## Article

## Reference

### Quantum Monte Carlo study of the ground state and low-lying excited states of the scandium dimer

### MATXAIN, Jon M., et al.

**Abstract**

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 3Sigmau and 5Sigmau states are calculated to be close in energy in all cases, but whereas DFT predicts the 5Sigmau state to be the ground state by 0.08 eV, DMC and CASPT2 calculations predict the 3Sigmau 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 3Sigmau state, which breaks with the assumption of a 5Sigmau ground state for scandium dimer, believed throughout the past decades.

### MATXAIN, Jon M., *et al* . Quantum Monte Carlo study of the ground state and low-lying excited states of the scandium dimer. *Journal of Chemical Physics* , 2008, vol. 128, no. 19, p. 194315

### DOI : 10.1063/1.2920480

Available at:

http://archive-ouverte.unige.ch/unige:3185

Disclaimer: layout of this document may differ from the published version.

**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 Gagliardi^{3}

1*Kimika Fakultatea, Euskal Herriko Unibertsitatea, P.K. 1072, 20080 Donostia, Euskadi (Spain)*

2*Donostia International Physics Center, 20080 Donostia, Euskadi (Spain)*

3*University of Geneva, 30 Quai Ernest Ansermet, CH-1211 Geneva 4, Switzerland*

!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}!

*states are calculated to be close in energy in all cases, but whereas DFT predicts the*

_{u}^{5}!

*state to be the ground state by 0.08 eV, DMC and CASPT2 calculations predict the*

_{u}^{3}!

*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*

_{u}^{3}!

*state, which breaks with the assumption of a*

_{u}^{5}!

*ground state for scandium dimer, believed throughout the past decades. ©*

_{u}*2008 American Institute of Physics.*#DOI:10.1063/1.2920480$

**I. INTRODUCTION**

Scandium dimer is one of the least known first row tran-
sition 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 ex-
ample 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}Bad-
ger’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 dis-
crepancy. 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 de-
pending on the experimental method employed, ranging
from 1.13 eV using a thermodynamically determined third-
law value, to 0.79 eV obtained from a LeRoy–Berstein cal-
culation 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 Sc_{2} is based on the ESR measurements of

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 Sc_{2} they predicted a ^{5}!* _{u}*
ground state with 1$

_{g}^{2}1$

_{u}^{1}1%

_{u}^{2}2$

_{g}^{1}configuration. The

^{5}!

*state was also claimed to be the ground state by Walch*

_{u}*et*

*al.*

^{10}on the basis of some CASSCF/CI!SD"calculations. Ac- cording to their calculations, the 1$

_{g}^{2}1$

_{u}^{1}1%

_{u}^{2}2$

_{g}^{1}was indeed the dominant configuration of the

^{5}!

*state. This combina- tion of experimental and theoretical works seemed to set on firm grounds that the ground state of the scandium dimer was a*

_{u}^{5}!

*state, and not a*

_{u}^{5}&

*state as predicted by Busby*

_{g}*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}!

*state, which is in agreement with the removal of the electron from the 1$*

_{g}*orbital of the*

_{u}^{5}!

*ground state’s domi- nant configuration of Sc*

_{u}_{2}.

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 ap-
peared in the literature reporting such properties. Harris

*et*

*al.*

^{9}predicted a bond length of 2.70 Å and a harmonic fre- quency 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 dis- sociation energy, one must accurately reproduce the

^{2}

*D*

→^{4}*F* transition energy of the scandium atom, known to be
1.44 eV, since the ground state dissociates to the ^{2}*D+*^{4}*F*
asymptote. The dissociation energy calculated by Åkeby *et*
*al.*is accurate enough and agrees with the experimental val-

a"Electronic mail: jonmattin.matxain@ehu.es.

0021-9606/2008/128"19!/194315/5/$23.00 **128, 194315-1** © 2008 American Institute of Physics

ues of Moskovits^{6} 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 experimen-
tal value.

However, according to the CASSCF results, a^{3}'* _{u}* state
was found to be more stable than both the

^{5}!

*by 0.295 eV and a low-lying*

_{u}^{3}!

*excited state by 0.291 eV. The IC-ACPF method, on the other hand, predicted the*

_{u}^{5}!

*to be more stable than the*

_{u}^{3}'

*by 0.391 eV. The authors attributed this discrepancy to the fact that CASSCF does not correctly de- scribe the state ordering because of the inability to treat dy- namical electron correlation, which was better accounted for by the IC-ACPF method. Suzuki*

_{u}*et al.*

^{15}performed multiref- erence configuration interaction calculations of the

^{5}!

*state, predicting frequencies in very good agreement with experi-*

_{u}ment !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 ex-
perimental value of 6.56 eV. However, the dissociation en-
ergy of Sc_{2}, along with the ^{2}*D*−^{4}*F*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}!

*, with an electronic configuration of*

_{u}1$_{g}^{2}1$_{u}^{1}!↓"1%_{u}^{2}!↑"2$_{g}^{1}!↑", 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, be-
ing the^{3}!* _{u}*described earlier by Papai and Castro the lowest-
lying excited state only 0.18 eV higher than the

^{5}!

