THE JOURNAL OF CHEMICAL PHYSICS 128, 194315 !2008"
Quantum Monte Carlo study of the ground state and low-lying excited
states of the scandium dimer
Jon M. Matxain,1,2,a! Elixabete Rezabal,1,2 Xabier Lopez,1,2 Jesus M. Ugalde,1,2 and
Laura Gagliardi3
1
Kimika Fakultatea, Euskal Herriko Unibertsitatea, P.K. 1072, 20080 Donostia, Euskadi (Spain)
Donostia International Physics Center, 20080 Donostia, Euskadi (Spain)
3
University of Geneva, 30 Quai Ernest Ansermet, CH-1211 Geneva 4, Switzerland
2
!Received 17 March 2008; accepted 15 April 2008; published online 22 May 2008"
A large set of electronic states of scandium dimer has been calculated using high-level theoretical
methods such as quantum diffusion Monte Carlo !DMC", complete active space perturbation theory
as implemented in GAMESS-US, coupled-cluster singles, doubles, and triples, and density functional
theory !DFT". The 3!u and 5!u states are calculated to be close in energy in all cases, but whereas
DFT predicts the 5!u state to be the ground state by 0.08 eV, DMC and CASPT2 calculations
predict the 3!u to be more stable by 0.17 and 0.16 eV, respectively. The experimental data available
are in agreement with the calculated frequencies and dissociation energies of both states, and
therefore we conclude that the correct ground state of scandium dimer is the 3!u state, which breaks
with the assumption of a 5!u ground state for scandium dimer, believed throughout the past
decades. © 2008 American Institute of Physics. #DOI: 10.1063/1.2920480$
I. INTRODUCTION
Scandium dimer is one of the least known first row transition metal dimers in spite of the numerous experimental
and theoretical works carried out in order to determine the
ground state and characterize its properties.1,2 A clear example of this is the absence of a sound experimental value
for the bond length, although an empirical estimation was
derived from Badger’s rule by Weisshaar, 2.29 Å, nearly two
decades ago. Unfortunately, as discussed by the author,3 Badger’s rule does not work accurately for transition metals, and
the error can be as large as "0.35 Å.
The dissociation energy has also been a matter of discrepancy. It was first evaluated by Verhaegen et al.4 in 1964,
using mass spectrometry techniques, %1.30 eV. This value
was then revised by Gingerich,5 who set it at 1.65 eV. A few
years later, in 1984, based on resonance Raman experiments,
Moskovits et al.6 measured a dissociation energy of
1.1" 0.2 eV. Finally, in 1989, Haslett et al.7 reevaluated the
dissociation energy for a number transition metal dimers,
including scandium. They obtained very different results depending on the experimental method employed, ranging
from 1.13 eV using a thermodynamically determined thirdlaw value, to 0.79 eV obtained from a LeRoy–Berstein calculation on resonance Raman data. These authors claimed
that 0.79 eV should be understood as a lower bound instead
of an accurate value. According to the experimental data
available, it seems reasonable to assume that the dissociation
energy ranges from 0.80 to 1.15 eV.
The vibrational frequency, however, is well accepted. It
was measured by Moskovits et al.,6 #e! = 238.91 cm−1.
The only experimental assignment of the electronic
ground state of Sc2 is based on the ESR measurements of
a"
Electronic mail: [email protected].
0021-9606/2008/128"19!/194315/5/$23.00
Knight et al.,8 who proposed a 5! ground state. However,
this proposal did not come from direct observation, but was
made in such a way to obtain agreement with the prediction
of a previous theoretical calculations by Harris et al.9 They
performed local spin density calculations on the first row
transition metal dimers, and for Sc2 they predicted a 5!u
ground state with 1$2g1$1u1%2u2$1g configuration. The 5!u
state was also claimed to be the ground state by Walch et
al.10 on the basis of some CASSCF/CI!SD" calculations. According to their calculations, the 1$2g1$1u1%2u2$1g was indeed
the dominant configuration of the 5!u state. This combination of experimental and theoretical works seemed to set on
firm grounds that the ground state of the scandium dimer was
a 5!u state, and not a 5&g state as predicted by Busby et al.11
In a related work, Knight et al.12 experimentally determined
that the ground electronic state of the dimer cation, Sc+2 , was
the 4!g state, which is in agreement with the removal of the
electron from the 1$u orbital of the 5!u ground state’s dominant configuration of Sc2.
