Theoretical Study of Internal Vibrational Relaxation and Energy

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Theoretical Study of Internal Vibrational Relaxation and Energy Transport in Polyatomic Molecules Sarah L Tesar, Valeriy M Kasyanenko, Igor V Rubtsov, Grigory I Rubtsov, and Alexander L Burin J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp309481u • Publication Date (Web): 17 Dec 2012 Downloaded from http://pubs.acs.org on December 24, 2012

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TheJournalofPhysicalChemistry

Theoretical Study of Internal Vibrational Relaxation and Energy Transport in Polyatomic Molecules Sarah L. Tesar,† Valeriy M. Kasyanenko,† Igor V. Rubtsov,† Grigory I. Rubtsov,‡ and Alexander L. Burin∗,† Department of Chemistry, Tulane University, New Orleans LA, 70118, and Institute for Nuclear Research of RAS, 60th October Anniversary st. 7a, Moscow, Russia 117312 E-mail: [email protected]

∗

To whom correspondence should be addressed of Chemistry, Tulane University, New Orleans LA, 70118 ‡ Institute for Nuclear Research of RAS, 60th October Anniversary st. 7a, Moscow, Russia 117312 † Department

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Abstract We attempted to theoretically characterize internal vibrational relaxation and energy relaxation pathways due to anharmonicity in polyatomic molecules. Energy transport dynamics have been modeled based on a generalization of Marcus electron transfer theory. Modifications have been made to our previously developed theory in order to improve the description of internal vibrational dynamics. We applied our method to several molecules studied experimentally by relaxation-assisted two-dimensional infrared spectroscopy (RA 2DIR). The theoretical predictions were found to be consistent with the majority of the experimental data.

1 Introduction Vibrational energy dynamics and relaxation at the molecular level has been extensively investigated for the past three decades since it can shed light on the molecular structure, dynamics and non-equilibrium kinetics. 1–3 Nearly thirty years ago, Stewart and McDonald experimentally studied the intramolecular vibrational energy relaxation (IVR) process using IR fluorescence spectroscopy, 2 concluding that the total density of states characterizes the dynamics of energy transport. More recent studies suggest that the local density of anharmonically coupled states is the relevant parameters. 4–6 The similarity of the problem to Anderson localization has been exploited 4 giving rise to quantum mechanical theoretical approaches such as local random matrix theory (LRMT) 7–9 and the Bose Statistics Triangle Rule (BSTR) model, 10 which attempt to characterize IVR by combining spectroscopic properties with statistical models. Other theoretical models have been developed based on the numerical solution of the Schrödinger equation 11,12 (see also References 13,14 for reviews on several theoretical studies of IVR) for the isolated molecule, combined quantum treatment of the molecule and classical treatment of solvent 15 and classical molecular dynamics studies . Novel theoretical efforts are required in order to gain a fundamental understanding of vibrational energy transport dynamics on a molecular level in polyatomic molecules which are too large for exact quantum mechanical treatment. However, the vibrational harmonic modes and anhar2 ACSParagonPlusEnvironment

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monic interactions can be well characterized using the most up to date quantum chemistry software. 16 In this paper we extend the Fermi Golden rule based approach 17 applying electron transfer theory 18–21 to the anharmonic transitions and then apply this consideration to the recently developed non-linear infrared spectroscopy studies of several polyatomic molecules (the Fermi Golden rule has been earlier used for solvent stimulated transitions in Refs. 15,22). Vibrational energy relaxation and transport on a molecular level can be experimentally studied by methods of non-linear infrared and Raman-probe spectroscopy. 23–30 Two-dimensional nonlinear infrared spectroscopy (2DIR) measures vibrational mode interaction and can be useful in determining structural characteristics of proteins, peptides and other molecules. 21,31–34 Recently developed relaxation assisted 2DIR (RA 2DIR) spectroscopy 33,34 uses vibrational relaxation and vibrational energy transport in molecules and the thermalization process on a molecular scale to generate stronger cross-peaks between modes separated by large distances. One vibrational mode of a target molecule can be excited by pump pulse and a probe pulse can be applied to a second mode in order to study its time dependent absorption spectrum. The initially excited mode relaxes into other modes which causes a shift in the probe absorption spectrum due to their anharmonic coupling. The excess energy propagates through the molecule and modes located nearby and strongly coupled to the probed mode become excited. This results in a frequency shift of the probed mode and therefore, enhancement of the cross peak amplitude between the pumped and probed modes. Finally, the absorption spectrum returns to equilibrium because the excitation energy is dissipated to the solvent. Therefore, RA 2DIR can visualize vibrational energy flow inside a molecule and report on energy transport rates. The energy transport and redistribution in polyatomic molecules occurring via intramolecular vibrational energy redistribution (IVR) processes can be formulated as transitions between different harmonic vibrational states. 35–38 Harmonic vibrational states of an n-atomic molecule can be represented by a sequence of integer population numbers, {n i }, that characterize the quantum states of independent normal modes having frequencies ω i (quantization energies, ωi , are all expressed in inverse centimeters, and we set h¯ = 1). The harmonic vibrational state of isolated molecule can



