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The variationally stable procedure for Nth-order perturbative-transition-matrix elements introduced by the authors (Phys. Rev. Lett. 61, 404 (1988)) is presented here in detail. Its key features are that it is noniterative, involves only two unknown functions regardless of the value of N, is stable even near intermediate-state resonances (due to the presence of energy numerators rather than energy denominators), and uses the inverse of the perturbation operator for Ngreater than or equal to3. Explicit formulas are presented for application to high-order multiphoton processes in atomic hydrogen. Numerical results for multiphoton-ionization cross sections (for two, three, and seven photons), for the frequency dependence of the nonlinear susceptibilities for harmonic generation (for the third, fifth, and seventh harmonics), and for harmonic-generation transition rates up to 11th order for lambda = 1064 nm are presented for the atomic hydrogen ground state and compared with results of others by more standard procedures. We also present detailed analyses of the Z scaling of our results for hydrogenic systems, of the use of the imaginary part of the appropriate 2N-photon nonlinear susceptibility to obtain N-photon-ionization cross sections, and of the relation of the variational principle presented here to variational principles for scattering processes of Nuttall andmore » Cohen (Phys. Rev. 188, 1542 (1969)) and of Schwinger (Phys. Rev. 72, 742 (1947)).« less

Variational principle for high-order perturbations with application to multiphoton processes for the H atom