New Basis Sets for Lanthanide and
Actinide Energy-consistent Small-core
Pseudopotentials
Xiaoyan Caoa b* (曹晓燕)
a Institut für Theoretische Chemie, Universität zu Köln, D-50939,Germany
b Biochemistry Department, Zhongshan University, Guangzhou, 510275, P.R.
China
* cao@thch.uni-bonn.de
phone: (00)49-228-732919
fax: (00)49-228-739066
March 3, 2016
Abstract.
The
optimization
of
generalized
contracted
Gaussian
(14s13p10d8f6g)/[6s6p5d4f3g] atomic natural orbital valence basis sets for
relativistic energy-consistent small-core lanthanide and actinide pseudopotentials
of the Stuttgart-Bonn variety is described. Corresponding Gaussian valence basis
sets using a segmented contraction scheme (14s13p10d8f6g)/ [10s8p5d4f3g] for
lanthanides and (14s13p10d8f6g)/ [10s9p5d4f3g] for actinides are also discussed.
The basis sets have been proven to be stable and reliable by atomic and molecular
calibration studies. Application to the electronic ground states of some selected
homonuclear lanthanide dimers shows that the derived molecular constants are the
best theoretical estimates available at present and have a satisfactory agreement
with experimental data. Remaining discrepancies in the vibrational constants of
La2, Ce2 and Pr2 are shown to be probably related to quite large Ar-matrix shifts.
1
Introduction
Theoretical chemistry investigations on systems containing f elements are
still a considerable challenge
1, 2, 3
. The presence of several open shells with
different main quantum numbers poses difficulties to theoretical work, i.e. in
neutral atoms and their ions often nf (n=4, 5 for lanthanides and actinides,
respectively) as well as (n+1)d, (n+2)s and possibly (n+2)p orbitals may be
partially occupied in the ground states or rather low-lying excited states. A 2S+1LJ
term arising from a fm subshell may lead to a spin S as large as 7/2 and an angular
momentum L as large as 12. Even more extreme values of S and L may result
from the coupling of the fm subshell to other partially occupied shells of s, p, or d
symmetry. In addition spin-orbit coupling usually splits each LS term into a large
number of energetically adjacent electronic J states. From the experimental point
of view the limited availability of most actinide elements as well as their
radioactivity are additional obstacles. Therefore, up to now the experimental
knowledge of the energy levels of free actinide atoms and ions is very limited and
most of the higher ionization potentials are unknown for essentially all actinide
elements4. For free lanthanide atoms and ions, the experimental knowledge is
much wider, but still far from being complete
5.
Despite the problems outlined above, first-principles methods, i.e.,
wavefunction-based ab initio schemes as well as density-functional approaches,
are presently at the edge of successfully dealing with systems containing one or
two f atoms. For rigorous wavefunction-based ab initio methods relativity has
been taken into account at the all-electron Dirac-Coulomb-Breit level and electron
correlation effects are included by means of coupled-cluster (CC) or configuration
interaction (CI) methods. H- or even i-functions in the one-particle basis sets are
usually necessary to yield accurate results
6 ,
7
. Such highly-correlated
state-of-the-art all-electron studies are presently only feasible for atoms by
exploiting their spherical symmetry, and to our knowledge, the method is only
applicable for some elements, i.e., only calculations for closed shell systems, one
or two electrons outside a closed shell or one or two holes inside a closed shell are
feasible. In order to be able to treat all f atoms and also to deal with molecular
2
systems, compromises have to be made with respect to the treatment of relativity
and electron correlation.
Among the most successful approaches in relativistic quantum chemistry is
the ab initio pseudopotential (PP) method8, where the explicit quantum chemical
treatment is restricted to the valence electrons and relativistic effects are only
implicitly accounted for by a proper adjustment of free parameters in the valence
model Hamiltonian. Several sets of such effective core potentials (ECPs), both of
the energy-consistent9,
10, 11
and the shape-consistent PP variety
12, 13, 14, 15, 16, 17
,
have been published for the f elements. Recently also model potentials (MPs)
18, 19
became available
.
About a decade ago the energy-consistent small-core PPs for all f elements
except for lanthanum and lutetium have been published 11, 9, which proved to yield
quite accurate results in numerous first-principles molecular applications.
Corresponding optimized valence basis sets have not been derived initially. Later,
preliminary
(12s10p8d8f6g)/[5s5p4d3f3g]
for
lanthanide
and
(12s11p10d8f)/[8s7p6d4f] for actinide atomic natural orbital (ANO) Gaussian
valence basis sets were generated and distributed through the MOLPRO PP and
basis set library20,
21, 22, 23, 24
. These basis sets were not completely satisfactory in
several respects, e.g., they were not flexible enough to describe accurately states
with different f occupations since they have been optimized only for a single
electronic configuration. In addition the variation of the exponents with the
nuclear charge was partly irregular leading to difficulties in studies of trends along
the series.
Recently, we published the energy-optimized (14s13p10d8f6g)/[6s6p5d4f3g]
ANO Gaussian valence basis sets for the whole lanthanide and actinide series
26
25,
. Slightly larger basis sets (14s13p10d8f6g)/[10s8p5d4f3g] for lanthanides and
(14s13p10d8f6g)/[10s9p5d4f3g] for actinides using a segmented contraction were
also presented
studies
27, 28
. This article summarizes the atomic and molecular calibration
25, 26, 27, 28, 29, 30, 31
and discusses the reliability and stability for the new
basis sets. An application to the studies on the electronic structure of some
selected lanthanide dimers (Ln2, Ln=La, Ce, Pr, Eu, Gd, Yb, Lu)
reviewed.