*state, and interestingly, with very similar properties:*

_{u}*R*

*=2.61 Å,*

_{e}#* _{e}*=256 cm

^{−1}for the triplet and

*R*

*=2.63 Å, #*

_{e}*=241 cm*

_{e}^{−1}for the quintet. In addition to these similarities, the electronic state resulting from the ionization of the 1$

*orbital of the*

_{u}5!* _{u}*and the

^{3}!

*states is the same, i.e., the*

_{u}^{4}!

*state experi- mentally found by Knight*

_{g}*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 experimen-
tal evidences pointing to the

^{2}

*D−*

^{4}

*F*dissociation asymptote and the formation of the

^{4}!

*ground state of Sc*

_{g}_{2}

^{+}by ejection of the electron from the 1$

*orbital of the Sc*

_{u}_{2}dominant configuration’s ground state.

In this work, we will examine the properties of both the

5!* _{u}*and the

^{3}!

*states of Sc*

_{u}_{2}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 ^{2}*D*and^{4}*F* states and the ionization energy!IE"

has also been calculated and confronted to accurately mea- sure experimental values. This constitutes a further check to the accuracy of the methods used throughout this study.

**II. METHODS**

Preliminary geometry optimizations and harmonic fre- quency calculations have been carried out at the B3LYP

!Refs. 24and25" level of theory. This functional was com-

bined 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 Wachters^{27} for the excited states,
one set of diffuse pure angular momentum*d* function, opti-
mized by Hay,^{28}and three sets of uncontracted pure angular
momentum *f* functions, including both tight and diffuse ex-
ponents, 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 ho-
mogeneous, isotropic electron-electron terms *u, isotropic*
electron-nucleus terms ( centered on the nuclei, and isotro-
pic 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 opti-
mized using variance minimization techniques.^{36,37} The
DMC method is one of the quantum Monte Carlo implemen-
tations 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. How-
ever, the stochastic evolution is never exact, and the solution
converges to the bosonic ground state.^{39}The fermionic sym-
metry 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

194315-2 Matxain*et al.* J. Chem. Phys.**128, 194315**"2008!

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 exci-
tations noniteratively.^{48} Finally, CASPT2 !Refs. 49–51" ap-
plies second-order perturbation theory on a multiconfigura-
tional 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 usingGAMESS-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 ^{2}*D-*^{4}*F* energy gap and the ionization energy of the scan-
dium atom, for which precise experimental data are avail-
able. This will serve as initial check to the accuracy of our
selected theoretical procedures.

**A. The**^{2}**D-**^{4}**F****energy gap and the IE**
**of the scandium atom**

TableI shows the relative energies between the lowest
two electronic states of scandium atom !^{2}*D-*^{4}*F*" and the IE

for the scandium atom calculated at the B3LYP, CCSDT, CASPT2, and DMC levels of theory along with their corre- sponding experimental data.

Experiments show the^{4}*F* state to be 1.43 eV higher in
energy than the ^{2}*D* 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 ^{4}*F* only 0.60 eV above the ^{2}*D* ground
state. This improvement is due to the use of a larger basis set.

Our calculated value of 0.94 eV for the ^{2}*D-*^{4}*F* energy gap
constitutes a substantial improvement toward a better match-
ing with the experimental mark. Remarkably, CCSD!T"cal-
culations overestimate the experimental^{2}*D-*^{4}*F*energy gap by

&0.17 eV. This failure can tentatively be ascribed to the
multiconfigurational character of states investigated. Indeed,
the T1 diagnostics^{56}of the^{2}*D*and^{4}*F*yield 0.026 and 0.019,
respectively. The critical estimated value for the multicon-
figurational character is 0.02. Multiconfiguration based
methods, such as CASPT2, remarkably perform better. Thus,
observe that the CASPT2 and DMC estimate for the scandi-

FIG. 1. Scheme of the molecular or- bitals included in the active space for the CASPT2 calculations.

TABLE I. The relative energy, in eV, between the lowest two electronic
states of scandium atom !^{2}*D-*^{4}*F"* and the IE, in eV, for scandium atom
calculated at the B3LYP, CASPT2, and DMC levels of theory.

2*D-*^{4}*F* IE

B3LYP 0.94 6.56

CCSDT 1.60 6.34

CASPT2 1.39 6.47

DMC 1.52"0.01 6.44"0.01

Expt. 1.43!Ref.55" 6.56!Ref.57"

um’s ^{2}*D-*^{4}*F* energy gap are 1.39 eV and 1.52 eV, only
0.04 eV and 0.09 eV from the experimental mark, respec-
tively. 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 scan- dium 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 fre- quencies 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 vi-
brational frequency corresponds to the

^{1}!

*state, with a value of 180.6 cm*

_{g}^{−1}, while the largest value of 320.2 cm

^{−1}corre- sponds to the

^{3}!

*state. Focusing on&E’s, DMC predicts the*

_{g}3!* _{u}* state and not the accepted

^{5}!

