The ground state solution of hydrogen molecule and ions are numerically obtained as an application of our scheme to solve many-electron multi-center potential Schrödinger equation by using one-dimensional hydrogen wavefunctions as basis functions. The all-electron sparse Hamiltonian matrix for the given system is generated with the standard order finite-difference method, then the electronic trial wavefunction to describe the ground state is constructed based on the molecular orbital treatment, and finally an effective and accurate iteration process is implemented to systematically improve the result. Many problems associated with the evaluation of the matrix elements of the Hamiltonian in more general basis and potential are circumvented. Compared with the standard results, the variationally obtained energy of H2
+ is within 0.1 mhartree accuracy, while that of H2 and H3
+ include the electron correlation effect. The equilibrium bond length is highly consistent with the accurate results and the virial theorem is satisfied to an accuracy of −V/T = 2.0.
Distinctive from conventional electronic structure methods, we solve the Schrödinger wave equations of the helium atom and its isoelectronic ions by employing one-dimensional basis functions to separate components. We use full two-electron six-dimensional operators and wavefunctions represented with real-space grids where the refinement of the latter is carried out using a residual minimization method. In contrast to the standard single-electron approach, the current approach results in exact treatment of repulsion energy and, hence, more accurate electron correlation within five centihartrees or better included, with moderate computational cost. A simple numerical convergence between the error to accurate results and the grid-spacing size is found. The obtained two-electron Schrödinger wavefunction that contains vast and elaborating information for the radial correlation function and common one-dimensional functions shows the electron correlation effect on one-electron distributions.
The solution of three-dimensional Schrödinger wave equations of the hydrogen atoms and their isoelectronic ions (Z = 1 − 4) are obtained from the linear combination of one-dimensional hydrogen wave functions. The use of one-dimensional basis functions facilitates easy numericalintegrations. An iteration technique is used to obtain accurate wave functions and energy levels.The obtained ground state energy level for the hydrogen atom converges stably to −0.498 a.u. The result shows that the novel approach is efficient for the three-dimensional solution of the wave equation, extendable to the numerical solution of general many-body problems, as has been demonstrated in this work with hydrogen anion. K E Y W O R D S ground state, hydrogen atom, iteration, one-dimensional basis function, Schrödinger wave equations
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