Complementary Version of Fermion Coupled Coherent States Method and Gram–Schmidt Algorithm: Theory and Applications for Electronic States of H 2 and H 1 2 Mohammadreza Eidi , Mohsen Vafaee, and Mitra Rooein In our previous report, we introduced a new version of Fer- mion coupled coherent states method (FCCS) which was espe- cially suited for simulating the first symmetric spatial electronic state of two-electron systems. In this manuscript, we report a complementary version for FCCS method to simulate both of the first symmetric and antisymmetric spatial electronic states of two-electron systems. Moreover, the Gram– Schmidt orthogonalization process is employed to reach the excited states of the system. We apply this FCCS method and the original coupled coherent state method to simulate the energy of different electronic states of H 2 and H 1 2 , respectively. The results for the energy of computed electronic states of H 2 and H 1 2 show a pretty good consistency with the exact values. V C 2017 Wiley Periodicals, Inc. DOI: 10.1002/jcc.25133 Introduction For the last two decades the coupled coherent states (CCS) method for solving time dependent Schr€ odinger equation has been developed with the aim to be able to address high dimensional quantum problems. [1–12] Owing to the fact that the CCS method originally uses asymmetric (nonsymmetric) grids of coherent states, it is not suitable for simulating multie- lectron (fermionic) systems. A modified version of CCS named Fermionic coupled coherent states (FCCS-I), which takes explic- itly into account the exchange symmetry of fermions, has been developed. [4,9] The CCS and the FCCS-I methods have been developed as a useful tool to paves the way for investigating the full dimen- sional electron dynamics in multi-electron systems and their different types of interactions with intense laser fields. For example, Shalashilin et al. employed the CCS method for 6D investigation of double ionization mechanisms of He atom. [7] Gue et al. explored the strong-field multiphoton double ioniza- tion of He exposed to an intense long-wavelength laser field using CCS method. [8] Kirrander and Shalashilin have investi- gated the electron dynamics in laser fields implementing fro- zen Gaussians on the base of the FCCS-I method. [9] Symonds et al. reported a version of CCS method which had the capa- bility of computing the high-order harmonic generation spec- trum of a one dimensional electron in a laser field. [11] Finally, Zhou and Chu have reported the full dynamics investigation of H 2 in intense linearly polarized laser fields employing the Heller’s Frozen Gaussians method. [10] Compared to the FCCS-I method which uses a Slater deter- minant for generating symmetric and antisymmetric coherent states, recently we have introduced another version of fermion coupled coherent states method (FCCS-II). [12] A striking feature of FCCS-II is that there is no need for applying any additional symmetrizing equation and consequently equations are pretty simpler and easier to solve. In addition, we do not need to exclude high-energy coherent states and thereby bias the CS grid to the regions with the lowest energy. Besides, grid refinements is not necessary anymore. In Ref. [12], we have only reported the new symmetrizing scheme and its applica- tions on simulations of the ground state of two-electron sys- tems such as H 2 and He. In this article, the FCCS-II method has been extended and introduced as complementary FCCS method. The presented approach has the ability of antisymme- trizing the CS grid as well as symmetrizing it for simulating both of the first antisymmetric and symmetric spatial elec- tronic states of two-electron systems. Moreover, we have explained in detail how to employ the Gram–Schmidt algo- rithm for achieving the excited states of the system. FCCS method has a potential application for full dimensional simulation of multielectron systems exposed to intense laser fields. Interaction of an ultrashort intense laser field with multi- electron systems can induce different types of phenomena such as high-order harmonic generation and nonsequential double ionization. [13,14] During the past two decades, there has been considerable interest in study of such phenomena. Notwithstanding the fact that Helium atom as one of the sim- plest atomic and molecular systems provides the only conceiv- able meeting ground for ab initio simulations of multiple ionization induced by intense laser fields and its related experiments, simulation of He exposed to an intense laser field with the most frequently used wavelength in experiments (near 800 nm) and comparison of simulation results with the experiments has not yet been accomplished. [14] This compari- son has been made for a linearly polarized laser field with a M. Eidi, M. Vafaee, M. Rooein Department of Chemistry, Tarbiat Modares University, P.O.Box 14115-175, Tehran, Iran E-mail: m.vafaee@modares.ac.ir Contract grant sponsor: Iran National Science Foundation (INSF); Contract grant number: 94003251 V C 2017 Wiley Periodicals, Inc. Journal of Computational Chemistry 2017, DOI: 10.1002/jcc.25133 1 FULL PAPER WWW.C-CHEM.ORG