The MCTDH Equations of Motion for Several Electronic States

5 The MCTDH Equations of Motion for Several Electronic States 

In the previous sections, the Born-Oppenheimer approximation was assumed, i.e. nuclear motion on a single PES was considered. However, in many simations, the dynamics of the nuclei need to be treated on several PESs corresponding to coupled electronic state. The coupling between the electronic states can be due to the presence of an external electric field or to internal vibronic interactions. There exists two different ways of treating several coupled electronic states with the MCTDH method [60], the single-set formulation and the multi-set formulation. 

In the single-set formulation, the electronic manifold is treated as one extra degree of freedom, noted . The 2k, coordinate then labels the electronic states, taking only discrete values 2 , = 1, 2. ct, where a is the number of electronic states. The number of SPFs for the electronic degree of freedom is set to the number of states = a. In this formulation, the MCTDH equations of motions Eqs.(4.34) and (4.35) are unchanged. This formulation is called single-set formulation because a single set of SPF is used to treat the dynamics in all the electronic states. 

In the multi-set formulation, in contrast, a different set of SPF is used for each electronic state. The wavefunction and the Hamiltonian is expanded in the set of electronie states la), a = 1. r  

Using again the simplest choice for the constraint operators = 0, the equations 

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