眉题：Lanthanide Diatomics
Molecular Structure of Diatomic Lanthanide Compounds
CAO Xiaoyan (曹晓燕) a,c , LIU Wenjian (刘文剑) b, DOLG Michael a
a
b
Institut für Physikalische & Theoretische Chemie, Bonn, 53115,Germany
Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, 44780 ,Germany
c Biochemistry Department, Zhongshan University, Guangzhou, 510275
Correspondence should be addressed to
Michael Dolg (email: dolg@thch.uni-bonn.de)
Abstract The molecular constants of selected diatomic Lanthanide compounds (LaH,
LaO, LaF, EuH, EuO, EuF, EuS, GdO, GdF, GdH, YbH, YbO, YbF, YbS, LuH, LuO,
LuF) have been calculated, by using relativistic small-core pseudopotentials and
optimized (14s13p10d8f6g)/ [6s6p5d4f3g] valence basis sets. The results are in
satisfactory agreement with available experimental data, with exception of YbO and
LuF. The reasons for the discrepancies in case of YbO are due to a complicated
mixing of configurations in the ground state, whereas in case of LuF the binding
energy estimated by experimentalists appears to be too low.
Keywords Molecular structure, Lanthanide elements, Pseudopotentials, valence basis
sets
1
Introduction
In the past three decades, the study of Lanthanide compounds has received
much attention.
However, the complex electronic structure of the Lanthanide
elements poses a considerable challenge for both experimental and theoretical work
[1-5].
Both relativistic and correlation effects should be included in theoretical
studies. Till now, relativistic ab initio pseudopotentials (PP) in combination with
high-level wavefunction-based correlation treatments and relativistic density
functional theory (DFT) methods are the two most effective methods for theoretical
lanthanide chemistry. Because of the high electron correlation in the 4f shell and its
weak but noticeable contribution to chemical bonding, 4f electrons should be treated
as valence electrons in accurate ab initio calculations. The 4s, 4p, 4d orbitals and the
4f orbitals are in the same spatial region, and should be treated as valence electrons
too in order to get a consistent description of electron interactions. That is why a
small-core PPs (28 core-electrons) are chosen [6]. The another problem is the choice
of appropriate valence basis sets. Because of the importance of g functions in
describing 4f-shell correlation effects, the authors of this paper energy-optimized
(14s13p10d8f6g)/ [6s6p5d4f3g] basis sets adopting a generalized contraction scheme
based on atomic natural orbitals1. In this contribution we present molecular constants
for selected diatomic lanthanide compounds derived with the small-core PPs and the
1
Xiaoyan Cao, Michael Dolg, Valence basis sets for relativistic energy-consistent small-core lanthanide pseudopotentials, available from
the authors and the MOLPRO basis set library (homepage http://www.theochem.uni-stuttgart.de)
2
optimized valence basis sets in correlated calculations, and compare our results to
available experimental data.
Methods
Molecular structure constants were calculated at the CI (SD)+Q (single reference
configuration interaction including single and double excitations and Davidson's
correction) level for GdH, GdO, GdF, EuH, EuO, EuS, YbO, EuF, YbF, YbS, and at
the CCSD (T) (coupled cluster singles, and doubles, with a perturbative treatment of
triples) level for LaH, LaO, LaF, LuH, LuO, LuF, YbH. 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 (equation
(1)-(3)). A spacing of 0.1 a0 between the points was used. Binding energies were
obtained by subtracting the molecular energy at the equilibrium distance from the
energy at a distance of 50 a0, in order to correct the size-inconsistency of CI
calculations. For CCSD (T) the binding energies were obtained by subtracting the
molecular energy at the equilibrium distance from the sum of the energies for two
separated atoms. The basis set superposition error (BSSE) was corrected by applying
the counter-poise scheme of Boys and Bernardi [7].
