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

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The **Eckart conditions**^{[1]} 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[edit | edit source]

Let be the coordinate vector of nucleus *A* ()
with respect to a special body-fixed frame with origin in the center of mass of the molecule, a so-called Eckart frame.^{[2]} The mass of nucleus *A* is *M*_{A}. We suppose that the potential energy surface has a (deep) minimum for (); this means that the molecule is assumed to have a well-defined geometry (a *semi-rigid* molecule). The equilibrium coordinates are expressed with respect to the Eckart frame.

We define *displacement coordinates* .
The displacement coordinates satisfy the *translational* Eckart conditions, if:

The *rotational* Eckart conditions for the displacements are:

where indicates a vector product.

## Separation of external and internal coordinates[edit | edit source]

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

playing the role of positive definite metric. That is, an inner product in the space is defined as

We will follow the second route.

The following 3*N*-dimensional vectors are **external coordinates**

where **t** and **s** are arbitrary triplets of real numbers (3-dimensional column vectors).
The subspace of **R**^{3N} 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 **R**_{ext} and the total configuration space
becomes

where **R**_{int} is the 3*N* - 6 dimensional orthogonal complement of **R**_{ext}. The elements of **R**_{int} 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 **R**_{int}:

because
is in the orthogonal complement of **T**. Now, since **t** is arbitrary,

which are the translational Eckart conditions. In a very similar manner,

because
is in the orthogonal complement of **S**. Finally, since **s** is arbitrary

which are the rotational Eckart conditions.

## Relation to the harmonic approximation[edit | edit source]

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

where **H** is a 3*N* x 3*N* symmetric matrix of second derivatives of the potential . **H** is the Hessian matrix of *V* in the equilibrium . The diagonal matrix **M** contains the masses on the diagonal.
The diagonal matrix 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 **R**_{ext}. The generalized orthogonalities:
applied to the "internal" and "external" (zero-eigenvalue) columns of **C**
are in fact the Eckart conditions.

## References[edit | edit source]

- ↑ C. Eckart,
*Some studies concerning rotating axes and polyatomic molecules*, Physical Review, vol. 47, pp. 552-558 (1935). - ↑ L. C. Biedenharn and J. D. Louck,
*Angular Momentum in Quantum Physics*, Addison-Wesley, Reading (1981) p. 535.

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).