Vibrational States and Nitrile Lifetimes of Cyanophenylalanine

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Vibrational States and Nitrile Lifetimes of Cyanophenylalanine Isotopomers in Solution Hari Datt Pandey, and David M. Leitner J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b06300 • Publication Date (Web): 09 Aug 2018 Downloaded from http://pubs.acs.org on August 14, 2018

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The Journal of Physical Chemistry

Vibrational States and Nitrile Lifetimes of Cyanophenylalanine Isotopomers in Solution Hari Datt Pandey and David M. Leitner* Department of Chemistry and Chemical Physics Program, University of Nevada, Reno, NV 89557, USA * [email protected]

Abstract Nitrile lifetimes and the structure of the vibrational state space of 4 isotopomers of cyanophenylalanine in solution are calculated. While the frequency of the nitrile of the 4 isotopomers decreases in the order

12

C14N,

12 15

C N,

13 14

C N, and

13

C15N, the lifetime varies non-

monotonically with the change in frequency. The vibrational properties of the molecules that control the lifetime are examined. The specific resonances that contribute to the lifetime are tuned by isotopic substitution and the magnitude of the anharmonic constants involved in the coupling of vibrations that mediate the lifetime of the nitrile vary with CN mass. The nature of the modes coupled to the nitrile varies as the frequency of the nitrile changes with isotopic substitution. For some CN frequencies the modes coupled to the CN are rather localized to the ring while at other frequencies the modes coupled to the CN are more delocalized. Comparison of the calculated frequencies and lifetimes with recent experimental measurements on these molecules is discussed.

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1. Introduction Predicting rates and pathways for vibrational energy flow through large molecules in solution remains challenging. One means to control energy transfer is isotopic substitution, as explored, e.g., in the pioneering work of Sibert, Reinhardt and Hynes on vibrational relaxation of CH and CD overtones of benzene.1-2 Indeed, rather dramatic changes in energy transfer kinetics can occur even with much more modest changes in relative masses due to isotopic substitution. Gai and coworkers, in their study of 4 isotopomers of cyanophenylalanine,3 found that while the frequency of the nitrile of the 4 isotopomers decreases by a few percent from 13 14

C N, to

12 14

C N,

12 15

C N,

13 15

C N, respectively, the lifetime can vary several fold and non-monotonically with

changes in the frequency. Control of vibrational lifetime by isotopic substitution could be exploited in the design of labels for time resolved IR spectroscopy in cases where the lifetime limits the detection time. Nitrile labels in particular are useful in probing structure and dynamics of biological molecules.4-7 The vibrational lifetime of the bright state generally depends on the availability of resonances that couple via typically low-order anharmonicity.8-9 The magnitude of the anharmonic coupling depends not only on its order but also on the nature of the modes that couple to the nitrile.10-11 Here we present a computational study of the isotopomers of cyanophenylalanine in solution, in which we calculate the lifetime of the nitrile and examine the properties that mediate it. That the structure of molecular spectra and the lifetimes of molecular vibrations are mediated by the local density of anharmonically coupled states has been appreciated at least since the work of Sibert, et al.,2 who organized dark states into two tiers, one that couples anharmonically to the bright state, and another that couples to the first tier of dark states. Indeed, spectra of large molecules are determined by coupling involving many such tiers of states.12-14


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With many tiers emerges the possibility of localization in the vibrational state space,15-16 as has been observed for molecules where the total density of states is quite large.15 The resulting longlived vibrations can be found at energies sufficient for chemical reactions, and mediate chemical reaction kinetics involving sizable molecules in gas and condensed phases and in biomolecules.17-30 Similarly, tier models have provided insights into the role of resonances and anharmonicity in vibrational energy relaxation and thermalization in molecules at interfaces between two materials and thermal conduction through the interface.31-36 Some of that work has examined the possibility of energy rectification in molecules, where there is a propensity for energy to flow in molecules preferentially from one chemical group to another in condensed phase or at an interface.37-44 The non-monotonic lifetime observed for the nitrile of the 4 isotopomers of cyanophenylalanine in solution by Gai and coworkers3 is dictated by shifts in the nitrile mode with isotopic substitution relative to other modes of the molecule to which it is anharmonically coupled. In a set of companion calculations, the vibrational frequencies of the 4 isotopomers of cyanophenylalanine and the smaller nitrile-toluene were computed and organized into states that are coupled by different order of the anharmonicity.