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change due to anharmonic interactions, which lead to transitions that modify population numbers and consequently redistribute the harmonic energy. The molecule then serves as a thermal bath for its own vibrational relaxation. 2,35,36 We investigated the energy evolution and pathways of an initially excited vibrational mode in time and space in an attempt to understand the nature of vibrational energy transport through polyatomic molecules. Our theoretical model describing vibrational relaxation is based upon a modification of the Marcus electron transfer theory applied to anharmonic transitions. 19 Initial parameters such as anharmonic and harmonic frequencies, anharmonic force constants, and the X-matrix of anharmonic couplings were determined from the density functional theory (DFT) anharmonic calculations performed with the Gaussian 09 software package. 16 All calculations were done using the B3LYP functional and the 6-311G++(d,p) basis sets following eariler computational studies. 21,39 Only third-order anharmonic interactions were considered because of the computational limitations in anharmonic calculations.

1

Anharmonic transition rates were cal-

culated by extending the nonadiabatic expression for electron transfer rate derived by Bixon and Jortner. 20 The pre-exponential factor for the adiabatic regime was estimated following Holstein. 18 Our calculations revealed that transitions are nonadiabatic, and therefore rates were calculated using the generalized Fermi Golden rule. 40 Similarly to our recently developed theoretical model, 39 non-equilibrium time-dependent vibrational mode populations, excited vibrational mode lifetimes, and time evolution of mode populations following excitation were found following the collision integral approach. 41 The paper is organized as following. In section 2 we develop in detail our theory for calculating vibrational transition rates of all modes and modeling intramolecular energy transport dynamics. In section 3 we apply this theory to three isomers of acetylbenzonitrile (AcPhCN) and 4-(4-oxo-piperidine-1-carbonyl)-benzonitrile (PBN) (Figs. 1 and 2) and investigate energy transport originating from excitation of the CN stretching mode. Transport originating from other mode excitations is studied for PBN. In conclusion we summarize the results and their relationship to the 1 The DFT anharmonic frequency calculation computes only fourth order anharmonic force constants containing at least two identical indices, and therefore a significant number of force constants are neglected.


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recent RA 2DIR experimental data obtained for the molecules dissolved in chloroform. Since this solvent is non-polar we do not expect it to strongly affect the high frequency mode dynamics and describe the solvent effect within the semi-empirical rate equation approximation, consistent with the experimental observations. CH3 O C N C

O C CH3 N C

(a)

O C CH3

N C (b)

(c)

Figure 1: Structures of the (a) ortho, (b) meta, and (c) para isomers of AcPhCN. The CN stretching mode of the cyano moiety is excited by the pump pulse, and the C=O stretching mode is probed.

O

C

C N

N

C O

Figure 2: Structure of PBN.

2 Theoretical Model The Hamiltonian of molecular vibrations and anharmonicity can be written in a standard form as 42 = H 0 + V, H

(1)

0 , describes harmonic vibrations where the zeroth order term, H 0 = H

N

ωi b†i bi ,

i=1


(2)


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and the perturbation, V, represents third-order anharmonic interactions √ 2 V= 24

N

Vi jk (b†i + bi )(b†j + b j )(b†k + bk ).

(3)

i=1, j=1,k=1

In the above equations, ωi represents vibrational frequencies of N normal modes, Vi jk are third order anharmonic force constants, and b†i , bi are creation and annihilation operators for normal mode i. All energy and frequency parameters are expressed in inverse centimeters similarly to Ref., 42 assuming h¯ = 1. The harmonic vibrational state of an n-atomic molecule can be represented by the sequence of integer population numbers, {n i }(i = 1, ...N), characterizing N = 3n − 6 quantum states of independent normal modes with frequencies ω i (quantization energies ωi ). Anharmonic interactions depend on the vibrational state and lead to transitions that change population numbers (n i , n j , nk ↔ ni + 1, n j − 1, nk − 1). Such transition changes the harmonic energy by Δ(0) i jk = (ωi − ω j − ωk ).

