Continuum basis

In Section 2.1 we derived the expression for the transition rate kfi (2.22) by expanding the time-dependent wavefunction P(t) in terms of orthogonal and complete stationary wavefunctions Fa [see Equation (2.9)]. For bound-free transitions we proceed in the same way with the exception that the expansion functions for the nuclear part of the total wavefunction are continuum rather than bound-state wavefunctions. The definition and construction of the continuum basis belongs to the field of scattering theory (Wu and Ohmura 1962 Taylor 1972). In the following we present a short summary specialized to the linear triatomic molecule. 

Equation (2.39) gives the nuclear Hamiltonian in the center-of-mass system. If wbc( ) represents the vibrational potential of the free BC molecule, the interaction potential Vi vanishes asymptotically [see (2.42)] and the full Hamiltonian becomes 

The translational and the vibrational motions are completely decoupled asymptotically. The eigenfunctions of Hq for a total energy E are there- 

In the following we denote by nmax the highest vibrational state that can be populated for a given total energy E, i.e., for which k% 0. Product states with k2 0 cannot be populated because of energy conservation they are called closed channels in contrast to the open channels, which are accessible. The total number of open states is Nopen = nmax + 1. 

In deriving (2.54) and (2.55) we used the following relations for the Dirac delta-function (Messiah 1972 appendix A) 

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In Section 4.1 we will use the time-independent continuum basis 4//(Q E,0), defined in Section 2.5, to construct the wavepacket in the excited state and to derive (4.2). Numerical methods are discussed in Section 4.2 and quantum mechanical and semiclassical approximations based on the time-dependent theory are the topic of Section 4.3. Finally, a critical comparison of the time-dependent and the time-independent approaches concludes this chapter. 

The asymptotic form of radial open-channel orbitals fps(r) is given by Eq. (8.2). Functions of this form can be represented as linear combinations of two independent continuum basis functions for each open channel. These basis functions must be regular at the coordinate origin, but have the asymptotic forms... 

The derivation given above of the stationary Kohn functional [ K] depends on logic that is not changed if the functions Fo and l< of Eq. (8.5) are replaced in each channel by any functions for which the Wronskian condition mm — m 0 = l is satisfied [245, 191]. The complex Kohn method [244, 237, 440] exploits this fact by defining continuum basis functions consistent with the canonical form cv() = I.a = T, where T is the complex-symmetric multichannel transition matrix. These continuum basis functions have the asymptotic forms... 

Introduced in the context of heavy-particle reactive collisions [440], the complex Kohn method has been successfully applied to electron-molecule scattering [341], It is accurate but computationally intensive, since continuum basis orbitals do not have the Gaussian form that is exploited in most ah initio molecular bound-state studies. The method has been implemented using special numerical methods [341] developed for these integrals. These numerical methods mitigate another practical... 

Thus GF is regular at the origin but is asymptotically proportional to the irregular function w r). It has the properties assumed for the second continuum basis function required for each open channel in the matrix variational method. 

The continuum basis of models of this type of microstructural evolution really amounts to a consideration of the competition between surface and elastic energies. Both the elastic and interfacial energies depend upon the shape and size of the second-phase particles, and may even depend upon the relative positions of... 

If one views predissociation as a time-dependent process, then it is possible to derive the relationship between the FWHM of the predissociated line (assuming no oscillator strength from the continuum basis functions) and the predissociation rate. At time t = 0, let the system be prepared with unit amplitude in the discrete state,... 

Holzapfel GA, Gasser TC (2001) A viscoelastic model for fiber-reinforced composites at finite strains continuum basis, computational aspects and applications. Comput Meth Appl Mech Eng 190 4379 403... 

A partial acknowledgment of the influence of higher discrete and continuum states, not included within the wavefunction expansion, is to add, to the tmncated set of basis states, functions of the fomi T p(r)<6p(r) where dip is not an eigenfiinction of the internal Flamiltonian but is chosen so as to represent some appropriate average of bound and continuum states. These pseudostates can provide fiill polarization distortion to die target by incident electrons and allows flux to be transferred from the the open channels included in the tmncated set. 

The most popular of the SCRF methods is the polarized continuum method (PCM) developed by Tomasi and coworkers. This technique uses a numerical integration over the solute charge density. There are several variations, each of which uses a nonspherical cavity. The generally good results and ability to describe the arbitrary solute make this a widely used method. Flowever, it is sensitive to the choice of a basis set. Some software implementations of this method may fail for more complex molecules. 

Solvent effects on chemical equilibria and reactions have been an important issue in physical organic chemistry. Several empirical relationships have been proposed to characterize systematically the various types of properties in protic and aprotic solvents. One of the simplest models is the continuum reaction field characterized by the dielectric constant, e, of the solvent, which is still widely used. Taft and coworkers [30] presented more sophisticated solvent parameters that can take solute-solvent hydrogen bonding and polarity into account. Although this parameter has been successfully applied to rationalize experimentally observed solvent effects, it seems still far from satisfactory to interpret solvent effects on the basis of microscopic infomation of the solute-solvent interaction and solvation free energy. 

The second approach to fracture is different in that it treats the material as a continuum rather than as an assembly of molecules. In this case it is recognised that failure initiates at microscopic defects and the strength predictions are then made on the basis of the stress system and the energy release processes around developing cracks. From the measured strength values it is possible to estimate the size of the inherent flaws which would have caused failure at this stress level. In some cases the flaw size prediction is unrealistically large but in many cases the predicted value agrees well with the size of the defects observed, or suspected to exist in the material. 

The partial wave basis functions with which the radial dipole matrix elements fLv constructed (see Appendix A) are S-matrix normalized continuum functions obeying incoming wave boundary conditions. 

Stener and co-workers [59] used an alternative B-spline LCAO density functional theory (DFT) method in their PECD investigations [53, 57, 60-63]. In this approach a normal LCAO basis set is adapted for the continuum by the addition of B-spline radial functions. A large single center expansion of such... 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

The K-matrix method is essentially a configuration interaction (Cl) performed at a fixed energy lying in the continuum upon a basis of "unperturbed funetions that (at the formal level) includes both diserete and eontinuous subsets. It turns the Schrodinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon afinite L basis set. 

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