Apparently, the density of states of

vibrations coupled by third and fourth order anharmonicity is similar for each of the isotopomers, differing more significantly only for states coupled by higher order anharmonicity in a way that appears consistent with the relative nitrile lifetimes of the isotopomers.3 However, we shall see that, while the number of vibrational states coupled by cubic anharmonicity in a sizable energy window is indeed similar for the 4 isotopomers, the specific resonances that contribute to the lifetime are tuned by isotopic substitution. Moreover, the anharmonic constants for the modes that mediate the lifetime of the nitrile vary significantly. Some of the coupled modes are quite


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localized to the ring and others more delocalized, i.e., the nature of the modes coupled to the nitrile also varies as the frequency of the nitrile changes with isotopic substitution. We examine these points in this study to determine the properties that control the lifetime of the nitrile. In the following section, we present computational methods used to calculate the nitrile vibrations, anharmonic coupling, and vibrational lifetimes for the 4 isotopomers of cyanophenylalanine. In Sec. 3 we present results for the lifetimes, comparison with experimental results, and analysis of the vibrations of cyanophenylalanine that control the lifetime. Conclusions are presented in Sec. 4.

2. Theoretical and computational methods Following Gai and coworkers,3 we refer to the isotopomers with 12C14N, 12C15N, 13C14N, and 13C15N as, respectively, molecule 1, 2, 3 and 4. The initial geometry of cyanophenylalanine was constructed using the Avogadro visualization package and optimized using molecular mechanics (MM) with the General Amber Force Field (GAFF). The MM optimized geometry was introduced into a semi-empirical method (PM6) optimization, followed by a Hartree-Fock level (HF) calculation with the 6-31G basis set applying the CPCM water model for the selfconsistent reaction field (SCRF). The HF optimized geometry was taken as an initial structure for a DFT/B3LYP/6-31G-level calculation, followed by DFT/B3LYP/6-31+G** using the same water model for the SCRF. Finally, the geometry, Hessian, normal modes, frequencies, and anharmonic constants were calculated using DFT/B3LYP/6-31+G** with basis set 6-31+G** in the presence of the same implicit water solvent model as previously calculated with a SCRF. An ultrafine integration grid for the two-electron integral calculation with accuracy 10−13 and very


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tight convergence criteria were applied throughout the electronic structure calculation process. All the DFT calculations were executed using the Gaussian-09 computational package.45 Using the isotopomers’ vibrational frequencies with anharmonic corrections for the vibrational modes and third-order anharmonic coupling constants we have located the resonances for the nitrile stretches via anharmonic interactions and calculated vibrational relaxation rates for those and other modes.

Since the experiments were done for the solute in an aqueous

environment, we included effects of the solvent in our calculations of the vibrational lifetimes. The relaxation rate for excess energy in the nitrile modes due to the anharmonic interactions with other modes of the molecule and with the solvent can be approximated by Fermi's golden rule (GR). We have checked that at the energies of interest for these molecules, the product of the average coupling and the local density of states for these solvated molecules is at least of order 1, so that a GR calculation provides a reasonable estimate for the relaxation rate.15 The relaxation rate due to excess energy in mode, α, with frequency ߱௔ , interacting via 3rd order anharmonic coupling to other modes can be expressed as the sum of "decay" and "collision" terms, W = Wd + Wc, where46-48 (1a)

(1b)

Γβ is the damping rate (in units cm-1) of mode β due to coupling to other modes and to the environment, nα is the population, which we take to be

, and

Φ αβγ the

coefficients of the cubic terms in the expansion of the interatomic potential in normal coordinates. Damping rates are often on the order of 1 ps-1 for sizable organic molecules.49-52


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Since the vibrational relaxation rate that we calculate influences the value of each, we calculate its value self-consistently, as is done, for example, in vibrational self-consistent field (VSCF) calculations of vibrational spectra.53 In the approach we adopt, described below, we use an initial estimate to the rate, the initial mode frequencies and the off-diagonal anharmonic constants, and solve for the rate iteratively.

