Wave functions illustrated contour plots

Fig. 12.7 Illustration of the rc-electron interaction of with Ni [3]. (Left) Contour plots of 7t orbitals from DFT calculations for in the gas phase and adsorbed on Ni(lOO). Solid and dashed lines indicate different phases of the wave function. Right) Schematic illustration of the k orbital interactions in the allylic configuration of the N -Ni adsorption system in terms of the atomic N2p and Ni34 orbitals. Reprinted with permission from ref [3]. Copyright 2004 Elsevier...

As reference data for comparison, the full quantum mechanical wave-functions of the coherent-type Gaussian form as in Eq. (6.106) are numerically propagated. The wavefunction is chosen to start at a point xi,X2) = (5.5,0.0) with zero initial momentum on the first diabatic state. The point of conical intersection is located at (xi,X2) = (2.5,0.0), the potential energy of which is exactly the same as the value at (xi, X2) = (5.5,0.0). This symmetric dynamics is performed to purely illustrate how the quantum wave functions are branched behind the conical intersection. The contour plots of the density of nuclear wavefunctions at selected times are given in Fig. 6.12 (solid curves). 

Consider the wave function for the S ground state of the helium atom. In the Hartree-Fock approximation, the two electrons move independently of each other, each in the field generated by the average motion of the electrons. The motion of one electron is therefore unaffected by the instantaneous position of the other, as illustrated in Figure 7.1. In this figure, we have plotted the Hartree-Fock wave function for the helium ground state with one electron fixed at a position 0.5 o from the nucleus, in a plane that contains the nucleus and the fixed electron. For clarity, combined surface and contour plots are presented, with the positions of the fixed electron and the nucleus marked by vertical bars. As expected, the Hartree-Fock contour plot consists of a set of concentric circles that are undistorted by the fixed electron - that is, the wave-function amplitude for one electron depends only on its distance to the nucleus, not on the distance to the other electron. 

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