Single-set formulation

The most direct way to accomplish this is to choose one extra degree of freedom, the Kgth say, to represent the electronic manifold.The coordinate then labels the electronic states, taking only discrete values = 1,2,. ..,(T, where a is the number of electronic states under consideration. The number of SPFs for such an electronic mode is set to the number of states, i.e. = a. The equations of motion (17) and (18) remain unchanged, treating nuclear and electronic modes on the same footing. This is called the single-set formulation, since only one set of SPFs is used for all the electronic states. 

The single-set formulation is to be preferred when the different electronic states have a similar form. The well known spin-boson model is in this category. When the motion on the different electronic states is very different, the multi-set formulation becomes the appropriate choice. The smaller number of SPFs needed for convergence then over-compensates the overhead of the multi-set formulation. For vibronically coupled systems, we have always chosen the multi-set formulation. 

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. 

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See also in sourсe #XX -- [ ]

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