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Extra info for Applications of Group Theory to Atoms, Molecules, and Solids

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The subscripts are as follows: i = 1 is r y1 , i = 2 is r z1 , i = 3 is r y2 , i = 4 is r z2 , i = 5 is r y3 , and i = 6 is r z3 . The matrix M is a 6 × 6 diagonal matrix whose elements are Mii = m B for i = 1, 2, 3, 4, and m A for i = 5 or 6, where m A and m B are the masses of atoms A and B respectively. 2) where F = M−1/2 F M−1/2 , η = M1/2 ξ. 1 The action table for the AB2 molecule ry rz 1 2 3 E C2 ry rz 1 2 3 −r y rz 2 1 3 (m A or m B ). The 6-vector η transforms under the symmetry operations in the same way as ξ .

Other sorts of transitions are possible. For example, transitions from the ground state to more than one excited state, absorption of more than one quantum of vibrational energy, or even transitions from one excited state to another. These other types of transitions tend to be weaker in intensity, but can yield valuable spectroscopic information. Whatever the nature of the transition, the selection rule can be determined group-theoretically if the symmetry of the transition operator, the initial state, and the final state are specified.

S = 18 m[(ω A1 )2 − (ω B1 )2 + (ω B2 )2 ], u = 18 m[(ω A1 )2 − (ω B1 )2 − (ω B2 )2 ], eigenvalues. The force constants can then be adjusted until the eigenvalues achieve a best fit to the observed frequencies. However, special constraints must be employed in order to ensure that the translation and rotation modes have zero frequencies. 3] rather than Cartesian coordinates; for example, “bending” (changes in the angles between adjacent sides) and “stretching” (changes in distances between atoms) coordinates.

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