Quasienergy time dependent

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

In this section we outline the coupled cluster-molecular mechanics response method, the CC/MM response method. Ref. [51] considers CC response functions for molecular systems in vacuum and for further details we refer to this article. The identification of response functions is closely connected to time-dependent perturbation theory [51,65,66,67,68,69,70], Our starting point is the quasienergy and we identify the response functions as simple derivatives of the quasienergy. For a molecular system in vacuum where Hqm is the vacuum Hamiltonian for the unperturbed molecule and V" is a time-dependent perturbation we have the following time-dependent Hamiltonian, H,... 

Projection of Eq. (13-105) onto the state (HF gives an equation for the time-dependent CC quasienergy [51], Q(t),... 

Observe, on the other hand, that we deviate from the notation of Christiansen et al. [13] in that we use calligraphic for the time-dependent quasienergy and Latin Q for the time-averaged quantity, the quasienergy. With time-averaging the time-dependent variational principle (18) reduces to... 

We can now give a quite general definition of molecular properties, applicable to static as well as time-dependent perturbations, namely they are derivatives of the (time-averaged) quasienergy at zero perturbation strength... 

Consider now the time average of the nth order correction to the time-dependent quasienergy. In shorthand-notation we write this as... 

Using the BCH-expansion the time-dependent quasienergy may be expanded in terms of the variational parameters... 

For time-dependent oscillatory perturbations, the response functions can be related to the derivatives of the time-averaged quasienergy (Christiansen et al. 1998 Saue 2002). The derivation, based on a time-dependent variational principle and the time-dependent HeUman-Feynman theorem, shows explicitly the symmetry of the response functions with respect to the permutation of aU the operators. 

Frequency-dependent higher-order properties can now by obtained as derivatives of the time-average of the quasienergy WIt- with respect... 

Second-order molecular properties can be defined as second derivatives of the (time-averaged) quasienergy Q with respect to frequency-dependent perturbation strengths b (wb) at zero perturbation (s=0)... 

In Sect. 8.2.1, the natural tunneling time of the system was defined as a function of the field-free tunneling splitting, see Eq. (8.3). In analogy, the quasienergy difference of Eq.(8.53) defines a field-dependent tunneling splitting, and... 

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