^{1}, Jiaxiang Wang

^{2}and Sabre Kais

^{1,a)}

### Abstract

We present a theoretical framework which describes multiply charged atomic ions, their stability within super-intense laser fields, and also lay corrections to the systems due to relativistic effects. Dimensional scaling calculations with relativistic corrections for systems: H, H^{−}, H^{2 −}, He, He^{−}, He^{2 −}, He^{3 −} within super-intense laser fields were completed. Also completed were three-dimensional self consistent field calculations to verify the dimensionally scaled quantities. With the aforementioned methods the system's ability to stably bind “additional” electrons through the development of multiple isolated regions of high potential energy leading to nodes of high electron density is shown. These nodes are spaced far enough from each other to minimize the electronic repulsion of the electrons, while still providing adequate enough attraction so as to bind the excess electrons into orbitals. We have found that even with relativistic considerations these species are stably bound within the field. It was also found that performing the dimensional scaling calculations for systems within the confines of laser fields to be a much simpler and more cost-effective method than the supporting D = 3 SCF method. The dimensional scaling method is general and can be extended to include relativistic corrections to describe the stability of simple molecular systems in super-intense laser fields.

We would like to thank the Army Research Offices for funding this project. Jiaxiang Wang would like to thank National Science Foundation (NSF) China for the support by Grant No. 10974056. We would also like to thank Professor Dudley Herschbach of stimulating discussion of the materials within this paper.

I. INTRODUCTION

II. RELATIVISTIC CORRECTIONS

A. Non-relativistic methodology

B. The relativistic mass gauge

C. Trajectory corrections

D. 1D particle in a box

E. Potential under consideration

III. METHODOLOGY FOR D = 3 CALCULATIONS

IV. DIMENSIONAL SCALING: CALCULATIONS AND CONSIDERATIONS

A. Methodology

B. Planar infinite-D Hamiltonian

V. RESULTS AND DISCUSSION

VI. CONCLUSION

### Key Topics

- Relativistic corrections
- 13.0
- Dirac equation
- 7.0
- Electric fields
- 7.0
- Relativistic effects
- 7.0
- Schroedinger equations
- 7.0

## Figures

Non-relativistic effective potential for a 1D particle in a box under different laser intensity, measured by α_{0}.

Non-relativistic effective potential for a 1D particle in a box under different laser intensity, measured by α_{0}.

Relativistic corrections to the effective potential for different laser fields.

Relativistic corrections to the effective potential for different laser fields.

Effective potential for two alternating electric fields superposed along and , respectively, with different colors.

Effective potential for two alternating electric fields superposed along and , respectively, with different colors.

Contour plots of dressed potential for (clockwise) α_{0} = 0, 25, 100, 50. Note both the shift in regime as α_{0} grows and the key below the plots for the interpretation of the intensity of the contours.

Contour plots of dressed potential for (clockwise) α_{0} = 0, 25, 100, 50. Note both the shift in regime as α_{0} grows and the key below the plots for the interpretation of the intensity of the contours.

Three dimensional plots of the potential energy, , as a function of x and z coordinates for the case α_{0} = 100. Left and right of above are two different angles of the same surface.

Three dimensional plots of the potential energy, , as a function of x and z coordinates for the case α_{0} = 100. Left and right of above are two different angles of the same surface.

Plots of both molecular energies and binding energy for (clockwise): with non-relativistic trajectory; , relativistic trajectory; , relativistic trajectory; and with non-relativistic trajectory. All may be read as yellow: Binding energy; blue: Hydrogen energy; purple: H^{−} energy.

Plots of both molecular energies and binding energy for (clockwise): with non-relativistic trajectory; , relativistic trajectory; , relativistic trajectory; and with non-relativistic trajectory. All may be read as yellow: Binding energy; blue: Hydrogen energy; purple: H^{−} energy.

Top: Probability distribution for H^{−}, a two electron system. Bottom: Probability distribution for He^{−}, a three electron system.

Top: Probability distribution for H^{−}, a two electron system. Bottom: Probability distribution for He^{−}, a three electron system.

The above displays the relationship between the system's geometry with respect to the electrons and the coordinates use in Eq. (40).

The above displays the relationship between the system's geometry with respect to the electrons and the coordinates use in Eq. (40).

Above are plots of the binding energies of, from left to right and top to bottom: H^{−}, H^{−2}, He^{−}, He^{−2}, and He^{−3}.

Above are plots of the binding energies of, from left to right and top to bottom: H^{−}, H^{−2}, He^{−}, He^{−2}, and He^{−3}.

Plots of binding energy comparisons for (clockwise): comparison plot of versus , non-relativistic trajectory; with relativistic trajectory m_{0} and m_{ r }, differences value of Hydrogen energy between use of m_{0} and m_{ r }; and a comparison of relativistic trajectory (both m_{0} and m_{ r }) with non-relativistic trajectory both using .

Plots of binding energy comparisons for (clockwise): comparison plot of versus , non-relativistic trajectory; with relativistic trajectory m_{0} and m_{ r }, differences value of Hydrogen energy between use of m_{0} and m_{ r }; and a comparison of relativistic trajectory (both m_{0} and m_{ r }) with non-relativistic trajectory both using .

Plots of binding energy for the H^{−} (left) and He^{−} (right) systems. Top, Non-normalized plots of the calculation data showing agreement between the methods. Bottom, -B.E./B.E._{ max } to emphasis the qualitative similarity between the methods as they share minima for the B.E. curves.

Plots of binding energy for the H^{−} (left) and He^{−} (right) systems. Top, Non-normalized plots of the calculation data showing agreement between the methods. Bottom, -B.E./B.E._{ max } to emphasis the qualitative similarity between the methods as they share minima for the B.E. curves.

Top row: Plots of the probabiltiy distribution for the corrected (left) and non-corrected (right) H^{−} system, directly below is a superimposition of the trajectory upon the probability density function plot to emphasis their relation. Bottom row: Contour plots of the H^{−} system, both corrected (left) and non-corrected (right), note the different scales on the vertical (z) axis and the more diffuse behavior of the corrected system.

Top row: Plots of the probabiltiy distribution for the corrected (left) and non-corrected (right) H^{−} system, directly below is a superimposition of the trajectory upon the probability density function plot to emphasis their relation. Bottom row: Contour plots of the H^{−} system, both corrected (left) and non-corrected (right), note the different scales on the vertical (z) axis and the more diffuse behavior of the corrected system.

The electronic elliptical contribution can be seen as the function whose major axis lies in the z-direction (vertical) and the magnetic contribution has an orthogonal orientation.

The electronic elliptical contribution can be seen as the function whose major axis lies in the z-direction (vertical) and the magnetic contribution has an orthogonal orientation.

Within the above plot, the total trajectory can be seen, the amplitude of the major and minor axes are mediated in value between those from the electronic and magnetic components and the orientation is set off by an angle whose value respects the same coefficients as the relative amplitudes.

Within the above plot, the total trajectory can be seen, the amplitude of the major and minor axes are mediated in value between those from the electronic and magnetic components and the orientation is set off by an angle whose value respects the same coefficients as the relative amplitudes.

## Tables

Results of Mulliken population analysis, note that there are four outer orbitals the table contains one of the four values.

Results of Mulliken population analysis, note that there are four outer orbitals the table contains one of the four values.

Differences present when relativistic considerations are undertaken for these multiply charged ions. The non-relativistic values were taken from Ref. 14.

Differences present when relativistic considerations are undertaken for these multiply charged ions. The non-relativistic values were taken from Ref. 14.

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