When an atom or a molecule is exposed to an electric field, the discrete electronic spectrum turns into a spectrum of resonances with complex energies due to the mixing of the bound wave function of each unperturbed stationary state with the asymptotic form of the Airy function. The complex energies are eigenvalues of a non-Hermitian Schrödinger equation whose solutions have a special outgoing-wave asymptotic form, which is derived rigorously. At least three coordinate transformations (rotation, translation, and a combination of these) regularize the dc-field-induced resonance function, the most efficient one being the long-honored rotation, f(x)=${\mathit{xe}}^{\mathit{i}\mathrm{\theta }}$, which causes the function to dissipate asymptotically as ${\mathit{e}}^{\mathrm{-}\gamma \mathit{x}3/2}$. This finding, obtained from first principles, explains previous results that have been obtained either via trial computations or via elaborate mathematical analyses of the spectrum. The theory is further developed toward the efficient computation of such resonances with ${\mathit{scrL}}^2$ functions. Starting from the resonance-wave-function form ${\mathrm{\Psi }}^{\mathrm{res}}$=a${\mathrm{\Psi }}_0$+${\mathit{bX}}_{\mathrm{as}}$, the computation of the localized ${\mathrm{\Psi }}_0$ is carried out on the real axis, and only the free-electron function belonging to ${\mathit{X}}_{\mathrm{as}}$ is rotated and optimized in the complex plane. Two applications are presented. The first involves a numerically solvable one-dimensional model of a shape resonance in a dc field for a large range of field strengths.

The second involves the hydrogen atom in its ground as well as in its first excited state. In the first example, ${\mathrm{\Psi }}_0$ is obtained numerically in an explicitly constructed effective potential containing partly the effect of the field. To this ‘‘dressed’’ ${\mathrm{\Psi }}_0$, the asymptotic part ${\mathit{X}}_{\mathrm{as}}$ is then added as a sum of an only ten back-rotated complex Slater-type orbitals (STO’s), whose coefficients and nonlinear parameter are optimized from the diagonalization of the full Hamiltonian until stability of the complex eigenvalue is achieved. In the second example, ${\mathrm{\Psi }}_0$ was chosen as the exact 1s for the ground state and as the two roots 2s±2${\mathit{p}}_0$ of the total Hamiltonian for the excited state. ${\mathit{X}}_{\mathrm{as}}$ was expressed in terms of angular momentum symmetry blocks up to l=8, with each symmetry expanded in terms of ten complex STO’s. No optimization of θ or other nonlinear parameters was done. Comparison with the exact results for the model potential and with published ones for the H atom shows very good agreement, thereby demonstrating the efficiency and reliability of the theory as well as its potential for treating N-electron systems.

• Received 13 August 1991

DOI:https://doi.org/10.1103/PhysRevA.45.349