Abdelkrim Boukahil

Accuracy of the coherent potential approximation for a one-dimensional Frenkel exciton system with a Gaussian distribution of fluctuations in the optical transition frequency

Abstract We investigate the accuracy of the coherent potential approximation (CPA) for a one-dimensional Frenkel exciton system with nearest-neighbor interactions and a Gaussian distribution of fluctuations in the optical transition frequency. The CPA values of the integrated density of states and the inverse localization length are shown to be in excellent agreement with the results of mode-counting studies carried out on arrays of 107–108 sites. We also consider the asymptotic behavior of the inverse localization length and show that it can be approximated by the reciprocal of the decay length of an eigenstate localized about a single, strongly perturbed site in an otherwise perfect lattice.

MATRIX ELEMENTS OF A HYPERBOLIC VECTOR OPERATOR UNDER SO(2,1)

We deal here with the use of Wigner–Eckart type arguments to calculate the matrix elements of a hyperbolic vector operator by expressing them in terms of reduced matrix elements. In particular, we focus on calculating the matrix elements of this vector operator within the basis of the hyperbolic angular momentum whose components , , satisfy an SO(2,1) Lie algebra. We show that the commutation rules between the components of and can be inferred from the algebra of ordinary angular momentum. We then show that, by analogy to the Wigner–Eckart theorem, we can calculate the matrix elements of within a representation where and are jointly diagonal.

THE NUCLEAR BORN–OPPENHEIMER METHOD APPLIED TO NUCLEAR COLLECTIVE MOTION

We deal with the application of the nuclear Born–Oppenheimer (NBO) method to the study of nuclear collective motion. In particular, we look at the description of nuclear rotations and vibrations. The collective operators are specified within the NBO method only to the extent of identifying the type of collective degrees of freedom we intend to describe; the operators are then determined from the dynamics of the system. To separate the collective degrees of freedom into rotational and vibrational terms, we transform the collective tensor operator from the lab fixed frame of reference to the frame defined by the principal axes of the system; this transformation diagonalizes the tensor operator. We derive a general expression for the NBO mean energy and show that it contains internal, collective and coupling terms. Then, we specify the approximations that need to be made in order to establish a connection between Bohrs collective model and the NBO method. We show that Bohrs collective Hamiltonian can be recovered from the NBO Hamiltonian only after adopting some rather crude approximations. In addition, we try to understand, in light of the NBO approach, why Bohrs collective model gives the wrong inertial parameters. We show that this is due to two major reasons: the ad hoc selection of the collective degrees of freedom within the context of Bohrs collective model and the unwarranted neglect of several important terms from the Hamiltonian.