Variation principle, Frenkel

In essentially all of the prior formulations of TDDFT a complex Lagrangian is used, which would amount to using the full expectation value in Eq. (2.9), not just the real part as in our presentation. The form we use is natural for conservative systems and, if not invoked explicitly at the outset, emerges in some fashion when considering such systems. A discussion of the different forms of Frenkel s variational principle, although not in the context of DFT, can be found in (39). 

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

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... 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

To obtain the set of coupled equations of motion for the coefficients and SPFs, the Dirac-Frenkel variational principle is used. Dividing the Hamiltonian into parts that act only on a given particle (separable or correlated term). 

As is often the case in any new development in quantum chemistry, the fundamental formalism has been around for some time, in this case TDCPHF based on Frenkel s variational principle and density matrices [7]. However, the detailed theory will not be our immediate concern, but rather how it has been put to use. 

Substituting again = + with the definition (106) of H and (107) with the definition (108) of the variational principle) one obtains instead of (73) the generalized coupled HF equations.125... 

The time-dependent Schrodinger equation may be obtained by rewriting Frenkel variation principle for the case of nonlinear Hamiltonian as follows ... 

Applying the Frenkel variational principle, we arrive at the following time-dependent HF/KS equation... 

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... 

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... 

It has been shown [22] that the time-dependent non- linear Schrodinger Eq. (3.1) can be obtained from the Hamilton principle when a suitable QM Lagrangian density is defined (see Appendix A.l), and that this Lagrangian density implies an extension of the time-dependent Frenkel s variational principle [23]. 

The regular wave function (t) > satisfies a generalized time-dependent Frenkel variational principles [24] for an arbitrary variation W... 

Within the Ehrenfest framework [12], valid for exact as well as variational approximate methods, either the time-variational principle of Frenkel [11],... 

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],... 

The second approach is based on a Hamiltonian with explicit time dependence, provided with an opportune form of the of the time-dependent variation principle (the Frenkel principle is generally used (Cammi et al. 1996)). From this starting point linear and non linear response functions are derived. 

The variational derivation of TDHF equations (Frenkel, 1934) relates immediately to the stationary state approach and the SCF method. The quantities needed for substitution in the variation principle (12.2.2) are (5V V ) and (69 H 9 ) the first follows from first principles (for example from Slater s rules, noting that d/8r behaves like a 1-electron operator) as... 

In the variational multi-configuration Gaussian wavepacket (vMCG) method [10, 11, 59-63] the basis functions follow coupled quantum trajectories whereby the mean positions and momenta are treated as variational basis-function parameters that evolve according to the Dirac-Frenkel principle applied to the TDSE. Each basis function directly simulates quantum phenomena in a rigorous way, and the method thus promises much faster convergence than classical-trajectory-based methods due to a better sampling of the phase space. 

The variational equations formulated in Section 12.2 may be used to obtain a /nu/time-dependent case in particular, spedal care is needed in using the Frenkel principle (McWeeny, 1983). We therefore proceed cautiously, writing the principle in the fiiller form (12.2.5a), namely... 

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