Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Eckart conditions

The Eckart conditions simplify the molecular Schrödinger equation  that arises in the second step of the  Born-Oppenheimer approximation. They allow the separation of the nuclear center of mass motion and (partly) the rotational motion from the internal vibrational motions. Although the rotational and vibrational motions of the nuclei in a molecule cannot be fully separated, the Eckart conditions minimize the (Coriolis) coupling between these motions.

Definition Eckart conditions
Let $$\mathbf{R}_A$$ be the coordinate vector of nucleus A ($$A=1,\ldots, N$$) with respect to a special body-fixed frame with origin in the center of mass of the molecule, a so-called Eckart frame. The mass of nucleus A is MA. We suppose that the potential energy surface $$V(\mathbf{R}_1,\ldots,\mathbf{R}_N)$$ has a (deep) minimum for $$\mathbf{R}^0_A$$ ($$A=1,\ldots, N$$); this means that the molecule is assumed to have a well-defined geometry (a semi-rigid molecule). The equilibrium coordinates $$\mathbf{R}^0_A$$ are expressed with respect to the Eckart frame.

We define displacement coordinates $$\mathbf{d}_A\equiv\mathbf{R}_A-\mathbf{R}^0_A$$. The displacement coordinates satisfy the translational Eckart conditions, if:

\sum_{A=1}^N M_A \mathbf{d}_A = 0. $$ The rotational Eckart conditions for the displacements are:

\sum_{A=1}^N M_A \mathbf{R}^0_A \times \mathbf{d}_A = 0, $$ where $$\times$$ indicates a vector product.

Separation of external and internal coordinates
Displacement coordinates that satisfy the Eckart conditions are usually referred to as internal molecular coordinates. To clarify this nomenclature we define first external molecular coordinates.

The nuclear configuration space R3N arising in the second step of the Born-Oppenheimer approximation is a linear space of dimension 3N. One can renormalize its basis and obtain so-called mass-weighted coordinates, or one can introduce a generalized inner product with

\mathbf{M} \equiv \operatorname{diag}(\mathbf{M}_1, \mathbf{M}_2, \ldots,\mathbf{M}_N) \quad\textrm{and}\quad \mathbf{M}_i\equiv \operatorname{diag}(M_i, M_i, M_i) $$ playing the role of positive definite metric. That is, an inner product in the space is defined as

\langle\mathbf{f}|\mathbf{d}\rangle \equiv \mathbf{f}^\mathrm{T}\, \mathbf{M}\, \mathbf{d} = \sum_{A=1}^N \sum_{\alpha=x,y,z} M_A f_{A\alpha} \,d_{A\alpha}. $$ We will follow the second route.

The following 3N-dimensional vectors are external coordinates

\mathbf{T} \equiv \operatorname{col}(\mathbf{t},\, \mathbf{t},\ldots, \mathbf{t}) \quad\textrm{and}\quad \mathbf{S}\equiv \operatorname{col}( \mathbf{s}\times\mathbf{R}_1^0,\, \mathbf{s}\times \mathbf{R}_2^0,\ldots,\mathbf{s}\times\mathbf{R}_N^0 ), $$ where t and s are arbitrary triplets of real numbers (3-dimensional column vectors). The subspace of R3N of all vectors T and S is of dimension 6, for it is easy to show that the 6 vectors obtained from unit vectors t and s [t=(1,0,0), t=(0,1,0), s = (1,0,0), etc.] are orthogonal and hence linearly independent. We write this 6-dimensional subspace as Rext and the total configuration space becomes

\mathbf{R}^{3N} = \mathbf{R}_\textrm{ext}\oplus\mathbf{R}_\textrm{int}, $$ where Rint is the 3N - 6 dimensional orthogonal complement of Rext. The elements of Rint are internal coordinates. We show that they obey the Eckart conditions. We also show that, conversely, vectors obeying the Eckart conditions are in the space Rint:

\langle\mathbf{T}| \mathbf{d}\rangle = \sum_{A=1}^{N} M_A \mathbf{t}^\textrm{T} \mathbf{d}_A = \mathbf{t}^\textrm{T} \sum_{A=1}^{N} M_A \mathbf{d}_A = 0, $$ because $$ \mathbf{d}\equiv \operatorname{col}(\mathbf{d}_1, \mathbf{d}_2,\ldots, \mathbf{d}_N) $$ is in the orthogonal complement of T. Now, since t is arbitrary,

\mathbf{t}^\textrm{T} \sum_{A=1}^{N} M_A \mathbf{d}_A = 0 \Longleftrightarrow \sum_{A=1}^{N} M_A \mathbf{d}_A = 0, $$ which are the translational Eckart conditions. In a very similar manner,

\langle\mathbf{S}| \mathbf{d}\rangle = \sum_{A=1}^{N} M_A (\mathbf{s} \times\mathbf{R}_A^0 )^\textrm{T}\, \mathbf{d}_A = \sum_{A=1}^{N} M_A \mathbf{s}^\textrm{T}\, \left(\mathbf{R}_A^0 \times \mathbf{d}_A\right) = \mathbf{s}^\textrm{T}\,\sum_{A=1}^{N} M_A \mathbf{R}_A^0 \times \mathbf{d}_A = 0 $$ because $$ \mathbf{d}\,$$ is in the orthogonal complement of S. Finally, since s is arbitrary

\mathbf{s}^\textrm{T} \sum_{A=1}^{N} M_A   \mathbf{R}_A^0 \times \mathbf{d}_A = 0 \Longleftrightarrow \sum_{A=1}^{N} M_A \mathbf{R}_A^0 \times \mathbf{d}_A = 0, $$ which are the rotational Eckart conditions.

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
 * $$ \mathbf{H}\mathbf{C} = \mathbf{M} \mathbf{C} \boldsymbol{\Phi},

$$ where H is a 3N x 3N symmetric matrix of second derivatives of the potential $$V(\mathbf{R}_1, \mathbf{R}_2,\ldots, \mathbf{R}_N)$$. H is the Hessian matrix of V in the equilibrium $$\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 the vectors T, introduced above,  are eigenvectors of eigenvalue zero. From the invariance of V under an infinitesimal rotation of all nuclei around s it can been shown that also the vectors S are eigenvectors of eigenvalue zero. Thus, the 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  Rext. The generalized orthogonalities: $$ \mathbf{C}^\mathrm{T} \mathbf{M} \mathbf{C} = \mathbf{I} $$ applied to the "internal" and "external" (zero-eigenvalue) columns of C are in fact the Eckart conditions.