Physics Letters A 376 (2012) 1584–1588

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Physics Letters A

www.elsevier.com/locate/pla

Quantum thermodynamics of (H2)x@C60 [x = 1–2]: A path integral Monte Carlostudy

Anthony Cruz, Gustavo E. López ∗

Department of Chemistry, Lehman College-CUNY, Bronx, NY 10468, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 November 2011Received in revised form 26 March 2012Accepted 27 March 2012Available online 30 March 2012Communicated by R. Wu

The thermodynamic properties of H2 and (H2)2 inside C60 were computed using the path integralformalism. In accordance with experimental data, H2@C60 is thermodynamically stable in a wide range oftemperatures due to energetic factors. Variations in the molecular hydrogen–fullerene interaction energywere considered in order to increase the stability of the monomeric system. For (H2)2@C60 no stablestates were observed in the temperature range studied or in any reasonable modification in the molecularhydrogen–fullerene interaction parameter. Modification of the attractive part of the molecular hydrogen–fullerene interaction stabilized the dimeric system.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

During the last few years researchers have undertaken the taskof studying small molecules interacting with fullerenes, primarilybecause of possible nanotechnological applications. In particular,the interaction of species in condensed phase with the externalsurface of fullerenes has been extensively considered [1]. Moreover,the properties of atoms and molecules entrapped inside fullereneshave been reported [2]. Particularly, Turro and coworkers [3–5]presented the synthesis and characterization of molecular hydro-gen inside fullerenes. Of particular interest to this study was thecharacterization of H2 inside C60. Their results showed that it waspossible to trap only one hydrogen molecule inside C60. Compu-tational studies [5] confirmed that the destabilization of multiplehydrogen molecules entrapped in C60 is because two componentsof the total energy are repulsive – interactions between the con-fined hydrogen molecules and between the hydrogen moleculesand the fullerene. Similar results were obtained [6,7] using ab-initio calculation. Kruse and Grimme [6] used accurate quantummechanical electronic structure techniques to show that only onehydrogen molecule is stable inside C60. Similarly, Korona et al. [7]used symmetry-adapted perturbation theory to show that H2@C60was a stable system, whereas, 2H2@C60 has a positive interactionenergy corresponding to energetically unstable species. All of theabove calculations were based on state-of-the-art temperature in-dependent calculations.

In the present study we have implemented path integralreplica-exchange Monte Carlo simulations to characterize the ther-

* Corresponding author. Tel.: +1 718 960 8678.E-mail address: gustavo.lopez1@lehman.cuny.edu (G.E. López).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.03.058

modynamic stability of one and two hydrogen molecules in-side C60. Our results are in agreement with previous studies [5–7],where only one H2 molecule is stable inside C60 due to ener-getic contributions. We also proposed possible modifications of thefullerene that will allow for an increment in the stability of H2inside C60. For the dimeric system neither temperature nor rea-sonable modifications in the H2–C60 interaction parameter wereencountered as to permit the stabilization of (H2)2 inside C60.Manipulation of the attractive part of the hydrogen–fullerene po-tential yielded a thermodynamically stable system.

2. Methods

In this study H2 was treated as a spherically symmetricmolecule, hence para-hydrogen was the simulated species. Themain reason for this approximation is that C60 provides a verysymmetric environment for H2; this implies that even numberedrotational quantum states can be adequately modeled. This ap-proximation has been extensively used [5,8–11] for confined H2 inisotropic environments.

For systems with a small mass, m, and at low temperaturesquantum effects are usually estimated by the Boer parameter (Λ),given by

Λ = h

σ√

mε(1)

where ε is the well depth, σ is the atomic size, and h is Planckconstant divided by 2π . For systems where the Boer parameter islarger than 0.1, quantum effects needed to be considered for anaccurate computational description. In the case of molecular hy-drogen Λ = 0.3, hence, inclusion of quantum effects were incorpo-rated in the used model. Specifically, the description of molecular

A. Cruz, G.E. López / Physics Letters A 376 (2012) 1584–1588 1585

hydrogen was based on the path integral formalism [12] in thecanonical ensemble, where a quantum particle was represented asa cyclic polymeric ring with P beads bound together by harmonicoscillators. Within this formalism the total potential energy, V , wasgiven as the summation of the internal (V int) and external (V ext)interactions:

