Nonlinear response function calculation

A similar translation scheme from the full quantum approach to a mixed quantum classical description has been used recently in Ref. [26-29] to calculate infrared absorption spectra of polypeptides within the amide I band (note that the translation scheme has been also used in the mentioned references to compute nonlinear response functions). 

While gas phase work on the h5q5erpolarizability of small molecules has been relatively free of problems concerned with the definitions of measured quantities and their formal relationship to computed quantities, the same cannot be said about solution studies of rather larger organic species. It is the latter that possess the very large nonlinear response functions that are of greatest interest. The prototype system for such studies has been 4-nitroaniline (pNA) and this review is mainly concerned with the relation between the measurements, in vacuo and in solution, of the hyperpolarizabilities of pNA and the closely related molecule, MNA (2-methyl, 4-nitroaniline) to ab initio and DFT calculations of these quantities. 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

Finally we note several future directions which should be studied (a) Our final results for the VER rate depend on a width parameter y. Unfortunately we do not know which value is the most appropriate for y. Nonequilibrium simulations (with some quantum corrections [39]) might help this situation, and they are useful to investigate energy pathways or sequential IVR (intramolecular vibrational energy redistribution) [40] in a protein, (b) This work is motivated by pioneering spectroscopic experiments by Romesberg s group. The calculation of the VER rate and the linear or nonlinear response functions, related to absorption or 2D-IR (or 2D-Raman) spectra [41—44], is desirable, (c)... 

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

Most of the calculations of the two-photon absorptivities at the ab initio level were performed using the response theory [61, 63, 65, 66, 86, 87, 112]. Recently, Salek et al. [102] presented the implementation of the density-functional theory for the linear and the nonlinear response functions. In particular, in their most recent paper they reported on calculations of the two-photon absorption cross sections in terms of the single residue of the quadratic response function [101]. 

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-... 

Equations (27) and (28) or alternatively Eq. (31) provide the most general formal expression for any type of 4WM process. They show that the nonlinear response function R(t3,t2,t 1), or its Fourier transform (cum + a>n + (oq,com + tu ,aim), contains the complete microscopic information relevant to the calculation of any 4WM signal. As indicated earlier, the various 4WM techniques differ by the choice of ks and ojs and by the temporal characteristics of the incoming fields E, (t), E2(t), and 3(t). A detailed analysis of the response function and of the nonlinear signal will be made in the following sections for specific models. At this point we shall consider the two limiting cases of ideal time-domain and frequency-domain 4WM. In an ideal time-domain 4WM, the durations of the incoming fields are infinitely short, that is,... 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

In this chapter, we developed a general theory of 4WM processes in terms of the nonlinear response function of the nonlinear medium R(t3,t2,t,). The response function is an intrinsic molecular property that contains all the microscopic information relevant to any type of 4WM process. The details of a particular 4WM experiment are contained in the external fields E t), E2(t), E3(t), and in the particular choice of the observable mode ks. The generated signal is calculated by convolving the response function with the external fields and choosing ks [Eqs. (27), (28), and (31)]. It is only at this stage that... 

Ri(tN,tm-i,...,t2,ti). Because the nonlinear response functions carry the complete d3mamical information of a given spectroscopic process, the response-function formalism allows us to decompose the computation of the polarization into the calculation of purely molecular quantities (i.e. Ri t3,t2,ti)) and subsequent time integrations. The characterization of nonlinear optical processes in terms of nonlinear response functions has been extensively discussed by Mukamel and collaborators.74... 

Only static and dynamic molecular properties involving electric dipole and quadrupole operators will be discussed below. However, electric properties related to higher-order electric multipole operators can also be determined in a similar manner to the properties described here, in terms of expectation values, linear and nonlinear response functions. Nevertheless, it should be kept in mind that although the same formalism is applied in the calculation of response functions involving octupole, hexadecapole, and higher moments, in practice it may... 

Luce T A and Bennemann K H 1998 Nonlinear optical response of noble metals determined from first-principles electronic structures and wave functions calculation of transition matrix elements P/rys. Rev. B 58 15 821-6... 

Finally, a comparison between real and calculated signal intensities can demonstrate the quality of calibration, even if a nonlinear calibration function correlates better with the observed response (Table 4). 

If a nonlinear dose-response function has been determined, it can be used with the expected exposure to estimate a risk. If an RfD or RfC is calculated, the hazard can be expressed as a Hazard Quotient (HQ), defined as the ratio of an exposure estimate over the RfD or RfC, i.e., HQ = Exposure/(RfD or RfC). 

INTRODUCTION. A standard and universal description of various nonlinear spectroscopic techniques can be given in terms of the optical response functions (RFs) [1], These functions allow one to perturbatively calculate the nonlinear response of a material system to external time-dependent fields. Normally, one assumes that the Born-Oppenheimer approximation is adequate and it is sufficient to consider the ground and a certain excited electronic state of the system, which are coupled via the laser fields. One then can model the ground and excited state Hamiltonians via a collection of vibrational modes, which are usually assumed to be harmonic. The conventional damped oscillator is thus the standard model in this case [1]. 

Here, / is the fraction of the delayed nonlinear response, and 7Z is the memory function of the stimulated Raman effect. Parameterization by 7Z(t) sin Qr)e rT is often sufficient for ultrashort pulses [7], This simple formula has the advantage of easy implementation that avoids explicit calculation of the convolution integral. Often, an even simpler, exponential memory function is used, 7Z(r) e rr in simulations (see e.g. [28]). If the real memory function is sufficiently complex, a numerical convolution approach must be used to calculate the convolution. This is e.g. the case in silica [29],... 

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

Li et al °° examined the properties of two organometallic tungsten-carbon complexes, tungsten pentacarbonyl pyridine (TPCP) and tungsten pentacarbonyl fraw -l,2-bis(4-pyridyl)-thylene (TPCB), that also had been studied experimentally. They considered the isolated monomers as well as dimers and studied the systems in solutions. They used density-functional methods in order to calculate the linear and nonlinear responses to electric fields, and the solvents were treated with a... 

Semi-empirical ZINDO SOS (sum over states) and ab initio quadratic response function (DDRPA) calculations on a series of D-A-substituted 7t-conjugated chromophores based on styryl benzothiazoles were used to aid in the design of dyes with high nonlinear optical properties <2004PCP495>. 

Kobko et al.200 have used a third order response function formalism with TDHF and TDDFT to assess different levels of theory for calculations of excited state structure and nonlinear optical responses in donor-donor and donor-acceptor Ji-conjugated molecules. They make suggestions for numerically efficient approximations. 

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