## Analytical expression of V_{(2,0)} for static crystal in NMR

The first-order quadrupole interaction is related to V_{(2,0)}:

In static NMR experiment, V_{(2,0)} is defined by:

Its analytical expression can be determined as follows:

(1) Select and copy the following green lines; then paste them into a cell of Mathematica, a software for numerical and symbolic calculations.

(2) Press "Ctrl-A" for select all; then
press "Shift-enter" for evaluate cells.

(Or in the menu bar, select Kernel > Evaluation > Evaluate Cells)

Using Mathematica-5 running with a 3-GHz processor, the analytical
expression of V_{(2,0)} is obtained in **5 seconds**.

(* V2pas is a row-matrix with 3 columns containing the 3 nonzero eigenvalues of the EFG expressed as a 2-nd rang spherical tensor, in eq unit *) V2pas = {{eta/2, Sqrt[3/2], eta/2}}; (* D2 is a reduced form (3 rows x 1 column) of the 2-nd order Wigner active rotation matrix *) D2 = { {Sqrt[3/8]*Sin[beta1]^2*E^(-2*I*alpha1)}, {(-1 + 3*Cos[beta1]^2)/2}, {Sqrt[3/8]*Sin[beta1]^2*E^(2*I*alpha1)} }; (* V20static is an expression *) V20static = FullSimplify[V2pas.ComplexExpand[D2]]; (* suppression of the double curve brackets {{}} of V20static *) Print[V20static[[1, 1]]]; Remove[V2pas, eta, D2, alpha1, beta1, V20static];