3
32, 33
are also
Methods
The method of relativistic energy-consistent ab initio pseudopotentials (PPs)
is described in detail elsewhere
10, 11
and will be outlined here only briefly. The
valence-only model Hamiltonian for an atom or ion with n valence electrons is
given as
H  
1
2
n

i 
i
n
1
ij
ij
r
 Va  Vso
Here i and j are electron indices. Va denotes a spin-orbit averaged relativistic
PP in a semilocal form for a core with charge Q
n
Va  
i
Q

ri
n
A
i
lk
exp(alk ri2)Pl
l,k
Pl is the projection operator onto the Hilbert subspace of angular momentum
l. The spin-orbit term Vso may be written as
Vso 
n
  2l
i
l  0,k
2
Blk exp(blk ri 2)PllisiPl
 1
The free parameters Alk, alk, Blk, and blk are adjusted to reproduce the
valence total energies of a multitude of low-lying electronic states of the neutral
atom and its ions34. Small-core PPs for lanthanides (actinides) have been used, i.e.,
the 1s-3d (1s-4d) shells were included in the PP core, while the 4f (5f) shell and
all the others with main quantum number larger than 4 (5) were treated explicitly.
The choice of a small core allows an accurate description of electronic states
arising from all important electronic configurations, e.g., the frozen-core errors
resulting mainly upon changes in the 4f (5f) occupation number are restricted to a
tolerable minimum. The reference data used to determine Va have been taken
from relativistic all-electron calculations using the so-called Wood-Boring (WB)
scalar-relativistic Hartree-Fock (HF) approach.
Spin-orbit operators Vso have been derived for the p, d, f symmetries by
adjusting fine-structure splittings obtained with the valence model spin-orbit
4
Hamitonian
to
all-electron
Dirac-Hartree-Fock
(AE)
(MCDHF)
state-averaged
data
multi-configuration
obtained
with
the
Dirac-Coulomb-Hamiltonian for the one-valence electron ions35.
The routine to generate the Gaussian (14s13p10d8f6g)/[6s6p5d4f3g] ANO
valence basis sets using a generalized contraction scheme, as well as using a
segmented contraction pattern (14s13p10d8f6g)/[10s8p5d4f3g] for lanthanides
and (14s13p10d8f6g)/[10s9p5d4f3g] for actinides has been described in detail
elsewhere25,
26
and will be summarized here only briefly. The primitive
(14s11p8d8f) sets of exponents were energy-optimized in HF calculations for the
energetically lowest LS states of the neutral atoms. Two diffuse d and p functions
were then added to describe the (n+1)d and (n+2)p orbitals, yielding the final
(14s13p10d8f) uncontracted basis sets. The contraction coefficients were obtained
from atomic natural orbitals of state-averaged complete active space
multi-configuration self-consistent field calculations (CASSCF) with subsequent
20
multi-reference configuration interaction calculations (MRCI)
. The density
matrices for both low-lying configurations fn+1s2 and fnd1s2 were generated and
averaged, yielding the contraction coefficients describing these two states on
equal
footing.
The
generalized
contraction
scheme
yields
the
(14s13p10d8f)/[6s6p5d4f] ANO valence basis sets. Sets of 6g functions with
exponents identical to the 6 largest (and most important) f exponents were chosen.
A generalized ANO contraction, as described above, was performed to obtain the
final (14s13p10d8f6g)/[6s6p5d4f3g] Gaussian ANO valence basis sets. The
segmented contraction pattern was chosen in such a way that without too much
loss in accuracy relatively compact basis sets result, yielding slightly larger
(14s13p10d8f6g)/[10s8p5d4f3g] lanthanide and (14s13p10d8f6g)/[10s9p5d4f3g]
actinide basis sets. The segmented contraction coefficients were taken from the
ANO contractions of the lowest energy pseudo-valence orbital in each symmetry
in the generalized contracted basis sets. The described pseudopotentials and basis
sets are available from Electronic Physics Auxiliary Publication Service (EPAPS)
as well as by writing to the authors. The data will also be available in the
upcoming revision of the MOLPRO 20 PP and basis set library
36
.
Using both the generalized and segmented contraction for the valence basis
sets and the small-core pseudopotential, calibration studies have been carried out
for the first to fourth ionization potentials of all lanthanide and actinide elements
5
25, 26, 27, 28
. State-averaged complete active space self-consistent field (CASSCF)
calculations with subsequent multi-reference averaged coupled-pair functional
(ACPF)37 calculations were performed. The active space in the CASSCF always
comprised all open-shell orbitals (4f, 5d, 6s for lanthanides and 5f, 6d, 7s, 7p for
actinides) as well as 6s for lanthanides and 7s for actinides if doubly occupied.
State averaging always included all components of the energetically lowest
LS-term in order to avoid symmetry breaking. In the ACPF calculations
excitations were allowed from all occupied valence orbitals. The MOLPRO20
program system was applied for all scalar-relativistic calculations.
The large uncontracted valence basis sets have been applied to extrapolate to
the basis set limit29,
31
. For the lanthanides (14s13p10d8f6g) basis sets were
augmented by a (2s2p2d2f2g) diffuse set as well as (8h) and (8h8i) sets. The g, h
and i exponents were chosen to be identical to the f exponents with the largest
expansion coefficient in the 4f orbital. For actinides slightly smaller basis sets
were applied, i.e., (14s13p10d8f6g) augmented by (6h) and (6h6i) sets, since the
addition of diffuse functions was found to lead to negligible changes in the results
for Ac. Again, the g, h and i exponents were chosen to be identical to the f
exponents having the largest expansion coefficients in the 5f orbital.
The dynamical correlation energies E(l) obtained from calculations with
basis sets up to g (l=4), h(l=5) and i (l=6) functions were found to be almost
linear in 1/l3, when l denotes the maximum angular quantum number in the basis
set 25, 29.
E(l)  al3  E()
The correlation coefficient was found to deviate by less than 10-4 from the
ideal value of one for all cases. The extrapolations 1/l3  0 yield the basis set
limit for the dynamical correlation energies E().
Spin-orbit corrections have been derived for lanthanides by a full CI within
the space of fractionally occupied orbitals (COSCI, complete open shell CI)
31
25, 26,
. In case of the actinides the radial shape of the two-component spinors deviates
much more from the scalar-relativistic orbitals than for the lanthanides. Therefore
a CASSCF within the space of fractionally occupied orbitals was performed,
keeping the 5s, 5p, 5d, 6s, 6p shell frozen in their scalar-relativistic form.
