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Chemistry LibreTexts

Coordinate systems and background

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  • Equations of motion with in the harmonic approximation {page.toc}}

    Mass-weighted Cartesian coordinates

    The kinetic energy is

    The potential energy is

    The fij's are constants given by

    and the set of coordinates qi are mass-weighted Cartesian coordinates.

    Newton's equations of motion can be written in the form

    Substitution of the kinetic and potential energy yields

    Solutions can have the form

    By substituting these solutions into the linear differential equations a set of algebraic equations results

    This set of equations forms a matrix known as the secular determinant.

    Cartesian coordinates

    The kinetic energy is

    The potential energy is

    The fij''s are constants given by

    The secular equations in Cartesian coordinates are

    An example of a three-body problem demonstrates the application of these equations.

    Normal coordinates

    The solutions represent oscillatory motion of each atom about its equilibrium position with amplitude Aik, frequency lk1/2/2p, and phase f. The amplitudes are different for each atom, but their frequencies and phases are identical. Thus, each atom reaches the maximum displacement and equilibrium position at the same time. Such a concerted motion is called a normal mode of vibration.

    Normal coordinates are related the mass-weighted Cartesian coordinates by the linear equations

    The coefficients are chosen so that the kinetic and potential energies have the form

    The eigenvalues lk are the same as in the mass-weighted Cartesian equations above. The coefficients lik have the property (l-1) ki = lik.

    Internal coordinates

    For non-linear molecules, six of the 3N coordinates can be removed by defining the problem in the center-of-mass coordinate system. There are three coordinates due to translation of the center of mass and three coordinates due to rotation.

    Instead of using the three Cartesian coordinates to describe the displacement of each atom, it is convenient to introduce a vector ra for each atom a whose components along the internuclear axis are the Cartesian displacements for each atom. This is known as a bond stretching internal coordinate. We can also define internal coordinates for valence angle bending, torsions, out-of-plane wags, and linear bends.

    For small motions we can define 3N - 6 internal coordinates St in terms of Cartesian coordinates

    The G matrix is the inverse mass matrix used to define the kinetic energy in internal coordinates. The G matrix is defined by the transformation

    so that the kinetic energy is

    The potential energy in internal coordinates is

    The transformation from Cartesian to internal coordinates and the generation of the G matrix can be illustrated by the non-linear three body system.

    As above for the other coordinate systems an assumed solution leads to a set of linear equations. In matrix form we can write this as

    |F - G-1l| = 0

    or multiplying both sides by G we have

    |GF - El| = 0.