(4)

The transition rate between quantum vibrational states can be approximated using Marcus classical theory. 43,44 The rates of anharmonic transitions for raising mode i population by 1 and lowering jk

mode j and k populations by 1, W ijk , (and backward, Wi ) can be expressed as W ijk

jk Wi

⎛ ⎞ ⎜⎜⎜ (Δi jk + λi jk )2 ⎟⎟⎟ ⎟⎠ . = κ exp ⎜⎝− 4λi jk kB T

(5)

⎞ ⎛ ⎜⎜⎜ (−Δi jk + λi jk )2 ⎟⎟⎟ ⎟⎠ , = κ exp ⎜⎝− 4λi jk kB T

(6)

where the thermal energy k B T is also expressed in inverse centimeters (room temperature is T = 300 K, so in our consideration k B T ≈ 208 cm−1 ). The reorganization energy, λi jk , can be separated into two parts, including contributions from the internal rearrangement of atoms and the solvent. We ignored the solvent contribution because


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bonding to the solvent is much weaker than covalent bonding within the molecule. The internal effective reorganization energy with frequency weight factors is defined as 40,44

λi jk =

N

λa ωa ω , a a=1 2k B T sinh 2kB T

(7)

where λa is the partial reorganization energy due to interaction with vibrational mode a, which can be expressed as 44

λa =

Viia − V j ja − Vkka

2

ωa

.

(8)

The effective transition matrix element, V i jk , also changes, decreasing due to the “polaron” like rearrangement of high frequency modes, taking the form

⎡⎢ ⎛⎜ N ωa ⎢⎢⎢⎢ ⎜⎜⎜ λa ωa tanh Vi jk∗ = Vi jk exp(−S ), S = ⎢⎢⎣1 − ⎜⎜⎝ 2ωa 4kB T 4kB T sinh a=1

ωa 4kB T

⎞2 ⎤ ⎟⎟⎟ ⎥⎥⎥ ⎟⎟⎟⎠ ⎥⎥⎥⎥ . ⎦

(9)

The pre-exponential factor is Equations 5 and 6 was determined as a combination of both the adiabatic and non-adiabatic regimes as κ=

κad κnad . κad + κnad

(10)

In the non-adiabatic regime the prefactor is defined as 44 κnad = cVi2jk∗

π , λi jk kB T

(11)

where Vi jk∗ is the effective anharmonic coupling 44 and c is the velocity of light in cm/s. The adiabatic prefactor was estimated using a generalized expression c κad = 2π

1 ω3a λa λi jk a=1 2kBT sinh N


ωa 2kB T

.

(12)


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Earlier the expression for the pre-exponential factor has been derived in Refs. 18,45,46 assuming all involved vibrations to be classical ω ≤ k B T . In order to remove the quantum mechanical high frequency modes, we artificially introduced the inverse hyperbolic sine factor in accordance with the definition of reorganization energy. This generalization estimates the pre-exponential factor in the adiabatic regime qualitatively. Our calculations revealed that transitions are nonadiabatic, κnad κad , and therefore the rates were calculated using the generalized Fermi Golden rule, 40 while the adiabatic prefactor does not affect the result. Internal relaxation times for each vibrational mode are determined in terms of anharmonic transition rates following the collision integral approach, 5,21 ni − ni,eq coll dni + Ii . =− dt τs

(13)

The equilibrium population number of mode i at temperature T of the environment is determined by the Bose-Einstein distribution, n i,eq =

exp

1 , ωi k T −1

and Iicoll stands for a sum of collision integrals

B

involving the mode i. It can be expressed as the difference between the processes of raising and reducing the population of mode i

Iicoll = −

N N

jk Wi ni (1 + n j )(1 + nk + δ jk ) − W ijk (1 + ni )n j (nk − δ jk )

j=1 k= j N N ij Wikj ni (n j − δi j )(1 + nk ) − Wk (1 + ni )(1 + n j + δi j )nk . −

(14)

j=1 k=1

The marginal case of the mode decay into itself, i → i, k, is ignored in Eq. (14) because the rates for such processes are always negligible due to poor energy conversation. 0 , Eq. (2)) rather The collision integral approximation uses the harmonic states (Hamiltonian H than the eigenstates of the full Hamiltonian, Eq. (1). This is justified when the lifetime of excited vibrations is long compared to the coherent oscillation period of vibrational modes. As it follows from the consideration below, the lifetimes of vibrational states, τ d ∼ 1 ps are indeed much longer than the oscillation periods ω −1 ∼ 0.01 ps. In addition, the phase coherence of the vibrational states 8 ACSParagonPlusEnvironment

