hilbert2 said:The closer the orbiting electron is to the nucleus, the higher the expectation value of its speed (result of position-momentum uncertainty principle), and relativistic effects are therefore more important in the case of core electrons.
hilbert2 said:When the electron is more localized near the nucleus (like in a core orbital), we know its position more precisely, and therefore its momentum is more uncertain according to the equation [itex]\Delta x \Delta p \geq \frac{\hbar}{2} [/itex] . When the momentum is more uncertain, the expectation value of speed (norm of velocity vector) must be higher. At high speeds, relativistic effects become important.
citw said:[tex]\text{if }\sqrt{\big<p^2\big>-\big<p\big>^2}\text{ is large, } \big<p\big>^2\text{ could be small, or }\big<p^2\big>\text{ could be large.}[/tex]
citw said:I realize this is something I should probably know intuitively, but why are core states (generally) treated relativistically in density functional calculations? What exactly makes these relativistic bound states rather than non-relativistic? I think this is some basic physics I'm forgetting.
DrDu said:Both their potential and kinetic energy are higher than that of the valence electrons. The high kinetic energy means that they have speeds near the speed of light so that relativistical corrections become important.
gadong said:The higher speed near the core is related to <KE> not dp, and can be understood via the virial theorem (http://en.wikipedia.org/wiki/Virial_theorem), which applies to both QM and CM. Briefly, there is a definite mathematical relationship between <KE> and the mean potential for a Coulombic potential.
Note that in a closed shell atom, the potential at a distance R from the nucleus, V(R), depends mainly on the charge that is located in the region between 0 and R (from Gauss' Law). So, deep core electrons in heavy atoms experience a largely unshielded nuclear potential.
I believe that hilbert2 is essentially answering the question of how core electrons come about, which is more closely related to the uncertainty principle.
The relativistic treatment of core electrons is a method used in Density Functional Theory (DFT) calculations to incorporate the effects of special relativity on the movement of these electrons. This is important because core electrons, which are those in the innermost shells of an atom, move at speeds close to the speed of light and thus must be treated relativistically to accurately describe their behavior.
The relativistic treatment of core electrons is important because it helps to accurately predict the electronic structure and properties of heavy elements, such as those found in the lower rows of the periodic table. Without considering the effects of relativity, DFT calculations may not accurately describe the behavior of these electrons and can lead to incorrect results.
In DFT calculations, the relativistic treatment of core electrons is typically included through the use of relativistic pseudopotentials. These are mathematical functions that approximate the effect of the core electrons on the valence electrons, taking into account their relativistic motion. This allows for more accurate calculations of the electronic structure of heavy elements.
One limitation of the relativistic treatment of core electrons in DFT is that it is only applicable for single-electron systems, meaning that it does not take into account the interactions between electrons. Additionally, the accuracy of the calculations can be affected by the choice of pseudopotential used, and there may be computational limitations for larger systems.
Yes, the relativistic treatment of core electrons can be used for all elements. However, it is most commonly used for heavy elements, such as those with atomic numbers greater than 100, where relativistic effects are more significant. For lighter elements, the effects of relativity are typically negligible and the non-relativistic treatment may be sufficient.