Molecular orbital contours, point

If we choose the oxygen 2s orbital for bonding and leave the 2orbital nonbonding (from the symmetry point of view the opposite choice or a mixed orbital would do just as well actually if the two arbitals are close in energy, they mix), the MOs of the water molecule can be constructed as shown in Figure 6-18. These MOs are compared with the calculated contour diagrams of the water molecular orbitals in Figure 6-19. 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

A molecular surface defined as the contour surface of individual molecular orbitals such as HOMO and LUMO, other frontier orbitals, or localized and delocalized orbitals [Mezey, 1991b]. In practice, it is the collection of all those points r of the space where the value of the electronic wavefunction (r) of the considered molecular orbital is equal to a threshold value m, i.e. 

The boundary surfaces in Figure 3.11 are for the electron density, the probability of an electron being at any point in space. The electron density is given by the square of the wavefunction. Points with the same electron density will have the same numerical value for the wavefunction, but the wavefunction may be positive or negative. For example, if the probability of finding an electron at a particular point was one-quarter, 0.25, then the wavefunction at that point would have the value plus one half, + 0.5, or minus one half, -0.5, since both (0.5)2 and (-0.5)2 are equal to 0.25. The sign of the wavefunction gives its phase. To represent the wavefunction itself we can use the same contours as for electron density, but we also need to indicate the phase of the wavefunction. In atomic and molecular orbital representations, we shall use colour to show differences in phase. Is orbitals are all one phase and so are shown in one colour. 2p orbitals have two lobes, which are out... 

If a molecule is rotated by changing the position of the viewer (left mouse button rotation) then the molecule s position in the molecular coordinate system has not changed and another contour plot can be requested without recomputing the wave function. That is, many orbitals can be plotted after a single point ab initio or semi-empirical calculation. Any contour map is available without recomputation of the wave function. 

Fig. 3. Contour plots (in the plane z = 1 bohr) of the position-space electron density for the occupied n-orbitals for trans-C2oH22- Red (/ iO and green (broken) contours denote regions in which the wavefunction has opposite phases. The labels mark the positions of the nuclei (projected onto the plane z = 1). Orbital rr, has the highest binding energy and orbital n o is the least strongly bound. Note also the orientation of the axes (the x-axis points along the chain and the z-axis perpendicular to the molecular plane), as well as the marked distances d and h to which reference is made in the text...

From a conceptual point of view, it appears that polymer quantum chemistry is an ideal field for cooperation between condensed matter physicists and molecular quantum chemists. There exists a common interpretation in the discussions concerning orbital energies, orbital symmetry, and gross charges by chemists and solid-state physicists. These physicists use terms less familiar to the chemist, such as first Brillouin zone, dependence of wave function with respect to wave vector k (the one-electron wave function is called an orbital by the chemist), Fermi surfaces, Fermi contours, and density of states (DOS). 

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