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formed in decay processes is neglected within the collision integral approach. The phase decoherence between vibrational states a and b having frequencies ωa , ωb takes place during the time τab ∼ |ωa − ωb |−1 , which is of the order of 0.1 ps even for adjacent vibrational states in AcPhCN (average interlevel spacing there is around 50 cm−1 as follows from our calculations). This time is shorter than the characteristic decay time τd ∼ 1 ps (see estimates below in Table 2 for CN and CO stretching modes). This justifies neglecting the phase coherence in the molecules of interest. For larger molecules where interlevel splitting is remarkably smaller the phase coherence can become important. In the long quasi-periodic polymers (e. g. 47–49 ) it can lead to ballistic transport of a wavepacket, which is beyond the scope of the present manuscript. Following the standard collision integral approach, 41 the correlations between different mode populations are ignored by making a mean field theory assumption, < n i n j >≈< ni >< n j >. Mean field theory is well justified for models on a Bethe lattice or Cayley tree, 50 which can be described as a cycle-free path in which each node is connected to z neighbors. Different branches or generations of the tree do not intersect with each other and therefore, transitions are statistically independent which allows us to ignore the correlations. This approach is applicable to the regime of strong anharmonic coupling which leads to delocalization of the vibrational state in the space of harmonic states. The delocalization is due to the resonances that occur where the anharmonic transition matrix element, V i jk , exceeds the transition energy |ωi − ω j − ωk |. We estimated the number of resonances per mode to be of the order of 1 for intermediate energy modes, ω ≤ 2000 cm −1 , in the AcPhCN molecule. This should be sufficient for the delocalization within the harmonic space. 50 The correlations for squared population numbers cannot be ignored. Assuming the state of each mode is close to a thermally equilibrated state at its temperature, the relation n2i = 2 ni 2 + ni can be used. Although this assumption cannot be proved, a mode is expected to equilibrate to a certain local internal temperature relatively fast as it is coupled to a large number of other modes. In this case, the effect of the n2i terms is relatively small. Applying the assumptions discussed above, collision integrals can be linearized with respect to



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the mode population number, n i , to get each mode relaxation time, τi , expressed as a variational derivative of the collision integral with respect to the mode population number δI 1 =− i . τi δni coll

(15)

The equilibrium population numbers are calculated at room temperature, k B T ≈ 208 cm−1 . Finally, vibrational dynamics induced by excitation of one mode in each molecule was modeled solving the collision integral equation (Equation 13) with initial conditions corresponding to the first excited state of the initially excited mode and thermal equilibrium for all other modes. For example, the CO stretching mode frequency was evaluated at different time delays after CN mode excitation, which models the experimental CN/CO cross peak amplitude. The time-dependent shift of the CO stretch frequency was calculated following Reference 42 using the time-dependent populations of each mode. Mode populations after each time evolution were multiplied by the X-matrix of anharmonic couplings determined by the anharmonic analysis using Gaussian. 16 Vibrational dynamics induced by excitation of the CO and Am-I modes in PBN were also studied. The time interval was determined as dt =

1 , 50τavg

(16)

in which τavg is the average of all vibrational mode relaxation rates. We assumed that all modes have the same equilibration time due to coupling to the solvent, and set the relaxation time to τ s = 14 ps in agreement with the experimentally determined asymptotic cross peak decay rates. 39,51 In reality, each vibrational mode has a different coupling strength to the solvent, however we believe this difference does not strongly affect the results. The latter assumption is questionable in highly polar solvents, such as water, where the solvent can interact strongly with the vibrational modes. For example, the nitrile stretch lifetime of the thiocyanate ion, SCN− , in D2 O is 20 − 50 ps depending on the ion concentration, but it is only 2 − 3 ps in H2 O 52 (see also Ref. 53 ). A similar trend takes place for the CN stretching mode in the cyanide ion, 54 CO stretching mode lifetimes in the water-soluble complex [RuCl 2 (CO)3 ]2 ). 55 We do not


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expect such a strong effect from non-polar solvents like chloroform used in the experiment under consideration (see detail in Ref. 21 ). This is approximately consistent with the recent investigation of the CO stretching mode relaxation in chlorobenzene and chloroform solvents. 56 One should notice that a remarkable effect of the replacement of solvent (chloroform to dichloromethane) on the lifetimes of CN and CO stretching modes in AcPhCN has not been seen in our preliminary studies (the results of this work will be reported separately).

3 Results and Discussion In this section, we report the results of our theoretical investigation of IVR in both the AcPhCN and PBN molecules and compare them to the related experimental data obtained using the RA 2DIR method. 21 For use in theoretical calculations, the geometries of both molecules were optimized with Gaussian 09 16 using density functional theory, the B3LYP functional and the 6-311+G(d,p) basis sets. Harmonic and anharmonic frequencies, third order anharmonic force constants, and anharmonic couplings of the X-matrix were calculated using the same computational method.

