有限基近似下计算 Dirac 方程的多势真空极化密度
Calculating the many-potential vacuum polarization density of the Dirac equation in the finite-basis approximation
AI总结：
本文提出在 Dirac 有限基近似下计算类氢原子多势真空极化密度的方法，支持任意径向核电荷分布，并通过强制电荷共轭对称性提升数值可靠性。
摘要：
在本工作中，我们提出一种高效且精确的计算方法，用于在 Dirac 方程的有限基近似内评估类氢原子的多势 $α\left(Zα\right)^{n\ge3}$ 真空极化密度。为证明该计算方法的性能，我们选择以单电子 $_{\\,\\,\\,92}^{238}\text{U}$ 原子为研究对象。总之，我们发现，要获得与我们对解析问题的认识相符的物理结果，必须先验地满足电荷共轭对称性。我们还注意到，最终数值结果与以往仅限于少数几种简单核分布模型的形式解析（及数值）评估高度一致。我们的技术可以在有限基组框架下求解径向 Dirac 方程的程序中高效实现和评估，并允许使用任意（径向）核电荷分布。所得非微扰真空极化密度的数值结果会自动计入扩展核尺寸效应。因此，该方法对于那些无法获得解析 Green 函数表达式或解析形式相对复杂的原子 Dirac 问题具有特殊重要性。此外，我们提出了一个真空极化密度公式，该公式强制满足电荷共轭对称性，并可用于相对论性基组违反该对称性的情形，正如大多数相对论性基组程序中所出现的情况。另外，我们已经证明，当 Dirac Hamiltonian 在时间反演对称性下对称时，真空极化四流的矢量分量为零。
英文摘要：
In this work, we propose an efficient and accurate computational method to evaluate the many-potential $α\left(Zα\right)^{n\ge3}$ vacuum polarization density of hydrogen-like atoms within the finite-basis approximation of the Dirac equation. To prove the performance of our computational method, we choose to work with the one-electron $_{\,\,\,92}^{238}\text{U}$ atom. In summary, we find that compliance with charge conjugation symmetry is a priori required to obtain physical results that are in line with our knowledge of the analytical problem. We also note that the final numerical results are found to be in excellent agreement with previous formal analytical (and numerical) evaluations that are limited to a few simple nuclear distribution models. Our technique can be efficiently implemented and evaluated in codes that solve the radial Dirac equation in the finite basis set framework and allows the use of arbitrary (radial) nuclear charge distribution. The obtained numerical results of the non-perturbative vacuum polarization density automatically account for the extended nuclear size effect. This method is hence of special importance for atomic Dirac problems whose analytical Green's functions expressions are not at hand or have relatively complicated analytical forms. Furthermore, we propose a vacuum polarization density formula that forces compliance with charge conjugation symmetry and can be used in cases where the relativistic basis violates this symmetry, as is the case in most relativistic basis set programs. In addition, we have shown that vector components of the vacuum polarization four-current vanish in the case where the Dirac Hamiltonian is symmetric under time-reversal symmetry.
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