Molecular properties mixed derivatives

Mixed derivatives refer to cross terms if the energy is expanded in more than one perturbation. There are many such mixed derivatives which translate into molecular properties, below are a few examples. 

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]... 

Several methods for the determination of -y(R) have been proposed [224,225]. One is the direct computation of the non-adiabatic coupling matrix element ( i l(B 2))(r) by finite difference techniques, which gives the derivative of y (cf. Eq. (10)). Another is by supposing that the diabatic states adiabatic states Xk and Xkt are (almost) pure linear combinations of the two monomer states. This approximation can be made at the orbital level or at the A-electron level (or at both levels simultaneously). Also mixing matrix elements of molecular properties over adiabatic states may be used. 

There are different grades of each of these materials according to the overall molecular weight and the degree of substitution. These polymers are used as components of systems with unique adhesive properties for example, in the manufacture of safety glass laminates (poly(vinyl butyral) and mixed derivatives) and of metal-to-metal adhesive (poly(vinyl formal) cured with phenolics and other resins). Reactions of poly(vinyl alcohol) with acids or anhydrides occur as normal esterifications, a route used to synthesize polymers and copolymers that cannot be readily formed by conventional polymerization (e.g., when the reactivity ratios of the monomers are not suitable). 

Atmospheric Reaeration. Interfacial properties and phenomena that govern oxygen concentrations in river systems include 1) oxygen solubility (temperature, partial pressure and surface dependency), 2) rate of dissolution of oxygen (saturation level, temperature and surface thin film dependency, i.e., ice, wind), and 3) transport of oxygen via mixing and molecular diffusion. A number of field and empirically derived mathematical relationships have been developed to describe these processes and phenomena, the most common of which is (32) ... 

Supported Ziegler-type neodymium surface species (54, see below) have been prepared by mixing molecular components composed of [Nd(naph)3] (derived from naphthenonic acids) and alkyl aluminium reagents such as Al2Et3Cl3, Al( Bu)3 and/or Al( Bu)2H at 50-60°C with silica (source QiLu Petrochemicals Co., China) [158-160]. Although the immobihzed neodymium species are iU-defined, the materials display interesting properties in butadiene polymerization (Section 12.4.1.2). 

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