6
State-averaging over all possible J-levels for a given nonrelativistic configuration
was performed. The spin-orbit calculations have been carried out with a modified
version of the GRASP relativistic atomic electronic structure code 35.
The molecular calculations were carried out with the MOLPRO program
package, using basis sets and methods as indicated in the text25,
26, 30
. The
spectroscopic constants were derived by fitting a fifth-degree polynomial in the
interatomic distance R times a factor 1/R for six points on the potential curve near
the equilibrium distance. The stability of the derived spectroscopic constants is
better 0.001 Å (Re), 1 cm-1 (e) and 0.01 eV (De). The basis set superposition error
(BSSE) was accounted for by the counterpoise method38.
Results and discussion
Before we discuss the results for atomic and molecular calibration
calculations in detail we want to emphasize that the newly developed ANO
valence basis sets exhibit errors in the total valence energies which are up to a
factor of 10 smaller than those for the previous basis sets. The errors, which were
evaluated by comparing algebraic atomic HF calculations to corresponding finite
difference results, stay below 8 milli-Hartree (0.2 eV). In addition, the variation of
the exponents along the lanthanide/actinide series is much more regular than that
for the previously used basis sets and therefore more reliable comparisons
between systems containing different lanthanide or actinide elements are possible.
Atoms
The ground state configurations and LSJ labels of all atoms and ions
considered here are listed in Tables 1 and 2 for lanthanides and actinides,
respectively. The LSJ labels assigned in the experimental compilations of atomic
energy levels 4, 5 were found to be consistent with our present AE MCDHF as well
as PP CASSCF/MRACPF results, with the exception of Pa2+, where we found a
2
H state 0.16 (0.26) eV below the 4I state in scalar-relativistic CASSCF (MRCI)
calculations. Both states yield J=11/2 as the lowest level in our spin-orbit
corrected calculations, which also is experimentally found to be the ground state.
Since unfortunately ACPF calculations did not converge for the 2H state, we used
the 4I state of Pa2+ as the lowest scalar-relativistic level. In addition, ACPF
7
convergence problems were encountered for Pa+, and limiting values of IP1 and
IP2 could not be derived. However, since the energy difference between Pa and
Pa2+, i.e., IP1+IP2, is 17.74 eV for a (14s13p10d8f6g)/[6s6p5d4f3g] ANO standard
basis set and 17.78 eV for our present basis set extrapolation, we assume that also
the individual results for IP1 and IP2 are quite similar. Therefore we use the values
obtained with the standard basis set in the following. The results for IP1 to IP4 of
all lanthanides and actinides are compiled in Tables 3 and 4, respectively.
Let us first check the applied valence basis sets. At the HF level the m.a.e. for
the lanthanides are <0.01 eV for IP1, IP2 and < 0.02 eV for IP3, IP4, respectively.
The corresponding mean relative errors (m.r.e.) are <0.1% in all cases. The largest
relative error (0.6%) occurs for IP1 of Gd; all other relative errors are less than
0.4%. For actinides the mean absolute errors (m.a.e.) are 0.02, 0.01, 0.03 and 0.03
eV, and the corresponding m.r.e. are 0.4, 0.1, 0.2 and 0.1%. The largest error of
2.1% occurs for IP1 of Lr, all other errors are less than 0.9%. The errors were
evaluated by comparing CASSCF calculations to corresponding finite difference
results. Since the basis set errors are very small at the HF level, an extrapolation
to the basis set limit is performed at the correlated level.
Differential electron correlation effects (SCF vs. ACPF) turn out to be very
important. For actinides they are at the most 1.5 eV for IP1, IP2 and 3 eV for IP3,
IP4. These effects are of the same magnitude as found for lanthanides for IP1 and
IP2, but by more than 1 eV smaller for IP3 and IP4. We attribute this to the more
diffuse character of the actinide 5f shell compared to the lanthanide 4f shell and to
the corresponding smaller electron-electron interaction in the shell. The large
correlation contributions indicate that the quality of the results mainly depends on
the treatment of electron correlation.
Quite accurate experimental data is available for IP1 of lanthanides
(maximum error bar 0.02 eV 5) and actinides (maximum error bar 0.12 eV; no
value available for Lr
4, 39, 40
). In case of the lanthanides IP2 (maximum error bar
0.08 eV for La to Yb, 0.4 eV for Lu) is also sufficiently accurate, whereas for
the actinides only values for Ac and U are at hand (maximum error bar 0.19 eV).
The higher ionization potentials of the lanthanides exhibit experimental
uncertainties of up to a few tenths of an electron volt (maximum error bars 0.4
eV for IP3, 0.7 eV for IP4). For actinides IP3 and IP4 are available only for Th
and U (maximum error bars 0.3 eV for IP3, 1.0 eV for IP4).
8
We now turn to the discussion of the results obtained for lanthanides, which
are listed in Table 3. The m.a.e. of the theoretical value for IP1 (0.15 eV) and IP2
(0.13 eV) obtained with the generalized contracted standard basis sets augmented
by diffuse functions (std.) are sufficiently small, so that a basis set extrapolation
does not really appear to be worthwhile. For selected elements (La, Yb, Lu), the
CCSD(T) leads to an improvement of the m.a.e. for IP1 (0.06 eV) and IP2 (0.13
eV) with respect to the ACPF values (0.16 eV; 0.15 eV). The results for IP3 and
IP4 at the basis set limit without (ext; 0.30 and 0.37 eV) and with (est; 0.35 and
0.38 eV) correction of PP errors exhibit m.a.e. of similar magnitude. Also here for
selected cases (IP3 of La, Eu, Yb, Lu; IP4 of La, Ce, Lu) the m.a.e. is considerably
reduced when changing from ACPF (0.37 eV) to CCSD(T) (0.06 eV). The series
of estimated values including a correction of PP errors show slightly larger m.a.e.
for IP3 and IP4 than the underlying series without correction. This is most likely a
result of an error compensation, i.e., the relativistic AE Wood-Boring method used
to generate the reference data for the PP generation tends to overestimate IP 3 and
IP4 by a few tenths of an electron volt, whereas the corresponding correlation
contributions are underestimated.
