equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. If the molecule possesses symmetries, the normal modes "transform as" an irreducible representation under its point group. The normal modes are determined by applying group theory, and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO2, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch: symmetric stretching: the sum of the two C–O stretching coordinates; the two C–O bond lengths change by the same amount and the carbon atom is stationary. Q = q1 + q2 asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the other decreases. Q = q1 − q2 When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are principally C–H stretching with a little C–N stretching; Q1 = q1 + a q2 (a << 1) principally C–N stretching with a little C–H stretching; Q2 = b q1 + q2 (b << 1) The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method. Newtonian mechanics Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere. By Newton's second law of motion this force is also equal to