V = V int + V ext (2)

V int has the form of a harmonic oscillator describing the interac-tion between adjacent beads nb and nb+1 in a given molecule i,separated by a distance Rnb

i :

V int = 1

2kc

P∑nb=1

N∑i=1

∣∣Rnbi − Rnb+1

i

∣∣2(3)

Here kc is the force constant given by

kc = Pm

β2h(4)

where m is the mass of the particle, β = 1/kT , k is Boltzmannconstant, and T is the absolute temperature. The external part ofthe potential energy consisted of two pair-wise 12-6 Lennard-Jones(L-J) potentials [5,13]:

V ext = 1

P

P∑nb=1

N∑i< j

4εi j

[(σi j

rnbi j

)12

−(

σi j

rnbi j

)6]

+ 1

P

N∑i=1

P∑nb=1

60∑k=1

V (H2)x–Ck

(Rnb

ik

)(5)

The first term describes the interaction between bead nb in par-ticle i and bead nb in particle j, separated by a distance rnb

i j (inthe case of one H2 molecule this term is zero). The numerical

values used for εi j and σi j were 35.60 K and 2.749 Å [5], respec-tively. The second term in Eq. (5) describes the interaction betweenbead nb in the ith quantum particle and the kth carbon atom inthe fullerene, separated by a distance Rnb

ik . The analytical form ofthis potential is given by

V (H2)x–Ck

ε(H2)x–Ck

= 4

[Rij

(σ(H2)x–Ck

R(H2)x–Ck

)12

− Aij

(σ(H2)x–Ck

R(H2)x–Ck

)6](6)

Here σ(H2)x–Ck is the usual L-J size parameter equal to 3.08 Å [5].The triple summation in Eq. (5) gives the total interaction en-ergy between molecular hydrogen and the fullerene, V H2–C60 . Thestrength of the interaction between molecular hydrogen and C60 isdetermined by the value of ε(H2)x–Ck , which is equal 27.92 K [5].The constants Rij and Aij are factors that were used to scale therepulsive and attractive part of the potential, respectively. In mostcases these values were set equal to one. In this study we con-sidered variations in the stability of the systems by increasingthe value of ε(H2)x–Ck by 10%, i.e., five systems were studied withvalues of ε(H2)x–Ck = 27.92 K + 2.792n K for n = 0,1,2,3, and 4.Experimentally, this might corresponds to variation in the elec-tron density of the fullerene caused by the solvent. For example,in polar solvents, fullerenes are electron deficient centers; hencethe interactions between the trapped molecules and the fullereneshould decrease. On the contrary, organic solvents increase the in-teraction between the trapped molecules and C60 because of elec-tronic donation of the solvent. Interestingly, Murata and cowork-ers [14] have shown that addition of three extra electrons to theC60-π system increases the interaction between encapsulated H2and C60. For the (H2)2@C60 system calculations were performedwith Aij = 1,2,2.5, and 3. In all calculations the carbon atoms inthe fullerene were treated as rigid species.

To obtain the total energy, Etot, of the system within the pathintegral formalism, the T -method was implemented. Namely,

Etot = − ∂

∂βln Q (N, β, V ) (7)

where Q (N, β, V ) is the canonical partition function given by

Q (N, β, V ) =(

mP

2πβh3

) 3P2

×∫

dP r exp

[−β

P∑i=1

(V int + 1

PV ext

)](8)

The energetic stability of the two systems considered here wascomputed using an approach similar to the one used in previousstudies [5–7,15]. Specifically, the difference in total energy (bindingenergy), �E , for (H2)x@C60 is defined by

�E = E(H2)xC60 − EC60 − x[EH2 ] (9)

where E X is the total energy of system X . Because of the absentof external fields, the change in total energy is equal to the changein internal energy, �E = �U .

