[Wien] Electron density at the nucleus (electron capture nuclear decay rate work)
Dear Prof. Blaha, Thank you very much for the detailed explanation regarding the treatment of the core 1s state of Be. I can now understand much better how?the calculation for the core state?is being done. If there is any paper or document describing the treatment of the core state in detail, then please give me the reference. As a beginner, I still have a few questions and shall be grateful for your reply. ? 1) My understanding was that the total potential (electrostatic and exchange) inside a lattice approximately looks like a Muffin tin potential. For example, in the interstitial region, the electrostatic fields of the neighboring ions should approximately cancel out producing approximately a constant potential (zero field) region. I understand there is some fuzziness or arbitrariness in the determination of Muffin tin radius RMT and that part has no physical significance, but the overall picture of Muffin tin potential inside a lattice should have physical significance. Are you also saying the same thing? ? 2) I did not know earlier that 1s wave function is seeing a continuous potential and?the potential?is continuing with a 1/r tail outside the Be sphere. However this would mean that 1s electrons are seeing the potential of a single Be ion outside RMT. But will not the potential outside the Be sphere be approximately constant because of the presence of other Be ions? If 1s and 2s electrons see different potentials outside the Be sphere, then they would not be orthogonal to each other. ? 3) I have done calculations by treating both 1s and 2s states of Be as valence states, but I have not really understood how the code actually handled the situation. I understand that both 1s and 2s electrons will see the same complete potential inside the Be sphere and there would be spherical harmonic solutions. But outside the Be sphere, 2s electrons are seeing the potential of the interstitial region and are treated as approximately plane waves with a matching boundary condition at RMT. ? How will 1s electrons be treated outside the Be sphere, when they are also considered as valence electrons? They should also see the same constant potential in the interstitial region, but probably because of their higher energy should not be treated as plane waves. There would be the question of boundary condition at RMT for 1s valence electrons also. ? The absolute value of the total electron density at R0 increases by 0.07% to 0.14% if 1s is treated as a core state compared to as a valence state for Be. However the change of total electron density for Be?at R0 for the compressed versus normal BeO lattice cases remain essentially the same whether we treat 1s as core or valence state. ? 4) The energy of 1s core state always increases due to the compression of BeO lattice. From the?physical point of view, I thought that the energy?was increasing because the electrons?were confined to a smaller region due to the compression. If the 1s electrons are spreading out, then what are the physical?reasons for the increase of the energy of 1s electrons. ? With best regards Amlan Ray Address Variable Energy Cyclotron Center 1/AF, Bidhan Nagar Kolkata - 700064 India? -- next part -- An HTML attachment was scrubbed... URL: http://zeus.theochem.tuwien.ac.at/pipermail/wien/attachments/20100427/5d353a99/attachment.htm
[Wien] Electron density at the nucleus (Electron capture nuclear decay rate work)
I have been reading all the messages about the electron density at the Be nucleus under compression and would like to say a few things. My background is in experimental nuclear physics and I am very interested to undertsand quantitatively the results of electron capture experiments in compressed material. WIEN2K is probably the best availabale code at this time for this purpose. Given my background, please excuse me if I make any incorrect statements. I shall be grateful if you would kindly point out my mistakes. ? 1) Let me start with the Physics justification for thinking why Be 1s wave function should satisfy boundary conditions at the muffintin radius RMT(Be). As I understand, in this model, 1s electrons are seeing scf-potential of the crystal only within the Be sphere. Outside the Be sphere, it should see the potential of the interstitial region. Since there is an abrupt change of potential at the muffintin radius RMT(Be), so the wave function inside and outside the Be sphere should be different and there should be a matching boundary condition at RMT(Be). If we assume that outside the Be sphere, the 1s wave function should be that of a free Be ion, then it should be matched with the core wave function inside the Be sphere at RMT(Be). As a gross oversimplification, I suggested that the 1s wave function outside RMT(Be) might be taken as zero, because I thought that would be relatively easy to implement.