Response function time-derivative

Following Refs. [51,68] we may identify the response functions as derivatives of the time-averaged quasienergy ... 

We obtain by expanding the Lagrangian in orders of the perturbation along with the time-averaged procedure [51] the response functions as derivatives of the time-averaged CC quasienergy L(t) r. Finally, we obtain the response equations from the stationary condition. In particular, the linear response function is given by... 

In this chapter, we develop efficient methods for the calculation of 4WM and SRF processes of large polyatomic molecules in condensed phases (e.g., solution, solid matrices, and glasses). The key quantity in the present formulation is the nonlinear response function R(t3,t2,ti)> which contains all the microscopic information relevant for any type of 4WM and SRF.6,11 12>19-2o>57 In Section II we introduce the nonlinear response function and derive the general formal expression for 4WM. The two ideal limiting cases of time-... 

Coupled cluster response functions were derived by Koch and Jprgensen (1990) starting from a time-dependent version of the transition expectation value Eq. (9.95) of Arponen (1983)... 

All these methods define response functions as derivatives of a perturbed time-dependent quasi-energy (Lowdin and Mukherjee, 1972 Langhoff et al, 1972 Kutzelnigg, 1992)... 

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... 

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. 

Owing to the coherence, we need to consider the macroscopic evolution of the field in a medium that shows a macroscopic polarization induced by the field-matter interaction. This will be done in three steps. First, the polarization induced by an arbitrary field will be calculated and expanded in power series in the field, the coefficients of the expansion being the material susceptibilities (frequency domain) or response function (time domain) of wth-order. Nonlinear Raman effects appear at third order in this expansion. Second, the perturbation theory derivation of the third-order nonlinear susceptibility in terms of molecular eigenstates and transition moments will be outlined, leading to a connection with the spontaneous Raman scattering tensor components. Last, the interaction of the initial field distribution with the created polarization will be evaluated and the signal expression obtained for the relevant techniques of Table 1. 

In the derivation of response functions one considers a molecule or an atom described by the time-independent Hamiltonian which is perturbed by an external one-electron perturbation V t e). 

As a consequence of the time-averaging of the quasienergy Lagrangian, the derivative in the last equation gives only a nonvanishing result if the frequencies of the external fields fulfill the matching condition Wj = 0. In fourth order Eq. (29) gives the cubic response function ... 

In the last three decades, density functional theory (DFT) has been extensively used to generate what may be considered as a general approach to the description of chemical reactivity [1-5]. The concepts that emerge from this theory are response functions expressed basically in terms of derivatives of the total energy and of the electronic density with respect to the number of electrons and to the external potential. As such, they correspond to conceptually simple, but at the same time, chemically meaningful quantities. 

All the methods used in this study are response methods. They deserihe the response of an ohservahle sueh as an eleetrie dipole moment /I or quadrupole moment to an external or internal perturhation, e.g., an eleetrie field or field gradient. Response funetions originated in various diseiplines in physies. In statistieal physies, they were used as time-eorrelation functions in the form of Green s functions [44,45]. Linderherg and Ohrn first showed the usefulness of this idea for quantum chemistry [46]. Since then response functions have been derived for many types of electronic wavefunctions. Four of these methods are employed here. 

The fluorescence quenching of Pe and derivatives has been investigated by fluorescence upconversion. Excitation was performed with the frequency-doubled output of a Ti Sapphire amplifier. The instrument response time was around 240 fs with 0.4 mm thick samples. The data were analysed by iterative reconvolution of the instrument response function with trial functions. For most samples, measurements were carried out at three different wavelengths (438, 475, and 490 nm). Global fits were done with all the available data. 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

The effects of coupling of the DTO and RB units in not only one- but also three-dimensional arrays are discussed below and molecular weight trends illustrated. A fundamental connection between relaxation times and normal mode frequencies, shown to hold in all dimensions, allows the rapid derivation of the common viscoelastic and dielectric response functions from a knowledge of the appropriate lattice vibration spectra. It is found that the time and frequency dispersion behavior is much sharper when the oscillator elements are established in three-dimensional quasi-lattices as in the case of organic glasses. 

Equation (108) is universal for any small amplitude perturbation method and may, in principle, be used as the starting point for the derivation of a response vs. time relation for a given perturbation function. It is easily verified [53] that substitution of s = ico into any expression of an operational impedance or admittance delivers the complex impedance or admittance as they are defined in Sect. 2.3.1. 

