Angular momentum normalization factor

N is a normalization factor which ensures that = 1 (but note that the are not orthogonal, i. e., 0 lor p v). a represents the orbital exponent which determines how compact (large a) or diffuse (small a) the resulting function is. L = 1 + m + n is used to classify the GTO as s-functions (L = 0), p-functions (L = 1), d-functions (L = 2), etc. Note, however, that for L > 1 the number of cartesian GTO functions exceeds the number of (27+1) physical functions of angular momentum l. For example, among the six cartesian functions with L = 2, one is spherically symmetric and is therefore not a d-type, but an s-function. Similarly the ten cartesian L = 3 functions include an unwanted set of three p-type functions. 

Oppenheimer approximation, hydrogen molecule, minimum basis set calculation, 546-550 Normalization factor, angular-momentum-... 

When the angular momentum of an atom is due entirely to the orbital motion of the electrons the value of the 0-factor is 1, and when it is due entirely to the spin of the electrons the value of the 0-factor is 2. For example, the normal state of the nitrogen atom is 4 f hence the 0-factor for the normal nitrogen atom is 2. 

The above interpretation is, in essence, the result of the limiting case of Eqs. (D.15) and (D.14), as may be confirmed by comparison between (D.47), (D.48) and (D.31). It will emerge that in the classical limit the total population nj will be replaced by the total probability W, and the normalizing factor (2J + l)-1/2 will be replaced by (4 r)-1/2. The latter replacement is due to the fact that, in passing from the quantum to the classical approach, the number of possible spatial orientations of the angular momentum, equalling 2J + 1, must be replaced by the full solid angle 4n. 

Equations (5.106) and (5.104) give the explicit formula for the normalized theta factor in the angular-momentum eigenfunctions. Using (5.106), we construct Table 5.1, which gives the theta factor in the angular-momentum eigenfunctions. 

These three functions are eigenfunctions of the angular momentum operator Li with eigenvalues wj = 0, 1. The normalization factor N is (3/4ir). Under the symmetry operations of Eq. 7.10(b) is totally symmetric, thus it is the 2po orbital that transforms like S" " in the site symmetry (Table 7.1). The functions 0(2p+i) and 0(2p i) are partners in the ir representation of C , and give rise to molecular functions of... 

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