Influence of Spin Multiplicity on the Melting of Na55+ - The Journal of

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Letter 55+

The Influence of Spin Multiplicity on the Melting of Na

José Manuel Vásquez-Pérez, Gabriel Ulises Gamboa, Daniel Mejia-Rodriguez, Aurelio Alvarez-Ibarra, Gerald Geudtner, Patrizia Calaminici, and Andreas M. Koster J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b01983 • Publication Date (Web): 09 Nov 2015 Downloaded from http://pubs.acs.org on November 11, 2015

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The Journal of Physical Chemistry Letters

The Influence of Spin Multiplicity on the Melting of Na55+ J. M. Vásquez-Pérez+, G. U. Gamboa++, D. Mejía-Rodríguez, A. Alvarez-Ibarra+++, G. Geudtner, P. Calaminici*, A. M. Köster* Departamento de Química, Cinvestav, Avenida Instituto Politécnico Nacional 2508 A.P. 14-740, México D.F. 07000, México +

Present address: CONACYT Research Fellow – Universidad Autónoma del Estado de Hidalgo, Carretera PachucaTulancingo Km 4.5, Mineral de la Reforma, Hidalgo, México

++

Present address: CONACYT Research Fellow – Universidad Autónoma de San Luis Potosí, Av. Manuel Nava 6, Zona Universitaria C.P. 78210, San Luis Potosí, México +++

Present Address: Centro Empresarial del Plástico, Adolfo Prieto 424 Col. Del Valle C.P. 03100 México, D.F.

ABSTRACT The influence of spin multiplicity on the melting of the Na55+ cluster has been investigated by means of all-electron KohnSham Born-Oppenheimer molecular dynamics simulations. Based on the quantitative agreement between the experimental and theoretical melting temperature and latent heat a detailed analysis of the cluster dynamics was performed. This analysis showed a significant structure deformation of the cluster which is inconsistent with the geometrical shell closing concept. In subsequent structure optimizations a high-spin ground state in perfect icosahedral symmetry was found for the Na55+ cluster. The Born-Oppenheimer molecular dynamics of this high-spin Na55+ cluster indicates a particular thermal stability of the icosahedral cluster structure. A new electronic mechanism, named sub-shell closing, is suggested as origin for this enhanced thermal stability of the icosahedral cluster structure. This mechanism is a natural extension of the common jellium model. By its nature, the sub-shell closing mechanism is general for finite systems and expected to be found in many other clusters for which the jellium model is applicable.

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With the rise of nanoscience the melting of finite sys-

with the abundance of cationic sodium clusters in mass

tems has attracted the attention of physicists, chemists

spectra23. Here a pronounced peak for Na59+ is observed

and material scientists among others. Recent experi-

indicating an enhanced stability due to the electronic

mental developments allow heat capacity measurements

shell closing with 58 electrons. Counter-intuitively, this

1

of small size selected clusters . With these new techniques

enhanced stability does not correspond to a higher melt-

the melting temperatures and latent heats of small metal

ing temperature. Instead, the less stable Na55+ cluster with

clusters can be determined with high accuracy2-7. So far

54 electrons shows a considerably higher melting temper-

three different experimental methods have been pro-

ature than the more stable Na59+ cluster with 58 electrons.

posed. Common to all of them is that they provide heat

In order to explain this discrepancy, the concept of geo-

capacity curves as function of temperature. Due to the

metrical shell closing has been invoked21. In the particular

finite size of the clusters a smooth caloric curve with a

case of Na55+ a closed shell icosahedra is assumed, based

maximum at the melting temperature is usually ob-

on the comparison of the measured photoelectron spectra

3

and calculated density of states10. However a closed-shell

served . Sodium clusters are extensively studied in these finite

Na55+ cluster will undergo Jahn-Teller distortion and,

system melting experiments due to their prominent role

therefore, cannot be perfectly icosahedral. At this point

in the development of cluster science concepts. The ca-

the question arises as how spin multiplicity can influence

nonical valence molecular orbitals (MOs) in sodium clus-

the Jahn-Teller distortion and, thus, the geometrical

ters possess a shell structure, very much the same as in

structure of the Na55+ cluster. Does this affect the cluster

free atoms. Each sodium atom of the cluster donates its

dynamics and its melting temperature? Can spin multi-

outermost valence electron to these MOs. They can be

plicity be the reason for the discrepancy between the

labeled by principal and angular quantum numbers as: 1S

magic numbers in cluster melting and cluster abundance

1P 1D 2S 1F 2P 1G 2D …8-12. The here appearing S, P, D, F

in mass spectrometry?

