Atomic orbitals angular form

In the alkaline earth atom ease, the polarized orbital pairs are formed by mixing the ns and np orbitals (aetually, one must mix in equal amounts of pi, p.i, and po orbitals to preserve overall S symmetry in this ease), and give rise to angular eorrelation of the eleetron pair. Use of an (n+l)s2 CSF for the alkaline earth ealeulation would eontribute in-out or radial eorrelation beeause, in this ease, the polarized orbital pair formed from the ns and (n+l)s orbitals would be radially polarized. 

For diatomic molecules, there is coupling of spin and orbital angular momenta by a coupling scheme that is similar to the Russell-Saunders procedure described for atoms. When the electrons are in a specific molecular orbital, they have the same orbital angular momentum as designated by the m value. As in the case of atoms, the m value depends on the type of orbital. When the internuclear axis is the z-axis, the orbitals that form a bonds (which are symmetric around the internuclear axis) are the s, pz, and dzi orbitals. Those which form 7r bonds are the px, p, dlz, and dyi orbitals. The cip-y2 an(i dxy can overlap in a "sideways" fashion with one stacked above the other, and the bond would be a 8 bond. For these types of molecular orbitals, the corresponding m values are... 

The analytical expressions of the various wave functions, for the ground state (n = 1) and for orbitals 2p and 3d, are listed in figure 1.15. The same figure also shows the form (actually, the contour surface) attained by the five ADs of sub-level 3d and, for comparative purposes, the three ADs of sublevel 2p and the AD s for the ground state (n = 1). The orientation of the atomic orbitals depends on angular factor xf/i and not on principal quantum number n. Orbitals of the same / thus have the same orientation, regardless of the value of n. 

In Eq. (14), /max is the maximum of the orbital angular momentum quantum numbers of the active electron in either the initial or final states, I nl, n l ) is the radial transition integral, that contains only the radial part of both initial and final wavefunctions of the jumping electron and a transition operator. Two different forms for this have been employed, the standard dipole-length operator, P(r) = r, and another derived from the former in such a way that it accounts explicitly for the polarization induced in the atomic core by the active electron [9],... 

The molecular orbitals (MOs) are formed by the linear combination of atomic orbitals (LCAO-MO method). For diatomic molecules, the component of the angular momentum (A) in the direction of the bond axis is now important. The energy states are expressed by the symbol... 

The pictures normally used to represent these angular forms are shown in Fig 4.4. Such diagrams of the shape of atomic orbitals can be interpreted in two ways ... 

Figure 4.7 shows the forms of other radial wavefunctions for orbitals with n up to three. It can be seen that functions for r orbitals are non-zero at the nucleus, whereas the other orbitals are zero there because of the centrifugal repulsion term. The other obvious feature is that the number of nodes in these functions increases with n. In fact, the number of radial nodes is equal to n - / — 1. If we recall that the angular functions can also have nodes, the number being /, then it is apparent that the total number of nodes in any atomic orbital is equal to n -1 this increases with the energy, as we have found with the wavefunctions for other systems. 

---