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

The **GF method**, sometimes referred to as **FG method**, is a classical mechanical method introduced by E. Bright Wilson to obtain certain *internal coordinates* for
a vibrating semi-rigid molecule, the so-called *normal coordinates* *Q*_{k}. Normal coordinates decouple the classical vibrational motions of the molecule and thus give an easy route to obtaining vibrational amplitudes of the atoms as a function of time. In Wilson's GF method it is assumed that the molecular kinetic energy consists only of harmonic vibrations of the atoms, *i.e.,* overall rotational and translational energy is ignored. Normal coordinates appear also in a quantum mechanical description of the vibrational motions of the molecule and the Coriolis coupling between rotations and vibrations.

It follows from application of the Eckart conditions that the matrix **G**^{-1} gives the kinetic energy in terms of arbitrary linear internal coordinates, while **F** represents the (harmonic) potential energy in terms of these coordinates. The GF method gives the linear transformation from general internal coordinates to the special set of normal coordinates.

## The GF method

A non-linear molecule consisting of *N* atoms has 3*N*-6 internal degrees of freedom, because
positioning a molecule in three-dimensional space requires three degrees of freedom and the description of its orientation in space requires another three degree of freedom. These degrees of freedom must be subtracted from the 3*N* degrees of freedom of a system of *N* particles.

The atoms in a molecule are bound by a potential energy surface (PES) (or a force field (chemistry)|force field) which is a function of 3*N*-6 coordinates.
The internal degrees of freedom *q*_{1}, ..., *q*_{3N-6} describing the PES in an optimum way are often non-linear; they are for instance *valence coordinates*, such as bending and torsion angles and bond stretches. It is possible to write the quantum mechanical kinetic energy operator for such curvilinear coordinates, but it is hard to formulate a general theory applicable to any molecule. This is why Wilson^{[1]} linearized the internal coordinates by assuming small displacements. The linearized version of the internal coordinate *q*_{t} is denoted by *S*_{t}.

The PES *V* can be Taylor expanded around its minimum in terms of the *S*_{t}. The third term (the Hessian matrix|Hessian of *V*) evaluated in the minimum is a force derivative matrix **F**. In the harmonic approximation the Taylor series is ended after this term. The second term, containing first derivatives, is zero because it is evaluated in the miminum of *V*. The first term can be included in the zero of energy.
Thus,

- .

The classical vibrational kinetic energy has the form:

where *g*_{st} is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivatives. Mixed terms generally present in curvilinear coordinates are not present here, because only linear coordinate transformations are used. Evaluation of the metric tensor **g** in the minimum **q**^{0} of *V* gives
the positive definite and symmetric matrix **G** = **g**(**q**^{0})** ^{-1}. **
One can solve the following two matrix problems simultaneously

since they are equivalent to the generalized eigenvalue problem

where and is the unit matrix. The matrix **L**^{-1} contains the *normal coordinates* *Q*_{k} in its rows:

Because of the form of the generalized eigenvalue problem, the method is called the GF method,
often with the name of its originator attached to it: **Wilson's GF method**. By matrix transposition in both sides of the equation and using the fact
that both **G** and **F** are symmetric matrices, as are diagonal matrices, one can recast this equation into a very similar one for **FG** . This is why the method
is also referred to as **Wilson's FG method**.

We introduce the vectors

which satisfy the relation

Upon use of the results of the generalized eigenvalue equation
the energy *E* = *T * + *V* (in the harmonic approximation) of the molecule becomes,

The Lagrangian *L* = *T* - *V* is

The corresponding Lagrange equations are identical to the Newton equations

for a set of uncoupled
harmonic oscillators. These ordinary second-order differential equations are easily solved yielding *Q*_{t} as function of time, see the article on harmonic oscillators.

## Normal coordinates in terms of Cartesian displacement coordinates

Often the normal coordinates are expressed as linear combinations of Cartesian displacement coordinates.
Let **R**_{A} be the position vector of nucleus A and **R**_{A}^{0}
the corresponding equilibrium position. Then
is by definition the *Cartesian displacement coordinate* of nucleus A.
Wilson's linearizing of the internal curvilinear coordinates *q*_{t} expresses the coordinate *S*_{t} in terms of the displacement coordinates

where **s**_{A}^{t} is known as a *Wilson s-vector*.
If we put the into a 3*N*-6 x 3*N* matrix **B**, this equation becomes in matrix language

The actual form of the matrix elements of **B** can be fairly complicated.
Especially for a torsion angle, which involves 4 atoms, it requires tedious vector algebra to derive the corresponding values of the . See for more details on this method, known as
the *Wilson s-vector method*, the book by Wilson *et al.*, or molecular vibration. Now,

In summation language:

Here **D** is a 3*N*-6 x 3*N* matrix which is given by (i) the linearization of the internal coordinates **q** (an algebraic process) and (ii) solution of Wilson's GF equations (a numeric process).

## Relation with Eckart conditions

From the invariance of the internal coordinates *S*_{t} under overall rotation and translation
of the molecule, follows the same for the linearized coordinates **s**^{t}_{A}.
It can be shown that this implies that the following 6 conditions are satisfied by the internal
coordinates,

These conditions follow from the Eckart conditions that hold for the displacement vectors,

See Eckart conditions|this article for more details.

## References

**Cited reference**

- ↑ E. B. Wilson, Jr.
*Some Mathematical Methods for the Study of Molecular Vibrations*, J. Chem. Phys. vol.**9**, pp. 76-84 (1941)

** Further references **

- E. B. Wilson, J. C. Decius, and P. C. Cross,
*Molecular Vibrations*, McGraw-Hill, New York, 1955 (Reprinted by Dover 1980). - D. Papoušek and M. R. Aliev,
*Molecular Vibrational-Rotational Spectra*Elsevier, Amsterdam, 1982. - S. Califano,
*Vibrational States*, Wiley, London, 1976.