Quasi-energy time-averaged

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... 

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

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. 

The first-order parameters and k" are determined by a variational condition on the second-order quasi-energy, the result of which is an equation identical to Eq. (70). Differentiating the time-averaged exchange-correlation quasi-energy with respect to the field strengths e, according to Eq. (157), we obtain... 

The quadratic response function is obtained as the third derivative of the time-averaged quasi-energy. The program is then to expand the energy to third order in the first-order parameters ... 

The quasi-ergodic hypothesis asserts the equivalence of the MD time average (9) with an ensemble average in an ensemble characterized by the constants of the dynamical motion, viz., the volume V, the number of particles N, and at least for most numerical applications, the energy (For an up-... 

Within the variational time-dependent approach of Sect. 3.1.1, the molecular response functions (3.13) are determined by expanding the time-dependent wave-function >/ (t) > and the time-averaged free-energy functional (3.10) in orders of the perturbation, and by imposing that the variational condition (3.9) is satisfied at the various order. The response functions are then identified by means of the Hellmann-Feynman theorem (3.11), as terms of the expansion of the quasi-free-energy. 

Let us consider the first order equations for the coupled-cluster Fourier amplitudes A (o) T a>). They are determined from the stationary of the second order time-averaged quasi-free energy G -a>, co) ... 

TDMP2) method of Hattig und Hefi (1995) were steps in this direction. The most recent attempt that in a way summarises all previous developments is the time-averaged quasi-energy method of Christiansen et al. (Christiansen et at, 19986). 

In the TDMP2 method, in the newest version of the QED method, called QED-MP2 Aiga and Itoh, 1996, and in the time-averaged QED method, the derivatives are taken of an MP2 time-dependent and relaxed quasi-energy Lagrangian... 

The same problem with the pole structure appears also for coupled cluster response functions, if one defines them as derivatives of a time-average quasi-energy Lagrangian including orbital relaxation, ft is therefore preferable also in the analytical derivative approach like in Section 11.4 to derive coupled cluster response functions as derivatives of a time-dependent quasi-energy Lagrangian without orbital relaxation... 

Boltzmann distribution 13 change of average z projection 17 change per collision 18-19 correlation functions 12, 25-7, 28 calculation 14-15 correlation times quasi-free rotation 218 various molecules 69 and energy relaxation 164-6 impact theory 92 torque 18-19, 27... 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

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