Perturbation theory 584 INDEX

The next lowest unperturbed energy level however, is four-fold degenerate and, consequently, degenerate perturbation theory must be used to determine its perturbation corrections. For simplicity of notation, in the quantities and we drop the index n, which has the value... 

Using the method described in Sect. 8.3.3.1 to obtain the electric field in a microtube resonator, the device sensitivity can be calculated with the perturbation theory for different modes. This provides more systematic insight into the operation of microtube resonators. First, bulk index sensing is considered. The sensitivity is proportional to the integrated optical field inside the liquid core region over that of the entire space, and can be expressed as72 ... 

Now, let us discuss the rate equations embodied in eq.(74). To do this, there is need of a statistical analysis. If the system is kept coupled to a thermostat at absolute temperature T, and assuming that w(i - >if) contains effects to all orders in perturbation theory, the rate of this unimolecular process per unit (state) reactant concentration k + is obtained after summation over the if-index is carried out with Boltzman weight factors p(if,T) ... 

Various reactivity indices have been derived for benzenoid hydrocarbons from the following purely topological approaches the Huckel model (HMO), first-order perturbation theory (PMO), the free electron MO model (FEMO), and valence-bond structure resonance theory (VBSRT). Since many of the indices that have been known for a long time (index of free valence Fr, self-atom polarizability ir , superdelocalizability Sr, Brown s index Z, cation localization energy Lr+, Dewar reactivity number Nt, Brown s para-localization energy Lp) have been described in detail by Streitwieser in his well-known volume [23] we will refer here only to some more recent developments. 

The basic ideas that are necessary for the first program stage are explained in Sections II, III, and IV. In Section II, we formulate the problem of how to analyze a system that has a gap in characteristic time scales. Our method is to use perturbation theory with respect to a parameter that is the ratio between a long time scale and a short time scale, which is a version of singular perturbation theory. The reason will be explained in Section II. In Section III, the concept of NHIMs is introduced in the context of singular perturbation theory. We will give an intuitive description of NHIMs and explain how the description is implemented, leaving the precise formulation of the NHIM concept to the literature in mathematics. In Section IV, we will show how Lie perturbation theory can be used to transform the system into the Fenichel normal form locally near a NHIM with a saddle with index 1. Our explanation is brief, since a detailed exposition has already been published [2]. 

Until now, we have discussed NHIMs in general dynamical systems. In this section, we limit our argument to Hamiltonian systems and show how singular perturbation theory works. In particular, we discuss NHIMs in the context of reaction dynamics. First, we explain how NHIMs appear in conventional reaction theory. Then, we will show that Lie permrbation theory applied to the Hamiltonian near a saddle with index 1 acmally transforms the equation of motion near the saddle to the Fenichel normal form. This normal form can be considered as an extension of the Birkhoff normal form from stable fixed points to saddles with index 1 [2]. Finally, we discuss the transformation near saddles with index larger than 1. 

The spectral transition energy hv corresponds to AE 2 in Eq. 74. A refinement of Eq. 82, for which a scaled ( reduced ) lineshape [e(i ) v] and a refractive index correction are used is discussed in [27] and [96a]. The first-order perturbation theory embodied in Eq. 82 is an excellent approximation for field strengths used in conventional spectroscopy. 

Index of chemical reactivity calculated from the perturbation theory as ... 

Second-order Moller-Plesset perturbation theory (MP2) is the computationally least expensive and most popular ab initio electron correlation method [4,15]. Except for transition metal compounds, MP2 equilibrium geometries are of comparable accuracy to DFT. However, MP2 captures long-range correlation effects (like dispersion) which are lacking in present-day density functionals. The computational cost of MP2 calculations is dominated by the integral transformation from the atomic orbital (AO) to the molecular orbital (MO) basis which scales as 0(N5) with the system size. This four-index transformation can be avoided by introduction of the RI integral approximation which requires just the transformation of three-index quantities and reduces the prefactor without significant loss in accuracy [36,37]. This makes RI-MP2 the most efficient alternative for small- to medium-sized molecular systems for which DFT fails. 

Classic perturbation theory considerations [55] predicts that changes in the effective refractive index of a guided mode A eff are related to the changes in the... 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

The QTS is a part of a reaction mechanism. These species may be found related to stationary arrangements of the external Coulomb sources. Solutions to eq.(8) coming as saddle points of at least index one (one imaginary frequency) are natural candidates to play the role of QTS. If the saddle point wave function has closed electronic shell structure, its electronic parity is positive. In this case one would expect a situation similar to the symmetry-forbidden electronic absorption bands. The intensity is borrowed from the excited states having the correct parity via couplings at second-order perturbation theory [21]. 

McRae87 88 has derived an expression for the solvent-induced frequency shift, from the second order perturbation theory, taking into account all the types of interactions suggested by Bayliss and McRae86. On the basis of a simple electrostatic model, the frequency shift, Av, is related to the refractive index and the static dielectric constant of the solvent by an equation consisting of four terms. The first term in the equation represents the contribution from dispersive interactions which give rise to a general red shift the second term represents the contribution from the solute dipole-induced solvent dipole interactions the third term accounts for the solute dipole-solvent dipole interactions and the fourth term represents the contribution from the... 

In the perturbation theory, we assume that the magnetic field vector hy of a surface plasmon supported by a general planar waveguide with and without the refractive index profile perturbation is described by Eq. 18. For the unperturbed and perturbed waveguide with permittivity profiles e(x) and s(x) = s(x) + Ss(x), respectively, this equation can be rewritten as ... 

Fig. 15 Sensitivity of the real propagation constant (—) and effective index (---) of a surface plasmon on a metal-dielectric interface to a bulk refractive index change as a function of wavelength calculated rigorously from eigenvalue equation and using the perturbation theory. Waveguiding structure gold-dielectric (nj = 1.32)...

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