CHEM 630P

Statistical Mechanics

Lecture Notes

Part 3: Real gases and liquids (Molecular dynamics)

by Guillaume Lamoureux

Department of Chemistry and BiochemistryConcordia UniversityMontreal, Quebec, CanadaEmail: guillaume.lamoureux@concordia.ca

Compiled on May 20, 2009

Latest version available at http://faculty.concordia.ca/glamoure/teaching.html

c© Concordia University

Real gases and liquids 1

1 Computer simulations: Molecular dynamics Reading:Chapter 3 of Allen andTildesley [1987]1.1 Basic method

The dynamics of a collection of N particles is usually described by Newton’sequations:

miri = Fi. (1)

Except for a few special cases, this system of 3N differential equations is non-integrable: it is not possible to obtain analytical formulae for the trajectoriesr(t) from the initial positions r(0) and initial velocities r(0). It is neverthelesspossible to generate a sequence of positions {r(δt), r(2δt), r(3δt), . . .} that obeyfinite-difference equations that are equivalent to Newton’s equations for δt→ 0. Wikipedia:

Finite difference methodFor instance, one could generate discrete trajectories from the equations

miri(t + δt)− 2ri(t) + ri(t− δt)

δt2= Fi(t). (2)

One would then compute ri(t + δt) from ri(t) and ri(t− δt):

ri(t + δt) = 2ri(t)− ri(t− δt) + δt2Fi(t)mi

. (3)

This equation is the Verlet formula, used for the first time in 1967 for theReference:Verlet, Phys. Rev. E 159,98–103 (1967).numerical integration of the motion of a large number of particles (864 particles,

actually).Naturally, the same finite-difference approach can be tried on Hamilton’s

equations (here in Cartesian coordinates):

ri =∂H

∂piand pi = −∂H

∂ri. (4)

For a Hamiltonian of the form

H = U({r}) +∑

i

p2i

2mi, (5)

Hamilton’s equations (4) are equivalent to Newton’s equations (1), with Fi =−∂U/∂ri. One could, for instance, generate discrete trajectories from the fol-lowing finite-difference equations:

ri(t + δt/2)− ri(t− δt/2)δt

= vi(t), (6)

vi(t + δt/2)− vi(t− δt/2)δt

=Fi(t)mi

. (7)

By substituting t with t + δt/2 in the first equation, we can compute ri(t + δt)from ri(t) and vi(t + δt/2), and vi(t + δt/2) from vi(t− δt/2):

ri(t + δt) = ri(t) + δtvi(t + δt/2), (8)

vi(t + δt/2) = vi(t− δt/2) + δtFi(t)mi

. (9)

2 1 Computer simulations: Molecular dynamics

Those equations are the leap-frog formulae, also commonly used for moleculardynamics simulations.

All molecular dynamics algorithms are based on a set of finite-differenceequations that generate discrete trajectories {r(0), r(δt), r(2δt), . . .} that areclose to the exact trajectories {r(t)}. Since the motion of a large number ofinteracting particles is usually chaotic, it is more accurate to say that the discreteWikipedia:

Chaos theorytrajectories are similar to the exact ones: however small δt is, there will be atime when the discrete trajectories will diverge from the exact ones.

1.2 Example: The harmonic oscillator

We will now apply the basic method to the one-dimensional harmonic oscillator.Wikipedia:Harmonic oscillator The Hamiltonian of a harmonic oscillator of mass m and spring constant k is

H = 12kx2 + 1

2p2/m, Hamilton’s equations are x = p/m and p = −kx, and thecorresponding Newton’s equation is mx = −kx. For initial conditions x(0) = 0and v(0) = v0, the exact trajectory is

x(t) =v0

ωsin(ωt), (10)

with ωdef=

√k/m. The period of oscillation is τ = 2π/ω and the amplitude of

oscillation is A = v0/ω. The total energy of the oscillator is 12mv2

0 . This exam-ple is simple but nevertheless captures the essential features of many types ofmotion encountered in molecular systems: bonds stretching, bond angles bend-ing, molecules rattling in the virtual “cage” formed by surrounding molecules,etc.