3.1 AcPhCN isomers The experimental 21 and computed anharmonic frequencies for CN and CO stretches in all three isomers are shown in Table 1. The calculated frequency values are similar to experimental values, but are all slightly larger. In fact, the calculated CN stretching frequencies are roughly 3% larger, while CO stretch frequencies are approximately 1 − 1.5% greater than experimentally observed frequencies. Also, the proximity of the acetyl group to the cyano group in the ortho isomer affects the CN frequency as evident from its deviation in value from the meta and para isomers. The experimental 21 and theoretical results for CN and CO mode lifetimes as well as energy transport times are summarized in Table 2. The through bond distance between the pumped CN mode and probed CO mode is also shown. The lifetimes of the CN stretching mode in the ortho, meta, and para isomers of AcPhCN were calculated as 1.2, 3.8, and 3.4 ps, respectively. These



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Table 1: Experimental and computed anharmonic frequencies (cm−1 ) for CN and CO stretching modes in AcPhCN isomers. o-AcPhCN m-AcPhCN p-AcPhCN exp. calc. exp. calc. exp. calc. CN stretch 2227.9 2296.1 2235.6 2300.5 2233.7 2300.8 CO stretch 1696.6 1714.3 1695.6 1721.6 1692.7 1719.6 Table 2: Experimental and theoretically calculated lifetimes, energy transport times (Tmax ), and CN/CO distances for AcPhCN isomers.

CN lifetime (ps) CO lifetime (ps) T max CN/CO (ps) CN-CO distance, AÅ

o-AcPhCN m-AcPhCN p-AcPhCN exp. calc. exp. calc. exp. calc. 3.4 ± 0.2 1.2 7.1 ± 0.3 3.8 7.2 ± 0.2 3.4 1.8 ± 0.2 1.05 1.2 ± 0.2 1.9 2.1 ± 0.2 2.0 5.4 ± 0.5 2.9 9.1 ± 0.7 6.7 10.1 ± 1 6.9 4.4 5.7 7.1

values agree reasonably well with the experimental values of 3.4, 7.1, and 7.2 ps, and a similar trend in both theoretical and experimental values is observed. The CN lifetime of the ortho isomer is at least twice as short compared to the meta and para isomers in both experimental and calculated methods. To understand the nature of this difference we considered anharmonic transitions responsible for CN lifetimes in ortho, meta and para isomers of AcPhCN molecule. We found that the significant part (around 75% in ortho isomer) of the decay of the CN stretching mode (mode 8) occurs with the excitation of pairs of skeletal modes 20 and 22. Mode 20 (calculated at 1210 cm−1 ) contains the substantial contribution of the C-C stretching mode where one of the carbons belongs to the cyano group (Fig. 1). Skeletal mode 20 is more strongly localized near the CN group in the ortho isomer compared to the meta and para isomers, which leads to the larger anharmonic interaction matrix element, V 8,20,22 = 13.5 cm−1 in the ortho isomer and 1.3 and −5.3 cm −1 in the meat and para isomers, respectively. The difference in the anharmonic interactions results in a faster decay of the CN stretching mode in the ortho isomer. The stronger localization of mode 20 in the ortho isomer near the CN mode results from the close proximity of the cyano and acetyl moieties. The results for the CO mode excited state lifetimes computed at 1.05, 1.9, and 2.0 ps agree


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reasonably with the experimental values of 1.8, 1.2 and 2.1 ps found for ortho, meta, and para AcPhCN, respectively. As follows from our analysis of mode decay channels the most efficient decay of the CO stretching mode (mode 9) occurs to modes 19 and 38, which both are delocalized skeletal modes (mode 39 instead of 38 for the para isomer). DFT calculations predict a larger anharmonic interaction matrix element for the ortho isomer, V 9,19,38 = 5.3 cm−1 , compared to 3.0 cm−1 for meta and 5.0 cm−1 for para isomers. Also, the transition energy is much closer to zero in the para isomer compared to the ortho isomer. The predicted trend does not agree with the experimental observations; the CO stretching mode shows the fastest decay in meta isomers. This could be because the Gaussian calculations are less accurate for low frequency modes such as significant mode 38 (calculated at 490 cm−1 ). While quantitatively the calculated lifetimes are smaller than experimental lifetimes by a factor of two, the qualitative agreement is encouraging. It is conceivable that the solvent contribute significantly to the reorganization energy, which can result in slower transition rates. A more accurate analysis of the solvent contribution could lead to a better quantitative agreement between the theory and experiment. Figure 3 shows both the experimentally determined 21 and modeled CN/CO cross peak amplitudes for o-, m-, and p-AcPhCN as a function of the waiting time, T . The cross peak amplitude at T = 0, is determined by the coupling of the modes involved (here CN and CO). CN mode excitation causes a shift of the CO mode frequency by the value of their off-diagonal anharmonicity (ΔCN/CO ). In the limit of small anharmonicities the cross-peak amplitude is proportional to the anharmonicity value. If the two modes are far from each other spatially, the shift is small and the resulting cross-peak amplitude is small as well. The CN mode relaxation results in excitation of other modes in the molecule, which can be coupled more strongly to the CO mode causing its larger frequency shift and larger cross-peak amplitude. Since the modes coupled the strongest to the CO mode are expected to be spatially close to the CO group, the cross-peak amplitude dependence on the waiting time reports on the dynamics of the energy transport in the molecule originating from relaxation of the initially excited mode (CN). Notice that the energy transport



