Quantum Monte Carlo study of the ground state and lowlying excitedstates of the scandium dimer
Jon M. Matxain,
1,2,a
Elixabete Rezabal,
1,2
Xabier Lopez,
1,2
Jesus M. Ugalde,
1,2
andLaura 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, CH1211 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 highlevel theoreticalmethods such as quantum diffusion Monte Carlo
DMC
, complete active space perturbation theoryas implemented in
GAMESSUS
, coupledcluster singles, doubles, and triples, and density functionaltheory
DFT
. The
3
u
and
5
u
states are calculated to be close in energy in all cases, but whereasDFT predicts the
5
u
state to be the ground state by 0.08 eV, DMC and CASPT2 calculationspredict the
3
u
to be more stable by 0.17 and 0.16 eV, respectively. The experimental data availableare in agreement with the calculated frequencies and dissociation energies of both states, andtherefore we conclude that the correct ground state of scandium dimer is the
3
u
state, which breakswith the assumption of a
5
u
ground state for scandium dimer, believed throughout the pastdecades. ©
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 experimentaland theoretical works carried out in order to determine theground state and characterize its properties.
1,2
A clear example of this is the absence of a sound experimental valuefor the bond length, although an empirical estimation wasderived from Badger’s rule by Weisshaar, 2.29 Å, nearly twodecades ago. Unfortunately, as discussed by the author,
3
Badger’s rule does not work accurately for transition metals, andthe 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 valuewas then revised by Gingerich,
5
who set it at 1.65 eV. A fewyears 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 thedissociation energy for a number transition metal dimers,including scandium. They obtained very different results depending on the experimental method employed, rangingfrom 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 claimedthat 0.79 eV should be understood as a lower bound insteadof an accurate value. According to the experimental dataavailable, it seems reasonable to assume that the dissociationenergy ranges from 0.80 to 1.15 eV.The vibrational frequency, however, is well accepted. Itwas measured by Moskovits
et al.
,
6
e
=238.91 cm
−
1
.The only experimental assignment of the electronicground 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 wasmade in such a way to obtain agreement with the predictionof a previous theoretical calculations by Harris
et al.
9
Theyperformed local spin density calculations on the ﬁrst rowtransition 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
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
g
2
1
u
1
1
u
2
2
g
1
was indeedthe dominant conﬁguration of the
5
u
state. This combination of experimental and theoretical works seemed to set onﬁrm grounds that the ground state of the scandium dimer wasa
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 determinedthat the ground electronic state of the dimer cation, Sc
2+
, wasthe
4
g
state, which is in agreement with the removal of theelectron from the 1
u
orbital of the
5
u
ground state’s dominant conﬁguration of Sc
2
.Therefore, it became widely accepted that the groundstate of scandium dimer was a
5
u
state. However, a numberof its properties, such as the bond length and dissociationenergy, along with the nature of the lowlying excited statesremained 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 highlevel CASSCFand ICACPF calculations,
13,14
predicted the bond length tobe 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
2
D
→
4
F
transition energy of the scandium atom, known to be1.44 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.
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2008
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128
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ues of Moskovits
6
and Haslett.
7
The calculated frequenciesof 261 cm
−
1
at the CASSCF level and 286 cm
−
1
at ICACPFlevel are in reasonably good agreement with the experimental value.However, according to the CASSCF results, a
3
u
statewas found to be more stable than both the
5
u
by 0.295 eVand a lowlying
3
u
excited state by 0.291 eV. The ICACPFmethod, on the other hand, predicted the
5
u
to be morestable than the
3
u
by 0.391 eV. The authors attributed thisdiscrepancy to the fact that CASSCF does not correctly describe the state ordering because of the inability to treat dynamical electron correlation, which wasbetter accounted forby the ICACPF 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 inagreement with that of Åkeby
et al.
However, they severelyunderestimated the dissociation energy, predicting a value of 0.6 eV.Several approximate density functionals have also beenused to evaluate the properties of the scandium dimer.
16–21
Ingeneral, these studies predicted bond lengths in the range of 2.55–2.70 Å and frequencies that are in good agreementwith the corresponding experimental marks. The calculatedionization energy of Sc was also in agreement with the experimental value of 6.56 eV. However, the dissociation energy of Sc
2
, along with the
2
D
−
4
F
energy difference of thescandium atom, was in all cases badly underestimated. Inthose cases where different electronic states were studied, thepredicted ground state was the