We will integrate the motion of this harmonic oscillator using the velocity-Verlet formulae:

v(δt/2) = v(0) + 12

F (0)m

δt, (11)

x(δt) = x(0) + v(δt/2)δt, (12)

v(δt) = v(δt/2) + 12

F (δt)m

δt. (13)

Like the Verlet and leap-frog equations, these equations are time-reversible and

Exercise:Prove that Eqs. (11) to (13)are invariant under thetransformation t→ −t.

produce stable trajectories for values of the integration step δt as large as τ/π(see Figure 1). Even more interesting is the fact that the discrete trajectories—even those that deviate considerably from the exact trajectory—fit almost per-fectly a continuous trajectory similar to the exact one, namely, a sinusoid ofdifferent frequency and amplitude. In other words, the velocity-Verlet formulaeused with a finite integration step are generating a motion that corresponds ex-actly to a modified Hamiltonian H ′(δt) similar to the original Hamiltonian H.Moreover, the modified frequency ω′(δt) and amplitude A′(δt) quickly convergeto the original values ω and A as the integration step gets smaller.

Real gases and liquids 3

Figure 1: Discrete trajectories generated with the velocity-Verlet algorithm,using gradually smaller integration steps: δt = τ/4 (in red), τ/16 (in green),τ/64 (in pink), and τ/256 (in blue). The cyan curve is the exact trajectory.The sinusoid in dashed line is adjusted on the δt = τ/4 trajectory (in red).

1.3 Evolution operator formalism

The evolution operator formalism was first applied to molecular dynamics byTuckerman, Berne, and Martyna, based on the idea that the temporal evolution

Reference:Tuckerman, Berne, andMartyna, J. Chem. Phys.97, 1990–2001 (1992).of a system can be formally represented by a time-propagation operator and

that, according to Trotter theorem, this operator can be decomposed into asequence of operators with an arbitrarily high accuracy.

Hamilton’s equations can be solved formally by introducing the Liouvilleoperator iL (also called Liouvillian), defined as the Poisson bracket with theHamiltonian of the system, H, a function of canonical variables {q} and {p}:

iLdef= {· · · ,H} =

∑i

(∂H

∂pi

∂

∂qi− ∂H

∂qi

∂

∂pi

). (14)

If we represent the state of the system by the vector Γ def= ({q}, {p}) (belongingto the phase space of the system), Hamilton’s equations (4) can be written inthe following compact form:

Γ = iLΓ. (15)

The formal solution to these equations of motion is

Γ(t) = eiLtΓ(0). (16)

The differential operator eiLt corresponds to a time propagator and representsthe following Taylor series:

Γ(t) = Γ(0) + t∂Γ∂t

∣∣∣∣0

+t2

2!∂2Γ∂t2

∣∣∣∣0

+t3

3!∂3Γ∂t3

∣∣∣∣0

+ · · · . (17)

4 1 Computer simulations: Molecular dynamics

Equation (16) does not allow much progress to be made on the analytical solu-tion of the equations of motion, and its exponential form is merely a re-writingof the trivial Eq. (17). However, it gives an interesting perspective on the formaloperation of advancing the state of a system in time. It is possible to break-downthe integration from 0 to t into a series of n steps of length δt = t/n:

eiLt =(eiLδt

)n. (18)

The action of eiLt on a phase vector Γ(0) can in principle be resolved step-by-step. The exact trajectory would nevertheless still require the exact formulafor operation eiLδt, which is not easier to find than that for eiLt. However, anapproximate formula will be more accurate if it decomposed into n steps than ifit used globally. In other words, one can get arbitrarily close to the exact globaloperator by decomposing an approximate formula f(t) into an arbitrarily largenumber of steps:

limn→∞

[f(t/n)]n = eiLt. (19)

Obviously, the form of expression f(δt) will have important practical conse-quences: for a given value of n, not all approximations are equally good atproducing trajectories similar to the exact ones.