Therefore, it became widely accepted that the ground
state of scandium dimer was a 5!u state. However, a number
of its properties, such as the bond length and dissociation
energy, along with the nature of the low-lying excited states
remained unclear. A handful of theoretical works have appeared in the literature reporting such properties. Harris et
al.9 predicted a bond length of 2.70 Å and a harmonic frequency of 200 cm−1. Åkeby et al. using high-level CASSCF
and IC-ACPF calculations,13,14 predicted the bond length to
be in the range of 2.5– 2.8 Å, and a dissociation energy of
1.04 eV. Recall that in order to accurately estimate the dissociation energy, one must accurately reproduce the 2D
→ 4F transition energy of the scandium atom, known to be
1.44 eV, since the ground state dissociates to the 2D + 4F
asymptote. The dissociation energy calculated by Åkeby et
al. is accurate enough and agrees with the experimental val-
128, 194315-1
© 2008 American Institute of Physics
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
194315-2
J. Chem. Phys. 128, 194315 "2008!
Matxain et al.
ues of Moskovits6 and Haslett.7 The calculated frequencies
of 261 cm−1 at the CASSCF level and 286 cm−1 at IC-ACPF
level are in reasonably good agreement with the experimental value.
However, according to the CASSCF results, a 3'u state
was found to be more stable than both the 5!u by 0.295 eV
and a low-lying 3!u excited state by 0.291 eV. The IC-ACPF
method, on the other hand, predicted the 5!u to be more
stable than the 3'u by 0.391 eV. The authors attributed this
discrepancy to the fact that CASSCF does not correctly describe the state ordering because of the inability to treat dynamical electron correlation, which was better accounted for
by the IC-ACPF method. Suzuki et al.15 performed multireference configuration interaction calculations of the 5!u state,
predicting frequencies in very good agreement with experiment !230 versus 239 cm−1" and a bond length of 2.8 Å in
agreement with that of Åkeby et al. However, they severely
underestimated the dissociation energy, predicting a value of
0.6 eV.
Several approximate density functionals have also been
used to evaluate the properties of the scandium dimer.16–21 In
general, these studies predicted bond lengths in the range of
2.55– 2.70 Å and frequencies that are in good agreement
with the corresponding experimental marks. The calculated
ionization energy of Sc was also in agreement with the experimental value of 6.56 eV. However, the dissociation energy of Sc2, along with the 2D − 4F energy difference of the
scandium atom, was in all cases badly underestimated. In
those cases where different electronic states were studied, the
predicted ground state was the 5!u state. However, it is
worth noting that Papai and Castro,16 using B3LYP, found an
open shell 3!u, with an electronic configuration of
1$2g1$1u!↓"1%2u!↑"2$1g!↑", only 0.22 eV higher in energy.
Similarly, Gutsev et al.,19 using BPW91, found at least three
states of the scandium dimer thermodynamically stable, being the 3!u described earlier by Papai and Castro the lowestlying excited state only 0.18 eV higher than the 5!u state,
and interestingly, with very similar properties: Re = 2.61 Å,
#e = 256 cm−1 for the triplet and Re = 2.63 Å, #e = 241 cm−1
for the quintet. In addition to these similarities, the electronic
state resulting from the ionization of the 1$u orbital of the
5
!u and the 3!u states is the same, i.e., the 4!g state experimentally found by Knight et al.12 for the dimer cation.
The 3!u triplet state was not considered in the high-level
multiconfigurational calculations performed by Åkeby et
al.,13,14 and therefore the only information available on this
state is the one provided by the DFT calculations. Given the
very similar physical properties of this state with respect to
those of the 5!u assumed ground state and the small energy
difference between the two states, it seems worthy to explore
further this triplet state at higher levels of theory. Recall that
3
!u state is also compatible with the established experimental evidences pointing to the 2D − 4F dissociation asymptote
and the formation of the 4!g ground state of Sc+2 by ejection
of the electron from the 1$u orbital of the Sc2 dominant
configuration’s ground state.
In this work, we will examine the properties of both the
5
!u and the 3!u states of Sc2 and a selected number of other
singlet, triplet, quintet, and septet states with the diffusion
quantum Monte Carlo method,22 which is known to be very
efficient and reliable to recover substantial large portions of
the electron correlation.23
Additionally, some properties of the scandium atom,
such as the first excitation energy, i.e., the energy difference
between the 2D and 4F states and the ionization energy !IE"
has also been calculated and confronted to accurately measure experimental values. This constitutes a further check to
the accuracy of the methods used throughout this study.
