# GF method

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The **GF method**, sometimes referred to as **FG method**, is a classical mechanical method introduced by [Edgar Bright Wilson](/source/Edgar_Bright_Wilson) to obtain certain *internal coordinates* for a [vibrating](/source/Molecular_vibration) semi-rigid molecule, the so-called *normal coordinates* *Q*k. [Normal coordinates](/source/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](/source/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](/source/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](/source/Degrees_of_freedom_(physics_and_chemistry)), 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 interaction among atoms in a molecule is described by a [potential energy surface](/source/Potential_energy_surface) (PES), which is a function of 3*N* − 6 coordinates. The internal degrees of freedom *s*1, ..., *s*3*N*−6 describing the PES in an optimal 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](/source/Curvilinear_coordinates), but it is hard to formulate a general theory applicable to any molecule. This is why Wilson linearized the internal coordinates by assuming small displacements.[1] The linearized version of the internal coordinate *s*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](/source/Hessian_matrix) of *V*) evaluated in the minimum is a force derivative matrix **F**. In the harmonic approximation the [Taylor series](/source/Taylor_series) is ended after this term. The second term, containing first derivatives, is zero because it is evaluated in the minimum of *V*. The first term can be included in the zero of energy. Thus,

- 2V \approx \sum_{s,t=1}^{3N-6} F_{st} S_s\, S_t.

The classical vibrational kinetic energy has the form:

- 2T = \sum_{s,t=1}^{3N-6} g_{st}(\mathbf{s}) \dot{S}_s\dot{S}_t ,

where *g**st* is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate [time derivatives](/source/Time_derivative). Mixed terms S_s\, \dot{S}_t 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 **s**0 of *V* gives the positive definite and symmetric matrix **G** = **g**(**s**0)**−1**. One can solve the two matrix problems

- \mathbf{L}^\mathrm{T} \mathbf{F} \mathbf{L} =\boldsymbol{\Phi} \quad \mathrm{and}\quad \mathbf{L}^\mathrm{T} \mathbf{G}^{-1} \mathbf{L} = \mathbf{E},

simultaneously, since they are equivalent to the [generalized eigenvalue problem](/source/Generalized_eigenvalue_problem)

- \mathbf{G} \mathbf{F} \mathbf{L} = \mathbf{L} \boldsymbol{\Phi},

where \boldsymbol{\Phi}=\operatorname{diag}(f_1,\ldots, f_{3N-6}) where *fi* is equal to 4{\pi}^{2}{\nu}_i^{2} ({\nu}_i is the frequency of [normal mode](/source/Normal_mode) *i*); \mathbf{E}\, is the unit matrix. The matrix **L**−1 contains the *normal coordinates* *Q*k in its rows:

- Q_k = \sum_{t=1}^{3N-6} (\mathbf{L}^{-1})_{kt} S_t , \quad k=1,\ldots, 3N-6. \,

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

- \mathbf{s} = \operatorname{col}(S_1,\ldots, S_{3N-6}) \quad\mathrm{and}\quad \mathbf{Q} = \operatorname{col}(Q_1,\ldots, Q_{3N-6}),

which satisfy the relation

- \mathbf{s} = \mathbf{L} \mathbf{Q}.

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

- 2E = \dot{\mathbf{s}}^\mathrm{T} \mathbf{G}^{-1}\dot{\mathbf{s}}+ \mathbf{s}^\mathrm{T}\mathbf{F}\mathbf{s}

- - = \dot{\mathbf{Q}}^\mathrm{T} \; \left( \mathbf{L}^\mathrm{T} \mathbf{G}^{-1} \mathbf{L}\right) \; \dot{\mathbf{Q}}+ \mathbf{Q}^\mathrm{T} \left( \mathbf{L}^\mathrm{T}\mathbf{F}\mathbf{L}\right)\; \mathbf{Q}

- - = \dot{\mathbf{Q}}^\mathrm{T}\dot{\mathbf{Q}} + \mathbf{Q}^\mathrm{T}\boldsymbol{\Phi}\mathbf{Q} = \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 + f_t Q_t^2 \big).

The Lagrangian *L* = *T* − *V* is

- L = \frac{1}{2} \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 - f_t Q_t^2 \big).

The corresponding [Lagrange equations](/source/Lagrange_equations) are identical to the Newton equations

- \ddot{Q}_t + f_t \,Q_t = 0

for a set of uncoupled harmonic oscillators. These ordinary second-order differential equations are easily solved, yielding *Q**t* as a function of time; see the article on [harmonic oscillators](/source/Harmonic_oscillator).

