Relativistic Quantum Mechanics

In the field of relativistic quantum mechanics, we elaborate on a relativistic fermionic theory based on the Dirac Hamiltonian, presenting a computational scheme for the approximation of positive-energy bound states in a basis set expansion of explicitly correlated Gaussians.

Albeit the basis set expansion technique optimized by application of the variational principle is paramount when calculating numerically eigenstates of non relativistic Hamiltonians, their generalization to relativistic Hamiltonians is not an intrinsically safe procedure. The unboundedness of the Dirac–Coulomb Hamiltonians plagues the variational approach by the appearance of spurious eigenvalues below the positive-energy ground state. These states survive if one takes the non-relativistic limit and they are characterized by a poorly represented kinetic energy. The existence of the problem is now well known and several techniques have been proposed for its alleviation. Particularly important are conditions that make the trial wave function kinetically balanced, by satisfying the kinetic energy correctness in the non-relativistic limit. While for wave functions approximated as products of one-fermion orbitals, variational stability can be ensured and these approaches are routinely employed in relativistic 4-component calculations, there is the need to develop an efficient explicitly correlated variant of the kinetic balance condition.

Thanks to this research project, it was possible to devise the exact kinetic-balance condition for 16-component explicitly correlated functions in order to ensure variational stability and avoid the variational collapse. The two fermions are treated explicitly in the simple laboratory fixed cartesian coordinates and we avoid having to transform the Dirac Hamiltonian to internal coordinates. Conversely, translationally invariant integrals are employed to ensure elimination of center-of-mass contributions from the expectation values.