# Convergence of Atomic Orbitals 2s and 2p with high Z number

1. Nov 13, 2012

### shihabdider

Hello all, this came up in my chemistry class when our prof. showed a graph of the size of atomic orbitals (or orbital energy) in relation to the Z number (or number of protons). He did this to show that as the z number increases the size of the orbitals also decrease (because effective nuclear charge increases and the orbitals get pulled closer to the nucleus). However, I noticed that at very high z numbers (90 and above) the 2s and 2p orbital lines converge into one continuous line. I just wanted to know why this happened. By convergence I refer to the fact that 2s and 2p have different energies due to the spin orbital coupling hence why the lines for 2s and 2p are separated, yet at high Z numbers the lines converge. This would mean the spin orbital coupling effect is being negated somehow.

My shot at the answer involves electron-electron interactions between higher level orbitals and the lower level orbitals. This is supported by the fact that the higher level orbitals do not experience this convergence, of if they do, not to the same extent (the lines don't merge to one continuous line). However my prof. wants mathematical justification for the answer to this question (which ironically is my own). Looking up the equation for energy of spin orbital splitting I got this:

Eso= Z^4/[2(137^2)n^3]*[j((j+1)-l(l+1)-s(s+1)/2l(l+1/2)(l+1)]

I do not know what the above equation actually means however ( I do not know what the variables are). I assume that Z is the Z number or number of protons, j I found to be l+s which is the total angular momentum, s must then be spin and l, angular momentum. However I still do not see what this equation tells you, (is it energy difference between two orbitals as a result of spin orbital coupling? If so how would you account for all the electrons in an orbital?) nor how you would use it to determine that the energy difference between orbitals 2s and 2p is zero at high Z numbers, according to this equation, Z number shouldn't even matter, since the only way Eso can be zero is when j((j+1)-l(l+1)-s(s+1) equals zero. I think I am using the wrong equation or am not understanding what this equation actually represents. If someone could help me in these two respects I would be very grateful.

2. Nov 13, 2012

### Bill_K

shihabdider, I'd suggest asking this over on the Atomic Physics forum.

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