Second-order response properties

In addition to expectation values and transition moments, we have a third class of important physical properties, which we refer to as second-order response properties. To develop some understanding for the meaning of and theoretical methods for studying these responses, let us investigate the response of a state 0> corresponding to H to an external time-independent one-cicetron perturbation (a//i)... 

Notice that in this MCSCF result, the multiplier of a is equal to the expectation value of the perturbation operator H,. We have thus obtained an analytical expression from which to determine the desired first- and second- order response properties. This analytical approach for determining the second-order properties is referred to as the coupled multiconfiguration Hartree-Fock (CMCHF) approach (Dalgaard and Jorgensen, 1978). 

The desired first- and second-order response properties of the state y> ire calculated by summing and respectively, over the index m... 

FIG. 5.1. All zeroth- and first-order diagrams Tor a second-order response property. 

Cybulski, S. M. and Bishop, D. M. (1994). Theory of relaxed density matrices Application to second-order response properties. Int. J. Quantum Chem., 49, 371-381. 

Magnetic and Other Second-order Response Properties... 

Second-order response properties, such as electric polarizabilities, magnetic susceptibilities, and atomic polar tensors, can be readily partitioned into either atomic or atom-pair contributions with the help of the theory of AIMs. The former partitioning is accomplished by taking derivatives of the pertinent first-order properties with respect to strengths of external perturbations, whereas the latter involves a somewhat more complicated (albeit more theoretically consistent) formalism. In general, the atomic and atom-pair contributions to the second-order response properties are the sums of the atomic basin and surface relaxation terms. ... 

Both 20 and n are formally defined in terms of second derivatives of the energy, i.e., as second-order response properties, but one can also rationalize these quantities by means of a two-step mechanism. We note that the external homogeneous field induces a current density in the atom or molecule. This... 

Linear response function approaches were introduced into the chemistry literature about thirty years ago Ref. [1,2]. At that time they were referred to as Green functions or propagator approaches. Soon after the introduction it became apparent that they offered a viable and attractive alternative to the state specific approaches for obtaining molecular properties as excitation energies, transition moments and second order molecular properties. 

Myers, R.H., Vining, G.G., Giovannitti-Jensen, A., and Myers, S.L. (1992), Variance Dispersion Properties of Second-Order Response Surface Designs . J. Qual. Technol., 24, pp. 1-11. 

Naively, one would expect that second hyperpolarizabilities y are theoretically and experimentally more difficult to obtain than first hyperpolarizabilities (3. From a computational point of view the calculation of fourth-order properties requires, according to the 2n + 1-rule, second-order responses of the wavefunction and thus the solution of considerably more equations than needed for j3 (cf. Section 2.3). However, unlike (3 the second dipole hyperpolarizability y has two isotropic tensor... 

Since the former theoretical predictions of push-pull substituted phthalocyanines as candidates for second-order NLO properties, some work has been devoted to prepare and study different substituted derivatives with the aim to establish the key structural parameters affecting the NLO response. Some revisions have already been done on the second-order NLO behavior of phthalocyanines [16, 17, 31]. For most of the unsymmetrically substituted push-pull compounds (planar conjugated 7T-electron systems in the XZ plane), 2 symmetry may be assumed due to the... 

Polyphosphazenes are suitable materials to be used as carriers for nonlinear optical (NLO) chromophores. Second order NLO properties have been studied for the polymer (128) and blends of (129) with the free chromophore (130) or the cyclophosphazene (131). All systems have glass transition temperatures higher than 135°C and a wide transparency window. The system (129)-(130) appears to exhibit the highest second-harmonic generation (SHG) response. For possible applications the SHG capability has to be enhanced. ... 

The principles of nonlinear optics and the main techniques used to evaluate the second-order NLO properties are briefly presented here. Major details can be found in more specialised reviews and books. At the molecular level, the interaction between polarisable electron density and the alternating electric field of the laser light beam (E) induces a polarisation response (Afi) that can be expressed following Equation 1.1 ... 

In general, the physical properties of an electron system are defined by referring to a specific perturbation problem and can be classified according to the order of the perturbation effect. For instance, the electric dipole moment is associated with the first-order response to an applied electric field (i.e. the perturbation), the electric polarizability with the second-order response, hyperpolarizabilities with higher-order terms. In addition to dipole moments, there is a number of properties which can be calculated as a first-order perturbation energy and identified with the expectation value... 

One may note that quasienergy derivatives dt listed above are symmetric with respect to the capital indices Ai... A . It must be emphasized that the indices must be thought of as a double index specifying a specific perturbation strength a, as well as a frequency cOa, and, as discussed previously, they must be permuted at the same time in order to retain symmetry. Further simplifications of the expressions for molecular properties are possible using the response equations, e.g. (48), (49) and (50), but generally at the expense of index symmetry and numerical precision [28,9]. As an example, using the response equation the expression for second-order molecular properties (52) can be rewritten as... 

Recently, the second-order NLO properties of new lanthanide complexes of the type [Ln(N03)3-L] (Ln = La, Gd, Dy, Yb, Y 19), where L is a rather rigid ter-pyridine-like ligand, have been determined by HRS, working with a nonresonant incident wavelength of 1.907 pm. The value of the quadratic hyperpolarizability Si,9i(HRS) increases by increasing the number of/-electrons, from 186 to 288 x 10 ° esu [95]. The dependence of the second-order NLO response on the nature of the lanthanide metal center suggests that/-electrons may contribute to the second-order NLO response [95]. 

---