HuKel Theory
(a) ethene H2C=CH2

A={{a-e,b},{b,a-e}}

eqn=Det[A]==0

Solve[eqn,e]

(b) 1,3 butadiene H2C=CH-CH=CH2

A={{a-e,b,0,0},{b,a-e,b,0},{0,b,a-e,b},{0,0,b,a-e}}

MatrixForm[A]

eqn=Det[A]==0

N[Solve[eqn,e]]

Expand[%]

(c) cyclo-butadiene HC=CH-CH=CH

ChemicalData["Cyclobutadiene"]

A={{a-e,b,0,b},{b,a-e,b,0},{0,b,a-e,b},{b,0,b,a-e}}

eqn=Det[A]==0

sol=Solve[eqn,e]

eqn=A.{c1,c2,c3,c4}==0

e=e/. sol[[3]]

Solve[eqn,{c1,c2,c3,c4}]