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dynamics is expected to be complicated and, in many cases, cannot be well represented with a few exponential components. The whole cross-peak amplitude kinetics is computed theoretically and compared to the experimental kinetics (Figure 3). In addition, the waiting time at which the cross peak reaches its maximum, referred to as the energy transport time (T max ), is used for comparison of the experimental and theoretical data.

(a) o-AcPhCN

(b) m-AcPhCN

(c) p-AcPhCN

Figure 3: Experimentally measured and modeled CN/CO cross peak amplitudes for (a) ortho, (b) meta, and (c) para isomers of AcPhCN. 21 Blue curves represent experimental results and red curves show the calculated cross peak amplitudes. The computed energy transport times of 2.9, 6.7, and 6.9 ps for o-, m-, and p-AcPhCN, respectively, reproduce the experimental trend of 5.4, 9.1, and 10.1 ps. These times are essentially determined by the CN mode decay times (see Table 2) representing the longest time of the internal vibrational relaxation forming the cross peak. There is a clear correlation between the energy transport time (CN stretching mode decay time) and the through-bond distance between the CN and CO groups. Such strong dependence of the energy transport on molecular structure can be useful in recognizing structures of different isomers and provide information on structural characteristics. The predicted cross peak decay is faster in all three isomers than the experimentally observed decay. The decay takes place because of the vibrational energy dissipation to the solvent. In our opinion the discrepancy between the experiment and the theory can be a consequence of the use of the oversimplified model for solvent interaction with vibrational modes. In reality low frequency modes with quantization energies matching the solvent acoustic waves energies decay really fast, while other modes decay much slower. A more accurate consideration of the interaction 14 ACSParagonPlusEnvironment

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of vibrations with the solvent can enhance the present agreement of the experiment and the theory.

3.2 PBN Experimental 51 and computed anharmonic frequencies for CN, CO, Am-I, and Am-II modes are compared in Table 3; as expected the computed anharmonic frequencies are ca. 2 − 3% greater than the experimental frequencies. The experimental 51 and theoretical results for mode lifetimes, energy transport times, and through-bond distances are summarized in Table 4. The calculated lifetimes of the CN, CO, and Am-I modes are 3.0, 0.9, and 0.8 ps, respectively, compared to the experimental values of 6.3, 0.6, and 0.85 ps. Overall, there is a reasonable agreement in the magnitude and the trend between the calculated and experimental 51 lifetimes. The calculated lifetimes for the CO and Am-I modes agree well with the experimental values, while the CN mode lifetime differs by a factor of two, similarly to that in the AcPhCN isomers. This deviation could be attributed to the assumptions of a constant solvent relaxation time, and to the absence of the solvent contribution to the reorganization energy. The lifetime of the Am-II mode was calculated as 1.0 ps, while the experimental lifetime was not reported. The calculated Am-II lifetime seems to be reasonable as it is similar to that in other compounds. 57 Figure 4 shows experimental and theoretical cross peak amplitude dependences on the waiting time for the CN/CO, CN/Am-I, and CN/Am-II cross peaks in PBN. The computed energy transport times (T max ) for the CN/CO, CN/Am-I, and CN/Am-II cross peaks are 11.5, 6.9, and 7.3 ps, compared to the experimental values of 10.5, 7.5, and 8.9 ps, respectively. The computed values reproduce well the difference in the energy transport time emphasizing the correlation of the energy transport time with the transport distance. Although the experimental data and theoretical predictions show strong agreement, there are some discrepancies. The modeled curve of the cross peak decays slower and in the experimental curve we see a “dip” at the tail at ca. 40 ps. Following the expectations of Reference, 58 we attribute this dip to a difference in frequency shift signs caused by coupling at early time delays and at the plateau (T > 50 ps). Theory does not fit the observed behavior well because of the oversimplified treatment of interactions with the solvent. Notice that 15 ACSParagonPlusEnvironment


