GF method

Wilson's GF method, sometimes referred to as FG method, is a classical mechanical method to obtain certain "internal coordinates" fora 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) which is a function of 3"N"-6 coordinates.The internal degrees of freedom "q"₁, ..., "q"_3"N"-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 [E. B. Wilson, Jr."Some Mathematical Methods for the Study of Molecular Vibrations", J. Chem. Phys. vol. 9, pp. 76-84 (1941)] 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 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,:$2V\; approx\; sum\_\{s,t=1\}^\{3N-6\}\; F\_\{st\}\; S\_s,\; S\_t$.

The classical vibrational kinetic energy has the form::$2T\; =\; sum\_\{s,t=1\}^\{3N-6\}\; g\_\{st\}(mathbf\{q\})\; dot\{S\}\_sdot\{S\}\_t\; ,$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:since they are equivalent to the generalized eigenvalue problem:where 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 factthat 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 methodis also referred to as Wilson's FG method.

We introduce the vectors:$mathbf\{s\}\; =\; operatorname\{col\}(S\_1,ldots,\; S\_\{3N-6\})quadmathrm\{and\}quadmathbf\{Q\}\; =\; operatorname\{col\}(Q\_1,ldots,\; Q\_\{3N-6\}),$which satisfy the relation:$mathbf\{s\}\; =\; mathbf\{L\}\; mathbf\{Q\}.$Upon use of the results of the generalized eigenvalue equationthe energy "E" = "T " + "V" (in the harmonic approximation) of the molecule becomes, :$2E\; =\; dot\{mathbf\{s^mathrm\{T\}\; mathbf\{G\}^\{-1\}dot\{mathbf\{s+mathbf\{s\}^mathrm\{T\}mathbf\{F\}mathbf\{s\}$

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

::The Lagrangian "L" = "T" - "V" is:The corresponding Lagrange equations are identical to the Newton equations :$ddot\{Q\}\_t\; +\; f\_t\; ,Q\_t\; =\; 0$for a set of uncoupledharmonic 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_A⁰the 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 the internal curvilinear coordinates "q"_"t" expresses the coordinate "S"_"t" in terms of the displacement coordinates :$S\_t\; =sum\_\{A=1\}^N\; sum\_\{i=1\}^3\; s^t\_\{Ai\}\; ,\; d\_\{Ai\}=\; sum\_\{A=1\}^N\; mathbf\{s\}^t\_\{A\}\; cdot\; mathbf\{d\}\_\{A\},\; quad\; mathrm\{for\}quad\; t\; =\; 1,ldots,3N-6,$where s_A^t is known as a "Wilson s-vector".If we put the $s^t\_\{Ai\}$ into a 3"N"-6 x 3"N" 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 $s^t\_\{Ai\}$. See for more details on this method, known asthe "Wilson s-vector method", the book by Wilson "et al.", or molecular vibration. Now, :$mathbf\{Q\}\; =\; mathbf\{L\}^\{-1\}\; mathbf\{s\}\; =\; mathbf\{L\}^\{-1\}\; mathbf\{B\}\; mathbf\{d\}\; equivmathbf\{D\}\; mathbf\{d\}.$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 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 translationof 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 internalcoordinates,:$sum\_\{A=1\}^N\; mathbf\{s\}^t\_\{A\}\; =\; 0quadmathrm\{and\}quad\; sum\_\{A=1\}^N\; mathbf\{R\}^0\_A\; imes\; mathbf\{s\}^t\_A=\; 0,\; quad\; t=1,ldots,3N-6.$These conditions follow from the Eckart conditions that hold for the displacement vectors, :$sum\_\{A=1\}^N\; M\_A;\; mathbf\{d\}\_\{A\}\; =\; 0\; quadmathrm\{and\}quadsum\_\{A=1\}^N\; M\_A;\; mathbf\{R\}^0\_\{A\}\; imes\; mathbf\{d\}\_\{A\}\; =\; 0.$See this article for more details.

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.
* S. Califano, "Vibrational States", Wiley, London, 1976.