Web Release Date: June 4,
Surface Jumping: Franck-Condon Factors and Condon Points in Phase Space
Department of Chemistry, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel, and Departments of Chemistry and Physics, Harvard University, Cambridge, Massachusetts 02138
Received: November 29, 2001
In Final Form: March 27, 2002
Abstract:
We generalize the concept of Tully-Preston surface hopping to include larger jumps in the case that the surfaces do not cross. Instead of identifying a complex hopping point, we specify a jump between two locales in phase space. This concept is used here to find propensity rules for the accepting vibrational mode(s) in a radiationless vibronic relaxation transition. A model inspired by the S_{2} S_{0} vibronic relaxation transition of the benzene molecule in which 30 modes of vibration compete for the electronic energy is studied within this approach. For this model, we show that almost all of the energy must go to a single C-H local stretching. The initial conditions for vibrations of this mode are a coordinate jump of the hydrogen atom toward the ring. All of the other modes undergo an almost vertical transition, in which the energy that they take is determined by their equilibrium displacement between the two surfaces. We observe that for a large energy gap the masses and frequencies become the defining parameters for choosing the accepting mode. Anharmonicities are very important when a competition between degenerate modes occurs. These conclusions are demonstrated by the specific model considered here but apply in general to any weak internal conversion process.
Molecular electronic transitions may be radiative or nonradiative. In either case, the process may be Franck-Condon enhanced or suppressed. The enhanced processes correspond to a crossing of Born-Oppenheimer potential energy surfaces in the classically accessible region, whereas Franck-Condon suppressed events have no such crossing. Examples include radiative processes in the wings of absorption or emission band envelopes and radiationless events for nested potential energy surfaces.
We focus here on intuition and procedures for realistic polyatomic processes. For the case of surface crossing, or avoided crossing, the Tully-Preston surface hopping^{1} picture has been of considerable value, permitting both intuition and simple procedures for calculating rates of electronic conversion. Some extensions of this approach and insight can be found in refs 2 and 3.
When the surfaces cross, trajectories can hop smoothly with little or no change in position or momentum at the time of the hop. But often surfaces do not cross. What then? Of course, the rate for such cases is generally lower because of the implied suppression of Franck-Condon (FC) factors. However, these suppressed events may be "the only game in town" or may be significant channels competing with others.^{4} Analytical continuation is sometimes used in these cases to recognize a complex jumping point in coordinate space,^{5,6} and quasiclassical models are also useful,^{7} but we want a more direct procedure that is applicable to many degrees of freedom. The approach we use is surface jumping.^{8}
Our paradigm in this paper is a radiationless transition between nested surface potentials as shown in Figure 1b. This situation can also arise radiatively, if we consider, say, the upper surface to be raised past the absorption maximum by the photon energy h as in Figure 1d. We have in mind a many-coordinate example.
Perhaps it is not obvious that any useful classical picture can emerge for this noncrossing situation, because the situation that we describe is classically forbidden. However, other classically forbidden processes have very useful classical descriptions, such as barrier tunneling, which involves trajectories in imaginary time or on the inverted potential energy surface.
Recently, some of us have outlined a procedure to recognize the jumping points in phase space in the noncrossing regime.^{8,9} We demonstrated the method by applying it to a two-dimensional harmonic oscillator, a model in which one can find the jumping point analytically.^{8} The results were encouraging. Identification of the jumping points was shown to allow for an easy derivation of propensity rules for the distribution of the electronic energy between competing vibrational modes. The treatment was limited in two ways: first by the assumption of harmonic potentials and second by the assumption of an allowed transition with no derivative coupling between the states. Here, we generalize the treatment to transitions between any two multidimensional, nonharmonic potentials and establish the numerical tools that are needed for the recognition of the jumping points in the general case. We also generalize the treatment so that it applies to internal conversion, that is, electronically forbidden transitions induced by derivative couplings, and study what is then the contribution of the promoting mode.