5
u
state. However, it isworth noting that Papai and Castro,
16
using B3LYP, found anopen shell
3
u
, with an electronic conﬁguration of 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 threestates 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:
R
e
=2.61 Å,
e
=256 cm
−
1
for the triplet and
R
e
=2.63 Å,
e
=241 cm
−
1
for the quintet. In addition to these similarities, the electronicstate 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 highlevelmulticonﬁgurational calculations performed by Åkeby
et al.
,
13,14
and therefore the only information available on thisstate is the one provided by the DFT calculations. Given thevery similar physical properties of this state with respect tothose of the
5
u
assumed
ground state and the small energydifference between the two states, it seems worthy to explorefurther this triplet state at higher levels of theory. Recall that
3
u
state is also compatible with the established experimental evidences pointing to the
2
D
−
4
F
dissociation asymptoteand the formation of the
4
g
ground state of Sc
2+
by ejectionof the electron from the 1
u
orbital of the Sc
2
dominantconﬁguration’s ground state.In this work, we will examine the properties of both the
5
u
and the
3
u
states of Sc
2
and a selected number of othersinglet, triplet, quintet, and septet states with the diffusionquantum Monte Carlo method,
22
which is known to be veryefﬁcient and reliable torecover 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 differencebetween the
2
D
and
4
F
states and the ionization energy
IE
has also been calculated and confronted to accurately measure experimental values. This constitutes a further check tothe 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.24and25
level of theory. This functional was combinedwith 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 diff use pure angular momentum
d
function, optimized by Hay,
28
and three sets of uncontracted pure angularmomentum
f
functions, including both tight and diff use exponents, as recommended by Ragavachari and Trucks.
29
Thisbasis set is denoted as TZVP+G
3
df
,2
p
. This basis set hasbeen claimed to be accurate for transition metals.
30
Quantum diffusion Monte Carlo
DMC
calculationswere performed at the previously optimized geometries. Forthe DMC calculations, the trial wave functions used in thiswork are written as a product of a Slater determinant and arecently developed Jastrow factor,
31
which is the sum of homogeneous, isotropic electronelectron terms
u
, isotropicelectronnucleus terms
centered on the nuclei, and isotropic electronelectronnucleus terms
f
, also centered on thenuclei. The determinantal part was calculated at the UHFlevel of theory, combined with the relativistic Stuttgartpseudopotentials and basis sets
ECP10MDF
,
32,33
motivatedby their earlier successful performance in DMC calculationsin a number of transition metal containing systems.
34,35
TheJastrow factor, containing up to 51 parameters,wasoptimized using variance minimization techniques.
36,37
TheDMC method is one of the quantum Monte Carlo implementations that, together with the variational Monte Carlomethod, is becoming widely used nowadays. Brieﬂy, inDMC,
22,38
the imaginarytime Schrödinger equation is usedto evolve an ensemble of electronic conﬁgurations towardthe ground state. Exact imaginarytime evolution would leadto the exact fermion groundstate wave function, provided ithas a nonzero overlap with the initial fermion state. However, the stochastic evolution is neverexact, and the solutionconverges to the bosonic ground state.
39
The fermionic symmetry is maintained by the ﬁxednode approximation, inwhich the nodal surface of the wave function is constrainedto be equal that of a guiding wave function. The ﬁxednodeDMC energy provides a variational upper bound on thegroundstate energy withan error that is second order in theerror in the nodal surface.
40,41
CCSD
T
and CASPT2 calculations were carried out forthe scandium’s ground and lowlying excited states and thelowestlying two states of its dimer. For both the CCSD
T
and CASPT2 calculations, the TZVPG
3
df
,2
p
basis set
1943152 Matxain
et al.
J. Chem. Phys.
128
, 194315
2008
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
described above were used. CCSD
T
is a coupledclustermethod,
42,43
which uses single and double excitations fromthe Hartree–Fock determinant,
44–47
and includes triple excitations noniteratively.
48
Finally, CASPT2
Refs.49–51
applies secondorder perturbation theory on a multiconﬁgurational wave function generated with the orbitals of the activespace
CASSCF
. For the scandium dimer calculations, theactive space was chosen to be that shown in Fig.1, whichmakes a total of six electrons in 18 molecular orbitals. Theeffect of the highest six orbitals has been observed to benegligible, and therefore a ﬁnal active space of six electronsin 12 orbitals was chosen.All the B3LYP, UHF, and CCSD
T
calculations werecarried out using the
GAUSSIAN03
Ref.52
package, theDMC calculations with the
CASINO
Ref.53
program and,CASPT2 calculations were carried out using
GAMESSUS
.
54
III. RESULTS
Before turning our attention to the characterization of theelectronic structure and physical properties of the ground andlowlying states of the scandium dimer, we wish to estimatethe
2
D