For actinides the theoretical values for IP1 frequently underestimate the
experimental values. For a 5fn 7s2  5fn 7s1 ionization process the errors are
almost constant at 0.3 eV. This behavior is similar to the one observed for the
lanthanides, although the errors are slightly larger for actinides, and mainly related
to the incomplete correlation treatment. Our theoretical values for IP2 of the
actinides are most likely by 0.1 to 0.2 eV below the unknown experimental values,
similar to the corresponding situation for the lanthanides. Unfortunately, only two
experimental values are available for calibration, i.e., 11.780.19 eV for Ac and
11.590.37 eV for U. The theoretical values (11.60 eV for Ac, 11.92 eV for U)
agree well with the experimental data. For IP3 and IP4 only two experimental
values are at hand, i.e., IP3, 18.330.05 eV; IP4, 28.650.02 for Th and IP3,
19.800.31 eV; IP4, 36.701.00 eV for U. For Th the theoretical values (IP3, 18.21
eV; IP4, 28.21 eV) are in reasonable agreement with the experimental data. The
ionization of Th3+ 5f1 was found to be a difficult case. Similar with the ionization
of Ce3+ 4f1, both require an especially extensive correlation expansion at the
CCSD(T) level. A graphical comparison of the quality of our results for Ce and Th
is given in figure 1. The very similar trends observed for the correlation
9
contributions in Ce and Th also support that errors in our predicted values for
actinides are of similar magnitude to those found for lanthanides, provided the
same electronic configurations are involved. For U a strong disagreement between
our values of 19.07(18.77) eV for IP3, 33.17(32.73) eV for IP4 without (with)
correction of PP errors and the experimental result (IP3, 19.800.31 eV; IP4,
36.701.00 eV) exists. Previous fully-relativistic DKS DFT calculations using
three different functionals with self-interaction and gradient corrections yielded
values between 18.77 and 18.94 eV for IP3, 32.68 and 32.87 eV for IP441 which
agree well with our present values. Refined experiments are recommended.
Besides for the calibration studies using basis sets obtained with generalized
contraction scheme, the (14s13p10d8f6g)/[10s8p5d4f3g] valence basis sets
obtained with a segmented contraction scheme have also been used to calculate
the first to fourth ionization potentials of all lanthanide and actinide elements for
calibration purposes27,
28
. The values agree with those of the generalized
contracted (14s13p10d8f6g)/[6s6p5d4f3g] basis sets within 0.06 eV at the
CASSCF and 0.20 eV at the ACPF level. The comparison shows that segmented
and generalized contracted basis sets have essentially the same quality.
Diatomic molecules
Molecular calibration calculations for some selected monohydrides,
monoxides, monofluorides and monosulfides of lanthanides and actinides were
performed using standard valence basis sets with a generalized and a segmented
contraction scheme, respectively
25,
26,
27,
28,
30
. Standard augmented
correlation-consistent valence quadruple-zeta basis sets (AVQZ in MOLPRO
were used for H, F, O, and S. The results are summarized in Table 5.
20
)
It is also
worth reviewing the applications on the study of electronic structure for Ln2
(Ln=La, Ce, Pr, Eu, Gd, Yb, Lu)
32, 33
, not only because good agreement with
experimental results was obtained but also since they are typical representatives of
the rather complex homonuclear lanthanide dimers containing partially occupied
4f shells on both atoms, i.e., 4f is empty (La2), partially occupied (Ce2, Pr2), half
filled (Eu2, Gd2) and completely filled (Yb2, Lu2). The results are summarized in
Table 6.
10
Hydrides, Oxides, Fluorides and Sulfides
At the Hartree-Fock level the BSSE is negligibly small, whereas at the CI and
CC level it slightly increases the bond lengths and depresses the binding energies.
The BSSE for the heavier lanthanide elements is usually larger than for the lighter
ones. For example, the BSSE correction to bond distances, binding energies and
vibrational frequencies of LuH and LaH is respectively: 0.032 Å, 0.011 Å; 0.29
eV, 0.09 eV; 71 cm-1, 10 cm-1. The large contributions in the heavier lanthanide
elements arise mainly from the more compact 4d and 4f subshells for heavier
elements, which have to be correlated in accurate work. Test calculations for LuH
yield counterpoise corrections of 0.009 Å, -0.12 eV, -23 cm-1 when Lu 4s-4f is
inactive, 0.016 Å, -0.18 eV, -37 cm-1 when Lu 4s-4d is inactive, and 0.032 Å,
-0.29 eV, -71 cm-1 when Lu 4s-4p is inactive. Note that, e.g., the
counterpoise-corrected CCSD(T) bond length for inactive Lu 4s-4f is 1.930 Å,
whereas the corresponding value for inactive Lu 4s-4p of 1.914 Å agrees very
well with the experimental result of 1.912 Å42.
The monoxides and monofluorides are the experimentally most extensively
investigated diatomics of the lanthanides. Our CCSD(T) results are in good
agreement with experimental data for LaO, LuO and LaF (errors in bond distances,
binding energies and vibrational frequencies are at most 0.015 Å, 0.16 eV, 6 cm-1).
Unfortunately a CCSD(T) treatment was not possible for LnX (Ln=Eu, Gd, Yb;
X=O, F). We attribute the larger errors for these molecules, at least partly, to the
deficiencies of the CI (SD)+Q correlation treatment. The electronic structure of
YbO is still an open problem and a challenge for future more accurate
investigations. The calculated bond distance (1.871 Å, PP) for the 4f14σ2σ2π4
1
Σ+ state is close to the DFT result (1.865 Å43), but deviates by about 0.06 Å
from the experimental value (1.807 Å44). The vibrational frequency (725 cm-1, PP)
agrees well with both DFT (725 cm-1
43
) and experimental data (699 cm-1
44
).