Spin-orbit effects were not considered in the present work, since only ground
states of Σ-symmetry were calculated . In the CI and CCSD (T) calculations,
excitations were allowed from the H 1s, first-row element 2s and 2p, second-row
element 2s, 2p, 3s, 3p, Lanthanide element 4d, 5s, 5p, 4f, 5d, 6s orbitals. Standard
3
augmented correlation-consistent valence quadruple-zeta basis sets (AVQZ in
MOLPRO) were used for H, F, O, and S.
E ( R) 
K

1 5
 C k ( R  Re ) k
R k 0
2E
R 2
1
2
(1)
(2)
R  Re
K

(3)
All calculations were done by using the MOLPRO[8] program system.
Results and discussions
All calculated and available experimental results for the molecules considered
here are shown in table 1.
Very little experimental information is available for the monohydrides of the
lanthanide elements (see table 1). Our results are in good agreement with available
experimental data (the largest errors are: bond distances 0.019Å, binding energies
0.06eV, and vibrational frequencies 7 cm-1). Our results agree well with the theoretical
results obtained from scalar-relativistic DFT, except for the binding energies which
tend to be larger by up to 1 eV in DFT.
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; 0.29eV, 0.09eV;
71cm-1, 10cm-1.
4
Table 1 Bond lengths Re(Å), binding energies De(eV), vibrational frequencies e(cm-1) for XH(X=La,Eu,Gd,Yb,Lu),XO(X=La,Eu,Gd,
Yb,Lu),XF(X=La,Eu,Gd,Yb,Lu),XS(X=Eu,YbS)
worka
LaH
EuH
GdH
YbH
LuH
LaO
EuO
GdO
YbO
LuO
LaF
EuF
GdF
YbF
LuF
EuS
YbS
2.104
2.116
1.911
1.924
2.041
2.072
1.882
1.914
1.836
1.841
1.874
1.879
1.785
1.791
1.862
1.871
1.784
1.794
2.027
2.034
2.081
2.088
1.956
1.963
2.022
2.034
1.908
1.923
2.410
2.417
2.352
2.373
1.878
2.033
2.053
1.891
1.912
1.869
1.826
1.88
≈1.89
1.825
1.812
1.865
1.807
1.802
1.790
2.031
2.027
1.996
1.962
1.987
2.016
1.920
1.917
2.42
2.39e,
2.51e
2.35
2.359
Re
this
this workb
2.016
2.027
分子
Ref.c
Expt.d
2.005
2.032
this worka
2.97
1.99
2.47
1.65
3.64
8.30
4.28
6.82
3.14
7.28
6.92
5.70
6.38
5.36
7.81
3.41
2.85
workb
2.88
3.72
1.92
2.34
2.77
1.49
1.50
3.35
3.82
8.13
7.96
4.16
5.17
6.62
7.72
2.93
4.53
6.90
7.42
6.83
7.30
5.61
6.24
7.17
5.22
5.32
7.50
7.55
3.31
3.52
2.55
2.81
≤ 1.55,
≤1.93
3.47
8.29
4.88
7.39
4.29
7.04
6.90
5.42
6.95
4.90, >
5.36,
4.83-4.
89
5.93
3.71
2.73
1581
1559
1312
1256
1577
1506
814
807
734
729
884
877
736
725
857
840
578
574
496
490
613
606
514
502
620
603
405
402
405
393
1682
1211
1249
1513
1500
768
813
705
688,
672
807
824
725
699
834
842
570
570
554
607
532
502
612
612
374
400f
379
367
De
this
Ref.c
Expt.d
e
this worka
this workb
1456
1446
Ref.c
Expt.d
1416
1294
1272
a
Results for LaH, YbH, LuH, LaO, LuO, LaF, LuF are from CCSD(T) calculations. The results for others are from CISD+Q calculations.
b
BSSE（basis set superposition errors）correction calculations.
c
Theoretical results. References are：LaH, LuH, LaO, LuO, LaF, LuF [9]；GdH, GdO, GdF [12]; YbH, YbF[13]; EuO, EuS, YbS[17].
d
References are：LaH [10]; YbH, LuH, LaO, LuO, LuF[11]; EuO [3,4,14]; GdO [3,15,16]; YbO[4]; LaF[11,18]; GdF[11,19]; YbF[11,20,21]; EuS[11,22];
YbS[22,23].
e
2.39 eV and 2.51 eV for EuS were derived from different interpolations.
f
Empirically interpolated results.