4
F
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 ourselected theoretical procedures.
A. The
2
D

4
F
energy gap and the IEof the scandium atom
TableIshows the relative energies between the lowesttwo electronic states of scandium atom
2
D

4
F
and the IEfor the scandium atom calculated at the B3LYP, CCSDT,CASPT2, and DMC levels of theory along with their corresponding experimental data.Experiments show the
4
F
state to be 1.43 eV higher inenergy than the
2
D
state.
55
As one may observe, B3LYPclearly yields incorrect results; it underestimates
E
byabout 0.5 eV. However, the B3LYP results presented heresubstantially improve earlier ones by Papai and Castro,
16
who calculated the
4
F
only 0.60 eV above the
2
D
groundstate. 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 gapconstitutes a substantial improvement toward a better matching with the experimental mark. Remarkably, CCSD
T
calculations overestimate the experimental
2
D

4
F
energy gap by
0.17 eV. This failure can tentatively be ascribed to themulticonﬁgurational 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 multiconﬁgurational character is 0.02. Multiconﬁguration basedmethods, such as CASPT2, remarkably perform better. Thus,observe that the CASPT2 and DMC estimate for the scandi
FIG. 1. Scheme of the molecular orbitals included in the active space forthe CASPT2 calculations.TABLE I. The relative energy, in eV, between the lowest two electronicstates of scandium atom
2
D

4
F
and the IE, in eV, for scandium atomcalculated at the B3LYP, CASPT2, and DMC levels of theory.
2
D

4
F
IEB3LYP 0.94 6.56CCSDT 1.60 6.34CASPT2 1.39 6.47DMC 1.52
0.01 6.44
0.01Expt. 1.43
Ref.55
6.56
Ref.57
1943153 States of the scandium dimer J. Chem. Phys.
128
, 194315
2008
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
um’s
2
D

