J. Phys. Chem. B 2008, 112, 213-218
213
Momentum and Velocity Autocorrelation Functions of a Diatomic Molecule Are Not Necessarily Proportional to Each Other† Jeremy M. Moix and Rigoberto Hernandez* Center for Computational Molecular Sciences & Technology, School of Chemistry & Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0430
Eli Pollak‡ Chemical Physics Department, Weizmann Institute of Science, 76100, RehoVoth, Israel ReceiVed: April 22, 2007; In Final Form: July 4, 2007
We present a computation of the classical momentum and velocity correlation functions of Br2 considered as an idealized molecular wire connecting dissipated lead atoms at each end of the dimer. It is demonstrated that coupling of the diatomic relative momentum to the leads may result in momenta that are not equal to the mass-weighted velocity. These differences show up in numerical simulations of both the average value and time correlations of the bond momentum and velocity. These observations are supported by analytical predictions for the average temperature of the diatomic. They imply that the “standard recipes” for modeling the system with a generalized Langevin equation are insufficient.
I. Introduction The generic model of a system bilinearly coupled via the spatial coordinates to a harmonic bath has recently received resurgent interest because it is useful in understanding chemical reactions in complex media.1-3 It has been noted that such coupling need not necessarily be effected only through spatial coupling but also through momentum coupling. For example, black body radiation may be modeled in terms of a Langevin equation whereby the coupling of the electromagnetic field is effected via a bilinear coupling of the system velocity to the field. Ford et al.,4 show that, for this case, a coordinate transformation leads back to the same Hamiltonian structure (detailed in eq 3 below) of the generic model of spatial bilinear coupling. Bao and co-workers5-7 have considered in recent years models of velocity coupling to both the coordinates and velocities of the bath. Makhnovskii and Pollak8 have shown that bilinear momentum coupling accounts for the phenomenon known as stochastic acceleration9,10 within an equilibrium context.11 The bilinear coupling model they used appears to the best of our knowledge for the first time in a paper by Cuccoli et al.,12 in which they consider certain aspects of the dynamics of a quantum harmonic oscillator coupled bilinearly through the momentum to the bath.13 Most recently, Ankerhold and Pollak14 have considered the bilinear momentum coupling model for both a harmonic oscillator as well as a parabolic barrier. They found that momentum coupling increases the tunneling probability, leading to the perhaps surprising result that dissipation may lead to the enhancement of quantum effects. It is thus of interest to understand what are the physical processes that can lead to momentum coupling and what effect does such coupling have on the system bath dynamics. For this †
Part of the “James T. (Casey) Hynes Festschrift”. * To whom correspondence should be addressed.
[email protected]. ‡ E-mail:
[email protected].
E-mail:
purpose, we will consider here the dynamics of an idealized molecular wire consisting of a diatomic Bromine molecule flanked by two metal atoms each of which is coupled spatially to a harmonic bath. The baths at each end emerge from the projection of all the modes associated with the remainder of the corresponding metal wire and electrode. The construction and characterization of molecular wires has been of significant interest for some time, particularly with a focus on the electron transport properties.15-18 The substitution of part of the wire with a homonuclear diatomic clearly would change the electronic properties, but it also permits the investigation of the effect of local vibrations in such systems. Perhaps surprisingly, the results of this paper demonstrate that for this model system the velocity correlation function is not proportional to the momentum correlation function. Arguably, the most important model used to analyze and understand the dynamics of atoms and molecules in the presence of a surrounding heat bath is the generalized Langevin equation (GLE).19,20 In its one-dimensional version for a particle with mass m
mq¨ +
dV(q) +m dq
∫t dt′γ(t - t′)q˘ (t′) ) F(t)
(1)
with the dot denoting differentiation with respect to the time t. The system coordinate q moves under the influence of a potential of mean force V(q), time dependent friction γ(t), and a Gaussian random force F(t) with zero mean. The time dependent friction is related to the random force via a fluctuation dissipation relation
mkBTγ(t - t′) ) 〈F(t)F(t′)〉.
(2)
The fluctuation dissipation relation is a reflection of the fact that the reduced dynamics is derived from Hamiltonian dynamics for the combined system and bath.21
10.1021/jp0730951 CCC: $40.75 © 2008 American Chemical Society Published on Web 09/07/2007
214 J. Phys. Chem. B, Vol. 112, No. 2, 2008
Moix et al.
