GF method
Encyclopedia
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 Qk. 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 atoms in a molecule are bound by a potential energy surface
(PES) (or a force field
) which is a function of 3N-6 coordinates.
The internal degrees of freedom q1, ..., q3N-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 linearized the internal coordinates by assuming small displacements. The linearized version of the internal coordinate qt is denoted by St.
The PES V can be Taylor expanded around its minimum in terms of the St. 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 minimum of V. The first term can be included in the zero of energy.
Thus,
.
The classical vibrational kinetic energy has the form:
where gst is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivative
s. 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 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
where fi is equal to 
(
is the frequency of normal mode i); 
is the unit matrix. The matrix L-1 contains the normal coordinates Qk 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,
a vibrating semi-rigid molecule
Semi-rigid molecule
A semi-rigid molecule is a molecule which has a potential energy surface with a well-defined minimum corresponding to a stable structure of the molecule...
, the so-called normal coordinates Qk. 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
Eckart conditions
The Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
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 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
Potential energy surface
A potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...
(PES) (or a force field
Force field (chemistry)
In the context of molecular modeling, a force field refers to the form and parameters of mathematical functions used to describe the potential energy of a system of particles . Force field functions and parameter sets are derived from both experimental work and high-level quantum mechanical...
) which is a function of 3N-6 coordinates.
The internal degrees of freedom q1, ..., q3N-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
Curvilinear coordinates
Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...
, 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 version of the internal coordinate qt is denoted by St.
The PES V can be Taylor expanded around its minimum in terms of the St. The third term (the Hessian
Hessian matrix
In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
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 minimum of V. The first term can be included in the zero of energy.
Thus,
.The classical vibrational kinetic energy has the form:
where gst is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivative
Time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t\,.-Notation:...
s. 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 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
where fi is equal to ( is the frequency of normal mode i); is the unit matrix. The matrix L-1 contains the normal coordinates Qk 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 Qt as a function of time; see the article on harmonic oscillatorHarmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
s.
Normal coordinates in terms of Cartesian displacement coordinates
Often the normal coordinates are expressed as linear combinations of Cartesian displacement coordinates.
Let RA be the position vector of nucleus A and RA0
the corresponding equilibrium position. Then
is by definition the Cartesian displacement coordinate of nucleus A.
Wilson's linearizing of the internal curvilinear coordinates qt expresses the coordinate St in terms of the displacement coordinates
where sAt is known as a Wilson s-vector.
If we put the
into a 3N-6 x 3N 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 vibrationMolecular vibrationA molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion...
. Now,
In summation language:
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 St under overall rotation and translation
of the molecule, follows the same for the linearized coordinates stA.
It can be shown that this implies that the following 6 conditions are satisfied by the internal
coordinates,
These conditions follow from the Eckart conditionsEckart conditionsThe Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
that hold for the displacement vectors,
See this articleEckart conditionsThe Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
for more details.
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