Numerical variational solution of hydrogen molecule and ions using one-dimensional hydrogen as basis functions

Abstract

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.

Density Functional Calculations on H2 Using 1s Slater Type Orbitals

“…Up to this point, this Journal has published an illustrative resource on DFT calculations by hand for the helium atom using the X-Alpha exchange functional on a single Gaussian orbital as the atomic orbital. However, because the test system chosen was the helium atom, the previous publication fails to capture the complexity of multicenter molecular system − where the calculations of the total energy components are more involved. Additionally, the application of the linear combination of atomic orbital (LCAO) framework to construct the molecular orbitals is missing. , Furthermore, DFT calculations on a molecular system will illustrate the notion of the potential energy surface and the Born–Oppenheimer approximation.…”

An efficient finite difference approach to solutions of Schrödinger equations of atoms in non-linear coordinates

“…Second, the presence of the central difference formula represents the first and second derivative accurate to O h 2 ( ) where h is the uniform grids interval. To achieve the required accuracy in the atomic potentials [43][44][45][46][47][48][49][50][51], the use of very fine uniform grids is necessary, causing a large amount of computation to represent the Hamiltonian matrix. Except for the heavy computations, the convergence is also disappointing.…”

“…Except for the heavy computations, the convergence is also disappointing. It may take many steps to converge however with unsatisfactory results [44,45,51]. It is well known that the potential energy is mainly near the nucleus.…”

Highly Accurate Potential Energy Curves for the Hydrogen Molecular Ion

“…The H 2 + system is of particular interest since, as it involves a real molecule, it is useful not only to provide benchmark values for testing numerical methods in general, [8] but also for and quantum-chemical methods, such as those based on perturbation theory, [9][10][11] or the variational theorem. [12,13] It has also been used to validate potential energy surfaces (PESs), [14,15] which are of the utmost importance in the computation of condensed matter properties. [16,17] The computed spectra and PESs of H 2 + have also been used to model covalent bonding, [18] and in experimental observations of molecular properties.…”