The contributing solvent effect due to the

surrounding water is about 4 cm-1,54-56 but the result for the lifetime is not very sensitive to that value; by varying the solvent effect from 1 to 8 cm-1, we find that the nitrile lifetimes differ within 10%, so that it is mainly intramolecular effects that control energy transfer from the nitrile. The self-consistent analysis was carried out as follows: There are 3N-6 self-consistent equations representing the damping rate coupled to one another. Accordingly, we have adopted a self-consistent system for damping rate of the nitrile, which for mode α is Γα (ωα ) − f ( Γα , ωα ) = 0

(2)

where,

(3)

where Γα(s) is the relaxation rate from mode α into the solvent, and the latter two terms are Wα , the intramolecular relaxation rate defined by Eq. (1). For the initial guess for Wα we use a nominal value of 1 ps-1, and for the solvent we adopt a range of values listed above. The vibrational relaxation rate is updated at each step until it converges with an error less then 0.0001%. The vibrational lifetime was found when the iterative solution of the simultaneous


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equations converged. In the general iterative scheme, the roots of Eq. (3) were obtained using the Newton-Raphson method. The calculated vibrational lifetimes of all the modes were introduced back into Eq. (1) to calculate the lifetime of the nitrile. We note that the frequencies used in Eq. (1) – (3) are the normal mode frequencies corrected by diagonal anharmonicity. The anharmonic frequencies of the isotopomers are listed in the SI.

In addition to diagonal

anharmonicity, the frequency may be shifted by off-diagonal anharmonicity, which to third order can couple a triple of modes and shift the frequency of each mode.15, 57-58 We have examined this shift to determine if there would be a noticeable effect if it were introduced into Eq. (1), and found the effect of the shift on the nitrile lifetime to be negligible. It is instructive to characterize the resonant couplings with the nitrile stretch for each isotopomer and compare them with each other. We have calculated the resonance distance, ߂߱, defined as ∆ω = ωα − ω β − ωγ , and the corresponding third order resonance parameter, TFR .59-60 For a particular triple of modes (or a pair, where one is an overtone), the value of TFR is computed as a ratio of the third-order anharmonic constant and resonance distance,59-60

TFR =

Φαβγ ∆ω

(4)

Modes are resonantly coupled if the value of TFR is roughly 1 or larger. We present results below for this measure of resonance coupling for the modes of the 4 isotopomers that contribute to the nitrile lifetime for an indication as to whether the coupling is resonant or not.


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3. Results and discussion We begin with the tier diagrams constructed for each of the isotopomers, shown in Fig. 1. The frequency of each of the nitrile stretches is indicated there. They are, respectively, 2284.0 cm-1, 2257.3 cm-1, 2232.2 cm-1 and 2202.6 cm-1, for molecules 1, 2, 3 and 4. These values are about 50 cm-1 higher than those found experimentally, which are, respectively, 2236.7 cm-1, 2210.1 cm-1 2183.5 cm-1, and 2156.1 cm-1.3 The calculated values lie over a range of 81.4 cm-1 from molecules 1 to 4, whereas the measured values lie over a very similar range of 80.6 cm-1. The first tier of states resonant with the nitrile stretch for the 4 isotopomers is indicated in Fig. 1. These are all combinations of modes and overtones that lie within +/- 50 cm-1 of the nitrile frequency that can be coupled by cubic anharmonic terms. We observe in Fig. 1 that there are 6 or 7 states in this window in each case. However, a closer look reveals that, despite the similarity in the number of states within the rather large window, the local density of states is not the same for all the molecules. The tier 1 states in some cases appear closer together in frequency than in others. For example, the local density of states in tier 1 is evidently smaller for molecule 1; the states of that tier are more spread out over the allotted interval than for the other isotopomers. As can be seen in Eq. (1), it is not enough to only elucidate the first tier, since the relaxation rate into the first tier depends on the rate of relaxation of states from the first into the second tier, etc. Using the same criterion for identifying potentially relevant states for relaxation from a given vibrational state, we calculate a second tier, consisting of all states that lie within +/- 50 cm-1 of the frequency combination of any state in the first tier.

The third tier is

constructed in the same fashion. For the second and third tiers, we only plot in Fig. 1 those states that lie within a window ≈ +/- 50 cm-1 around the nitrile stretch frequency.