In order to obtain information about the thermodynamic stabil-ity of hydrogen molecules inside C60, the change in Helmholtz freeenergy, �A, was computed using thermodynamic integration [16].To implement this method the potential energy was written as

V = V int + λV ext (10)

where λ is a scaling factor. Within this formalism, �A was ob-tained using

�A =1∫

0

⟨∂V

∂λ

⟩λ

dλ (11)

In the case of only one H2 molecule interacting with C60, 〈 ∂V∂λ

〉λ =〈V H2–C60 〉λ .

In order to circumvent the endpoint singularity problem usu-ally encountered in thermodynamic integration in the non-bondedinteractions represented by L-J potentials, a linear mixing soft-corepotential [17] was implemented. Previous studies have shown [17]that this soft-core potential effectively allows for proper thermo-dynamic integration even when using a relatively small number ofintegration points. The calculation of the change in Helmholtz freeenergy was reduced to the evaluation of a one-dimensional inte-gral, which was solved numerically using six points in the Gauss–Legendre integration algorithm:

1∫0

⟨∂V

∂λ

⟩λ

dλ =6∑

i=1

W (i)

⟨∂V

∂λ

⟩i

(12)

where W (λ) are the weights for a particular value of λ. As re-cently discussed by Holden and Freeman [18], this procedure,coupled with proper sampling schemes (as discussed below), al-lows for significant convergence in the numerical integration ofEq. (11). The entropic contribution to the stability was obtainedfrom −T �S = �A − �U . The number of beads representing thequantum particle was adjusted in order to obtain proper conver-gence of the computed thermodynamic variables, i.e. a total of60 beads were used at all temperatures. Various sampling schemeswere implemented in order to circumvent quasi-ergodicity prob-lems. Specifically, every bead was sequentially moved followingthe standard Monte Carlo (SMC) procedure with an acceptanceprobability of p = min[1, β�V ], where �V is the difference inenergy between configurations being sampled. Displacement and

1586 A. Cruz, G.E. López / Physics Letters A 376 (2012) 1584–1588

Fig. 1. (A) Average change in Helmholtz free energy, (B) average change in internalenergy, and (C) average change in entropy multiple by temperature as a function oftemperature for H2@C60 using various values of the hydrogen–fullerene interparticlepotential, ε(H2)i –Ck . Error bars (one standard deviation) are within the width of theline, which was generated by multiple single point calculations. (For interpretationof the references to color, the reader is referred to the web version of this Letter.)

rotational moves around the centroid of the quantum particlewere performed using the SMC procedure. Every ten SMC moves areplica-exchange movement was attempted based on a transitionprobability that allows exchanging tandem temperature walkers,p = min[1, (β1 − β2)�V ]. The temperature separation was chosensuch that the acceptance ratio of interchanges was 20% or higher.A total of 109 warm-up steps were taken, and data was collectedfor 109 steps. The step size was adjusted in order to obtain a 50%of acceptance for the moves. All simulations were performed us-ing codes developed in our laboratory, which have been previouslyused [19–22] to study temperature dependent problems in quan-tum systems in condensed phase.

3. Results and discussion

The results obtained from this study are presented in Figs. 1through 4 and Tables 1 and 2. In Table 1, the change in internal en-ergy obtained in this study at the lowest temperature considered(T = 5 K) for H2@C60 and (H2)2@C60 are compared with resultsof three quantum mechanical calculations using temperature inde-pendent techniques (T = 0 K). All results are in qualitative agree-ment in that H2@C60 is energetically stabilized, whereas (H2)2@C60is energetically unstable. Our results are in closer agreement withthe interaction energies calculated using diffusion Monte Carlo(DMC) techniques. The quantitative differences between the twosets of results are due to the use of completely different theo-retical approaches, i.e. DMC results do not reflect thermal effectsthat cause increment in energy differences as the temperature in-creases.

The black curve in Fig. 1A shows 〈�A〉 as a function of tem-perature for H2@C60, with ε(H2)x–Ck = 27.92 K. It can be observedthat at all temperatures the change in Helmholtz free energy isnegative, hence the system is thermodynamic stable. Moreover,as the temperature decreases the thermodynamics stability of thesystem increases. From Fig. 1B it can be observed that the stabi-

Fig. 2. (A) Snapshot configuration of H2@C60 and (B) sampled configurations of thecentroid of H2@C60, at T = 300 K. The color code used measures the deviation be-tween the centroid of H2 and the center of mass of C60, i.e., red color is used todescribe a deviation of 0.0, white for deviations of 0.5, and blue for deviations 1.5.Variations in tonalities are used for intermediate values. (C) and (E) are the sameas (A), but for T = 100 K and T = 10 K, respectively. (D) and (F) are the same as(B) but for T = 100 K and T = 10 K, respectively. (For interpretation of the refer-ences to color in this figure legend, the reader is referred to the web version of thisLetter.)

