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Microscopic pressure-cooker model for studyingmolecules in confinementRuben Santamariaa, Ludwik Adamowicza & Hortensia Rosas-Acevedob

a Department of Chemistry and Biochemistry, University of Arizona, Tucson, AZ, USAb FES-Zaragoza, UNAM, D. F. MexicoPublished online: 17 Nov 2014.

To cite this article: Ruben Santamaria, Ludwik Adamowicz & Hortensia Rosas-Acevedo (2014): Microscopic pressure-cookermodel for studying molecules in confinement, Molecular Physics: An International Journal at the Interface Between Chemistryand Physics, DOI: 10.1080/00268976.2014.968649

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Molecular Physics, 2014http://dx.doi.org/10.1080/00268976.2014.968649

RESEARCH ARTICLE

Microscopic pressure-cooker model for studying molecules in confinement

Ruben Santamariaa,∗, Ludwik Adamowicza and Hortensia Rosas-Acevedob

aDepartment of Chemistry and Biochemistry, University of Arizona, Tucson, AZ, USA; bFES-Zaragoza, UNAM, D. F. Mexico

(Received 9 July 2014; accepted 18 September 2014)

A model for a system of a finite number of molecules in confinement is presented and expressions for determining thetemperature, pressure, and volume of the system are derived. The present model is a generalisation of the Zwanzig–Langevinmodel because it includes pressure effects in the system. It also has general validity, preserves the ergodic hypothesis, andprovides a formal framework for previous studies of hydrogen clusters in confinement. The application of the model isillustrated by an investigation of a set of prebiotic compounds exposed to varying pressure and temperature. The simulationsperformed within the model involve the use of a combination of molecular dynamics and density functional theory methodsimplemented on a computer system with a mixed CPU–GPU architecture.

Keywords: pressure and temperature effects; molecular dynamics; thermodynamic states; electronic structure; confinedmolecules

Introduction

The behaviour of particles under the effects of pressure andtemperature is of great interest. There are several modelswhich can be applied to study this behaviour [1,2]. The mod-els usually assume periodic boundary conditions to avoidphysical limits regarding the system size, and introduce ad-ditional degrees of freedom in the system to mimic the roleof a thermostat and a barostat. Still, pressure and temper-ature can be introduced to the system in the simulation byincluding the interaction with an additional set of particlessurrounding the particles of interest. In practice, a system ofparticles placed under confinement and subject to the effectsof temperature requires the presence of confining walls lim-iting its spatial expansion. The walls need to be in contactwith a heat reservoir to maintain a constant temperature.Thus, the container has to be allowed to exchange energywith the reservoir, while keeping the pressure of the systemat certain value. At the microscopic level, the surroundingmedium and the container in the model need to have atomicstructures to allow the atoms of the two systems to interactand exchange energy. For instance, as the energy transferfrom the heat-bath particles to the container makes the con-tainer atoms to vibrate, this vibrational motion propagatesto the confined atoms. Achieving the thermodynamic equi-librium of the system in the container usually takes time.After the system is equilibrated, the confined particles ap-pear under different thermodynamic conditions before theenergy transfer took place. The model describing this be-haviour resembles heating of a pressure cooker.

∗Corresponding author. Email: rubensantamaria@email.arizona.edu

The purpose of this work is to build a ‘pressure cooker’model and to derive appropriate equations describing themotion of molecules in a confined system under pressureand temperature. One of the purposes of constructing such amodel is to simulate the behaviour of prebiotic compoundsin the Miller–Urey experiment [3]. By assuming that theelectrons in the system are described by quantum mechan-ics and the nuclei by Newtonian theory, a Lagrangian of theZwanzig–Langevin type is proposed for the motion of theparticles forming the system. Then, by following standardrules of analytical mechanics, the Lagrangian equations ofmotion are obtained. The model provides a formal frame-work for the previous studies of the hydrogen gas underthe effects of pressure and temperature [4–11]. Contrary toother proposed models, the present model includes a rep-resentation of the medium surrounding the system of theconfined particles by a direct inclusion of such propertiesas the structure and rigidity of the container, the viscos-ity of the fluid forming the surrounding medium, etc., de-scribed in terms of mechanical and statistical mechanicsaxioms.

The Lagrangian

The formalism used here starts with an expression for themolecular Lagrangian. Care needs to be exercised in dealingwith the fact that the Lagrangian is related to the Hamilto-nian through a Legendre transformation. This is importantbecause the Hamiltonian is used to represent the particles of

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the heat reservoir surrounding the system according to thestatistical mechanics theory [12]. Furthermore, it needs tobe ensured that the erogodic hypothesis is preserved in themodel. As the model is to be applied to a system with a finitenumber of particles, the interactions of the atoms formingthe container with the atoms of the confined system needto be included and properly described in the simulation.Taking into account all the interactions raises the compu-tational cost of the simulation. This cost increases as moreparticles are added to the system. In spite of that, the modelis of general validity and it is applicable to any number ofparticles.

