Benzene has been of considerable interest both spectroscopically and theoretically.

C6H6

Much effort has been devoted to observing and understanding how energy is transferred out of initially excited overtone states involving the CH stretching modes.

The energy transfer out of the CH stretching modes is rapid, occurring on a subpicosecond timescale.

Sibert and co-workers (1982-1984) explained these results as a consequence of a cubic coupling involving the CH stretch and levels containing two quanta of CCH wag.

This sort of efficient 2:1 Fermi resonant stretch bend mechanism for CH modes occurs in a wide range of other molecules.

For example:

Breathing Mode (993 cm-1) Kekule Mode (1146 cm-1) Twist Mode (1365 cm-1)

Movies courtesy of http://www.cchem.berkeley.edu/ChemResources/Vibrations/index.html

Considerably less study has been devoted to an understanding of how and on what timescale energy is transferred amongst the ring modes.

Such low frequency modes dominate the vibrational state density and play a role in the irreversible nature of the decay out of C-H modes.

Nicholson & Lawrance, Flinders University, Chem. Phys. Lett. 1995

Analysis of high-resolution dispersed fluorescence spectra suggests that IVR amongst the ring modes in S0 benzene up to 8200 cm^-1 is much slower than that involving the excited CH overtones.

The instrument limited linewidths of 1.0-1.3 cm^-1, implying an IVR contribution of 0.5 cm^-1, indicate that the IVR rate has an upper limit of 0.047 ps^-1 (IVR lifetime > 21 ps).

To model this behaviour and determine what role is played by potential energy and kinematic couplings.

Trajectories
Ensembles of classical trajectories are generated by integrating the equations of motion using a vectorised version of the "velocity Verlet" integrator.

State Selection
Initial states are specified by excitations corresponding to local CH mode or normal ring mode.

Analysis
The ensemble averaged local and normal mode energies are monitored as a function of time.

Most previous theoretical studies have involved only a consideration of the modes in planar benzene.

In the present study, in addition to the in plane modes, we also consider the effect of out of plane modes.

At the first level of approximation we assume Morse terms for the C-H stretches and a quadratic force-field for the remaining internal modes.

The quadratic part of the force-field is largely based on the ab initio calculations of Pulay et al.

In terms of internal coordinates, coupling between modes can be thought of as essentially kinematic.

A recent study by Maslen et al. reported some information on the PES of benzene up to quartic terms but it seems that most such potential coupling terms are small.They found that:

"the dominant potential coupling is CH <-> CH potential coupling which includes diagonal anharmonicity and also anharmonic coupling between the CH stretch normal modes to produce CH stretch local modes. This accounts for over 80% of the potential coupling."

symmetry    Mode        PFB Set I   PFB Set II  Experiment  Clarke et al.  This work   
e2u (op,d)  16           402                     398                        401.1       
e2g (d)     6            607         607         606         606            606.1       
            bend                                                               
a2u (op)    11           667                     673                        662.7       
b2g (op)    4            701                     707                        699.9       
e1g (op.d)  10           843                     846                        838.5       
e2u (op,d)  17           969                     967                        963.5       
b2g (op)    5            996                     990                        990.1       
a1g         1            983         993         993         993            993.5       
            breathe                                                                 
b1u         12           997        1010        1010        1009           1009.1      
e1u (d)     18          1036        1036        1037        1033           1033.5      
b2u         15          1162        1145        1146        1140           1139.9      
            Kekule                                                                    
e2g (d)     9           1183        1185        1178        1179           1179.1      
b2u         14          1297        1307        1309        1305           1305.5      
a2g         3           1365        1358        1350        1479           1350.5      
            twist                                                                 
e1u         19          1482        1485        1482        1479           1479.3      
e2g         8           1607        1604        1599        1602           1602.2
            CC stretch      
b1u         13          3051        3052        3057        3135           3135.9      
e2g (d)     7           3061        3061        3056        3145           3145.6      
e1u (d)     20          3080        3080        3064        3163           3164.4      
a1g         2           3095        3096        3073        3178           3179.2

Decay of Excitation Energy in CH Local Mode

Decay of Probability of CH Local Mode Excitation

CH overtone lifetimes (lifetimes in fs)

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Ring Mode Decay

Ring mode energy decay lifetimes

Preliminary classical trajectory calculations on a full-dimensional model of benzene, indicate that, whereas IVR out of excited CH overtones is on a subpicosecond timescale, IVR involving the ring modes is on a considerably slower timescale, involving many picoseconds (> 29 ps).

This slow IVR is in agreement with recent experiments which suggest a lower limit on the IVR lifetime of 21 ps.

Further studies are proposed with an extended model in order to determine the role of higher order potential couplings. These may either enhance or suppress the existing kinematic couplings