http://www.files.chem.vt.edu/chem-ed/data/fourier.html

If  is an orthonormal wavefunction ∫* dτ = 1 ∫*H dτ ≥ Eo  is either o or an excited state with eigenvalue > Eo ∫* (H – Eo) dτ = ∫* H dτ – Eo∫*  dτ = ∫* H dτ – Eo  = Σiciφi ∫* (H – Eo) dτ = ∫ (Σici* φi * ) (H – Eo) (Σiciφi) dτ Harry Kroto 2004 2

Since the φi’s are eigenfunctions of H … ie ∫φi*Hφidτ = Ei ∫* (H – Eo) dτ = ∫ (Σici* φi * ) (H – Eo) (Σiciφi) dτ = Σici* ci (Ei – Eo) All the ci* ci are positive numbers and by definition Ei ≥ Eo ∫* (H – Eo) dτ ≥ 0 ∫*H dτ ≥ Eo Thus the Variation Principle states that no calculated wavefunction can have an eigenvalue greater than the experimentally determined eigenvalue 3

Herzberg and Monfils obtained 36 113.6 I just got 36 117.1 Herzberg and Monfils obtained 36 113.6 Kolos and Wolniewicz calculated 36 118 Herzberg between 36 116.3 – 36 118.3 4

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Since the φi’s are eigenfunctions of H … ie ∫* (H – Eo) dτ = ∫ (Σici* φi * ) (H – Eo) (Σiciφi) dτ Since the φi’s are eigenfunctions of H … ie ∫φi*Hφidτ = Ei ∫* (H – Eo) dτ = ∫ (Σici* φi * ) Σi (Ei – Eo) (Σiciφi) dτ = Σici* ci (Ei – Eo) ∫* (H – Eo) dτ ≥ 0 ∫*H dτ ≥ Eo 6

¾ (109 677.58) = 82 258.19 ¾ R 118 375.3 82 258.19 36 117.1 7

¾ (109 677.58) = 82 258.19 ¾ R 118 375.3 82 258.19 36 117.1 8

Herzberg and Monfils obtained 36 113.6 ¾ (109 677.58) = 82 258.19 ¾ R I get 36 117.1 Herzberg and Monfils obtained 36 113.6 Kolos and Wolniewicz calculated 36 118 Herzberg between 36 116.3 – 36 118.3 9

Men’n” Franck Condon II Harry Kroto 2004 10