Quantum localization, dephasing and vibrational energy flow in a trans-formanilide (TFA)–H2O complex

Chemical Physics 374 (2010) 111–117

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Quantum localization, dephasing and vibrational energy flow in a trans-formanilide (TFA)–H2O complex Johnson K. Agbo a,b,c,d, Amber Jain a,e, David M. Leitner a,* a

Department of Chemistry and Chemical Physics Program, University of Nevada, Reno 89557, United States Department of Chemistry, Sterling College, Sterling, KS 67579, United States c Department of Physics, Sterling College, Sterling, KS 67579, United States d Department of Mathematics, Sterling College, Sterling, KS 67579, United States e Department of Chemistry, University of Wisconsin-Madison, Madison, WI 53706, United States b

a r t i c l e

i n f o

Article history: Received 19 May 2010 In final form 5 July 2010

Dedicated to Professor Horst Köppel on the occasion of his 60th birthday. Keywords: Hydrogen bonding Energy flow Random matrices RRKM theory

a b s t r a c t Recent stimulated emission pumping-population transfer spectroscopic studies are providing measurements of energy barriers to hydrogen bond rearrangements involving biological molecules and water. To determine the kinetics of hydrogen bond rearrangements we need in addition information about energy flow in the biomolecule–water complex. We address the problem of quantum energy flow in one such complex system using a random matrix approach. We report here calculations of energy flow in the peptide trans-formanilide (TFA) that account for the hydrogen bonding of a water molecule to one of two sites on the peptide. Coupling to the water is found to enhance energy flow in the peptide. At energies near the hydrogen bond rearrangement barrier the rate of energy flow in TFA is nevertheless sufficiently sluggish to have a significant impact on the kinetics of water shuttling between hydrogenbonding sites. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Large molecules, such as those encountered in biology, exhibit complex dynamics and spectra arising from, among other factors, coupling among many vibrational degrees of freedom. Many features of the complex spectroscopy and kinetics of large molecules, or even small ones that are highly excited, can be captured by studies of ensembles of molecules selected to capture the size, magnitude of coupling, and selection rules for a given molecule of interest [1–7]. This random matrix approach can be extended to incorporate coupling of the molecular vibrations to the solvent environment [8], such as the coupling of the vibrations of a biomolecule to water. As we shall see below, coupling to its environment affects energy flow in the biomolecule, and ultimately kinetics of conformational change or hydrogen bond rearrangement. In this article, we apply a random matrix approach to study how energy flows quantum mechanically among the vibrational degrees of freedom of a peptide and how the energy flow is affected by hydrogen bonding to a water molecule. We furthermore address how energy flow in the peptide influences kinetics of hydrogen bond rearrangements between the peptide and water.

* Corresponding author. E-mail address: [email protected] (D.M. Leitner). 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.07.005

Recent experiments combining stimulated emission pumping (SEP) with population transfer (PT) spectroscopy [9] are providing information about barriers for transfer of a water molecule between hydrogen-bonding sites on biomolecules. A recent example is measurement of the barrier for shuttling of a water molecule between two hydrogen-bonding sites on trans-formanilide (TFA) [10]. The TFA–H2O complex (Fig. 1) in its most stable isomer has a modest binding energy of about 1990 cm1 [10]. The SEP-PT measurements have revealed that the barrier to transfer of the water molecule between hydrogen-bonding sites on TFA lies between about 750 and 988 cm1, with the bounds being essentially the same for the transfer from the CO to the NH site and for the reverse direction [10]. We recently [11] adopted these barriers to estimate rates of shuttling of the water molecule with Rice– Ramsperger–Kassel–Marcus (RRKM) theory [12,13]. We concluded, after exploring several possible dynamical corrections to the RRKM theory rate, that the actual hydrogen bond rearrangement rate of the TFA–H2O complex was not slower than the RRKM theory prediction but it could be faster if some of the vibrational modes of TFA were merely spectator modes and not active in reaction. Here we present calculations of rates of energy flow in TFA at energies near the barrier to hydrogen bond rearrangement. Using harmonic vibrational frequencies and estimates for anharmonic force constants fit to earlier [11] ab initio calculations of the TFA and TFA–H2O potential energy surfaces, we adopt a random matrix

