Dirac-Frenkel time-dependent variational

Computational strategies can be based on variational procedures using the Dirac-Frenkel time-dependent variational principle (TDVP). Introducing a shorthand notation so that... 

Assuming the simplest choice for the constraints, i.e 3 = 0, the MCTDH equations of motion can be derived by inserting the MCTDH ansatz of Eq. (4.22) into the Dirac-Frenkel time-dependent variational principle of Eq. (4.20). Other choices are possible for the constraints, see Ref. [28] for details. We first introduce the projector on the space spanned by the SPFs of the k particle... 

Equations of motion for the time-dependent coefficients Aj time-dependent single particle functions, and time-dependent Gaussian parameters A K s) = aj c s), f r]jK s can be derived via the Dirac-Frenkel variational principle [1], leading to... 

Note, that other than in the standard form Eq. (5.1) not only the coefficients, but both the basis functions (SPFs) and the MCTDH expansion coefficients, are taken to be time-dependent in Eq. (5.2). Inserting this ansatz in the Dirac-Frenkel variational principle leads to equations of motion (EOM) for the SPF and the coefficients... 

Let us begin a short tour of variational methods writing the moving wavepacket as a generic function of the vectors of coordinates Q and of the time-dependent parameters a i/ (Q, a)). The basic instrument is the Dirac-Frenkel TD variational principle [19, 20],... 

A simple application of the very general approach used in earlier sections leads to the time-dependent generalization of Hartree-Fock theory. The time-dependent Hartree-Fock (TDHF) equations (Dirac, 1929) were first formulated variationally by Frenkel (1934) they are also widely used in nuclear physics (see e.g. Thouless, 1%1) under the name random-phase approximation (RPA). Since the equations describe response to a perturbation, as in Section 11.9 but now time-dependent, they will... 

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