General formalism

Wtoljoditesoeiation results when the energy eigenstates reached by photon absorption Sire iji the continuum. When the spectmm is continuous, we have to use the scattering 

The portion of the wave packet excited to a continuous segment of the spectrum is given by 

Because we now have an integral over E in the expansion, the normalization of the constituent states E, n) is different than that of Em). It is given by 

We see that it is the interaction potential E(R, r) that couples the motion of the A atbnTto the motion of the BC diatomic. In its absence the two free fragments A and BG described by the free Hamiltonian 

The e ) eigenstates of hr are often called channels and a channel is said to be open if E — en 0 it is said to be closed if E — en 0. When a channel is open, the solution of Eq. (2,43), written in the coordinate representation, 

The coupling tensors Q, D, o, and J are tensors P of rank 2 [Hael, Mehl, Spil]. For each of the corresponding interactions, A, the tensor, can be separated into an isotropic part Pf, an antisymmetric part P[ and a traceless symmetric part P . For simplicity of notation, the index A is not carried along in the next six equations. In the principal axes system XYZ of the symmetric part of the coupling tensor, the generic coupling tensor P = Pij, where i,j = X, Y, Z, is written as 

Equation (3.1.15) is written in Cartesian coordinates. Because the tensorial properties of the interactions lead to an angular dependence of the resonance frequency, a representation in spherical coordinates is preferred [Spil], 

It is always observed, even under fast molecular reorientation. For the different interactions A the expressions for R are listed in Table 3.1.1. 

To first order, the resonance frequency is determined by the isotropic and the symmetric parts only and not by the antisymmetric part of P [Anel]. Because the trace of the symmetric part P is zero, two parameters are sufficient for its determination. They are the largest principal value 3 or the anisotropy A, 

Using the generic coupling tensor representing the different interactions A, the generic Hamilton operator is expressed in the notation of irreducible spherical tensors 

---

The variational procedure for the states xi can be implemented in many different ways a very general formalism introduces complex parameters z(t). 

Becker, P.J. and Coppens, P. (1974) Extinction within the limit of validity of the Darwin transfer equations. I. General formalisms for primary and secondary extinction and their application to spherical crystals, Acta Crystallogr., A, 30, 129-147. 

These two seemingly distinct approaches of thermodynamic integration and perturbation can be seen as the limiting cases of a more general formalism in which the transformation between the two states proceeds at a finite rate. Seen in this light, one might also hope to obtain free energies from a transformation that converts the initial to the final state neither infinitely slowly (as in thermodynamic... 

Stevens formalism turned out to be very powerful, and works easily as long as only the ground 2S+1Lj multiplet ofthe lanthanide ion is considered. As such, it has been widely used in studies on EPR properties of lanthanide-based inorganic systems [6, 22], while it is not well suited for optical spectroscopy. Indeed, when starting to include excited multiplets the Stevens formalism becomes much too involved. This is the reason why a more general formalism, developed by Wybourne [3], is of widespread use in optical studies - naturally dealing with excited multiplets - and... 

Xue Y, Datta S, Ratner MA (2002) First-principles based matrix Green s function approach to molecular electronic devices general formalism. Chem Phys 281(2—3) 151—170... 

Durand P (1983) Direct determination of effective Hamiltonians by wave-operator methods. I. General formalism. Phys Rev A 28 3184... 

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

As an introduction to the peculiar properties of the spin Hamiltonians, we first give a short summary of the theory of spin relaxation in liquids where the problem is in fact a Brownian motion one. Then we consider the many-spin problem in solids and apply the general formalism of the theory of irreversible processes developed by Prigogine and his co-workers. We also analyse some aspects of the recent work of Caspers and Tjon on this subject. Finally, we indicate the special interest of spin-spin relaxation phenomena in connection with non-Markovian processes. 

Canuto, V. M. andM. S. Dubovikov (1996a). A dynamical model for turbulence. I. General formalism. Physics of Fluids 8, 571-586. 

Alhassid, Y., and Shao, B. (1992a), Algebraic Eikonal Approach to Electron-Molecule Scattering. I. Generalized Formalism, Phys. Rev. A 46, 3978. 

The preceding analysis yields a general formalism with which fluorescence excitation spectra of molecules in small particles can be theoreti-... 

Delarue, M. and Qrland, H. (2000). General formalism for phase combination and phase refinement a statistical thermodynamics approach in reciprocal space. Actu Crystallogr. A 56, 562-574. 

Yamamoto M. The visco-elastic properties of network structure. 1. General formalism. J Phys Soc Jpn 1956 11 413-421. 

In recent years, there appear some new functionals where the current of electrons instead of their density is used as a basic variable [13]. SRPA equations for this case can be straightforwardly obtained from the general formalism given in Sec. 2. 

The procedure of developing a semi-empirical parameterization can be generally formalized in terms of Eq. (2) as follows. A set of experimental energies 5(C QF5) corresponding to different chemical compositions C, molecular geometries Q, and electronic states with specific values of S and T is given. In the case when a response to an external field is to be reproduced the latter can be included into the coordinate set Q. Developing a parameterization means to find certain (sub)set of parameters w which minimizes the norm of the deviation vector with the components... 

Discussion. DD has many extensions that go beyond what has been discussed above. An important extension is the universal DD sequence [80] that combats all forms of coupling to a bath, namely, both AN and PN. The general formalism presented above has not yet addressed this issue, but attempts are currently being made to generalize the treatment. 

M.E. Coltrin, RJ. Kee, and F.M. Rupley. Surface Chemkin A Generalized Formalism and Interface for Analyzing Heterogeneous Chemical Kinetics at a Gas-Surface Interface. Int. J. Chem. Kinet., 23 1111-1128,1991. 

The theories referred to above, dating from around 1974, have been extended since then, as regards both refinements of the basic models and other types of mechanism, e.g. the important case of the transport of ions facilitated by their complexing with membrane-soluble carriers [109, 111]. A quite general formalism has been proposed by Rangarajan et al. [112]. 

One of the simplest methods to generalize formal kinetics is to treat reactant concentrations as continuous stochastic functions of time, which results in a transformation of deterministic equations (2.1.1), (2.1.40) into stochastic differential equations. In a system with completely mixed particles the macroscopic concentration n (t) turns out to be the average of the stochastic function Cj(<)... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

For solid surfaces with crystalline structure, we can apply diffraction techniques for analysis. In a diffraction experiment the sample surface is irradiated with electrons, neutrons, atoms, or X-rays and the angular distribution of the outgoing intensity is detected. The analysis of diffraction patterns is a formidable task and in the first subsection we only introduce a simple case, which, nevertheless, contains the main features. A more general formalism for the interested reader is described in the Appendix. 

Application of this general formalism to the rotation matrix R(e) G SO (3) yields... 

---