One molecule per unit cell

In the special case of crystals with one molecule per unit cell eqns (3.97) have the simpler form 

In this case the Davydov splitting is absent, and when molecular terms are degenerate, a splitting occurs which is analogous to the Bethe splitting (see Section 2.1). 

If only two excited states contribute, the quantities M(k) can be given explicitly  

The wavefunctions corresponding to crystal excited states have the form 

Comparing (3.99) and (3.100) we can see that the contribution of the second excited intramolecular state to the excitonic energy, which is near to the energy of the first excited intramolecular state, becomes relevant only if the following inequality holds 

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In order to demonstrate the NDCPA a model of a system of excitons strongly coupled to phonons in a crystal with one molecule per unit cell is chosen. This model is called here the molecular crystal model. The Hamiltonian of... 

To verify effectiveness of NDCPA we carried out the calculations of absorption spectra for a system of excitons locally and linearly coupled to Einstein phonons at zero temperature in cubic crystal with one molecule per unit cell (probably the simplest model of exciton-phonon system of organic crystals). Absorption spectrum is defined as an imaginary part of one-exciton Green s function taken at zero value of exciton momentum vector... 

Since plane waves are delocalised and of infinite spatial extent, it is natural to perform these calculations in a periodic environment and periodic boundary conditions can be used to enforce this periodicity. Periodic boundary conditions for an isolated molecule are shown schematically in Fig. 8. The molecular problem then becomes formally equivalent to an electronic structure calculation for a periodic solid consisting of one molecule per unit cell. In the limit of large separation between molecules, the molecular electronic structure of the isolated gas phase molecule is obtained accurately. 

The crystal structure of Pb2Li7 is hexagonal with one molecule per unit cell, a = 4.751 and c = 8.589 A. Figure 9.40 shows a projection of the unit cell. The heights along c are ... 

The crystal structure of Ni2Al3 is hexagonal, with one molecule per unit cell, D3d, C 3m, a0 = 4.036, and c0 = 4.901 A. The structure is the same as La203 (Section 5.5.10) with different spacings. A projection of the cell is shown in Figure 9.41a. Aluminum atoms alone are in A positions, with Ni and A1 in B and C positions. These layers are filled. The heights along c0 are ... 

How many lines there are There are as many lines as there are orbitals in the unit cell. Each line is a band, generated by a single orbital in the unit cell. In the case of CO, there is one molecule per unit cell, and that molecule has well-known 4o, lx, 5unit cell has four Ni atoms. Each has five 3d, one 4s, and three 4p basis functions. We see some, but not all, of the many bands these orbitals generate in the energy window shown in Fig. 7. 

Rh (111) C6Hg+CO c(2V/3x4) rect Benzene is coadsorbed with CO, each with one molecule per unit cell, both centered over 3-fold hep sites, benzene is parallel to and 2.25 0.05 A above the surface. CO is 1 to the surface and the metal-carbon spacing is 1.50 0.05 A. The benzene molecule has an in-plane Kekule distortion, with alternating long and short bonds. LEED/14/... 

A crystal with only one molecule per unit cell does not possess translational vibrations. 

In a subsequent paper, Munn [98] showed that the frequency-dependent local-field tensors accounted for the shift of the poles of the linear and nonlinear susceptibilities from the isolated molecular excitation frequencies to the exciton frequencies. The treatment also described the Davydov splitting of the exciton frequencies for situations where there is more than one molecule per unit cell as weU as the band character or wave-vector dependence of these collective excitations. In particular, the direct and cascading contributions to x contained terms with poles at the molecular excitation energies, but they canceled exactly. Combining both terms is therefore a prerequisite to obtaining the correct pole structure of the macroscopic third-order susceptibility. Munn also demonstrated that this local field approach can be combined with the properties of the effective or dressed molecule and can be extended to electric quadrupole and magnetic dipole nonlinear responses [96]. 

Applying this general expression (2.44) to the case of a crystal with one molecule per unit cell and taking into account only one molecular excited state /, one obtains... 

As a second example of the application of the above described theory we consider crystals with one molecule per unit cell. We wish to consider polaritonic states with frequencies near to the nondegenerate electronic excitation of the molecule, where the transition from the ground to the excited state is dipole allowed. 

As was shown by Born and Huang (4) in cubic crystals with one molecule per unit cell the tensor Q"f is reduced to the scalar QtJ = (An/ yv)dlJ where v is the volume of the unit cell, if we ignore spatial dispersion. Moreover, a - = aSij so that eqn (5.3) yields the tensor Ai3 = Adij where A = [1 — (47ra/3v). Substituting this expression into eqn (5.6) we obtain c,3 = eSl3 where... 

For crystals having one molecule per unit cell the polarization per unit volume... 

In this approximation (the additive refraction or mean polarizability approximation) eqn (5.42) with the addition of eqn (5.37) fully determines the dependence of the dielectric constant tensor on the impurity concentration c. The optical properties of mixed crystals with large impurity concentrations are discussed in Section 5.6. As we did above for crystals with one molecule per unit cell, we shall discuss here the case of small values of c when we can ignore terms of the order of c2, c3, etc. in the expansion of the tensor (5.42) in powers of c. Then we obtain ... 

Crystals with one molecule per unit cell are most easily analyzed. Here eqn (6.24) is of the form... 

For a crystal with a one molecule per unit cell, these terms are ... 

In discussing infrared absorption spectra, we refer, first of all, to the papers by Dows and Schettino (54) and of Schettino and Salvi (55). Dows and Schettino (54) investigated the CO2 crystal spectrum in the frequency region corresponding to the combination tone of the intramolecular vibrations v and 1/3 u /y3 Ri 3720 cm-1). Schettino and Salvi (55) measured the infrared (IR) spectra of N2O and OCS crystals. The CO2 and N2O molecules are linear, have no permanent dipole moments, and form a simple cubic lattice upon crystallization. This lattice has four molecules per unit cell, which are oriented along the axes of a tetrahedron. The OCS molecule is also linear, but it forms a crystal of the trigonal system with one molecule per unit cell. 

In the low-temperature phase IV, NH4CI crystals belong to the cubic class Td and have one molecule per unit cell. Though these are not molecular crystals,... 

In later studies, the Raman spectra and corresponding infrared spectra indicated that the primary differences between the I and I/j forms of native cellulose were in the pattern of hydrogen bonding. Furthermore, the Raman spectra of the two forms raise questions as to whether the structures can possess more than one molecule per unit cell since there is no evidence of any correlation field splittings of any of the bands in the spectra of the two forms. 

The linear azide anions of RbNa and CsNa take random orientations (or rotate about the central nitrogen) to possess a spherical symmetry at high temperature. Then the tetragonal structure (a form) transforms to a cubic (j3 form) with one molecule per unit cell [9]. 

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