and G shells can accommodate 2, 6, 10, 14 and 18 elec-

To gain insight into these questions we performed first-

trons, respectively. The closing of these shells according

principle all-electron Born-Oppenheimer molecular dy-

to the Aufbau principle accounts for particularly stable

namics (BOMD) simulations of Na55+ in its low-spin sin-

clusters8, low polarizabilities13,14 and chemical inertness15,16.

glet and high-spin quintet state and compared our results

These molecular orbital shells have also been observed in

with experimental data. The excellent agreement between

17,18

other metal clusters, e.g. copper clusters

basis for the so-called superatom concept

, and form the

19,20

experiment and theory for the cluster melting tempera-

in chemistry.

ture and latent heat permits the here reported detailed

Surprisingly, the trends in the melting temperatures of

analysis of the influence of the spin multiplicity on the melting of Na55+.

small sodium clusters cannot be straightforwardly explained by their electronic shell structures. In particular, the relative high melting temperature21,22 of Na55+ at

The presented first-principle all-electron BOMD

around 290 K is not obviously related to its electronic

simulations were performed with the linear combination

shell structure. The 54 electrons of Na55+ give rise to the

of Gaussian-type orbital (LCGTO) density functional the-

following electronic shell occupation: 1S2 1P6 1D10 2S2 1F14

ory (DFT) program deMon2k24. All calculations were car-

2P6 1G14. Thus, 4 electrons are missing for closing the 1G

ried out in the framework of auxiliary density functional

shell which occurs in Na59+. This is in complete agreement

theory25 employing a double-zeta valence polarization

2


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(DZVP) basis and GEN-A2 auxiliary function set26. For the

Na55+ cluster is clearly seen from the temperature depend-

exchange-correlation contribution the generalized gradi-

ency of the so-called Lindemann parameter34, depicted in

ent approximation from Perdew, Burke and Ernzerhof

Figure 1b. The sharp change in this distance-fluctuation

(PBE) was used27. In order to avoid spin contamination in

criterion at around 280 K is a characteristic indicator for a

open-shell calculations the spin-restricted open-shell

cluster melting process. In accordance with the smooth

Kohn-Sham method, as implemented in deMon2k, was

caloric curve and the abrupt change in the Lindemann

employed. The here used electronic structure method

parameter, we find from our BOMD simulations a latent

yields optimized bond lengths and energies for small so-

heat of around 20 kcal/mol (Figure 1c) in excellent agree-

dium clusters that deviate by less than 3 pm and 1

ment with the experimental reported values of around 19

kcal/mol from experimental reference data

28-31

kcal/mol21,22. These data are reasonable stable with respect

, respective-

ly.

to the recorded trajectory lengths, i.e. the same qualita-

The BOMD trajectory runs were carried out in the ca-

tive results were obtained by using only half of the steps

nonical ensemble at various temperatures between 150

from each trajectory for the multiple histogram analysis.

and 500 K applying a Nosé-Hoover chain thermostat32

Thus, at first glance it seems that our all-electron BOMD

with seven thermostats in the chain. For both spin multi-

simulation reproduces with good accuracy the experi-

plicities, singlet and quintet, 9 trajectories, each of 125 ps

mental data for the melting of Na55+. This is in agreement

length, were recorded with a step size of 2 fs. The BOMD

with previous pseudo-potential plane wave results, how-

simulations were started from the optimized, nearly ico-

ever on the neutral Na55 cluster35. The excellent agreement

sahedral, Na55+ singlet minimum structure and the perfect

between theory and experiment permits a detailed analy-

icosahedral Na55+ quintet minimum structure. No spin-flip

sis of the BOMD trajectories in order to gain insight into

was permitted during the BOMD simulations. The linear

the cluster dynamics. The visual inspection of these tra-

and angular momentum of the cluster were initialized to

jectories reveals immediately that the Na55+ cluster rear-

zero and conserved during the simulations. The internal

ranges rapidly, even at very low temperatures. This fluc-

energies and heat capacities were calculated from the sin-

tional nature of Na55+ is confirmed by the rather high Lin-

glet trajectories employing the multiple histogram meth-

demann parameter value (see Figure 1b) even before the

od33. Local energy calculations, structure optimizations

melting transition at 280 K. Thus, the maximum in the

and frequency calculations were also performed with the

caloric curve is not reflecting a transition of a solid-like,

local density approximation as well as hybrid functionals

i.e. fixed cluster, to a liquid-like cluster as in other metals,

and larger basis sets. We have found no qualitative

e.g. Al clusters36. Instead, it indicates only a qualitative

changes in the here reported results by employing these

change in the distance fluctuations as seen by the Linde-

alternative computational methodologies.

