The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by
Edgar Bright Wilson to obtain certain ''internal coordinates'' for a
vibrating
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
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
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, 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 (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, 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 ''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
A Hessian is an inhabitant of the German state of Hesse.
Hessian may also refer to:
Named from the toponym
*Hessian (soldier), eighteenth-century German regiments in service with the British Empire
**Hessian (boot), a style of boot
**Hessian f ...
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 ''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 s
0 of ''V'' gives the positive definite and symmetric matrix G = g(s
0)
−1.
One can solve the two matrix problems
:
simultaneously, since they are equivalent to the
generalized eigenvalue problem
:
where
where ''f
i'' is equal to
(
is the frequency of normal mode ''i'');
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 a function of time; see the article on
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'':
\v ...
s.
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
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
At is known as a ''Wilson s-vector''.
If we put the
into a (3''N'' − 6) × 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
A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz ...
. Now,
:
which can be inverted and put in summation language:
:
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.
These quantities are related by a variety of matrices. For clarity, we provide the coordinate systems and their interrelations here.
The relevant coordinates are:
*
Cartesian coordinates for each atom
*
Internal coordinates for each atom
*
Mass-weighted Cartesian coordinates
*
Normal coordinates
These different coordinate systems are related to one another by:
*
, i.e. the matrix
transforms the Cartesian coordinates to (linearized) internal coordinates.
*
i.e. the mass matrix
transforms Cartesian coordinates to mass-weighted Cartesian coordinates.
*
i.e. the matrix
transforms the normal coordinates to mass-weighted internal coordinates.
*
i.e. the matrix
transforms the normal coordinates to internal coordinates.
Note the useful relationship:
These matrices allow one to construct the G matrix quite simply as
Relation to 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,
:
References
Further references
*
*
*{{cite book , first1=E. B. , last1=Wilson , first2=J. C. , last2=Decius , first3=P. C. , last3=Cross , title=Molecular Vibrations , location=New York , origyear=1955 , publisher=Dover , date=1995 , isbn=048663941X , url-access=registration , url=https://archive.org/details/molecularvibrati00wils
Spectroscopy
Molecular physics
Quantum chemistry