Questions that we address in this paper are as follows: Where will the jump between the two surfaces take place? What is the best system of coordinates to describe the process? What is the sensitivity of these predictions to the value of various parameters? The importance of the absolute value of the frequency and the reduced mass of a mode in the determination of its propensity as an accepting mode was previously discussed^{10,11} with implications for the isotope effect for nonradiative decay.^{11} Here, we supplement these early studies by considering the transition in phase space.
As an example, we apply our approach to a model inspired by a complex physical system, the 30 modes problem of the S_{2} S_{0} transition of the benzene molecule. As detailed below, the model imitates some properties of the benzene molecule yet differs from it in some other features. While it is not a complete description of the benzene molecule, much can be learned from it nevertheless. Even for this simplified model of the S_{2} S_{0} transition, it is not trivial to determine which modes or combination of the 30 vibrational modes of our system would be first excited during the quantum jump. With the new surface-jumping approach, we are able to do so with relative ease.
The outline of the paper is as follows: Surface jumping is defined and analyzed in section II. In section III, we apply the method to a model of an S_{2} S_{0} transition inspired by the benzene molecule and study the sensitivity of the results to different conjectures regarding the surface potentials. Section IV concludes and summarizes.
2.1. Quantum-Mechanical Treatment in Coordinate Space.
The probability for an allowed transition from the vibronic state
i to a vibronic manifold j, where i, j refer to electronic states,
is given by
The rate of the transition is given by the Fermi golden rule:
The common way to determine which vibrations are most likely to be excited is to calculate the different FC factors and the densities of final states for all possible combinations of different divisions of the energy quanta between the modes.^{12} This, however, would demand an enormous computational effort and can be regarded as impossible for large molecules especially when the energy that is transferred between the degrees of freedom is large. Efficient ways to calculate FC factors are limited to the harmonic approximation.^{13} Moreover, when the potential energy surface cannot be treated as separable, the eigenstates themselves are of mixed character and many will share roughly the same FC intensity, without revealing the mechanism or geometry of the jump between surfaces. Indeed, this can happen even for separable surfaces, in that many different final state FC factors could be comparable in size, reflecting the fact that the "jump" was not along any one of the separable coordinates.
2.2. Quantum-Mechanical Treatment in Phase Space. Our approach to overcome the difficulties presented in the previous section is to consider the transition in phase space. The donor state is represented by its Wigner function, the acceptor state by a classical energy hypersurface in phase space, and the transition itself is determined by the overlap between the two.
In the Wigner representation, the total FC factors squared multiplied by the final density of states are expressed as an overlap integral in phase space. Our method for the derivation of propensity rules is based on recognizing the points in phase space in which the nuclear integrand peaks. For weak transitions, the integrand that we study tends to be exponentially small, and the dominant region in phase space where this integrand peaks may often be exponentially dominant over the rest of the integral. For convenience, we use an abridged form (q,p) for the nuclear positions and momenta for the set of normal modes ({q_{k}_{}},{p_{k}_{}}), k = 1,..., d.
In the Wigner phase-space representation, takes the form
For allowed transitions, we derived eqs 10, and 12 above by replacing the trace over the product of the initial and final density matrices by a phase-space integral over their respective Wigner functions. How do we generalize the approach to forbidden transitions, such as internal conversion? The new feature is that the nuclear integral as defined above in eq 4 now contains a derivative with respect to the coordinate of the promoting mode. Nevertheless, eq 10 easily generalizes to include this case by replacing the Wigner transform of the initial-state density matrix with the Wigner transform of an effective density matrix defined for the derivative of the initial state wave function. Thus, we include the derivative or any other transition operator in a redefinition of the initial state. In addition, the rate is obtained by a sum over all of the promoting modes allowed by symmetry.