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V¼

N XY m

Fig. 1. TFA–H2O complex, where the water molecule is shown here hydrogen bonded to the C@O. A random matrix approach is adopted to locate the energy threshold where vibrational states of isolated TFA are localized as well as the rates of energy flow in TFA hydrogen bonded to water at energies below and somewhat above the localization threshold energy. The extent and rate of energy flow in TFA influences the kinetics of hydrogen bond rearrangement in the complex.

approach to calculate thresholds for vibrational localization of TFA and examine the influence of coupling between TFA and the attached water molecule on this threshold. We find that the rate of energy flow among vibrational states of TFA is much slower than energy relaxation to the modes that are active in the reaction. As a result, the space of states that are active in the reaction is restricted and the reaction rate is greater than the rate predicted by RRKM theory, as we discuss below. In the following section we summarize the random matrix method used to calculate the vibrational localization transition for TFA and how it is affected by coupling to water. In Section 3, we present results for the localization transition of TFA, rates of energy transfer to the TFA–H2O modes of the complex, and how coupling to the latter affect energy flow in TFA. We then discuss rates of water shuttling between binding sites. Concluding remarks are presented in Section 4. 2. Computational methods 2.1. Quantum energy flow in a molecule Molecular vibrations, such as the vibrations of a peptide, can be described by a quantum mechanical system of coupled nonlinear oscillators, which in turn are coupled to a solvent environment. We begin by explicitly addressing the oscillator system. The corresponding classical system ultimately explores the entire phase space subject to specific constraints, such as constant energy, though there are conditions where transport in the phase space, via Arnold diffusion, may be very slow in many-dimensional systems [14,15]. The quantum mechanical oscillator system, on the other hand, can exhibit a localization transition to a relatively small number of states on the energy shell [8]. We summarize the random matrix approach that locates the energy of this transition for a specific molecule in this subsection. While quantum localization strictly exists only in the absence of coupling to the environment [8,16,17], remnants of the transition in this case still influence the rate of energy flow among vibrational states of the molecule, as discussed in Section 2.2. We define a generic coupled N-nonlinear oscillator Hamiltonian by

H ¼ H0 þ V; where

H0 ¼

N X

a¼1

ea ðn^a Þ;

ð1aÞ

þ

Um ðbþa Þma ðba Þma ;

ð1bÞ

a¼1

 and m ¼ fm 1 ; m2 ; . . .g. The zero-order Hamiltonian H0 consists of a sum over the energies of each of the nonlinear oscillators, ea, where each oscillator has frequency xa(na) = ⁄1 oea/ona, and nonlinearity x0a ðna Þ ¼ h1 @ xa =@na , and the number operator is defined by ^ a ¼ bþ n a ba . For most molecular vibrations the nonlinearity satisfies h  jx0a ðna Þj  xa ðna Þ. The coefficients of V, {Um}, are typically small compared to ⁄xa and generally decay on average exponentially with larger m [18–20]. The low-order terms in V couple states close to one another in the vibrational quantum number space. We can thus picture [8] the topology of the state space as an N-dimensional lattice where each lattice site is identified by the number of vibrational quanta in each mode. For example, a zero-order state for a 3-mode molecule could be identified by the lattice site (215), where one mode has two vibrational quanta, the second mode has one quantum, and the third mode has five vibrational quanta. These states are coupled locally by matrix elements arising from low-order anharmonic terms in the potential. We are interested in whether the coupling between sites is sufficiently large for the molecule to explore all states on the energy shell, or whether it is so small that the molecule remains localized to a relatively small number of states. While the energy at each site has a definite value that depends on the frequencies of all the oscillators, it is useful in exploring general properties of quantum energy flow to take these energies to be random from a distribution determined by the distribution of oscillator frequencies. Matrix elements arising from the anharmonic coupling in Eq. (1), which couple states locally in the quantum number space, also have specific values, but to explore general properties of energy flow in the vibrational state space we can take these to either be represented by a ‘‘typical” value, or we can take them as random from a distribution of values that characterizes anharmonic coupling of a particular order. For this reason we refer to this approach as Local Random Matrix Theory (LRMT) [21]. It is based on analysis of a random matrix ensemble, where the particular molecule of interest is one member of an ensemble of similar molecules, all of which have zero-order energies and coupling terms falling within a similar range. An important constraint on members of the ensemble is that they obey selection rules, as introduced to a variety of random matrix ensembles [4,5,22], in this case due to the local anharmonic coupling between states. The problem of vibrational energy flow in molecules, where zero-order states are sites on a lattice with random energies and coupled locally to one another, resembles the problem of singleparticle quantum transport on a many-dimensional disordered lattice. Theoretical approaches to address the condensed phase problem and the emergence of Anderson localization [23] can thus be brought to bear on describing vibrational energy flow in molecules. Exploiting this connection and building on the approach of Abou-Chacra, Anderson and Thouless (AAT) to address the condensed phase problem [24], Logan and Wolynes [8] identified criteria for localization of vibrational states, which occurs at a critical value of order one, of the product of the anharmonic matrix elements and local density of resonantly coupled states. The criterion for energy to flow over the energy shell in the vibrational state space is akin to a percolation threshold; if the average coupling between sites on the energy shell, or the density of resonantly coupled states, is too small then transitions between states are confined to a local cluster on the energy shell, while sufficiently large coupling between sites, or sufficiently high density of resonantly coupled states, permits transitions among all states on the energy shell.

