The Eckart conditions, named after
Carl Eckart
Carl Henry Eckart (May 4, 1902 – October 23, 1973) was an American physicist, physical oceanographer, geophysicist, and administrator. He co-developed the Wigner–Eckart theorem and is also known for the Eckart conditions in quantum mechanics ...
, simplify the nuclear motion (rovibrational) Hamiltonian that arises in the second step of the
Born–Oppenheimer approximation
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
. 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
In chemistry and molecular physics, fluxional (or non-rigid) molecules are molecules that undergo dynamics such that some or all of their atoms interchange between symmetry-equivalent positions. Because virtually all molecules are fluxional in so ...
, which is a molecule with a
potential energy surface
A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; ...
''V''(R
1, R
2,..R
''N'') that has a well-defined minimum for R
''A''0 (
). 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
''A''0 = (''R''
''A''10, ''R''
''A''20, ''R''
''A''30) 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'' (
). 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
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
.
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
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigi ...
, that is, when all ''N'' displacement vectors are zero.
Separation of external and internal coordinates
The ''N'' position vectors
of the nuclei constitute a 3''N'' dimensional linear space R
3N: the ''configuration space''. The Eckart conditions give an orthogonal direct sum decomposition of this space
:
The elements of the 3''N''-6 dimensional subspace R
int 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 R
ext 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 R
ext. To that end we introduce the following 6 vectors (i=1,2,3):
:
An orthogonal, unnormalized, basis for R
ext 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 R
ext, so that it is an internal vector.
We obtain a basis for the internal space by defining 3''N''-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 ''q''
r depends on the vectors
. For instance, ''q''
r 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
:
Now, under translation
:
Clearly,
is invariant under translation if and only if
:
because the vector
is arbitrary. So, the translational Eckart conditions imply the translational invariance of the vectors belonging to internal space and conversely. Under rotation we have,
:
Rotational invariance follows if and only if
:
The external modes, on the other hand, are ''not'' invariant and it is not difficult to show that they change under translation as follows:
:
where ''M'' is the total mass of the molecule. They change under infinitesimal rotation as follows
:
where I
0 is the inertia tensor of the equilibrium molecule. This behavior shows
that the first three external modes describe the overall translation of the molecule, while
the modes 4, 5, and, 6 describe the overall rotation.
Vibrational energy
The vibrational energy of the molecule can be written in terms of coordinates with respect to the Eckart frame as
:
Because the Eckart frame is non-inertial, the total kinetic energy comprises also centrifugal and Coriolis energies. These stay out of the present discussion. The vibrational energy is written in terms of the displacement coordinates, which are linearly dependent because they are contaminated by the 6 external modes, which are zero, i.e., the d
''A'''s satisfy 6 linear relations. It is possible to write the vibrational energy solely in terms of the internal modes ''q''
r (''r'' =1, ..., 3''N''-6) as we will now show. We write the different modes in terms of the displacements
:
The parenthesized expressions define a matrix B relating the internal and external modes to the displacements. The matrix B may be partitioned in an internal (3''N''-6 x 3''N'') and an external (6 x 3''N'') part,
:
We define the matrix M by
:
and from the relations given in the previous sections follow the matrix relations
:
and
:
We define
:
By using the rules for block matrix multiplication we can show that
:
where G
−1 is of dimension (3''N''-6 x 3''N''-6) and N
−1 is (6 x 6).
The kinetic energy becomes
:
where we used that the last 6 components of v are zero. This form of
the kinetic energy of vibration enters Wilson's
GF method The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain ''internal coordinates'' for a vibrating semi-rigid molecule, the so-called ''normal coordinates'' ''Q''k. Norm ...
. It is of some interest to point out that the potential energy in the harmonic approximation can be written as follows
:
where H is the Hessian of the potential in the minimum and F, defined by this equation, is the F matrix of the
GF method The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain ''internal coordinates'' for a vibrating semi-rigid molecule, the so-called ''normal coordinates'' ''Q''k. Norm ...
.
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
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
:
where H is a 3''N'' × 3''N'' symmetric matrix of second derivatives of the potential
. H is the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
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 vectors T = (t, ..., t) are in the kernel of H.
From the invariance of ''V'' under an infinitesimal rotation of all nuclei around s, it can be shown that also the vectors S = (s x R
10, ..., s x R
N0) are in the kernel of H :
:
Thus, six columns of C corresponding to eigenvalue zero are determined algebraically. (If the generalized eigenvalue problem is solved numerically, one will find in general six 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 constant
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
s, 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" (non-zero eigenvalue) and "external" (zero-eigenvalue) columns of C are equivalent to the Eckart conditions.
References
Further reading
The classic work is:
*
More advanced book are:
*
*{{cite book , first=S. , last=Califano , title=Vibrational States , publisher=Wiley , location=New York-London , year=1976 , isbn=0-471-12996-8
Molecular physics
Quantum chemistry