The description of the ‘pressure cooker’ model at themolecular level starts with writing a Lagrangian of theZwanzig–Langevin type for the system consisting of a heatbath and a container holding some number of particles [13].The Lagrangian describes all particles of the system and ac-counts for their individual motions and for all inter-particleinteractions:

L({qij , xi, sk}) =∑

i

mi

2x2

i +∑

i

∑j

mij

2q2

ij +∑

i

m′i

2s2i

−∑i,j

[mij ω2

ij

2q2

ij − cij qij xi + c2ij

2mij ω2ij

x2i

]

−∑

j

kj

2

(xj − x0

j

)2 − U ({xi, sj }). (1)

Lagrangian L({qij , xi, sk}) involves three types of par-ticles described by the qij, xi, and si coordinates (in thediscussion which follows these particles will be called qij

particles, xi particles, and si particles, respectively). Theparticles of the first type belong to the heat bath (we alsorefer to these particles as medium or fluid particles). Theparticles of the second type belong to the container (orcage), and the particles of the third type belong to the con-fined system, which is being studied. The first three termsin L represent the kinetic energies of the three types ofparticles and the next terms represent the inter-particle in-teractions. The number of particles of the fluid thermal bathis assumed to be very large and they are assumed to vibrateharmonically. These particles are described by coordinatesqij, where the pair of indices ij indicates that the ith particleof the container is involved in the interaction with the jthparticle of the fluid. The mass of the ijth particle of the fluidis mij. It should be noted that mij is different from mji, mij �=mji. The fluid particles vibrate with angular frequency ωij.The particles of the fluid interact with the particles of thecontainer and the interaction is represented by a bilinearform involving coefficients cij and expressed in units of en-ergy per square distance. The position vector xi correspondsto the ith particle of the container. This particle has massmi. Additional terms are introduced to the Lagrangian todescribe harmonic vibrations of the cage particles around

their equilibrium positions, x0j . This part of the Lagrangian

will be discussed later. The third type of particles are parti-cles located inside the cage. They are described by positionvariables si. The mass of the ith particle located inside thecage is m′

i . The interaction between particles described byposition vectors xi and si is represented by the interactionpotential U({xj}, {sk}). This potential is determined usingab-initio quantum-mechanical calculations using the time-independent Schrodinger equation:

Hψ({xj , sk}) = E ψ({xj , sk}) ;

U ({xj (t), sk(t)}) = E| {xj (t),sk (t)} . (2)

The Hamiltonian H includes all x- and s-type par-ticles with their corresponding interactions. The Born–Oppenheimer approximation is assumed in this case. Thenuclei move and interact being surrounded by an electroncloud. The electronic energy, E, has parametric dependenceon the instantaneous positions {xj, sk} and plays the role ofthe interaction potential for the nuclei. The particles locatedinside the cage do not directly interact with the fluid parti-cles; however, their state indirectly depends on the state ofthe fluid particles through the interaction with the particlesof the container. Forces Fxj

and Fskacting on the xj and

sk particles are obtained by differentiating the interactionpotential:

Fxj= −∂U ({xi(t), sk(t)})

∂xj

;

Fsk= −∂U ({xi(t), sj (t)})

∂sk

. (3)

Summarising, the Lagrangian, L({qij , xi, sk}), de-scribes the container, which holds the particles of thestudied system, submerged in a heat bath. It representsthe ‘pressure-cooker’ model with the walls of the cookerformed by the molecular cage.

Equations of motion

The following steps are involved in deriving the equationsof motion for the pressure-cooker model, according to therules of analytical mechanics. The equations of motion forthe xi particles are obtained by the following differentiation[14]:

dπi

dt= ∂L

∂xi

; πi = ∂L∂xi

. (4)

Similar equations of motion are derived for the qij andsi particles. By using Lagrangian (1), these equations have

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the following form:

mixi =∑

j

(cij qij − c2

ij

mijω2ij

xi

)

− ki

(xi − x0

i

) − ∂U ({xj , sk})∂xi

,

m′k sk = −∂U ({xi, sj })

∂sk

; mij qij + mij ω2ij qij = cij xi .

(5)

These are second-order differential equations and theirnumber depends on the number of degrees of freedom inthe system. There are three types of equations. The firsttype includes equations corresponding to the particles ofthe cage. Every particle of the cage interacts with sev-eral particles of the fluid and the particles confined in-side the cage. The second type includes equations for thepressurised particles located inside the cage. The trajec-tory of a confined particle directly depends on the otherpressurised particles and on the particles of the cage. Italso indirectly depends on the particles of the fluid. Theequations of the third type describe the motion of the fluidparticles. For simplicity, we assume that each fluid particleinteracts only with a single particle of the cage. The nextstep involves finding the solution of the equations describ-ing the motion of the qij particles and use this solution tosolve the equations describing the motion of the xi parti-cles. The method of parameters variation is applied in thistask and the solution which is obtained has the followingform [15]:

qij (t) − cij

mij ω2ij

xi(t)

=[qij (0)− cij

mij ω2ij

xi(0)

]cos(ωij t)+ kij (0)

mij ωij

sin(ωij t)−

− cij

mij ω2ij

∫ t

0cos[ωij (t − t ′)] xi(t

′) dt ′. (6)

The values of the particle positions, qij(t) and xi(t), attime t = 0 define the initial conditions for the coordinatesand the momenta (kij = mij qij ) of the heat-bath particles.Note that the term containing x2

i (t) in Equation (1) is trans-formed into a term proportional to xi (due to differentiationwith respect to time) in the equation of motion (5), and isabsorbed into Equation (6) because the fluid particles arecoupled to the cage particles. It is due to this term that theinteraction of the cage particles with the fluid particles istaken into consideration and produces a viscosity term atthe statistical level. Next, using the above expression in the

equation for xi(t), (5), we have

mixi =∑

j

{[cij qij (0) − c2

ij

mij ω2ij

xi(0)