II. METHODS
Preliminary geometry optimizations and harmonic frequency calculations have been carried out at the B3LYP
!Refs. 24 and 25" level of theory. This functional was combined with the triple zeta quality basis set, given by Schäfer
et al.26 supplemented with one diffuse s function, two sets of
p functions optimized by Wachters27 for the excited states,
one set of diffuse pure angular momentum d function, optimized by Hay,28 and three sets of uncontracted pure angular
momentum f functions, including both tight and diffuse exponents, as recommended by Ragavachari and Trucks.29 This
basis set is denoted as TZVP+ G!3df , 2p". This basis set has
been claimed to be accurate for transition metals.30
Quantum diffusion Monte Carlo !DMC" calculations
were performed at the previously optimized geometries. For
the DMC calculations, the trial wave functions used in this
work are written as a product of a Slater determinant and a
recently developed Jastrow factor,31 which is the sum of homogeneous, isotropic electron-electron terms u, isotropic
electron-nucleus terms ( centered on the nuclei, and isotropic electron-electron-nucleus terms f, also centered on the
nuclei. The determinantal part was calculated at the UHF
level of theory, combined with the relativistic Stuttgart
pseudopotentials and basis sets !ECP10MDF",32,33 motivated
by their earlier successful performance in DMC calculations
in a number of transition metal containing systems.34,35 The
Jastrow factor, containing up to 51 parameters, was optimized using variance minimization techniques.36,37 The
DMC method is one of the quantum Monte Carlo implementations that, together with the variational Monte Carlo
method, is becoming widely used nowadays. Briefly, in
DMC,22,38 the imaginary-time Schrödinger equation is used
to evolve an ensemble of electronic configurations toward
the ground state. Exact imaginary-time evolution would lead
to the exact fermion ground-state wave function, provided it
has a nonzero overlap with the initial fermion state. However, the stochastic evolution is never exact, and the solution
converges to the bosonic ground state.39 The fermionic symmetry is maintained by the fixed-node approximation, in
which the nodal surface of the wave function is constrained
to be equal that of a guiding wave function. The fixed-node
DMC energy provides a variational upper bound on the
ground-state energy with an error that is second order in the
error in the nodal surface.40,41
CCSD!T" and CASPT2 calculations were carried out for
the scandium’s ground and low-lying excited states and the
lowest-lying two states of its dimer. For both the CCSD!T"
and CASPT2 calculations, the TZVP-G!3df , 2p" basis set
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194315-3
J. Chem. Phys. 128, 194315 "2008!
States of the scandium dimer
FIG. 1. Scheme of the molecular orbitals included in the active space for
the CASPT2 calculations.
described above were used. CCSD!T" is a coupled-cluster
method,42,43 which uses single and double excitations from
the Hartree–Fock determinant,44–47 and includes triple excitations noniteratively.48 Finally, CASPT2 !Refs. 49–51" applies second-order perturbation theory on a multiconfigurational wave function generated with the orbitals of the active
space !CASSCF". For the scandium dimer calculations, the
active space was chosen to be that shown in Fig. 1, which
makes a total of six electrons in 18 molecular orbitals. The
effect of the highest six orbitals has been observed to be
negligible, and therefore a final active space of six electrons
in 12 orbitals was chosen.
All the B3LYP, UHF, and CCSD!T" calculations were
carried out using the GAUSSIAN03 !Ref. 52" package, the
DMC calculations with the CASINO !Ref. 53" program and,
CASPT2 calculations were carried out using GAMESS-US.54
III. RESULTS
Before turning our attention to the characterization of the
electronic structure and physical properties of the ground and
low-lying states of the scandium dimer, we wish to estimate
the 2D- 4F energy gap and the ionization energy of the scandium atom, for which precise experimental data are available. This will serve as initial check to the accuracy of our
selected theoretical procedures.
A. The 2D- 4F energy gap and the IE
of the scandium atom
Table I shows the relative energies between the lowest
two electronic states of scandium atom ! 2D- 4F" and the IE
for the scandium atom calculated at the B3LYP, CCSDT,
CASPT2, and DMC levels of theory along with their corresponding experimental data.
