[Wien] Electron density at the nucleus (Electron capture nuclear decay rate work)

[Amlan Ray]

Dear Prof. Marks,
I am writing in reply to your suggestion dated April 19, 2010 on the above subject. The RMT(Be) was always larger than RMT(O). I used RMT(Be)=1.45 BU and RMT(O)=1.23 BU. Later on, I used up to RMT(Be)=1.58 BU and RMT(O)= 1.1 BU. As RMT(Be) is increased from 1.45 to 1.58 for BeO(Normal case), the 1s electron density at Be nucleus increases very slightly by 0.0158% and the total electron density at the Be nucleus increases by 0.014%. However when the calculation is repeated for the compressed BeO, keeping RMT(Be)=1.58 or less, the 1s electron density at the Be nucleus always decreases and 2s electron density at Be nucleus always increases, the net result is about 0.1% increase of the total electron density at the nucleus due to about 9% volume compression of BeO lattice, against the experimental number of 0.6%. 
 
My problem is regarding the reduction of 1s electron density at Be nucleus due to the compression. As per your suggestion, I have checked out the leakage of 1s electron charge from the Be muffintin sphere. I subtracted out the total 2s valence charge in Be sphere (CHA001) from the total charge (CTO001) in Be sphere to obtain the 1s charge in Be sphere. So CTO001-CHA001 = 1s charge in Be sphere. 
I kept RMT(Be)=1.45 BU fixed for both the uncompressed and compressed cases and studied the 1s charge leakage from Be sphere. I find
1) for 9% volume compression of BeO, leakage of 1s charge from Be sphere = 0.01%; reduction of 1s electron density at Be nucleus due to compression = 0.148%.
2) for 16.6% volume compression of BeO, leakage of 1s charge from Be sphere=0.018%; reduction of 1s electron density at Be nucleus due to compression = 0.265%.
3) for 28.4% volume compression of BeO, leakage of 1s charge from Be sphere = 0.033%; reduction of 1s electron density at Be nucleus due to compression = 0.466%.
 
If I fix RMT(Be)=1.58 BU for all the calculations, then 
1) for 9% volume compression of BeO, leakage of 1s charge from Be sphere = 0.004%; reduction of 1s electron density at Be nucleus due to compression = 0.15%. 
2) for 16.6% volume compression of BeO, 1s charge leakage from Be sphere =0.011%; reduction of 1s electron density at Be nucleus due to compression = 0.265%. 
 
So as the compression on BeO lattice is increased, Hartree potential increases and the character of the 1s electron wave function of Be changes. The 1s wave function becomes more defused and spread out and so the leakage from the Be sphere increases with the compression. Since the free atom 1s wave function becomes more defused and spread out due to the compression, the 1s electron density at Be nucleus decreases. Now if a boundary condition such as 1s wave function must be zero at RMT(Be) is put on, then the compression will not cause the spread out of the wave function. WIEN2K suggests that we should use a smaller value of RMT(Be) when BeO lattice is compressed. This should increase the 1s electron density at the nucleus, if the wave function is constrained to be zero at RMT(Be). The absolute percentage of the 1s charge leakage from Be sphere might be small, but it tends to increase very quickly with the compression. I think if the 1s wave function
 is constrained to be zero at RMT=1.45 or 1.58, then that can influence the change of 1s electron density at Be nucleus under compression by the fraction of a percent. 
 
In the case of 2s valence electrons of Be, I find that when RMT(Be) is kept fixed at 1.45 BU, then 2s valence charge in Be sphere increases from 0.1943 to 0.2213 for 16.6% volume compression of BeO. The 2s electron density at Be nucleus also increases due to the compression. 2s electron wave function satisfies an appropriate boundary condition at RMT and that may be the part of the reason for the increase of 2s electron density under compression. 
 
So my suggestion is to kindly consider putting a boundary condition such as 1s Be wave function = 0 at RMT(Be). Such a boundary condition should affect the character of the wave function and hence the change of 1s electron density at the nucleus due to the compression. 
 
With best regards
                                                                                         Amlan Ray
Address
Variable Energy Cyclotron Center
1/AF, Bidhan Nagar
Kolkata - 700064
India

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