Suppose that you wanted to synthesize a polyene that has its lowest excitation energy at a wavelength of 600 nm. What would the structure of molecule be, according to free electron molecular orbital theory?
Free electron molecular orbital therby approximates polyene absorption as a particle in a box. The best results are obtained using not the literal additive length of the bonds but rather a = 1.4(k+2) Å, where k is the number of double bonds.
hc/l = DE = (h2/8ma2)[(k+1)2-k2] = (h2/8ma2)(2k+1)
Substituting for a,
DE = [h2/8m(1.4Å)2][(2k+1)/(k+2)2].
Solving for l,
l = (8m(1.4Å)2/h2)(hc)[(k+2)2/(2k+1)] = [(8m*(1.4x10-10 m)*c)/h][(k+2)2/(2k+1)].
The pertinent mass is the mass of an electron, 9.11x10-31 kg; c=3.00x108 m s-1; h = 6.63x10-34 J/s.
l = [(8*9.11x10-31 kg*(1.4x10-10 m)2*3.00x108 m s-1)/6.63x10-34 Js][(k+2)2/(2k+1)] = (6.46x10-8 m)[(k+2)2/(2k+1)] = (64.6 nm)[(k+2)2/(2k+1)].
Using guess and check methodology,
k l л/nm 2 206 14 570 15 602 16 634 The structure must have 15 double bonds: CH2=CH-(CH=CH)13-CH=CH2