 # Question & Answer: Molecule PO has the following properties in its ground electronic state: w_e = 1233.34 cm^-1….. Molecule PO has the following properties in its ground electronic state: w_e = 1233.34 cm^-1, w_e x_e = 6.56 cm^-1, B = 0.734 cm^-1 Be sure to account for the anharmonic correction, w_ex_e, in the following calculations. Quote your answer to the following questions to 2 decimal places. a) What is its zero point energy? cm^-1 b) What is energy of the v = 1 vibrational level? cm^-1 c) Where would its IR fundamental occur? cm^-1 d) Where would the first overtone transition be found? cm^-1 e) How far would the band center of the first hot-bane (i.e. v = 1 to v = 2) be shifted from the band center of the fundamental? (Include the sign + or -1 to indicate the direction of the shift to higher or lower cm^2) cm^2

Hooke’s law states that the force tending to restore an atom to its equilibrium position is proportional to the displacement of that from its equilibrium position i.e restoring force

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Question & Answer: Molecule PO has the following properties in its ground electronic state: w_e = 1233.34 cm^-1…..
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F= k(r – re) (r-re) – displacement

A system which does not obey Hooke’s law is called an anharmonic oscillator .

P.M . Morse suggested an emperical expression for energy of an anharmonic oscillator

E= Deq[ 1- exp{a(req- r)}]2 Deq – dissociation energy a- constant (req- r)- displacement

Using the Morse energy the Schrodinger eq can solved to cal. the energy of vibrational levels can be given as

Ev = (v+1/2)we – (v+1/2)2wexe we- oscillation frequency xe- anharmonicity const

a. Zero point energy(E0) = 1/2we(1- 1/2xe) = 1/2×1233.34 – 1/4x 6.56 = 615.03 cm-1

b. Ev = (1+1/2)x 1233.34 – (1+ 1/2)26.56= 1835.25 cm-1

c. v= 0 to v= 1   v = +1

E = EV=1 – Ev = 0 = we(1 – 2xe) cm-1 we- IR fundamental

line near we is IR fundamental i.e 1233.34cm-1

d. v= 0 to v =2 , v = +2

E = 2we(1- 3xe)cm-1

line near 2we is first overtone

e. v=1 to v=2 , v = +1   E= we(1- 4xe)cm-1

E= Deq[1- exp{a(req – r )}]2 req – internuclear dist