## 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**A0 the corresponding equilibrium position. Then \mathbf{x}_A \equiv \mathbf{R}_A -\mathbf{R}_A^0 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

- S_t =\sum_{A=1}^N \sum_{i=1}^3 s^t_{Ai} \, x_{Ai}= \sum_{A=1}^N \mathbf{s}^t_{A} \cdot \mathbf{x}_{A}, \quad \mathrm{for}\quad t = 1,\ldots,3N-6,

where **s**At is known as a *Wilson s-vector*. If we put the s^t_{Ai} into a (3*N* − 6) × 3*N* matrix **B**, this equation becomes in matrix language

- \mathbf{s} = \mathbf{B} \mathbf{x}.

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 s^t_{Ai}. See for more details on this method, known as the *Wilson s-vector method*, the book by Wilson *et al.*, or [molecular vibration](/source/Molecular_vibration). Now,

- \mathbf{s} = \mathbf{L} \mathbf{Q} = \mathbf{L} \mathbf{l}^{tr} \mathbf{q} = \mathbf{B} \mathbf{M}^{-1/2} \mathbf{q} \equiv \mathbf{D} \mathbf{q},

which can be inverted and put in summation language:

- Q_k = \sum_{A=1}^N \sum_{i=1}^3 D^k_{Ai}\, d_{Ai} \quad \mathrm{for}\quad k=1,\ldots, 3N-6.

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

## Matrices involved in the analysis

There are several related coordinate systems commonly used in the GF matrix analysis.[2] These quantities are related by a variety of matrices. For clarity, we provide the coordinate systems and their interrelations here.

The relevant coordinates are:

- \mathbf{x}: Cartesian coordinates for each atom
- \mathbf{s}: Internal coordinates for each atom
- \mathbf{q}: Mass-weighted Cartesian coordinates
- \mathbf{Q}: Normal coordinates

These different coordinate systems are related to one another by:

- \mathbf{s} = \mathbf{B}\mathbf{x}, i.e. the matrix \mathbf{B} transforms the Cartesian coordinates to (linearized) internal coordinates.
- \mathbf{x} = \mathbf{M}^{-1/2}\mathbf{q}, i.e. the mass matrix \mathbf{M}^{1/2} transforms Cartesian coordinates to mass-weighted Cartesian coordinates.
- \mathbf{q} = \mathbf{l} \mathbf{Q}, i.e. the matrix \mathbf{l} transforms the normal coordinates to mass-weighted Cartesian coordinates.
- \mathbf{s} = \mathbf{L}\mathbf{Q}, i.e. the matrix \mathbf{L} transforms the normal coordinates to internal coordinates.

Note the useful relationship:

\mathbf{L}=\mathbf{B} \mathbf{M}^{-1/2} \mathbf{l}.

These matrices allow one to construct the **G** matrix quite simply as

\mathbf{G} = \mathbf{B} \mathbf{M}^{-1} \mathbf{B}^{\rm T}.

## Relation to Eckart conditions

Main article: [Eckart conditions](/source/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,

- \sum_{A=1}^N \mathbf{s}^t_{A} = 0\quad\mathrm{and}\quad \sum_{A=1}^N \mathbf{R}^0_A\times \mathbf{s}^t_A= 0, \quad t=1,\ldots,3N-6.

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

- \sum_{A=1}^N M_A\; \mathbf{d}_{A} = 0 \quad\mathrm{and}\quad \sum_{A=1}^N M_A\; \mathbf{R}^0_{A} \times \mathbf{d}_{A} = 0.

## References

1. Wilson, E. B. Jr. (1941). "Some Mathematical Methods for the Study of Molecular Vibrations". *[J. Chem. Phys.](/source/Journal_of_Chemical_Physics)*. **9** (1): 76–84. [Bibcode:1941JChPh...9...76W](https://ui.adsabs.harvard.edu/abs/1941JChPh...9...76W). [doi:10.1063/1.1750829](https://doi.org/10.1063/1.1750829)

1. Califano, S. (1976). *Vibrational states*. London: Wiley. ISBN 0-471-12996-8. [OCLC 1529286](https://www.worldcat.org/oclc/1529286)

## Further references

- Califano, S. (1976). *Vibrational States*. New York-London: Wiley. ISBN 0-471-12996-8.
- Papoušek, D. & Aliev, M. R. (1982). *Molecular Vibrational-Rotational Spectra*. [Elsevier](/source/Elsevier). ISBN 0-444-99737-7.
- Wilson, E. B.; Decius, J. C.; Cross, P. C. (1995 [1955]). [*Molecular Vibrations*](https://archive.org/details/molecularvibrati00wils). New York: Dover. ISBN 0-486-63941-X.

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Adapted from the Wikipedia article [GF method](https://en.wikipedia.org/wiki/GF_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/GF_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.