted for the cases considered here. 4. Conclusions A consistent set of theoretical values for IP3 and IP4 of all lanthanide atoms has been derived from relativistic ab initio pseudopotential calculations using the CASSCF/ACPF approach together with basis set extrapolation techniques. With the exception of IP4 of Gd the theoretical results range between 97.3% and 101.0% of the experimental values, which themselves bear error bars of up to 1.8%. Corresponding CCSD(T) calculations for
single-reference cases lead to a signi®cant improvement over the ACPF results with errors below 0:5%. The experimental IP4 of 44:0 0:7 eV for Gd, partly derived on the basis of intrapolated estimates, appears to be too low by almost 1 eV and should be remeasured. The correct value is expected to be 44:85 0:25 eV. The dierential correlation contributions accounted for by higher angular momentum functions decrease drastically, and nearly monotonously from Ce to Lu, i.e., whereas the inclusion of g, h and even i functions appears to be mandatory for Ce; results of an even better quality can be obtained for Lu with a basis set restricted to at most g functions. Acknowledgements The ®nancial support of Fonds der Chemischen Industrie is acknowledged. References [1] P. Pyykk o, Inorg. Chim. Acta 139 (1987) 243. [2] M. Pepper, B.C. Bursten, Chem. Rev. 91 (1991) 719. [3] K. Balasubramanian, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 18, Elsevier, Amsterdam, 1994, p. 29. [4] M. Dolg, H. Stoll, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 22, Elsevier, Amsterdam, 1996, p. 607. [5] M. Dolg, in: P.V.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollman, H.F. Schaefer III, P.R. Schreiner (Eds.), Encyclopedia of Computational Chemistry, Wiley, Chichester, 1998, p. 1478. [6] G. Schreckenbach, P.J. Hay, R.L. Martin, J. Comput. Chem. 20 (1999) 70. [7] M. Dolg, P. Fulde, W. K uchle, C.-S. Neumann, H. Stoll, J. Chem. Phys. 94 (1991) 3011. [8] M. Dolg, P. Fulde, H. Stoll, H. Preuû, R.M. Pitzer, A. Chang, Chem. Phys. 195 (1995) 71. [9] N.M. Edelstein, P.G. Allen, J.J. Bucher, D.K. Shuh, C.D. So®eld, N. Kaltsoyannis, G.H. Maunder, M.R. Russo, A. Sella, J. Am. Chem. Soc. 118 (1996) 13115. [10] M. Dolg, H. Stoll, H.-J. Flad, H. Preuû, J. Chem. Phys. 97 (1992) 1162. [11] W. Liu, M. Dolg, L. Li, J. Chem. Phys. 108 (1998) 2886. [12] S.A. McDonald, R.F. Rice, R.W. Field, C. Linton, J. Chem. Phys. 93 (1990) 7676. [13] W.C. Martin, R. Zalubas, L. Hagan, Atomic Energy Levels ± The Rare Earth Elements, NSRDS-NBS 60, Washington, DC, 1978.
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Chemical Physics Letters 349 (2001) 489±495 www.elsevier.com/locate/cplett
Basis set limit extrapolation of ACPF and CCSD(T) results for the third and fourth lanthanide ionization potentials Xiaoyan Cao a
a,b
, Michael Dolg
a,*
Institut f ur Physikalische und Theoretische Chemie, Universitat Bonn, Wegelerstr. 12, D-53115 Bonn, Germany b Department of Biochemistry, Zhongshan University, 510275 Guangzhou, People's Republic of China Received 31 August 2001; in ®nal form 8 October 2001
Abstract Theoretical values for the third and fourth ionization potentials of the lanthanide elements from La to Lu are presented. The results are based on large-scale multi-reference averaged coupled-pair functional calculations starting from a complete active space reference wavefunction and single-reference coupled-cluster calculations. Relativistic eects were accounted for by small-core energy-consistent ab initio pseudopotentials. The results obtained for optimized uncontracted valence basis sets including up to i functions were extrapolated to the basis set limit. The importance of higher angular momentum basis functions in ab initio calculations for energy dierences between con®gurations with dierent 4f occupation numbers is discussed. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The chemistry of lanthanides and actinides received much attention among theoretical chemists in recent years [1±6]. Several lanthanide compounds possess low-lying electronic states arising from lanthanide metals with dierent oxidation states and correspondingly dierent 4f occupation numbers. Experimentally and theoretically studied examples are the competition between Ce 4f 1 and 4f 0 in Ce C8 H8 2 [7±9] or between Yb 4f 13 and 4f 14 in YbO [10±12]. Since for most molecules energy dierences between electronic states arising from dierent 4f occupation numbers have not been
*
Corresponding author. Fax: +49-228-73-9066. E-mail address: [email protected] (M. Dolg).