As is well-known, the GLE may be derived from a Hamiltonian model in which the system is spatially bilinearly coupled to a harmonic bath22,23
H)
p2 2m
+ V(q) +
1 2
∑j
(
(
pj2
+ mjωj2 xj -
mj
))
cj mjωj2
2
q
. (3)
Each harmonic bath mode is described in terms of a mass mj, frequency ωj, and bilinear coupling coefficient cj. Hamilton’s equations of motion for the j-th bath mode x˘ j ) pj/mj, p˘ j ) -mjωj (xj - cj/mjωj2 q) may be readily solved in terms of the initial conditions of the j-th bath mode and the history of the coordinate q(t). Inserting the implicit solution into Hamilton’s equations of motion for the system,
q˘ ) p˘ ) -
dU(q)
+
dq
∑j
p m
(4)
(
cj xj -
cj mjωj2
)
q,
atoms is that of a Lennard-Jones potential. The vibration of the dimer is generally described as a simple harmonic oscillator, though anharmonic effects will also be considered. The center-of-mass motion of the particle gives rise to the usual diffusive correlation functions. As such, a simpler model, in which a monomer would be trapped along the molecular wire, would not readily provide evidence for momentum coupling. Thus, the presence of the internal vibration in the dimer, which can be hierarchically separated from the other motions, is one of the keys needed for this system to exhibit momentum coupling. A. The Cartesian Frame. Specifically, we consider a symmetric sequence of 4 particles, “ABCD”, where the dimer is represented by BC and the edge atoms, A and D, are coupled to distinct dissipative baths. The resulting Hamiltonian in the Cartesian coordinate system can thus be written as
H ) uABCD(pA, pB, pC, pD) + Vbond(qBC) + VLJ(qAB) + VLJ(qAC) + VLJ(qAD) + VLJ(qBD) + VLJ(qCD) + Vwall(qA) + N
(5)
Vwall(qD) +
∑
leads to the GLE (eq 1) with the identification of the time dependent friction function as
γ(t) )
∑j
cj2 mmjωj2
cos(ωjt).
(6)
From this model it is clear that the system momentum correlation function is proportional to the system velocity correlation function
〈p(t)p(t′)〉 ) m2〈q˘ (t)q˘ (t′)〉.
(7)
When considering the dynamics of a realistic system interacting with a bath, one models the dynamics in terms of the GLE using a few simple steps.24,25 One first constructs the potential of mean force acting upon the system by computing the force on the system ∂V(q,x)/∂q averaged over all (thermal) bath configurations. Then keeping the system coordinate fixed25 (or approximating it as locally harmonic),24 one computes the force autocorrelation function and uses the fluctuation dissipation relation (eq 2) to obtain the time dependent friction function. Alternatively, one can use the velocity autocorrelation function and derive from it the friction function.24 Evidently, a necessary condition for this type of modeling to be useful is that the momentum correlation function of the system be proportional to the velocity correlation function, as in eq 7. The central result of the present paper is the demonstration of a system of interest in which this proportionality does not hold. II. The Model System In order to probe the presence of momentum coupling, we consider an idealized model for a molecular wire composed of a diatomic molecule. Each of the leads of the wire at the ends of the dimer is modeled by a single metal atom whose interaction with the rest of the structure is represented by a dissipative bath. The latter could arise from the projection of structures as diverse as an atomic wire or an STM electrode. The dimer interacts with each bath only through its interaction with the edge atoms of the leads. The interaction between the dimer and the edge
[
]
1
1 pj,A2 + (ωjxj,A - ωj-1cjqA)2 + 2 2 N 1 1 pk,D2 + (ωkxk,D - ωk-1ckqD)2 , (8) 2 k)1 2
∑ j)1
[
]
where the kinetic energy operator will be discussed below, and the difference vectors qRR′ (≡ qR′ - qR) are used for notational simplicity. The walls are included to prevent dissociation of the system, although neither their presence nor their location changes the central result of this paper. Each is accounted for by a steeply increasing split harmonic potential energy function given by Vwall(qi) ) (kBT/l02)(qi - l0)2 for qi beyond l0 and Vwall(qi) ) 0 otherwise, where l0 is chosen to be just inside (0.1 Å) the respective wall. The form of the force constant ensures that at a given temperature the edge atoms will be repelled at the specified location of the wall on average. The vibrational potential of the internal diatomic can be specified by the usual expansion