It is important to recognize the unique relationship that exists between the responses to an impulse and step change in concentration. The derivative of the step response (Eq. 2.14) is identical to the impulse response (Eq. 2.4), and the integral of the impulse response is identical to the step response. This reciprocity is an important property of linear systems in general. The reader should now appreciate that under linear conditions, the time dependence of any concentration profile can be treated by adding the response functions for its component impulses. 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

For a linear system essentially the same information can be deduced from either a pulse or step response measurement. (Since the pulse is the time derivative of the step function, the response to the pulse will be the derivative of the step response.) Both methods are widely used, and the choice is therefore dictated by experimental convenience rather than by fundamental theoretical considerations. 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

We derive the response functions using the Ehrenfest theorem for a time-independent one-electron operator Q... 

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,... 

In this chapter we surveyed the theoretical analysis of resonant multidimensional spectroscopies generated by the interaction of 3 fs pulses with a Frenkel exciton system. Closed expressions for the time-domain third-order response function derived by solving the NEE are given in terms of various exciton Green functions. Alternatively, the multidimensional time-domain signal can be calculated starting from the frequency domain the third-order... 

Generally speaking, the equilibrium FDT establishes a link between the dissipative part tBA(t. tr) of the linear response function %BA(t, 0 and the symmetrized equilibrium correlation function CBa MO = ([A(f ),B(f)]+) (or the derivative dCBA(t, t )jet with t < t the earlier time). 

Before entering the discussion of the classical limit, we will propose another formulation of the FDT as given by Eq. (37) or Eq. (38) in the time-domain. It provides the expression of the dissipative part of the linear response function in terms of the derivative of the symmetrized correlation function, an expression which becomes particularly simple in the classical limit [34]. 

One deduces from Eq. (49) that the expression of the response function Xba(MO involves the derivative of the correlation function CBA(t,t ) with respect to the earlier time t ... 

Terms of higher order in the field amplitudes or in the multipole expansion are indicated by. . . The other two tensors in (1) are the electric polarizability ax and the magnetizability The linear response tensors in (1) are molecular properties, amenable to ab initio computations, and the tensor elements are functions of the frequency m of the applied fields. Because of the time derivatives of the fields involved with the mixed electric-magnetic polarizabilities, chiroptical effects vanish as a> goes to zero (however, f has a nonzero static limit). Away from resonances, the OR parameter is given by [32]... 

The response functions of systems of noninteracting particles, on the other hand, are functional derivatives of the density with respect to the time-dependent single-particle potential v. ... 

Figure 2 shows the spectral response functions (5,(r), Eq. 1) derived firom the spectra of Fig. 1. In order to adequately display data for these varied solvents, whose dynamics occur on very different time scales, we employ a logarithmic time axis. Such a representation is also useful because a number of solvents, especially the alcohols, show highly dispersive response functions. For example, one observes in methanol significant relaxation taking place over 3-4 decades in time. (Mdtiexponential fits to the methanol data yield roughly equal contributions from components with time constants of 0.2, 2, and 12 ps). Even in sinqrle, non-associated solvents such as acetonitrile, one seldom observes 5,(r) functions that decay exponentially in time. Most often, biexponential fits are required to describe the observed relaxation. This biexponential behavior does not reflect any clear separation between fast inertial dynamics and slower diffusive dynamics in most solvents. Rather, the observed spectral shift usually appears to sirrply be a continuous non-exponential process. That is not to say that ultrafast inertial relaxation does not occur in many solvents, just that there is no clear time scale separation observed. Of the 18 polar solvents observed to date, a number of them do show prominent fast components that are probably inertial in origin. For example, in the solvents water [16], formamide, acetoniuile, acetone, dimethylformamide, dimethylsulfoxide, and nitromethane [8], we find that more than half of the solvation response involves components with time constants of 00 fs.

The derivation of formulae for the frequency-dependent nonlinear susceptibilities of nonlinear optics from the time-dependent response functions can be found in a number of sources, (Bloembergen, Ward and New, Butcher and Cotter, Flytzanis ). Here it is assumed that the susceptibilities can be expressed in terms of frequency-dependent quantities that connect individual (complex) Fourier components of the polarization with simple products of the Fourier components of the field. What then has to be shown is how the quantities measured in various experiments can be reduced to these simpler parameters. 

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