(But I agree it was a?wrong boundary condition.)??However ?my main point is that the core wave function inside and outside the Be sphere should be different and there should be boundary conditions at RMT(Be). ? 2) I think whether compression would delocalize 1s wave function?should depend on the boundary condition applied. If the only boundary condition is that the core wave function would be zero at infinity, then of course, it will delocalize under compression. But probably there should be boundary conditions at RMT(Be). ? 3) I certainly agree that the tail of 1s wave function would experience more attractive potential when BeO is compressed. But I think that would affect the core wave function outside the Be sphere. It is not clear to me how that would affect the core wave function inside the Be sphere, particularly near the nucleus. The potential inside and outside the Be sphere is different and the wave functions should, in general, be different with a matching boundary condition at RMT(Be). ? 4) I certainly agree that the?contraction of 2s orbital would drive 1s orbital into expansion. But the reduction of 1s electron density at the nucleus is essentially independent of the muffintin radius used. I have done calculations of normal and compressed BeO cases keeping RMT(Be) the same in both the cases and have also done calculations by reducing RMT(Be) for the compressed case only. The change of 1s electron density at the nucleus remains the same always. The change of valence electrons in Be sphere is only 0.01 electrons and I can vary this number by adjusting RMT(Be). But that did not affect the change of 1s electron density at the nucleus. s-valence electrons in Be sphere can be made smaller?for the compressed case by adjusting RMT(Be), but still the result did not change. So I think that the effect of 2s orbital contraction on 1s electron density at the nucleus is probably very small. ? 5) I know about three experiments (done by different people) where the increase of electron capture rate by nuclei under compression?was seen and the effect is much more than expected from valence electrons. ? With best regards Amlan Ray Address Variable Energy Cyclotron Center 1/AF, Bidhan Nagar Kolkata - 700064 India -- next part -- An HTML attachment was scrubbed... URL: http://zeus.theochem.tuwien.ac.at/pipermail/wien/attachments/20100423/1bc4d143/attachment.htm
[Wien] Electron density at the nucleus (Electron capture nuclear decay rate work)
The construction of atomic spheres with a certain RMT is only a mathematical trick to obtain nicely represented wave functions and potentials in a convenient way. Of course there is a weak dependency of results on RMT, because series expansions converge better or worse with different RMTs, but there's no physics in it. RMT(Be). As I understand, in this model, 1s electrons are seeing scf-potential of the crystal only within the Be sphere. Outside the Be sphere, it should see the potential of the interstitial region. Since there is an abrupt change of potential at the muffintin radius RMT(Be), so the wave function inside and outside the Be sphere should be different and there should be a matching boundary condition at RMT(Be). No, the 1s electron sees the (spherical) potential not only inside RMT, but the potential is continued outside with a 1/r tail. (There is only ONE 1s wavefunction on a radial grid reaching to infinity.) Of course one can discuss this approximation, but as you have shown yourself, treating the 1s state as valence, where it sees the accurate non-spherical potential everywhere, does NOT change anything qualitatively (there is a limited basis set for the Be-s functions when you include 1s, but that does not matter for this purpose). However my main point is that the core wave function inside and outside the Be sphere should be different and there should be boundary conditions at RMT(Be). From the above it should be clear, that there is only ONE 1s function. For a core state, however, we make the approximation that the core-density outside the sphere is added as a constant smeared out over the whole interstitial. Also this is an approximation (and the code gives WARNINGS if the core leakage is too large), but again, your test with 1s as valence (where this is not done) proves that there is no real problem. PS: In the next release it will be possible to Fourieranalyze the leaking core density and get a correct charge distribution even with sizable core-leakage. -- - Peter Blaha Inst. Materials Chemistry, TU Vienna Getreidemarkt 9, A-1060 Vienna, Austria Tel: +43-1-5880115671 Fax: +43-1-5880115698 email: pblaha at theochem.tuwien.ac.at -
