Eckart conditions

In chemistry and physics, the Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion (rovibrational) Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external (rotation and translation) motions from the internal (vibration) motions. Although the rotational and vibrational motions of the nuclei in a molecule cannot be fully separated, the Eckart conditions minimize the coupling between these two.

Definition of Eckart conditions

The Eckart conditions can only be formulated for a semi-rigid molecule, which is a molecule with a potential energy surface V(R₁, R₂,..R_N) that has a well-defined minimum for R_A⁰ (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle A=1,\ldots, N} ). These equilibrium coordinates of the nuclei—with masses M_A—are expressed with respect to a fixed orthonormal principal axes frame and hence satisfy the relations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{A=1}^N M_A\,\big(\delta_{ij}|\mathbf{R}_A^0|^2 - R^0_{Ai} R^0_{Aj}\big) = \lambda^0_i \delta_{ij} \quad\mathrm{and}\quad \sum_{A=1}^N M_A \mathbf{R}_A^0 = \mathbf{0}. }

Here λ_i⁰ is a principal inertia moment of the equilibrium molecule. The triplets R_A⁰ = (R_A1⁰, R_A2⁰, R_A3⁰) satisfying these conditions, enter the theory as a given set of real constants. Following Biedenharn and Louck we introduce an orthonormal body-fixed frame, the Eckart frame,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec\mathbf{F} = \{ \vec{f}_1, \vec{f}_2, \vec{f}_3\} . }

If an observer is tied to the Eckart frame, which—following the molecule—rotates and translates in space, (s)he would see the molecule in its equilibrium geometry, provided the nuclei are at the points,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{R}_A^0 \equiv \vec\mathbf{F} \cdot \mathbf{R}_A^0 =\sum_{i=1}^3 \vec{f}_i\, R^0_{Ai},\quad A=1,\ldots,N . } .

Let the elements of R_A be the coordinates with respect to the Eckart frame of the position vector of nucleus A (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle A=1,\ldots, N} ). Since we take the origin of the Eckart frame in the instantaneous center of mass, the following relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_A M_A \mathbf{R}_A = \mathbf{0} }

holds. We define displacement coordinates

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{d}_A\equiv\mathbf{R}_A-\mathbf{R}^0_A} .

Clearly the displacement coordinates satisfy the translational Eckart conditions,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{A=1}^N M_A \mathbf{d}_A = 0 . }

The rotational Eckart conditions for the displacements are:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{A=1}^N M_A \mathbf{R}^0_A \times \mathbf{d}_A = 0, }

where × indicates a cross product. These rotational conditions follow from the specific construction of the Eckart frame, see Biedenharn and Louck, loc. cit., page 538.

Finally, notice that an Eckart frame becomes a principal axes frame in the case that the molecule is a rigid rotor, that is, when all N displacement vectors are zero.

Separation of external and internal coordinates

The N position vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{R}_A} of the nuclei constitute a 3N dimensional linear space R^3N: the configuration space. The Eckart conditions give an orthogonal direct sum decomposition of this space

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{D}} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^A_r} 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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^A_r} . 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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. }

Overall translation and rotation

The vibrational (internal) modes are invariant under translation and infinitesimal rotation of the equilibrium (reference) molecule if and only if the Eckart conditions apply. This will be shown in this subsection.

An overall translation of the reference molecule is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{R}_{A}^0 \mapsto \vec{R}_{A}^0 + \vec{t} } '

for any arbitrary 3-vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{t}} . An infinitesimal rotation of the molecule is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{R}_A^0 \mapsto \vec{R}_A^0 + \Delta\phi \; ( \vec{n}\times \vec{R}_A^0) }

where Δφ is an infinitesimal angle, Δφ >> (Δφ)², and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n}} is an arbitrary unit vector. From the orthogonality of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{Q}_r} to the external space follows that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^A_r} satisfy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{A=1}^N \vec{q}^{\,A}_r = \vec{0} \quad\mathrm{and}\quad \sum_{A=1}^N \vec{R}^0_A\times \vec{q}^A_r = \vec{0}. }

Now, under translation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_r \mapsto \sum_A\vec{q}^{\,A}_r \cdot(\vec{d}^A - \vec{t}) = q_r - \vec{t}\cdot\sum_A \vec{q}^{\,A}_r = q_r. }

