Nodal patterns

The Most Elementary Molecular Orbital Models Contain Symmetry, Nodal Pattern, and Approximate Energy Information... 

Fjj,v Cy i 8i Zy Sjj y Cy j as initial guesses for the Cy i. Using only the one-eleetron part of the Hamiltonian to determine initial values for the LCAO-MO eoeffieients may seem like a rather severe step it is, and the resultant Cy i values are usually far from the eonverged values whieh the SCF proeess eventually produees. However, the initial Cy i obtained in this manner have proper symmetries and nodal patterns beeause the one-eleetron part of the Hamiltonian has the same symmetry as the full Hamiltonian. 

The closed-shell singlet ( A ) has no electrons in the tt orbitals, and the bond lengths indicate only a small amount of delocalization of the six tt electrons into the empty p-TT orbital at the carbenic carbon. In both A", 13A", and 23A" the tt SOMO has the nodal pattern expected for a pentadienyl NBMO mixed... 

M. E. Kellman Following up on Stuart Rice s point regarding scars of periodic orbits, are there abrupt changes in reaction rates as nodal patterns of the vibrational wave functions change in the regular regime, for example in DCO, where the local-mode nodal pattern breaks down ... 

R. Schinke No, we did not clearly see abrupt changes in the dissociation rate as nodal patterns of the underlying wave functions change. The rates fluctuate strongly with energy, and since there is no clear-cut assignment, it is difficult to recognize any schematic trends. 

Because the excited vibrational wavefunctions usually have a complicated nodal pattern (examples are given in Figure 14.4) it is rather difficult to draw specific conclusions about the time dependence of the Cif(t). 

Figure 15. Radial charge density plot for the resonant p-type virtual orbital for dilation angles 9 = 0.0 and 6 = 90pt (0.42 radians) in e-Be scattering. The role of optimal theta in the accumulation of electron density near the nucleus is clearly seen. In the inset, the maximum is seen to occur at rmaz — 2.5 a.u., very close to that for the rmax of the outer valence 2s orbital, seen in fig. 14- Though a cursory look at the nodal pattern identifies this as a 4P orbital, the dominant contribution to the charge density distribution is mainly of 2p-iype.

Figure 19. A plot of the charge density obtained from a linear combination of all the resonant amplitudes in e-Mg scattering withx(r) = Xi(r)+Xn(r)+Xni(r), where the individual amplitudes are considered to describe different parts of the same resonant wave packet. The nodal pattern for 8 = 0.0 favors its identification as the lowest Sp-iype unoccupied orbital of Mg. The role of optimal theta (8, = 0.12 radians) in shifting the electron density near the nucleus is clearly seen.

For a second example, let s return to acetylene on Pt(lll), specifically in the twofold and fourfold geometries.29 In the twofold geometry, we saw earlier (from the decomposition of the DOS) that the most important acetylene orbitals were irff and it. These point toward the surface. Not surprisingly, their major interaction is with the surface z2 band. But t and 7r interact preferentially with different parts of the band, picking out those metal surface orbitals which have nodal patterns similar to those of the adsorbate. Diagram 116 shows this in the twofold geometry at hand the trff orbital interacts better with the bottom of the surface z2 band and the ir0 with the top of that band. 

Fig. 11. Nodal patterns and nuclear magnetic moment alignments for a-type lower-energy triplet states for a traditional X-H-Y hydrogen bond.

Four different nodal patterns are illustrated in Fig. 11 for a complex with a traditional X-H...Y hydrogen bond. How would wavefunctions with such patterns contribute to the reduced two-bond spin-spin... 

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