measured accurately, the spectroscopic data available for lanthanide atoms and ions may serve as suitable reference for calibration purposes. Of considerable interest are the third and fourth ionization potentials (IP3 and IP4 ), where, with a few exceptions, the 4f occupation changes by one electron. Unfortunately the experimentally determined values for some cases bear large error bars, i.e., up to several tenths of an electron volt [13]. Much of the data result from a combination of actually measured energy dierences and extra- or intrapolated data as well as estimated values [14,15]. The accurate calculation of these IPs is a considerable challenge, since both relativistic eects as well as electron correlation eects are large (1 eV and more). Previous results for lanthanides [16±18] and actinides [19] were obtained both with energy-
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 2 1 1 - 8
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consistent relativistic ab initio pseudopotentials (PP) and a wavefunction-based correlated treatment using standard basis sets also suitable for molecular calculations as well as with relativistic all-electron gradient-corrected density functional methods. Here we present an investigation of differential electron correlation eects in ab initio PP calculations for IP3 and IP4 of the lanthanides using large uncontracted basis sets and basis set extrapolation techniques to approach the experimental values as close as possible. 2. Method The method of relativistic energy-consistent ab initio PPs applied in the present study is described in detail elsewhere [20,21]. Small-core PPs in semilocal form were used for La to Lu [17,22], i.e., the 1s±3d shells were included in the PP core, while all shells with main quantum number larger than 3 were treated explicitly. The free parameters of the valence model Hamiltonian were adjusted to reproduce the valence total energies of a multitude of low-lying electronic states of the neutral atom and its ions. The reference data for the scalar-relativistic terms have been taken from relativistic all-electron calculations using the socalled Wood±Boring Hartree±Fock approach, whereas the multi-con®guration Dirac±Hartree± Fock approach based on the Dirac±Coulomb Hamiltonian was used for the spin±orbit part. The valence orbitals were expanded in optimized (14s13p10d8f6g) Gaussian basis sets [17], to which a (2s2p2d2f2g) diuse set was added. The resulting (16s15p12d10f8g) basis set was augmented by (8h) and (8h8i), respectively, using the same exponents for h and i as for g. The M O L P R O [23±27] program system was applied for all scalar-relativistic calculations using D2h symmetry. The averaged coupled-pair functional (ACPF) approach [28] based on a state-averaged complete active space multi-con®guration self-consistent ®eld (CASSCF) wavefunction was used to account for electron correlation eects. All orbitals belonging to partly occupied shells were active. The state average was taken over all components of the lowest energy LS state of each
con®guration in D2h symmetry in order to avoid symmetry breaking. The considered LS states and active open shells are listed in Tables 1 and 2. In addition we report coupled-cluster results with single and double excitations as well as perturbative triple excitations (CCSD(T)) for La, Ce, Eu, Gd, Yb and Lu. For each state the reference wavefunction was a single-determinant symmetrybroken Hartree±Fock solution. Excitations were allowed from all occupied orbitals in the ACPF and CCSD(T) calculations. Spin±orbit contributions were derived from complete con®guration interaction calculations within all open-shell orbitals (COSCI) with and without spin±orbit potential and then added to the scalar-relativistic results. All CASSCF, ACPF and CCSD(T) values reported here include spin±orbit corrections. Modi®ed versions of the ®nite-dierence programs MCHF [29] and GRASP [30] were applied exploiting the spherical symmetry. CASSCF/ACPF and CCSD(T) results are reported for the uncontracted (16s15p12d10f8g8h8i) basis sets truncated after f (l 3), g (l 4), h (l 5) and i (l 6) functions, respectively. Empirically the total valence energies (or valence correlation energies) obtained with truncation at l > 3 were found to be linear in 1=l3 with high accuracy and extrapolated to l ! 1 in order to estimate the basis set limit. The correlation coef®cient of the linear regression was in all cases larger than 0.9999, i.e., the worst correlation was 0.99991 for the ACPF energy of La2 , whereas the deviation from 1 is even one or two orders of magnitude smaller for the heavier elements of the series. Other extrapolation formulas were tried, but all were found to yield lower accuracy. 3. Results and discussion The current results for IP3 and IP4 are compared to experimental data in Tables 1 and 2, respectively. The mean absolute errors (m.a.e.'s) with respect to the available experimental values [13] are reported for the ®rst (La±Gd) and second (Tb±Lu) halves of the lanthanide series in order to distinguish between cases where the 4f shell is up to half-®lled and more than half-®lled, respec-
X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
491
Table 1 IP3 of the lanthanides (in eV) Ln
State
CAS
Initial La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
1
d f2 f3 f4 f5 f6 f7 f 7 d1 f9 f 10 f 11 f 12 f 13 f 14 f 14 s1
Final 1
2
D3=2 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 9 D2 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1 S0 2 S1=2 3
f1 f2 f3 f4 f5 f6 f7 f8 f9 f 10 f 11 f 12 f 13 f 14
S0 F5=2 3 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 7 F6 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1 S0 2
m.a.e. La±Gd m.a.e. Tb±Lu
ACPF
Expt.
