Hamiltonian Bloch-modified

Here the Bloch-modified Hamiltonian HB is obtained by integration by parts of the kinetic energy integral,... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

Some supplementary remarks to the theory of Penn might be appropriate here. There are additional effects which are of relevance if a more quantitative theory of the photoemission process from an adsorbate-covered surface is envisaged. The first point is that the Anderson model as applied to chemisorption is a clearly oversimplified model to describe real metal-adsorbate systems. Besides overlap effects due to the nonorthogonality of the states k) and a), there are several interaction effects which are neglected in the Hamiltonian, Eq.(5). The adsorbed atom, for instance, may act as a scattering centre for the metal electrons and thus modify the Bloch wave functions characteristic of the free substrate. This can be accounted for by adding a term... 

In Section 2 of this paper a brief account of the 3D q -HO is given, while in Section 3 the appUcation of the periodic orbit theory of Balian and Bloch to metal clusters is briefly described. The predictions of the 3D g-HO model axe compared to the restrictions imposed by the theory of Balian and Bloch in Section 4, while in Section 5 a modified Hamiltonian for the 3D q-HO is introduced, allowing for full agreement with the theory of Balian and... 

Bloch. Numerical details concerning the modified 3D g-HO Hamiltonian are given in Section 6, together with a study of supershells in the firamework of this Hamiltonian. In Section 7 a variational method leading from the usual harmonic oscillator to the Morse oscillator is introduced, while in Section 8 this method is applied for deriving the modified Hamiltonian introduced in Section 5 firom the original 3D g-HO Hamiltonian. Finally in Section 9 a discussion of the present results and plans for future work are given. 

In the quantum theory the key elements are the Hamiltonian and the Bloch wave functions. Nanosolid densification modifies the wave functions slightly as, in this case, no chemical reaction occurs. 

---