4
F
energy gap are 1.39 eV and 1.52 eV, only0.04 eV and 0.09 eV from the experimental mark, respectively. The experimental value for the IE was determined tobe 6.56 eV.
57
B3LYP exactly predicts this value, but thissuccess must be ascribed to a lucky cancellation of errorsrather 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 errorsof about 0.1 eV.Taking into account these results, we can conclude thatDMC results are as conﬁdent as CASPT2 results and bothaccurately predict the calculated properties of scandiumatom.
B. Ground state and excited statesof scandium dimer
We have characterized ten electron states of the scandium dimer. Namely, two septets, four quintets, three triplets,and four singlets. TableIIshows the dominant electronicconﬁguration of the considered states.All these states have been optimized and harmonic frequencies calculated at the B3LYP level of theory, and thensingle point calculations performed using DMC method. Theobtained bond lengths, frequencies, relative energies, anddissociation energies of the two lowestlying states are givenin TableII. Observe that bond length and frequency valuesmay signiﬁcantly differ depending on the electronic state.Thus, bondlength 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 valueof 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 groundstate, 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 by0.07 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 correctlyreproduce the wellestablished experimental values, whichare a vibrational frequency of
=238.9 cm
−
1
Ref.6
and adissociation energy
D
e
=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 tohave a vibrational frequency of 259.9 cm
−
1
, a dissociationenergy of 0.93 eV, and a bond length of 2.58 Å, which are inagreement 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 abond length of 2.57 Å.In order to further discriminate between both states andcheck the DMC results, CCSD
T
and CASPT2 calculationshave 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 diagnostic
55
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 thepredicted equilibrium structures are very similar to theB3LYP ones, as one can note in Fig.2. The CASPT2 resultsconﬁrm the DMC prediction that the
3
u
state is more stableby 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 scandiumdimer has been studied using B3LYP, DMC, CCSD
T
, andCASPT2. B3LYP predicted a
5
u
state as the ground state, inagreement with previous studies. However, highlevel methods such as DMC and CASPT2 predict a
3
u
to be morestable than the
5
u
by around 0.15 eV. This triplet state is an
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 lowlying electronic states of the scandium dimer, in eV. Forthe lowestlying two state dissociation energies
D
e
are given, in eV.
R
e
and
e
are calculated at the B3LYP leveland
E
and
D
e
at the DMC level of theory.State Dominant Conﬁguration
R
e
e
E D
e
7
u
4
s
g
1
↑
3
d
g
1
↑
u
2
↑
g
1
↑
4
s
u
*
,1
↑
2.447 255.3 1.57
0.01
¯
7
g
4
s
g
1
↑
3
d
g
1
↑
u
2
↑
g
2
↑
2.231 314.4 2.46
0.01
¯
5
u
4
s
g
2
↑↓
3
d
g
1
↑
u
2
↑
4
s
u
*
,1
↑
2.582 259.9 0.17
0.01 0.93
5
g
4
s
g
2
↑↓
3
d
g
1
↑
u
2
↑
g
1
↑
2.363 280.5 0.86
0.01
¯
5
g
4
s
g
2
↑↓
u
2
↑
g
2
↑
2.466 239.6 1.56
0.01
¯
5
u
4
s
g
1
↑
3
d
g
1
↑
u
2
↑
g
1
↑
4
s
u
*
,1
↓
2.451 268.7 1.94
0.01
¯
3
u
4
s
g
2
↑↓
3
d
g
1
↑
u
2
↑
4
s
u
*
,1
↓
2.570 273.3 0.00
0.01 1.10
3
g
4
s
g
2
↑↓
3
d
g
2
↑↓
u
2
↑
2.356 320.2 2.55
0.01
¯
1
g
g
2
4
s
g
2
3
d
4
s
u
*
,2
3.163 180.6 0.49
0.01
¯
1
g
g
2
4
s
u
3
↑
4
s
u
*
,1
↓
2.551 266.1 0.77
0.01
¯
1943154 Matxain
et al.
J. Chem. Phys.
128
, 194315
2008
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
open shell triplet, with very similar bond length, frequency,and dissociation energy to the
5
u
state. Both states have thesame orbital occupancy, the only difference is that while inthe quintuplet the
u
orbital has an electron with alpha spin,in the triplet this electron has a beta spin. Our calculationsshow that the ground state of scandium dimer is the
3
u
state, and not the
5
u
state as was thought. New experimentswould 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 ScienceFoundation
Grant No. 20021111645/1
. The SGI/IZOSGIker UPV/EHU
supported by Fondo Social Europeo andMCyT
is gratefully acknowledged for generous allocationof computational resources.
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.FIG. 2. Potential energy curves for the
5
u
top curve
and the
3
u
bottomcurve
electronic states of Sc
2
. Internuclear distances are in Å and energiesin hartree, at the CASPT2 level of theory. The zero energy has been set at
−
1519.475 hartree.
1943155 States of the scandium dimer J. Chem. Phys.
128
, 194315
2008
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