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
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  Quantum Monte Carlo study of the ground state and low-lying 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, 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 theoreticalmethods such as quantum diffusion Monte Carlo  DMC  , complete active space perturbation theoryas implemented in GAMESS-US , coupled-cluster 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 tran-sition 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 ex-ample 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 Bad-ger’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 dis-crepancy. 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 de-pending on the experimental method employed, rangingfrom 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 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. Ac-cording 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 combina-tion 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 domi-nant 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 low-lying excited statesremained 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 CASSCFand IC-ACPF 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 dis-sociation 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. THE JOURNAL OF CHEMICAL PHYSICS 128 , 194315  2008  0021-9606/2008/128  19   /194315/5/$23.00 © 2008 American Institute of Physics 128 , 194315-1 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp  ues of Moskovits 6 and Haslett. 7 The calculated frequenciesof 261 cm − 1 at the CASSCF level and 286 cm − 1 at IC-ACPFlevel are in reasonably good agreement with the experimen-tal 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 low-lying 3  u excited state by 0.291 eV. The IC-ACPFmethod, 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 de-scribe the state ordering because of the inability to treat dy-namical electron correlation, which wasbetter accounted forby the IC-ACPF method. Suzuki et al. 15 performed multiref-erence configuration interaction calculations of the 5  u state,predicting frequencies in very good agreement with experi-ment  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 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 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, be-ing the 3  u described 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: 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 experi-mentally found by Knight et al. 12 for the dimer cation.The 3  u triplet state was not considered in the high-levelmulticonfigurational 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 experimen-tal 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 mea-sure experimental values. This constitutes a further check tothe 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-binedwith 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, opti-mized by Hay, 28 and three sets of uncontracted pure angularmomentum f  functions, including both tight and diff use ex-ponents, 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 ho-mogeneous, isotropic electron-electron terms u , isotropicelectron-nucleus terms   centered on the nuclei, and isotro-pic electron-electron-nucleus 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,wasopti-mized using variance minimization techniques. 36,37 TheDMC method is one of the quantum Monte Carlo implemen-tations that, together with the variational Monte Carlomethod, is becoming widely used nowadays. Briefly, inDMC, 22,38 the imaginary-time Schrödinger equation is usedto evolve an ensemble of electronic configurations towardthe ground state. Exact imaginary-time evolution would leadto the exact fermion ground-state wave function, provided ithas a nonzero overlap with the initial fermion state. How-ever, the stochastic evolution is neverexact, and the solutionconverges to the bosonic ground state. 39 The fermionic sym-metry is maintained by the fixed-node approximation, inwhich the nodal surface of the wave function is constrainedto be equal that of a guiding wave function. The fixed-nodeDMC energy provides a variational upper bound on theground-state 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 low-lying excited states and thelowest-lying two states of its dimer. For both the CCSD  T  and CASPT2 calculations, the TZVP-G  3 df  ,2  p  basis set 194315-2 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 coupled-clustermethod, 42,43 which uses single and double excitations fromthe 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 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 GAMESS-US . 54 III. RESULTS Before turning our attention to the characterization of theelectronic structure and physical properties of the ground andlow-lying states of the scandium dimer, we wish to estimatethe 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 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 corre-sponding 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 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 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 multicon-figurational 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 or-bitals 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  194315-3 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, respec-tively. 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 scan-dium 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 fre-quencies 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 lowest-lying states are givenin TableII. Observe that bond length and frequency valuesmay 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  g state, with a valueof 180.6 cm − 1 , while the largest value of 320.2 cm − 1 corre-sponds 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 accor-dance 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 well-established 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 ex-trapolation pointed out a bond length of 2.5 Å. The previ-ously 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 fre-quency 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 car-ried out with the T1 diagnostic 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 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, high-level meth-ods 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 low-lying electronic states of the scandium dimer, in eV. Forthe lowest-lying 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 ¯ 194315-4 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 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 ScienceFoundation  Grant No. 20021-111645/1  . 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