248 Walter Gordy 



where /, m and ^7 are the spin, magnetic quantum number, and magnetic 

 moment (in nm units) of the nucleus, f^j is the nuclear magneton, r is the radius 

 vector from the nucleus to the electron, and d is the angle between r and H. 

 Although the nucleus may be regarded as located at a fixed point within the 

 molecule or crystal, the electron definitely cannot be so regarded. Hence, 

 to find the averaged or effective (A/r)eff acting on the electron in a molecular 

 orbital ip, we must average the above quantity over the orbital ip. Thus 



{^H\s^i^-^[^J^,^w{^J^Oco^''0-\)^*dT . (15) 



Since the coordinates are separable, we can write this equation as 



Av 



(A//)eff = (^- j /^2/^Z^73/^ <3 C0S2 - 1 >^„ (16) 



where 



/'■'K 



ipr* d7 



m 



and (3 cos^ — 1)^^ = WeO cos^ — \)fe* dr . 



To attack such a problem one can assume, as is usually done in other calculations 

 of molecular orbitals, that ip is a linear combination of atomic orbitals, ip^, 

 tPi, etc. We then readily get a part of the solution for we already know, at 



least to a fair approximation, \^/ and (3cos-0— 1>av for electrons 



various kinds of atomic orbitals. Expressions for these to various degrees of 

 approximation together with couphng constants actually measured are available 

 in standard texts on atomic spectra (17, 18). There is more to the problem than 

 this, however. Although an overlap or cross temi of the forni y>,lllr^)(3 cos- 

 — 1)^)* may possibly be neglected, an electron in an atomic orbital of atom B 

 might have a significant interaction with the nucleus of an adjacent atom A. 

 It is thus necessary to include terms of the form : 



Jv.,(;l-J(3cos2 0„,-l)v',r/T, (17) 



where /•„;, and O^j, are the coordinates of an electron on atom B referred to 

 the nucleus of atom A as the origin. The values of these terms are sensitive 

 functions of the hybridization of the atomic orbitals and of the direction of the 

 projections of the major lobes of the hybridized orbitals. As we get greater 

 skill in the procedure, these orientation-dependent couphng terms should give 

 additional information about orbitals of radicals. Expressed in convenient 

 numerical units equation (16) becomes: 



A// (in gauss) = 5.05 ^ (4) (3 cos^ d - 1>av, (18) 



^ X'' / AV 



where /Uj is in nm units and r is in A. 



