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arXiv:2610.03217

Two-center Dirac equation in a Gaussian basis set

Dávid Ferenc

quant-phSubmitted Oct 2, 2026• Announced Oct 5, 2026
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Abstract

A method for solving the two-center Dirac equation using a Gaussian-product basis set tailored to the axial symmetry of diatomic systems is presented. For the ground state of H$_2^+$ at an internuclear distance $R=2.0~a_0$, we obtain a relativistic energy with relative precision at the $10^{-18}$ level. The properties of the basis set allow for an analytic Fourier transform of the wave function. As a first application we evaluate, for the first time, the zero-potential contribution to the one-loop self-energy for H$_2^+$ in Feynman and Coulomb gauges. Although this contribution is not separately gauge invariant, its calculation provides a proof of principle for the use of the present framework in prospective QED calculations.

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A method for solving the two-center Dirac equation using a Gaussian-product basis set tailored to the axial symmetry of diatomic systems is presented. For the ground state of H2+_2^+2+​ at an internuclear distance R=2.0 a0R=2.0~a_0R=2.0 a0​, we obtain a relativistic energy with relative precision at the 10−1810^{-18}10−18 level. The properties of the basis set allow for an analytic Fourier transform of the wave function. As a first application we evaluate, for the first time, the zero-potential contribution to the one-loop self-energy for H2+_2^+2+​ in Feynman and Coulomb gauges. Although this contribution is not separately gauge invariant, its calculation provides a proof of principle for the use of the present framework in prospective QED calculations.
Solutions of the Dirac equation for an electron in the Coulomb potential are obtained using operator invariants of the equation, namely the Dirac, Johnson-Lippmann and recently found new invariant. It is demonstrated that these operators are the spin invariants. The generalized invariant is constructed and the exact general solution of the Dirac equation are found. In particular, the explicit expressions of the bispinors corresponding to the three complete sets of the invariants, their eigenvalues and quantum numbers are calculated. It is shown that the general solution of one center Coulomb Dirac equation contains free parameters. Changing one or more of these parameters, one can transform one solution of the Dirac equation into any other. It is shown for the first time that these invariants determine electron spatial probability amplitude and spin polarization in each quantum state. Electron probability densities and spin polarizations are explicitly calculated in the general form for several electron states in the hydrogen-like energy spectrum. Spatial distributions of these characteristics are shown to depend essentially on the invariant set, demonstrating, in spite of the accidental degeneracy of energy levels, physical difference of the states corresponding to different spin invariants.
In this preliminary Brief Report we explore a method for examining relativistic effects in one-electron molecules that avoids the evaluation of multicenter integrals. Here the Dirac equation for a simplified version of the H2+{{\mathrm{H}}_{2}}^{+}H2​+ ion is solved exactly for the ground-state Schr\"odinger and Dirac energy levels. The model consists of representing the electron-proton Coulomb interaction as a projection operator onto a linear subspace of Hilbert space.
With the use of a simple similarity transformation which brings the radial wave equations of the Dirac-Coulomb problem into a form nearly identical to those of the Schr\"odinger and Klein-Gordon equations, we derive simplified solutions to the Dirac-Coulomb equation for both the bound and continuum states following the familiar standard procedure adopted in the derivation of the conventional solutions. We show that to obtain the desired form of the second-order radial equations we can still work with a first-order partial differential equation rather than with the second-order Dirac equation widely employed in the derivation of the simplified solutions, and thus we can avoid the task of reducing the solutions of the second-order equations to those of the original Dirac equation. The transformed Dirac-Coulomb radial equations are so simple that one can apply the WKB method to them in the same way that one applies the WKB approximation to the Schr\"odinger radial equation, without making the further approximations commonly invoked, and the Sommerfeld-Dirac discrete spectrum follows immediately. For small Z\ensuremath{\alpha}, we also present approximate expressions for both the transformed and the original Dirac-Coulomb wave functions, which are valid for all energies and certainly valid in the quasirelativistic approximation which takes account of relativistic effects. The structure of these approximate expressions also suggests one of the methods to obtain relativistic approximate wave functions from the Pauli-spinor wave function for more general central potentials arising in atoms, molecules, and solids.
Exact cylindrical solutions of the free Dirac equation provide natural single-particle modes for a broad class of axially symmetric relativistic fermion systems, including electron vortices, twisted-particle scattering, rotating matter, and cylindrical field quantization, but are commonly formulated in different internal bases. We derive the general regular positive-energy cylindrical solution at fixed energy, transverse and longitudinal momenta, and total angular momentum, and show how the commonly used spin-polarized, separation, helicity, and transverse-helicity modes are embedded in the resulting two-dimensional solution space. A conserved transverse operator resolves the residual doublet into two symmetry-defined branches. Their parameter-dependent eigenspaces admit a gauge-invariant quantum-geometric characterization, with opposite Berry curvatures, a common quantum metric, and saturation of the two-level metric--curvature relation. We further derive symmetry constraints on branch conversion and the corresponding reduced two-state dynamics. The resulting framework connects mode classification, symmetry resolution, quantum geometry, and dynamics across distinct cylindrical Dirac settings.
It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.