^{1,a)}, Zengxiu Zhao

^{2}, Thomas Brabec

^{2}and D. M. Villeneuve

^{3}

### Abstract

High harmonic radiation is produced when atoms or molecules are ionized by an intense femtosecond laser pulse. The radiated spectrum has been shown experimentally to contain information on the electronic structure of the molecule, which can be interpreted as an image of a single molecular orbital. Previous theory for high harmonic generation has been limited to the single-active-electron approximation. Utilizing semisudden approximation, the authors develop a theory of the recombination step in high harmonic generation and tomographic reconstruction in multielectron systems, taking into account electron spin statistics and electron-electron correlations within the parent molecule and the ion. They show that the resulting corrections significantly modify the theoretical predictions, and bring them in a better agreement with experiment. They further show that exchange contributions to harmonic radiation can be used to extract additional information on the electronic wave function.

I. INTRODUCTION

II. THEORY

A. Harmonic generation in many-electron systems

B. Molecular orbital tomography

C. Self-consistent tomographic reconstruction

D. Determinantal expansion

E. Rigid orbital approximation (“Koopmans’ case”)

III. NUMERICAL IMPLEMENTATION AND TECHNICAL DETAILS

IV. NUMERICAL RESULTS AND DISCUSSION

A. Singlet molecules with nondegenerate HOMO: LiH and

B. Triplet molecules:

C. Singlet molecule with spatially degenerate HOMO:

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Wave functions
- 79.0
- Ionization
- 25.0
- Nonlinear optics
- 14.0
- Ground states
- 13.0
- Orthogonalization
- 10.0

## Figures

(Color) Three-dimensional numerical simulation of the recollision of the electron wave packet with a molecular ion . The wave packet is generated by ionization of with a sine laser pulse (, is parallel to the molecular axis). The ground state population is projected out at , with the residual renormalized to unity. The combined ionization and excitation probability at is 16.4%. The real part of the continuum wave function is shown at The coordinates are in bohrs. The color bar is in units of . The recolliding wave packet, approaching from the right, is indistinguishable from a plane wave at this scale. Deviations from the plane wave character appear as the electron approaches the nuclei, indicated by the diamonds. The scattered-wave character fully develops only after the electron has left the immediate vicinity of the nuclei, and can no longer recombine.

(Color) Three-dimensional numerical simulation of the recollision of the electron wave packet with a molecular ion . The wave packet is generated by ionization of with a sine laser pulse (, is parallel to the molecular axis). The ground state population is projected out at , with the residual renormalized to unity. The combined ionization and excitation probability at is 16.4%. The real part of the continuum wave function is shown at The coordinates are in bohrs. The color bar is in units of . The recolliding wave packet, approaching from the right, is indistinguishable from a plane wave at this scale. Deviations from the plane wave character appear as the electron approaches the nuclei, indicated by the diamonds. The scattered-wave character fully develops only after the electron has left the immediate vicinity of the nuclei, and can no longer recombine.

(Color) Components of the dipole field in , calculated in the single-determinant approximation [Eqs. (14), (35), and (36)]. Experimental internuclear separation in is used in the calculations. The isosurfaces are at , , and (bohr). The cut is along the plane. (a) The component. The component is obtained by rotation around the axis. (b) The component.

(Color) Components of the dipole field in , calculated in the single-determinant approximation [Eqs. (14), (35), and (36)]. Experimental internuclear separation in is used in the calculations. The isosurfaces are at , , and (bohr). The cut is along the plane. (a) The component. The component is obtained by rotation around the axis. (b) The component.

(Color) (a) The experimentally recovered orbital from Ref. 15. (b) Calculated Hartree-Fock HOMO of . (c) Predicted measurement using the multielectron theory for high harmonic generation— [Eq. (50)], integrated along the direction. Images (b) and (c) were processed with a Fourier bandpass filter, matching the experimental range of harmonics used to obtain panel (a).

(Color) (a) The experimentally recovered orbital from Ref. 15. (b) Calculated Hartree-Fock HOMO of . (c) Predicted measurement using the multielectron theory for high harmonic generation— [Eq. (50)], integrated along the direction. Images (b) and (c) were processed with a Fourier bandpass filter, matching the experimental range of harmonics used to obtain panel (a).

(Color) Components of the recombination dipole field in , calculated in the minimal determinantal approximation [Eqs. (56)–(58)]. Experimental internuclear separation in is used. The isosurfaces are at , , and (bohr). (a) The component. The cut is along the plane. (b) The component. The cut is along the plane. (c) The component. [(d)–(f)] Views of the dipole field components [(a)–(c)] along the main diagonal. The panels (d)–(f) are not at the same scale as panels (a)–(c). The components of the dipole field can be obtained by a rotation of around the axis.