1 Vbond(qvib) ) µvibω2qvib2 - k(3)qvib3 + O(qvib4) 2
(9)
where the reduced mass µ corresponds to that of the particles involved in the vibration qvib at the harmonic frequency ω, the anharmonic correction k(3) must be specified,26 and higher-order terms are neglected at the current order of approximation. The Lennard-Jones potential has the usual form
VLJ(r) ) 4
[(σr ) - (σr ) ]. 12
6
(10)
The last two terms in eq 8 correspond to the distinct baths spatially dissipating the edge atoms which due to symmetry are assumed to be described by the same friction kernel. For simplicity, we will further assume that the friction kernel is Ohmic, and consequently, the equations of motion for the edge atoms can be written as Langevin equations,
mA
dVA dV(q b) - mAγVA + ξA )dt dqA
(11)
mD
dVD dV(q b) - mDγVD + ξD, )dt dqD
(12)
Momentum and Velocity Autocorrelation Functions
J. Phys. Chem. B, Vol. 112, No. 2, 2008 215
where V(q b) is the sum of all the potential terms specified in eq 8, and the two random forces, ξA and ξD, are uncorrelated with each other but are connected to the friction γ through the fluctuation-dissipation theorem, as usual. Thus, the dissipation is completely specified by the friction and the temperature. The kinetic energy operator takes on the usual form for the four atom subsystem
uABCD(pA, pB, pC, pD) )
pA2 pB2 pC2 pD2 + + + 2mA 2mB 2mC 2mD
(13)
where the masses must be specified. In this frame, the momenta are diagonal, and hence, the projection of the system onto any of their corresponding conjugate positions leads to a Langevin system of the usual kind. The key point, however, is the fact that, in other frames, the momenta may be coupled and thus lead to momentum coupling in the projected representation. B. The Dimer Vibration Frame. The dynamics of the system is independent of the frame of reference chosen to describe the system. Differences may arise though, when one attempts to substitute the full dynamics with a reduced generalized Langevin-like description. Specifically, canonical coordinate transformations also lead to a transformation of the kinetic energy, which due to the transformation may no longer be diagonal. Ignoring the off-diagonal terms when considering the reduced representation may lead to error. A specific example of this occurs when one is particularly interested in the dimer vibration in the four atom model system described above. Typically, one is interested in the vibrational dynamics of the diatomic molecule, which are most reasonably represented through the diatomic bond coordinate (RBC ) qC - qB). A natural reduced representation would then include only the relative diatomic distance RBC and associated velocity. In the absence of the bath coupling, the Wilson G-matrix27 for the internal vibrations of the ABCD tetratomic (with explicit masses, but constrained to be linear) is
[
1 1 1 + MA MB MB 1 1 1 + G) -M M M B B C 1 0 MC
0 -
1 MC
1 1 + MC MD
]
.
metal
σ (Å)
(eV)
lattice constant (Å)
Br Ag Al Au Cu Ir Ni Pd Pt Rh
3.80 2.574 2.551 2.569 2.277 2.419 2.220 2.451 2.471 2.396
0.0126 0.351 0.408 0.458 0.415 0.830 0.529 0.465 0.694 0.687
not available 4.0857 4.0496 4.0782 3.6146 3.8392 3.5240 3.8903 3.9236 3.8032
a All of the values are taken from ref 28 except for those of bromine which is taken from ref 29. The lattice constants correspond to those of the FCC bulk metal taken from ref 31.
From Hamilton’s equations of motion, the desired momentum pBC in terms of computable quantities is given by
µAB µCD + R˙ CD MB MC µAB µCD 1 µBC M 2 M 2 B C
R˙ BC + R˙ AB pBC )
(16)
where the reduced masses µij ≡ MiMj/(Mi + Mj) are defined as per the usual convention. Equation 16 is clearly not equivalent to the naive choice of the momentum as the mass-weighted bond velocity µBCR˙ BC. The differences that arise in using this naive momentum instead of the correct canonical momentum in computing standard observables such as the average values and time correlations are nontrivial. In particular, as demonstrated in the Appendix
〈pBC2〉 µBC2〈R˙ BC2〉
g1
(17)
where the equality holds only in the limit that the edge atoms A and D are absent.