The result for internal conversion is
For the final state Wigner function, _{F}(q,p), a formal
expression is obtained that substantially simplifies the calculation. For relaxation processes, the final state (usually a quasi-continuum manifold of states) is defined by energy conservation
to be given by the density matrix (_{F} - E). We define (q,p)
to be the Wigner transform of this delta-function density and
get
2.3. Surface Jumping. The Wigner function of the quasi-continuum final state (q,p) can be expanded as an asymptotic
power series in .^{14-17}
We are looking for the phase-space point(s) (q*,p*) where
the integrand
The geometric interpretation of the problem is demonstrated in Figure 2. The solid inner ellipses represent the contours of the Wigner function, here a Gaussian, in some two-dimensional space. The outer dashed curve is the energy surface constraint H_{F} = E. The geometric assignment is to find the points where the highest contour of the surface, _{I}(q,p), meets the constraint hypersurface, E = H_{F}(q,p). As demonstrated in Figure 2, the strength of the extremal points can vary. Panel a shows the case in which the point of maximum of the Wigner function under the energy constraint is a strong maximum. In this case, there is a very rapid decrease of the Wigner function as one moves away from the extremum point on the energy constraint hypersurface. We can refer to the point as a true jumping point. Panel b stands as an example for a weak extremum. In this diagram, the Wigner function contour and the energy constraint hypersurface have a very similar curvature. A decisive jumping point is not well-defined.
2.4. Numerical Considerations. The identification of the
jumping point reduces in this formalism to the mathematical
problem of finding the maximum of a multidimensional
nonlinear objective function under a nonlinear constraint. Simple
geometric considerations show that at all of the extremum points
the contours of the constant initial Wigner function are tangent
to the constraint hypersurface
The model considered here is motivated by the S_{2} S_{0}
transition in the benzene molecule.^{20-28}
3.1. The Model. The following properties of benzene are imitated by our model: It has one aromatic ring of six carbons and six hydrogens. The configuration of the ground S_{0} electronic state is hexagonal and belongs to the D_{6}_{h}_{} symmetry group. The molecule has 30 modes of vibrations, which we number according to Wilson^{32} (some details are given below). The vertical energy gap between the S_{0} and S_{2} state is 0.228 eV.^{33} The ground-state potential energy surface is taken from ref 34 (see below). The equilibrium position at S_{0} is = 6.47 bohr, = 5.02 bohr, and q_{i}_{} = 0 for all other normal modes by symmetry. The equilibrium position at S_{2} is sometimes considered as free parameters and sometimes taken as in S_{1}, = 6.63 bohr and = 5.01 bohr.^{33} Harmonic frequencies on S_{2} are taken from ref 35.
Our model differs from benzene by the following properties: The potential energy surfaces that we use show no conical intersection. The transition is assumed to be a direct transition from S_{2} to S_{0} not going through S_{1}. The model assumes a planar molecule on the S_{2} surface.
Normal modes that have special importance in the rest of
the paper are depicted in Figure 3. Six in-plane C-H stretching
modes are each an orthogonal linear combination of the six local
C-H in-plane stretching modes, s_{i}_{}:
3.2. A Harmonic Approximation for Both PES. Taking
the initial state of S_{2} to be the ground-state wave function for
the normal mode k of a harmonic oscillator, we have
The integral over the nuclear degrees of freedom giving the transition strength for internal conversion differs from the FC factor squared for an allowed transition by an additional polynomial in the integrand multiplying the Gaussian initial Wigner function. With this additional factor, the transition probability would vanish for a zero excitation of the promoting mode, and therefore, the promoting mode must have, at least, some minimal excitation.
The jumping point for an allowed transition is found by maximizing _{I}(q,p), while the jumping point for internal conversion is found by maximizing '(q,p), both under the same constraint: H_{F}(q,p) = E. It can be shown that when surface jumping occurs these two procedures give the same quantum jump. For large excitations of the promoting mode, the behavior of the Wigner function is dominated by the exponent and the influence of the polynomial is negligible. For small excitations of the promoting mode, there is, as mentioned above, a minimal amount of energy that must be transferred to the promoting mode of vibration, yet this hardly affects the quantum jump. Thus, we maximize _{I}(q,p) and not '(q,p). Therefore, we look for minima of W defined in eq 29 under the constraint H_{F}(q,p) = E.