Fig. 3. (A) Average change in Helmholtz free energy, (B) average change in inter-nal energy, and (C) average temperature times the change in entropy, as a functionof temperature for (H2)2@C60 using various values of the attractive part of thehydrogen–fullerene interparticle potential, Aij . The black, red, green and blue curvescorrespond to Aij equal 1.0, 2.0, 2.5, and 3.0, respectively. Error bars (one standarddeviation) are within the width of the line, which was generated by multiple singlepoint calculations. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this Letter.)

lization is due to energetic contributions (�U < 0), with the ener-getic stabilization being approximately temperature independent.The entropic contribution decreases the thermodynamic stabilityat all temperatures (−T �S > 0) but not in a significant manner.Table 2 shows absolute thermodynamic values for the energeticof H2@C60. It can be seen that the stabilization of the systemfor temperatures below 25 K (see for example T = 25 K) is dueto a decrease in the components of the internal energy – de-crease in the average H2-fullerene internal energy, 〈V H2–C60 〉, and

A. Cruz, G.E. López / Physics Letters A 376 (2012) 1584–1588 1587

Fig. 4. Same as Fig. 2 but for (H2)2@C60 at T = 100 K ((A) and (B)) and T = 10 K((C) and (D)). (For interpretation of the references to color, the reader is referred tothe web version of this Letter.)

Table 1Comparison of interactions energies calculated in this work at T = 5 K with thosefrom three recent studies using temperature independent (T = 0 K) quantum me-chanical techniques. All energies are in cm−1.

System This work DMC (Ref. [5]) DFT (Ref. [7]) MP2 (Ref. [6])

H2@C60 −1087.12 −1498.71 −1618 −2553(H2)2@C60 +573.43 2878.02 +2065 +210

the average hydrogen internal energy, 〈V int〉, as the temperaturedecreases. This reduction in energy is due to the increment inthe radius of gyration of the quantum particle as the tempera-ture decreases (see column 6 in Table 2), i.e., increment in thesize of the quantum particle increases the attraction between H2and C60. This size-temperature dependence is depicted in configu-ration snapshots shown in Figs. 2A, C, and E for T = 300 K, 100 K,and 10 K, respectively. Figs. 2B, D, and F show the positions sam-pled by the centroid of the quantum particle at the temperaturespreviously mentioned. At low temperatures the centroid of thequantum particle is located predominantly in the center of thefullerene, whereas as the temperature increases regions far fromthe center of the fullerene are sampled. When Table 2 is exam-ined at temperature higher than 25 K (see for example 50 K) itcan be observed that the system has a positive value for the in-ternal energy because of the increment in the hydrogen–fullereneand hydrogen internal energies. However, the systems remain en-ergetically stable when compared to the energetic of isolated H2.These stability measurements are in agreement with the experi-mental work by Turro et al. [2–4], where H2@C60 was synthesizedat ambient temperature.

Because the stabilization of molecular hydrogen is primarilycontrolled by the magnitude of the interaction of H2 with thefullerene, increments between 10% and 50% in the numerical value

of ε(H2)x–Ck were considered. Fig. 1A shows the average changein Helmholtz free energy as a function of temperature at variousvalues of ε(H2)x–Ck . It can be observed that the system becomesthermodynamically more stable as the value of ε(H2)x–Ck increases.Specifically, increments of 10% in ε(H2)x–Ck caused increments instabilization of approximately 1 kJ/mol. From Figs. 1A and B itcan be seen that the stabilization comes from a decrease in �U .Although the entropic contribution increases with temperature(Fig. 1C), it does not affect the stabilization of the system. Theseresults are in qualitative agreement with Ref. [7], where overes-timation of the energies of the noncovalent interactions typicalof MP2 based approaches yield more stable complexes. Moreover,these results are in agreement with electrochemical experimentaldata [14] that suggest that increments in the H2–C60 interaction byaddition of electron to the outer C60-π system stabilize the com-plex.

