Rotational overlap matrix element

The second type of matrix element may be called a "rotational overlap matrix element , because it depends only on the angular coordinates, and because it is responsible for the transfer of rotational motion from the parent ABC to the products AB and C. 

Overlap matrix elements in the x,y-plane involve the additional complication of the rotations of coordinates to allow the alignment of the primitive function with the locations of the sources in the unit cell n. Thus, one finds for the diagonal elements and ... 

In the molecular orbital description of the states of formaldehyde the active nonbonding orbital is mainly a pi(0) orbital. The inr states involve p (0) and Py C) orbitals. To have a significant value, the spin-orbit coupling matrix elements must include orbitals that have significant overlap. The effective matrix element for this particular interaction is of the type (py(0), (Rp (0)). The operator (R, rotates the p, orbital on the o.xygen into strong overlap with the Py orbital on oxygen and this is the portion of (R that can effectively mix the As and Mi states. The other components of (R are not so effective. 

The matrix elements for 5 needed for the spin projection methods discussed below involve multiple products of the a- overlap matrix and can be computed readily for UHF wavefunctions. Since the Hartree-Fock energy is invariant to unitary transformations among occupied orbitals of the same spin, the a and P orbitals can be rotated so that the a-fi overlap matrix is diagonal. These are termed corresponding orbitals, and their use greatly simplifies expressions for matrix elements ofS. ... 

Each of the MOs in Fig. 10-3 is symmetric or antisymmetric for some of the operations that apply to a tetrahedron. 02 is symmetric for rotations about the z axis by 27t/3, and also for reflection through the xz plane. This same reflection plane is a symmetry plane for 03 and 04, but neither of these MOs shows symmetry or antisymmetry for rotation about the z axis. Each MO contains one p AO and, perforce, has the symmetry of that AO. We shall refer to these as p-type MOs. Note that hydrogen Is AOs lying in the nodal plane of a p AO do not mix with that p AO in formation of MOs. This results from zero interaction elements in H, which, in turn, results from zero overlap elements in S. Note also that the MO 02 is the MO that we anticipated earlier on the basis of inspection of the matrix H. Because of phase agreements between the hydrogen Is AOs and the adjacent lobes of the p AOs, 0j, 02, and 03 are C-H bonding MOs. 

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