*state to be the ground state, which is predicted to lie 0.17 eV higher in energy.*

_{u}B3LYP predicts the quintet to be the ground state, in accor-
dance 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 *D** _{e}*=0.79–1.13 eV.

^{7}For bond lengths, direct measurements are not available, and the indirect ex- trapolation pointed out a bond length of 2.5 Å. The previ- ously assumed ground state, the

^{5}!

*state, is calculated to have a vibrational frequency of 259.9 cm*

_{u}^{−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}!

*candidate also fulfills these requirements, being the fre- quency 273.3 cm*

_{u}^{−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}!

*. CCSD!T"calculations have been car- ried out with the T1 diagnostic*

_{u}^{55}to check if a multiconfigu- rational treatment is necessary. The energy difference be- tween these two states is decreased to 0.03 eV at this level of theory, but for both states, T1&0.04. which mean that mul- ticonfigurational calculations are required for the proper de- scription of these states. Therefore, CASPT2 geometry opti- mizations 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}!

*state is more stable by 0.16 eV. According to these results, we predict the*

_{u}^{3}!

*state to be the ground state of scandium dimer, and not the*

_{u}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 meth-
ods such as DMC and CASPT2 predict a

^{3}!

*to be more stable than the*

_{u}^{5}!

*by around 0.15 eV. This triplet state is an*

_{u}TABLE II. Equilibrium distances*R** _{e}*, in A

*ˆ*, 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*D** _{e}*are given, in eV.

*R*

*and#*

_{e}*are calculated at the B3LYP level*

_{e}and&Eand*D** _{e}*at the DMC level of theory.

State Dominant Configuration *R** _{e}* #

*&E*

_{e}*D*

_{e}7&* _{u}* $!4s"

_{g}^{1}!↑"$!3d"

_{g}^{1}!↑"%

_{u}^{2}!↑")

_{g}^{1}!↑"$!4s"

_{u}^{*}

^{,1}!↑" 2.447 255.3 1.57"0.01 ¯

7!* _{g}* $!4s"

_{g}^{1}!↑"$!3d"

_{g}^{1}!↑"%

_{u}^{2}!↑")

_{g}^{2}!↑" 2.231 314.4 2.46"0.01 ¯

5!* _{u}* $!4s"

_{g}^{2}!↑↓"$!3d"

_{g}^{1}!↑"%

_{u}^{2}!↑"$!4s"

_{u}^{*}

^{,1}!↑" 2.582 259.9 0.17"0.01 0.93

5&* _{g}* $!4s"

_{g}^{2}!↑↓"$!3d"

_{g}^{1}!↑"%

_{u}^{2}!↑")

_{g}^{1}!↑" 2.363 280.5 0.86"0.01 ¯

5!* _{g}* $!4s"

_{g}^{2}!↑↓"%

_{u}^{2}!↑")

_{g}^{2}!↑" 2.466 239.6 1.56"0.01 ¯

5&* _{u}* $!4s"

_{g}^{1}!↑"$!3d"

_{g}^{1}!↑"%

_{u}^{2}!↑")

_{g}^{1}!↑"$!4s"

_{u}^{*}

^{,1}!↓" 2.451 268.7 1.94"0.01 ¯

3!* _{u}* $!4s"

_{g}^{2}!↑↓"$!3d"

_{g}^{1}!↑"%

_{u}^{2}!↑"$!4s"

_{u}^{*}

^{,1}!↓" 2.570 273.3 0.00"0.01 1.10

3!* _{g}* $!4s"

_{g}^{2}!↑↓"$!3d"

_{g}^{2}!↑↓"%

_{u}^{2}!↑" 2.356 320.2 2.55"0.01 ¯

1!* _{g}* $

_{g}^{2}!4s"$

_{g}^{2}!3d"$!4s"

_{u}^{*}

^{,2}3.163 180.6 0.49"0.01 ¯

1'* _{g}* $

_{g}^{2}!4s"%

_{u}^{3}!↑"$!4s"

_{u}^{*}

^{,1}!↓" 2.551 266.1 0.77"0.01 ¯

194315-4 Matxain*et al.* J. Chem. Phys.**128, 194315**"2008!

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$

*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*

_{u}^{3}!

*state, and not the*

_{u}^{5}!

*state as was thought. New experiments would be interesting in order to confirm our prediction.*

_{u}**ACKNOWLEDGMENTS**

This research was funded by Euskal Herriko Unibertsi- tatea!The University of the Basque Country", Eusko Jaurlar-

itza!the Basque Government", and the Ministerio de Educa-

cion y Ciencia. L.G. thanks the Swiss National Science Foundation !Grant No. 20021-111645/1". The SGI/IZO- SGIker UPV/EHU!supported by Fondo Social Europeo and MCyT" is gratefully acknowledged for generous allocation of computational resources.

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FIG. 2. Potential energy curves for the^{5}!* _{u}*!top curve"and the

^{3}!

*!bottom curve"electronic states of Sc*

_{u}_{2}. 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.