[Wien] Electron density at the nucleus (Electron capture nuclear decay rate work)
I'd have to recheck how the Fe-Isomershift core contributions change under pressure, but the longer I think about the problem, the more I understand that the Be-1s density gets more delocalized under compression. If the neighbors are far away, the Be 1s orbital sees for long time a kind of Z/r potential and only at rather large distances the potential bends over and gets attractive again because there is a neighboring atom. If you compress, the potential gets attractive at smaller distances, i.e. the localized 1s orbital will expand a bit, because it sees a more attractive potential at its tail. Also the contraction of the 2s orbital drives the 1s electron into expansion: More 2s charge in the core region weakens the attractive potential for the 1s orbital. Certainly, GGA-DFT potentials have their shortcuts and we know that the core states are not localized enough, but I believe qualitatively the behavior is correct. For sure I believe, that WIEN2k is one of the very few programs which can deal with such problems because of the numerical all electron basis set. Of course if DFT is the problem,. Pavel Novak schrieb: let me comment. I do not recommend to use the Lundin-Eriksson functional. While the contact hyperfine field for 3d atoms is improved, we realized that it violates important sum rule for the exchange-correlation hole, which is imposed by the density functional theory. This brings several shortcomings e.g. incorrect energy of the core states. I doubt that any local or semilocal Vxc can provide reliable value of the core density at the nucleus. For bcc Fe Akai and Kotani (Hyperfine Interactions, 120/121, 3, 1999 obtained good contact field using the optimised effective potential method, but this is computationally expensive. Regards Pavel Novak On Wed, 21 Apr 2010, Laurence Marks wrote: A few comments, and perhaps a clarification on what Peter said. Remember that while Wien2k is more accurate than most other DFT codes, it still has approximations with the form of the exchange-correllation potential and in how the core wavefunctions are calculated. Hacking by applying unphysical constraints so it will match experiments is wrong. (Remember the story of the graduate student who matched all properties of silicon by tuning the parameters of the DFT calculation for each one so it was right.) I would instead suggest that you look at better functionals for the core wavefunctions, see Novak et al, Phys. Rev. B 67, 140403(R) (2003) as well as the papers that cite it and the earlier paper by U. Lundin and O. Eriksson, Int. J. Quantum Chem. 81, 247 (2001) and papers that cite this. If you ask Peter or Pavel really nicely they may be able to provide the code that uses this functional but you will almost certainly have to do some coding work. This might not explain your experimental results, and if it does not either the experiments are wrong or we just don't have good enough theory yet for what you are measuring, probably the latter. -- - Peter Blaha Inst. Materials Chemistry, TU Vienna Getreidemarkt 9, A-1060 Vienna, Austria Tel: +43-1-5880115671 Fax: +43-1-5880115698 email: pblaha at theochem.tuwien.ac.at -
[Wien] Electron density at the nucleus (Electron capture nuclear decay rate work)
There is no physics involved in constraining the 1s wavefuction to zero at an arbitrary radius RMT. It is anyway constrained to be zero at r=infinity and only this is meaningful. It seems pretty clear that the results are as they are, whether you like it or not. If you want to cheat the results, you could do a frozen core, i.e. after init_lapw you do: x lapw0-- creates a potential from superposed atomic densities x lcore-- create the core density rm case.inc -- remove the input file to prevent recalculation of core states run_lapw Of course, for consistency you should never change sphere sized when you compare densities. Amlan Ray schrieb: 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
[Wien] Electron density at the nucleus (Electron capture nuclear decay rate work)