Experiments show the 4F state to be 1.43 eV higher in
energy than the 2D state.55 As one may observe, B3LYP
clearly yields incorrect results; it underestimates &E by
about 0.5 eV. However, the B3LYP results presented here
substantially improve earlier ones by Papai and Castro,16
who calculated the 4F only 0.60 eV above the 2D ground
state. This improvement is due to the use of a larger basis set.
Our calculated value of 0.94 eV for the 2D- 4F energy gap
constitutes a substantial improvement toward a better matching with the experimental mark. Remarkably, CCSD!T" calculations overestimate the experimental 2D- 4F energy gap by
&0.17 eV. This failure can tentatively be ascribed to the
multiconfigurational character of states investigated. Indeed,
the T1 diagnostics56 of the 2D and 4F yield 0.026 and 0.019,
respectively. The critical estimated value for the multiconfigurational character is 0.02. Multiconfiguration based
methods, such as CASPT2, remarkably perform better. Thus,
observe that the CASPT2 and DMC estimate for the scandiTABLE I. The relative energy, in eV, between the lowest two electronic
states of scandium atom ! 2D- 4F" and the IE, in eV, for scandium atom
calculated at the B3LYP, CASPT2, and DMC levels of theory.
2
B3LYP
CCSDT
CASPT2
DMC
Expt.
D- 4F
0.94
1.60
1.39
1.52" 0.01
1.43 !Ref. 55"
IE
6.56
6.34
6.47
6.44" 0.01
6.56 !Ref. 57"
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194315-4
J. Chem. Phys. 128, 194315 "2008!
Matxain et al.
TABLE II. Equilibrium distances Re, in Â, harmonic vibrational frequencies #e, in cm−1, and relative energies
&E, in eV, with respect to the ground state, of ten low-lying electronic states of the scandium dimer, in eV. For
the lowest-lying two state dissociation energies De are given, in eV. Re and #e are calculated at the B3LYP level
and &E and De at the DMC level of theory.
State
Dominant Configuration
Re
#e
&E
De
2.447
2.231
255.3
314.4
1.57" 0.01
2.46" 0.01
¯
¯
,1
2.582
2.363
2.466
2.451
259.9
280.5
239.6
268.7
0.17" 0.01
0.86" 0.01
1.56" 0.01
1.94" 0.01
0.93
¯
¯
¯
,1
2.570
2.356
273.3
320.2
0.00" 0.01
2.55" 0.01
1.10
¯
3.163
2.551
180.6
266.1
0.49" 0.01
0.77" 0.01
¯
¯
&u
!g
,1
$!4s"1g!↑"$!3d"1g!↑"%2u!↑")1g!↑"$!4s"*u !↑"
1
1
2
2
$!4s"g!↑"$!3d"g!↑"%u!↑")g!↑"
!u
&g
5
!g
5
&u
$!4s"2g!↑ ↓ "$!3d"1g!↑"%2u!↑"$!4s"*u !↑"
$!4s"2g!↑ ↓ "$!3d"1g!↑"%2u!↑")1g!↑"
$!4s"2g!↑ ↓ "%2u!↑")2g!↑"
,1
$!4s"1g!↑"$!3d"1g!↑"%2u!↑")1g!↑"$!4s"*u !↓"
3
!u
!g
$!4s"2g!↑ ↓ "$!3d"1g!↑"%2u!↑"$!4s"*u !↓"
$!4s"2g!↑ ↓ "$!3d"2g!↑ ↓ "%2u!↑"
!g
'g
$2g!4s"$2g!3d"$!4s"*u
,1
$2g!4s"%3u!↑"$!4s"*u !↓"
7
7
5
5
3
1
1
,2
um’s 2D- 4F energy gap are 1.39 eV and 1.52 eV, only
0.04 eV and 0.09 eV from the experimental mark, respectively. The experimental value for the IE was determined to
be 6.56 eV.57 B3LYP exactly predicts this value, but this
success must be ascribed to a lucky cancellation of errors
rather than to a high accuracy. CCSD!T", on the other hand,
underestimates the IE by 0.22 eV, predicting a value of
6.34 eV. DMC and CASPT2 behave much better, with errors
of about 0.1 eV.
Taking into account these results, we can conclude that
DMC results are as confident as CASPT2 results and both
accurately predict the calculated properties of scandium
atom.
B. Ground state and excited states
of scandium dimer
We have characterized ten electron states of the scandium dimer. Namely, two septets, four quintets, three triplets,
and four singlets. Table II shows the dominant electronic
configuration of the considered states.