We first consider a harmonic approximation for the Hamiltonian of the lower electronic surface:
3.3. Harmonic Excited PES and Anharmonic Ground
Electronic Potential. In this subsection, we repeat the analysis
of the previous subsection with the same initial harmonic state
on the excited electronic surface but this time with the most
recent anharmonic force field for the ground-state potential
surface:
In Table 1, we show the points found by the numerical minimization of W under the constraint H_{F} = E. The points that have the lowest value of W, highest propensity, have an almost 6-fold degeneracy with W 17. These points correspond to the same small position excitation of the totally symmetric C-C stretching mode and different specific combinations of the six C-H stretching modes position excitations.
The data in the literature led us to perform the analysis in the normal modes of vibrations framework, yet to decode the meaning of the points, we have transformed the coordinates from normal modes to local modes using the inverse matrix of the transformation, eq 24. In Table 2, we present the same points as in Table 1 in local-mode coordinates. The physical meaning of the points is now obvious. The six points with the highest propensity refer to six equal points of local-mode excitation of C-H stretching with an addition of a very small excitation of the totally symmetric C-C stretching mode, which is the only significantly displaced mode within the 30 modes of the benzene.
3.4. Local vs Normal Coordinates. The best choice of coordinates for the description of molecular spectroscopy depends on the exact process that has to be described. Low-energy vibrational excitations such as IR absorption spectroscopy are usually described in terms of normal-mode oscillators, while high-energy processes such as dissociation are best described within a local-mode framework. It is clear that dissociation of a molecule occurs by breaking one local mode between two atoms; this is inconveniently described by high excitation of several bonds between atoms in the normal-mode description.
In our analysis, the excitation of a local C-H mode seems
to have its origin in the structure of the surface potential. When
the local coordinates are used, the surface potential for the six
in-plane C-H stretching modes has the form
We summarize our conclusions so far as follows: Inclusion of anharmonic effects for the ground electronic state reduces the dimensionality of the transition from an 11-dimensional hyperspace to small regions surrounding six degenerate points. The points with the highest propensity describe a single excitation of a local mode of C-H stretching and another considerably smaller simultaneous excitation of the totally symmetric C-C stretching mode.
3.5. Sensitivity to the Energy Gap between the Two Surfaces. How sensitive are the results to the energy gap between the two potential energy surfaces? What would have changed if the energy gap was smaller or bigger? To answer that, we have taken several different hypothetical values for the energy gap and repeated the calculation for each such value, while keeping all other parameters, such as the displacements between the modes, fixed. The results are depicted in Figure 4, which shows the magnitude of the jump in different directions and the value of the Wigner function at the jumping point as a function of the energy gap between the states. We have found that most of the accepting modes undergo a vertical or an almost vertical transition, which is insensitive to the available energy. There is only one mode, here local C-H stretching, of which the excitation depends on the energy gap. This mode is excited by all of the available energy, given by the energy gap minus the energy that is required for a vertical transition of the other modes. The transition probability, estimated by the value of (e^{-}^{W}) at the jumping point, decreases exponentially with an increase of the energy gap between the surfaces. This feature is ascribed to the fact the enlargement of the gap between the surfaces leads to a larger quantum jump between them.
3.6. Influence of the Displacements between the Two Surfaces. The benzene molecule is hexagonal in the ground electronic state and, therefore, belongs to the D_{6}_{h}_{} symmetry group. The only modes that can have nonzero displacements under this symmetry are the totally symmetric breathing modes, that is, q_{1} and q_{2}. If, however, the molecule is distorted on the upper surface, S_{2}, more modes become totally symmetric and can in principle be displaced. The possibility of an extreme change also in the frequency of these modes is not considered here.^{40} How do the displacements, both for a symmetric and for a distorted upper surface, influence the jumping point?
