Perturbation expansion of variational principles at arbitrary order

When perturbation theory is applied to a quantity for which a variational principle holds (eigenenergies of Hamiltonian, Hartree-Fock or density-functional-theory, etc.), different variation-perturbation theorems can be derived. A general demonstration of the existence of variational principles for an even order of perturbation, when constraints are present, is provided here. Explicit formulas for these variational principles for even orders of perturbation, as well as for the ‘‘2n+1 theorem,’’ to any order of perturbation, with or without constraints, are also exhibited. This approach is applied to the case of eigenenergies of quantum-mechanical Hamiltonians, studied previously by other methods.