What are the main forms of energy?
Kinetic energy
Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy is the movement energy of an object. Kinetic energy can be transferred between objects and transformed into other kinds of energy.
Eckart conditions
The vibrational energy of the molecule can be written in terms of coordinates with respect to the Eckart frame as
Because the Eckart frame is non-inertial, the total kinetic energy comprises also centrifugal and Coriolis energies. These stay out of the present discussion. The vibrational energy is written in terms of the displacement coordinates, which are linearly dependent because they are contaminated by the 6 external modes, which are zero, i.e., the d's satisfy 6 linear relations. It is possible to write the vibrational energy solely in terms of the internal modes "q" ("r" =1, ..., 3"N"-6) as we will now show. We write the different modes in terms of the displacements
The parenthesized expressions define a matrix B relating the internal and external modes to the displacements. The matrix B may be partitioned in an internal (3"N"-6 x 3"N") and an external (6 x 3"N") part,
We define the matrix M by
and from the relations given in the previous sections follow the matrix relations
and
We define
By using the rules for block matrix multiplication we can show that
where G is of dimension (3"N"-6 x 3"N"-6) and N is (6 x 6).
The kinetic energy becomes
where we used that the last 6 components of v are zero. This form of
the kinetic energy of vibration enters Wilson's GF method. It is of some interest to point out that the potential energy in the harmonic approximation can be written as follows
where H is the Hessian of the potential in the minimum and F, defined by this equation, is the F matrix of the GF method.