Matrix orthogonal 97 - labeled

The entity of all nine (real-valued) components Ry defines the rotation R, where the first or left index (i) labels the row and the second or right index (y) labels the column of the corresponding entry of the matrix. Orthogonal matrices are defined by the requirement that the inverse transformation is given by the transposed matrix. 

The Clebsch Gordan coefficients for given values of y) and j2 form a square matrix labelled by the j, m values one way and by mu m2 the other. This matrix is always real and orthogonal, so that the inverse transformation to (5.77) is... 

Step 1 Choose a balanced (mean orthogonal) design at random label the design matrix D. 

Often in science, one associates transformation properties under coordinate transformations with the sub-index labels of a matrix M = R. And, one can associate such behaviour separately for each member of the index pair. Thus, the human in charge has the freedom to construct relationships between two different coordinate systems, each linked to one of the indices. Generally, R is real orthogonal but not symmetric. 

Here j labels the diabatic electronic state, while k is the free rotor quantum numbers. As should be clear from eq. (16) the doorway state has exactly the same distribution. It is however much more interesting to look at the corresponding partition with respect to the adiabatic states. This may be easily derived in terms of the orthogonal matrix U(9) which diagonalizes the electronic Hamiltonian, eq. (5). 

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