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More formally, localization of vibrational states of a molecule depends on the self-energy of the states. The imaginary part is finite for extended states and proportional to the energy transfer rate from that state, while it is infinitesimally small for a localized state. To calculate the real and imaginary parts of the self-energy, following the approach of AAT, it is convenient to approximate the topology of the vibrational state space as a Cayley tree. This approximation becomes better justified as the number of oscillators in the system becomes large, the regime of interest when addressing the coupled vibrations of biomolecules. Starting with the Feenberg renormalized perturbation series (RPS), only the lowest order terms for the real and imaginary parts of the site self-energy are needed for a state space with a Cayley tree topology. The lowest order terms in the Feenberg RPS for the real, Ej, and imaginary, Dj, parts of the site self-energy are given by

Ej ðEÞ ¼

Dj ðEÞ ¼

KQ X

jV Q;jk j ðE  ek  Ek Þ

Q ;k–j

ðE  ek  Ek Þ2 þ ðDk ðEÞ þ gÞ2

KQ X

jV Q;jk j2 ðDk ðEÞ þ gÞ

Q;k–j

ðE  ek  Ek Þ2 þ ðDk ðEÞ þ gÞ2

2

;

ð2aÞ

;

ð2bÞ

local subset of states on the energy shell, the number of which can be estimated with the theory [26]. Above the transition, T(E) > 1, energy flows among all states of the energy shell. Schofield and Wolynes [27,28] have argued that energy flow in the vibrational state space both just above and well beyond the transition can be described by a random walk in quantum number space, a picture that has been supported by numerical calculations over a wide range of time scales [29–33]. The state-tostate energy transfer rate can be estimated by LRMT from the most probable value of the imaginary part of the site self-energy, which is given by [25]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X Dmp ðEÞ ¼ p 1  T 1 ðEÞ hjV Q ðEÞj2 iqQ ðEÞ;

TðEÞ 

rffiffiffiffiffiffiffi 2p X hjV Q ðEÞjiqQ ðEÞ P 1; 3 Q

ð3Þ

while quantum transport is localized in the vibrational state space at energy E when T(E) is less than 1. In Eq. (3), qQ is the local density of states that lie a distance Q away in quantum number space, and h|VQ|i is the average coupling matrix element to such states. In practice, for the calculations presented here, we calculate qQ using the vibrational frequencies for all the modes of the molecule, and determine from them density of states that lie a distance Q in quantum number states from a particular state on the energy shell, and then take an average of this density at energy E. To obtain h|VQ|i we take an average over the coupling matrix elements estimated with Eq. (6). If T(E) < 1, transitions only occur among a relatively small and

ð4Þ

and is related to the most probable energy transfer rate in the vibrational state space by k(E) = 2Dmp(E)/ h. Below we use Dmp to represent the most probable rate. Eq. (4) goes over to a golden rule-like expression when T(E)  1 through a crossover region just above the transition. The golden rule expression, i.e., average value of imaginary part of the site self-energy is