f
g
h
i
Extr.
18.19 18.34 19.80 19.89 19.87 21.10 22.47 19.93 17.98 19.06 18.46 17.95 18.92 20.37 19.97
18.56 18.49 20.06 20.58 20.97 22.18 23.42 20.36 19.96 21.18 21.17 21.09 22.12 23.43 20.71
18.69 19.48 20.99 21.46 21.80 23.02 24.13 20.50 20.85 21.99 21.91 21.77 22.72 23.99 20.79
18.76 19.76 21.26 21.71 22.05 23.25 24.34 20.58 21.13 22.25 22.15 22.01 22.94 24.16 20.81
18.78 19.88 21.38 21.83 22.15 23.36 24.45 20.60 21.26 22.37 22.27 22.13 23.05 24.27 20.81
18.82 20.05 21.54 21.98 22.30 23.44 24.58 20.65 21.42 22.53 22.41 22.28 23.18 24.37 20.82
1.85 3.90
1.22 1.47
0.54 0.85
0.33 0.65
0.24 0.55
0.14 0.42
19.18 20.20 21.62 22.1 0.3 22.3 0.4 23.4 0.3 24.92 0.10 20.63 0.10 21.91 0.10 22.8 0.3 22.84 0.10 22.74 0.10 23.680.10 25.05 0.03 20.96 0:15 0:10
Results from CASSCF and multi-reference ACPF calculations in comparison to experimental data (expt. [13]). The results were obtained with relativistic energy-consistent ab initio pseudopotentials and a (16s15p12d10f8g8h8i) valence basis set truncated after the angular quantum number l denoted by the labels f, g, h and i. The results of an extrapolation to the basis set limit using 1=l3 (extr.) are also given. All results were corrected for spin±orbit interaction contributions. The m.a.e. in eV is given for the ®rst (La±Gd) and second (Tb±Lu) half of the lanthanide series. The averaged error bars of the experimental values are also given if larger than 0.01 eV.
Table 2 IP4 of the lanthanides (in eV) Ln
State
CAS
Initial La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
6
p f1 f2 f3 f4 f5 f6 f7 f8 f9 f 10 f 11 f 12 f 13 f 14
Final 1
S0 F5=2 3 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 7 F6 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1 S0 2
m.a.e. La±Gd m.a.e. Tb±Lu For explanations cf. Table 1.
5
p p6 f1 f2 f3 f4 f5 f6 f7 f8 f9 f 10 f 11 f 12 f 13
2
P3=2 S0 2 F5=2 3 H4 4 I9=2 5 I4 6 H5=2 7 F0 8 S7=2 7 F6 6 H15=2 5 I8 4 I15=2 3 H6 2 F7=2 1
ACPF
Expt.
f
g
h
i
Extr.
