DETERMINATION OF NORMAL COORDINATES OF AN n-ATOMIC MOLECULE.
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Date
1967
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Ohio State University
Abstract
The molecular energy expression involving cartesian displacement coordinates $s_{i\alpha}(i=1,\ldots, n; \alpha = x,y,z$) has two structures, first recognized by Wigner [G$\bar{\rm o}$ttinger Nachrichten (1930), p. 133]. One is a permutation structure: group operators R move equilibrium position $r^{0}_{i}$ to $r^{0}_{i}=Rr^{0}_{i}=$, which can be viewed as a permutation of the labels i.j of equivalent atoms. The second is a vector structure: $Rs_{\alpha}=\Sigma R_{\alpha\beta^{S};\beta}$ where $R_{\alpha \beta}={\rm D}^{(v)}(R)_{\alpha\beta}$ is a rotation matrix. The representation of operators R whose basis is the displacement coordinates (suitably ordered) is the direct product $\Delta \times D^{(v)}$ of a permutation matrix $\Delta$ and the vector representation $D^{(v)}$. If the molecule possesses non-equivalent atoms, the permutation matrix is partially reduced. Similarity transformation of $\Delta \times D^{(v)}$ by $u\times v$, where u fully reduces $\Delta$ and v (possibly the identity) fully reduces $D^{(v)}$, produces an at least partially reduced representation, the diagonal blocks of which are direct products of two irreducible representations of the group. Final transformation by a Wigner matrix W completes the reduction of the representation. The normal mode coordinates are given by the composite transformation $(u\times v)W$.
Description
The research was sponsored by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation.
Author Institution: Oak Ridge National Laboratory, Oak Ridge, Tennessee, and the University of Tennessee
Author Institution: Oak Ridge National Laboratory, Oak Ridge, Tennessee, and the University of Tennessee