Thanks, interesting approach from quantum mechanics as operators. I have seen that it works also if we take into account that $\nabla^{\mu}\nabla_{\nu}\phi=\nabla_{\nu}\nabla^{\mu}\phi$ for a scalar field in the first term, the term $\nabla_{\mu}\nabla_{\nu}\nabla^{\mu}\phi$ becomes $\nabla_{\mu}\nabla^{\mu}\nabla_{\nu}\phi$.