A few comments, and perhaps a clarification on what Peter said. Remember that while Wien2k is more accurate than most other DFT codes, it still has approximations with the form of the exchange-correllation potential and in how the core wavefunctions are calculated. Hacking by applying unphysical constraints so it will match experiments is wrong. (Remember the story of the graduate student who matched all properties of silicon by tuning the parameters of the DFT calculation for each one so it was right.) I would instead suggest that you look at better functionals for the core wavefunctions, see Novak et al, Phys. Rev. B 67, 140403(R) (2003) as well as the papers that cite it and the earlier paper by U. Lundin and O. Eriksson, Int. J. Quantum Chem. 81, 247 (2001) and papers that cite this. If you ask Peter or Pavel really nicely they may be able to provide the code that uses this functional but you will almost certainly have to do some coding work. This might not explain your experimental results, and if it does not either the experiments are wrong or we just don't have good enough theory yet for what you are measuring, probably the latter. On Wed, Apr 21, 2010 at 6:48 AM, Peter Blaha pblaha at theochem.tuwien.ac.at wrote: There is no physics involved in constraining the 1s wavefuction to zero at an arbitrary radius RMT. It is anyway constrained to be zero at r=infinity and only this is meaningful. It seems pretty clear that the results are as they are, whether you like it or not. If you want to cheat the results, you could do a frozen core, i.e. after init_lapw you do: x lapw0 ? ?-- creates a potential from superposed atomic densities x lcore ? ?-- create the core density rm case.inc -- remove the input file to prevent recalculation of core states run_lapw Of course, for consistency you should never change sphere sized when you compare densities. Amlan Ray schrieb: 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
[Wien] Electron density at the nucleus (Electron capture nuclear decay rate work)
let me comment. I do not recommend to use the Lundin-Eriksson functional. While the contact hyperfine field for 3d atoms is improved, we realized that it violates important sum rule for the exchange-correlation hole, which is imposed by the density functional theory. This brings several shortcomings e.g. incorrect energy of the core states. I doubt that any local or semilocal Vxc can provide reliable value of the core density at the nucleus. For bcc Fe Akai and Kotani (Hyperfine Interactions, 120/121, 3, 1999 obtained good contact field using the optimised effective potential method, but this is computationally expensive. Regards Pavel Novak On Wed, 21 Apr 2010, Laurence Marks wrote: A few comments, and perhaps a clarification on what Peter said. Remember that while Wien2k is more accurate than most other DFT codes, it still has approximations with the form of the exchange-correllation potential and in how the core wavefunctions are calculated. Hacking by applying unphysical constraints so it will match experiments is wrong. (Remember the story of the graduate student who matched all properties of silicon by tuning the parameters of the DFT calculation for each one so it was right.) I would instead suggest that you look at better functionals for the core wavefunctions, see Novak et al, Phys. Rev. B 67, 140403(R) (2003) as well as the papers that cite it and the earlier paper by U. Lundin and O. Eriksson, Int. J. Quantum Chem. 81, 247 (2001) and papers that cite this. If you ask Peter or Pavel really nicely they may be able to provide the code that uses this functional but you will almost certainly have to do some coding work. This might not explain your experimental results, and if it does not either the experiments are wrong or we just don't have good enough theory yet for what you are measuring, probably the latter. --
[Wien] electron density at the nucleus (Electron capture nuclear decay rate work)
Hi, I must admit that I don't know the physics of electron capture measurements, but a few thoughts: a) Electron density at the nucleus ??? What kind of nucleus ?? A point nucleus (r=0) or a nucleus of finite size ?? Do you need the density at r=0 or an average over the volume of the nucleus or a larger area (Tomson radius as used for Hyperfine fields in M?ssbauer) ??? b) The printed density :RTO is not for r=0, but for the first radial mesh point r0 (PS: Of course you can change R0 to a smaller value, but be aware to use a meaningful value or the outer regions will have a very crude mesh). c) The relativistic wf. at the nuclues diverge for a point-nucleus. Try to use NREL instead of RELA in case.struct. A non-relativistic