Mode expansion

Andersen H C and Chandler D 1970 Mode expansion in equilibrium statistical mechanics I. General theory and application to electron gas J. Chem. Phys. 53 547... 

Chandler D and Andersen H C 1971 Mode expansion in equilibrium statistical mechanics II. A rapidly convergent theory of ionic solutions J. Chem. Phys. 54 26... 

Andersen H C and Chandler D 1971 Mode expansion in equilibrium statistical mechanics III. 

To account for photochemical processes, we adopt a simple model that was proposed by Seidner and Domcke for the description of cis-trans isomerization processes [164]. In addition to the normal-mode expansion above, they introduced a Hamiltonian exhibiting torsional motion. The diabatic matrix elements of the Hamiltonian are given as... 

J. Ctyroky, J. Homola, M. Skalsky, Modelling of surface plasmon resonance waveguide sensor by complex mode expansion and propagation method, Opt. Quantum Electron. 29, 301-311 (1997). 

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta and R. Kaji, Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis. Journal of Lightwave Technology 16, 2040-2045 (1998). 

Solution (15) is written in the form of a normal mode expansion with eigenvalues —X D and eigenfunctions represented by the spatial parts of Eq. (15). Of course, the above solution must fulfil the appropriate initial and boundary conditions. 

When the size parameter x is sufficiently small, that is, when the particle is small compared with the wavelength of light, only the leading term in the normal mode expansion for the spherical harmonic functions is needed. In this case Eq. (76) reduces to Rayleigh s result, Eq. (47), for the ratio of the scattered irradiance to the incident irradiance. 

If we input the mode expansion into this wave equation we arrive at the wave equation... 

H2O in its electronic ground state is best described by a local mode expansion (Child and Halonen 1984 Child 1985 Halonen 1989). For the purpose of this chapter it suffices to consider a simple two-dimensional model in which the bending angle is frozen at its equilibrium value 104° and the oxygen atom is assumed to be infinitely heavy. For an exact three-dimensional treatment see Bacic, Watt, and Light (1988), for example. The approximate two-dimensional Hamiltonian reads... 

Fig. 13.3. Left-hand side Contour plots of the modulus square of the lowest four anti-symmetric eigenfunctions of H2O in the electronic ground state, multiplied with the X — A transition dipole function. They have a node on the symmetric stretch line. The assignment rnn ) is based on the local mode expansion (13.5)...

From the preceding paragraph, the reader will note that many assumptions are involved in transition-state theory. Alternative derivations exhibit differing hypotheses. In a quicker but perhaps less intuitive derivation, translation in the reaction coordinate is treated formally as the low-frequency limit of a vibrational mode. Expansion of the vibrational partition function given in Section A.2.3 then yields Q = Q (k T/hv), which is substituted into equation (A-24), to be used directly in equation (66), thereby producing equation (69) when v = 1/t. The decay time thus is identified as the reciprocal of the small frequency of vibration in the direction of the reaction coordinate. 

In the second equality we have expanded the coordinate deviation <5x in normal modes coordinates, and expressed the latter using raising and lowering operators. The coefficients are defined accordingly and are assumed known. They contain the parameter a, the coefficients of the normal mode expansion and the transformation to raising/lowering operator representation. Note that the inverse square root of the volume Q of the overall system enters in the expansion of a local position coordinate in normal modes scales, hence the coefficients scale like... 

Fourier series mode expansions for the multipolar electric displacement and magnetic field operators may be written in terms of fhe creation and destruction operators as... 

E.S.C. Ching, P.T Leung, A. Maassen van den Brink, W. Suen, S.S. Tong, K. Young, Quasinormal-mode expansion for waves in open systems. Rev. Mod. Phys. 70 (1998) 1545. A. Settimi, S. Severini, B.J. Hoenders, Quasi-normal-modes description of transmission properties for photonic bandgap structures, J. Opt. Soc. Am. B 26 (2009) 876. 

Calculation of a perturbed distribution function can be approached in various ways (1) direct solution of the Boltzmann equation for the distribution function in the perturbed system, (2) distribution-difference methods, (3) local calculations, and (4) normal-mode expansion methods. 

In the normal-mode expansion method, the perturbed distribution is expressed in terms of the normal modes, or eigenfunctions, of the unperturbed reactor (9). This classic approach to perturbation calculations will not be reviewed in this work. 

The expansion (5.47) for the phase-space density implies a similar decomposition (mode expansion) for the phase-space correlation function... 

A versatile interfacial and film rheometer has been developed in our laboratory (7—10). In this technique, a curved, spherical cap-shaped fluid interface or liquid film is formed at a capillary tip and the interfacial tension (IFT) of the single interface or the film tension of the film can be determined by measuring the capillary pressure of the interface or film (Fig. 1). The IFT or film tension is related to the capillary pressure and the radius of the interface or film curvature by the Young-Laplace equation. The IFT and film tension can be measured not only in equilibrium, but also in dynamic conditions as well. The automated apparatus makes it possible to change the interfacial or film area in virtually any mode (expansion or contraction) at various rates (Fig. 2). This instrument is now made available through our laboratory. 

This feature of incorrectly giving the behavior of h(r) for r < d is common to many types of approximation schemes, for example, the mode expansion. This is a formal theory of the equilibrium properties of fluids that is not based on cluster expansion methods but whose results can be expressed in terms of cluster integrals. In the mode expansion, a useful strategy was to redefine or optimize the perturbation for rself-consistent F-ordered result. Moreover, it can also be usefully applied even to approximations that do not suffer from the difficulty of giving physically unreasonable values of h for roptimization method will be discussed more fully in the next section. 

The second-order, optimized mode expansion of Andersen and Chandler, which proves to be identical to this, has been evaluated for ionic solutions. The results confirm that (104) is a highly accurate approximation for the 1-1 electrol) es except at very low concentrations where our AB2 correlation is important. For dipolar spheres, (104) was evaluated by Verlet and Weis, who were led to its consideration along with the LIN result (99) on somewhat different grounds from ours. It is a reasonably good approximation, but not as good as the Fade result given by Eq. (38). However, if one adds the third-order... 

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