mann parameter. This has also been seen in BOMD simulations of other sodium cluster37. Thus, a fixed icosahedral structure does not exist in the depicted temperature

The results of the singlet Na55+ BOMD simulations are

ranges of Figure 1. To analyze if a fluctional icosahedral

collected in Figure 1. As this figure shows, the obtained

like droplet of Na55+ exists in the studied temperature

caloric curve (a) is smooth with a well-defined maximum

ranges the prolate deformation coefficient38 of Na55+ as a

at around 280 K. This is in good agreement with the ex-

function of temperature is depicted in Figure 1d. For an

perimental melting temperature of around 290 K. That

icosahedral structure this coefficient is 1. As Figure 1d

this maximum is indeed related to a phase change in the

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Based on this qualitative analysis of the electronic struc-

shows the prolate deformation coefficient for the singlet +

Na55 is already below the melting temperature well above

ture of the Na55+ cluster we have optimized also its triplet,

1 and shows a sharp increase at the melting point. Inspec-

quintet and septet ground states. Table I lists the relative

tion of the BOMD trajectories indicates a deformation of

energies of these optimized minima along with their

+

the Na55 cluster droplet towards an ellipsoid. Certainly,

HOMO-LUMO energy differences. As this table shows,

this is in disagreement with the geometrical shell closing

we find indeed a high-spin quintet as ground state for the

concept.

Na55+ cluster. Varying the functional and basis set does

To gain further insight into this discrepancy we plot in

not alter the energetically ordering of the Na55+ spin mul-

Figure 2 the canonical Kohn-Sham MOs of Na55+ that de-

tiplicities in Table I. The only significant difference arises

fine its electronic shells. Even though the singlet Na55+

from spin contamination if the unrestricted Kohn-Sham

shows the expected Jahn-Teller distortion its MOs are

method is used. In this case the relative energy difference

almost degenerated according to the irreducible represen-

between triplet and quintet reduces to 0.5 kcal/mol. In

tations of the Ih group that are given in parentheses in

any case, the quintet remains the ground state multiplici-

Figure 2, too. Because the maximum degeneracy in Ih is

ty for Na55+. To the best of our knowledge this is the first

five, electronic shells with more than five MOs (1F and 1G)

time that a high-spin ground state is predicted for the

must split into two (or more) sub-shells. For the Na55+ the

Na55+ cluster.

splitting of the 1G shell is particularly important because

This unexpected change in the ground state multiplicity

of its partially filled occupation. In the case of the singlet

of Na55+ has a direct effect on the density of states (DOS)

Na55+ the upper 1G subshell (Gg irreducible representation

of the cluster. In Figure 3 the DOS for the occupied mo-

in Ih) with four MOs shows a very small HOMO-LUMO

lecular orbitals of the singlet (top) is compared with the

splitting as depicted in the middle of Figure 2. As a result,

corresponding DOS of the quintet (bottom). The sub-

the T = 0 K structure of Na55+ is nearly icosahedral. How-

shell closing of the quintet yields a characteristic shoulder

ever, in the BOMD simulations this splitting increases by

in the frontier orbital DOS at around -4.4 eV. The shoul-

distorting the cluster into an ellipsoidal droplet as shown

der is due to the half occupation of the 1G Gg sub-shell by

by the temperature dependent prolate deformation coef-

the α-spin manifold (see Figure 2). Consequently, in the

ficient in Figure 1d. A possibility to avoid this deformation

singlet DOS this shoulder is absent. Because DOS struc-

in the BOMD simulation would be a high-spin quintet

tures are often reproduced by photoelectron spectra this

+

ground state for the Na55 as shown by the orbital occupa-

qualitative difference may bridge directly to experiment.