However the PP binding energy (2.93 eV) underestimates the experimental value
(4.29 eV
44
), whereas DFT (4.65 eV
43
) slightly overestimates it. In the case of
LuF the theoretical results of 7.50 eV (PP) and 7.55 eV (DKH-DFT) indicate that
the estimated experimental value of 5.93 eV 42 is substantially too low. Very little
experimental information is available for the monohydrides of the lanthanide
elements (see Table 5). The PP results are in good agreement with available
11
experimental data (the largest errors are: bond distances 0.019 Å, binding energies
0.06 eV, and vibrational frequencies 7 cm-1), as well as DFT results, except for the
binding energies that tend to be larger by up to 1 eV in DFT. The PP results of
EuS, YbS are in quite good agreement with experiments; the errors of equilibrium
distances, binding energies and vibrational frequencies are 0.1 Å, 0.4 eV, 2 cm-1
for EuS and 0.01 Å, 0.18 eV, 26 cm-1 for YbS.
We now turn to the discussion of the results obtained for actinides. At the
SCF level, the counterpoise corrections (CPC) of the BSSE are almost negligible
(Re0.001 Å, -e1 cm-1, -De0.02 eV). At the CCSD(T) level if excitations
from Ac 5d are allowed the significant CPCs are obtained for AcH, AcO, and AcF,
especially for De (0.015 Å, 24 cm-1, 0.53 eV) . Relatively small CPCs are
obtained if the Ac 5d shell is kept frozen (0.011 Å, 14 cm-1, 0.08 eV) as it has
been done in the basis set derivation. However, the corrected results show good
agreement with those obtained from a relaxed Ac 5d shell. The uncontracted basis
set leads to a substantially smaller CPC for De (0.016 Å, 17 cm-1, 0.16 eV).
The basis sets generated including Ac 5d correlation, yields corrections of similar
magnitude (0.017 Å, 14 cm-1, 0.16 eV); however, it tends to yield by 0.05 eV
too low De after CPC both at the SCF and CCSD(T) level. We attribute this to a
loss of flexibility of the Ac basis set in the valence region, since some ANOs tend
to be more compact to account for the correlation of the Ac 5d shell. In contrast to
Ac more consistent CPCs are obtained for ThO, i.e., the CCSD(T) results obtained
with a Th basis set optimized without correlation of 5d (1.839/1.845 Å, 898/891
cm-1, 9.58/9.38 eV without/with CPC) are almost identical to those using a Th
basis set optimized with correlation of 5d (1.840/1.845 Å, 897/890 cm-1, 9.56/9.37
eV). We attribute the higher sensitivity of Ac with respect to the optimization of
the basis set to the very diffuse character of the Ac 5f and 6d shells and that the
corresponding ANO contractions derived from Ac 5f1 7s2 and 6d1 7s2 less
efficiently correlate the quite compact 5d shell.
The experimental information for diatomics of the actinides is very limited.
For ThO counter-poise corrected CCSD(T) values , including also a correction for
the fine-structure splitting of the Th 6d2 7s2 3F ground state (Re=1.845 Å, =891
cm-1, D0=8.96 eV), are in excellent agreement with the experimental data
(Re=1.840 Å, =896 cm-1, D0=9.00  0.09 eV
Laerdahl et al.
47
42, 45, 46
). For AcH and AcF
performed fully relativistic DHF and MP2 calculations using
12
large uncontracted basis sets for AcH and AcF. Their results compare quite
favorably to our data. Hong et al.48 presented DFT results for AcH, AcO and AcF
using the ZORA as well as the DKH Hamiltonians and the gradient-corrected
Becke exchange and Perdew correlation functionals. The agreement with the
present data is satisfactory for AcH and AcF, whereas slightly larger deviations are
observed for AcO (Table 5).
Molecular calibration studies were also done for basis sets using a segmented
contraction scheme 27, 28. At the HF and CCSD(T) level the results agree well with
those of the generalized contracted basis sets, i.e., the differences amount up to at
most 0.002 Å, 0.05 eV, 13 cm-1 for Re, De and e, respectively. Therefore, we
conclude that the basis sets obtained from both contraction patterns are reliable
tools for electronic structure calculations on lanthanide/actinide systems.
Dimers
The PP calculations show that the ground states of homonuclear dimers
composed of lanthanide elements with ground state valence sub-configuration 5d1
6s2 (6s2 for Pr, Eu, Yb), are most likely La2 1g+ g2 u4, Ce2 1g+, 1u-, 3g-, 3u+,
1
6g, 36u 4f14f1g2u4 , Pr2 5g+ , 5u-, 510g 4f24f2 g2 u4, Eu2
u+ 4f74f7g2u2,
15
Gd2 19g- 4f74f7g2u1g1u2, Yb2 1g+ 4f144f14g2u2 and Lu2 3g- 4f144f14g2u2u2
32, 33
. At present it is difficult to clarify the ground state for Ce2 and Pr2, since all
candidates listed above are virtually degenerate (below 20 cm-1 at the MRCI+Q
level) and the same spectroscopic constants were obtained (within 210-4 Å and
0.04 cm-1 for Re and e). For Ce2 the practical degeneracy of singlet and triplet
states is attributed to the core-like properties of the 4f orbitals (<r>4f 0.55 Å),
which lead due to the long bond distance (Re 2.6 Å) to a negligible spin
correlation for the two 4f electrons localized on the two Ce atoms. In contrast to
Ce2 the state average CASSCF calculations for Pr2 yield the lowest triplet and
singlet states at term energies of 300 cm-1 and 400 cm-1, respectively. This
13
different behavior for Pr2 may be attributed to the increasing number of 4f
electrons (four 4f electrons for Pr2 and two 4f electrons for Ce2), and the
concomitant increased multi-reference character of the wavefunction. Whereas the
two unpaired electrons in Ce2 are essentially localized in 4f  orbitals and direct
coupling is not mediated by other occupied shells of  symmetry, in Pr2 the four
unpaired electrons are in 4f , 4f  orbitals and the direct coupling is possible via
the occupied valence shells having partly the same symmetry. For La2, Ce2 and Pr2
the analysis of the valence orbitals for the ground state exhibits a weak
contribution (3%) of the 4f shell to chemical bonding. Therefore, the 4f electrons
should be treated as valence electrons in highly accurate calculations.