]

× cos(ωij t) + cij kij (0)

mij ωij

sin(ωij t)

}

−∑

j

{c2ij

mij ω2ij

∫ t

0xi(t

′) cos[ωij (t − t ′)] dt ′}

− ki(xi − x0i ) − ∂U ({xj , sk})

∂xi

. (7)

The above expression describes the coupling betweenthe motions of the particles of the container and the parti-cles of the fluid. In this point, we only have to solve twotypes of equations of motion, i.e. the one given above andthe equation of motion for the sk particles given by Equa-tion (5). However, the above expression is complicated andrequires simplification. This is accomplished by combiningthe effects due to all heat-bath particles into just two terms.Since we are not interested in describing every particle ofthe heat reservoir, a collective approach is used to repre-sent the effects of the fluid particles. In this approach, wedefine the so-called ‘noisy source’, Gi, and the so-calleddissipative kernel, Ki, both of statistical nature:

Gi(t) =∑

j

cij

{kij (0)

mij ωij

sin(ωij t)

+[qij (0) − cij

mij ω2ij

xi(0)

]cos(ωij t)

},

Ki(t − t ′) = 1

mi

∑j

c2ij

mij ω2ij

cos[ωij (t − t ′)]. (8)

The stochastic force, Gi, introduces: (1) a dependenceof the fluid particles and the cage particles on the initialconditions, and (2) a set of frequencies that make Gi be aso-called ‘coloured noise’ force. The dissipative kernel, Ki,is a function of the coupling coefficients, cij. It correlatesthe past time, t′, with the present time, t, by introducing adependence on the time difference, t − t′. The Ki functionis the kernel of the integral over time (see Equation (7)) andcan be considered as a memory function. The equation ofmotion for the xi particles is now written in terms of thekernel, Ki, and the stochastic force, Gi, as

mixi = −ki (xi − x0i ) − ∂U ({xj , sk})

∂xi

−mi

∫ t

0Ki(t − t ′) xi(t

′) dt ′ + Gi(t). (9)

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This expression describes the behaviour of the ith par-ticle of the cage immersed in the heat bath. The particleinteracts with the other cage particles and with the parti-cles located inside the cage. Equation (9) is still difficultto solve and a further simplification needs to be made. Thesimplification is based on a model describing the responseof a capacitor to a set of subsequent electric square pulses.This model is applied to describe a particle with short timeintervals of mobility due to thermal fluctuations of the fluid.Accordingly, the memory function is represented as a set ofsubsequent square pulses appearing at regular time inter-vals. In this case, the speed is an exponential pulse functionthat rapidly decays with time due to the fluid viscosity ef-fects. The convolution of such a function, represented by theintegral in Equation (9), results in a ‘smeared’ form of theparticle’s velocity with the proportionality factor beingthe friction coefficient, ξ i:

mi

d2xi(t)

dt2= −ki [xi(t) − x0

i ] − ∂U ({xi, sk})∂xi

+Gi(t) − mi ξi

dxi(t)

dt,

m′k sk = −∂U ({xi, sj })

∂sk

. (10)

The first of the above equations is the Langevin equa-tion. The set of particles xi describes a Brownian body withan atomic structure holding a set of particles inside. Thelast two terms of the equation have a non-potential originbecause they are statistical terms representing the net effectthe fluid particles have on the container. The force termcontaining ξ i introduces the local friction of the medium.The force Gi represents random interactions exerted by themedium on the ith particle of the container. Such termsfavour the exchange of energy with the fluid, but make theequation of motion of particle xi time irreversible due to therandom terms that are unpredictable in nature.

Integrating the equations of motion

The two above-described differential equations are cou-pled and need to be solved in an iterative (self-consistent)manner. When this is done, the coordinates xi(t1), xi(t2),. . .provide the trajectory of the ith cage particle interact-ing with all other particles of the system. Similarly, thecoordinates sk(t1), sk(t2), . . .provide the trajectory of thekth particle located inside the cage and interacting with thecage particles and other particles located inside the cage.Integration of the equation for xi(t) leads to [16–18]:

xi(tn+1) = xi(tn) + c1 vi(tn) δt + c2Fi(tn)

mi

δt2 + Xin(δt),

vi,1/2(tn) = c0 vi(tn) + (c1 − c2)Fi(tn)

mi

δt + V in (δt),

vi(tn+1) = vi,1/2(tn) + c2Fi(tn+1)

mi

δt. (11)

Fi defines the net force exerted on the ith particle be-longing to the cage. This force is due to the interaction of theith particle with the other particles of the cage and with theparticles located inside the cage. The force is determinedfollowing the ab-initio calculation of the potential as: Fi =−∂U({xj, sk})/∂xi. The terms Xn and Vn are the stochasticposition and the stochastic speed of the ith particle, respec-tively. These quantities, obtained by integrating the forceGi, depend on the interaction of the particle with the heatbath. The coefficients ci, with i = 0, 1, 2, depend on thetime step, δt, and on the friction coefficient of the medium,ξ i = ξ . They can be calculated in a recursive manner. Forsmall x = ξδt, we have [17]

c0 = e−x = 1 − x+1

2x2 − 1

6x3 + 1

24x4 − 1

120x5 + · · ·

c1 = 1 − c0

x= 1 − 1

2x + 1

6x2 − 1

24x3 + 1

120x4 + · · ·

c2 = 1 − c1

x= 1

2− 1

6x + 1

24x2 − 1

120x3 + · · · (12)

