Quantifying Dirac hydrogenic effects via complexity measures

The primary dynamical Dirac relativistic effects can only be seen in hydrogenic systems without the complications introduced by electron-electron interactions in many-electron systems. They are known to be the contraction towards the origin of the electronic charge in hydrogenic systems and the nodal disappearance (because of the raising of all the nonrelativistic minima) in the electron density of the excited states of these systems. In addition we point out the (largely ignored) gradient reduction of the charge density near and far from the nucleus. In this work we quantify these effects by means of single (Fisher information) and composite [Fisher-Shannon complexity and plane, López-Ruiz, Mancini, and Calbet (LMC) complexity] information-theoretic measures. While the Fisher information measures the gradient content of the density, the (dimensionless) composite information-theoretic quantities grasp twofold facets of the electronic distribution: The Fisher-Shannon complexity measures the combined balance of the gradient content and the total extent of the electronic charge, and the LMC complexity quantifies the disequilibrium jointly with the spreading of the density in the configuration space. Opposite to other complexity notions (e.g., computational and algorithmic complexities), these two quantities describe intrinsic properties of the system because they do not depend on the context but are functionals of the electron density. Moreover, they are closely related to the intuitive notion of complexity because they are minimum for the two extreme (or least complex) distributions of perfect order and maximum disorder.

Figure 1

Dependence of the ground-state hydrogenic LMC (left) and Fisher-Shannon (right) complexity measures on the nuclear charge $Z$.

Figure 2

LMC relative ratio ${\zeta }_{\text{LMC}}$, for circular states with $n⩽3$ (left) and Fisher-Shannon relative ratio ${\zeta }_{\text{FS}}$, for the ground state and the excited states $(n,l,j,m_j)=(2,0,\frac{1}{2},\frac{1}{2})$ and $(3,1,\frac{1}{2},\frac{1}{2})$ (right).

Figure 3

Radial density $D^i\left(r\right)$ and radial Fisher information kernel $I_{\text{kernel}}^i\left(r\right)$ in the Dirac ($i=D$) and Schrödinger ($i=S$) settings for the hydrogenic states $n=5,l=2,j=\frac{1}{2}$ (left) and $n=6,l=1,j=\frac{1}{2}$ (right) with nuclear charge $Z=50$. Atomic units have been used.

Figure 4

Contribution of $g\left(r\right)$ and $f\left(r\right)$ to the total probability density for the hydrogenic state $n=5,l=2,j=2.5$ with nuclear charge $Z=90$. Atomic units have been used.

Figure 5

Radial density $D^i\left(r\right)$ and radial Fisher information kernel $I_{\text{kernel}}^i\left(r\right)$ in the Dirac ($i=D$) and Schrödinger ($i=S$) settings for the ground state (left) and the circular state $n=5$ (right) with nuclear charge $Z=50$. Atomic units have been used.

Figure 6

Dependence of the LMC complexity on $m_j$ for $Z=19$ (left) and $Z=90$ (right).

Figure 7

Dependence of the Fisher-Shannon complexity on $m_j$ for $Z=19$ (left) and $Z=90$ (right).

Figure 8

LMC (left) and Fisher-Shannon (right) complexities for excited $s$ states in $n$ with $Z=55$.

Figure 9

LMC (left) and Fisher-Shannon (right) complexities for some excited states in $n$ and $l$ with $Z=90$ and $m_j=j$ as a function of the energy (a.u.).

Figure 10

LMC (left) and Fisher-Shannon (right) complexities for some excited states in $n$ and $k$ with $Z=90$ and $m_j=j$ as a function of the relativistic quantum number $k$.

Figure 11

LMC or disequilibrium-Shannon (left) and Fisher-Shannon (right) information planes of hydrogenic states with $n⩽6$, $m_j=j=l+\frac{1}{2}$, and $Z=90$. Atomic units have been used.