Operator iL can be divided into parts iLq and iLp, with

iLq =∑

i

∂H

∂pi

∂

∂qiand iLp = −

∑i

∂H

∂qi

∂

∂pi. (20)

Even though operators iLq and iLp do not commute, it is possible to approxi-Wikipedia:Commutator mately factor propagator eiLδt using the Baker–Campbell–Hausdorff expansion.Wikipedia:Baker-Campbell-Hausdorffformula

For two noncommuting variables A and B, the first terms of this expansion are

eAδteBδt = exp{δt

[A + B + 1

2δt[A,B] + 112δt2[(A−B), [A,B]] + · · ·

]}. (21)

We observe an interesting symmetry alternance upon exchanging the roles of Aand B:

eBδteAδt = exp{δt

[A + B − 1

2δt[A,B] + 112δt2[(A−B), [A,B]]− · · ·

]}.(22)

We can group each of the terms of the expansion into a symmetric part Fs thatdoes not change sign upon permutation of A and B, and an antisymmetric partFa that does change its sign:

Fs(δt)def= A + B + 1

12δt2[(A−B), [A,B]] + · · · , (23)

Fa(δt)def= 1

2δt[A,B] + · · · . (24)

We can then re-write the product of Eqs. (21) and (22) as

eBδt/2eAδteBδt/2 = e(F ′s−F ′

a)δt/2e(F ′s+F ′

a)δt/2. (25)

For this product to represent a single integration step, δt has been replaced byδt/2. To shorten the notation, we are using F ′

sdef= Fs(δt/2) and F ′

adef= Fa(δt/2).

Real gases and liquids 5

By applying the Baker–Campbell–Hausdorff formula one more time—this timeto the right-hand-side of the previous equation—we get

eBδt/2eAδteBδt/2 = exp{δt

[F ′

s + 12δt[F ′

s , F′a] + 1

12δt2[[F ′s , F

′a], F

′a] + · · ·

]}.(26)

If A stands for iLq and B stands for iLp, we get

eiLpδt/2eiLqδteiLpδt/2 = exp{δt

[iL + iL2(δt2) +O(δt4)

]}, (27)

with

iL2(δt2) = 148δt2[(iLq − iLp), [iLq, iLp]] + 1

16δt2[iL, [iLq, iLp]]

= 124δt2[(2iLq + iLp), [iLq, iLp]]. (28)

In other words, the sequence of three operators on the left-hand-side of Eq. (27) Exercise:Derive Eqs. (26) and (28).

corresponds to a propagator eiL′δt, where iL′ is the original Liouvillian iL mod-ified by a term iL2 that proportional to the square of the integration step,followed by a term O(δt4). We also notice that, due to its bilateral symmetry, Wikipedia:

Big O notationthis propagator is time-reversible by construction:

Γ(δt) = eiLpδt/2eiLqδteiLpδt/2Γ(0), (29)e−iLpδt/2e−iLqδte−iLpδt/2Γ(δt) = Γ(0). (30)

For a Hamiltonian of the form (5), the Liouville operator is written

iL =∑

i

(ri ·

∂

∂ri+

Fi

mi· ∂

∂vi

)= iLr + iLv, (31)

with

iLr =∑

i

ri ·∂

∂riand iLv =

∑i

Fi

mi· ∂

∂vi. (32)

Knowing that

ea(∂/∂x)x =∞∑

n=0

an

n!∂n

∂xnx = x + a, (33)

we can calculate step-by-step the effect the sequence of operations eiLvδt/2eiLrδteiLvδt/2

has on state vector Γ = ({r}, {v}). The effect of operator eiLvδt/2 on vector({r(0)}, {v(0)}) is

eiLvδt/2

[{r(0)}{v(0)}

]=

[{r(0)}{v(0) + 1

2δtF(0)/m}

]def=

[{r(0)}{v(δt/2)}

]. (34)

The effect of operator eiLrδt on vector ({r(0)}, {v(δt/2)}) is

eiLrδt

[{r(0)}{v(δt/2)}

]=

[{r(0) + δtv(δt/2)}{v(δt/2)}

]def=

[{r(δt)}{v(δt/2)}

]. (35)