hDðEÞi ¼ p where the absence of coupling to the environment corresponds to the limit g ? 0, which we consider in this section; we address finite g in Section 2.2. Q represents the distance in quantum number space between two states. It corresponds to the total number of quanta lost and gained in all oscillators in making a transition from the state labeled by j to states labeled by k; for example, cubic coupling can give rise to at most an exchange of 3 quanta among all the oscillators, in which case Q = 3, whereas quartic coupling can give rise to at most an exchange of 4 quanta among the oscillators, in which case Q = 4. |VQ,jk| is a matrix element coupling state j to state k, which lies a distance Q away; and KQ is the number of such states. To establish criteria for energy flow in the vibrational state space the probability distribution of Dj must be solved self-consistently in the limit g ? 0. In one approach to the self-consistent analysis, which we follow here, only the most probable value for the imaginary part of the self-energy, Dmp, is found self-consistently [8,25]. In a second approximate approach, the average inverse participation ratio is solved self-consistently [26]. In both approaches, the eigenstates of H are assumed localized, so that Dmp must be infinitesimally small and the average value for the inverse participation ratio must be finite. A range of molecular parameters is then identified for which Dmp is in fact infinitesimally small, or, similarly, where the average inverse participation ratio is finite. The results for both self-consistent approaches are nearly the same, differing only by a constant of order one. Details of the analysis, including incorporation of higher-order anharmonic terms and contributions of higher-order terms in the Feenberg RPS, can be found in Ref. [25]. Solving for Dmp we find that vibrational energy flow is unrestricted when [8,25]

TðEÞ > 1;

Q

X hjV Q ðEÞj2 iqQ ðEÞ:

ð5Þ

Q

The average value is finite at any energy, but when T(E) < 1 it is a poor estimate for the transition rate between states on the energy shell, since in the averaging there are many states for which D is 0, and a very small number of states on the energy shell for which D is very large. (In the limit N ? 1 the latter are of measure 0.) Indeed, when T(E) < 1, the most probable value of D vanishes, i.e., Dmp = 0 in the absence of coupling to the environment. The average coupling terms in Eqs. (3)–(5) can be calculated from a set of anharmonic constants obtained from a given potential energy surface. Alternatively, we may assume a scaling relation for the average anharmonic interactions. On average, one finds that coupling between states that lie a ‘‘distance” Q from one another P in quantum number space, where Q = aDna, and Dna is the occupation number difference in mode a between the two vibrational states, decreases with Q roughly as Vjk  CQ, where C > 1. Gruebele and coworkers have found empirically that for modest-sized organic molecules [34]

V jk ¼

Y

RDa na ;

ð6aÞ

a

where

Ra 

a1=Q ðxa ta Þ1=2 : b

ð6bÞ

In Eq. (6) ta is the number of quanta in mode a, Dna is the occupation number difference in mode a between two vibrational states, and a and b are constants. For Vjk expressed in cm1, a and b are chosen to be 3050 and 270, respectively. This approach can be used to estimate anharmonic force constants to any order. As a check, we have found that for TFA it gives average cubic anharmonic terms in good agreement with those obtained from the electronic structure calculations [11]. 2.2. Effects of coupling to water on localization in peptide The most probable rate of energy transfer in the vibrational state space of a peptide is proportional to Dmp(E), which we have until now expressed in the absence of interactions with water molecules. We take this coupling into account by considering finite g. Contributions to g in the solid state problem usually include inelastic scattering due to electron–phonon interactions, which in the vibrational problem addressed here would correspond to

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population relaxation contributions to the dephasing rate that arise from the coupling of vibrational states of the cofactor to its environment; g can also represent rates of pure dephasing [8,35,36]. Here, to illustrate effects of coupling of the peptide to its environment, we use g to represent the rate of population relaxation of modes of the peptide strongly coupled to the water. We then examine how such coupling affects energy flow among the vibrational states of the peptide. Defining l(E) = Dmp(E) + g, the statistical self-consistent analysis summarized above yields [8,25]

Dmp ðEÞ ¼ TðEÞlðEÞð1 þ TðEÞl2 ðEÞ=hDðEÞi2 Þ1 ;

ð7Þ

which holds for any value of T(E). We take the average of the imaginary part of the site self-energy, hD(E)i, which is proportional to the mean energy transfer rate, as the golden rule result, which is given by Eq. (5). While Eq. (7) is a result of the statistical self-consistent analysis for the coupled many-oscillator model given by Eq. (1), where some of the oscillators are also coupled to the environment, this solution has the general form expected for the lifetime of a vibrational state that is coupled to a tier of other states, which in turn couples to another tier of states [37], the latter mimicking the coupling of some of the vibrational states of the molecule to the environment. For this state space structure, we would expect that the lifetime, as exhibited by Eq. (7), generally decreases with increasing coupling of cofactor modes to the environment until such coupling becomes so large that the lifetime turns over, essentially a motional narrowing regime, at which point the lifetime increases with increasing coupling of the intermediate tier of states to the environment. These variations are illustrated in Fig. 2. For convenience, following Logan and Wolynes [8], we define dimensionless variables, D0mp ðEÞ and g0 , as

