Tensor mixed

For nonaxial EFG tensors, mixing of the nuclear mj basis functions occurs even for B being oriented along This results from the contributions of the shift operators and fi in the Hamiltonian described by (4.29). These contribu-... 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

In the mixed quantum-classical molecular dynamics (QCMD) model (see [11, 9, 2, 3, 5] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. The full quantum system is first separated via a tensor product ansatz. The evolution of each part is then modeled either classically or quan-tally. This leads to a coupled system of Newtonian and Schrbdinger equations. 

Similar to vectors, based on the transfomiation properties of the second tensors the following three types of covariant, contravariant and mixed components are defined... 

The electron-electron dipolar term, Ho, equals S1.D.S2. The tensor D is completely anisotropic and only mixes T-states with one another. It is therefore dropped. The nuclear Zeeman term, tlzi =... 

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

A value of 0 = 0° corresponds to a pure ground state, and 6 = 90° to a pure 3,2 ground state. Since the d orbital rotation matrix elements are different for the d and d -y orbitals, this will lead to a variation of the local g tensor of the Fe" site with the mixing angle d ... 

According to Eqs. (8) and (9), the g tensor of [4Fe-4S] clusters depends on four ferrous gi, 2, gi, gi) and two ferric gs,gs) local g tensors. This holds true even if the mixed-valence pair is fully delocalized Ca = Cb = i Owing to the low-lying excitation energies of the... 

The major contribution to the components of the D tensor as well as the deviations of the g values from 2.0023 arises from the mixing of ligand field states by SOC other contributions to D result from direct spin-spin coupling, which mixes states of the same spin S. The D tensor and the g matrix both carry chemical information as they are related to the strength and symmetry of the LF, which is competing and counteracting to the effects of SOC. Details on the chemical interpretation of the parameters by quantum chemical means is found in Chap. 5. 

They are called contravariant, covariant and mixed tensors, respectively. A useful mixed tensor of the second rank is the Kronecker delta... 

Tensors of higher rank are defined in the same way, for example, a mixed tensor of rank four is... 

Multiplication of Am by Bn yields a mixed tensor AmBn = C , called the outer product, which may be formed with tensors of any rank or type,... 

Let m = q in the mixed tensor = Alkh, defined before, and write... 

This result shows that, by its transformation properties, Aljkl is equivalent to a covariant vector of rank two. This process of summing over a pair of contravariant and covariant indices is called contraction. It always reduces the rank of a mixed tensor by two and thus, when applied to a mixed tensor of rank two, the result is a scalar ... 

The quantities bij, 6U, and 6 are respectively called the components of covariant, contravariant or mixed tensors of the second order, if they transform according to the formulae... 

There are different paths to achieving surface specificity. One can exploit optical susceptibilities and resonances that are nonzero only at the surface or only for the molecular species of interest adsorbed on the surface. Examples include the use of second-order nonlinear mixing processes such as second harmonic generation7-9 for which the nonlinear susceptibility tensor is nonzero only where inversion symmetry is broken. Spectroscopic techniques with very high selectivity for molecular resonances such as surface-enhanced infrared or Raman spectroscopy10-12 may also be used. 

From a theoretical perspective, since the designation of the lab-fixed axes is arbitrary, what is relevant is the relative orientation of the polarizations of the excitation and scattered light. Thus the line shape for excitation light polarized along axis p, and scattered light polarized along axis q (p or q denote X, Y, or Z axes in the lab frame) is called Ipq(co). When p = q this is lyy, and when p q this is IVH. Mixed quantum/classical formulae for Ipq(co) are identical to those for the IR spectmm, except mPi is replaced by apqP which is the pq tensor element of the transition polarizability for chromophore i. Thus we have, for example [6],... 

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