tion on the right side of Figure 2. Even though a high-spin

In fact, the experimental photoelectron spectrum10 of

multiplicity is unusual for a simple sodium cluster ground

Na55+ possess a narrow peak for the valence electrons with

state it might be favorable in the particular case of Na55+

a characteristic shoulder, just like our quintet DOS. So far

due to sub-shell closing. Note that such high-spin multi-

this structure of the photoelectron spectrum has been

plicities have been already suggested for Li39,40 and Cs41

used as an argument for the spherical geometry of the

clusters. As the canonical Kohn-Sham MO diagram of the

cluster. Our analysis here goes one step further and con-

right side of Figure 2 shows the α-spin manifold in the

nects this structure in the photoelectron spectrum with

Na55+ quintet fills completely the 1G shell, whereas the β-

the electronic structure of the Na55+ cluster. As Figure 3

spin manifold fills only the Hg sub-shell of this 1G shell.

shows the sub-shell closing in the high-spin quintet structure is responsible for the shoulder in the DOS. This sug-

4


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gests that the observed shoulder in the photoelectron

speculate that the experimental data are indeed for the

spectrum originates from a high-spin quintet state. The

electronically excited singlet Na55+ cluster. To clarify this

associated sub-shell closing, that stabilizes this state, is

point, a more extended sampling for the quintet Na55+

also stabilizing the highly symmetric Ih geometrical struc-

cluster at the here used level of theory is needed. Algo-

ture because it avoids Jahn-Teller distortion. What re-

rithmic improvements to make such a sampling possible

mains to be seen is how this stabilization of the highly

are currently under development in our laboratory.

symmetric Ih structure by sub-shell closing influences the

The melting behavior of sodium clusters has been dis-

cluster dynamics.

cussed extensively in the literature (see 1, 21, 35, 42 and

To this end we performed BOMD simulations of the

references therein). It is generally agreed that the excep-

Na55+ quintet employing the same settings as for the sin-

tional high melting temperature of Na55+ is related to a

glet. Figure 4 compares the Lindemann parameter and the

high symmetrical geometric structure. By this relation the

prolate deformation coefficients of the singlet (black dots)

cluster is classified as a geometrical magic number43. We

and quintet (blue triangles) Na55+ in the relevant tempera-

show here that for the special case of Na55+ the high sym-

ture range. As the prolate deformation coefficient shows

metric structure arises from an underlying electronic

+

the quintet Na55 is at lower temperatures more spherical.

mechanism that favors energetically a high-spin quintet

To illustrate this in terms of the corresponding cluster

state. Note that in the corresponding BOMD simulations

structures we compare in Figure 5 four snapshots along

these icosahedral structures occur below and above the

+

the first 50 ps of the quintet and singlet Na55 BOMD tra-

melting temperature suggesting that the underlying elec-

jectory at 250 K. As this figure shows the quintet struc-

tronic mechanism influences both, the solid and liquid

tures remain more spherical along the trajectory than the

states.

corresponding singlet structures, which distort into ellipsoids. A difference is also seen in the potential energies

The present study suggests a new electronic mecha-

along the trajectories that are flat for the quintet but in-

nism, named sub-shell closing, as one origin for the en-

crease in the singlet simulations. This indicates that the

hanced thermal stability of Na55+. It is expected that this

sub-shell closing stabilizes higher symmetric structures in

mechanism is also active in other clusters and responsible

the BOMD simulations. A more detailed analysis of all

for highly symmetric structures, albeit with unexpected

trajectories shows that the quintet Na55+ oscillates be-

spin multiplicities. One promising candidate is the re-

tween spherical and distorted cluster structures on a

cently studied Ag12V+ cluster that shows a perfect cub-

roughly 100 ps time scale. Therefore, our sampling for the

octahedral symmetry and triplet multiplicity44. Certainly,

+

quintet Na55 is not sufficient for a multiple histogram

there are much more systems with sub-shell closing that

analysis. This is the reason why we cannot report a heat

still have to be identified.

