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                    uV = Y e-Ev/kT                 (5.16)
                         v

where v is the vibrational quantum number and E  is the
                                              v
energy of the v-th vibrational state. The. molecule can be treated as an harmonic oscillator having a molecular binding force described by Hooke's law, with the corresponding potential


                    V(r) =   k(r - r )2            (5.17)
                             2      e

where r is the separation distance between the atoms, k is the force constant, and re the equilibrium position. The oscillatory energy eigenvalues corresponding to (5.17) are


               Ev= (v + 1)hw, v=O, 1, 2, . . .     (5.18)

where


       w = Vk7jY, the classical vibrational frequency

       = M1m2/(m1 + M2), the reduced mass

When (5.18) is used in (5.16), the partition function becomes


      -(v + 1)h1//kT      1
uV       e              =   csch((I w/kt)          (5.19)


The expression (5.19) should be very accurate for small