Q space

All equations of two variables, such as equation (A2.1.12). are necessarily integrable because they can be written in the fonn dy/dx = fix, y), which detennines a unique value of the slope of the line tln-ough any point (x, y). Figure A2.1.4 shows a set of non-intersecting lines in V-Q space representing solutions of equation (A2.1.12F... 

In Eq. (11b), we observed that since the crude adiabatic basis is used S = 0, for kzQ. Therefore the degeneracy is lifted at first order in the Q-space only, which is therefore used to identified the branching space. The first-order result is... 

Next, the full-Hilbert space is broken up into two parts—a finite part, designated as the P space, with dimension M, and the complementai y part, the Q space (which is allowed to he of an infinite dimension). The breakup is done according to the following criteria [8-10] ... 

Scattering data over the extended range of q-space were also collected from Nafion samples containing a wide range of water contents = 0.05—0.84). While systematic shifts in the intensities and positions of the structure maxima (i.e., the crystalline and ionomer peaks) were observed and consistent with those found by others, the general shapes of the scattering profiles for all samples were quite similar. This observation supported the assumption that the swelling process simply involves a dilution of the... 

The Debye-Waller factor, Eq. 4, describes how the uncertainty in real space (u) determines the range of S(Q) in Q space. Now the exact converse happens with respect to the resolution of the measurement in Q space. If the Q resolution of the instrument is AQ, the PDF will have an envelope exp(- r ( AQ) ), and the oscillations in the PDF decay. Therefore in order to determine the PDF up to large distances it is important to use an instrument with high Q resolution. Since the PDF method was initially applied to glasses and liquids in which atomic correlation decays quickly with distance, this point was not... 

The ( )j(q)Qm (Q) is a set of linearly independent functions the Qm(Q) functions are not orthogonal in Q-space for arbitrary electronic states the overlap integrals Jd Q Qm(Q) Q m (Q) are the well known Franck-Condon factors. The hypothesis is that an arbitrary quantum molecular state is given by the linear superposition jus as in the general case ... 

Figure 9.1 displays the computed J(t), and 2(0 for the first 100 fs, according to Ref. [30]. The J(t) plot is in excellent agreement with the equivalent multiconfiguration time- dependent Hartree results shown in Figure 6 of Ref [58], with any small differences arising from the use of only 176 basis states in the Q-space, for the reasons discussed earlier. 

Figure 9.3 shows the P2(t) curves for both the maximized and minimized initial states, where all 176 Q-space states are included in the optimization. The decay of... 

First, the S2 excitation of pyrazine under pulse train action is considered. The pyrazine vibronic structure description is that of Refs [29, [30]], that is, the Q-space (in the context of the QP-algorithm presented in Section 9.3.1) consists of the 176 states in S2 with the largest FCF to the ground vibrational state of the Sq electronic state (as discussed in Section 9.3.1). Characteristic examples of the Ps (t) populations produced by pulse trains with different parameters are... 

In Reference [35], numerical examples of perturbative Sq - S2 excitation and the S2 IC dynamics for the / -carotene are discussed, too. The absence of reliable potential surfaces for this system motivated the use of a minimal two-dimensional model [66], which utilizes a Morse potential in each dimension. All three electronic surfaces Sq, and S2 involved in this example assume the same 2D potential form however, these potentials are shifted to each other. More importantly, in Ref. [35], each potential has 396 bound states in each electronic state within this model, while additionally the S2 and electronic states are coupled by linear coupling. Thus, the Q-space and P-space, as introduced in the context of the QP-algorithm in Section 1.3.1, consist of the S2 and 5 bound states, respectively. 

One thus identifies the adiabatic states of 9.33, with the 0 ) bound states of a Q space within the PT projection formalism given in Section 9.2 these state become resonances, since they are superpositions of the I q) and IA2) states, which interact with continuum states c,E 1) in the P space. Following Section 9.2, it follows that the E - QhQ matrix in such a case of two overlapping resonances case is given by... 

The total binding energy can be rewritten as a sum over real space rather than q-space vectors by changing the order of their summation in . From eqn (6.57) the band-structure energy is given by... 

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