6 1 Computer simulations: Molecular dynamics

Finally, a second application of operator eiLvδt/2 gives

eiLvδt/2

[{r(δt)}{v(δt/2)}

]=

[{r(δt)}{v(δt/2) + 1

2δtF(δt)/m}

]def=

[{r(δt)}{v(δt)}

]. (36)

Forces {F(δt)} are computed from the force law of the system, using the newpositions {r(δt)}. We recognize the three operations from the velocity-Verletalgorithm introduced in Eqs. (11) to (13):[

{r(0)}{v(0)}

]eiLvδt/2

−→[{r(0)}{v(δt/2)}

]eiLrδt

−→[{r(δt)}{v(δt/2)}

]eiLvδt/2

−→[{r(δt)}{v(δt)}

].(37)

1.4 Modified (“shadow”) Hamiltonian

Operator iL2 from Eq. (28) has all the properties of a Poisson bracket andthere exists a Hamiltonian H2(δt2) such that iL2 = {· · · ,H2}. Poisson bracketssatisfy the Jacobi identity:Wikipedia:

Jacobi identity {A, {B,C}}+ {C, {A,B}}+ {B, {C,A}} = 0, (38)

where A, B, and C are any noncommuting variables. Making use of the notationDA

def= {· · · , A}, we can re-write this identity in an “operator” form, for whichWikipedia:Differential operator

the third variable (C) is implicit:

[DA, DB ] = D{B,A}. (39)

The original Liouvillian can be written iL = DH and for a Hamiltonian of theform H = U({q}) + K({p}), one can also write iLq = DK and iLp = DU .Operator iL2 can be reformulated in a similar way:

iL2 = 124δt2[(2DK + DU ), [DK , DU ]]

= 124δt2[D2K+U , D{U,K}]

= 124δt2D{{U,K},2K+U}. (40)

By direct comparison, we have

H2 = 124δt2{{U,K}, 2K + U}

= 124δt2

∑i,j

(2∂K

∂pi

∂2U

∂qi∂qj

∂K

∂pj− ∂U

∂qi

∂2K

∂pi∂pj

∂U

∂qj

). (41)

Because the consecutive terms of the Baker–Campbell–Hausdorff expansion arecalculable from relatively simple symbolic algorithms, the modified Hamiltonian

Reference:Munthe-Kaas and Owren,Phil. Trans. Royal Soc. A357, 957–981 (1999).

Reference:Reinsch, J. Math. Phys. 41,2434–2442 (2000).

can be expanded to any order with the help of symbolic calculation software suchas Maple or Mathematica. It is therefore in principle possible to analyze theaccuracy of any integration algorithm by calculating the successive terms of theexpansion of the modified Hamiltonian H ′ = H+H1(δt)+H2(δt2)+H3(δt3)+· · ·,and by examining their physical impact on the trajectories.

We can re-visit the harmonic oscillator example introduced in Section 1.2to understand the origin of the systematic deviations of the numerical solution

Real gases and liquids 7

Figure 2: Modified frequencies ω′(δt) for the velocity-Verlet algorithm. Weactually present the combination (ω′/ω)2 − 1. The dashed line corresponds to112δt2ω2, the first term of correction H2.

obtained with the velocity-Verlet algorithm (see Fig. 1). The Hamiltonian isH = 1

2kx2 + 12p2/m and, according to Eq. (41), the first term of the modified

Hamiltonian expansion is

H2(δt) = 12

(− 1

12δt2k/m)kx2 + 1

2

(16δt2k/m

)p2/m. (42)

The sum H + H2 corresponds to the Hamiltonien of a modified harmonic oscil-lator H ′ = 1

2k′x2 + 12p2/m′, which has a frequency

ω′(δt) =√

k′/m′ = ω√(

1− 112δt2ω2

) (1 + 1

6δt2ω2). (43)

Down to the order δt2, the ratio of the squared frequencies is given by

(ω′/ω)2 − 1 = 112δt2ω2. (44)

This function is presented as a dashed line on Fig. 2, together with the actualmodified frequencies ω′(δt) obtained from fitting a sinusoid on each discretetrajectory from Fig. 1. We notice that the correction H2 explains deviationsfrom the exact frequency for integration steps as long as one tenth of the naturalperiod of oscillation. For longer integration steps, one would have to considerthe higher-order corrections H4, H6, and so on.