Expression (10), which describes the behaviour of theparticles located inside the cage, is integrated using thestandard velocity-Verlet integrator procedure:

sk(tn+1) = sk(tn) + vsk(tn) δt − 1

2m′k

∂U ({xi, sk})∂sk

δt2,

vsk(tn+1) = vs

k(tn) − 1

2m′k

[∂U ({xi(tn), sk(tn)})

∂sk

+ ∂U ({xi(tn+1), sk(tn+1)})∂sk

]δt. (13)

The speed of the particle sk is sk = vsk . The superscript

s is added to distinguish the speed of an sk particle from thespeed of an xi particle. The interaction potential, U, and,henceforth, the force on the particle sk are computed on thefly in the dynamic simulation. No stochastic terms appearin the last set of equations, as the particles inside the cagehave no direct contact with the fluid.

The total energy is computed by adding the kinetic andpotential energies of the individual particles. According tothe equipartition theorem, the kinetic energy of the parti-cles of the fluid in the thermodynamic equilibrium is pro-portional to the temperature. Therefore, in the simulationperformed at constant temperature, the total kinetic energyof the fluid particles is approximately constant and, in prin-ciple, can be removed from the expression for the totalenergy of the system, as it only shifts the energy value by aconstant. Also, as the fluid particles are weakly coupled tothe cage particles, the contribution to the total energy from

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the interactions of these two types of particles is neglected.With that the total energy is

Etot =∑

i

1

2mix

2i +

∑i

1

2m′

i s2i

+∑

i

ki

2(xi − x0

i )2 + E({xj , sk}). (14)

The kinetic energies of the cage particles and the parti-cles located in the cage are described by the first two termsin the above equation. The following terms correspond tothe potential energy contributions. The first of these termsis the harmonic vibrational potential energy and the secondone is the ab-initio energy obtained from Equation (2). Thetotal energy is not conserved with time due to the stochas-tic random terms (which reflect the fluctuation-dissipationnature of the medium [19]) in the equations of motion forthe xi particles. The approach is valid in a general case andapplies equally to small and large systems of particles.

Distribution function of the cage particles in the fluid

Expression (10) contains random terms that introduce thefluctuation-dissipation effects of the fluid on the cage parti-cles. These stochastic contributions should be dependent onthe distribution function compatible with the macroscopiccharacteristics of the fluid. Thus, the next task is to deter-mine the distribution function from which the random termsof Equation (10) can be determined. This problem was, infact, first discussed by Chandrasekhar back in 1945 [20].The probability distribution function in the phase space,W(r, v, t + δt), depends on time. The distribution functionat time t + δt is deduced from the probability distributionfunction at an earlier time t by integration. The time evolu-tion of the system is described as a Markoff process. Thus,we have

W (r, v, t + δt) =�

W (r − δr, v − δv, t)

×p(r − δr, v − δv; δr, δv) d(δr) d(δv). (15)

Function p in the above equation is a transition prob-ability of W evolving from time W(t) to W(t + δt). Thetransition probability is a function of the velocity and, byassuming Dirac delta functions for the positions, the tran-sition probability becomes a product and has the followingform:

p(r, v; δr, δv) = p(r, v; δv) δ(�x − vx�t)

× δ(�y − vy�t) δ(�z − vz�t). (16)

Using this form of p in Equation (15) leads to integralsover the velocity components:

W (r, v, t + δt) =�

W (r − vδt, v − δv, t)

×p(r − vδt, v − δv; δv) d(δv).

By expanding the functions in power series (this is con-sistent with the Markoff process), it is possible to transformthe integral equation into a boundary-value problem of par-tial differential equations:

D

DtW = ξ divv(Wv) + ξ

kBT

m∇2

vW ;

D

Dt= ∂

∂t+ v · ∇r + F · ∇v. (17)

The solution of this equation, together with the secondmoments of the position and velocity of a Brownian parti-cle, leads to the following bivariate probability distributionfunction:

W (Xn(δt), Vn(δt)) =1

2π√

ac − b2e−(aX2

n−2bXnVn+cV 2n )/2(ac−b2), a = 〈V 2

n (δt)〉 ;

b = 〈Xn(δt) Vn(δt)〉 ; c = 〈X2n(δt)〉. (18)

Constants a, b, and c correspond to the second momentsof the Brownian particle and are given in terms of thefriction constant ξ , temperature T of the fluid, and massm of the Brownian particle. In case x = ξδt 1, the secondmoments are expanded in power series:

〈X2n(δt)〉 =

2kBT

3mx δt2

[1−3x

4+7x2

20−x3

8+31x4

840−3x5

320+ · · ·

],

〈V 2n (δt)〉 =2kBT

mx

[1 − x + 2x2

3− x3

3+ 2x4

15− 2x5

45+ · · ·

],

〈Xn(δt) Vn(δt)〉 =kBT

mx δt

[1 − x + 7x2

12− x3

4+ 31x4

360− x5

40+ · · ·

].

