Continuum pseudo

In a large molecule fhe vibrational and rofafional levels associated wifh any elecfronic sfafe become so exfremely congested af high vibrational energies fhaf fhey form a pseudo-continuum. This is illusfrafed for Sq, Si and fhe lowesf excited friplef sfafe T, lying below Sj,... 

An alternative and complementary use of CFD in fixed bed simulation has been to solve the actual flow field between the particles (Fig. lb). This approach does not simplify the geometrical complexities of the packing, or replace them by the pseudo-continuum that is used in the first approach. The governing equations for the interstitial fluid flow itself are solved directly. The contrast is thus between the interstitial flow field type of simulation and the superficial flow... 

Fio. 1. Comparison of (a) pseudo-continuum and (b) interstitial CFD approaches to packed-tube simulation. 

Figure 5.2 Hydrogen exposed to an electromagnetic field pulse with peak intensity 5.0 1013 W/cm2 and carrier frequency a> = 0.6 a.u. 16 eV/ft. Upper panel Population of the ground state computed without complex rotation (line with circles and line with triangles, respectively, 50 and 100 atomic states for each angular symmetry) and with complex rotation (black solid line, 20° and 30 states). The energy width associated with each rotated pseudo-continuum state explains the improved description. Lower panel The vector potential describing the pulse.

In practical implementations of complex scaling, the Hamiltonian is regularly discretized in finite space, for example, in a box of radius R. This yields a discrete pseudo-continuum with energies that fulfill Ek e w for Z = 0 and approaches it with increasing k and R for Z 0. If exterior complex scaling is made in such a finite box, Eq. (15) is adjusted to... 

We can now return to the question of the representation of an outgoing wave with a complex rotated pseudo-continuum. An interesting example is the outgoing component of a resonance state, for example, a doubly excited state in a two-electron system, for example, with the Hamiltonian... 

It is evident that the pseudo-continuum states are able to give a good representation of the outgoing wave. In fact, this representation is even superior with complex rotation, as indicated in Figure 5.2. The reason for this... 

Figure 5.5 The figure shows the projection of a box normalized pseudo-continuum state onto rotated analytical continuum states with outgoing, plane wave asymptotic behavior, i.e., F0 in Eq. (31). The x-axis shows the energy of the analytical continuum states. The half width of the distribution increases with increasing 9. More specifically, the width of F0 m 2 coincides quite well with twice the absolute value of the imaginary part of the energy of the box normalized pseudo-continuum state, lm(En). This particular case corresponds to the i = 0 channel with the scaling angle 0 = 5° and a box state with Re(En) = 2.0 a.u.

Figure 5.8 Energy eigenvalues (dots) of the discretized and complex rotated one-particle Dirac Hamiltonian with s-symmetry. Bound states that are well represented in the box and on the grid are lying on the x-axis. The low-energy pseudo-continuum states are rotated with an angle of approximately 20 (where 0 is the complex rotation angle) down from the real axis, as indicated with the straight solid line.

Atabek, et al, (1980) have pointed out that this conclusion is incorrect when 12 l, V2- Then it would not be proper to prediagonalize the interaction between the two discrete states. Indeed level 2, after its interaction with the continuum of state 3, must be considered as a pseudo-continuum with respect... 

Equations 3.3 show the interesting result that the bed resistivity takes on a pseudo-continuum behavior for particles smaller than 65 pm. 

The pseudo-continuum couples strongly to the ionization continuum (on the left of Figure 18.9), and autoionizes, producing electrons with kinetic energy, which are lost during the delay before extraction. This results in a loss of intensity for the ZEKE signal. 

The pseudo-continuum and interloper are strongly coupled, with the transition to the pseudocontinuum being strongly favoured. This results... 

Deuterium lamps are commonly used for UV spectroscopy. They give a moderately intense continuum of UV radiation from 360 nm down to 160 nm, with a weaker pseudo-continuum with strong line emission superimposed throughout the visible (Fig. 14.4b). Deuterium lamps with quartz envelopes generate ozone, a toxic chemical hazard formed via ionisation of O2 molecules. 

Section 4.1 via Section 4.1.2 formally illustrates vapor-Uquid equilibria vis-a-vis distillation in a closed vessel along with bubble-point and dew-point calculations for multicomponent systems. How vapor-liquid equilibrium is influenced by chemical reactions in the liquid phase is treated in Section 5.2.1.2, where two subsections, 5.2.1.2.1 and 5.2.1.2.2, deal with reactions influencing vapor-Uquid equilibria in isotopic systems. We next encounter open systems in Chapter 6. The equations of change for any two-phase system (e.g. a vapor-Uquid system) are provided in Section 6.2.1.1 based on the pseudo-continuum approach for the dependences of species concentrations... 

The solution of the problem of flxed-bed adsorption of a strongly adsorbed species i present in an inert mobile feed liquid is at hand when the concentration Ca of the desired species in the liquid phase (moles of species i per unit volume of the liquid phase) is obtained as a function of time t and spatial coordinates x, y, z. Complexity of the fluid dynamics in a packed bed and the presence of convective dispersion require, however, considerable reduction in this goal for practical purposes. In the "pseudo-continuum approach (Lee et al. (1977a) see Section 6.2.1.1), only the axial (mean flow direction, z-coordinate) variation is retained, i.e. we look only for a... 

Figure 7.1.1. (a) Movement of the color front in the packed adsorbent bed with time and the corresponding concentration of the coloring species at the bed outlet (b) packed-bed schematic and its idealization in pseudo-continuum approach for adsorption from a liquid. (After Lightfbot et aL, 1962.)... 

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