Question

. The rotational motion of molecules has an effect on the equilibrium separation of the nuclei, a phenome- non known as bond stretching. To model this effect, con- sider a diatomic molecule with reduced mass u, oscilla- tor frequency wo, and internuclear separation Ro when the angular momentum is zero. The effective potential energy for nonzero values of l is then (see Section 8.5) ħ2 ont OBRO Veff = { uwz(- Ro)2 + ece + 1) sileib in I DEOQC1103 2ur? 3119 (a) Minimize the effective potential Uef(r) to find an equation for the equilibrium separation of the nuclei, Re, when the angular momentum is l. Solve this equation approximately, assuming € < uw Rz/. Near the corrected equilibrium point, Re, the effec- tive potential again is nearly harmonic and can be writ- ten approximately as 10 nited Veff ~ į uw?(r - Re)? + Uosi asuisi va Find expressions for the corrected oscillator frequency w and the energy offset Uo by matching Ueff and its first two derivatives at the equilibrium point Re. Show that the fractional change in frequency is given by on Aw3l(l + 1)ħ? 1 olun brei't add this w 2u?w RT s essid

Answer 1

. At equilibrium separation, R a separation, R. . dueffl =0 dr tre => Mwo? (Re- Ro) - l (1+1)h². MRP => Re = Ro t l (2+1) ħ + )" too l << MwoRo the second term on the right represents a small correction. So, let's approximate it by taking R a Ro > Rex Rot ell + 1) th ( 1 ) l (2+1) h27a + The value of Ueff at Re is the energy offset Uo: U = Ueff (R) = 1/2 MW? M? WER l(4+1) ħ² - 2UR? - Teletnh? Trece+10th? + 1] | ZuR 1 M² wordt 1 * 1(2+1) ? a lleti) t" 2MR? QMR² The new Curvature at mwi = d'Ueff eq. point is: = mw? + 31 (2+1) h we = dre? Ire 7 Wi' - w * LA AR,1 aw ² + 32 (2+1) 2² MRT M?R?

:. The difference between w & Wo is very small. We can write wer-w² = (we-wo) (Wet wo) e QWoAW => = 2 Wo 1 2 Wo A w 31 (2+1)ħ² 2M²w ² Rod (ans)