Quasi-energy

The Cauchy moments have been derived in Ref. [4] for CC wavefunctions, using the time-dependent quasi-energy Lagrangian technique [I]. In Section 2.1 we recapitulate the important points of that derivation and use it in Section 2.2 to derive the CC3-specific formulas. 

Apparently, the time evolution of the IEq) component of 2i(t)) is governed by a quasi-energy of Eq - Q2(0I> whereas the time evolution of the I q) component of U2(0) is governed by a quasi-energy of + f22(0l- hi such a case, one says that the two levels are Autler-Towns split by an amount equal to 2 i22(0l ... 

The study of quantum systems in a periodic field and the introduction of the concept of quasi-energy are also of much interest a brief article on this subject is placed in the next volume. Here we restrict ourselves to a reference to the review paper [13]. 

Reactions having Q-states provide mechanistic pathways to achieving the electronic interconversion which can be modulated by external electromagnetic fields such as, for instance, microwaves as well as thermal blackbody radation, due to the quasi energy degeneracy. In non-stationary situations, the operator... 

Note that in the multichannel case (i.e., when the quasi-energy states are broa- dened due to the interaction with more than one n continuum) Eq. (9.54) does rnity necessarily hold, and the numerator of Eq. (9.53) does not become a purely real function. Hence there is no real E value that would cause it to vanish. In this casff EIT will not occur, although a substantial dip in the absorption cross section may be observed. 

For an alternative formulation of TDDFT response equations starting with a definition of response properties as quasi-energy derivatives, the reader is referred to [50] and references provided therein. 

Samuelides, M., Fleckinger, R., Touziller, L. and Bellissard, J. (1986). Instabilities of the quantmn rotator and transition in the quasi-energy spectrum, Europhys. Lett. 1, 203-208. 

This half-scrap could be generalized for a /i-multi photon process (n > 2) in a multilevel system, with the use of the full quasi-energies and Floquet states (calculated numerically). 

The topology of the quasi-energy surfaces thus shows which appropriate delays and peak amplitudes induce desired atomic population and photon transfers. In the adiabatic regime, these loops can be classified into topologically inequivalent classes. If the evolution is adiabatic, all paths of a given class lead to the same end effect. This property underlies the robustness of the process. 

With the technique combining the rotating wave transformations and contact transformations developed in Section III.C, one can treat accurately the dynamical resonances and construct approximately the quasi-energies. If we take into account the first two dynamical resonances by appropriate RWTs [associated with path (b)], one obtains the following explicit expression for the dressed energy surfaces ... 

For the parameters 8 = 2Qq and <2max = O0, corresponding to the path (a) on the surfaces in Fig. 19, we show in Fig. 20 the solution of the semiclassical Schrodinger equation (321). It features a STIRAP-like process inducing a complete population transfer for this choice of the delays. Two zones of the quasi-energy spectrum associated with the surfaces of Fig. 19 are pictured as a function of time in Fig. 21b. We notice that the state 11 0,0) is... 

In this chapter, we will not be concerned with the detailed expressions of the response functions that we find for the standard electronic structure methods in theoretical chemistry. However, we will briefly outline the basic elements in two alternative formulations of response tlieory, namely the polarization propagator and the quasi-energy derivative approaches. 

One somewhat displeasing detail in the approximate polarization propagator methods discussed in the previous section is the fact that concern needs to be made as to which formulation of wave mechanics that is used. This point has been elegantly resolved by Christiansen et al. in their quasi-energy formulation of response theory [23], in which a general and unified theory is presented for the evaluation of response functions for variational as well as nonvariational electronic structure methods. 

Keywords coupled cluster, CCSD, CC3, response theory, quasi-energy Lagrangian, time-dependent... 

The time-dependent ground-state coupled cluster wavefunction for such a system is conveniently parameterized in a form, where the oscillating phase factor caused by the so-called level-shift [43 5, 88, 89] or time-dependent quasi-energy W(r, e) (vide infra) is explicitly isolated [42 6, 90, 91] ... 

The energy, which in the time-dependent case is no longer a constant of motion, has to be generalized to the time-dependent quasi-energy ... 

Frequency-dependent higher-order properties can now be obtained as derivatives of the real part of the time-average of the quasi-energy W j- with respect to the field strengths of the external perturbations. To derive computational efficient expressions for the derivatives of the coupled cluster quasi-energy, which obey the 2n-(-1- and 2n-(-2-rules of variational perturbation theory [44, 45, 93], the (quasi-) energy is combined with the cluster equations to a Lagrangian ... 

The time-average of tlie quasi-energy Lagrangian... 

The imaginary part of Wit, e) is connected to the time derivative of the wavefunction norm and does not contain information about physical observables. For normalized wavefunctions and bra and ket state conjugated to each other the quasi-energy is real [44. 45]. However, the CC energy is calculated from a projection expression and in is general not real. Thus with CC wavefunctions it is important to consider only the real part of the quasi-energy [44, 65, 92]. 

The last equation is a variation principle for the coupled cluster quasi-energy and wavefunction within oscillating harmonic external fields. If one inserts a perturbation and Fourier expansion as ansatz for the cluster amplitudes... 

Table 2. Partial derivatives of the Lagrange functional for the coupled cluster quasi-energy...

In Harfrcc-Fock theory, the quasi-energy and its time average are defined as follows... 

The periodic perturbation Eq. (155) induces a change in the quasi-energy, which is expanded in orders of the perturbation ... 

The response function of order n is then recovered by differentiating the time-averaged perturbed quasi-energy with respect to the frequency-dependent... 

The frequency associated with A is set equal to minus the sum of the perturbing frequencies so that the time-averaged quasi-energy does not vanish. 

To calculate the perturbed quasi-energy Eq. (156), we must first determine the perturbed orbital-rotation parameters K(r), which are likewise expanded in orders of the perturbation. 

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