(Color) Components of the recombination dipole field in , calculated in the minimal determinantal approximation [Eqs. (56)–(58)]. Experimental internuclear separation in is used. The isosurfaces are at , , and (bohr). (a) The component. The cut is along the plane. (b) The component. The cut is along the plane. (c) The component. [(d)–(f)] Views of the dipole field components [(a)–(c)] along the main diagonal. The panels (d)–(f) are not at the same scale as panels (a)–(c). The components of the dipole field can be obtained by a rotation of around the axis.

(Color) Components of the recombination dipole field in , calculated in the minimal determinantal approximation [Eq. (64)]. Internuclear separation of is used in the calculations. The isosurfaces are at , , and (bohr). Cuts in panels (a)–(d) are along the plane. (a) The contribution to . This is the only term accounted for in the single active electron model. (b) The complete many-electron component. (c) The SAE contribution to . (d) The complete component. (e) The component. The SAE and many-electron results coincide for this term. The contribution is antisymmetric with respect to reflection—cf. panels (c) and (f) of Fig. 4. The complementary RDF can be obtained by a rotation of around the axis.

(Color) Components of the recombination dipole field in , calculated in the minimal determinantal approximation [Eq. (64)]. Internuclear separation of is used in the calculations. The isosurfaces are at , , and (bohr). Cuts in panels (a)–(d) are along the plane. (a) The contribution to . This is the only term accounted for in the single active electron model. (b) The complete many-electron component. (c) The SAE contribution to . (d) The complete component. (e) The component. The SAE and many-electron results coincide for this term. The contribution is antisymmetric with respect to reflection—cf. panels (c) and (f) of Fig. 4. The complementary RDF can be obtained by a rotation of around the axis.

## Tables

Dyson and exchange correction orbitals for LiH. [Orbital expansion coefficients in terms of Hartree-Fock MOs (Eqs. (35) and (36)). The molecule is oriented along the axis. Experimental is used. The origin is at the center of mass of LiH. Coefficients of the Dyson orbital are dimensionless. The exchange orbital coeeficients are in bohrs. The and components of the exchange correction orbital are zero by symmetry and are not shown. The Hartree-Fock permanent electronic dipole moment of LiH is ].

Dyson and exchange correction orbitals for LiH. [Orbital expansion coefficients in terms of Hartree-Fock MOs (Eqs. (35) and (36)). The molecule is oriented along the axis. Experimental is used. The origin is at the center of mass of LiH. Coefficients of the Dyson orbital are dimensionless. The exchange orbital coeeficients are in bohrs. The and components of the exchange correction orbital are zero by symmetry and are not shown. The Hartree-Fock permanent electronic dipole moment of LiH is ].

Dyson and exchange correction orbitals for . See Table I for notation and units. (At experimental . The contribution is related to by symmetry, and is not shown. MOs which do not contribute to either or are not shown).

Dyson and exchange correction orbitals for . See Table I for notation and units. (At experimental . The contribution is related to by symmetry, and is not shown. MOs which do not contribute to either or are not shown).

Dyson and exchange correction orbitals for the component of the recombination dipole field [, Eq. (56)] in . ( exchange correction orbital vanishes and is not shown. MOs giving negligible contributions to and are not shown. Composition of the orbital contributing to the component of the RDF can be obtained by interchanging and in all sub- and superscripts.) See Table I for notations.

Dyson and exchange correction orbitals for the component of the recombination dipole field [, Eq. (56)] in . ( exchange correction orbital vanishes and is not shown. MOs giving negligible contributions to and are not shown. Composition of the orbital contributing to the component of the RDF can be obtained by interchanging and in all sub- and superscripts.) See Table I for notations.

Dyson and exchange correction orbitals for the component of the recombination dipole field [, Eq. (64)] in . (The exchange correction orbital vanishes and is not shown. MOs giving negligible contributions to both and are not shown. Composition of the orbital contributing to the component of the RDF can be obtained by interchanging and in all sub- and superscripts). See Table I for notation and units.

Dyson and exchange correction orbitals for the component of the recombination dipole field [, Eq. (64)] in . (The exchange correction orbital vanishes and is not shown. MOs giving negligible contributions to both and are not shown. Composition of the orbital contributing to the component of the RDF can be obtained by interchanging and in all sub- and superscripts). See Table I for notation and units.

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