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,

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

Read more about this topic: Eckart Conditions

Famous quotes containing the words separation, external and/or internal:

“A separation situation is different for adults than it is for children. When we were very young children, a physical separation was interpreted as a violation of our inalienable rights....As we grew older, the withdrawal of love, whether that meant being misunderstood, mislabeled or slighted, became the separation situation we responded to.”
—Roger Gould (20th century)

“All in all, the creative act is not performed by the artist alone; the spectator brings the work in contact with the external world by deciphering and interpreting its inner qualifications and thus adds his contribution to the creative act. This becomes even more obvious when posterity gives its final verdict and sometimes rehabilitates forgotten artists.”
—Marcel Duchamp (1887–1968)

“Personal change, growth, development, identity formation—these tasks that once were thought to belong to childhood and adolescence alone now are recognized as part of adult life as well. Gone is the belief that adulthood is, or ought to be, a time of internal peace and comfort, that growing pains belong only to the young; gone the belief that these are marker events—a job, a mate, a child—through which we will pass into a life of relative ease.”
—Lillian Breslow Rubin (20th century)