The Eckart conditions, named after Carl Eckart, simplify the nuclear motion (rovibrational) Hamiltonian that arises in the second step of the Born–Oppenheimer approximation. They make it possible to approximately separate rotation from vibration. Although the rotational and vibrational motions of the nuclei in a molecule cannot be fully separated, the Eckart conditions minimize the coupling close to a reference (usually equilibrium) configuration. The Eckart conditions are explained by Louck and Galbraith and in Section 10.2 of the textbook by Bunker and Jensen, where a numerical example is given.
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,..RN) that has a well-defined minimum for RA0 (). These equilibrium coordinates of the nuclei—with masses MA—are expressed with respect to a fixed orthonormal principal axes frame and hence satisfy the relations
Here λi0 is a principal inertia moment of the equilibrium molecule. The triplets RA0 = (RA10, RA20, RA30) 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, . If we were tied to the Eckart frame, which—following the molecule—rotates and translates in space, we would observe the molecule in its equilibrium geometry when we would draw the nuclei at the points, . Let the elements of RA be the coordinates with respect to the Eckart frame of the position vector of nucleus A (). Since we take the origin of the Eckart frame in the instantaneous center of mass, the following relation
holds. We define displacement coordinates
. Clearly the displacement coordinates satisfy the translational Eckart conditions,
The rotational Eckart conditions for the displacements are:
where indicates a vector product. These rotational conditions follow from the specific construction of the Eckart frame, see Biedenharn and Louck, loc. cit., page 538.
Finally, for a better understanding of the Eckart frame it may be useful to remark that it 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 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
The elements of the 3N-6 dimensional subspace Rint 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 Rext 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 Rext. To that end we introduce the following 6 vectors (i=1,2,3):
An orthogonal, unnormalized, basis for Rext is,
A mass-weighted displacement vector can be written as
For i=1,2,3,
where the zero follows because of the translational Eckart conditions. For i=4,5,6
where the zero follows because of the rotational Eckart conditions. We conclude that the displacement vector belongs to the orthogonal complement of Rext, so that it is an internal vector.
We obtain a basis for the internal space by defining 3N-6 linearly independent vectors
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,
The physical meaning of qr depends on the vectors . For instance, qr 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,
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
'
for any arbitrary 3-vector . An infinitesimal rotation of the molecule is given by
where Δφ is an infinitesimal angle, Δφ >> (Δφ)², and is an arbitrary unit vector. From the orthogonality of to the external space follows that the satisfy