Unfortunately there are not many experimental results to be compared. The
theoretical results are in reasonable overall agreement with the available
experimental data. A notable exception is the e values of La2, Ce2 and Pr2. In
order to get some idea of possible matrix effects in the experimental work (Raman
spectroscopy in Ar-matrix49,
50
), a simple linear complex between a single Ar
atom and the lanthanide dimers Ln2 was geometry-optimized at the CCSD(T)
level
32, 33
. Compared to free La2 (206 cm-1), Ce2 (205 cm-1), Pr2 (198 cm-1), Gd2
(136 cm-1), and Lu2 (120 cm-1) the substantially higher frequencies (228 cm-1 for
La2, 223 cm-1 for Ce2 cm-1, and 216 cm-1 for Pr2) were found in the Ar-Ln2
complexes (Ln=La, Ce, Pr). In contrast to this, the Ar-Gd2 and Ar-Lu2 complexes
exhibit vibrational frequencies (136 cm-1 for Gd2 and 123 cm-1 for Lu2) which are
almost the same as for the free Gd2 and Lu2 systems.
Conclusion
Optimized atomic natural orbital valence basis sets of polarized valence
quadruple-zeta quality describing both low-lying configurations fn+1 s2 and fn d1 s2
14
on equal footing were presented and tested for lanthanide/actinide small core
energy-consistent ab initio pseudopotentials. The new basis sets yield
significantly lower total valence energies than the ones previously in use. In
atomic test calculations, mean absolute errors of less than 0.30 eV are obtained for
IP1 and IP2 of lanthanides, as well as for IP1 of actinides. For IP3 and IP4 of
lanthanides the mean absolute errors amount up to 0.60 eV. Calculations with
the extended basis sets and extrapolation techniques reproduce IP3 and IP4 with an
accuracy of 0.34 eV. For actinides the accuracy of IP2-IP4 is estimated to be better
than 3%. In molecular test calculations it was found that the basis set
superposition errors are considerably smaller, proving the reliability of the new
basis sets. An application to the studies of electronic structure for some selected
lanthanide dimers (La2, Ce2, Pr2, Eu2, Gd2, Yb2, Lu2) shows the derived
spectroscopic constants are in reasonable agreement with experimental data,
except for the vibrational frequencies of La2, Ce2, and Pr2. Model calculations
point to possible large positive matrix shifts of e for La2, Ce2, and Pr2, but
normal behavior for Gd2 and Lu2. It is concluded that La2 and Ce2 exhibit stronger
bonding than Pr2, Gd2 and Lu2. Weak contributions to chemical bonding of 4f
orbitals (ca. 3%) are found for La2, Ce2, and Pr2, whereas no contributions are
observed for Gd2 and Lu2. For the latter systems a relatively large contribution
from the 6p orbital (23% for Gd2, 32% for Lu2) is detected.
Acknowledgements. The author is grateful to Prof. Michael Dolg for reading
the manuscript. The financial support of Fonds der Chemischen Industrie is
acknowledged.
Key words: Lanthanides, Actinides, Basis sets, Pseudopotentials, Ionization
potentials, Spectroscopic constants, Lanthanide dimer
15
Figure captions:
Fig. 1
Error with respect to experimental values of IP4 of Ce (36.76 0.01
eV) and Th (28.650.02 eV) for PP CCSD(T) results including spin-orbit
corrections. The Th results include a correction for PP errors. The highest angular
momentum l included in the uncontracted basis set is indicated on the abscissa;
their location is defined by 1/l3. The error of a linear extrapolation to the CCSD(T)
limit based on the basis sets including g, h, and i functions are given in
parentheses.
16
Table 1: Electronic ground states and configurations for the lanthanides Lnn+ (n=0-4).
M1+
M
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
d1s2
f1d1s2
f3s2
f4s2
f5s2
f6s2
f7s2
f7d1s2
2
D3/2
G4
4
I9/2
5
I4
6H
5/2
7F
0
8
S7/2
9
D2
1
d2
f1d2
f3s1
f4s1
f5s1
f6s1
f7s1
f7d1s
M2+
3
d1
F2
H7/2
5
I4
6
I7/2
7H
2
8F
1/2
9
S4
10
D5/2
f3
f4
f5
f6
f7
f7d1
7
H8
6I
17/2
5I
8
4H
13/2
3F
4
2S
1/2
1S
0
f9
f10
f11
f12
f13
f14
f14s1
4
f2
M3+
2
D3/2
H4
4
I9/2
5
I4
6H
5/2
7F
0
8
S7/2
9
D2
3
M4+
1
f1
f2
f3
f4
f5
f6
f7
S0
F5/2
3
H4
4
I9/2
5I
4
6H
5/2
7
F0
8
S7/2
2
p5
p6
f1
f2
f3
f4
f5
f6
2
P3/2
S0
2
F5/2
3
H4
4I
9/2
5I
4
6
H5/2
7
F0
1
1
Tb
Dy
Ho
Er
Tm
Yb
Lu
f9s2
f10s2
f11s2
f12s2
f13s2
f14s2
f14d1s2
6
H15/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
2D
3/2
f9s1
f10s1
f11s1
f12s1
f13s1
f14s1
f14s2
17
6
H15/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
2S
1/2
f8
f9
f10
f11
f12
f13
f14
7
F6
6H
15/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
f7
f8
f9
f10
f11
f12
f13
8
S7/2
7F
6
6H
15/2
5I
8
4I
15/2
3H
6
2F
7/2
Table 2: Electronic ground states and configurations for the actinides Ann+ (n=0-4).