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The tier diagram is useful to explain the iterative calculation we use to solve Eq. (1), in a manner similar to that used to calculate spectral line widths by Stuchebrukhov and Marcus for tert-butyl acetylene and a Si substitute.13 We take an initial guess for the lifetime of the states in tier 3 to be 1 ps-1 to calculate the lifetimes of the states in tier 2 using the iterative scheme described in Sec. 2. We then do the same for the states in tier 1, taking the lifetimes for the states calculated in tier 2 and using the same iterative scheme, and finally, the lifetimes for the modes of tier 1 are used in Eq. (1) to calculate the nitrile lifetime. Using Eq. (1), we thereby find the lifetime of the nitrile of molecule 1 to be 4.5 ps, close to the experimental value of 4.0 ps. For the isotopomers, the calculated lifetime is 2.4, 2.0 and 3.7 ps respectively, for molecules 2, 3 and 4. The experimental values were found to be 2.2, 3.4 and 7.9 ps, respectively.3 For the two isotopomers with 12C the lifetimes are within 15% of the measured results, though for the other two, with

13

C, the values are off by about a factor of 2.

The results of the calculations and experiments are listed in Table 1. We observe a nonmonotonic trend as the mass increases, with the longest lifetimes found for

12

C14N and

13 15

C N.

The experiments also indicate that molecules 1 and 4 exhibit the longest lifetimes. We now turn to examination of the resonances that mediate the vibrational lifetimes of these isotopomers, followed by discussion of similarity and differences between the computational and experimental results. To quantify the Fermi resonances involving the nitrile, we have calculated the third order Fermi resonance (TFR) parameter, defined by Eq. (4). Consider first molecule 1. In Table 2 we see that for this molecule the value of the TFR for all combinations and overtones that couple with the nitrile is 0.3 or smaller, so that all couplings are off-resonance. The dominant coupling with the nitrile of molecule 1, which contributes most to establishing the lifetime, is to the modes


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with frequency 1508 cm-1 and 745 cm-1. The atomic displacements of these modes are illustrated in Fig. 2. There we see that both modes are in part localized to the ring, with significant displacements in the direction of the nitrile.

The lower frequency mode also exhibits

displacements over other parts of the molecule, not only the ring. The ring mode of frequency 1508 cm-1 also couples to a more delocalized ring mode of frequency 797 cm-1, the combination of which makes a smaller contribution to the nitrile lifetime. Another ring mode that couples to the nitrile and contributes to the lifetime has frequency 1610 cm-1 and couples to a more delocalized mode of frequency 710 cm-1. Similar contributions are made by modes of frequency 1216 cm-1 and 1020 cm-1, partially localized to the ring, which are very similar to modes that mediate the lifetime of some of the other isotopomers, discussed below. Concerning the pathway for energy flow from

12 15

C N of molecule 2, energy flow is

mediated to a large extent via the Fermi resonance coupling to modes with frequency 1509 cm-1 and 743 cm-1, for which we find the TFR to be about 1.7 due to the magnitude of the third order anharmonic coupling of about 8.5 cm-1 and a resonance distance of 5 cm-1. These modes are very similar to those that mediate the lifetime of molecule 1, but in this case they are resonant with the nitrile, giving rise to a shorter lifetime. Again, the mode with frequency 1509 cm-1 is localized to the ring and the 743 cm-1 mode is more delocalized over the molecule. For molecule 3, there are some 3 pairs of modes that are resonant or nearly so, with a TFR of at least 0.7. The modes contributing most to the lifetime, for which the cubic anharmonic constant is largest, is again a pair of ring modes but also delocalized in other parts of the molecule. They have frequency 1216 cm-1 and 1020 cm-1. Other significant pairs of modes that regulate the nitrile lifetime involve the ring mode of frequency 1508 cm-1 and more delocalized modes of frequency 724 cm-1 and 713 cm-1. These pairings are very similar to those that regulate the lifetime of


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molecules 1 and 2. Since there are several resonant interactions with the nitrile the lifetime is relatively short, in our calculations shorter than for any of the other isotopomers, somewhat shorter than the lifetime for molecule 2. Finally, the nitrile of molecule 4 exhibits resonant coupling with two ring modes of frequency 1180 cm-1 and 1021 cm-1, with the TFR 3.6 due to anharmonic coupling of magnitude 5.3 cm-1 and separation in energy of 1.5 cm-1. Still, this anharmonic coupling is smaller than the magnitude of the anharmonic coupling constant contributing most to the lifetime of the other isotopomers.