(14)
For this linear system the kinetic energy is given explicitly by
uint(pAB, pBC, pCD) )
TABLE 1: Lennard-Jones Parameters Used in Simulations for the Nine Metals and the Bromine Atoma
pAB2 pCD2 (pAB - pBC)2 + + + 2MA 2MD 2MB (pCD - pBC)2 , (15) 2MC
where the center of mass contribution has been ignored since it separates and has no effect on the dynamics. Clearly, the last two terms represent the coupling of the BC bond momentum to its neighbors. This coupling is found regardless of whether the system is dissipated by a bath. If one then wants to model the motion of the diatomic in terms of a reduced GLE description, one must take this coupling into account. Resorting to a spatial coupling model as in eq 1, will be, as we shall also demonstrate below, insufficient. In order to demonstrate the implications of this observation, the canonical momentum, pBC must be expressed in this frame.
III. Numerical Results The bromine dimer is characterized by a harmonic vibration with a force constant taken from the experimental frequency (315 cm-1). The remaining interactions are summarized in Table 1 and consist of only van der Waals terms with the respective parameters for the nine transition metals and the bromine atoms taken from refs 28 and 29, respectively. The metal-dimer interactions are obtained from the Lorentz-Berthelot combining rules, although the main conclusions are largely unaffected by the details of the potential. The flanking atoms are coupled to separate heat baths using Langevin forces with a friction constant of 3.2 × 10-4/fs as taken from ref 30 for the surface diffusion of a copper atom on a copper surface. However, the particular choice of mechanism leading to the thermalization of the flanking atoms does not change the qualitative conclusions shown here. Indeed, although not shown, separate simulations have been performed in which each atom instead has been thermalized by an independent Nose-Hoover chain of thermostats with a relaxation time between 0.5 and 2 ps. The results for that model were analogous to those shown below as obtained using the Langevin thermalization mechanism. Simulations are performed to measure the autocorrelation functions of pBC and R˙ BC over 40 ps sampling every 4 fs following a lengthy equilibration of 10 ns at 350 K. The walls
216 J. Phys. Chem. B, Vol. 112, No. 2, 2008
Moix et al.
Figure 1. Average momentum squared and bond velocity squared. The solid and dashed lines represent the analytical predictions in the Appendix. The circles and crosses correspond to numerical results for nine different metal atoms.
bounding the system are typically placed at positions that are one lattice spacing of the bulk metal away from the edge atoms.31 Note that this value is different for each of the metals. The exact location of the walls should not change the average values of the observables but can influence the correlations as will be seen below, presumably as it affects the degree of coupling between the particles by way of modifying the frequency of their interactions. Simulations are performed for the set of nine metal atoms surrounding the bromine dimer and the ratio of the dimensionless 〈pBC2〉 to 〈µBC2R˙ BC2〉 averaged over 800 trajectories is displayed in Figure 1 along with the analytic predictions of eqs 19 and 20. As expected, the momentum scales linearly with increasing mass of the edge atoms while the bond velocity remains constant. These results seem to indicate that the temperature of the diatomic may become arbitrarily large and that an appropriate apparatus for extracting this energy could be constructed in violation of basic thermodynamic principles. A detailed discussion of this scenario is given in ref 8 which demonstrates that this is not the case. Briefly, the associated renormalization of the effective mass (eq 18) of the diatomic offsets any possible gain by increasing the work required to extract this energy. The time correlations of the momentum and velocity functions may show different behavior as well. Figure 2 displays two autocorrelation functions of the Au-Br system averaged over 5000 trajectories with the wall at one and two lattice spacings from the edge atoms. In both panels, one notes the differences between the velocity and momentum correlation functions. Changing the location of the walls does not change this qualitative effect. However, the long time dynamics does change with the location of the wall by way of the coupling to the edge atoms. The short-to-intermediate time behavior is primarily determined by the interaction of the dimer with the dissipated neighbors. The comparison between the top to bottom panels of Figure 2 shows that a doubling of the distance between the outside atoms and the wall preserves both the 0-time values as well as the differences in the decay of the correlation function between the squares of the momentum and velocity. Does anharmonicity in the dimer vibration in the Au-Br system lead to an additional decoherence in the velocity thereby suppressing the differences seen in Figure 2? The answer is no as is shown in Figure 3. The parameter for the third-order term
Figure 2. Momentum coupling in the gold-bromine system. The black lines show the momentum autocorrelation function multiplied by a factor of 1/(µkBT), and the red line is the velocity autocorrelation function multiplied by a factor of µ/(kBT). The wall is located at one lattice spacing from the gold atom in the bottom panel and two lattice spacings in the top. The insets show the respective short time behavior where it is clear that the zero time values are the same in both cases.
