Stuck deriving the Hylleraas variational method

[alxm]

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:
$$\frac{\partial^2\psi}{\partial r_1^2}+\frac{2}{r_1}\frac{\partial\psi}{\partial r_1}+\frac{\partial^2\psi}{\partial r_2^2}+\frac{2}{r_1}\frac{\partial\psi}{\partial r_2}+2\frac{\partial^2\psi}{\partial r_{12}^2}+\frac{4}{r_{12}}\frac{\partial\psi}{\partial r_{12}}+\frac{r_1^2-r_2^2+r_{12}^2}{r_1r_{12}}\frac{\partial^2\psi}{\partial r_1\partial r_{12}}+\frac{r_2^2-r_1^2+r_{12}^2}{r_1r_{12}}\frac{\partial^2\psi}{\partial r_1\partial r_{12}} +(\frac{1}{r_1}+\frac{1}{r_2}-\frac{1}{2r_{12}})\psi=-\frac{\lambda}{4}\psi$$
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:"

$$\int^\infty_0dr_1\int^\infty_0dr_2\int^{r_2+r_1}_{|r_2-r_1|}dr_{12} \{r_1r_2r_{12} [(\frac{\partial\psi}{\partial r_1})^2+(\frac{\partial\psi}{\partial r_2})^2+ 2(\frac{\partial\psi}{\partial r_{12}})^2] +r_2(r_1^2-r_2^2+r^2_{12})\frac{\partial\psi}{\partial r_1}\frac{\partial\psi}{\partial r_{12}}+ r_1(r_2^2-r_1^2+r^2_{12})\frac{\partial\psi}{\partial r_2}\frac{\partial\psi}{\partial r_{12}}\\ -[r_{12}(r_1+r_2)-\frac{r_1r_2}{2}]\psi^2\}=\lambda$$
With the normalization condition:
$$\frac{1}{4}\int^\infty_0dr_1\int^\infty_0dr_2\int^{r_2+r_1}_{|r_2-r_1|}dr_{12}r_1r_2r_{12}\psi^2=1.$$

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...

 

[0xDEADBEEF]

Well did you try what the paper said? Take the second term and calculate its Euler Lagrange equation?
http://en.wikipedia.org/wiki/Calculus_of_variations

 

[charng]

Hello there, thank you for reaching out. The Hylleraas variational method is a powerful tool for solving the Schrodinger equation for multi-electron atoms. I can understand how you might be stuck in the derivation process, as it can be quite complex. Let me try to explain it in a simpler manner.

The goal of the Hylleraas method is to find a variational wavefunction that will minimize the energy of the system. This wavefunction is parametrized in terms of r1, r2, and r12, as you mentioned. Now, in order to find this wavefunction, we need to solve the Schrodinger equation, which is a second-order differential equation. When we insert the parameters into the Schrodinger equation, we get the equation that you mentioned in your post (equation 5).

Now, in order to make the equation simpler, we can multiply both sides by the density function r1r2r12. This makes the equation self-adjoint, which means that the left and right sides are equal when integrated over all space. This is the step you mentioned where you got confused.

After multiplying by the density function, we can rewrite the equation in a more convenient form, by using the chain rule and some algebraic manipulation. This results in the equation that you mentioned in your post (equation 6).

Now, the variational problem that Hylleraas set up is to find the wavefunction that minimizes the energy. This is done by finding the minimum of the integral on the left side of equation 6, subject to the normalization condition on the right side. This is a variational problem, and the solution to this problem will give us the optimal wavefunction.

I hope this helps to clarify the derivation process for you. It is a complex method, but with patience and persistence, I am sure you will be able to understand it fully. Good luck with your research!