D0mp ðEÞ ¼ Dmp ðEÞ=hDðEÞi;

ð8aÞ

0

g ðEÞ ¼ gðEÞ=hDðEÞi:

ð8bÞ 0

Eq. (7) is quadratic in g with solution [8]

g0 ðEÞ ¼

  1=2 1  ½1  4T 1 ðEÞD02 mp ðEÞ 2D0mp ðEÞ

 D0mp ðEÞ:

ð9Þ

Since both D0mp ðEÞ and g0 are real and non-negative, when 0 6 T 6 2 and if 0 6 g0 6 gmax = [2  T]/2T1/2 we choose the negative sign, while for g0 P gmax we choose the positive sign. For T P 2 only

the positive sign is correct. We plot in Fig. 2 results for the most probable energy flow rates as a function of dephasing rate at various T(E), calculated with Eq. (9).

3. Results and discussion We begin with the vibrational localization threshold for TFA calculated using Eq. (3). The cubic coupling terms entering this equation were evaluated with Eq. (6) after previously fitting that relation to ab initio calculations [11]; the higher-order anharmonic contributions were also evaluated using Eq. (6). We plot T(E), computed with Eq. (3) and the average energy flow rate in the vibrational state space, hDi, computed with Eq. (5), in Fig. 3. From the T(E) curve we deduce that the ergodicity threshold of TFA is about 1450 cm1, as reported previously [11]. This result implies that below 1450 cm1 energy flow in isolated TFA is limited to a few isolated resonances, whereas above it, the flow rate can be estimated using Eq. (4). The average rate of energy flow in the vibrational state space, hDi, provides a poor estimate for the energy transfer rate in isolated TFA at energies below 1450 cm1 since the most probable energy transfer rate is 0 below the ergodicity threshold. The presence of the loosely bound water molecule in the TFA– H2O complex will smear the energy flow threshold of TFA, in the sense that energy flow in TFA is no longer restricted due to the coupling to the water molecule. As seen in Fig. 2, the rate of energy flow in TFA, as a result of coupling to water, depends not only on the value of T(E), but also on the dephasing rate, g, due to energy transfer from the vibrational states of TFA to states of the TFA– H2O complex. To estimate the rate of the latter, we have calculated the energy transfer rate from modes of TFA to the modes of the complex. The frequencies of the six intermolecular vibrational modes of TFA–H2O(C@O) and TFA–H2O(N–H), which are taken as those modes with the largest projection onto the water displacement coordinates, are listed in Table 2 of Ref. [11]. We use Eq. (5) to calculate the rate of energy transfer from states of TFA to the complex. For the latter we use the intermolecular modes and estimate the rate of energy transfer via cubic anharmonic terms. This is, of course, only a rough approximation to the rate of energy transfer from TFA to the complex, since higher-order terms may also contribute, but in a large molecule the cubic terms often account for much of the rate [25,38,39]. We find that the rate of energy flow from TFA to the complex occurs by

Fig. 2. Most probable energy flow rates as a function of dephasing rate, g, for values of the transition parameter, T(E), of, from bottom to top, 0.2, 0.3, 0.5, 0.8, and 1.6.

J.K. Agbo et al. / Chemical Physics 374 (2010) 111–117

115

Fig. 3. The transition parameter, T(E), is plotted as a function of energy in TFA (solid curve). The average value of the energy flow rate among the vibrational states of TFA, given by the golden rule expression (Eq. (5)), is also plotted (dashed line). As discussed in the text, the golden rule rate is often an inappropriate estimate for the energy transfer rate when T(E) is less than or not far above 1. The most probable energy transfer rate in TFA is found to be much smaller than the average rate, as discussed in the text.