(19)

The distribution function, W, described by Equations(18) and (19) determines the stochastic terms, Xn and Vn,used in Equations (11) for describing the time evolution ofthe particles forming the container. The distribution func-tion, W, is not only consistent with the Langevin equation ofmotion (10), but also it is compatible with both the equipar-tition theorem, which involves the speed of the particle inthe fluid, and with the diffusion equation, which describesthe position of this particle as a function of time [20].

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Expressions (19) show that the friction is the mainmechanism through which the particles dissipate energyinto the thermal energy. Conversely, the thermal fluctuationscan be transferred to the internal energy of the particles.The two processes complement each other and an equilib-rium between them can be established. When the numberof particles of the cage is small (compared to the numberof particles of the fluid, which is usually the case) and theinteraction between the cage and the fluid is sufficientlyweak that it can be neglected, the distribution function ofthe fluid particles is that of the canonical ensemble (this canbe shown by relating the Lagrangian (1) with the Hamilto-nian of the fluid particles) [21,22]. In short, the distributionfunction, W, can be used to compute the probabilistic mo-bility of the particles forming the container.

Molecular container

In the following, we take a fullerene cage formed by 180He atoms as the container. The equation of motion of thecage atoms is given by Equation (10). We then examinethe behaviour of the atoms of the cage in the fluid by usingthe bivariate distribution function. The analysis is simplifiedby eliminating the systematic forces in Equation (10), whichare independent of the fluid:

mi

d2xi(t)

dt2= −ki [xi(t) − x0

i ] − mi ξi

dxi(t)

dt+ Gi(t).

(20)

The spring forces (described by the first term of theabove equation) keep the helium atoms in place. Withoutthese forces, the helium atoms would diffuse in the fluid.Figure 1, inset A, shows how the temperature, T, of thecage changes with time. The temperature of the fullereneatoms is computed in the simulation according to the ki-netic theory, namely, from the velocities of the atoms ofthe cage. The inset A shows that the mechanical temper-ature is consistent with the statistical temperature whichappears in the distribution function. Due to the inertia ofthe particles and low velocities of the fluid particles, ittakes some time to reach the thermodynamic equilibriumin the system at a given temperature. Figure 1, inset B,shows similar plots as in A but for different friction co-efficients, ξ . It can be noticed that, as expected, the equi-librium temperature is achieved faster for higher frictioncoefficients. In all cases, the fluctuations of the temperatureare within 10% of the equilibrium temperature. The param-eters T and ξ characterise the macroscopic properties of thefluid. Thus, the influence of these parameters on the dynam-ics of the particles of the cage submerged in the fluid can bedescribed.

Thermodynamic observables: volume, temperature,and pressure

It is necessary to define such thermodynamic observables astemperature, volume, and pressure to determine the thermo-dynamic state of the particles confined in the cage (Brown-ian body). The temperature of the container was discussedpreviously. It appears in the distribution function and thusits statistical value is determined. This value can be verifiedby comparison with the value obtained from the mechanicalconsideration.

The next observable to be determined is the volume. It isdetermined from the geometrical description of the particlesconfined in the cage. As the fullerene cage is spherical, thevolume of the confined atoms is

V = 4πR3conf/3 ; Rconf = Rpart + [Rcage − Rpart]/2.

(21)

The average radius of the sphere containing the con-fined particles is Rconf and the radius of the fullerene cage isRcage. The quantity Rpart is the radius of the confined atomsconsidered as point particles. The radius of the confine-ment, Rconf, is Rpart plus a correction for the electron cloudssurrounding the nuclei of the confined atoms located at theinterface of the cage and the confined system.

The last quantity to be determined is the pressure. Thisquantity is a tensor and it is determined using the mechani-cal approach. In our case, the fullerene cage has a sphericalsymmetry and the expression for the pressure is simplified.The kinetic contribution to the pressure, Pk, is due to themotion of the confined particles and is computed as PkV =NkBT, where N is the number of confined atoms, T theequilibrium temperature, V the volume of the confinement(discussed above), and kB is the Boltzmann constant. Thestatic contribution, Ps, to the pressure is due to the inter-actions between the confined particles and it is calculatedfrom the changes of the potential energy of the confinedparticles with respect to the volume:

Ps = −(∂Econf/∂V )T ; Econf = Etot − Ecage. (22)

The energy, Ecage, is the energy of the empty cage andEtot is the energy of the whole system, i.e. the atoms of thecage and the confined atoms. The pressure increases whenthe radius of the fullerene shrinks.

Equation of state

The thermodynamic observables, P, V, and T, are the fun-damental quantities which appear in the equation of state ofthe system. Several calculations are required to formulatethe equation of state. They involve the determination of thefollowing quantities: the confinement energy in terms of thevolume, V, with maintaining the equilibrium temperature

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Figure 1. Variation of the temperature with time. In inset A, the cage reaches the equilibrium at temperatures 200, 300, and 400 K.These temperatures are inserted in the distribution function (Equation (18)). In inset B, the fluid friction coefficient is changed from 5.0to 0.5 ps−1. The higher the friction coefficient, the faster the equilibrium temperature is achieved. In both the insets, the radius of the180-atom fullerene cage is 8.33 A and the spring constant is 2.00 N/cm = 0.459 Hartree/A.

fixed, and the time-averaged P , V , and T . The use of thesequantities in the equation of state allows to preserve theergodic hypothesis. Yet, it is possible to generate a fit of theEconf vs. V data and to propose an analytical expression ofEconf in terms of V [23,24] such as

