Eckart Conditions - Separation of External and Internal Coordinates

Separation of External and Internal Coordinates

The N position vectors of the nuclei constitute a 3N dimensional linear space R3N: the configuration space. The Eckart conditions give an orthogonal direct sum decomposition of this space

$\mathbf{R}^{3N} = \mathbf{R}_\textrm{ext}\oplus\mathbf{R}_\textrm{int}.$

The elements of the 3N-6 dimensional subspace R_int are referred to as internal coordinates, because they are invariant under overall translation and rotation of the molecule and, thus, depend only on the internal (vibrational) motions. The elements of the 6-dimensional subspace R_ext are referred to as external coordinates, because they are associated with the overall translation and rotation of the molecule.

To clarify this nomenclature we define first a basis for R_ext. To that end we introduce the following 6 vectors (i=1,2,3):

$\begin{align} \vec{s}^A_{i} &\equiv \vec{f}_i \\ \vec{s}^A_{i+3} &\equiv \vec{f}_i \times\vec{R}_A^0 .\\ \end{align}$

An orthogonal, unnormalized, basis for R_ext is,

$\vec{S}_t \equiv \operatorname{row}(\sqrt{M_1}\;\vec{s}^{\,1}_{t}, \ldots, \sqrt{M_N} \;\vec{s}^{\,N}_{t}) \quad\mathrm{for}\quad t=1,\ldots, 6.$

A mass-weighted displacement vector can be written as

$\vec{D} \equiv \operatorname{col}(\sqrt{M_1}\;\vec{d}^{\,1}, \ldots, \sqrt{M_N}\;\vec{d}^{\,N}) \quad\mathrm{with}\quad \vec{d}^{\,A} \equiv \vec{\mathbf{F}}\cdot \mathbf{d}_A .$

For i=1,2,3,

$\vec{S}_i \cdot \vec{D} = \sum_{A=1}^N \; M_A \vec{s}^{\,A}_i \cdot \vec{d}^{\,A} =\sum_{A=1}^N M_A d_{Ai} = 0,$

where the zero follows because of the translational Eckart conditions. For i=4,5,6

$\, \vec{S}_i \cdot \vec{D} = \sum_{A=1}^N \; M_A \big(\vec{f}_i \times\vec{R}_A^0\big) \cdot \vec{d}^{\,A}=\vec{f}_i \cdot \sum_{A=1}^N M_A \vec{R}_A^0 \times\vec{d}^A = \sum_{A=1}^N M_A \big( \mathbf{R}_A^0 \times \mathbf{d}_A\big)_i = 0,$

where the zero follows because of the rotational Eckart conditions. We conclude that the displacement vector belongs to the orthogonal complement of R_ext, so that it is an internal vector.

We obtain a basis for the internal space by defining 3N-6 linearly independent vectors

$\vec{Q}_r \equiv \operatorname{row}(\frac{1}{\sqrt{M_1}}\;\vec{q}_r^{\,1}, \ldots, \frac{1}{\sqrt{M_N}}\;\vec{q}_r^{\,N}), \quad\mathrm{for}\quad r=1,\ldots, 3N-6.$

The vectors could be Wilson's s-vectors or could be obtained in the harmonic approximation by diagonalizing the Hessian of V. We next introduce internal (vibrational) modes,

$q_r \equiv \vec{Q}_r \cdot \vec{D} = \sum_{A=1}^N \vec{q}^A_r \cdot \vec{d}^{\,A} \quad\mathrm{for}\quad r=1,\ldots, 3N-6.$

The physical meaning of q_r depends on the vectors . For instance, q_r could be a symmetric stretching mode, in which two C—H bonds are simultaneously stretched and contracted.

We already saw that the corresponding external modes are zero because of the Eckart conditions,

$s_t \equiv \vec{S}_t \cdot \vec{D} = \sum_{A=1}^N M_A \;\vec{s}^{\,A}_t \cdot \vec{d}^{\,A} = 0 \quad\mathrm{for}\quad t=1,\ldots, 6.$

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