We first take the potentials from section 3.3, implement our maximization procedure, and search for jumping points for different displacements of q_{1}(C-C) and q_{2}(C-H). Figure 5 displays the nonzero coordinates of the jumping point with the highest propensity and the value of the Wigner function at this point versus the displacement of the C-C bond length. From the graph, it is easy to see that the main change of the excitation is in the C-C totally symmetric direction. The dependence is linear with a slope of almost 1.1. Changes of the C-H local excitation and of the value of the Wigner function at the jumping point are small and nonlinear. In Figure 6, we display the coordinate of the jumping point with the highest propensity and the value of the Wigner function at this point versus the displacement of the C-H bond length. Again, the excitation of the displaced mode, here the totally symmetric C-H stretching normal mode, is linearly proportional to the displacement (as in a vertical transition). The totally symmetric C-C is not affected at all, while the local C-H stretching mode, which is the mode that undergoes the jump, is again slightly, nonlinearly affected. Note that in this framework the normal and local C-H stretching act like almost different directions in space.
It is very likely that a distortion of the conformation of the benzene molecule takes place in the excited state because of a pseudo-Jahn-Teller (pseudo-JT) effect. Additional modes that become totally symmetric under the new symmetry can have nonzero displacements. Just as an example, we consider here the case in which the molecule remains planar but belongs to the D_{2}_{h}_{} symmetry group. The modes that will have the most significant distortion will be the modes q_{6} and q_{8}, which correspond to ring deformations (see Figure 3). In Figure 7, we display the nonzero coordinates of the jumping point with the highest propensity and the value of the Wigner function at this point versus the difference between the angles of the benzene ring, that is, a nonzero displacement of q_{6}. Again, we see that the excitation of a displaced mode, here q_{6}, linearly depends on the displacement. Excitations of the other modes do not significantly change. Figure 8 plots the coordinates of the jumping point with the highest propensity and the value of the Wigner function at this point versus the difference between the C-C bond lengths. This change of the bond lengths induces displacements in q_{8} and q_{1} and as a result a change in the coordinates of the jumping point. Even for a moderate change in the symmetry, a noticeable amount of the energy is transferred to the new modes that are displaced.
When one direction in phase space dominates the quantum jump, excitations in other directions are proportional to the displacement, as if in a vertical transition. A change of symmetry can strongly influence the jumping point.
3.7. Two Anharmonic PES. Having shown in the previous sections the separability in the subspace of C-H stretching modes of the ground surface potential, we study in this subsection the influence of the anharmonicity of the upper surface on the jumping point within the effective one-dimensional problem of a local C-H mode. Figure 9 displays the one-dimensional ground and excited electronic surface potentials in the local-mode representation. Relative to the symmetric harmonic potential, the anharmonic potential is softened on its dissociation part and has a sharper slope on its close-approach part. Applying a harmonic approximation for both surfaces gives the value of the Wigner function at the jumping point of W 32. Taking the ground surface potential to be anharmonic gives the value of W 17. Taking into account the anharmonicities of the excited state makes both the wave function and the Wigner function wider on the dissociation side of the potential and narrower on the close-approach side. Consequently, the value of the Wigner function at the jumping point, W(q*,p*), gets a value between the two extremes of 32 W 17.
For a quantitative analysis of this property and to make the
calculation with an anharmonic potential that has a closed form
expression for the initial Wigner function, we use a Morse
approximation for the excited potential surface:
The wave function of the Morse oscillator is a combination
of the associated Laguerre polynomials, and the Wigner function
is a combination of the modified spherical Bessel function of
the third kind (MacDonald function).^{43,44} The Wigner function
of the ground vibrational state is
Another interesting feature that arises from the reduction of the problem to one-dimensional Morse potential regards the direction of the sudden change in the local C-H stretch. One may conclude that the wave packet lands on the ground electronic surface at the close-approach side of the potential.
