  We construct Lie algebras arising from commutators of the harmonic
Hamiltonian and the perturbed anharmonic Hamiltonian. From there we form a very
specific element of the associated Lie group and transform the unperturbed
Hamiltonian into the perturbed Hamiltonian minus perturbation energies. An
explicit formula is given for the first order perturbation energy of all
Hamiltonians of the form H = a^{\dagger}a + 1/2 + \lambda x^n.
