Variational vs perturbative relativistic energies for small and light atomic and molecular systems

Abstract

Variational and perturbative relativistic energies are computed and compared for two-electron atoms and molecules with low nuclear charge numbers. In general, good agreement of the two approaches is observed. Remaining deviations can be attributed to higher-order relativistic, also called non-radiative quantum electrodynamics (QED), corrections of the perturbative approach that are automatically included in the variational solution of the no-pair Dirac--Coulomb--Breit (DCB) equation to all orders of the $\alpha$ fine-structure constant. The analysis of the polynomial $\alpha$ dependence of the DCB energy makes it possible to determine the leading-order relativistic correction to the non-relativistic energy to high precision without regularization. Contributions from the Breit--Pauli Hamiltonian, for which expectation values converge slowly due the singular terms, are implicitly included in the variational procedure. The $\alpha$ dependence of the no-pair DCB energy shows that the higher-order ($\alpha^4 E_\mathrm{h}$) non-radiative QED correction is 5~\% of the leading-order ($\alpha^3 E_\mathrm{h}$) non-radiative QED correction for $Z=2$ (He), but it is 40~\% already for $Z=4$ (Be$^{2+}$), which indicates that resummation provided by the variational procedure is important already for intermediate nuclear charge numbers.

Evaluation of the Bethe logarithm: from atom to chemical reaction

“…In this work, we describe the implementation of Schwartz' method [13] in the QUANTEN computer program aiming at a general applicability for small polyatomic and polyelectronic systems. QUANTEN is an in-house developed, general-purpose molecular physics platform for theoretical developments in precision physics and spectroscopy, and includes by now computation of single-and multi-state non-adiabatic corrections to the non-relativistic energy [31,32], pre-Born-Oppenheimer energy [33], energy lower bounds [34], perturbative relativistic and QED corrections [7,30], and variational relativistic energies [35][36][37][38].…”

The Bethe–Salpeter QED Wave Equation for Bound-State Computations of Atoms and Molecules

“…This section provides a brief overview of the practical aspects of solving the no-pair Dirac–Coulomb or Dirac–Coulomb–Breit equation with explicitly correlated trial functions. − Explicitly correlated, i.e., two-particle, basis functions make it possible in practice to converge the energy to a precision where comparison of the 16-component results with precise and accurate perturbative computations (nonrelativistic QED) established in relation with precision spectroscopy is interesting and has been unexplored until recently. For the sake of this comparison, we focus on atoms and molecules of light elements, but in principle, the theoretical and algorithmic framework presented in this work is not limited to low Z systems (unlike finite-order nrQED).…”

“…We have implemented the kinetic balance condition in an operator form, i.e., as a “metric” − ,, H KB false[ 16 false] = X 12 [ 16 ] † H false[ 16 false] X 12 false[ 16 false] goodbreak0em.5em⁣ and goodbreak0em.5em⁣ I KB false[ 16 false] = X 12 [ 16 ] † X 12 false[ 16 false] and detailed operator expressions can be found in previous work. − …”

“…For calculating the matrix elements of eq , the positive-energy projection of the Hamiltonian must be carried out. The matrix representation of the scriptL ++ false[ 16 false] projector is constructed by using the positive-energy eigenstates of the noninteracting two-particle Hamiltonian, H 1 [4] ⊡ I [4] + I [4] ⊡ H 2 [4] , represented as a matrix over the actual basis space. − , Selection of the “positive-energy” two-electron states can be realized approximately by “cutting” the noninteracting spectrum based on some energetic condition, , or more precisely, by rotating the spectrum to the complex plane via complex rescaling of the electron coordinates. This complex rotation (CR) allows us to distinguish three different “branches” of the noninteracting two-electron system (positive-, Brown–Ravenhall, and negative-energy states), in principle, for any finite rotation angle .…”

Evaluation of the Bethe Logarithm: From Atom to Chemical Reaction

“…In this work, we describe the implementation of the Schwartz method in the QUANTEN computer program aiming at general applicability for small polyatomic and polyelectronic systems. QUANTEN (QUANTum mechanical treatment of Electrons and atomic Nuclei) is an in-house developed, general-purpose molecular physics platform for theoretical developments in precision physics and spectroscopy, and includes by now computation of single- and multistate nonadiabatic corrections to the nonrelativistic energy, , pre-Born–Oppenheimer energy, energy lower bounds, perturbative relativistic and QED corrections, , and variational relativistic energies. − …”