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Here I saw that if an element above atomic no 137 has to exist, it must have electron speed greater than speed of light. My question is , has this calculation been done keeping in mind Einstein's relativity? {I am just asking, I have not done this calculation.}

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    $\begingroup$ Electrons, under quantum mechanics, do not have 'speeds' in the Bohr sense. But, if you asking if relativistic quantum mechanics is possible, it most certainly is. $\endgroup$
    – Jon Custer
    Dec 3 '15 at 21:43
  • $\begingroup$ The article you cite is bad, and those calculation couldn't be relativistic $\endgroup$
    – Mithoron
    Dec 3 '15 at 22:07
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    $\begingroup$ Possible duplicate of The last element's atomic number $\endgroup$
    – Mithoron
    Dec 3 '15 at 22:10
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    $\begingroup$ Already for elements in the fifth (or even earlier?) period, calculations of the inner electronic structure make no sense at all if relativity is not taken into account. $\endgroup$
    – Karl
    Dec 3 '15 at 22:20
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    $\begingroup$ @DavePhD Geez, he's talking about Bohr. With Dirac you'd have oscillating ground state not "exceeding c". and with more precise analysis you have problem at about 173, but of another kind en.wikipedia.org/wiki/… $\endgroup$
    – Mithoron
    Dec 3 '15 at 22:38
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Yes, the value 137 occurs even considering Einstein's theory of special relativity.

137 comes from considering Sommerfeld or Dirac relativistic theories, when the nucleus is modeled as a point.

See equation 1 of A new method for solving the Z > 137 problem and for determination of energy levels of hydrogen-like atoms

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