Cyclic unit vectors

Now the vector dJ, which describes the eigenvibrations of the dipole with frequency wo, as well as the rotation of the molecule with frequency fi, may be expanded over cyclic unit vectors eo, i see A.l. As a result we obtain [95] ... 

Passing over to a space-fixed coordinate system with z J (Fig. 1.5(d)) and expanding the unit vector dJ over cyclic unit vectors, we obtain three components. Two of these, namely d 1 and d+1, similarly to (1.7) and (1.8), correspond to P- and i -types of transitions, with frequencies wo— and (jJo + Q. respectively. In addition, a new component d° appears which corresponds to Unear osciUations along the 2-axis with frequency cjo. This component of the vector d is connected with a Q-type transition, as a result of which we obtain a change in the electronic state of the molecule, but no change in its rotational quantum number A = J — J" = 0. 

It is interesting to note that the procedure, as described above, of expansion over cyclic unit vectors of the transition dipole moment d, which has its own oscillation frequency ljq, and rotates together with the... 

In physical calculations containing angular momenta, cyclic coordinates are frequently employed [379]. Cyclic unit vectors are introduced according to the following rules ... 

Any vector A may be resolved over cyclic unit vectors eM ... 

Considering the orthogonality properties of the cyclic unit vectors ... 

Within a phase factor and amplitude, c 1 = e(2> is the vectorial part of the complex description of right and left circularly polarized radiation. The basis unit vectors c 1 c 1, and c 3 form the 0(3) cyclic permutation relations ... 

Let us now consider the second mechanism, namely, the appearance of the electronic contribution gj due to the interaction with the paramagnetic electronic states. In particular, the singlet terms 1II and of one parity (either u u or g - g) interact because of the non-zero matrix elements of the electron-rotation operator [—l/(2/iro)](J+L- + J L+), where // is the reduced mass, ro is the internuclear distance (in atomic units) and the cyclic components of the vectors are defined in the same way as in [267] = Lx iLy, = Jx iJy connecting the x and y... 

The bracket in the second equation in (5) denotes a cyclic sum, and we use units (henceforth in this article) with c = 1. Combining the definition of /p as the four-dimensional curl of a 4 vector, as in Eq. (4), Maxwell s equations in terms of the vector potential are ... 

The authors identified BTD (54) from the literature and FIVXUD (55) from a CAVEAT vector search of the Cambridge Crystallographic Database as reasonable mimetic scaffolds that could fulfill these objectives. The 6,5-bicyclic unit BTD was selected for incorporation into cyclic mimetic structures in both possible enantiomeric forms (56-58). Disappointingly, these... 

Escobedo and Chen [37] have used this technique to study linear and cyclic alkanes (using a united-atom force field [39]). In order to test the efficiency of these algorithms, a model system consisting of one isolated linear alkane molecule at T=400K was simulated. A half-chain end-to-end vector autocorrelation function was used as a measure of the relaxation of the chain. This function was measured in the simulation as a function of number of sites regrowing in every RCB2 move (Fig. 15). The dependence of the autocorrelation function on the choice of was considered (Fig. 16). 

In (4.53), the index fi labels all AOs in the reference primitive unit cell (p = 1,2,..., M) and Rn is the translation vector of the direct lattice (for the reference primitive cell Rn = 0). The summation in (4.53) is snpposed to be made over the infinite direct lattice (in the model of the infinite crystal) or over the inner primitive translations ii of the cyclic cluster (in the cyclic model of a crystal). In the latter case, the sum of the two inner translation vectors fi -I- R = Ri may appear not to be the inner translation of the cychc cluster. However, the subtraction of the translation vector A of the cyclic cluster as a whole (in the cyclic model the vector A is... 

The use of localized orbitals for the cluster calculation is an efficient approach for defective crystals. To connect the perfect crystal localized orbitals and molecular cluster one-electron states the molecular cluster having the shape of a superceU was considered [699]. Such a cluster differs from the cyclic cluster by the absence of PBC introduction for the one-electron states. Evidently, the molecular cluster chosen is neutral and stoichiometric but its point symmetry can be lower than that of the cyclic cluster. Let the locaUzed orthogonal crystalline orbitals (Wannier functions) be defined for the infinite crystal composed of snpercells. The corresponding BZ is L-times reduced (the snpercell is supposed to consist of L primitive unit cells). The Wannier functions W r — Ai) are now introdnced for the supercells with the translation vectors and satisfy the following equation ... 

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