Econf(V ) = −(9B0V0/η2) [1 − η(1 − x)] exp[η(1 − x)],

x = (V/V0)1/3 ; η = 3(B ′0 − 1)/2. (23)

The parameters B0, B ′0, and V0 are obtained by fit-

ting the above analytical expression to the numerical data

Econf vs. V . Hence, the quantities V and Econf in the ana-lytical expression represent the time-averaged volume andtime-averaged energy of the confined system, respectively.The parameters B0, B ′

0, and V0 are interpreted as the isother-mal bulk modulus and its first pressure derivative at zeropressure, while V0 is usually considered the zero-pressurevolume [23]. The static pressure, Ps, is obtained by applyingEquation (22):

Ps = [3B0(1 − x)/x2] exp[η(1 − x)]. (24)

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By using this procedure, it was possible, for example,to determine the thermodynamic state of a small numberof confined particles, like hydrogen clusters, resulting in anexcellent agreement with the experimental measurementsperformed on the macroscopic solid [10]. It is importantto mention that the expressions for P , V , and T are basedon the Newtonian and statistical mechanics. This makesthe confinement method independent of the number of theconfined atoms/molecules. In this respect, the method is ofa general applicability. Extending the model and permittingthe particles to escape through pores in the fullerene cagewould make the model more realistic and versatile in deter-mining the stability of a set of confined particles at differentpressures and temperatures.

Ionising radiation

One of the key ingredients in the Miller–Urey experimentis the presence of ultraviolet (UV) radiation (see Figure 2).Making the glass flask holding the molecules of the inves-tigated compounds transparent to the UV radiation allowsfor the radiation to electronically excite the molecules andremove electrons from them, which resulted in breaking ofchemical bonds and in chemical transformations of the sys-tems. In the present simulations, the effect of the radiation isconsidered in an implicit manner by stochastically breakinga pair of bonds located close to each other at some randommoments of time. In such an approach, the interaction of themolecules with the UV electromagnetic field does not needto be explicitly included in the Lagrangian (Equation (1)).After the radiation-striking event and an ejection of an elec-tron, the charge and multiplicity of the system change from(charge = −1, multiplicity = 2) to (0,1), assuming that onlyone electron is removed in the process. In the simulation,the system is permitted to recapture the lost electron (after afew time steps of the simulation) from the surrounding en-vironment (see Table 1). The stochastic breaking of bondsin a given region mimics the net effect of the UV radia-tion. This is done without an explicit consideration of theinteraction of a photon with the electrons of the moleculewhich, if the photon energy is sufficiently high, may causea bond breakage.

Ab-initio method

In order to perform a molecular dynamics simulation, a pro-cedure for determining the time-independent wave equa-tion given by expression (2) needs to be established. Forthis, we apply the density functional theory (DFT) method,in particular, the generalised gradient version because itis sufficiently accurate for the molecular energy evalua-tion, yet computationally efficient and simple to apply. Weuse the functionals of Becke for the exchange energy [26]and Lee–Yang–Parr for the correlation energy [27], as theyhave shown the correct behaviour and good accuracy in

calculations of confined clusters. The calculations are per-formed using the standard 6-31G Gaussian basis set. Theuse of this basis keeps the computational cost of the calcu-lation at a modest level. The dispersion forces play a minorrole for pressurised molecules and the long-range interac-tions are insignificant in comparison with the short-rangeones.

In the DFT method, an electronic wave function is con-structed in the form of a single determinant using theDFT molecular orbitals. These are determined by self-consistently solving the Kohn–Sham one-electron equa-tions with a threshold convergence of about 10−6 Hartreein the density matrix and the energy. The DFT calculationsare performed using either open- or closed-shell methodand the spin-restricted approach. To mimic an absorptionof a photon by a molecule and an ejection of an electron,the −1 charge and the spin multiplicity 2 of the system arechanged to zero charge and spin multiplicity 1. The initialcharge and multiplicity of −1 and 2 are usually recoveredafter some time steps in the simulation (see Table 1). Thesoftware package used in the simulation is written in For-tran language and is implemented on a CPU–GPU computerworkstation. The package uses TeraChem [28], written inCUDA language, for computing the DFT electronic energy.In principle, any other electronic-structure program pack-age, such as, for example, NWChem [29] or Gamess [30],can be used for this purpose. The advantage of employingTeraChem is the possibility to use the graphics process-ing units (GPUs), as this package is implemented on thatplatform. This greatly accelerates the calculation. In thepresent calculations, a single Tesla-K20c GPU card is used.In essence, the CUDA external program for calculating theDFT energy, which runs on GPUs, is included as a sub-routine in the Fortran code, which runs on the CPU. Suchan approach eliminates the need to modify the source codeof the external program which does the electronic structurecalculation. The software and the script to run it will bedescribed and made available elsewhere.

