Computer Physics Communications 178 (2008) 571–577 www.elsevier.com/locate/cpc Calculation of atomic hydrogen and its photoelectron spectra in momentum space T.F. Jiang Institute of Physics, National Chiao-Tung University, Hsinchu 30010, Taiwan Received 19 September 2007; received in revised form 12 November 2007; accepted 29 November 2007 Available online 23 December 2007 Abstract An essential modification to the kernel in the numerical calculation of hydrogenic momentum wave functions, is presented in this paper. Using only 256 grid points, the calculated eigenvalues, eigenfunctions and the oscillator strengths are shown to be in excellent agreement with the exact analytic results. The reliable pseudocomplete set of momentum space eigenfunctions is then applied to the time-dependent calculation of intense laser pulse on the hydrogen atom. With the advantage of having no boundary reflection during the time evolution, like that inherent in the coordinate space method, the photoelectron spectra of above-threshold-ionization (ATI) are elucidated for four cases. Some of which are not feasible or very difficult to solve with the coordinate space method. Generalization of the method to single-active electron systems is straightforward. Due to the good accuracy with a reasonably small-sized basis set, applications to the currently interested intense case of laser pulse on atom or molecule are expected.  2007 Elsevier B.V. All rights reserved. PACS: 31.15.-p; 03.65.Ca; 32.70.Cs Keywords: Intense laser on atom; Momentum space; Photoelectron spectra; Above threshold ionization 1. Introduction An atomic electron under intense laser fields may absorb many photons and photoionize. The above-threshold-ionization (ATI), first observed in 1979, is a phenomenon in which elec- tron absorbs excess photons than is necessary to ionize [1]. Since then, the ATI has attracted much attention and inter- est [2,3]. Theoretically, the electron spectra can be calculated by solving the time-dependent Schrödinger equation, but this method is often limited by reflection from the boundary when the equation is solved in coordinate space. The trouble is less severe for shorter pulse where a large spatial region can be cho- sen such that the electron is still within the boundary when the laser pulse ends [4]. For stronger fields, the boundary re- flection is still a problem even for few-cycle pulses. From the complementary principle of quantum mechanics, a wave func- tion distributed in a large coordinate space corresponds to a wave function localized in momentum space. Using the mo- E-mail address: tfjiang@faculty.nctu.edu.tw. mentum wave functions as a basis set to study atoms/molecules in laser fields, one would be able to avoid the trouble of bound- ary reflection. We have previously shown that the low-order ATI phenomenon is described rather well by the momentum space method [5]. In this paper, we find a more efficient and accurate momen- tum space method than our previous work. Although the for- mulation of Schrödinger equation in momentum space is well known [6], the number of analytic eigenfunctions is infinite and cannot be used directly in the simulation of time-dependent problems. There were efforts in developing the momentum space solution: Lande invented a regularization method for the Coulomb kernel singularity [7], Ivanov and Mitroy [8] de- signed the iteration codes for the expansion of the kernel, Nor- bury et al. [9] applied the specific basis function forms to the bound states, and Tang et al. [10] used the Bystrom method for the Coulomb kernel related integration and solved some of the bound states. For numerical calculations within a finite range, we find that the earlier formalism must be modified. Our key correction to the Coulomb kernel in practical compu- 0010-4655/$ – see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2007.11.019