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at large time delays there is a small but measurable residual temperature increase in the sample, which causes a small frequency shift for the modes of the solute. Both intramolecular coupling and solvent contributions influence the solute frequencies. 59 Since the Am-II mode is located at approximately the same distance from the CN mode as the Am-I mode, we expect to see similar T max values. Indeed, the calculated CN/Am-II energy transport time, 7.3 ps, is only slightly larger than that for CN/Am-I, 6.9 ps. Furthermore, following the conclusions drawn from the experimental results in Reference, 51 the calculated T max values provide the evidence that the low frequency modes strongly coupled to the Am-II mode are located at the piperidine ring while those coupled to the Am-I mode reside on the phenyl ring side of the amide group. Table 3: Experimental and computed anharmonic frequencies (cm−1 ) for CN, CO, Am-I, and Am-II vibrational modes in PBN. PBN exp. calc. CN stretch 2234 2301.5 CO stretch 1722 1758.7 Am-I stretch 1640 1674.5 Am-II stretch 1438 1468.9

(a) CN/CO

(b) CN/Am-I

(c) CN/Am-II

Figure 4: Experimentally measured and modeled PBN cross peak amplitudes for (a) CN/Am-I, (b) CN/CO, and (c) CN/Am-II modes. Blue curves represent experimental results and red curves show the calculated cross peak amplitudes. We also calculated the energy transport times to the Am-I and Am-II modes following CO excitation and the energy transport time to the Am-II mode following Am-I excitation. The calculation 16 ACSParagonPlusEnvironment

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Table 4: Experimental and theoretically calculated lifetimes, energy transport times (Tmax ), and CN/CO, CN/Am-I, CN/Am-II distances for PBN. PBN exp. calc. CN lifetime, (ps) 6.3 ± 0.1 3.0 CO lifetime, (ps) 0.6 ± 0.15 0.9 Am-I lifetime, (ps) 0.85 ± 0.15 0.8 Am-II lifetime, (ps) — 1.0 T max (CN/CO), (ps) 10.5 ± 0.6 11.5 T max (CN/Am-I), (ps) 7.5 ± 0.7 6.9 T max (CN/Am-II), (ps) 9.0 ± 0.6 7.3 T max (CO/Am-I), (ps) 3.5 ± 0.6 4.3 T max (CO/Am-II), (ps) 2.4 ± 0.4 2.3 T max (Am-I/Am-II), (ps) 1.9 ± 0.2 1.6 CN-CO distance, A 11 CN-Am-I distance, A 6.5 CN-Am-II distance, A 6.5 results are shown in Figure 5 together with the experimental data for the CO/Am-II cross peak. The experimental T max values for the CO/Am-I and Am-I/Am-II cross peaks are given in Table 4.

(a) CO/Am-I

(b) CO/Am-II

(c) Am-I/Am-II

Figure 5: Experimentally measured (blue lines) and modeled (red lines) (a) CO/Am-I, (b) CO/AmII and (c) Am-I/Am-II cross peak amplitudes in PBN. The calculated transport times to different modes after excitation of CO or Am-I reproduce quite well the experimental values. Following CO excitation, we calculated energy transport times to the Am-I and Am-II modes to be 4.3 and 2.3 ps, respectively. Since the modes coupled strongly to the Am-II mode are on the piperidine ring side of the amide bond, it is not surprising that the transport time from CO to Am-II is shorter than that from CO to Am-I. The shortest transport time, 17 ACSParagonPlusEnvironment


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1.6 ps, was found for Am-I/Am-II, which is expected due to the closeness of these modes. In order to compare energy transport in the PBN molecule originating from different mode excitations, the T max values are shown in a schematic representation in Figure 6. We calculated

Figure 6: Schematic representation of energy transport times for various initially excited modes in PBN. Arrows represent energy transport from the initially excited mode to the probed mode, and corresponding T max values are shown. that after CN excitation, it takes 11.5 ps for the energy to travel from the CN to the CO mode. By adding the energy transport times from other mode excitations, we can compare the total energy transport time over the length of the molecule. For example, it takes 6.9 ps for excess energy to reach Am-I from CN, and 4.3 ps to reach Am-I from CO. Therefore, the time for energy to travel through the molecule is 11.2 ps, which is close to the CN/CO energy transport time of 11.5 ps. Furthermore, it takes 7.3 ps to reach Am-II from CN and 2.3 ps to reach Am-II from CO, which totals 9.6 ps to cross the length of the molecule. Finally, adding the transport times for CN/Am-I, CO/Am-II, and Am-I/Am-II, the total energy transport time over the molecule is 10.2 ps. Let us use the transport times in Fig. 6 to analyze the energy transport mechanism in the PBN molecule. We consider two possible dependencies of transport time on the distance including the direct proportionality t ∝ x (e. g. ballistic transport) and the random walk dependence, t ∝ x 2 . 47–49 In the first case the total energy transport time should be approximately equal to the sum of the √ 2 transport times of all consecutive steps i ti = ttot , while in the second case we expect i ti = √ 2 ttot . Our case is intermediate so we have i ti < ttot < i ti , therefore the regime is intermediate 18 ACSParagonPlusEnvironment