49.05 35.23 37.58 39.24 39.44 39.59 40.93 43.00 35.98 38.09 39.37 38.96 38.49 39.58 41.25
49.62 34.55 37.31 39.05 39.68 40.16 41.69 43.54 37.44 39.50 41.00 41.06 41.05 42.19 43.83
49.85 35.60 38.40 40.08 40.64 41.12 42.55 44.31 38.44 40.44 41.86 41.83 41.74 42.79 44.40
49.93 35.86 38.71 40.38 40.93 41.40 42.83 44.56 38.79 40.75 42.15 42.11 42.02 43.04 44.59
49.96 35.98 38.85 40.52 41.06 41.53 42.95 44.68 38.94 40.90 42.29 42.25 42.15 43.17 44.72
50.01 36.15 39.04 40.70 41.23 41.64 43.12 44.83 39.15 41.08 42.46 42.42 42.32 43.33 44.83
1.40 3.68
1.21 1.63
0.42 0.85
0.26 0.58
0.27 0.44
0.33 0.27
49.95 0.06 36.76 0.01 38.98 0.02 40.4 0.4 41.1 0.6 41.4 0.7 42.7 0.6 44.0 0.7 39.37 0.10 41.4 0.4 42.5 0.6 42.7 0.4 42.7 0.4 43.56 0.10 45.25 0.03 0:39 0:29
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X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
tively. The m.a.e.'s given under the entry CAS also indicate the importance of dynamical electron correlation eects. It is rather obvious that electron correlation is less important for the ®rst half of the series (CAS m.a.e. IP3 1.85 eV, IP4 1.40 eV) than for the second half (3.90 eV, 3.68 eV). However, due to the more diuse character of the 4f shell at the beginning of the series and the higher number of empty 4f orbitals accepting virtual excitations, electron correlation contributions are more dicult to account for the light lanthanide elements. Using standard generalized [17] or segmented [18] contracted basis set including up to g functions at the spin±orbit corrected CASSCF/ACPF level for the evaluation of IP3 and IP4 leads to disappointingly large underestimations of the experimental values by up to 5% and 2%, respectively. Notably only IP4 of Gd is overestimated by about 1.5%. We want to demonstrate here by an extrapolation to the basis set limit that these errors are mostly due to the limitations in the electron correlation treatment, and to a much lesser extent in the applied PPs. Our extrapolated results for IP3 have m.a.e. of 0.13 eV (La±Gd) and 0.42 eV (Tb±Lu). These results are considerably better than those reported previously for calculations using augmented standard basis sets including up to h functions (La±Gd 0.39 eV, Tb±Lu 0.69 eV [17]). The m.a.e. for La±Gd even lies within the mean experimental error bar of 0.15 eV, whereas for Tb±Lu the mean experimental error bar of 0.10 eV is signi®cantly below our m.a.e. For IP4 the present m.a.e.'s of 0.32 eV (La±Gd) and 0.27 eV (Tb±Lu) are both below the mean experimental
error bar of 0.39 eV and 0.29 eV, respectively, as well as signi®cantly smaller than the theoretical m.a.e. obtained previously (La±Gd 0.39 eV, Tb±Lu 0.76 eV [17]). Recently it has been demonstrated that the IP4 of Ce can be calculated at the CCSD(T) level with an error of less than 0.06 eV (0.3%) with respect to experimental data (36.82 eV, expt. 36:76 0:01 eV), using a (slightly dierent) basis set extrapolation technique and the PP of the present work [17]. Similarly, an error of )0.02 eV ()0.1%) was obtained for IP3 of La (19.16 eV, expt. 19.18 eV) [17]. The atomic states of the lanthanide ions required to calculate IP3 and IP4 mostly need a multi-reference treatment, however, besides La and Ce at the beginning of the series, single-reference CCSD(T) calculations are also applicable for some IPs of a few heavier lanthanide ions, i.e., for Eu, Gd, Yb and Lu. Our corresponding results for IP3 and IP4 are listed in Table 3. The estimated CCSD(T) basis set limits for Lu, i.e., IP3 20.92 eV (expt. 20.96 eV; error )0.2%), and IP4 45.33 eV (expt. 45:25 0:03 eV; error 0.2%), have a similar accuracy as the results for La and Ce reported previously. The extrapolated CCSD(T) result for IP3 of Eu (25.02 eV, expt. 24:92 0:10 eV; error 0.4%) is also signi®cantly improved with respect to the corresponding ACPF value. The same is true for IP3 of Yb (25.00 eV, expt. 25:05 0:03 eV; error )0.2%). Disregarding Gd for the moment, the m.a.e. of the extrapolated ACPF results (Tables 1 and 2) for the systems reported in Table 3 is 0.37 eV. At the CCSD and CCSD(T) levels the m.a.e.'s are 0.21 and 0.06 eV, respectively. The signi®cant improvement upon the perturbative
Table 3 Selected CCSD(T) results for IP3 and IP4 of the lanthanides (in eV) Ln La La Ce Eu Gd Yb Lu Lu
IP3 IP4 IP4 IP3 IP4 IP3 IP3 IP4
f
g
h
i
Extr.
Expt.