For that reason, though the coupling is resonant, when this rather small

anharmonicity is introduced into Eq. (1) one finds the lifetime to be relatively long, comparable to what we find for the

12 14

C N nitrile, and about twice as long as for the other isotopomers.

Other contributions to the energy transfer rate from the nitrile for this isotopomer, and the others, can be found in Table 2. In addition to calculation of the lifetimes themselves, we examined modes of the molecule that regulate the lifetime. For 12C14N of molecule 1, the coupling is off-resonance, the only such case among the 4 isotopomers. This is the reason for the relatively long lifetime for the nitrile of molecule 1. Coupling of the nitrile to the partial ring modes of molecules 2, 3, and 4 are all resonant. For molecules 2 and 3 there are combinations of ring modes and more delocalized ring modes that resonantly couple to

12 15

C N and

13 14

C N, respectively. This is why

the lifetime of the nitrile is shorter than for molecule 1, where the modes that couple to it are similar, but off-resonance. The situation for molecule 4 is different. Coupling to

13 15

C N is

resonant, but it is by a combination of modes that are only partially localized to the ring, so that the anharmonic constant is relatively small. For the other isotopomers, at least one of the modes that couple to the nitrile is localized to the ring, giving rise to a somewhat larger anharmonic


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constant. That is the reason for the relatively long lifetime of the nitrile in molecule 4, and we believe that this is why in the experiment, too, a relatively long lifetime is observed3 for the 13 15

C N isotopomer. It may also be that, with a small detuning in the first tier not captured in our calculations,

energy transfer from the nitrile of molecule 4 into the first tier is off-resonance. This could explain the shorter computed lifetime compared to experiment.

In general, quantitative

agreement between calculation and experiment depends on the precise positioning of the computed vibrational frequencies. Better agreement would require a more accurate, higher level ab initio approach than that carried out here. Nevertheless, the ab initio calculations reveal how the nature of the modes that are coupled, whether more or less localized to the ring, also influence the lifetime. Even if the vibrational frequencies are computed imperfectly, trends seen in the experiments, notably the relatively long lifetime of molecule 4, can be captured in the computational work, suggesting the important role played by the rather delocalized modes coupled to the nitrile of molecule 4 in establishing the lifetime. Since the vibrational states of the second tier of molecule 4 appear somewhat offresonance, too, it could be that the mechanism for energy transfer from the nitrile of molecule 4 is in fact off-resonance through more than one tier, an example of vibrational superexchange.13, 15, 61

Calculation of energy transfer by vibrational superexchange, which is the quantum

mechanical equivalent of dynamical tunneling,62 would require a treatment beyond first-order time-dependent perturbation theory.13, 15, 61 Because of the resonance structure we found for the cyanophenylalanine isotopomers this was not necessary for our calculations, but we anticipate for some vibrations of isotopomers of a variety of molecules such an approach would be needed.


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4. Conclusions For certain applications, where the vibrational lifetime limits the detection time, the utility of a vibrational label in time-resolved IR experiments may depend at least in part on the lifetime of the mode that is probed. The lifetime, in turn, is mediated by the local density of states coupled to the vibration that is probed by primarily low-order anharmonicity, the nature of the vibrational modes that are coupled, and by the coupling of the vibration to the solvent. Those properties can be tuned by isotopic substitution. Here we have calculated the vibrational lifetime of the nitrile mode of 4 isotopomers of cyanophenylalanine and examined how the lifetime is affected by isotopic substitution. The nitrile of the 4 isotopomers we examined were

12

C14N,

12 15

C N, 13C14N, 13C15N, referred to as molecules 1 to 4, respectively. The longest lifetimes were

found for

12

C14N and

13 15

C N and the shortest for

12 15

C N and

13 14

C N. The lifetime is mainly

controlled by the tuning of resonances between the nitrile and vibrational modes that are at least partially localized to displacements of the ring.

