Generalized variational principle for excited states using nodes of trial functions.

The familiar variational principle provides an upper bound to the ground-state energy of a given Hamiltonian. This allows one to optimize a trial wave function by minimizing the expectation value of the energy. This approach is also trivially generalized to excited states, so that given a trial wave function of a certain symmetry, one can compute an upper bound to the lowest-energy level of that symmetry. In order to generalize further and build an upper bound of an arbitrary excited state of the desired symmetry, a linear combination of basis functions is generally used to generate an orthogonal set of trial functions, all bounding their respective states. However, sometimes a compact wave-function form is sought, and a basis-set expansion is not desirable or possible. Here we present an alternative generalization of the variational principle to excited states that does not require explicit orthogonalization to lower-energy states. It is valid for one-dimensional systems and, with additional information, to at least some n-dimensional systems. This generalized variational principle exploits information about the nodal structure of the trial wave function, giving an upper bound to the exact energy without the need to build a linear combination of basis functions. To illustrate the theorem we apply it to a nontrivial example: the 1s2s (1)S excited state of the helium atom.