When two hydrogen molecules are incorporated inside thefullerene it is observed that the change in Helmholtz free energyas a function of temperature is always positive, independently ofthe value of ε(H2)x–Ck . This can be observed for ε(H2)x–Ck = 27.92 Kin the black curve of Fig. 3A. Both the energetic factor (Fig. 3B)and the entropic contribution (Fig. 3C) caused destabilization ofthe system. The main reason for this destabilization is a significantincrement in the repulsive forces between the fullerene and thehydrogen molecules. This result is in accordance with experimen-tal data [2–4] were two hydrogen molecules enclosed in C60 couldnot be synthesized.

Additional calculations for this system were performed wherethe attractive part of the H2–Ck interaction was increased, i.e. Aijwas set equal to 1.0, 2.0, 25, and 3.0 reduced units. Fig. 3A showsthe change in Helmholtz free energy as a function of tempera-ture for the values of Aij mentioned above. As stated, for Aij = 1.0the system is not thermodynamically stable in the range of tem-peratures studied. When Aij equals 2.0, 2.5, and 1.5 the systembecomes thermodynamically stable at all temperatures considered.As in the case of H2@C60, the reason for this stabilization is due toenergetic contributions (Fig. 3B). Similarly to the monomeric sys-tem, destabilized the dimeric system due to entropic effects arenot significant. Figs. 4A and C show snapshots at T = 100 K andT = 10 K, respectively, for the stable Aij = 3.0 system, whereasFigs. 4B and D show the sampled regions. As in the monomericsystem, as the temperature decreases the system becomes moredelocalized, and the sampled configuration space is localized inparticular regions of the C60. These results suggest that in orderto accommodate two hydrogen molecules in C60, the attraction be-tween the fullerene and H2 need to be increased, and the systemmust be keep at relatively low temperature. Modification of the ex-ternal area of the fullerene by adding metals and/or electrons [14]could increase the attraction between H2 and C60.

4. Conclusion

By performing path integral computer simulations the thermo-dynamic stability of H2 and (H2)2 entrapped in C60 was deter-

Table 2Average values for: average hydrogen internal energy, 〈V int〉, average hydrogen–fullerene energy, 〈V H2–C60 〉, average potential energy, 〈V 〉, average total energy, 〈Etot〉, andaverage radius of gyration, 〈Rg〉 for H2@C60.

T (K) 〈V int〉 (kJ mol−1) 〈V H2–C60 〉 (kJ mol−1) 〈V 〉 (kJ mol−1) 〈Etot〉 (kJ mol−1) 〈Rg〉 (bohr)

300 70.06(2) −7.92(2) 62.79(2) 66.89(1) 0.072(3)200 47.23(1) −8.60(3) 38.49(2) 41.25(1) 0.10(1)100 23.43(2) −8.81(1) 14.61(2) 16.11(3) 0.20(2)

50 11.52(3) −9.01(3) 2.49(1) 3.46(8) 0.39(2)25 5.40(4) −9.05(2) −3.64(3) −2.80(1) 0.63(2)10 1.75(1) −9.07(1) −7.32(1) −6.59(3) 0.84(3)

5 0.62(2) −9.10(2) −8.50(2) −7.87(1) 0.88(2)

1588 A. Cruz, G.E. López / Physics Letters A 376 (2012) 1584–1588

mined. One hydrogen molecule is thermodynamically stable at allthe temperatures considered. The stabilization is due to energeticcontributions to the change in Helmholtz free energy. Modificationin the interparticle parameter that describes the H2–C60 interac-tion caused an increase in the stability of the system. In the caseof (H2)2@C60 no stable structures were obtained in the tempera-ture range and values of ε(H2)x–Ck considered. Increments in theattractive part of the H2–C60 potential allowed stabilization of thesystem due to an increment in energetic effects. At present, entrap-ment of quantum particles in various cages is being considered indetail.

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