Perturbation Theory Treatment of Quadrupole Term

First- and second-order energy corrections then are  

In the electron-spin quantization system (x, y, z ), the quadrupole interaction matrix is  

We next transform the P matrix, using the a,b,c-matrix and compute P33, F3 = Pn2 + P232, and F4 = (Pxx - P22)2 + 4P,22, simplifying the expressions using the relations  

P33 = 3(c3533 + b3S23 + a3SX3)2P — P + (c353l + b3S2X + Ct3SXX)2Pt] 

F3a = (c3S33 + b3S23 + a3SX3)2—(c3S33 + b3S23 + a3SX3)4 

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