energy transfer from one of the TFA modes into one or a pair of modes of the complex to which it is resonantly coupled. For example, we find decay of the 228 cm1 mode of TFA into the 115 cm1 mode of the complex, where the water molecule is hydrogen bonded to the C@O, by a 2:1 Fermi resonance. In other cases energy flows from one mode of TFA into another mode of TFA and a mode of the complex, as in the decay of the 355 cm1 mode of TFA. We plot the results for energy transfer rates from specific TFA modes to the complex as a function of energy in Fig. 4. Decay from only three modes is shown, since for all others the energy transfer rate was found to be much smaller. These energy transfer rates are relatively slow compared to the golden rule estimates for the energy flow rate within TFA itself, plotted in Fig. 3. We find that energy transfer to the complex is faster when the water molecule is hydrogen bonded to C@O than when it is bonded to N–H, which

is likely just a result of the particular resonance structure for these complex isomers. When the water molecule is hydrogen bonded to the C@O the energy transfer rate from TFA is about 0.2 ps1, whereas it is about a fifth to one-quarter this rate when the water molecule is hydrogen bonded to the N–H. We use these rates as estimates to the dephasing rate, g, then calculate with Eq. (9) the most probable rate of energy transfer in TFA, Dmp, as a function of energy in the peptide. For the water molecule hydrogen bonded to the C@O, we use the faster of the two energy transfer rates plotted in Fig. 4, since for only a very small fraction of states at a given energy considered are both modes that are plotted in Fig. 4 actually populated. In fact, since both modes of TFA–H2O(C@O) relax at about the same rate, one could choose either one for the relaxation rate. The energy-dependent dephasing rate, g, used for TFA–H2O(C@O) is then taken from the upper curve in Fig. 4, whereas g used for TFA–H2O(N–H) is

Fig. 4. Rates of energy transfer from modes of TFA to TFA–H2O modes of the complex as a function of energy in TFA. Filled symbols correspond to decay from TFA into TFA– H2O(C@O), while the open symbols correspond to energy relaxation into TFA–H2O(N–H). For the former case, the TFA modes are 355 cm1 (filled circles) and 228 cm1 (filled squares), and for the latter case it is the 196 cm1 (open squares) mode of TFA. Energy transfer rates from all other modes of TFA were computed to be much smaller over this range of energy.

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taken from the lowest curve. Values of T(E) and hD(E)i used in Eq. (9) are those that are plotted in Fig. 3. Results for Dmp in TFA as a function of energy for the two isomers are plotted in Fig. 5. The most probable rate of energy flow in TFA is about an order of magnitude faster when the water molecule is hydrogen bonded to the C@O than when it is hydrogen bonded to the N–H, which results from the significantly larger value of g for the former structure. It is useful to compare the rates of energy transfer from TFA to the complex, which are plotted in Fig. 4, with the rates of energy flow within TFA, plotted in Fig. 5. When the water is hydrogen bonded to the N–H, the rate of energy flow from TFA to the complex is much faster than the rate of energy flow within TFA to energies of about 1000 cm1; by 1200 cm1 these rates are both about 0.05 ps1. This means that excess energy in TFA, in particular the 198 cm1 mode, will relax into the modes of the complex before being redistributed into other modes of TFA. A similar conclusion can be drawn for TFA–H2O(N–H). Up to about 1000 cm1, the rate of energy transfer from TFA to the complex, about 0.2 ps1, is much greater than the rate of energy flow within TFA, which by 1000 cm1 is still a factor of 4 slower than the rate of energy flow to the intermolecular modes. By 1150 cm1 these two rates start to become comparable. In previous work on TFA–H2O [11], we calculated rates of water shuttling between the two hydrogen-bonding sites beginning with RRKM theory and examining possible dynamical corrections that could slow down the rate. We found that the shuttling rate is not slower than the RRKM theory estimate for the rate, and concluded that we cannot rule out the possibility that reaction rates at energies just above the barrier to water shuttling are faster than that predicted by RRKM theory. We are now in a position to reach a more definite and quantitative conclusion, but we first summarize our previous arguments: Near the barrier energy the calculated value of T(E) for TFA is 0.1, well below the threshold value of 1. This result suggests that, despite coupling to water, many vibrational states of the complex might be spectator states during the course of reaction, in the sense that energy flow among them, albeit possible, is slow. When many modes of a reactant are inactive the reaction rate can be much faster than the RRKM theory estimate assuming all modes are active [7,40–42]. Above 1200 cm1, where the rate of energy flow within TFA is comparable or even faster than the rate of energy flow from TFA