An increase of the Wigner function at the jumping point is obtained with an increase of the anharmonicity of the lower surface. For a fixed anharmonicity of the lower surface, a decrease of the Wigner function at the jumping point is obtained with an increase of the anharmonicity of the upper surface. The anharmonicity on the upper surface gives correction to the two extreme approximations of harmonic-harmonic and anharmonic-harmonic potentials. Anharmonicity can induce small excitation of nondisplaced modes due to changes in the center of the initial wave packet.
3.8. Duschinsky Rotation. Consider the impact of a possible
Duschinsky rotation that couples q_{14} and q_{15}. Figure 3 displays
diagrams of these two modes as they appear on the ground
electronic state. Suppose that the new normal modes on the
excited state are mixed according to
The rotation in two dimensions and its influence on the jumping point is demonstrated in Figure 11. The contours of the initial Wigner function on the excited electronic surface and the constraint on the lower surface, H_{F} = E, are plotted by solid and dashed lines, respectively. The implementation of the Duschinsky rotation is done by rotating the inner ellipse by the angle . A larger difference between the widths of the Wigner function would increase the effect of the rotation. The effect has to be considered in position as well as in momentum space. However, in our calculations, no momentum excitation is found. We first examine the case = 90, that is, q'_{14} = q_{15} and q'_{15} = -q_{14}, and find a new couple of points with high propensity with a large excitation of q_{14} and small excitations of the totally symmetric C-C and C-H stretching, q_{1} and q_{2}. The value of W at these points is 16.8, very close to the value of the points with the highest propensity found without the Duschinsky rotation. The new point that we have found for the extreme rotation is used as an initial point for a local minimum search for different angles of rotation. In Figure 12, we display this local minimum, which is found in our calculations, and the value of the Wigner function at this point versus , the rotation angle.
A new jumping point with significant propensity develops only for angles of rotation above 65. For smaller rotations, the point that originates from a rotation exists as a local minimum but has a very high value of W, which makes the probability of decaying through this channel negligible.
Duschinsky effect can cause, in general, a change of the direction of the quantum jump, but for the model considered here, the angles for rotation needed for this feature to appear are nonphysical.
The mechanism of surface jumping complements the mechanism of Tully-Preston's surface hopping by extending it to Franck-Condon suppressed transitions. In this paper, the surface jumping approach to nonvertical transitions was developed into a general "ready to use" tool. The formalism was first extended to include forbidden transitions, in particular, internal conversion. This results in an additional factor in the FC integrand, a polynomial of the position and momentum of the promoting mode of vibration. In most cases, the influence of the polynomial term on the direction of the jump can be neglected. More generally, the maximization procedure with this additional term is mathematically equivalent to the consideration of a decay not from the ground vibrational state but from a vibrationally excited state. This deserves further study in the future. A numerical prescription for analyzing the jump was developed for transitions between any general potential energy surfaces, including, in particular, distorted and anharmonic surfaces. The surface jumping method allows a simple determination of the accepting modes even for systems with a very large dimension.
The surface jumping method for nonvertical transitions was then applied to recognizing accepting modes in a complex model inspired by the S_{2} S_{0} transition in benzene. The transition takes place through nonradiative internal conversion and a large energy of 0.228 hartree is released to the vibrational degrees of freedom of the ground S_{0} state. We note that experiments have determined the decay rate of the vibrationally excited S_{2} state to the S_{0} electronic state to be at the scale of tens of femtoseconds. This supports the suggestion that the decay occurs via conical intersections between the S_{2}, S_{1}, and S_{0} surfaces. Here, we have ignored the conical intersections and concentrated on a direct quantum jumping process. Future work should address the general question of competing tunneling mechanism, that of surface jumping and that of tunneling through a barrier to get to a conical intersection from an initial vibrationless state. The model incorporates an exact, state of the art potential surface for the ground electronic state but a simplistic treatment of the excited electronic state. Dependence on the excited-state features is tested by treating as free parameters the energy gap, displacements, and anharmonicities. We observe that for a large energy gap the masses and frequencies become the defining parameters for choosing the accepting mode, while for smaller energy gaps the displacements are more important. Anharmonicities are very important when a competition between degenerate modes occurs. These conclusions are demonstrated by the specific model considered here but apply in general to any weak internal conversion process.