5
The monoxides of the lanthanide elements are the experimentally most
extensively investigated diatomic molecules calculated here. Our CCSD(T) results are
in good agreement with experimental data for LaO and LuO (errors in bond distances,
binding energies and vibrational frequencies are 0.015 Å, 0.16 eV, 6cm-1 for LaO and
0.004 Å, 0.14 eV, 2cm-1 for LuO). Unfortunately a CCSD(T) treatment was not
possible for EuO, GdO and YbO. 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 still is an open problem and a challenge for future more accurate
investigations [5,13]. Our calculated bond distance (1.871 Å) for the 4f14σ2σ2π4
1
Σ+ state is close to the DFT result (1.865 Å), but deviates by about 0.06 Å from the
experimental value (1.807 Å). The present vibrational frequency (725 cm-1) agrees
well with both DFT (725 cm-1) and experimental data (699 cm-1). Whereas our
binding energy (2.93 eV) underestimates the experimental value (4.29 eV), DFT (4.65
eV) overestimates by it.
Our calculated bond lengths and vibrational frequencies for the monofluorides are
in satisfactory agreement with experimental data (errors smaller than 0.02 Å, 10cm-1)
and DFT results. With the notable exception of LuF this is also true for the binding
energies (errors smaller than 0.7 eV). In case of LuF the theoretical results of 7.50 eV
(this work) and 7.55 eV (DKH-DFT) indicate, that the experimental value of 5.93 eV
is substantially too low.
The calculated 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, 0.4eV, 2cm-1 for EuS and 0.01 Å, 0.18eV, 26cm-1 for YbS.
7
The Mulliken population analysis of the ground state SCF wavefunction for the
monoxides shows a weak contribution of 4f orbitals to chemical bonding at the
beginning of the Lanthanide series compared to the end. For example, the charge
distributions
for
LaO
and
LuO
are
respectively:
La0.87+5p5.974f0.215d0.996s0.95O0.87-2s1.912p4.93;
Lu1.06+4f14.015d0.816s0.986p0.13O1.06-2s1.902p5.11. Therefore the 4f electrons should be
treated as valence electrons in accurate calculations. It was concluded from the
previous studies that larger errors would arise for the light lanthanides than for the
heavy lanthanides if the 4f electrons were put into the PP core [5].
Comparing with the previously used basis sets (12s10p8d8f)/[5s5p3d3f], the
BSSE produced much larger corrections of the final results for the old basis sets than
for the new ones. For example, the BSSE corrections for GdH obtained at the
CI(SD)+Q level (diffuse function were added, exponents s, 0.01, 0.005; p, 0.06, 0.02;
d, 0.30, 0.10; f, 0.1) from the new and old basis sets are respectively, equilibrium
distance (Å), 0.009, 0.052; binding energies (eV), 0.08, 0.35; vibrational frequencies
(cm-1), 12, 113. It is definitely concluded that the new basis sets are more reliable than
the old ones.
Conclusion
The molecular constants of selected diatomic Lanthanide compounds have been
calculated, by using relativistic small-core pseudopotentials and optimized
8
(14s13p10d8f6g)/ [6s6p5d4f3g] valence basis sets. The calculated results for LuH,
YbH, LaO, LuO, EuO, GdO, LaF, LuF EuF, GdF, YbF, EuS, YbS are in good
agreement with the experimental results. The BSSE corrections are much smaller than
those for the old basis sets, which proves the reliability of the new basis sets.
Therefore the derived valence basis sets provide a valuable tool for further studies of
lanthanide compounds.
The present work as well as previous all-electron density functional studies
indicates that the currently used estimate of the binding energy of LuF is too low and
should be corrected to ~7.5 eV. The complicated mixing of configurations in the
ground state of YbO is still a challenge for further more accurate investigations.
9
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