to the intermolecular modes, which overlap the reaction coordinate for hydrogen bond rearrangement [11], we would expect RRKM theory to provide a good estimate for the water shuttling rate. At these energies, Dmp is sufficiently large that the modes of TFA are no longer merely spectator modes in the reaction, and all modes of the complex should be considered active modes in an RRKM theory calculation. Since we previously concluded that the water shuttling rate constant is not limited by energy flow into and out of the activated complex, we conclude that the rate of reaction is neither faster (due to a limited number of active vibrational states) nor slower (due to sluggish energy flow into and out of the reaction coordinate) than the RRKM theory estimate. At lower energy, however, particularly at energies between the barrier energy of 750–988 cm1 and 1100 cm1, we have found energy flow within TFA to be much slower than energy dissipation from TFA to the intermolecular modes of the complex, and most modes of TFA are indeed spectator modes. At these energies the active modes in a RRKM theory calculation are the six intermolecular modes, which are all strongly coupled to each other and are listed in Table 2 of Ref. [11], and the one or two modes of TFA identified in this study and shown in Fig. 4 to deposit energy to these modes. We have carried out RRKM theory calculations using only these active modes for both isomers, and we find the rate to be one order of magnitude greater than our previously calculated RRKM theory results [11], where we assumed all modes to be active. For example, we calculated previously [11] a water shuttling rate of 30 ns1 at 1000 cm1 of total energy in the TFA–H2O complex assuming a reaction barrier of 750 cm1. If only the six intermolecular modes and the two modes of TFA relatively strongly coupled to the modes of the complex (for the C@O isomer) are active we calculate the rate using RRKM theory to be 300 ns1 at 1000 cm1. We calculate a similar rate at this energy for the N–H isomer using the seven active modes. While there are no kinetic measurements on this system to verify this conclusion, we note that recent measurements of the fragmentation rate of a similar complex, tryptamine–H2O, has been observed to occur at a rate that is three orders of magnitude faster than the RRKM theory estimate [43]. This discrepancy was attributed to a large number of inactive, spectator modes [43], which we have shown to exist in TFA–H2O at energies near the barrier to hydrogen bond rearrangement.

Fig. 5. The most probable rate of energy flow among the vibrational states of TFA as a function of energy in TFA, when the water molecule is hydrogen bonded to the C@O (triangles) and the N–H (circles).

J.K. Agbo et al. / Chemical Physics 374 (2010) 111–117

117

4. Concluding remarks

References

We have applied a random matrix approach to calculate rates of energy flow in a peptide hydrogen bonded to a water molecule. Random matrix theory, traditionally used to describe statistical properties of energy levels, is applied here to provide a statistical description of the vibrational eigenstates of a molecule. In particular, we adopt an ensemble of random Hamiltonians that incorporates selection rules for anharmonic coupling among vibrational modes to locate the energy of the localization transition of the vibrational states of a peptide. The peptide interacts with a water molecule, so that energy transfer among the vibrational states of the peptide is possible at energies below the localization transition of the bare peptide. The rate of energy flow in the peptide, TFA, plays a critical role in the kinetics of hydrogen bond rearrangement of the peptide–water complex. Because the threshold for vibrational energy localization in TFA of about 1450 cm1 lies above the hydrogen bond rearrangement barrier of 750–988 cm1, we find that the rate of energy flow within TFA when it is coupled to the water molecule is well below estimates obtained using a golden rule calculation, at least at energies near the reaction barrier. Well above the reaction barrier energy, above at least 1200 cm1, we found that coupling to the water molecule gives rise to energy flow in TFA that is sufficiently fast for most modes to be active in the hydrogen bond rearrangement. In this case, because we previously concluded that the actual reaction rate is not slower than the RRKM theory estimate [11], we predict that above 1200 cm1 of energy in the complex RRKM theory should provide a good estimate for the rate. However, at lower energies, below 1100 cm1 the rate of energy flow in TFA is much slower than the relaxation rate from TFA modes into the intermolecular modes, so that most modes of TFA are not active in the reaction. Accounting for only active modes, we predict the rate to be an order of magnitude larger than the RRKM theory estimate.

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Acknowledgements Support from NSF Grant CHE-0910669 is gratefully acknowledged. We thank Ken Jordan and Evgeniy Myshakin for providing the vibrational modes of TFA and TFA–H2O used here. One of us (DML) has enjoyed working with Horst Köppel on applications of random matrix theory to describe and interpret complex spectra of highly excited molecules, and it is a pleasure to dedicate this paper to him on the occasion of his 60th birthday.