Atomic orbitals angular components

The electronic Hamiltonian commutes with both the square of the angular momentum operator r and its z-component and so the three operators have simultaneous eigenfunctions. Solution of the electronic Schrddinger problem gives the well-known hydrogenic atomic orbitals... 

Hence, = I + 1 if k > 0 and = I — 1 if k < 0. Consequently, in the Dirac-Pauli representation and have definite parity, (—1) and (—1) respectively. It is customary in atomic physics to assign the orbital angular momentum label I to the state fnkm.j- Then, we have states lsi/2, 2si/2) 2ri/2, 2p3/2, , if the large component orbital angular momentum quantum numbers are, respectively, 0,0,1, ,... while the corresponding small components are eigenfunctions of to the eigenvalues 1,1,0,2,. 

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... 

Fig. IV-3.—Diagram representing orientations of the spin vectors of the two electrons and the orbital angular momentum vectors of the two electrons in the extreme Paschen-Back effect for an atom with two 2 electrons. The two spins orient themselves separately in the vertical magnetic field, as do also the two orbital angular momentum vectors. Each electron spin can assume orientations such that the component of angular momentum along the field direction is represented by the quantum number m, + or — and each orbital angular momentum may orient itself in such a way that the component of the orbital angular momentum along the field direction is represented by the quantum number m +1, 0, or —1.

In chapter 6 we described the theory of molecular electronic states, particularly as it applies to diatomic molecules. We introduced the united atom nomenclature for describing the orbitals, and pointed out that this was particularly useful for tightly bound molecules with small intemuclear distances, like H2. We also discussed the more conventional nomenclature for describing electronic states, which is based upon the assumption that the component of electronic orbital angular momentum along the direction of the intemuclear axis is conserved, i.e. is a good quantum number. The latter description is therefore appropriate for molecules in electronic states which conform to Hund s case (a) or case (b) coupling. 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used... 

The valeuce bands, which arise from p atomic orbitals, have sixfold degeneracy and contain at 0 K the six p-orbital Se valence electrons. Due to spiu-orbit coupliug, this degeneracy is split at k = 0 into a fourfold degenerate J = 312 band and a twofold degenerate J = 1/2 split-off valence band where Jis the total unit cell angular momeutum. For ki= = 3/2 band splits into two doubly degenerate components the heavy hole... 

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