Linear and nonlinear responses

Equations (5.4.2) cannot be solved in integral form unless they are linearized by setting M (t) = Mq. The linear response is obtained for small excitation amplitudes (cf. eqns (5.3.4)). In this case, the solution becomes 

So far the magnetization has been treated for a single NMR frequency Hq without consideration of the chemical shift and a space-dependent spin density. The amplitude Afo of the thermal equilibrium magnetization is the sum over all magnetization components which differ in space and frequency. 

For an inhomogeneous sample, the linear transverse magnetization response (5.4.5) is rewritten with the help of (5.4.6) and using (t) = y (/), 

This equation is fundamental for the explanation of many imaging methods. It describes the linear part of the complex transverse magnetization response in the presence of arbitrary magnetic-field gradient modulation G(t) and quadrature rf excitation jc(r). When the excitation is a delta pulse x(t) = Sit), the response yi ( ) is given by the kernel (cf. Section 4.2.1) of the outer integral in (5.4.7), 

In the last step, the frequency dependence of the spin density and the exponential Bloch decay have been replaced by a normalized spectroscopic FID sit,r). This is advantageous when arbitrary spectra Sio),r) need to be considered. Thus, the impulse response measured in the presence of magnetic-field gradients, maps the Fourier transform of the spin density Mo(r), as long as the signal decay introduced by the FID can be neglected. 

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From such a treatment, we may derive explicit expressions for the nonlinear radiation in tenns of the linear and nonlinear response and the excitation conditions. For the case of nonlinear reflection, we obtain an irradiance for the radiation emitted at the nonlinear frequency of... 

NLO chromophores. Indeed, a close relation often exists between the linear and nonlinear responses of 7i-conjugated organic compounds which are known to present substantial NLO responses [2-4],... 

Linear and nonlinear responses of the three isomers of nitroaniline... 

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

The constitutive equations discussed previously contain both linear and nonlinear response parameters. Both have to be evaluated experimentally. The first five to ten terms... 

Mikkelsen). Linear and nonlinear response functions have been implemented at the MCSCF level by Mikkelsen et al. [13], for a spherical cavity, and by Cammi et al. [14] and by Frediani et al. [15] for the PCM solvation models. 

Olsen J, Jorgensen P (1985) Linear and nonlinear response functions for an exact state and for an MCSCF state. J Chem Phys 82 3235-3264... 

D.C. Langreth, in J.T. Devreese, V.E. van Doren (Eds.), Linear and nonlinear response theory with applications, Linear and Nonlinear Electron Transport in Solids, Plenum Press, New York, 1975, p. 3. 

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... 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

Despite the discouraging performance of the SOS approach for the electronic contributions to properties of more extended systems, the SOS formulas are frequently used for reasons of interpretation. Strongly absorbing states in the linear absorption spectrum may be identified as main contributors to the linear and nonlinear response properties, and truncated SOS expressions can be formed with this consideration in mind. In certain cases, molecules may have a single strongly dominant one-photon transition, and so-called two-states-models (TSM) can then be applied. The two states in question are obviously the ground state 0) and the Intense excited state /). 

From the point of view of a computational chemist, one of the most appreciated strengths of the polarization propagator approach is that, although being generally applicable to many fields in physics, it also delivers efficient, computationally tractable formulas for specific applications. Today we see implementations of the theory for virtually all standard electronic structure methods in quantum chemistry, and the implementations include both linear and nonlinear response functions. The double-bracket notation is the most commonly used one in the literature, and, in analogy with Eq. (5), the response functions are defined by the expansion... 

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. 

In early years of quantum chemistry, several theoretical papers were devoted to calculations of linear and nonlinear responses of molecules to the electric field perturbations using the Uncoupled Hartree-Fock (UCHF) method. In comparison with the CI ansatz, the UCHF is less accurate in the description of electronic structure of molecules. Since this method was of some interest in computations of NLO properties we present this method in Section 5. 

An instructive approach to the linear and nonlinear response is the perturbative analysis of the Bloch equations in the frequency domain [Houl]. The linear response of the transverse magnetization... 

The supennolecule approach is used to study the linear and second-order nonlinear susceptibilities of the 2-methyl-4-nitroaniline ciystal. The packing effects on these properties, evaluated at the time-dependent Hartree-Fock level with the AMI Hamiltonian, are assessed as a function of the size and shape of the clusters. A simple multiplicative scheme is demonstrated to be often reliable for estimating the properties of two- and three-dimensional clusters from the properties of their constitutive one-dimensional arrays. The electronic absorption spectra are simulated at the ZINDO level and used to rationalize the linear and nonlinear responses of the 2-methyl-4-nitroaniline clusters. Comparisons with experiment are also provided as well as a discussion about the reliability of the global approach. 

Linear and Nonlinear Responses to a Homogeneous Electric Field Comparing Eqs.(12.16) and (12.17) we get... 

Due to the homogeneous character of the electric field, this formula pertains exclusively to the interaction of the molecular dipole (the permanent dipole plus the induced linear and nonlinear response) with the electric field. 

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