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Vibrational Spectroscopy:

Absorption(Diatomic Molecules)

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Born Oppenheimer

𝜓ሺ𝑅𝐴,𝑅𝐵,𝑟𝐴,𝑟𝐵ሻ== 𝜓𝑛𝑢𝑐𝑙𝑒𝑎𝑟 ሺ𝑅𝐴,𝑅𝐵ሻ.𝜓𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐 (𝑟𝐴,𝑟𝐵)

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𝑉ሺ𝑟ሻ= 𝐷𝑒൫1− 𝑒−𝛼(𝑟−𝑟𝑒)൯2 Morse Potential

𝐸= ൬𝑣+ 12൰ℎ𝝂𝒐 −ቂቀ𝑣+ 12ቁℎ𝝂𝒐ቃ𝟐𝟒𝑫𝒆

𝛼= ඨ𝐾2𝐷𝑒 𝜈𝑜 = 12𝜋ඨ𝐾𝜇

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𝜓𝑛ሺ𝑥ሻ= ට 12𝑛𝑛!ቀ2𝑚𝜔ℎ ቁ

14 𝑒−𝜋𝑚𝜔 𝑥2ℎ 𝐻𝑛(ට2𝜋𝑚𝜔ℎ 𝑥))

𝐻𝑛ሺ𝑥ሻ= (−1)𝑛𝑒𝑥2 𝑑𝑛𝑑𝑥𝑛 𝑒−𝑥2)

𝜇𝑥,𝑦,𝑧𝑖𝑗 = න𝜓𝑖∗𝜇𝑥,𝑦,𝑧𝜓𝑗𝑑𝜏 ≠ 0

𝑥.𝐻𝑛ሺ𝑥ሻ= 12𝐻𝑛+1ሺ𝑥ሻ+ 𝑛.𝐻𝑛−1ሺ𝑥ሻ Δv=±1

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Rotational Spectroscopy:

Absorption

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𝐽22𝐼𝜓 = ℎ2𝐽(𝐽+ 1)8𝜋2𝐼 𝜓

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𝐸= 𝐵𝐽ሺ𝐽+ 1ሻ 𝐵= ℎ28𝜋2𝐼 ∆𝐸= 𝐸ሺ𝐽+ 1ሻ− 𝐸(𝐽) = 2𝐵ሺ𝐽+ 1ሻ

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Centrifugal Distortion

𝐸= 𝐵𝐽ሺ𝐽+ 1ሻ− 𝐷𝐽2(𝐽+ 1)2 𝐷= 4𝐵3𝜔2

∆𝐸= 𝐸ሺ𝐽+ 1ሻ− 𝐸ሺ𝐽ሻ= 2𝐵ሺ𝐽+ 1ሻ− 4𝐷(𝐽+ 1)3

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Roto-Vibrational Coupling

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𝐸= 𝐵𝐽ሺ𝐽+ 1ሻ+ሺ𝐴− 𝐵ሻ𝐾2

Prolate A>B=C(cigar shaped)

Oblate A<B=C(frisbee shaped)