1.5 Stability and accuracy

The efficiency of the velocity-Verlet algorithm is ultimately limited by a reso-nance phenomenon that destabilizes the discrete trajectories {r(0), r(δt), r(2δt), . . .}if δt is greater than a critical value.

8 1 Computer simulations: Molecular dynamics

Let us analyze the stability of the velocity-Verlet method using again theharmonic oscillator example. For a one-dimensional harmonic oscillator of nat-ural frequency ω =

√k/m, one application of the velocity-Verlet propagator

corresponds to the following recurrence formulae (written in matrix form):Wikipedia:Recurrence relation [

ωx(t + δt)v(t + δt)

]=

[1 0− 1

2ωδt 1

] [1 ωδt0 1

] [1 0− 1

2ωδt 1

] [ωx(t)

v(t)

]=

[1− 1

2ω2δt2 ωδt−ωδt

(1− 1

4ω2δt2)

1− 12ω2δt2

] [ωx(t)

v(t)

](45)

It is a special case of the transformations from Eqs. (11) to (13), made simplerfrom the fact that F = −kx. The three elementary steps of the algorithmare composed as a single linear application, that will be denoted S. In order tohave unitless matrix elements, we have converted position x to units of speed, bymultiplying it by the frequency ω. The system follows a stable dynamics uponiteration of the linear application S if its eigenvalues λ have an amplitude smallerthan 1. The two eigenvalues are the roots of the characteristic polynomialdet(S− λ1):

λ = 1− 12ω2δt2 ± i 1

4ωδt√

1 + 14ω2δt2. (46)

We can easily show that the stability condition |λ| < 1 holds for ωδt < 2. Theoscillator could therefore be integrated with an integration step as large as τ/π,the natural period of oscillation divided by π. For comparison, the red curve ofFig. 1 corresponds to δt = τ/4, which is slightly shorter that the stability limitτ/π. The green curve, significantly more accurate than the red one, correspondsto a time step of τ/16. For molecular dynamics simulations of diffusive motion(“diffusive” meaning “for which the exact trajectories of individual particles arenot important”; by contrast to “ballistic”), it is common to use an integrationstep about twenty times shorter than the oscillation period of the fastest motionin the system—something more like the green curve of Fig. 1.

Exercise:Estimate the oscillationperiods of various chemicalbonds: C H, O H,C C, C C, C C, etc.

1.6 Non-Hamiltonian systems

1.6.1 Constant-temperature simulations

The evolution operator formalism can be extended to systems that have a con-served total energy E = H (the “Hamiltonian”), but whose equations of motioncannot be obtained from Hamilton’s equations. This is for instance the case ofsystems where the individual particles are coupled to a heat bath that acts asa thermostat and that simulates the canonical ensemble (NVT) instead of themicrocanonical ensemble (NVE). A popular constant-temperature scheme is theNose–Hoover thermostat, which has the following equations of motion:

Reference:Hoover, Phys. Rev. A 31,1695–1697 (1985)

miri = Fi −miriη, (47)

Qη =∑

j

mj r2j −NfkBT. (48)

Real gases and liquids 9

Here, η is an additional dynamical variable describing the state of the heatbath. Its dynamics is described by Eq. (48), where the left-hand side has theusual “ma” form, an inertia parameter Q times a double time-derivative η,and the right-hand side corresponds to the “force” driving variable η. Theterm 1

2

∑j mj r

2j corresponds to K, the total instantaneous kinetic energy of the

particles which, according to the equipartition theorem, should be equal to 12kBT

Wikipedia:Equipartition theorem

times the number of degrees of freedom of the system, Nf . (For unconstrainedsystems, Nf = 3N .) If the kinetic energy of the particles is too high, thegeneralized force will be positive and variable η will increase. If the kineticenergy of the particles is too low, the force will be negative and η will decrease.Since the “velocity” η plays the role of a variable friction coefficient in Eq. (47), Wikipedia:

Drag (physics)a positive value of η will slow down the particles, and a negative value will speedthem up. Note that we recover Newton’s equations if η is zero—which wouldhappen if the inertia factor Q was infinitely large.

