Beyond Euler angles: exploiting the angle-axis parametrization in a multipole expansion of the rotation operator

Area of Science:

• Physics
• Quantum Mechanics
• Computational Chemistry

Background:

• Euler angles present computational challenges and indirect links to experimental parameters.
• Angle-axis representations, particularly quaternions, are effective alternatives in rf pulse design.

Purpose of the Study:

• To demonstrate the advantages of angle-axis parameters in representing rotations.
• To simplify the analysis of experimental parameters in rotation dynamics.

Main Methods:

• Utilized multipole operator expansion of the rotation operator D(Phi, n).
• Applied Clebsch-Gordan expansion to rotation matrices D(MM')(J)(Phi, n).
• Analyzed coefficients involving spherical harmonics Y(lambdamu)(n) and Gegenbauer polynomials C(2J-lambda)(lambda+1)(cosPhi/2).

Main Results:

• Clebsch-Gordan expansion coefficients isolate spherical harmonics of the rotation axis (n) and Gegenbauer polynomials of the rotation angle (Phi).
• Coherence order changes are isolated in Clebsch-Gordan coefficients.
• Rotation angle is isolated in Gegenbauer polynomials.

Conclusions:

• The Clebsch-Gordan expansion provides direct access to experimental parameters.
• This method simplifies the interpretation of rotation dynamics in physical systems.
• Angle-axis parameters offer a more intuitive and computationally tractable approach compared to Euler angles.