Figure 3. Momentum coupling in the anharmonic gold-bromine system. The black lines show the momentum autocorrelation function, and the red line is the mass weighted velocity autocorrelation function. The wall is located at one lattice spacing from the gold atom.
k(3) in eq 9 is taken from ref 26 with the walls placed one lattice spacing from the edge atoms. As compared with Figure 2, the anharmonic correction is seen to have very little effect on either the momentum or velocity correlations. Both the average values and the initial decay remain in agreement with the harmonic case. IV. Discussion There are some noteworthy features of the momentum coupling results. Suppose that the potential between the edge atoms and the dimer is zero as would be the case for example if the respective edge atoms have been moved to (∞. In this limit, the kinetic energy terms in both the original frame and the vibrational frame remain unchanged. So in what sense does
Momentum and Velocity Autocorrelation Functions it mean that the diatomic in the latter frame has a renormalized momentum, and how is it that it can be arbitrarily renormalized depending on the masses of the edge atoms to which it is not coupled? The change comes from our attempt to describe the motion through a reduced equation of motion. If the edge atoms are moved to (∞, then we will find in the full numerical simulation that indeed the momentum and velocity correlation functions are proportional to each other. When the edge atoms interact with the diatomic, we would find in the full numerical simulation that the two correlation functions are not proportional to each other and this would indicate that the reduced description will not be accurate unless the momentum coupling is taken into account. If instead, the edge atoms are coupled to the dimer (because they are sufficiently close to each other), then they not only affect the zero-time averages, but they also affect the correlations. It is also instructive to consider the limit in which the motion between a given dimer atom and its neighboring edge atom is very fast compared to the dimer vibration. In this case, the effective dimer vibration is no longer that of the undressed dimer. Hence, the mass and frequency must be appropriately scaled. V. Conclusions This article presents an analysis and associated numerical results providing evidence for the importance of momentum coupling in reduced dimensional representations when one is calculating correlation functions in a frame in which the kinetic energy is not diagonal. Specifically, the vibrations of an idealized diatomic molecule, with each atom coupled to a distinct bath, have been simulated. This system was seen to exhibit a momentum autocorrelation function whose measurement depends on whether one uses the correct choice of the canonical momentum rather than the naive choice of the massweighted internal vibrational velocity. These differences imply that when one attempts to describe a many body problem in terms of a reduced generalized Langevin equation description it is not sufficient to merely compute velocity correlation functions and postulate that they provide the spatial friction function. It is necessary to compute simultaneously the correct analogous momentum correlation functions and check whether they are indeed proportional to the velocity correlation functions in the frame of the reduced coordinates chosen for the observation and projection. If yes, then a reduced spatial GLE is sufficient. If not, one must introduce momentum coupling into the GLE to ensure that the reduced description correctly reflects the full dynamics. Insofar as the model discussed in this work is applicable to the dynamics of a diatomic within an idealized molecular wire, the present results serve as a warning that the dynamics of such a system when viewed using projected coordinates may be coupled through position and momentum. We have shown in this paper that when one considers reduced motion in terms of bond coordinates one should expect that momentum coupling cannot be ignored. There are other possible scenarios for the introduction of momentum coupling, such as the coupling to random magnetic fields which lead to stochastic acceleration. These remain a topic for future investigations. VII. Appendix: 〈pBC2〉 vs 〈R4 BC2〉 The kinetic energy of the ABCD linear molecule in the center of mass frame of reference is
J. Phys. Chem. B, Vol. 112, No. 2, 2008 217
T(p) )
pAB2 pCD2 (pAB - pBC)2 (pCD - pBC)2 + + + 2MA 2MD 2MB 2MC
(
pAB - pBC
)
) (
µAB MB
2
2µAB
pCD - pBC
+
)
µCD MC
2
2µCD
+
pBC2 2MAB,CD
where the µij’s are the reduced masses that is µAB ) MAMB/ (MA + MB), etc. and
MAB,CD )
(MA + MB)(MC + MD) (MA + MB + MC + MD)
.