Clearly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^A_r} is invariant under translation if and only if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_A \vec{q}^{\,A}_r = 0, }

because the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{t}} is arbitrary. So, the translational Eckart conditions imply the translational invariance of the vectors belonging to internal space and conversely. Under rotation we have,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_r \mapsto \sum_A\vec{q}^{\,A}_r \cdot \big(\vec{d}^A - \Delta\phi \; ( \vec{n}\times \vec{R}_A^0) \big) = q_r - \Delta\phi \; \vec{n}\cdot\sum_A \vec{R}^0_A\times\vec{q}^{\,A}_r = q_r. }

Rotational invariance follows if and only if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_A \vec{R}^0_A\times\vec{q}^{\,A}_r. }

The external modes, on the other hand, are not invariant and it is not difficult to show that they change under translation as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} s_i &\mapsto s_i + M \vec{f}_i \cdot \vec{t} \quad \mathrm{for}\quad i=1,2,3 \\ s_i &\mapsto s_i \quad \mathrm{for}\quad i=4,5,6, \\ \end{align} }

where M is the total mass of the molecule. They change under infinitesimal rotation as follows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} s_i &\mapsto s_i \quad \mathrm{for}\quad i=1,2,3 \\ s_i &\mapsto s_i + \Delta \phi \vec{f}_i \cdot \mathbf{I}^0\cdot \vec{n} \quad \mathrm{for}\quad i=4,5,6, \\ \end{align} }

where I⁰ is the inertia tensor of the equilibrium molecule. This behavior shows that the first three external modes describe the overall translation of the molecule, while the modes 4, 5, and, 6 describe the overall rotation.

Vibrational energy

The vibrational energy of the molecule can be written in terms of coordinates with respect to the Eckart frame as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T_\mathrm{vib} = \sum_{A=1}^N M_A \dot{\mathbf{R}}_A\cdot \dot{\mathbf{R}}_A = \sum_{A=1}^N M_A \dot{\mathbf{d}}_A\cdot \dot{\mathbf{d}}_A. }

Because the Eckart frame is non-inertial, the total kinetic energy comprises also centrifugal and Coriolis energies. These stay out of the present discussion. The vibrational energy is written in terms of the displacement coordinates, which are linearly dependent because they are contaminated by the 6 external modes, which are zero, i.e., the d_A's satisfy 6 linear relations. It is possible to write the vibrational energy solely in terms of the internal modes q_r (r =1, ..., 3N-6) as we will now show. We write the different modes in terms of the displacements

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} q_r = \sum_{Aj} d_{Aj}& \big( q^A_{rj} \big) \\ s_i = \sum_{Aj} d_{Aj}& \big( M_A \delta_{ij} \big) =0 \\ s_{i+3} = \sum_{Aj} d_{Aj}& \big( M_A \sum_k \epsilon_{ikj} R^0_{Ak} \big)=0 \\ \end{align} }

The parenthesized expressions define a matrix B relating the internal and external modes to the displacements. The matrix B may be partitioned in an internal (3N-6 x 3N) and an external (6 x 3N) part,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}\equiv \begin{pmatrix} q_1 \\ \vdots \\ \vdots \\ q_{3N-6} \\ 0 \\ \vdots \\ 0\\ \end{pmatrix} = \begin{pmatrix} \mathbf{B}^\mathrm{int} \\ \cdots \\ \mathbf{B}^\mathrm{ext} \\ \end{pmatrix} \mathbf{d} \equiv \mathbf{B} \mathbf{d}. }

We define the matrix M by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} \equiv \operatorname{diag}(\mathbf{M}_1, \mathbf{M}_2, \ldots,\mathbf{M}_N) \quad\textrm{and}\quad \mathbf{M}_A\equiv \operatorname{diag}(M_A, M_A, M_A) }

and from the relations given in the previous sections follow the matrix relations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}^\mathrm{ext} \mathbf{M}^{-1} (\mathbf{B}^\mathrm{ext})^\mathrm{T} = \operatorname{diag}(N_1,\ldots, N_6) \equiv\mathbf{N}, }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}^\mathrm{int} \mathbf{M}^{-1} (\mathbf{B}^\mathrm{ext})^\mathrm{T} = \mathbf{0}. }

We define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{G} \equiv \mathbf{B}^\mathrm{int} \mathbf{M}^{-1} (\mathbf{B}^\mathrm{int})^\mathrm{T}. }

