Eckart Conditions - Definition of Eckart Conditions

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 . 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

$\sum_{A=1}^N M_A\,\big(\delta_{ij}|\mathbf{R}_A^0|^2 - R^0_{Ai} R^0_{Aj}\big) = \lambda^0_i \delta_{ij} \quad\mathrm{and}\quad \sum_{A=1}^N M_A \mathbf{R}_A^0 = \mathbf{0}.$

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,

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,

$\vec{R}_A^0 \equiv \vec{\mathbf{F}} \cdot \mathbf{R}_A^0 =\sum_{i=1}^3 \vec{f}_i\, R^0_{Ai},\quad A=1,\ldots,N$ .

Let the elements of R_A be the coordinates with respect to the Eckart frame of the position vector of nucleus A . Since we take the origin of the Eckart frame in the instantaneous center of mass, the following relation

$\sum_A M_A \mathbf{R}_A = \mathbf{0}$

holds. We define displacement coordinates

Clearly the displacement coordinates satisfy the translational Eckart conditions,

$\sum_{A=1}^N M_A \mathbf{d}_A = 0 .$

The rotational Eckart conditions for the displacements are:

$\sum_{A=1}^N M_A \mathbf{R}^0_A \times \mathbf{d}_A = 0,$

where 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.

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