1. The problem statement, all variables and given/known data The ground, 1st excited state and 2nd excited state wave functions for a vibrating molecules are: Ψ0(x) = a e-α2/2 Ψ1(x) = b (x+d) e-α2/2 Ψ2(x) = c (x2 + ex + f) e-α2/2 respectively, with constants: a, b, c, d, e and f. Use the fact that these wave functions are part of an orthogonal set of functions to find out the constants, d, e and f ONLY. 2. Relevant equations Given above 3. The attempt at a solution I used these equations: ∫(from x = -infinite to +infinite) Ψ0 Ψ1 dx = 0 b/c of orthogonality--> from which I got d = 0 ∫(from x = - infinite to + infinite) Ψ0 Ψ2 dx = 0 b/c of orthogonality--> from which I got f = -1/2α ...so, to solve for e, I decided to use ∫ (from x = -infinite to + infinite) Ψ2 Ψ2 dx = 1 (normalization)...except, I could not figure out how to get e: c2∫(x2 + ex - 1/(2α))2e-αx2 dx = 1 ∫(x2 + ex - 1/(2α))2e-αx2 dx = 1/c2 ∫(x4 + 2ex3 -x2/α + e2x2 - ex/α +1/(4α2))e-αx2 = 1/c2 1/c2 = (3!!)(1/2α)2(π/α)½ + 2e/α2 - 1/(2α)(π/α3)½ +e2/2 (π/α3)½ + 1/(4α2)(π/α)½ ...and I am not sure if I doing right? If someone could point out the mistake or clarify, I would appreciate it!