Problem 1.20 from Modern Quantum Chemistry, Intro to electronic structure theory

ClearAll[a, b, d, o,c, omega, s, minE]

$Assumptions=(And@@(#>0&)/@{ a, b, d}) ;

c := {Cos[#], Sin[#]} & ;
o := {{a, b}, {b, d}} ;
omega := c[#] . o . c[#] & ;
omega[θ] // FullSimplify // DisplayForm
s = Solve[D[omega[t], t] == 0, {t}] // FullSimplify

ss = (s /. C[1] -> 0) 
minE = (omega[t] /. ss) // FullSimplify // DeleteDuplicates ;
minE // DisplayForm