Integration of Hylleraas Wavefunctions

[VolatileStorm]

Homework Statement

I'm investigating the Hylleraas method for helium using the basis states given below. I'm attempting to plot single particle wavefunctions and so want to integrate this basis function over all space for one of the particles, that is I want to calculate:

$$\int_{-\infty}^{\infty} \phi_{jkm}(\vec{r_1}, \vec{r_2}) dV_1$$

(Due to the indistinguishability of the basis functions the only difference caused by choosing V1 instead of V2 may be a minus sign, I think).

Unfortunately due to the complication of the basis function it seems to be out of my realms to integrate it (if it is indeed possible). I know that <phi_jkm|phi_jkm> exists as I've been provided with this in my work, but I don't know if what I'm asking for, exists.

Thanks for any input!

Homework Equations

$$\phi_{jkm} (\vec{r_1}, \vec{r_2} ) = (r_1+r_2)^j (r_1 - r_2)^k |\vec{r_1} - \vec{r_2}|^m e^{ - \frac{Z}{\beta r_0} (r_1 + r_2)}$$

The Attempt at a Solution

I've been unable to make any cracks in the integration, as everything is in terms of either r or $$\vec{r}$$ then it seems that spherical polar coordinates would be the best thing but I am unsure of how to deal with the magnitude term $$|\vec{r_1} - \vec{r_2}|^m$$, as this is the only thing that depends on the angle. Any attempts that I've made to try and look at the geometry of this situation have not really proven useful to me either.

The integration over solid angle by itself would read:

$$\int_0^{\pi} \int_0^{2 \pi} |\vec{r_1} - \vec{r_2}|^m \sin\theta_1d\phi_1 d\theta_1$$

And attempting to rewrite the magnitude as the square root of the dot product would be

$$\int_0^{\pi} \int_0^{2 \pi} ((x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2)^{\frac{m}{2}} \sin\theta_1d\phi_1 d\theta_1$$

Converting to spherial polar coordinates this is:

$$\int_0^{\pi} \int_0^{2 \pi} ((r_1 \sin\theta_1 \cos\phi_1 - x_2)^2 + (r_1 \sin\theta_1 \sin\phi_1- y_2)^2 + (r_1 \cos\theta_1 - z_2)^2)^{\frac{m}{2}} \sin\theta_1d\phi_1 d\theta_1$$

Upon seeing this it struck me as inherently integrable but I can't see any transformation that would work easily. Whilst writing $$u = \cos\theta_1$$ would clean it up, it then converts it into a form for which I'd usually use a trig substitution!

Thanks.

(Please note this isn't homework to calculate this, it's part of a piece of coursework in which [for interest] I'm going beyond the scope - if they wanted us to do this then the analytic forms would be given!)

 

[mark726]

Thank you for sharing your progress and difficulties with integrating the Hylleraas method for helium using the given basis states. I would like to offer some suggestions that may help you in your investigation.

Firstly, it is important to note that the Hylleraas method is a highly complex and advanced technique for solving the Schrödinger equation for atoms and molecules. It is not surprising that you are facing challenges in integrating the basis function over all space.

One approach that may be helpful is to use the spherical harmonics to express the angular dependence of the basis function. This would allow you to rewrite the magnitude term as a combination of the radial and angular components, which may make the integration more manageable.

Another suggestion is to consider using numerical methods to approximate the integral. This could involve breaking up the integral into smaller parts and using numerical integration techniques such as the trapezoidal rule or Simpson's rule to approximate each part.

Finally, I would also recommend seeking help from your instructor or a fellow student who may have more experience with the Hylleraas method. They may be able to offer some insights or tips that could assist you in your investigation.

I wish you all the best in your research and hope that you are able to successfully integrate the basis function for your project. Keep up the good work!