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

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

MATXAIN, Jon M., et al.


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

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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,1,2Xabier Lopez,1,2Jesus M. Ugalde,1,2and Laura Gagliardi3

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

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

3University 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 inGAMESS-US, coupled-cluster singles, doubles, and triples, and density functional theory!DFT". The3!uand5!ustates 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 the3!uto 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 the3!ustate, 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$


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,3Bad- 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 Verhaegenet al.4in 1964, using mass spectrometry techniques, %1.30 eV. This value was then revised by Gingerich,5who 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, Haslettet al.7reevaluated 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 Moskovitset 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

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$g21$u11%u22$g1 configuration. The 5!u state was also claimed to be the ground state by Walch et al.10on the basis of some CASSCF/CI!SD"calculations. Ac- cording to their calculations, the 1$g21$u11%u22$g1was indeed the dominant configuration of the 5!u state. This combina- tion of experimental and theoretical works seemed to set on firm grounds that the ground state of the scandium dimer was a5!ustate, and not a5&gstate as predicted by Busbyet al.11 In a related work, Knightet al.12experimentally determined that the ground electronic state of the dimer cation, Sc2+, was the 4!g state, which is in agreement with the removal of the electron from the 1$uorbital of the5!uground state’s domi- nant configuration of Sc2.

Therefore, it became widely accepted that the ground state of scandium dimer was a5!ustate. 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. Åkebyet 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 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-

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 Moskovits6 and Haslett.7The calculated frequencies of 261 cm−1at the CASSCF level and 286 cm−1at IC-ACPF level are in reasonably good agreement with the experimen- tal value.

However, according to the CASSCF results, a3'u state was found to be more stable than both the5!uby 0.295 eV and a low-lying3!uexcited 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 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. Suzukiet al.15performed multiref- erence configuration interaction calculations of the5!ustate, predicting frequencies in very good agreement with experi-

ment !230 versus 239 cm−1"and a bond length of 2.8 Å in

agreement with that of Åkebyet 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.16–21In 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 Sc2, along with the 2D4Fenergy 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,16using B3LYP, found an open shell 3!u, with an electronic configuration of

1$g21$u1!↓"1%u2!↑"2$g1!↑", only 0.22 eV higher in energy.

Similarly, Gutsevet al.,19using BPW91, found at least three states of the scandium dimer thermodynamically stable, be- ing the3!udescribed earlier by Papai and Castro the lowest- lying 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!uand the3!ustates is the same, i.e., the 4!gstate experi- mentally found by Knightet al.12 for the dimer cation.

The3!utriplet 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!uassumedground 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 2D−4F dissociation asymptote and the formation of the4!gground state of Sc2+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!uand the3!ustates of Sc2and 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 2Dand4F 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.


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.26supplemented with one diffusesfunction, two sets of p functions optimized by Wachters27 for the excited states, one set of diffuse pure angular momentumd function, opti- mized by Hay,28and three sets of uncontracted pure angular momentum f functions, including both tight and diffuse ex- ponents, as recommended by Ragavachari and Trucks.29This 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,31which 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",32,33motivated by their earlier successful performance in DMC calculations in a number of transition metal containing systems.34,35The 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.39The 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 Matxainet 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


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 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. The2D-4Fenergy gap and the IE of the scandium atom

TableI 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 corre- sponding experimental data.

Experiments show the4F 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 match- ing with the experimental mark. Remarkably, CCSD!T"cal- culations overestimate the experimental2D-4Fenergy gap by

&0.17 eV. This failure can tentatively be ascribed to the multiconfigurational character of states investigated. Indeed, the T1 diagnostics56of the2Dand4Fyield 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 !2D-4F" and the IE, in eV, for scandium atom calculated at the B3LYP, CASPT2, and DMC levels of theory.

2D-4F 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 2D-4F 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!gstates, respectively. Similarly, the smallest calculated vi- brational frequency corresponds to the1!gstate, with a value of 180.6 cm−1, while the largest value of 320.2 cm−1 corre- sponds to the3!gstate. 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 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 De=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!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 fre- quency 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 and5!u. CCSD!T"calculations have been car- ried out with the T1 diagnostic55to 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 the3!ustate 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.


A large set of different electronic states of scandium dimer has been studied using B3LYP, DMC, CCSD!T", and CASPT2. B3LYP predicted a5!ustate as the ground state, in agreement with previous studies. However, high-level meth- ods such as DMC and CASPT2 predict a 3!u to be more stable than the5!uby around 0.15 eV. This triplet state is an

TABLE II. Equilibrium distancesRe, in Aˆ, 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 energiesDeare given, in eV.Reand#eare calculated at the B3LYP level

and&EandDeat the DMC level of theory.

State Dominant Configuration Re #e &E De

7&u $!4s"g1!↑"$!3d"g1!↑"%u2!↑")g1!↑"$!4s"u*,1!↑" 2.447 255.3 1.57"0.01 ¯

7!g $!4s"g1!"$!3d"g1!"%u2!")g2!" 2.231 314.4 2.46"0.01 ¯

5!u $!4s"g2!↑↓"$!3d"g1!"%u2!"$!4s"u*,1!" 2.582 259.9 0.17"0.01 0.93

5&g $!4s"g2!↑↓"$!3d"g1!↑"%u2!↑")g1!↑" 2.363 280.5 0.86"0.01 ¯

5!g $!4s"g2!↑↓"%u2!")g2!" 2.466 239.6 1.56"0.01 ¯

5&u $!4s"g1!"$!3d"g1!"%u2!")g1!"$!4s"u*,1!" 2.451 268.7 1.94"0.01 ¯

3!u $!4s"g2!↑↓"$!3d"g1!"%u2!"$!4s"u*,1!" 2.570 273.3 0.00"0.01 1.10

3!g $!4s"g2!↑↓"$!3d"g2!↑↓"%u2!↑" 2.356 320.2 2.55"0.01 ¯

1!g $g2!4s"$g2!3d"$!4s"u*,2 3.163 180.6 0.49"0.01 ¯

1'g $g2!4s"%u3!↑"$!4s"u*,1!↓" 2.551 266.1 0.77"0.01 ¯

194315-4 Matxainet al. J. Chem. Phys.128, 194315"2008!


open shell triplet, with very similar bond length, frequency, and dissociation energy to the5!ustate. Both states have the same orbital occupancy, the only difference is that while in the quintuplet the$uorbital 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 the5!ustate as was thought. New experiments would be interesting in order to confirm our prediction.


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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56T. J. Lee and P. R. Taylor, Int. J. Quantum Chem., Quantum Chem.

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FIG. 2. Potential energy curves for the5!u!top curve"and the3!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.




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