Orientational quantum number

As mentioned in Section Wl, an electron has magnetism associated with a property called spin. Magnetism is directional, so the spin of an electron is directional, too. Like orbital orientation, spin orientation is quantized Electron spin has only two possible orientations, up or down. The spin orientation quantum number )... 

The principal quantum number n is the most important determinant of the radius and energy of the electron atomic orbital. The orbital shape quantum number I determines the shape of the atomic orbital. When / = 1, the atomic orbital is called an s orbital there are two s orbitals for each value of n, and they are spherically symmetric in space around the nucleus. When I = 2, the orbitals are called the p orbitals there are six p orbitals, and they have a dumbbell shape of two lobes that are diametrically opposed. When I = 3 and 4, we have 10 d orbitals and 14 f orbitals. The orbital orientation quantum number m controls the orientation of the orbitals. For the simplest system of a single electron in a hydrogen atom, the most stable wave function Is has the following form ... 

An electron has a spin S of a single unpaired electron in an atomic or molecular orbital has two possible orientations (parallel or antiparellel) with respect to an applied magnetic field. These are characterized by the spin orientation quantum numbers (sometimes referred to as M,). For S = Sj can be... 

The magnetic quantum number (mi) is an integer from -I through 0 to +1. It prescribes the orientation of the orbital in the space around the nucleus and is sometimes called the orbital-orientation quantum number. The possible values of an orbital s magnetic quantum number are set by its angular momentum quantum number that is, I sets the possible values of m,. An orbital with I = 0 can have only ni/ = 0. However, an orbital with / = 1 can have any one of three m/ values, - 1, 0, or -fl thus, there are three possible orbitals with / = 1, each with its own orientation. Note that the number of possible m/ values equals the number of orbitals, which is 2/ -f 1 for a given I value. 

Fig. 2.26 Dependence of M on the orientational quantum number M for transitions with AM = 0, 1...

The magnetic quantum number is also known as the orientational quantum number. It designates the orientation of orbitals in space relative to each other and distinguishes orbitals within a subshell from each other. It is called the magnetic quantum number because the presence of a magnetic field can result in the appearance of additional lines among those emitted by electronically excited atoms (i.e., in atomic emission spectra). The possible values of m in a subshell with azimuthal quantum number Z are given by m, = +1, +(Z - 1),..., 0,..., -(Z - 1), -Z. As examples, for Z = 0, the only possible value of m, is 0, and for I = 3,m, may have values of 3, 2,1,0, -1, -2, -3. 

The orbital angular momentum quantum number 1 can take the values 0,1,2,3,... (also know as azimuthal quantum number) and the magnetic quantum number m must be in —/, — / + 1,..., / (also known as orientational quantum number). The eigenfunctions can be efficiently constructed through the definition of ladder operators, which is standard in nonrelativistic quantum mechanics and therefore omitted here. The general expression for the spherical harmonics reads [70]... 

In the absence of external fields, the rotational energies do not depend on M, as Eq. (23) implies, and all levels are (27-1- l)-fold degenerate. However, when an external field is applied, this degeneracy is lifted, and the energy depends on the space orientation quantum number M. A similar condition holds for symmetric and asymmetric tops. For rotational absorption of radiation, the selection rule... 

As apparent from Eqs. (29) and (31), the energy levels increase with K for a prolate rotor (A> B) and decrease with K for an oblate rotor (C < B). There are 7 -I-1 different rotational levels for each J value since the energy does not depend on the sign of K. The rotational levels for / 2 3 are illustrated in Fig. 7. Furthermore, in the absence of external fields each level is (27 - - l)-fold degenerate in the space orientation quantum number M. For absorption of radiation, the important selection rules are... 

---