4-component Relativistic Calculations in a Multiwavelet Basis with Improved Convergence

Abstract

In this contribution we revive an approach to solving the Dirac equation originally proposed by Kutzelnigg which makes use of the squared Dirac operator $\hat{\mathfrak{D}}^{2}$. This approach holds the promise to avoid the negative energy solution because the negative energy spectrum is now ``folded" on the positive energy side and at the same time provides a convex equation, which is amenable to a minimization process and increased precision in the final result. The $\hat{\mathfrak{D}}^{2}$ yields an equation similar to the non-relativistic one, yet in a four-component framework, where Multiwavelet tools and algorithms developed for the non-relativistic case can be employed with minor modifications. On the other hand, the use of Multiwavelets is here essential to achieve the full potential of the approach. We implemented and validated this approach for one- and two-electron systems with increasing nuclear charge. Numerical tests were performed to gauge the actual precision of the approach with respect to either analytical reference values when possible or numerical results obtained with the \textit{GRASP} code otherwise.

Figures (5)

• Figure 1: Relative error in the total energy of the H atom computed with MW, with respect to the GRASPgrasp_ref reference value. Each graph represents a choice of derivative operator (ABGV or BS) and atom placement (at a dyadic point or not). Each line in the graph represents a given choice of algorithm ($\Phi_{\hat{\mathfrak{D}}}$ or $\Phi_{\hat{\mathfrak{D}^2}}$) and expectation value ($\hat{\mathfrak{D}}$ or $\hat{\mathfrak{D}^2}$). Each point is the requested precision (from MW4 to MW8) of the calculation.
• Figure 2: Relative error in the total energy of the Ne^9+ atom computed with MW, with respect to the GRASPgrasp_ref reference value. Each graph represents a choice of derivative operator (ABGV or BS) and atom placement (at a dyadic point or not). Each line in the graph represents a given choice of algorithm ($\Phi_{\hat{\mathfrak{D}}}$ or $\Phi_{\hat{\mathfrak{D}^2}}$) and expectation value ($\hat{\mathfrak{D}}$ or $\hat{\mathfrak{D}^2}$). Each point is the requested precision (from MW4 to MW8) of the calculation.
• Figure 3: Relative error in the total energy of the He atom computed with MW, with respect to the GRASPgrasp_ref reference value. Each graph represents a choice of derivative operator (ABGV or BS) and atom placement (at a dyadic point or not). Each line in the graph represents a given choice of algorithm ($\Phi_{\hat{\mathfrak{D}}}$ or $\Phi_{\hat{\mathfrak{D}^2}}$) and expectation value ($\hat{\mathfrak{D}}$ or $\hat{\mathfrak{D}^2}$). Each point is the requested precision (from MW4 to MW8) of the calculation.
• Figure 4: Relative error in the total energy of the Hg$^+78$ atom computed with MW, with respect to the GRASPgrasp_ref reference value. Each graph represents a choice of derivative operator (ABGV or BS) and atom placement (at a dyadic point or not). Each line in the graph represents a given choice of algorithm ($\Phi_{\hat{\mathfrak{D}}}$ or $\Phi_{\hat{\mathfrak{D}^2}}$) and expectation value ($\hat{\mathfrak{D}}$ or $\hat{\mathfrak{D}^2}$). Each point is the requested precision (from MW4 to MW8) of the calculation.
• Figure 5: Relative error in the total energy of the Ar^16+ atom computed with MW, with respect to the GRASPgrasp_ref reference value and using a modified speed of light $c' = 30.83309764$ in order to achieve the same order of magnitude in the relativistic correction found in the Hg atom. Each graph represents a choice of derivative operator (ABGV or BS) and atom placement (at a dyadic point or not). Each line in the graph represents a given choice of algorithm ($\Phi_{\hat{\mathfrak{D}}}$ or $\Phi_{\hat{\mathfrak{D}^2}}$) and expectation value ($\hat{\mathfrak{D}}$ or $\hat{\mathfrak{D}^2}$). Each point is the requested precision (from MW4 to MW8) of the calculation.