18.81 49.68 34.93 23.79 43.55 23.95 20.78 44.26
18.98 49.91 36.11 24.55 44.34 24.58 20.87 44.87
19.06 49.99 36.42 24.78 44.59 24.76 20.89 45.07
19.09 50.02 36.53 24.89 44.71 24.88 20.90 45.21
19.14 50.07 36.74 25.02 44.86 25.00 20.92 45.33
19.18 49.950.06 36.760.01 24.920.10 44.00.7 25.050.03 20.96 45.250.03
For further explanations and involved electronic states cf. Tables 1 and 2.
X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
inclusion of triples emphasizes the importance of such excitations. Most likely the lack of higher excitations is also the main remaining source of error in the CASSCF/ACPF results reported here. In principle such higher excitations can be included in CASSCF/ACPF by enlarging the active orbital space, but from the present ACPF wavefunctions it is not obvious to us as to how to do this in a systematic way and still keep the calculation size feasible for our current hardware and software. It is noteworthy, however, that the extrapolated CCSD(T) result for the IP4 of Gd (44.86 eV, expt. 44:0 0:7 eV; error 1.9%) shows a signi®cant disagreement with experimental data. Since already the ACPF IP4 , which is typically below the corresponding CCSD(T) value, is by almost 1 eV larger than the experimental value, we suppose that the experimental result is too low by roughly this amount. Besides the comparison with experiment the in¯uence of higher angular momentum functions on the dynamical correlation contributions is interesting. We use the CASSCF (0%) and extrapolated CASSCF/ACPF (100%) values as reference points and list correlation contributions obtained with uncontracted basis sets including functions up to a given angular momentum in Table 4. Due
493
to the special electronic con®gurations involved, the results for La have to be considered separately, whereas the values are changing monotonously with increasing nuclear charge for the heavier elements of the series. With an spdf (spdfg) basis set 59.1% (79.6%) and 59.3% (83.4%) of the correlation contributions to IP3 and IP4 , respectively, are recovered for La. An even faster convergence is observed for Lu, where an spdf (spdfg) basis set yields 86.6% (95.8%) and 71.9% (87.7%). These results already demonstrate the need for g basis functions for accurate investigations. The situation is signi®cantly worse for the lighter lanthanide elements with a partially occupied 4f shell. The most dicult case appears to be Ce: an spdf (spdfg) basis set recovers 8.4% (66.3%) of the correlation contributions to IP3 , whereas for IP4 )74.4% (40.5%) are accounted for, i.e., the correlation contributions calculated with an spdf basis set lead to signi®cantly worse results than the CASSCF treatment. The inclusion of higher angular momentum functions, i.e., at least h functions, appears to be mandatory when ab initio evaluations of energy dierences between Ce(III) and Ce(IV) compounds are attempted. Finally, a discussion of possible errors in the results of this work applies. First of all, the PPs for
Table 4 Percentage (%) of dierential electron correlation contributions to IP3 and IP4 of the lanthanides Ln La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
IP3
IP4
spdf
+g
+h
+i
>i
spdf
+g
+h
+i
>i
59.1 8.4 14.7 33.0 45.1 46.0 45.3 60.9 57.6 61.0 68.2 72.5 75.1 76.5 86.6
20.5 57.9 53.3 42.2 34.4 36.0 33.7 18.6 25.8 23.4 19.1 15.6 14.1 14.1 9.2
10.7 16.3 15.6 12.0 10.0 10.0 10.0 11.0 8.0 7.5 6.0 5.7 5.1 4.1 2.4
3.2 7.4 6.9 5.5 4.4 4.6 4.9 2.9 3.7 3.5 3.1 2.7 2.6 2.8 0.3
6.4 9.9 9.5 7.3 6.1 3.3 6.1 6.6 4.9 4.6 3.7 3.5 3.1 2.5 1.5
59.3 )74.4 )18.6 )13.0 13.0 27.9 35.0 29.3 46.1 47.1 52.9 60.7 66.9 69.7 71.9
24.1 114.9 74.7 70.0 54.2 46.9 39.0 42.2 31.5 31.6 27.6 22.5 17.8 16.0 15.8
8.4 28.8 21.4 21.1 15.9 13.6 12.6 13.5 11.0 10.3 9.3 8.1 7.4 6.7 5.3
3.0 13.2 9.5 9.2 7.3 6.1 5.8 6.7 4.6 4.9 4.5 3.9 3.4 3.5 3.6
5.1 17.5 13.0 12.8 9.6 5.5 7.6 8.3 6.7 6.2 5.7 4.9 4.5 4.1 3.3
The percentages are recovered with basis sets containing up to f functions (spdf), as well as by addition of g (+g), h (+h), i (+i) and higher (> i) angular momentum functions. 0% and 100% refer to the entries CAS and ACPF, extr., respectively, in Tables 1 and 2.