The total energy of the system corresponds to U({r}) +∑

i p2i /2mi, the

energy of the particles, plus a contribution from the heat bath:

H = U({r}) + NfkBTη +∑

i

p2i

2mi+

p2η

2Q. (49) Exercise:

Show that Eqs. (47) and(48) do not correspond toHamilton’s equations usingH from Eq. (49).

The term NfkBT corresponds to the net quantity of heat transferred from thereservoir to the particles, and p2

η/2Q corresponds to the kinetic energy of thereservoir. The non-Hamiltonian Liouville operator for Nose–Hoover equations(47) and (48) is

iL =∑

j

pj

mj· ∂

∂rj+

pη

Q

∂

∂η+

∑j

(Fj −

pη

Qpj

)· ∂

∂pj+ G

∂

∂pη

=∑

j

vj ·∂

∂rj︸ ︷︷ ︸iLr

+∑

j

Fj

mj· ∂

∂vj︸ ︷︷ ︸iLv

+ η∂

∂η−

∑j

ηvj ·∂

∂vj︸ ︷︷ ︸iLη

+G

Q

∂

∂η︸ ︷︷ ︸iLη

, (50)

with

Gdef=

∑j

p2i

mi−NfkBT. (51)

Note that iL does not correspond to {· · · ,H} as it was defined for Hamilto-nian systems. In the spirit of the velocity-Verlet algorithm, we can decomposeoperator eiLδt as following:

Reference:Martyna, Tuckerman,Tobias, and Klein, Mol.Phys. 87, 1117–1157 (1996).e(iLη+iLη)δt/2eiLvδt/2eiLrδteiLvδt/2e(iLη+iLη)δt/2 = exp

{δt[iL+O(δt2)]

}.(52)

The three operators in the middle are as in Eq. (27), and have the same effecton a phase vector Γ = ({r}, {v}, η, η) [cf. Eqs. (34), (35) and (36)]. Operatore(iLη+iLη)δt/2 at the beginning and the end of each integration step is decom-posed as following:

eiLηδteiLηδteiLηδt = exp{

12δt[iLη + iLη +O(δt2)]

}. (53)

10 1 Computer simulations: Molecular dynamics

Knowing that ea(∂/∂x)x = x + a [cf. Eq. (33)] and that

eax(∂/∂x)x =∞∑

n=0

an

n!

(x

∂

∂x

)n

x = eax, (54)

each of the nine operations has the following effect on a phase vector Γ:

eiLηδt/4 : η ← η + 14δtG({v})/Q, (55)

eiLηδt/2 : vi ← vi e−ηδt/2, (56)η ← η + 1

2δtη, (57)

eiLηδt/4 : η ← η + 14δtG({v})/Q,

eiLvδt/2 : vi ← vi + 12δtFi({r})/mi, (58)

eiLrδt : ri ← ri + δtvi, (59)eiLvδt/2 : vi ← vi + 1

2δtFi({r})/mi,

eiLηδt/4 : η ← η + 14δtG({v})/Q,

eiLηδt/2 : vi ← vi e−ηδt/2,

η ← η + 12δtη,

eiLηδt/4 : η ← η + 14δtG({v})/Q.

This general procedure can be applied to many other kinds of extendedsystems, that describe more robust thermostatting schemes, constant-pressure

Reference:Martyna, Tuckerman,Tobias, and Klein, Mol.Phys. 87, 1117–1157 (1996). schemes, and so on.

Reference:Parrinello and Rahman,Phys. Rev. Lett. 451196–1199 (1980).

Real gases and liquids 11

References

M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Clarendon,Oxford, 1987.

Guillaume Lamoureux. Developpement d’une fonction potentielle pour la dy-namique moleculaire de biomolecules. PhD thesis, Universite de Montreal,2005.