M1+
M
Ac
Th
Pa
U
Np
d1s2
d2s2
f3d1s2
f3d1s2
f4d1s2
2
D3/2
3
F2
4
K11/2
5
K6
6L
11/2
s2
d2s1
f2s2
f3s2
f4d1s
M2+
1
S0
F3/2
3
H4
4
I9/2
7L
5
4
s1
f1d1
f2d1
f4
f5
M3+
2
S1/2
G4
4
I11/2
5
I4
6H
5/2
1
1
f1
f2
f3
f4
S0
F5/2
3
H4
4
I9/2
5I
4
2
M4+
p5
2
f1
f2
f3
P3/2
S0
2
F5/2
3
H4
4I
9/2
f4
f5
f6
f7
f8
f9
f10
f11
f12
f13
I4
H5/2
7
F0
8
S7/2
7F
6
6H
15/2
5I
8
4I
15/2
3H
6
2F
7/2
p6
1
1
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
f6s2
8
f6
F0
F1/2
7
1
9
fs
f7
S7/2
S4
7
1
2
7
2
9
8
fds
fs
f8
D2
S7/2
9
2
9
1
6
7
fs
fs
f9
H15/2
H8
10
2
10
1
5
6
f s
f s
f10
I8
I17/2
11
2
11
1
4
5
f s
f s
f11
I15/2
I8
12
2
12
1
3
4
f s
f s
f12
H6
H13/2
13
2
13
1
2
3
f s
f s
f13
F7/2
F4
14
2
14
1
1
2
f s
f s
f14
S0
S1/2
14
1
2
14
2
2
1
f ps
f s
f14s1
P1/2
S0
2+ 2
Our calculations found a Pa H11/2 ground state 26.
f7s2
7
f6s1
8
18
7
F0
S7/2
7
F6
6
H15/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
2S
1/2
8
f5
f6
f7
f8
f9
f10
f11
f12
f13
f14
6
H5/2
7
F0
8
S7/2
7
F6
6H
15/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
5
6
Table 3: First to fourth ionization potentials of the lanthanides (in eV). PP CASSCF/ACPF results,
corrected for spin-orbit interaction, obtained with generalized contracted standard basis sets
augmented by diffuse functions (std.) and by extrapolation to the basis set limit (ext.) are
compared to experimental data (Exp. [5]). Additional theoretical bases set limit estimates were
corrected for PP errors (est.). The mean absolute errors (m.a.e.) are listed in the last line.
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
m.a.e
.
std.
5.58
5.52
5.35
5.40
5.45
5.49
5.51
6.00
5.76
5.80
5.83
5.89
5.95
5.97
5.23
0.15
IP1
Exp.
5.58
5.54
5.46
5.53
5.55
5.64
5.67
6.15
5.86
5.94
6.02
6.11
6.18
6.25
5.43
std.
11.06
11.13
10.62
10.78
10.92
11.02
11.15
12.15
11.75
11.60
11.62
11.72
11.80
11.81
13.81
0.13
IP2
Exp.
11.06
10.85
10.55
10.73
10.90
11.07
11.24
12.09
11.52
11.67
11.80
11.93
12.05
12.18
13.90
ext.
18.82
20.05
21.54
21.98
22.30
23.44
24.58
20.65
21.42
22.53
22.41
22.28
22.73
24.37
20.82
0.30
19
IP3
est.
18.79
20.05
21.47
21.86
22.22
23.33
24.49
20.35
21.30
22.40
22.35
22.23
22.71
24.44
20.77
0.37
Exp.
19.18
20.20
21.62
22.10
22.30
23.40
24.92
20.63
21.91
22.80
22.84
22.74
23.68
25.05
20.96
ext.
50.01
36.15
39.04
40.70
41.23
41.64
43.12
44.83
39.15
41.08
42.46
42.42
41.80
43.33
44.87
0.34
IP4
est.
49.96
36.03
38.84
40.46
40.91
41.37
42.85
44.24
38.82
40.74
42.13
42.06
41.52
43.18
44.86
0.38
Exp.
49.95
36.76
38.98
40.40
41.10
41.40
42.70
44.00
39.37
41.40
42.50
42.70
42.70
43.56
45.25
Table 4: First to fourth ionization potentials of the actinides (in eV). PP CASSCF/ACPF results at
the basis set limit corrected for spin-orbit interaction (ext.) and also for PP errors (est.) compared
to experimental data (Exp. [4, 39, 40]).
IP1
IP2
ext.
est.
Exp.
ext.
est.
Ac
5.17
5.13
5.17a
11.60
11.60
Th
6.25
6.23
6.31
12.11
12.15
Pa
5.81
5.76
5.90a
11.96
11.96
U
6.06
6.01
6.19
11.63
11.92
Np
5.98
5.98
6.27
11.35
11.56
Pu
5.71
5.73
6.03
11.50
11.48
Am
5.71
5.69
5.97
11.71
11.68
Cm
5.68
5.59
5.99
12.17
12.42
Bk
5.90
5.90
6.20
11.96
11.95
Cf
5.96
5.96
6.28
12.03
12.06
Es
6.07
6.04
6.37
12.20
12.19
Fm
6.18
6.13
6.50a
12.38
12.35
Md
6.25
6.23
6.58
12.47
12.46
No
6.33
6.31
6.65
12.58
12.58
Lr
4.78
4.71
14.25
14.23
m.a.e.
0.24
0.27
Additional Exp. values (in eV) IP2 Ac 11.780.19a, U
19.800.31; IP4 Th 28.650.02, U 36.701.00.
Due to convergence problems for Pa+ the results for IP1
(14s13p10d8f6g)/[6s6p5d4f3g] ANO basis set 26.
a Semiempirical estimates.
20
IP3
ext.
