Wigner rotation matrices for second-rank spherical tensor

A. Wigner active rotation matrix

For static crystal, the orientation of the static magnetic field B₀ in the principal-axis system (主軸坐標系) of the EFG tensor (X_PAS, Y_PAS, Z_PAS) is described with the Euler angles α, β, and γ (see EasySpin, Wikipedia, Wolfram MathWorld).

The component V_(2,k) of the second-rank spherical tensor V in the obs coordinate frame is:

The Wigner matrix for active rotation is:

Reference

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B. Wigner passive rotation matrix

For static crystal, the orientation of the principal-axis system (主軸坐標系) of the EFG tensor (X_PAS, Y_PAS, Z_PAS) in the static magnetic field B₀ (x_obs, y_obs, z_obs) is described with the Euler angles α, β, and γ.

The component V_(2,k) of the second-rank spherical tensor V in the obs coordinate frame is:

The Wigner matrix for passive rotation is:

Reference

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   The Wigner passive rotation matrix presented in this book is D^{(2, passive,W)}(α, β, γ).

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   The three angles α, β, and γ are the Euler angles by which the laboratory frame can be brought into coincidence with the principal axis system.

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C. Conclusion

Both Equations 1 and 2 represent a passive rotation of the EFG tensor:
- from the PAS coordinate frame to the obs coordinate frame for Equation 1,
- from the obs coordinate frame to the PAS coordinate frame for Equation 2.

The Euler angles α, β, and γ in Equation 1 correspond to the Euler angles γ, β, and α in Equation 2, respectively.

Sometimes, Equation 2 is written in the following way:

The matrix D^{(2,passive,W)}(α, -β, γ) in Equation 3 is the transpose of D^{(2,passive,W)}(γ, β, α) defined in Equation 2.

Reference

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