capacity curve for this system. Nevertheless, the more

The driving force for this new electronic mechanism

continuous Lindemann parameter increase and the differ-

arises from the splitting of the well-known electronic

ence in the prolate deformation coefficient with respect to

shells into sub-shells due to the finite degeneracy of point

the singlet Na55+ suggest that the melting temperatures for

group irreducible representations. The closing of these

the singlet and quintet Na55+ differ significantly. On the

sub-shells by one spin manifold produces highly symmet-

other hand, our results for the singlet are in good agree-

ric structures in high-spin states. Note that this effect is

ment with experiment. Based on these results one can

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not taken into account in the common jellium model. This is the reason for its failure to describe the highly symmetric structure of Na55+. In this respect the here described mechanism can be seen as a natural extension of the jellium model. The presented all-electron BOMD simulations show good quantitative agreement with the available experimental data. The trajectory analysis reveals the fluctional nature of Na55+. The cluster rearranges rapidly well below its melting temperature. Therefore, the melting of this cluster is not a transition between solid- to liquid-like structures. This is different to other metal clusters, e.g. aluminium clusters. The comparison of the singlet and quintet Na55+ BOMD trajectories shows that spin multiplicity has a direct influence on the cluster dynamics and its melting temperature. This underlines the importance of first-principle all-electron BOMD simulations for cluster science in the future.

AUTHOR INFORMATION Corresponding Authors * [email protected]; [email protected]

Funding Sources Support from CONACYT through grants CB-130726 and CB179409.

ACKNOWLEDGMENT JMVP, GUG, PC and AMK gratefully acknowledge support from CONACYT through grants CB-130726 and CB-179409. DMR thanks CONACYT for a Ph.D. fellowship (47922). The BOMD simulations were performed at the WESTGRID of Compute Canada.

ABBREVIATIONS HOMO, Highest Occupied Molecular Orbital; LUMO, Lowest Unoccupied Molecular Orbital.

REFERENCES (1) Aguado, A.; Jarrold, M.F. Melting and Freezing of Metal Clusters. Annu. Rev. Phys. Chem. 2011, 62, 151-172. (2) Schmidt, M.; Kusche, R.; Kronmüller, W.; Issendorff, B. v.; Haberland H. Experimental Determination of the Melting Point and Heat Capacity for a Free Cluster of 139 Sodium Atoms. Phys. Rev. Lett. 1997, 79, 99–102.

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The Journal of Physical Chemistry Letters and Spin Moment in AgnV+ Clusters. J. Am. Chem. 2014, 136, 8229-8236.