calc. for Be should be ok. d) The core electrons are solved in the scf-potential of the crystal (V_c + V_xc) in an central field approximation (free atom) with a spherical potential only within the atomic sphere (numerical solution of the Dirac equation). The only problem: The Be 1s state is not completely confined within RMT, which may cause some artefacts. Use spheres for Be as large as possible (or treat it as valence as you did anyway. But the difference in the resutls is suspicious...) e) Usually M?ssbauer studies under pressure can be well described by our calculations. Regards Peter Blaha Am 07.04.2010 12:26, schrieb Amlan Ray: Dear Stefaan, Thank you for your detailed message suggesting to check several things. I have now done those calculations and let me discuss the results and my thoughts. Regarding the question whether the 1s electron density at the nucleus should increase because of the compression of the beryllium atom, I still think it should increase (although WIEN2K is predicting that it should decrease) and the presence of the 2s electrons should not reverse this result and cause a decrease. 1s and 2s states are orthogonal and the total electron density at the nucleus is simply the sum of the two as WIEN2K code is also showing. However 2s electrons would certainly screen 1s electrons and as a result of the compression of 2s orbitals and the corresponding increase of 2s electron density at the nucleus, this screening effect would also increase. But I think this would be a higher order effect and cannot reverse the increase of 1s electron density at the nucleus due to the compression effect. For example, J. Bahcall (Phys. Rev. 128, 1207 (1962)) and Hartree and Hartree (Proc. R. Soc. London ser. A 150, 9 (1935)) found from their calculations that even if both the 2s electrons are removed from a beryllium atom (making 2s electron density at the nucleus =0), then also 1s electron density at the nucleus changes by only about a few tenth of a percent. This is because when 1s electron is very close the nucleus, it sees only the bare nucleus. In our case, WIEN2K is predicting only 10% increase of 2s electron density at the nucleus and so its effect on 1s electron density at the nucleus would be much smaller. On the other hand, the compression of 1s orbital should definitely increase the electron density at the nucleus. So the prediction of WIEN2K (decrease of 1s electron density at the nucleus due to compression) is still puzzling to me. Regarding the calculations I have done 1) R0 =0.0001BU = 5.29 Fermi in my calculation. So it is nuclear distance. I tried to make it smaller, but the code didi not take any smaller value and made it again = 0.0001 BU. 2) I performed calculations making both 1s and 2s valence states (-8 Ry input). For 9% volume compression of BeO lattice, it predicts 0.033% increase of total electron density at the nucleus. When I kept 1s as a core state, then the corresponding increase of total electron density at the nucleus was 0.09%. However if I compress BeO lattice volume by 15%, and treat both the 1s and 2s states as valence states, the corresponding increase of total electron density at the nucleus = 0.18%. On the other hand, if 1s is treated as a core state, then for 15% volume compression, the increase of total electron density at the nucleus= 0.15%.The experimental result regarding the increase of electron density at the nucleus due to 9% compression of BeO lattice is about 0.6% -0.8% (W..K Henseley et al., Science, 181, 1164 (1973) and L.g. Liu et al., Earth. Planet. Sci. Lett. 180, 163 (2000)). (However Liu's result of 0.8% is for amorphous beryllium hydroxide.) Mossbauer isomer shift is proportional to the difference of contact densities, but the electron capture rate is directly proportional to the electron density at the nucleus. I do not know if anyone has studied Mossbauer isomer shift under the effect of compression of the atom. I would like to know how WIEN2K is doing 1s state wavefunction calculation under compression. What is the relevant subroutine to look at and any reference about the wavefunction calculation? Is it solving Schrodinger equation under both Coulomb
[Wien] electron density at the nucleus (Electron capture nuclear decay rate work)
