Linear and nonlinear response functions

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... 

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. 

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... 

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... 

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)... 

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. 

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. 

Ruzsinsky et alP have addressed the polarizability and the second hyperpolarizability of H2 chains using a DFT scheme that includes charge-transfer to correct standard local or semilocal density functionals for their overestimations of the linear and nonlinear responses to external electric fields. In this model, charge is transferred between H2 units paired up at equal distances from but on opposite sides of the chain center. All symmetric pairs of H2 units, not just the one for the chain ends, are included. 

This transfer is driven by the external electric field and opposed by the chemical hardness of each H2 unit. Since self-interaction-free approaches (like the HF method) increase the chemical hardness of an H2 unit in comparison with semilocal density functionals, they reduce the charge transfer and the related linear and nonlinear responses. 

The hardness h are intimately related to the linear and nonlinear electronic responses as shown explicitly in Equation 24.18. In particular, h is simply the inverse of the linear polarizability it is well known in chemistry that a hard atom has a low polarizability. The nonlinear terms hn/, could allow to better quantify the hardness/softness and polarizability relations (see Section 24.2.2). Note that for an atom in a molecule, the contribution of a2 has to be considered as well in Equation 24.12 through Equation 24.18. On the other hand, Equation 24.18 shows that all the polarizabilities can be formulated in terms of the linear one, if the derivatives hn, which are function of p, are known ... 

In the example given, the constitutive equation used is a multimode Phan Tien Tanner (PTT). It requires the evaluation of both linear and nonlinear material-response parameters. The linear parameters are a sufficient number of the discrete relaxation spectrum 2, and r]i pairs, which are evaluated from small-strain dynamic experiments. The values of the two nonlinear material-response parameters are evaluated as follows. Three semiarbitrary initial values of the two nonlinear parameters are chosen and the principal normal stress difference field is calculated for each of them using the equation of motion and the multimode PTT. They are compared at each field point (i, j) to the experimentally obtained normal stress difference and used in the following cost function F... 

All the linear and nonlinear optical properties introduced above are therefore expressed in terms of linear, quadratic and cubic response functions. They can be computed with high efficiency using analytical response theory [9] with a variety of electronic structure models [8],... 

Sophy KB, Pal S (2003) Density functional response approach for the linear and nonlinear electric properties of molecules, J Chem Phys, 118 10861-10866... 

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]. 

This equation holds for any time-independent one-electron operator Q. In particular, it holds for the spin-averaged excitation operators Ep in the expansion of K(f) in Eq. (41). Collecting these operators in the column vector q, we arrive at a set of nonlinear equations from which the time-dependence of k(t) may be determined. In the following, we shall use these equations to determine the first- and second-order terms in Eq. (46) and thereby the linear, quadratic, and cubic response functions. 

Experimental NMR data are typically measured in response to one or more excitation pulses as a function of the time following the last pulse. From a general point of view, spectroscopy can be treated as a particular application of nonlinear system analysis [Bogl, Deul, Marl, Schl]. One-, two-, and multi-dimensional impulse-response functions are defined within this framework. They characterize the linear and nonlinear properties of the sample (and the measurement apparatus), which is simply referred to as the system. The impulse-response functions determine how the excitation signal is transformed into the response signal. A nonlinear system executes a nonlinear transformation of the input function to produce the output function. Here the parameter of the function, for instance the time, is preserved. In comparison to this, the Fourier transformation is a linear transformation of a function, where the parameter itself is changed. For instance, time is converted to frequency. The Fourier transforms of the impulse-response functions are known to the spectroscopist as spectra, to the system analyst as transfer functions, and to the physicist as dynamic susceptibilities. 

---