(24) Köster, A. M.; Geudtner, G.; Calaminici, P.; et al. deMon2k Version 3, The deMon developers, Cinvestav, Mexico-City 2011. See also: www.demon-software.com. (25) Köster, A. M.; Reveles, J. U.; del Campo, J. M. Calculation of the Exchange-Correlation Potential with Auxiliary Function Densities. J. Chem. Phys. 2004, 121, 3417-3424. (26) Calaminici, P.; Janetzko, F.; Köster, A. M.; Mejia-Olvera, R.; Zuniga-Gutierrez, B. Density Functional Theory Optimized Basis Sets for Gradient Corrected Functionals: 3d Transition Metal Systems. J. Chem. Phys. 2007, 126, 044108. (27) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865– 3868. (28) Martin, S.; Chevaleyre, J.; Valignat, S.; Perrot, J. P.; Broyer, M.; Cabaud, B.; Hoareau, A. Autoionizing Rydberg States of the Na2 Molecule. Chem. Phys. Lett. 1982, 87, 235-239. (29) Verma, K. K.; Bahns, J. T.; Rajaei-Rizi, A. R.; Stwalley, W. C.; Zemke, W. T. First Observation of Bound-Continuum Transitions in the Laser-Induced A 1Σu+ - X 1Σg+ Fluorescence of Na2. J. Chem. Phys. 1983, 78, 3599-3613. (30)Hilpert, K. Mass Spectrometric Study of Sodium Clusters under Equilibrium Conditions with a Knudsen Cell. Ber. Bunsenges. Phys. Chem. 1984, 88, 260-262. (31) McHugh, K. M.; Eaton, J. G.; Lee, G. H.; Sarkas, H. W.; Kidder, L. H.; Snodgrass, J. T.; Manaa, M. R.; Bowen, K. H. Photoelectron Spectra of Alkali Metal Cluster Anions: Nan=2-5-, Kn=2-7-, Rbn=2-3- and Csn=2-3-. J. Chem. Phys. 1989, 91, 3792-3793. (32) Gamboa, G. U.; Vásquez-Pérez, J. M.; Calaminici, P.; Köster, A. M. The influence of thermostats on the calculation of heat capacities from Born-Oppenheimer molecular dynamics simulations. Int. J. Quant. Chem. 2010, 110, 2172-2178. (33) Ferrenberg, A. M.; Swendsen, R. H. New Monte Carlo technique for studying phase transitions. Phys. Rev. Lett. 1988, 61, 2635-2638; ibid 1989, 63, 1195-1198. (34) Beck, T. L.; Doll, J. D.; Freeman, D. L. The quantum mechanics of cluster melting. J. Chem. Phys. 1989, 90, 5651-5656. (35) Chacko, S.; Kanhere, D.G.; Blundell, S. A. First-principles calculations of melting temperatures for free Na clusters. Phys. Rev. B 2005, 71, 155407. (36) Vásquez-Pérez, J. M.; Calaminici, P.; Köster, A. M. Heat Capacities from Born-Oppenheimer Molecular Dynamics Simulations: Al27+ and Al28+. Comp. Theor. Chem. 2013, 1021, 229-232. (37) Rytkönen, A.; Häkkinen, H.; Manninen, M. Melting and Octupole Deformation of Na40. Phys. Rev. Lett. 1998, 80, 39403943. (38) Kavita, J.; Kanhere, D. G.; Blundell, S. A. Thermodynamics of tin clusters. Phys. Rev. B 2003, 67, 235413. (39) Gardet, G.; Rogemond, F.; Chermette, H. Density Functional Theory Study of some Structural and Energetic Properties of Small Lithium Clusters. J. Chem. Phys. 1996, 105, 9933-9947. (40) Ashman, C.; Khanna, S.N.; Pederson, M. R. Dynamical Effects on the Photo-Detachment Spectra of Li4-. Chem. Phys. Lett. 2002, 351, 289-294. (41) Aguado, A. Discovery of Magnetic Superatoms and Assesment of van der Waals Dispersion Effects in Csn Clusters. J. Chem. Phys. C 2012, 116, 6841-6851. (42) Calvo, F.; Spiegelmann, F. Mechanisms of Phase Transitions in Sodium Clusters: From Molecular to Bulk Behavior. J. Chem. Phys. 2000, 112, 2888-2908. (43) Noya, E. G.; Doye, J. P. K.; Wales, D. J.; Aguado, A. Geometric Magic Numbers of Sodium Clusters: Interpretation of the Melting Behaviour. Eur. Phys. J. D 2007, 43, 57-60. (44) Medel, V. M.; Reber, A. C.; Chauhan, V.; Sen, P.; Köster, A. M.; Calaminici, P.; Khanna, S. N. Nature of Valence Transition

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Figure Captions: Figure 1: Heat capacity (a), Lindemann parameter (b), internal energy (c) and prolate deformation coefficient (d) of the singlet Na55+ cluster as a function of temperature from the BOMD simulations. See text for details. Figure 2: Molecular orbitals of Na55+ assigned to the electronic shells and irreducible representations (in parentheses) of the Ih group. The corresponding orbital occupation for the singlet and quintet is also shown in the middle and on the right, respectively. Figure 3: Density of states for the closed-shell singlet Na55+ (top) and the open-shell quintet Na55+ (bottom). The discrete orbital energy spectrum is broadened by 0.06 eV to obtain the envelope curves. Figure 4: Comparison of Lindemann parameter (left) and prolate deformation coefficient (right) from the singlet (black dots) and quintet (blue triangle) Na55+ BOMD simulations. Figure 5: Trajectory snapshots from the quintet (Q) and singlet (S) BOMD simulations of Na55+ at 250 K. The first 50 ps are shown. The corresponding potential energy curves are given above the cluster structures. The black points indicate the positions of the snapshots.

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Figure 1: J. M. Vásquez-Pérez, et al.

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Figure 2: J. M. Vásquez-Pérez, et al. 10 ACS Paragon Plus Environment

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Table I:

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Relative energies and HOMO-LUMO energy differences for the optimized singlet, triplet, quintet and septet Na55+ minimum structures.

Multiplicity

∆E [kcal/mol]

HOMO-LUMO [eV]

1

02.3

0.04

3

01.7

0.12

5

00.0

0.27

7

13.4

0.08

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Influence of Spin Multiplicity on the Melting of Na55+ - The Journal of

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