For the model considered here, we found that the C-H modes undergo the jump. We showed that the jump takes place in the local C-H modes: Because the energy gap between the states is large compared to the vibrational energy scale and the ratio of the harmonic frequencies between the surfaces does not differ very much from 1 (0.7 < / < 1.2), the modes with the largest frequency and smallest reduced mass are the modes that undergo the jump. The local C-H in-plane stretching modes take almost all of the electronic energy, while all of the other modes decay almost vertically.
This finding can be interpreted also within the well-known "most probable escape path" principle of the tunnelling phenomena. Because in tunneling the competition is between channels with exponentially small probabilities, there is usually only one channel that dominates. The picture of one mode that undergoes quantum jump while the other modes decay vertically was demonstrated in several ways in the paper, for example, by altering the energy gap between the surfaces, by adding anharmonicities, and by changing the displacement of various modes.
Some of the issues to be considered in the future include transitions from thermal distributions, rotations, and an implementation of the method to other molecules. Application to dissociation, the coupling of the vibrational space of the molecule to additional degrees of freedom of a medium, the dynamics of the molecular wave packet after the quantum jump between the surfaces takes place, and the calculation of the rate using the phase-space method could also be studied within this approach.
B.S. gratefully acknowledges useful discussions with Prof. Benjamin Scharf. This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and by the National Science Foundation through a grant for the Institute for Theoretical Atomic and Molecular Physics (ITAMP) at the Harvard-Smithsonian Center for Astrophysics.
* To whom correspondence should be addressed. E-mail addresses: bsegev@bgumail.bgu.ac.il or heller@physics.harvard.edu.
Ben-Gurion University of the Negev.
Harvard University.
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38. See also: Halonen, L. Chem. Phys. Lett. 1982, 87, 221
39. Rashev, S. J. Phys. Chem. A 2001, 105, 6499.
40. Remarkable changes of the frequencies of q_{6} and q_{8} (mostly in the
triplet state but maybe also in the singlet) were considered, for example,
in: Scharf, B. Chem. Phys. Lett. 1979, 68, 242
41. Handbook of Physical and Chemistry Constants; Clarl, S. P., Jr., Ed.; Springer: Berlin, 1990.
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131.
43. Lee, H. W. Phys. Rep. 1995, 259, 147.
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1769-1789.
q_{1} |
q_{2} |
q_{7} |
q_{13} |
q_{20} |
q_{7a} |
q_{20a} |
W |
0.08 |
-0.31 |
-0.44 |
-0.31 |
-0.44 |
0 |
0 |
17.00 |
0.08 |
-0.31 |
-0.44 |
0.31 |
0.44 |
0 |
0 |
17.00 |
0.08 |
-0.32 |
0.22 |
0.31 |
-0.22 |
-0.38 |
-0.38 |
17.02 |
0.08 |
-0.32 |
0.22 |
-0.31 |
0.22 |
-0.38 |
0.38 |
17.02 |
0.08 |
-0.32 |
0.22 |
-0.31 |
0.22 |
0.38 |
-0.38 |
17.02 |
0.08 |
-0.32 |
0.22 |
0.31 |
-0.22 |
0.38 |
0.38 |
17.02 |
^{a} The points are given in the normal-mode representation.
s_{1} |
s_{2} |
s_{3} |
s_{4} |
s_{5} |
s_{6} |
W |
-0.76 |
0 |
0 |
0 |
0 |
0 |
17.00 |
0 |
0 |
0 |
-0.76 |
0 |
0 |
17.00 |
0 |
-0.76 |
0 |
0 |
0 |
0 |
17.02 |
0 |
0 |
-0.76 |
0 |
0 |
0 |
17.02 |
0 |
0 |
0 |
0 |
-0.76 |
0 |
17.02 |
0 |
0 |
0 |
0 |
0 |
-0.76 |
17.02 |