The results presented here suggest that it should be possible to design computationally long-lived labels for appropriate molecular systems, using the possibility of isotopic substitution in the design strategy. As a general rule, we expect longer lifetimes to be achieved when loworder anharmonic coupling is off-resonant and involves somewhat more delocalized modes of the molecule, as illustrated by the results for the isotopomers of cyanophenyalanine. More quantitatively, the anharmonic coupling and local resonance structure need to be determined, which we have carried out for this system, with trends in the frequencies and lifetimes that provide insights into those found experimentally.


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Supporting Information Available The vibrational frequencies computed for the isotopomers.

Acknowledgements The authors thank Matthew Tucker for helpful discussions. Support from NSF grant CHE1361776 is gratefully acknowledged.


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Tables

Table 1: Computed and experimental frequencies and lifetimes for the 4 isotopomers. The frequencies are listed in cm-1 and the lifetimes in ps. The experimental values are from Ref. 3.

Isotopomer

Calc. Freq.

Expt. Freq.

Calc. Lifetime

Expt. Lifetime

1

2284.0

2236.7

4.0

4.5

2

2257.3

2210.1

2.4

2.2

3

2232.2

2183.5

1.0

3.4

4

2202.6

2156.1

3.7

7.9

Table 2: Dominant vibrational modes of the isotopomers of cyanophenylalanine participating in vibrational energy relaxation of the nitrile mode. Units for frequencies, anharmonic constants are cm-1.

12

C 14N frequency = 2284.0 cm-1 (Molecule 1)

ωγ

Contribution to relaxation rate (ps-1)

Φαβγ

∆ω

TFR

Localization

1507.97

744.95

0.38

-8.86

31.06

0.285

(ring, ring+alanine)

1610.44

709.83

0.15

-6.09

36.3

0.168


1020.38

1216.47 0.06

-5.45

47.13

0.116


1507.97

796.96

2.43

20.95

0.116

(ring, alanine+ring)

1137.98

1137.98 0.02

-1.56

8.01

0.195

(ring+alanine, ring+alanine)

1610.44

638.59

-1.48

34.94

0.042


1137.98

1168.11 0.01

1.06

22.12

0.048


ωβ

0.04 0.01


20

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12

C 15N frequency= 2257.3 cm-1 (Molecule 2)

1509.43

742.71

0.85

-8.49

5.12

1.658


1021.83

1220.16 0.12

-5.92

15.26

0.388


1509.43

708.86

0.04

-7.99

38.96

0.205


1509.43

723.76

0.03

-5

24.07

0.208

(ring, alanine)

1611.12

637.88

0.02

-1.55

8.26

0.188


1137.5

1137.5

0.01

-1.61

17.74

0.091

(ring+alanine, ring+alanine)

1021.83

1220.39 0.01

1.28

15.03

0.085


13


1019.79

1215.63 0.36

-5.38

3.18

1.695


1507.76

723.84

0.33

-4.76

0.64

7.414

(ring, alanine)

1507.76

713.35

0.32

-7.58

11.13

0.68


1507.76

743.35

0.17

-8.36

18.87

0.443


1019.79

1183.81 0.03

-5.26

28.64

0.184

(ring,ring)

1019.79

1219.53 0.02

1.57

7.08

0.222


1019.79

1182.66 0.02

-3.56

29.79

0.12


13


1020.86

1180.24 0.39

-5.26

1.46

3.617

(ring, ring)

1020.86

1180.61 0.18

-3.56

1.09

3.279


1507.25

715.66

0.13

-7.58

20.35

0.372


1507.25

741.57

0.03

-8.36

46.26

0.181


1507.25

723.94

0.03

-4.76

28.63

0.166

(ring, alanine)

1020.86

1218.92 0.02

-5.38

37.22

0.145


1743.49

482.04

-0.1

22.97

0.004

(alanine, delocalized)

0.001


21


Page 22 of 24

Figures

Figure 1. Diagram depicting the first three tiers of states coupled sequentially from the bright state. These are energy levels for states coupled by cubic anharmonicity starting with the nitrile for the isotopomer indicated above each tier diagram.


22

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Figure 2. Under the name of each of the four isotopomers are the images indicating the displacements corresponding to the modes coupled to the nitrile that predominantly mediate the lifetime. For other important contributions see text.


23


Page 24 of 24

TOC Graphic


24

Vibrational States and Nitrile Lifetimes of Cyanophenylalanine

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