Electron correlation of one-dimensional H 2 in intense laser fields: Time-dependent extended Hartree-Fock and time-dependent density-functional-theory approaches Nam A. Nguyen and André D. Bandrauk Laboratoire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, QC J1K 2R1, Canada Received 21 September 2005; published 8 March 2006 Ionization and high-order harmonic generation of the one-dimensional 1D H 2 molecule in intense ul- trashort laser fields are investigated using several current approximations for electron dynamics. Single- and double-ionization probabilities are compared with exact results. It is found that for the ground state X 1  g + , time-dependent extended Hartree-Fock gives generally comparable results except in the plateau region. The adiabatic local density approximation and time-dependent optimized effective potential with self-interaction correction TDKLI methods underestimate the ionization probabilities with no plateau and knee for double ionization contrary to the exact results. For the triplet excited state A 3  u + , where exchange is important, the TDKLI results agree well with the exact results. The exact double-ionization probabilities suggest the need for accurate pair-correlation functions. DOI: 10.1103/PhysRevA.73.032708 PACS numbers: 33.80.Eh, 31.15.Ew I. INTRODUCTION Due to rapid advancement of laser technology, the field of research encompassing atoms, molecules, and laser interac- tion has been progressing at a fast pace. Highly nonlinear, nonperturbative phenomena, such as above-threshold ioniza- tion ATI and high-order harmonic generation HOHG, have been discovered for atoms 1,2. These studies have also been extended to molecules 3 where the extra degree of freedom of nuclear motion results in nonlinear phenomena such as charge resonance enhance ionization CREI4–7. At the same time, research fields exploiting the laser-matter interaction emerged such as quantum control of chemical and physical processes 8,9 and attosecond science 10,11. Many intense field phenomena can be explained based on a quasiclassical recollision model 12,13. This model basi- cally consists of three steps: 1 an electron leaves the atom or molecule and enters the ionization continuum; 2 it is then accelerated by the strong field and gains energy; 3 the ionized electron is driven back and combines with the parent ion, thus knocking out or exciting a second electron. One of the most studied phenomena based on this model, both ex- perimentally and theoretically, is the correlation of electrons in intense fields, as manifested through the double ionization of atoms 14. In this paper we investigate single and double ionizations and HOHG of molecules. Specifically, the one-dimensional 1D H 2 molecule is chosen as a prototype for a two-electron molecule with bonding and antibonding orbitals. The elec- tron correlation effect of 1D H 2 has been studied in Refs. 15,29 by using an exact numerical integration of the time- dependent Schrödinger equation TDSE. In this paper ap- proximate methods are used to solve the TDSE and are com- pared to the exact results. One of the earliest approximations of electron dynamics in intense fields was the time-dependent Hartree-Fock TDHF method by Kulander et al. 16; and it was applied afterwards to H 2 and H 3 + 17. The model failed, however, to account for the electron correlation effect due to the fact that the two electrons reside in the same orbital, and conse- quently suppression of recollision occurred in such a model. Improvement of TDHF was put forward by Horbatsch et al. 18 who used two separate orbitals to represent the two electrons. However, their models treated the electrons as dis- tinguishable particles and still omitted the exchange interac- tion between the electrons. Finally, van Leeuwen, Tolley, and Pindzola 19–22 presented 1D models where the exchange interaction was taken into account. Their models are gener- alized here to the 1D H 2 molecule with fixed nuclei. Note that in terms of the multiconfiguration time-dependent HF approach, which has been applied to 1D quantum dots in strong fields 23, our approach is limited to a two- configuration, or unrestricted HF system. Another approximation to describe a multielectron system is the density functional theory DFT24 and its time de- pendent version, i.e., time-dependent density functional theory TDDFT25,26. TDDFT replaces a system of inter- acting particles with a system without electron interaction that has the same one-electron density as the exact interact- ing one. TDDFT is in principle exact, but in practice one needs to approximate the exchange-correlation potential, v xc r , t. A previous application of TDDFT to intense field ionization of small clusters showed the sensitivity of results to the exchange-correlation potentials 27,28. In this paper various forms of v xc r , t are applied to the ionization and HOHG of 1D H 2 and results are compared to the exact ones 15,29. We remark that the DFT of 1D fermions with con- tact interaction has shown the applicability of the local den- sity approximation due to the inherent self-interaction free nature of this system 30. II. METHODOLOGY It was previously found 31,32 that at long wavelength, 1D and three-dimensional 3D models of H 2 gave ionization properties that are often similar, reflecting the dominance of 1D laser induced long electron trajectories at such low wave- lengths 12,13. The 1D time-dependent electronic Hamil- PHYSICAL REVIEW A 73, 032708 2006 1050-2947/2006/733/0327087/$23.00 ©2006 The American Physical Society 032708-1