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but it is closer to the linear transport time dependence on the intermode distance. We do not believe this is ballistic transport because it is too slow. Indeed, , our energy transport rate can be estimated as 134 m s−1, which is almost an order of magnitude smaller than the ballistic energy transport rate through alkane chains, v ≈ 950 m s−1 , reported by Wang et al. 47 Moreover, the theory cannot handle the true ballistic transport since it neglects phase coherence of different modes. Instead this can be a random walk affected by the energy dissipation to the solvent. In this case the cross peaks are determined by nearly unidirectional pathways. Although the probability for such pathways is lower than for typical random walks, the energy reaches the target mode faster following these pathways and therefore dissipation losses are smaller. This makes them dominating pathways and the energy transport becomes similar to the ballistic transport. Indeed, one can qualitatively model energy transport through the PBN molecule using the one dimensional diffusion equation, with the energy absorption term describing the energy dissipation to the solvent as ∂2 P P ∂P =D 2 − , ∂t τs ∂x

(17)

where P(x, t) is the time-dependent energy density along the molecule, D is the energy diffusion coefficient and τ s ≈ 14 ps is the relaxation time due to the interaction with the solvent. If we take the initial condition in the delta-function form P(x, 0) = δ(x), corresponding to the local optical excitation of some mode then the time dependent solution of Eq. (17) reads t x2 1 P(x, t) = √ e− τ s − 2Dt . Dt

(18)

The energy transport time to a point x can be determined as the time when P(x, t) reaches its maximum. This problem can be resolved analytically and the expression for the maximum reads

T max (x) =

At very long solvent relaxation times, τ s >

x2 D

1 2

2x2 D ,

+

1 4

+

2x2 Dτ s

.

(19)

we have the standard random walk behavior with 19

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T max ≈

x2 D,

while in the opposite case the linear dependence on transport distance, x, should be ob τs , as described above. Since the characteristic transport times are comserved with T max (x) ≈ x 2D parable with half of the solvent relaxation time, the intermediate regime takes place as expected. Assuming that the second regime takes place and using estimated velocity of energy transport, v ≈ 134 m s−1 , and the solvent relaxation time τ s ≈ 14 ps one can estimate the energy diffusion cov2 τ s 2

≈ 12.56 Å2 ps−1 . This diffusion coefficient describes the energy random walk √ as a set of hops during the mode decay time τd ∼ 1 ps to the length of order of three Dτd ∼ 3.5 Å.

efficient as D =

This size corresponds to the typical distance between anharmonically coupled vibrational modes in polyatomic molecules. The latter picture can be tested in gas phase measurements where the energy dissipation is negligible and a distance dependence of energy transport time should be closer to the random walk behavior, t ∼ x2 . Modifications of solvents affecting the energy dissipation time for the solute can be also used to verify the theory.

4 Conclusions A theoretical model has been presented that uses a generalized Marcus approach to describe vibrational relaxation and energy transport dynamics due to anharmonicity in polyatomic molecules. Our analysis was restricted to third-order anharmonicity and ignored the solvent contribution to reorganization energy in anharmonic transition rate calculations. The computed data were consistent with the results from RA 2DIR experiments qualitatively, however vibrational lifetimes deviated by less than a factor of two. The modeling of different isomers of AcPhCN and PBN polyatomic molecules was able to reproduce reasonably well the observed energy transport times and supported the correlation between the energy transport time and bond distance found through RA 2DIR experiments. The energy transport through the PBN molecule can be interpreted as a random walk in the presence of a significant energy dissipation to the solvent. Further applications of the theory can be useful in determining structural characteristics, rec-


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ognizing different isomers of various molecules, and can aid in interpreting spectroscopic data. Further developments of the theory may include an introduction of the solvent contribution to the reorganization energy and a more accurate analysis of energy dissipation to solvent with the use of mode energy dependent relaxation rates. Hopefully this development will improve the agreement between the experiment and the theory.

Acknowledgments This work is supported by the NSF EPSCoR LA-SIGMA program (EPS-1003897), and the NSF award CHEM 0750415.

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Theoretical Study of Internal Vibrational Relaxation and Energy

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