(18)
Note that as written in the second line, the coupling of the diatomic molecule to the atoms A and D causes a renormalization of the effective mass of the diatomic molecule. The reduced mass µBC is replaced by the reduced mass MAB,CD which is always larger than µBC. The momentum coupling causes an increase of the effective mass of the diatomic molecule. The thermal average of any operator O(p) which is a function of the momenta only is by definition
∫ dp O(p)e-βT(p) . 〈O〉β ) ∫ dp e-βT(p) It is then a matter of algebra and Gaussian integration to show that
β〈pBC2〉µAB β〈pAB2〉 )1+ µAB M 2 B
β〈pBC2〉µCD β〈pCD2〉 )1+ µCD M 2 C
β〈pBC2〉 MAB,CD ) . µBC µBC From Hamilton’s equations of motion we have that
R˙ BC )
pBC - pAB pBC - pCD + . MB MC
It then follows with some straightforward algebra that
βµBC〈R˙ BC2〉 ) 1.
(19)
We thus have the central result of this Appendix, which is that
〈pBC2〉 µBC2〈R˙ BC2〉
)
MAB,CD g 1. µBC
(20)
For a symmetric system, that is MA ) MD, MB ) MC, one readily finds that this ratio is just (MA + MB)/MB. Acknowledgment. This work has been supported by grants from the Israel Science foundation, the German-Israel Foundation for Basic Research, and the Einstein Center of the Weizmann Institute of Science. The computational facilities at the CCMST have been supported under NSF grant CHE 0443564. R.H. is the Goizueta Foundation Junior Professor and
218 J. Phys. Chem. B, Vol. 112, No. 2, 2008 was supported by the Alexander von Humboldt-Foundation during the preparation of this work. It is a pleasure to acknowledge the pioneering work of Casey Hynes on generalized Langevin equations, which underlies the theory presented in this paper. References and Notes (1) Hynes, J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. 4, pp 171-234. (2) Hynes, J. T. Annu. ReV. Phys. Chem. 1985, 36, 573. (3) Pollak, E.; Talkner, P. Chaos 2005, 15, 026116. (4) Ford, G. W.; Lewis, J. T.; O’Connell, R. F. Phys. ReV. A 1988, 37, 4419. (5) Bai, Z.-W.; Bao, J.-D.; Song, Y.-L. Phys. ReV. E 2005, 72, 061111. (6) Bao, J.-D.; Zhuo, Y.-Z. Phys. ReV. E 2005, 71, 010102. (7) Bao, J.-D.; Zhuo, Y.-Z.; Oliveira, F. A.; Ha¨nggi, P. Phys. ReV. E 2006, 74, 061111. (8) Makhnovskii, Y. A.; Pollak, E. Phys. ReV. E 2006, 73, 041105. (9) Fermi, E. Phys. ReV. 1949, 75, 1169. (10) Sturrock, P. A. Phys. ReV. 1966, 141, 186. (11) Cole, D. C. Phys. ReV. E 1995, 51, 1663. (12) Cuccoli, A.; Fubini, A.; Tognetti, V.; Vaia, R. Phys. ReV. E 2001, 64, 066124. (13) Leggett, A. J. Phys. ReV. B 1984, 30, 1208. (14) Ankerhold, J.; Pollak, E. Phys. ReV. E 2007, 75, 041103. (15) Yanson, A. I.; Bollinger, G. R.; van den Brom, H. E.; Agraı¨t, N.; van Ruitenbeek, J. M. Nature 1998, 395, 783.
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