17.39
18.30
17.73
19.07
19.92
21.37
22.34
20.36
21.93
22.84
23.06
23.66
24.69
26.05
21.52
est.
17.37
18.21
18.65
18.80
19.66
21.04
21.75
20.10
21.55
22.50
22.68
23.13
24.30
25.76
21.50
IP4
ext.
44.99
28.45
31.24
33.17
34.27
35.43
37.26
39.06
36.52
38.12
39.52
40.16
40.60
41.96
44.12
est.
44.78
28.21
30.91
32.76
33.84
34.95
36.77
37.66
35.99
37.64
38.85
39.30
40.03
41.54
43.64
11.590.37; IP3 Th 18.330.05, U
and IP2 of Pa were obtained with a
Table 5 Bond lengths Re(Å), binding energy De(eV), vibrational frequencies e (cm-1) for XH
(X=La, Eu, Yb, Lu, Ac), XO (X=La, Eu, Gd, Yb, Lu, Ac, Th), XF (X=La, Eu, Gd, Yb, Lu, Ac) and
XS
(X=Eu,
Yb)
from
pseudopotential
(PP)
calculations
using
basis
sets
(14s13p10d8f6g)/[6s6p5d4f3g] in comparison to experimental data
Re (Å)
a
De(eV)
b
a
LaH
AcH
EuH
GdH
YbH
LuH
PP
2.016/2.027
2.135/2.150
2.104/2.116
1.911/1.924
2.041/2.072
1.882/1.914
Exp.
2.032
LaO
AcO
EuO
GdO
YbO
LuO
ThO
1.836/1.841
1.938/1.945
1.874/1.879
1.785/1.791
1.862/1.871
1.784/1.794
1.839/1.845
1.826
LaF
AcF
EuF
GdF
YbF
2.027/2.034
2.113/2.122
2.081/2.088
1.956/1.963
2.022/2.034
2.027
1.962
2.016
6.92/6.83
7.57/7.16
5.70/5.61
6.38/6.24
5.36/5.22
LuF
1.908/1.923
1.917
7.81/7.50
2.053
1.912
1.89
1.812
1.807
1.790
1.840
PP
2.97/2.88
3.34/3.02
1.99/1.92
2.47/2.34
1.65/1.49
3.64/3.35
8.30/8.13
7.81/7.28
4.28/4.16
6.82/6.62
3.14/2.93
7.28/6.90
9.16/8.96d
b
Exp .
1.55, 1.93
3.47
8.29
4.88
7.39
4.29
7.04
9.000.09
8.870.15
8.790.13
814/807
773/762
734/729
884/877
736/725
857/840
898/891
813
688, 672
824
699
842
896
578/574
538/530
496/490
613/606
514/502
570
620/603
612
EuS
2.410/2.417
2.39(2.51)c
3.41/3.31
3.71
405/402
YbS
2.352/2.373
2.359
2.85/2.55
2.73
405/393
The notation …/… refers to without/with counterpoise correction of the basis set
superposition error.
a Results for LaH, AcH, YbH, LuH, LaO, LuO, ThO, LaF, AcF, and LuF are from CCSD(T)
calculations. The results for others are from CISD+Q calculations; f elements 4s(for Ac, Th
5s), 4p(for Ac, Th, 5p) and O, F, S 1s frozen in CCSD(T) and CI calculations. Ref [30, 26,
25].
b references are: LaH [51]; YbH, LuH, LaO, LuO, LuF [42]; EuO[44, 52, 53]; GdO[52, 54,
55]; YbO [44]; LaF[42, 56]; GdF[42, 57]; YbF[42, 58, 59]; EuS[42, 60]; YbS[60, 61];
ThO[45, 46, 42].
c 2.39 and 2.51 eV for EuS were derived from different interpolations.
d D , The values have been corrected for molecular (0.03 eV) and atomic (Th 0.38 eV, O 0.01
0
eV) spin-orbit energy lowerings.
e empirically interpolated results.
400e
367
21
6.90
e (cm-1)
PP
Exp.b
1456/1446
1386/1362
1294/1272
1581/1559
1312/1256
1249
1577/1506
1500
a
5.42
6.95
4.90, >5.36,
4.83-4.89
5.93
607
502
Table 6 Bond lengths Re (Å), vibrational constants e (cm-1), and binding energies De (eV) for
selected lanthanide dimers.
Metal
La
State
g2u4, 1g+
Re(Å)a
2.70  0.03
2.80
De(eV)b
2.31  0.13
2.52  0.22
e (cm-1)c
186  13
236  0.8
Ref.
PPs
Exp.
Ce
4f1 4f1g2u4, 3u+
2.62  0.02
1.73  0.41
2.47  0.22
201  13
245.4  4.2
PPs
Exp.
Pr
4f2 4f2g2u4, 510g
2.555
1.19  0.16
1.31  0.30
213
244.9  1.2
PPs
Exp.
Eu
4f74f7g2u2, 15u+
4.878
0.080
0.300.17
27
35
PPs
Exp.
Gd
4f74f7g2u1g1u2, 19g-
2.88  0.02
1.38  0.18
1.784  0.35
149  2
138.7  0.4
PPs
Exp.
Yb
4f144f14g2u2, 1g+
4.549
4.19
0.092
0.170.17
25
22
PPs
Exp.
4f144f14g2u2u2, 3g-
3.07  0.03
123  1
121.6  0.8
PPs
Exp.
1.40  0.12
1.43  0.34
a The estimated experimental bond lengths for La are from Ref. [62]
2
Lu
b
Experimental values are from Ref. [63]. Spin-orbit corrections of -0.16 eV (La2), +0.13 eV
(Ce2), -0.10 eV (Pr2), -0.22 eV (Gd2) and -0.30 eV (Lu2) where added to the scalar-relativistic
results 33.
c The
experimental values for La2, Ce2, Pr2, Eu2, Gd2, Yb2, and Lu2 are from Ref. [50], [64],
[64], [65], [65], [66], and [67], respectively.
22
Figure 1
23
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