Perturbed equations

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is  

The conditions on the interface express the continuity of the mass and heat fluxes and the equilibrium of all acting forces (Landau and Lifshitz 1959). In the frame of reference associated with the interface they are  

In the case when capillary flow undergoes small perturbations, the governing parameters Jj can be presented as a sum of their basic values, corresponding to the stationary flow Jj, plus small perturbations /  

In capillary flow with a distinct meniscus separating the regions of pure liquid and pure vapor flows, it is possible to neglect the change in densities of the phases and assume po and pu are constant. For flow of incompressible fluid (p = const., p = 0, duildx = 0) the substitution of (11.8) in Eqs. (11.1-11.3) leads, in a linear approximation, to the following system of equations 

We obtain from (11.9-11.11) the equations for small perturbations of velocity, pressure, temperature and enthalpy, as well as the specific volumetric rate of heat absorption. Assuming that hi = Cp. 7) we arrive at 

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In the MPPT/MBPT method, once the reference CSF is chosen and the SCF orbitals belonging to this CSF are detennined, the wavefiinction T and energy E are detennined in an order-by-order maimer. The perturbation equations determine what CSFs to include and their particular order. This is one of the primary strengdis of this technique it does not require one to make fiirtlier choices, in contrast to the MCSCF and Cl treatments where one needs to choose which CSFs to include. 

Here, x) >) is an aibitiaiy function of nuclear coordinates. It cannot be detemiined from Eq. (24) alone, but has to be determined from higher ordered perturbation equations. 

By assumption, the mass ratio = m/M is a small parameter. Thus, rescaling the Schrbdinger equation properly in time and potential transforms it into the singularly perturbed equation... 

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations... 

Solution of the second-order time-dependent perturbation equations. 

These are zero-, first-, second-, th-order perturbation equations. The zero-order equation is just the Schodinger equation for the unperturbed problem. The first-order equation contains two unknowns, the first-order correction to the energy, Wi, and the first-order correction to the wave function, 4< i. The th-order energy correction can be calculated by multiplying from the left by 4>o and Integrating, and using the turnover rule ( o Ho, ) = (, Ho o)... 

From this it would appear that the ( - l)th-order wave function is required for calculating the th-order energy. However, by using the turnover rule and the nth and lower-order perturbation equations (4.32), it can be shown that knowledge of the nth-order wave function actually allows a calculation of the (2n-i-l)th-order energy. 

Starting from the second-order perturbation equation (4.32), analogous formulas can be generated for the second-order corrections. Using intermediate normalization... 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

Stoning, /. disturbance perturbation interruption turbulence (Geol.) deformation trouble, derangement, disorder, storungsfrei, a. undisturbed, trouble-free. Stdrungsgleichung, /. perturbation equation. Storzel, m. see Kandisstbrzel. 

Chapter 11 consists of following Sect. 11.2 deals with the pattern of capillary flow in a heated micro-channel with phase change at the meniscus. The perturbed equations and conditions on the interface are presented in Sect. 11.3. Section 11.4 contains the results of the investigation on the stability of capillary flow at a very small Peclet number. The effect of capillary pressure and heat flux oscillations on the stability of the flow is considered in Sect. 11.5. Section 11.6 deals with the study of capillary flow at a moderate Peclet number. 

Here, for better estimates, perturbation equations with terms up to obtained for a... 

As a simple introduction to PMO theory suppose we consider the bond formation between the two-atom, two-orbital system shown in Figure 4.8. The energy gained on forming the bond A—B is given by the second-order perturbation equation... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

The combination of TDDFT with a QM/MM approach is, in principle, straightforward. The surrounding system of point charges modifies the electrostatic potential of the system, which enters the perturbation equations through the Kohn-Sham Hamiltonian Hks. This causes a change in the excitation wavelenghts which reflects the influence of the environment. 

The generalization of the Fukui functions to nonlinear and nonlocal chemical responses is done in Refs. [26,32] by using N derivatives and the KS perturbation equations. In this section, we propose a brief survey of a complementary derivation based on the concept of the internal charge transfer A introduced above. A more detailed discussion, including computational schemes, will be presented elsewhere. 

The above perturbation equations are all based on the assumption of a large energy gap between the interacting configurations. When the two configurations become degenerate, (18) and (22) are no longer applicable. In that case SE is just equal to P (24), and the two states, yG and yE, are described by... 

The decay or possible growth of a given mode will be determined by the eigenvalues of the Jacobian matrix associated with the perturbation equations above these in turn are determined by the trace and determinant. For a given n the trace has the form... 

Although all the exciton levels are not equally allowed, they are important for energy migration or exciton delocalization. Using time dependent perturbation equation, time period for transfer tw is given as... 

A.B. Vasilieva and V.F. Butuzov, Asymptotic Expansion for Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973 (in Russian). 

Here we will skip the notation details, as the relation established to the Coupled Perturbed frame allow us the shortcut of passing the references to the comprehensive works devoted to the analytic derivatives of molecular energy [9]. The recent advances in the analytic derivatives and Coupled Perturbed equations into multiconfigurational second order quasi-degenerate perturbation theory is the premise of further development in the ab initio approach of vibronic constants of JT effects [10]. 

Rates of non-adiabatic intramolecular electron transfer were calculated in Ref. [331] using a self-consistent perturbation method for the calculation of electron-transfer matrix elements based on Lippman-Schwinger equation for the effective scattering matrix. Iteration of this perturbation equation provides the data that show the competition between the through-bond and through-space coupling in bridge structures. 

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