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. F o r c e = − k Q {\displaystyle \mathrm {Force} =-kQ\!} By Newton's second law of motion this force is also equal to a reduced mass, μ , times acceleration. F o r c e = μ d 2 Q d t 2 {\displaystyle \mathrm {Force} =\mu {\frac {d^{2}Q}{dt^{2}}}} Since this is one and the same force the ordinary differential equation follows. μ d 2 Q d t 2 + k Q = 0 {\displaystyle \mu {\frac {d^{2}Q}{dt^{2}}}+kQ=0} The solution to this e...
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