494
X. Cao, M. Dolg / Chemical Physics Letters 349 (2001) 489±495
Ce to Yb have not been adjusted to ions with charges larger than +1 and errors of a few tenths of an electron volt might result for IP3 and IP4 . Second, the Breit interaction, which increases most of the IP3 and IP4 by one- or two-tenths of an electron volt, was not considered in the reference data for the PP adjustment. Third, the COSCI calculations used here to evaluate spin±orbit contributions do not account for a coupling with dynamic electron correlation eects. Finally, the empirical extrapolation scheme used to determine the basis set limit might be a source of small errors. Based on the good agreement of the extrapolated ACPF and especially CCSD(T) results with the experimental values we estimate the sum of these errors in IP3 and IP4 to amount to at most 1%. Within this accuracy the present values are the best available set of theoretical data for IP3 and IP4 of the lanthanides. For many lanthanides more reliable experimental values are desirable and would allow a more conclusive testing of the applied PPs, basis sets and correlation methods. More re®ned extrapolation techniques, or an explicit inclusion of the interelectronic distance in the wavefunction [31] in connection with an all-electron treatment based on the best relativistic Hamiltonian at hand, i.e., the Dirac±Coulomb±Breit Hamiltonian [32], might yield improved theoretical results which may both guide further experimental investigations and also help in the calibration of approximate computational schemes such as eective core potentials. However, to our knowledge such studies have not been attempted for the cases considered here. 4. Conclusions A consistent set of theoretical values for IP3 and IP4 of all lanthanide atoms has been derived from relativistic ab initio pseudopotential calculations using the CASSCF/ACPF approach together with basis set extrapolation techniques. With the exception of IP4 of Gd the theoretical results range between 97.3% and 101.0% of the experimental values, which themselves bear error bars of up to 1.8%. Corresponding CCSD(T) calculations for
single-reference cases lead to a signi®cant improvement over the ACPF results with errors below 0:5%. The experimental IP4 of 44:0 0:7 eV for Gd, partly derived on the basis of intrapolated estimates, appears to be too low by almost 1 eV and should be remeasured. The correct value is expected to be 44:85 0:25 eV. The dierential correlation contributions accounted for by higher angular momentum functions decrease drastically, and nearly monotonously from Ce to Lu, i.e., whereas the inclusion of g, h and even i functions appears to be mandatory for Ce; results of an even better quality can be obtained for Lu with a basis set restricted to at most g functions. Acknowledgements The ®nancial support of Fonds der Chemischen Industrie is acknowledged. References [1] P. Pyykk o, Inorg. Chim. Acta 139 (1987) 243. [2] M. Pepper, B.C. Bursten, Chem. Rev. 91 (1991) 719. [3] K. Balasubramanian, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 18, Elsevier, Amsterdam, 1994, p. 29. [4] M. Dolg, H. Stoll, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 22, Elsevier, Amsterdam, 1996, p. 607. [5] M. Dolg, in: P.V.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollman, H.F. Schaefer III, P.R. Schreiner (Eds.), Encyclopedia of Computational Chemistry, Wiley, Chichester, 1998, p. 1478. [6] G. Schreckenbach, P.J. Hay, R.L. Martin, J. Comput. Chem. 20 (1999) 70. [7] M. Dolg, P. Fulde, W. K uchle, C.-S. Neumann, H. Stoll, J. Chem. Phys. 94 (1991) 3011. [8] M. Dolg, P. Fulde, H. Stoll, H. Preuû, R.M. Pitzer, A. Chang, Chem. Phys. 195 (1995) 71. [9] N.M. Edelstein, P.G. Allen, J.J. Bucher, D.K. Shuh, C.D. So®eld, N. Kaltsoyannis, G.H. Maunder, M.R. Russo, A. Sella, J. Am. Chem. Soc. 118 (1996) 13115. [10] M. Dolg, H. Stoll, H.-J. Flad, H. Preuû, J. Chem. Phys. 97 (1992) 1162. [11] W. Liu, M. Dolg, L. Li, J. Chem. Phys. 108 (1998) 2886. [12] S.A. McDonald, R.F. Rice, R.W. Field, C. Linton, J. Chem. Phys. 93 (1990) 7676. [13] W.C. Martin, R. Zalubas, L. Hagan, Atomic Energy Levels ± The Rare Earth Elements, NSRDS-NBS 60, Washington, DC, 1978.
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