By using the rules for block matrix multiplication we can show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbf{B}^\mathrm{T})^{-1} \mathbf{M} \mathbf{B}^{-1} = \begin{pmatrix} \mathbf{G}^{-1} && \mathbf{0} \\ \mathbf{0} && \mathbf{N}^{-1} \end{pmatrix}, }

where G^-1 is of dimension (3N-6 x 3N-6) and N^-1 is (6 x 6). The kinetic energy becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T_\mathrm{vib} = \dot{\mathbf{d}}^\mathrm{T} \mathbf{M} \dot{\mathbf{d}} = \dot{\mathbf{v}}^\mathrm{T}\; (\mathbf{B}^\mathrm{T})^{-1} \mathbf{M} \mathbf{B}^{-1}\; \dot{\mathbf{v}} = \sum_{r, r'=1}^{3N-6} (G^{-1})_{r r'} \dot{q}_r \dot{q}_{r'} }

where we used that the last 6 components of v are zero. This form of the kinetic energy of vibration enters Wilson's GF method. It is of some interest to point out that the potential energy in the harmonic approximation can be written as follows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2V_\mathrm{harm} = \mathbf{d}^\mathrm{T} \mathbf{H} \mathbf{d} = \mathbf{v}^\mathrm{T} (\mathbf{B}^\mathrm{T})^{-1} \mathbf{H} \mathbf{B}^{-1} \mathbf{v} = \sum_{r, r'=1}^{3N-6} F_{r r'} q_r q_{r'}, }

where H is the Hessian of the potential in the minimum and F, defined by this equation, is the F matrix of the GF method.

Relation to the harmonic approximation

In the harmonic approximation to the nuclear vibrational problem, expressed in displacement coordinates, one must solve the generalized eigenvalue problem

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{H}\mathbf{C} = \mathbf{M} \mathbf{C} \boldsymbol{\Phi}, }

where H is a 3N x 3N symmetric matrix of second derivatives of the potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\mathbf{R}_1, \mathbf{R}_2,\ldots, \mathbf{R}_N)} . H is the Hessian matrix of V in the equilibrium Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}_1^0,\ldots, \mathbf{R}_N^0} . The diagonal matrix M contains the masses on the diagonal. The diagonal matrix ${\boldsymbol {\Phi }}$ contains the eigenvalues, while the columns of C contain the eigenvectors.

It can be shown that the invariance of V under simultaneous translation over t of all nuclei implies that vectors T = (t, ... , t) are in the kernel of H. From the invariance of V under an infinitesimal rotation of all nuclei around s it can been shown that also the vectors S = (s x R₁⁰, ..., s x R_N⁰) are in the kernel of H :

$\mathbf {H} {\begin{pmatrix}\mathbf {t} \\\vdots \\\mathbf {t} \end{pmatrix}}={\begin{pmatrix}\mathbf {0} \\\vdots \\\mathbf {0} \end{pmatrix}}\quad \mathrm {and} \quad \mathbf {H} {\begin{pmatrix}\mathbf {s} \times \mathbf {R} _{1}^{0}\\\vdots \\\mathbf {s} \times \mathbf {R} _{N}^{0}\end{pmatrix}}={\begin{pmatrix}\mathbf {0} \\\vdots \\\mathbf {0} \end{pmatrix}}$

Thus, 6 columns of C corresponding to eigenvalue zero, are determined algebraically. (If the generalized eigenvalue problem is solved numerically, one will find in general 6 linearly independent linear combinations of S and T). The eigenspace corresponding to eigenvalue zero is at least of dimension 6 (often it is exactly of dimension 6, since the other eigenvalues, which are force constants, are never zero for molecules in their ground state). Thus, T and S correspond to the overall (external) motions: translation and rotation, respectively. They are zero-energy modes because space is homogeneous (force-free) and isotropic (torque-free).

By the definition in this article the non-zero frequency modes are internal modes, since they are within the orthogonal complement of R_ext. The generalized orthogonalities: $\mathbf {C} ^{\mathrm {T} }\mathbf {M} \mathbf {C} =\mathbf {I}$ applied to the "internal" (non-zero eigenvalue) and "external" (zero-eigenvalue) columns of C are equivalent to the Eckart conditions.

References

The classic work is:

• E. Bright Wilson, Jr., J. C. Decius and Paul C. Cross, Molecular Vibrations, Mc-Graw-Hill (1955). Reprinted by Dover (1980).

More advanced book are:

• D. Papoušek and M. R. Aliev, Molecular Vibrational-Rotational Spectra, Elsevier (1982).
• S. Califano, Vibrational States, Wiley, New York-London (1976). ISBN 0 471 12996 8