All these states have been optimized and harmonic frequencies calculated at the B3LYP level of theory, and then
single point calculations performed using DMC method. The
obtained bond lengths, frequencies, relative energies, and
dissociation energies of the two lowest-lying states are given
in Table II. Observe that bond length and frequency values
may significantly differ depending on the electronic state.
Thus, bond-length values range from a minimum value of
2.231 Å to a maximum value of 3.163 Å, for the 7!g and
1
!g states, respectively. Similarly, the smallest calculated vibrational frequency corresponds to the 1!g state, with a value
of 180.6 cm−1, while the largest value of 320.2 cm−1 corresponds to the 3!g state. Focusing on &E’s, DMC predicts the
3
!u state and not the accepted 5!u state to be the ground
state, which is predicted to lie 0.17 eV higher in energy.
B3LYP predicts the quintet to be the ground state, in accordance with previous DFT calculations,16–20,58 only by
0.07 eV, however, as was observed for the atomic case,
DMC results are more trustable than B3LYP for this case.
The next excited states appear to be the calculated singlets,
followed by the rest of quintets, septets, and triplets.
The predicted ground state’s properties must correctly
reproduce the well-established experimental values, which
are a vibrational frequency of # = 238.9 cm−1 !Ref. 6" and a
dissociation energy De = 0.79– 1.13 eV.7 For bond lengths,
direct measurements are not available, and the indirect extrapolation pointed out a bond length of 2.5 Å. The previously assumed ground state, the 5!u state, is calculated to
have a vibrational frequency of 259.9 cm−1, a dissociation
energy of 0.93 eV, and a bond length of 2.58 Å, which are in
agreement with the experimental values. However, the 3!u
candidate also fulfills these requirements, being the frequency 273.3 cm−1, the dissociation energy 1.1 eV, and a
bond length of 2.57 Å.
In order to further discriminate between both states and
check the DMC results, CCSD!T" and CASPT2 calculations
have been carried out on the two states under consideration,
namely, 3!u and 5!u. CCSD!T" calculations have been carried out with the T1 diagnostic55 to check if a multiconfigurational treatment is necessary. The energy difference between these two states is decreased to 0.03 eV at this level of
theory, but for both states, T1 & 0.04. which mean that multiconfigurational calculations are required for the proper description of these states. Therefore, CASPT2 geometry optimizations have been carried out for both states, and the
predicted equilibrium structures are very similar to the
B3LYP ones, as one can note in Fig. 2. The CASPT2 results
confirm the DMC prediction that the 3!u state is more stable
by 0.16 eV. According to these results, we predict the 3!u
state to be the ground state of scandium dimer, and not the
5
!u, as was previously believed.
IV. CONCLUSIONS
A large set of different electronic states of scandium
dimer has been studied using B3LYP, DMC, CCSD!T", and
CASPT2. B3LYP predicted a 5!u state as the ground state, in
agreement with previous studies. However, high-level methods such as DMC and CASPT2 predict a 3!u to be more
stable than the 5!u by around 0.15 eV. This triplet state is an
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194315-5
J. Chem. Phys. 128, 194315 "2008!
States of the scandium dimer
17
FIG. 2. Potential energy curves for the 5!u !top curve" and the 3!u !bottom
curve" electronic states of Sc2. Internuclear distances are in Å and energies
in hartree, at the CASPT2 level of theory. The zero energy has been set at
−1519.475 hartree.
open shell triplet, with very similar bond length, frequency,
and dissociation energy to the 5!u state. Both states have the
same orbital occupancy, the only difference is that while in
the quintuplet the $u orbital has an electron with alpha spin,
in the triplet this electron has a beta spin. Our calculations
show that the ground state of scandium dimer is the 3!u
state, and not the 5!u state as was thought. New experiments
would be interesting in order to confirm our prediction.
ACKNOWLEDGMENTS
This research was funded by Euskal Herriko Unibertsitatea !The University of the Basque Country", Eusko Jaurlaritza !the Basque Government", and the Ministerio de Educacion y Ciencia. L.G. thanks the Swiss National Science
Foundation !Grant No. 20021-111645/1". The SGI/IZOSGIker UPV/EHU !supported by Fondo Social Europeo and
MCyT" is gratefully acknowledged for generous allocation
of computational resources.
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