Wavefunctions plane wave

As a simple example of a QM/MM Car-Parinello study, we present here results from a mixed simulation of the zwitterionic form of Gly-Ala dipeptide in aqueous solution [12]. In this case, the dipeptide itself was described at the DFT (BLYP [88, 89 a]) level in a classical solvent of SPC water molecules [89b]. The quantum solute was placed in a periodically repeated simple cubic box of edge 21 au and the one-particle wavefunctions were expanded in plane waves up to a kinetic energy cutoff of 70 Ry. After initial equilibration, a simulation at 300 K was performed for 10 ps. 

Space-coordinate density transformations have been used by a number of authors in various contexts related to density functional theory [26,27, 53-64, 85-87]. As the free-electron gas wavefunction is expressed in terms of plane waves associated with a constant density, these transformations were introduced by Macke in 1955 for the purpose of producing modified plane waves that incorporate the density as a variable. In this manner, the density could be then be regarded as the variational object [53, 54]. Thus, explicitly a set of plane waves (defined in the volume V in and having uniform density po = N/V) ... 

In Eq. (4.3) the plane wave is multiplied by the internal rotational wavefunction Yjmji )- Multiplying Eq. (A.l) by this function, we obtain... 

The surface states observed by field-emission spectroscopy have a direct relation to the process in STM. As we have discussed in the Introduction, field emission is a tunneling phenomenon. The Bardeen theory of tunneling (1960) is also applicable (Penn and Plummer, 1974). Because the outgoing wave is a structureless plane wave, as a direct consequence of the Bardeen theory, the tunneling current is proportional to the density of states near the emitter surface. The observed enhancement factor on W(IOO), W(110), and Mo(IOO) over the free-electron Fermi-gas behavior implies that at those surfaces, near the Fermi level, the LDOS at the surface is dominated by surface states. In other words, most of the surface densities of states are from the surface states rather than from the bulk wavefunctions. This point is further verified by photoemission experiments and first-principles calculations of the electronic structure of these surfaces. 

Irrespective of whether the photon is considered as a plane wave or a wavepacket of narrow radial extension, it must thus be divided into two parts that pass each aperture. In both cases interference occurs at a particular point on the screen. When leading to total cancellation by interference at such a point, for both models one would be faced with the apparently paradoxical result that the photon then destroys itself and its energy hv. A way out of this contradiction is to interpret the dark parts of the interference pattern as regions of forbidden transitions, as determined by the conservation of energy and related to zero probability of the quantum-mechanical wavefunction. 

Bound states are readily included in the line shape formalism either as initial or final state, or both. In Eq. 6.61 the plane wave expression(s) are then replaced by the dimer bound state wavefunction(s) and the integration(s) over ky and/or kjj2 are replaced by a summation over the n bound state levels with total angular momentum J n or J . The kinetic energy is then also replaced by the appropriate eigen energy. In this way the bound-free spectral component is expressed as [358]... 

The first term in this formula describes the initial state of the system before the scattering, the wavefunction of the incident particle being a plane wave ... 

The expansion of the plane wave into partial waves yields F m(K)I m(f) components. If these are multiplied by loo V 71) t ie orthogonality condition for spherical harmonics then leads to the result that only the (/ = 0)-component remains. Hence, there is no dependence on k in equ. (4.74b). Similarly, in equ. (4.75b) only the ( f = l)-component is proved. Note, in addition, the typical overlap property of these integrals if a Coulomb wave with Z = 2 were used for the continuum electron the result of the integration would be proportional to Z — Zeff and would vanish for Zeff = 2. In other words, the dependence on Z — Zeff reflects the fact that the final and initial wavefunctions belong to different sets of... 

The expansion into partial waves is of importance for the desired wavefunction of an emitted electron, because it provides a classification into individual angular momenta ( which refer to the centre of mass of the atom. As a starting point, the expansion of a plane wave will be considered. If a certain origin is selected and the direction of k is chosen to agree with the z-axis (the quantization axis see... 

At this stage, the formalism can be implemented in a computer program. The applications described below [15-21] rely on the expansion of the electronic wavefunctions in terms of a large number of plane waves, as well as on the replacement of nuclear bare potentials by accurate norm-conserving pseudopotentials. The Local Density Approximation was used, with the Ceperley and Alder data for the exchange-correlation energy of the homogeneous electron gas. 

Using a plane wave representation for the electron wavefunction with 163 grid points and approximately 800 independent electronic and molecular configurations from the path integral molecular dynamics trajectories, we have also computed the density of states for the electron under different supercritical conditions of the solvents and the corresponding steady-state optical absorption spectra. The latter were computed within the dipolar approximation from the following expression within the Frank-Condon approximation ... 

Elementary quantum mechanics showed that a plane wave exp (iK r) has the same dependence on space and time as the wavefunction of a particle with momentum hK. 

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