Dynamics history

In this work, we consider a set of 14 H2O, 8 NH3, and 8 CH4

molecules (114 atoms in total) confined in a cage of a spher-ical fullerene formed by 180 helium atoms (see Figure 1).This system models the experimental set-up used in theMiller–Urey experiment, where a mixture of H2O, NH3, andCH4 was confined to a glass flask and the chemistry occur-ring between these systems was studied under the effects ofthe temperature and pressure. The reason for using inert Heatoms to form the fullerene cage is to minimise the chemicalinteraction between the container and the molecules con-fined in the cage. Also, using helium minimises the numberof electrons associated with the container, what speeds upthe calculation. In the Miller–Urey experiment, the coolingof the compounds located inside the flask was done with the

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Figure 2. The model used in the Miller–Urey experiment which reproduces the pristine conditions on planet Earth and allows to elucidatethe synthesis of organic molecules from prebiotic compounds under the effects of temperature and ionising radiation [25]. The figures onthe right show the number of 14 H2O, 8 NH3 and 8 CH4 molecules located in the 180-atom fullerene cage.

help of an externally attached coil filled with cold water (acondensation column). In the case of the computer model,as the electrons of the He atoms are strongly localised, theenergy transfer through the walls of the cage and the coolingof the molecules located inside the cage are expected to beslow. However, as the simulation involves the distributionfunction (18), the cooling can be accelerated by applyingthis function to the atoms located close to the surface of the

fullerene cage (the threshold distance to determine whichatoms this applies to is 4.0 A).

In Table 1, we describe the main events of the dy-namics simulation. These include the interaction of theconfined molecules with the radiation, the time and therate of shrinking the container, the equilibrium temperaturein different time intervals, the periods of the cage cool-ing, etc. In particular, the temperatures imposed on the

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Table 1. Main events in the dynamics of the container and confined atoms.∗

Step Events

0 Set initial conditionsNumber of confined atoms 114Number of container atoms 180Method RO-BLYP/6-31g

1–5009 Time step (fs) 1.0Equilibrium temperature (K) 400Spring constant (Hartree/A2) 0.459

Friction constant (ps−1) 5.0Charge, multiplicity −1, 2

5010–8469 Cage shrinkingShrinking factor 0.95

Reference radius (A) 10.0New reference radius (A) 9.50

8470–8479 Striking radiationCharge, multiplicity 0, 1

Broken bonds OH, NHIonisation energy (eV) 3.6154

8480–9009 Electron uptakeCharge, multiplicity −1, 2Energy change (eV) −0.1356

9010–9199 Cage shrinkingShrinking factor 0.95

Reference radius (A) 9.50New reference radius (A) 9.02

9200 Striking radiationCharge, multiplicity 0, 1

Broken bonds CH, OHIonisation energy (eV) 3.0470

9201–10572 Cooling container atomsEquilibrium temperature (K) 300

Charge, multiplicity −1, 2Friction constant (ps−1) 7.0

10573–10583 Striking radiationCharge, multiplicity 0, 1

Broken bonds CH, OHIonisation energy (eV) 3.6520

10584–11783 Electron uptakeCharge, multiplicity −1, 2Energy change (eV) 0.2677

11784–11793 Striking radiationCharge, multiplicity 0, 1

Broken bonds OH, OHIonisation energy (eV) 4.7307

11794–11900 Electron uptakeCharge, multiplicity −1, 2Energy change (eV) 0.6022

11901–13000 Cooling container and confined atomsThreshold distance to the cage (A) 4.0

13001–16000 Cooling container and confined atomsEquilibrium temperature (K) 200

∗The events are correlated with the curves of Figure 3.

particles forming the container are 400, 300, and 200 K,with friction constants of 5 and 7 ps−1, and constant springfactor of 0.459 Hartree/ A2. The initial and final valuesof the temperature and the friction constant are used to de-termine when to start the heating and cooling processes at

the different stages of the simulation. The events describedin Table 1 correlate with the curves shown in Figure 3(a),where the temperatures of the cage atoms and the confinedmolecules are given in terms of time. Figure 3(b) shows thetime dependency of the energy, Econf (Equation (22)), of

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Figure 3. Inset A in the figure shows the temperature variation of the cage atoms and the confined atoms with time. The relevant eventsof the dynamics, like cooling of the cage atoms, and the effect of incident radiation on the confined atoms are indicated with symbols andcolours. The equilibrium temperatures of the cage atoms are also given. The events are correlated with those of Table 1. Inset B in thefigure shows the variation of the energy of the confined atoms with time. The units of time, temperature, and energy are fs, K, and Hartree,respectively.

the confined molecules. Both figures show sharp changesat the moments when the radiation strikes the system. Onecan also see sharp temperature changes with the shrink-ing and cooling of the cage. The intervals of time where

thermal equilibrium of the container occurs are [1000–4500] and [5500–9000] with T = 400 K, [9500–13000] withT = 300 K, and [13500–16000] with T = 200 K. The tem-perature of the molecules confined inside the cage exhibits

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more radical changes, specially in the intervals of the radi-ation striking. After the shrinking and cooling stages of thecontainer, the temperature of the confined molecules slowlychanges. In effect, the thermal equilibrium is achieved ata faster pace. It is only during the final cage cooling thatthe confined molecules and the cage atoms fully equilibrateand reach the same temperature.

According to the second law of thermodynamics,�Q = �U at a constant volume. This means that theenergy of the confined molecules depicted in Figure 3(b)increases as more energy transfers to them from the cageatoms. An increase of the cage volume should reduce boththe energy and temperature. However, as in the simulation,the temperature is lowered by cooling the fluid (this isdone by lowering the temperature in the distributionfunction), the cooling is transferred to the container (dueto fluctuation-dissipation effects) and eventually to themolecules confined in the cage.

