Fictitious-time wave-packet dynamics. II. Hydrogen atom in external fields

In the preceding paper [T. Fabčič, J. Main, and G. Wunner, Phys. Rev. A 79, 043416 (2009)] “restricted Gaussian wave packets” were introduced for the regularized Coulomb problem in the four-dimensional Kustaanheimo-Stiefel coordinates, and their exact time propagation was derived analytically in a fictitious-time variable. We now establish the Gaussian wave-packet method for the hydrogen atom in static external fields. A superposition of restricted Gaussian wave packets is used as a trial function in the application of the time-dependent variational principle. The external fields introduce couplings between the basis states. The set of coupled wave packets is propagated numerically, and eigenvalues of the Schrödinger equation are obtained by the frequency analysis of the time autocorrelation function. The advantage of the wave-packet propagation in the fictitious-time variable is that the computations are exact for the field-free hydrogen atom and approximations from the time-dependent variational principle only stem from the external fields. Examples are presented for the hydrogen atom in a magnetic field and in crossed electric and magnetic fields.

Figure 1

Sketch of the manifold $M$ of approximation of the trial wave function $\psi \left(\mathbf{z}\right)$. The variational evolution of the trial function denoted by the arrow with the white arrowhead is obtained as the projection of the exact time evolution $-iH\psi$, denoted by the arrow with the black arrowhead, onto the tangent space $T_{\psi }M$ of the manifold $M$ in the point $\psi$.

Figure 2

(Color online) Fictitious-time evolution of the state [Eq. (33)] with ${\rho }_0=6.0,z_0=0$ and a nonzero initial mean momentum. The wave function is plotted for different values of the dimensionless fictitious time $\tau$. The initial wave packet gradually becomes delocalized. Lengths are given in scaled atomic units $n_{\text{eff}}{a}_0$ with $a_0$ the Bohr radius [see Eq. (2)].

Figure 3

(Color online) Real part of the autocorrelation function $C^{\pm }\left(\tau \right)=⟨{\psi }_0^{\pm }\left(0\right)|{\psi }_0^{\pm }\left(\tau \right)⟩$ for the GWP [Eq. (33)] with the center ${\rho }_0=6.0,z_0=0$. Signal of the projected state with (a) even parity and (b) odd parity. The fictitious time $\tau$ and the signal $C\left(\tau \right)$ are in dimensionless units.

Figure 4

(Color online) Spectra with (a) even and (b) odd $z$ parities extracted from the autocorrelation function $C^{\pm }\left(\tau \right)=⟨{\psi }_0^{\pm }\left(\tau =0\right)|{\psi }_0^{\pm }\left(\tau \right)⟩$ computed from the evolution of the wave function [Eq. (33)] plotted in Fig. 2. The amplitudes (red lines in the upper panels) are given by the magnitude of overlap between the initial wave function and the respective eigenstates. For comparison the positions of the numerically exact eigenvalues obtained from a diagonalization are plotted with blue lines in the lower panels of the figures. The related eigenenergies and the magnetic field strength follow simply from Eq. (4). The effective quantum number $n_{\text{eff}}$ and the overlap matrix elements are in dimensionless units.

Figure 5

(Color online) Effective quantum numbers $n_{\text{eff}}$ at the field-free ionization threshold $E=\alpha =0$ for states of (a) even and (b) odd parities. The propagation involves $N=90$ basis states. The variational results (red lines in the upper panels) are in excellent agreement with the exact time-independent results (blue lines in the lower panels). The effective quantum number $n_{\text{eff}}$ and the overlap matrix elements are in dimensionless units.

Figure 6

(Color online) Spectra with (a) even and (b) odd $z$ parities of Hamiltonian (2) with $\alpha =0.5$, $\beta =0.05$, and $\zeta =0.01$ obtained from the propagation of two different 3D GWPs. Green and red lines (upper panels in the figures): ${\mathbf{x}}_0=\left(6,0,0\right)$, ${\mathbf{p}}_0=\left(0,\pm 1/\sqrt{2},1/\sqrt{2}\right)$, respectively. The eigenvalues are extracted from the autocorrelation function by the Fourier transformation. The peak positions agree very well with the numerically exact eigenvalues of the effective quantum number marked by blue lines in the lower panels of the figures. The related eigenenergies and the field strengths follow from Eq. (4). The effective quantum number $n_{\text{eff}}$ and amplitudes are in dimensionless units.