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A two-dimensional helium atom (2D-helium) is a real subject for current studies, particularly regarding a hot topic of negatively charged excitons (trions) in semiconducting monolayers. The present study considers a 2D-helium-like atom in a homogeneous magnetic field. We are able to rewrite its Schrödinger equation into a polynomial form concerning dynamic variables. This form is useful for utilizing the algebraic calculation by annihilation and creation operators, enabling the successful application of the Feranchuk-Komarov (FK) operator method to obtain numerically exact solutions (energies and wave functions) for this system. The polynomialization of the equation allows obtaining analytical expressions of all matrix elements, which saves the computational resources significantly. Numerical results for the case without a magnetic field are comparable to other calculations. Moreover, the precise separation of the center-of-mass motion, as provided in this study, leads to an equation for the relative motion of the electrons in a magnetic field, incorporating all previously neglected terms. This result is useful for further study of trions where the electron effective mass is comparable with the hole effective mass. Additionally, we provide a FORTRAN program designed to solve the problems above. Program Title: CHeAMF CPC Library link to program files: https://doi.org/10.17632/mp8tf2dz67.1 Licensing provisions: BSD 3-clause Programming language: FORTRAN90 Nature of problem: The Schrödinger equation for a 2D-helium-like atom in a homogeneous magnetic field is transformed into a polynomial form using the Levi-Civita transformation twice. This transformation results in a structure more conductive to applying algebraic methods based on annihilation and creation operators. Consequently, we employ the FK operator method [1] to obtain numerically exact solutions, ensuring that the calculated energies converge to a high level of precision, up to 15 decimal places in this study. This method is developed to cover a broad range of magnetic field intensities, extending up to 0.1 a.u. (2.35 × 10 4 Tesla). Moreover, it is applicable not only for the ground state but also for highly excited states. Solution method: The modified FK operator method, as introduced in reference [2], has been developed and applied to obtain precise numerical solutions for a 2D-helium-like atom. Concurrently, algebraic techniques have been employed to compute matrix elements. Subsequently, we transformed the Schrödinger equation into a linear matrix equation, which we solved using the 'dsygvx.f' subroutine from the LAPACK library [3]. This subroutine has been optimized for improved accuracy by employing real*16 variables instead of real*8. Furthermore, we have incorporated an optimal free parameter into the FORTRAN program, enhancing convergence speed. Additional comments including restrictions and unusual features: Operating system: Linux. RAM: at least 4 GByte per core. We recommend using the gFortran compiler for this program. The runtime varies from a few minutes to hours, depending on the required precision. Particularly for strong magnetic fields (γ ≥ 0.01 a.u.) or when working with excited levels, the runtime may extend to several hours to achieve a precision of 16 decimal places. In such cases, it is advisable to have at least 60 GB of RAM per core. [1] I. Feranchuk, A. Ivanov, Van-Hoang Le, A. Ulyanenkov, Non-perturbative Description of Quantum Systems, Springer, Switzerland, 2015, https://doi.org/10.1007/978-3-319-13006-4. [2] Thanh-Xuan H. Cao, Duy-Nhat Ly, Ngoc-Tram D. Hoang, Van-Hoang Le, High-accuracy numerical calculations of the bound states of a hydrogen atom in a constant magnetic field with arbitrary strength, Comput. Phys. Commun. (2019), https://doi.org/10.1016/j.cpc.2019.02.013. [3] Netlib.org. LAPACK: Linear Algebra PACKage, Subroutine dsygvx.f, https://netlib.org/lapack/explore-3.1.1-html/dsygvx.f.html. [ABSTRACT FROM AUTHOR]

We develop a simple and effective method for solving the Schrödinger equation of a hydrogen atom in a constant magnetic field with arbitrary strength. Energies are obtained not only for the ground and low-lying states but also for highly excited states with precision from 12 up to 20 decimal digits. The calculations are performed for an entire range of magnetic field intensity up to 9. 4 × 1 0 8 Tesla, the strongest field ever observed. The strong point of the development of the method is the construction of an anharmonic oscillator model for a hydrogen atom in a constant magnetic field via the Kustaanheimo–Stiefel transformation. This model allows the use of purely algebraic calculations and the Feranchuk–Komarov (FK) operator method for effectively solving the Schrödinger equation. The advantages of the basis set in this work are also discussed to extend its application to other problems, such as multi-electron atoms in a constant magnetic field. We also provide a program written by FORTRAN for the solutions mentioned above. Program Title: CHAMF Program Files doi: http://dx.doi.org/10.17632/s5shscgfcj.1 Licensing provisions: BSD 3-clause Programming language: FORTRAN77 Nature of problem: The Schrödinger equation for a hydrogen atom in a magnetic field is rewritten into that of a four-dimensional anharmonic oscillator by using the Kustaanheimo–Stiefel transformation. The later has a structure more convenient for applying the algebraic method based on annihilation and creation operators. Therefore, the FK operator method [1] can be applied for obtaining high-accuracy numerical solutions which means the obtained energies converge to a given precision (up to twenty decimal digits in this work). The method is developed for an entire range of magnetic field intensity up to 9. 4 × 1 0 8 Tesla and for not only ground state but also highly excited states. Solution method: The FK operator method is modified and used for the high-accuracy numerical solutions. This method combines with the algebraic technique which is useful for calculating all the needed matrix elements. Thus, the Schrödinger equation becomes the symmetric generalized eigenvalue problem which can be solved by the subroutine dsygvx.f of the LAPACK library [2]. For the better accuracy with quadruple precision, we modify this subroutine and its dependence with using the real*16 number instead of the real*8. Moreover, a free parameter introduced by the FK operator method can be used for increasing the speed of convergence. The optimal choice of this free parameter is implemented in the program. Additional comments including restrictions and unusual features: Operating system: Linux. RAM: at least 1 GByte per core. The gfortran compiler is recommended for this program. The running time is from a centisecond to a few minutes depending on the required precision. However, for the super-strong magnetic fields (γ ≥ 200), the running time may reach to two hours for the precision of twelve decimal digits, and RAM is needed to be at least 60 GB. [1] I. Feranchuk, A. Ivanov, Van-Hoang Le, A. Ulyanenkov, Non-perturbative Description of Quantum Systems, Springer, Switzerland, 2015. doi:10.1007/978-3-319-13006-4. [2] Netlib.org. LAPACK: Linear Algebra PACKage, Subroutine dsygvx.f. URL http://www.netlib.org/lapack/lapack-3.1.1/html/dsygvx.f.html [ABSTRACT FROM AUTHOR]