The simulations show that, in the case of the prebi-otic compounds, the breaking of bonds due to the tem-perature effects is, as expected, very sporadic. Also, thebroken bonds are immediately restored particularly whenhigh pressure is applied. The simulations show the forma-tion of hydrogen-bonded networks of water and ammoniamolecules, while the methane molecules show hydropho-bic character as they do not participate in the networks. Ineffect, they are removed from the central region of the con-tainer and stay for most of the time close to the containerwalls. The simulations reveal that the ionising radiationis the main cause of the bond breaking by far exceedingthe bond breaking resulting from the temperature effect atthe confined conditions studied in this work. In conclu-sion, it is only due to the radiation that the formation ofnew compounds is possible. However, some preliminaryresults concerning simulation of a confined system, whereSH2 molecules are introduced to the prebiotic soup, leadto somewhat different results. Future work will need toinvolve changing the densities or the numbers of the con-fined molecules and their chemical composition. Additionof light metals as elements of the confined molecules willalso be considered. The use of other simulation approachessuch as the quantum-mechanics/molecular-mechanics(QM/MM) scheme will also be considered. They may allowto reduce the number of electrons involved in the containerby simultaneously increasing the number of confined atoms.Finally, the work presented here is expected to be of helpfor the development of new force fields that take the effectsof temperature and pressure into account. Efforts in thisdirection have already started by other researchers [31–33].

AcknowledgementsWe acknowledge Prof. Bokhimi for the use of the Tesla K20c GPUhardware located in his laboratory.

References[1] W.G. Hoover, Phys. Rev. A 31, 1695 (1985).[2] M.E. Tuckerman, Y. Liu, G. Ciccotti, G.J. Martyna, J. Chem.

Phys. 115, 1678 (2001).[3] S.L. Miller, Science 117, 528 (1953).[4] J. Soullard, R. Santamaria, S.A. Cruz, Chem. Phys. Lett.

391, 187 (2004).[5] R. Santamaria, J. Soullard, Chem. Phys. Lett. 414, 483

(2005).[6] J. Soullard, R. Santamaria, J. Jellinek, J. Chem. Phys. 128,

064316 (2008).[7] R. Santamaria, J. Soullard, J. Jellinek, J. Chem. Phys. 132,

124505 (2010).[8] J. Soullard, R. Santamaria, D. Boyer, J. Phys. Chem. A115,

9790 (2011).[9] R. Santamaria, J.A. Mondragon-Sanchez, X. Bokhimi, J.

Phys. Chem. A 116, 3673 (2012).[10] R. Santamaria, X. Bokhimi, J. Soullard, J. Jellinek, J. Phys.

Chem. A 117, 5642 (2013).[11] R. Santamaria, J. Soullard, X. Bokhimi, P.R. Martınez-

Alanis, J. Chem. Phys. 140, 194301 (2014).[12] D.A. McQuarrie, Statistical Mechanics (University Science

Books, Mill Valley, CA, 2000).[13] R. Zwanzig, J. Stat. Phys. 9, 215 (1973).[14] H. Goldstein, Classical Mechanics (series in Physics), 2nd

ed. (Addison-Wesley, Manila, 1980), p. 672.[15] W.E. Boyce and R.C. DiPrima, Ecuaciones Diferenciales,

3rd ed. (Limusa, Mexico City, 1980), pp. 159–162.[16] W.F. van Gunsteren, H.J.C. Berendsen, Mol. Phys. 45, 637

(1982).[17] M.G. Paterlini and D.M. Ferguson, Chem. Phys. 236, 243

(1998).[18] E. Vanden-Eijnden and G. Ciccotti, Chem. Phys. Lett. 429,

310 (2006).[19] R. Kubo, Rep. Prog. Phys. 29, 255 (1966).[20] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).[21] S.E. Feller, Y. Zhang, R.W. Pastor, and B.R. Brooks, J.

Chem. Phys. 103, 4613 (1995).[22] L.D. Landau and E.M. Lifshitz, Statistical Physics (course

of theoretical physics), 3rd ed. (Elsevier Butterworth-Heinemann, Oxford, 2005), Vol. 5, part 1.

[23] P. Vinet, J.H. Rose, J. Ferrante, and J.R. Smith, J. Phys. 1,1941 (1989).

[24] J.H. Li, S.H. Liang, H.B. Guo, and B.X. Liu, Appl. Phys.Lett. 87, 194111 (2005).

[25] Figure modified after https://cdn-assets.answersingenesis.org/img/articles/ee/v2/miller-urey-experiment.jpg.

[26] A.D. Becke, Phys. Rev. A 138, 3098 (1988).[27] C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988).[28] I.S. Ufimtsev and T.J. Martinez, J. Chem. Theory Comput.

5, 2619 (2009).[29] M. Valiev, E.J. Bylaska, N. Govind, K. Kowalski, T.P.

Straatsma, H.J.J. Van Dam, D. Wang, J. Nieplocha, E. Apra,T.L. Windus, and W.A. de Jong, Comput. Phys. Commun.181, 1477 (2010).

[30] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert,M.S. Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A.Nguyen, S. Su, T.L. Windus, M. Dupuis, J.A. Montgomery,J. Comput. Chem. 14, 1347 (1993).

[31] P. Ren, J.W. Ponder, J. Phys. Chem. B 108, 13427(2004).

[32] O. Borodin, G.D. Smith, T.D. Sewell, D. Bedrov, J. Phys.Chem. B 112, 734 (2008).

[33] V.A. Harmandaris, G. Floudas, K. Kremer, Macromolecules44, 393-402 (2011).

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