Wigner rotation

The interaction energy can be written as an expansion employing Wigner rotation matrices and spherical hamionics of the angles [28, 130], As a simple example, the interaction between an atom and a diatomic molecule can be expanded hr Legendre polynomials as... 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

XII. The Adiabatic-to-Diabatic Transformation Matrix and the Wigner Rotation Matrix... 

Xn. THE ADIABATIC-TO-DIABATIC TRANSFORMATION MATRIX AND THE WIGNER ROTATION MATRIX... 

The obvious way to form a similarity between the Wigner rotation matrix and the adiabatic-to-diabatic transformation mabix defined in Eqs. (28) is to consider the (unbreakable) multidegeneracy case that is based, just like Wigner rotation matrix, on a single axis of rotation. For this sake, we consider the particular set of T matrices as defined in Eq. (51) and derive the relevant adiabatic-to-diabatic transfonnation matrices. In what follows, the degree of similarity between the two types of matrices will be presented for three special cases, namely, the two-state case which in Wigner s notation is the case, j =, the tri-state case (i.e.,7 = 1) and the tetra-state case (i.e.,7 = ). 

It is expected that for a certain choice of paiameters (that define the x matrix) the adiabatic-to-diabatic transformation matrix becomes identical to the corresponding Wigner rotation matrix. To see the connection, we substitute Eq. (51) in Eq. (28) and assume A( o) to be the unity matrix. 

C2H-molecule (1,2) and (2,3) conical intersections, 111-112 H3 molecule, 104-109 Wigner rotation matrix and, 89-92 Yang-Mills field, 203-205 Aharonov-Anandan phase, properties, 209 Aharonov-Bohm effect. See Geometric phase effect... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 91-92 Multiple independent spawning (MIS), direct molecular dynamics, non-adiabatic coupling, 402... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Time-dependent ground state (TDGS), molecular systems, component amplitude analysis, near-adiabatic limit, 220-224... 

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