Dear Stefaan, Thank you for your detailed message suggesting to check several things. I have now done those calculations and let me discuss the results and my thoughts. ? Regarding the question whether the 1s electron density at the nucleus should increase because of the compression of the beryllium atom, I still think it should increase (although WIEN2K is predicting that it should decrease)?and the presence of the 2s electrons should not reverse this result and cause a decrease. 1s and 2s states are orthogonal and the total electron density at the nucleus?is simply the sum of the two as WIEN2K code is also showing. However 2s electrons would certainly screen 1s electrons and as a result of the compression of 2s?orbitals and the corresponding increase of 2s electron density at the nucleus, this screening effect would also increase. But I think this would be a higher order effect and cannot reverse the increase of 1s electron density at the nucleus due to the compression effect. For example, J. Bahcall (Phys. Rev. 128, 1207 (1962)) and Hartree and Hartree (Proc. R. Soc. London ser. A 150, 9 (1935)) found from their calculations that even if both the 2s electrons are removed from a?beryllium atom (making 2s electron density at the nucleus =0), then also 1s electron density at the nucleus changes by only about a few tenth of a percent. This is because when 1s electron is very close the nucleus, it sees only the bare nucleus. In our case, WIEN2K is predicting only 10%?increase of?2s electron density at the nucleus and so its effect on 1s electron density at the nucleus would be much smaller. On the other hand, the compression of 1s orbital should definitely increase the electron density at the nucleus. So the prediction of WIEN2K (decrease of 1s electron density at the nucleus due to compression)? is still puzzling to me. ? Regarding the calculations I have done 1) R0 =0.0001BU = 5.29 Fermi in my calculation. So it is nuclear distance. I tried to make it smaller, but the code didi not take any smaller value and made it again = 0.0001 BU. 2) I performed calculations making both 1s and 2s valence states (-8 Ry input). For 9% volume compression of BeO lattice, it predicts 0.033% increase of total electron density at the nucleus. When I kept 1s as a core state, then the corresponding increase of total electron density at the nucleus was 0.09%. However if I compress BeO lattice volume by 15%, and treat both the?1s and 2s states as valence states, the corresponding increase of total electron density at the nucleus = 0.18%. On the other hand, if 1s is treated as a core state, then for 15% volume compression, the increase of total electron density at the nucleus= 0.15%.The experimental result regarding the increase of electron density at the nucleus due to 9% compression of BeO lattice is about 0.6% -0.8%?(W..K Henseley et al., Science, 181, 1164 (1973) and L.g. Liu et al., Earth. Planet. Sci. Lett. 180, 163 (2000)). (However Liu's result of 0.8% is for amorphous beryllium hydroxide.) ? Mossbauer isomer shift is proportional to the difference of contact densities, but the electron capture rate is directly proportional to the electron density at the nucleus. I do not know if anyone has studied Mossbauer isomer shift under the effect of compression of the atom. ? I would like to know how WIEN2K is doing 1s state wavefunction calculation under compression. What is the relevant subroutine to look at and any reference about the wavefunction calculation? Is it solving Schrodinger equation under both Coulomb and Hartree potential? N. Aquino et al. have performed (Phys. Lett A307, 326 (2003))?density functional calculations of a single compressed He atom placed in a spherical box. They have also recently completed such density functional calculations for compressed Li atom placed in a spherical box. It is found from their calculations that the 1s state electrons of a compressed He or Li atom very quickly start looking like a Thomas-Fermi atom where the electrons are in a box of radius equal to the mean radius of 1s?electrons and the electrons can be treated as free particles. The kinetic energy of 1s electrons increases as the inverse square of the radius of mean distance of 1s electrons from the nucleus (Thomas-Fermi atom result). ? I find if I take the increase of 1s electron energy (due to the compression) from WIEN2K or TB-LMTO calculations (both give similar results) and then apply simple Thomas-Fermi model of atom assuming that the increase of the energy of 1s electrons (mostly kinetic energy increase) is due to the reduction of 1s orbital volume, then I can get a number for the increase of electron density in the box. If this is interpreted as the increase of electron density at the nucleus, then I get reasonable agreement with the experimental numbers of Henseley et al. (regarding BeO)?and also with our experimental results regarding the increase of
