GF method

Wilson's GF method is a classical mechanical method to obtain linear internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Q_k. These normal coordinates can be used to obtain the classical motions (vibrational amplitude as function of time) for the harmonic vibrations of the atoms in the molecule. They also appear in a quantum mechanical description of the rotational and vibrational motions of the molecule.

The GF method

A (non-linear) molecule consisting of N atoms has 3N-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 3N degrees of freedom of a system of N particles.

The atoms in a molecule are bound by a potential energy surface (PES)—also known as force field—which is a function of 3N-6 coordinates. The internal degrees of freedom q 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 linearized the internal coordinates by assuming small displacements. The linearized versions of the internal coordinates q _k are denoted by S_t. For infinitesimal nuclear displacements, certain fixed linear combinations of the coordinates S_t coincide by definition with certain q _k. For growing nuclear displacements the two will start to deviate.

The PES V can be Taylor expanded around its minimum in terms of the S_t. The third term (the 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 functions as zero of energy. The classical kinetic energy has the form:

$2T=\sum _{s,t=1}^{3N-6}g_{st}(\mathbf {q} ){\dot {q}}_{s}{\dot {q}}_{t}$,

where g_st is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivatives. Evaluation of the metric tensor g in the minimum q⁰ of V gives the positive definite and symmetric matrix G = g(q⁰)^-1. One can solve the following two matrix problems simultaneously

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

since they are equivalent to the generalized eigenvalue problem

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

where ${\boldsymbol {\Phi }}=\operatorname {diag} (f_{1},\ldots ,f_{3N-6})$ and $\mathbf {E} \,$ 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 vector

$\mathbf {s} =\operatorname {col} (S_{1},\ldots ,S_{3N-6}).$

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

$2E=\sum _{st}{\dot {S}}_{t}(\mathbf {G} ^{-1})_{ts}{\dot {S}}_{s}+\sum _{st}S_{t}\mathbf {F} _{ts}S_{s}\equiv {\dot {\mathbf {s} }}^{\mathrm {T} }\mathbf {G} ^{-1}{\dot {\mathbf {s} }}+\mathbf {s} ^{\mathrm {T} }\mathbf {F} \mathbf {s}$

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

$={\dot {\mathbf {s} }}^{\mathrm {T} }(\mathbf {L} ^{\mathrm {T} })^{-1}\;\;\mathbf {L} ^{-1}{\dot {\mathbf {s} }}+\mathbf {s} ^{\mathrm {T} }(\mathbf {L} ^{\mathrm {T} })^{-1}\;\;{\boldsymbol {\Phi }}\;\;\mathbf {L} ^{-1}\mathbf {s}$

$={\dot {\mathbf {Q} }}^{\mathrm {T} }{\dot {\mathbf {Q} }}+\mathbf {Q} ^{\mathrm {T} }{\boldsymbol {\Phi }}\mathbf {Q} =\sum _{t=1}^{3N-6}{\big (}{\dot {Q}}_{t}^{2}+f_{t}Q_{t}^{2}{\big )}.$

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 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, 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_A⁰ the corresponding equilibrium position. Then $\mathbf {d} _{A}\equiv \mathbf {R} _{A}-\mathbf {R} _{A}^{0}$ is by definition the Cartesian displacement coordinate of nucleus A. Wilson's linearizing of internal curvilinear coordinates q expresses the coordinates S in terms of the displacement coordinates

$S_{t}=\sum _{A=1}^{N}\sum _{i=1}^{3}s_{Ai}^{t}\,d_{Ai}=\sum _{A=1}^{N}\mathbf {s} _{A}^{t}\cdot \mathbf {d} _{A},\quad \mathrm {for} \quad t=1,\ldots ,3N-6.$

If we put the $s_{Ai}^{t}$ into a 3N-6 x 3N matrix B, this equation becomes in matrix language

$\mathbf {s} =\mathbf {B} \mathbf {d} .$

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 $s_{Ai}^{t}$. See for details the book by Wilson et al. Now,

$\mathbf {Q} =\mathbf {L} ^{-1}\mathbf {s} =\mathbf {L} ^{-1}\mathbf {B} \mathbf {d} \equiv \mathbf {D} \mathbf {d} .$

In summation language:

$Q_{k}=\sum _{A=1}^{N}\sum _{i=1}^{3}D_{Ai}^{k}\,d_{Ai}\quad \mathrm {for} \quad k=1,\ldots ,3N-6.$

Here D is a 3N-6 x 3N 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 q under overall rotation and translation of the molecule, follows the same for the linearized coordinates s. It can be shown that this implies that the following 6 conditions are satisfied by the internal coordinates,

$\sum _{A=1}^{N}\mathbf {s} _{A}^{r}=0\quad \mathrm {and} \quad \sum _{A=1}^{N}\mathbf {R} _{A}^{0}\times \mathbf {s} _{A}^{r}=0,\quad r=1,\ldots ,3N-6.$

These conditions are much like the usual Eckart conditions for vectors in an 3N-6 dimensional "internal" subspace of the 3N dimensional configuration space:

$\sum _{A=1}^{N}M_{A}\;{\tilde {\mathbf {s} }}_{A}^{r}=0\quad \mathrm {and} \quad \sum _{A=1}^{N}M_{A}\;\mathbf {R} _{A}^{0}\times {\tilde {\mathbf {s} }}_{A}^{r}=0,$

the main difference being the mass weighting in the second set of equations.

The reason for this difference is that Wilson, Decius, and Cross define the linearized coordinate S_r as a normal inner product:

$S_{r}=\sum _{A=1}^{N}\mathbf {s} _{A}^{r}\cdot \mathbf {d} _{A}.$

It is very common to define in configuration space a generalized inner product with the matrix:

$\mathbf {M} \equiv \operatorname {diag} (\mathbf {M} _{1},\ldots ,\mathbf {M} _{N})\quad \mathrm {with} \quad \mathbf {M} _{A}\equiv \operatorname {diag} (M_{A},M_{A},M_{A}),$

playing the role of metric (overlap) matrix. With the use of the generalized inner product the coordinate S_r would be expanded as

$S_{r}=\sum _{A=1}^{N}{\tilde {\mathbf {s} }}_{A}^{r}\cdot \mathbf {M} _{A}\cdot \mathbf {d} _{A}=\sum _{A=1}^{N}\;M_{A}\;{\tilde {\mathbf {s} }}_{A}^{r}\cdot \mathbf {d} _{A}.$

Clearly the two definitions are related via

$\mathbf {M} _{A}\;{\tilde {\mathbf {s} }}_{A}^{r}=\mathbf {s} _{A}^{r},$

which explains the different form of the Eckart conditions.

References

Cited reference

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.