Eckart conditions

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.

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 (${\displaystyle A=1,\ldots ,N}$). 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

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,

Let the elements of R_A be the coordinates with respect to the Eckart frame of the position vector of nucleus A (${\displaystyle A=1,\ldots ,N}$). Since we take the origin of the Eckart frame in the instantaneous center of mass, the following relation

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