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Hi, first post here.
I've been trying (out of personal interest, not homework) to re-derive http://www.springerlink.com/content/mm61h49j78656107/" relatively famous calculation on the ground state of Helium from 1929. And I'm stuck at one point.
What Hylleraas did, was to parametricize the wavefunction in terms of r_1,r_2,r_{12} - the scalar electron-nuclear and electron-electron distances, which works since it's spherically symmetrical in the ground state. You insert these into the electronic Hamiltonian, do a little work with the chain rule, etc, and arrive at:
\frac{\partial^2\psi}{\partial r_1^2}+\frac{2}{r_1}\frac{\partial\psi}{\partial r_1}+\frac{\partial^2\psi}{\partial r_2^2}+\frac{2}{r_1}\frac{\partial\psi}{\partial r_2}+2\frac{\partial^2\psi}{\partial r_{12}^2}+\frac{4}{r_{12}}\frac{\partial\psi}{\partial r_{12}}+\frac{r_1^2-r_2^2+r_{12}^2}{r_1r_{12}}\frac{\partial^2\psi}{\partial r_1\partial r_{12}}+\frac{r_2^2-r_1^2+r_{12}^2}{r_1r_{12}}\frac{\partial^2\psi}{\partial r_1\partial r_{12}}<br /> +(\frac{1}{r_1}+\frac{1}{r_2}-\frac{1}{2r_{12}})\psi=-\frac{\lambda}{4}\psi<br />
Equation (5) in the original paper. Where lambda is the energy (Hylleraas used units of R*h=half a Hartree). So far, I'm all good. It's the next step is where I run into trouble. To quote the text (translated):
"This equation is self-adjoint after multiplying with the density function r_1r_2r_{12} and is the Euler equation of a variational problem. This variational problem is:"
<br /> \int^\infty_0dr_1\int^\infty_0dr_2\int^{r_2+r_1}_{|r_2-r_1|}dr_{12}<br /> \{r_1r_2r_{12}<br /> [(\frac{\partial\psi}{\partial r_1})^2+(\frac{\partial\psi}{\partial r_2})^2+<br /> 2(\frac{\partial\psi}{\partial r_{12}})^2]<br /> +r_2(r_1^2-r_2^2+r^2_{12})\frac{\partial\psi}{\partial r_1}\frac{\partial\psi}{\partial r_{12}}+<br /> r_1(r_2^2-r_1^2+r^2_{12})\frac{\partial\psi}{\partial r_2}\frac{\partial\psi}{\partial r_{12}}\\<br /> -[r_{12}(r_1+r_2)-\frac{r_1r_2}{2}]\psi^2\}=\lambda<br />
With the normalization condition:
<br /> \frac{1}{4}\int^\infty_0dr_1\int^\infty_0dr_2\int^{r_2+r_1}_{|r_2-r_1|}dr_{12}r_1r_2r_{12}\psi^2=1.<br />
Equation (6) in the original paper. So, both sides are multiplied with psi and r1r2r12 and integrated so that the normalization condition can be applied to the right side, giving the energy. (The later problem to is to fit the basis-function parameters to minimize the left side; the integrals can be manually evaluated for the basis used)
My problem here is that I still don't quite follow how he got from (5) to (6). The potential terms are easy, but the first ones..? E.g. How did \frac{\partial^2\psi}{\partial r_1^2} turn into (\frac{\partial\psi}{\partial r_1})^2 etc? I've looked around quite a bit for a more detailed derivation, but I can't find one. (Closest thing was in Bethe's "QM of one and two-electron atoms" which after a hand-waving reference to using Green's theorem, skipped to the final result)
So. Can anyone give me a clue here? I've stared at this thing so long it's probably something really simple...
I've been trying (out of personal interest, not homework) to re-derive http://www.springerlink.com/content/mm61h49j78656107/" relatively famous calculation on the ground state of Helium from 1929. And I'm stuck at one point.
What Hylleraas did, was to parametricize the wavefunction in terms of r_1,r_2,r_{12} - the scalar electron-nuclear and electron-electron distances, which works since it's spherically symmetrical in the ground state. You insert these into the electronic Hamiltonian, do a little work with the chain rule, etc, and arrive at:
\frac{\partial^2\psi}{\partial r_1^2}+\frac{2}{r_1}\frac{\partial\psi}{\partial r_1}+\frac{\partial^2\psi}{\partial r_2^2}+\frac{2}{r_1}\frac{\partial\psi}{\partial r_2}+2\frac{\partial^2\psi}{\partial r_{12}^2}+\frac{4}{r_{12}}\frac{\partial\psi}{\partial r_{12}}+\frac{r_1^2-r_2^2+r_{12}^2}{r_1r_{12}}\frac{\partial^2\psi}{\partial r_1\partial r_{12}}+\frac{r_2^2-r_1^2+r_{12}^2}{r_1r_{12}}\frac{\partial^2\psi}{\partial r_1\partial r_{12}}<br /> +(\frac{1}{r_1}+\frac{1}{r_2}-\frac{1}{2r_{12}})\psi=-\frac{\lambda}{4}\psi<br />
Equation (5) in the original paper. Where lambda is the energy (Hylleraas used units of R*h=half a Hartree). So far, I'm all good. It's the next step is where I run into trouble. To quote the text (translated):
"This equation is self-adjoint after multiplying with the density function r_1r_2r_{12} and is the Euler equation of a variational problem. This variational problem is:"
<br /> \int^\infty_0dr_1\int^\infty_0dr_2\int^{r_2+r_1}_{|r_2-r_1|}dr_{12}<br /> \{r_1r_2r_{12}<br /> [(\frac{\partial\psi}{\partial r_1})^2+(\frac{\partial\psi}{\partial r_2})^2+<br /> 2(\frac{\partial\psi}{\partial r_{12}})^2]<br /> +r_2(r_1^2-r_2^2+r^2_{12})\frac{\partial\psi}{\partial r_1}\frac{\partial\psi}{\partial r_{12}}+<br /> r_1(r_2^2-r_1^2+r^2_{12})\frac{\partial\psi}{\partial r_2}\frac{\partial\psi}{\partial r_{12}}\\<br /> -[r_{12}(r_1+r_2)-\frac{r_1r_2}{2}]\psi^2\}=\lambda<br />
With the normalization condition:
<br /> \frac{1}{4}\int^\infty_0dr_1\int^\infty_0dr_2\int^{r_2+r_1}_{|r_2-r_1|}dr_{12}r_1r_2r_{12}\psi^2=1.<br />
Equation (6) in the original paper. So, both sides are multiplied with psi and r1r2r12 and integrated so that the normalization condition can be applied to the right side, giving the energy. (The later problem to is to fit the basis-function parameters to minimize the left side; the integrals can be manually evaluated for the basis used)
My problem here is that I still don't quite follow how he got from (5) to (6). The potential terms are easy, but the first ones..? E.g. How did \frac{\partial^2\psi}{\partial r_1^2} turn into (\frac{\partial\psi}{\partial r_1})^2 etc? I've looked around quite a bit for a more detailed derivation, but I can't find one. (Closest thing was in Bethe's "QM of one and two-electron atoms" which after a hand-waving reference to using Green's theorem, skipped to the final result)
So. Can anyone give me a clue here? I've stared at this thing so long it's probably something really simple...
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