ADDITION OF TWO ANGULAR MOMENTA

The addition of two angular momenta (formula of the type (10.4)) may be directly generalized to cover the case of an arbitrary number of momenta. However, in such a case it is not enough to adopt the total momentum and its projection for the complete characterization of the wave function of coupled momenta. Normally, the quantum numbers of intermediate momenta must be exploited too. Moreover, these functions depend on the form (order) of the coupling between these momenta. The relationships between the functions, belonging to different forms of coupling of their momenta, may be found with the aid of transformation matrices. 

Later, we will also encounter angular momentum from molecular rotation and I rom nuclear spins. Consider the general case of vector addition of two angular momenta, which we will denote as Ji and J2 ... 

To sununarize, we have shown that the addition of two angular momenta characterized by quantum numbers f and ji results in a total angular momentum whose quantum number J has the possible values... 

In the notation of Section 11.4, we are dealing with the addition of two angular momenta with quantum numbers ji = I and 72 = to give eigenfunctions with total angular-momentum quantum numbers / = 1 and / = 0. The coefficients in (11.57) to (11.60) correspond to the coefficients C in (11.33) and are examples of Clebsch-Gordan coefficients.]... 

Here we consider the nature of the angular momentum states that result from the vector addition of two angular momenta Jj and J2 to form a resultant J,... 

Two coupling coefficient matrices for addition of three angular momenta... 

Finally, we consider the addition of spin angular momenta. We start with the case of two spin-1/2 particles, as the most common situation encountered in the study of solids ... 

Fig,3.5. Vector model for the addition of the angular momenta of two electrons in L-S coupling. 

The general Zeeman hamiltonian thus will contain two independent g values. A convenient separation of the product multiplicity may be based on spherical coupling coefficients. To this aim the U x U product space is put into correspondence with the space that results from the addition of two j — 3/2 angular momenta. Their symmetrized square yields P and F products, which subduce Tl and A2 + Tt + T2 resp. 

To add more than two angular momenta, we apply (11.39) repeatedly. Addition of y, = 1 and ji = 2 gives the possible quantum numbers 3,2, and 1. Addition of y s to each of these values gives the following possibilities for the total-angular-momentum quantum number ... 

For systems of two or more particles orbital and spin angular momenta are added separately by the rules of vector addition. Integral projections of the shorter vector on the direction of the longer vector, are added to the long vector, to give the possible eigenvalues of total angular momentum. 

Let us consider au atom with two s electrons, with different total quantum numbers for example, a beryllium atom with one valence electron in a 2s orbital and the other in a 3s orbital, in addition to the two electrons in the K shell. The orbital angular momenta of the two valence electrons are zero (h = 0, k = 0), and accordingly the resultant angular momentum is zero (L = 0). Each of the two electrons has spin quantum number (si = sz = ), and each spin angular mo-... 

To understand the formation of a triple bond between two CH fragments it is noted that no more than two p-electrons can be directed along the C-C (z) axis, allowing the formation of a linear H(s)1C(sp)1C(sp)1H(s)1 molecule. The remaining pair of p-electrons circulate in the xy-plane with opposite angular momenta. There is no barrier to rotation. The conventional description of a triple bond in terms of one a and two 7r interactions is inconsistent, not only for the reasons already discussed, but also because it violates the conservation of total angular momentum when assigning 2 pairs of p-electrons to px and py orbitals, in addition to the p-density in the sp hybrid orbitals. There is a total of only four p-electrons (l = 1) in the system. 

---