Plane-wave representation systems

For the purposes of this review it is convenient to focus attention on that class of molecules in which the valence electrons are easily distinguished from the core electrons (e.g., -n electron systems) and which have a large number of vibrational degrees of freedom. There have been several studies of the photoionization of aromatic molecules.206-209 In the earliest calculations either a free electron model, or a molecule-centered expansion in plane waves, or coulomb functions, has been used. Only the recent calculation by Johnson and Rice210 explicitly considered the interference effects which must accompany any process in a system with interatomic spacings and electron wavelength of comparable magnitude. The importance of atomic interference effects in the representation of molecular continuum states has been emphasized by Cohen and Fano,211 but, as far as we know, only the Johnson-Rice calculation incorporates this phenomenon in a detailed analysis. 

Nano-scale and molecular-scale systems are naturally described by discrete-level models, for example eigenstates of quantum dots, molecular orbitals, or atomic orbitals. But the leads are very large (infinite) and have a continuous energy spectrum. To include the lead effects systematically, it is reasonable to start from the discrete-level representation for the whole system. It can be made by the tight-binding (TB) model, which was proposed to describe quantum systems in which the localized electronic states play an essential role, it is widely used as an alternative to the plane wave description of electrons in solids, and also as a method to calculate the electronic structure of molecules in quantum chemistry. 

From the point of view of general methodology, several comments are in order. First, the appearance of the Fourier-Bessel transform in the stmcture function [Eq. (20)] reflects on the breakdown of translational invariance, which is prevalent in the case of the bulk. Second, the different symmetries of spherically projected structure functions for the finite system and of plane wave structures for the bulk system are crucial for a proper representation of the cluster excitations. Third, the discrete eigenvectors k n are determined by the boundary conditions. Fourth, the energies kin) are discrete. However, the complete spectrum for a fixed value of n containing 1 = 0, 1, 2,... branches would form a continuous smooth curve. 

Plane waves are often considered the most obvious basis set to use for calculations on periodic systems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear mmhination of plane waves which differ by reciprocal lattice vectors ... 

Figure 15. Sketch illustrating the different types of model systems and their representation within the three common types of ab initio modeling methods. A 3D system is a crystal, a 2D system is a slab, a ID system is polymer-like, and a OD system is a molecule. For periodic calculations, infinitely repeating directions in a model are indicate by black lines. Plane wave calculations require the system to be repeated in all three directions. In this case, slab models are created within a supercell containing vacuum space. Periodic LCAO based slabs are oidy repeated along the two lateral directions, and are finite in the third direction. Clusters may be approached by all three modehng methods.

We consider a vector plane wave of unit amplitude propagating in the direction (/ g, ttg) with respect to the global coordinate system. Passing from spherical coordinates to Cartesian coordinates and using the transformation rules imder coordinate rotations we may compute the spherical angles / and a of the wave vector in the particle coordinate system. Thus, in the particle coordinate system we have the representation... 

We assume that in (4.38) and (4.39), all velocities are measured with respect to the same coordinate system (at rest in the laboratory) and the particle velocity is normal to the shock front. When a plane shock wave propagates from one material into another the pressure (stress) and particle velocity across the interface are continuous. Therefore, the pressure-particle velocity plane representation proves a convenient framework from which to describe the plane Impact of a gun- or explosive-accelerated flyer plate with a sample target. Also of importance (and discussed below) is the interaction of plane shock waves with a free surface or higher- or lower-impedance media. 

In the final step of the EDA, the wave function of the molecule relaxes to its optimal form yielding the orbital interaction term AEoa. This term can be further partitioned into contributions by the orbitals belonging to different irreducible representations of the point group of the interacting system. Thus, it is possible to give energy contributions of the a and tt bonding contributions to a bond that has a mirror plane. More details about the EDA method in the framework of DFT can be found in a recent review article. ... 

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