Anharmonic coupling Darling-Dennison

Lehmann, K. K. (1983), On the Relation of Child and Lawton s Harmonically Coupled Anharmonic-Oscillator Model and Darling-Dennison Coupling, J. Chem. Phys. 79, 1098. 

Figure la illustrates some of the dynamics in the state space of the [ns = 1, nres = 11, / = 0] polyad illuminated at t = 0 via the (0, 1, 0, 8°, 0 )° ZOBS [2]. The four panels of this figure depict the survival probability of the ZOBS (upper left), die probability of transfer to the two most important first-tier states [coupled by the bend Darling-Dennison interaction upper right (0, 1, 0, 6+2, 2-2)0 and lower left (0, 1, 0, 6°, 2°), and the transfer probability to one of the dynamically most remote dark states in the polyad (1, 0, 0, 0°, 6°)° (lower right), which occurs via a minimum of four anharmonic coupling steps ( 1, 244 followed by 44,55 three times). 

Besides Fermi interactions, another class of anharmonic couplings among vibrational levels exist and they are often referred to as Darling-Dennison interaction [77]. They typically arise in conjunction with a normal-mode molecular picture, since the associated matrix elements can be written as... 

Although we will not discuss in detail this particular aspect of anharmonic resonances, it is important to note that Darling-Dennison couplings are automatically included by the action of the Majorana operator. A practical way to convince ourselves of this inclusion is to diagonalize (either numerically or in closed form) the Hamiltonian matrix explicitly for the first two polyads of levels and then to convert, in normal-mode notation, the vibrational states obtained. As discussed in Ref. 11, the Hamiltonian (4.38) can also be written (neglecting Cj2 and Cj2 interactions) as... 

Here //normal - 12 12 1 1 normul basis Hnormal is diagonal while Cj and C2 are not, so they break the dynamic symmetry. The parameter A is directly related to the local-mode anharmonicity. Consequently, we expect to obtain strong Darling-Dennison normal-mode couplings in the presence of very anharmonic local modes. 

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