# Overlap integral of hydrogen molecule

• hicetnunc
In summary: You need to use spherical coordinates, with the z-axis pointing towards the center of the nucleus, and the x- and y-axes pointing outwards.
hicetnunc
Homework Statement
Show that the overlap integral of a 1s orbital and a 2p orbital from hydrogen forming a ##\sigma## bond is
$$S = \frac{R}{a_0} \Big[ 1 + \frac{R}{a_0} + \frac{1}{3} \Big( \frac{R}{a_0} \Big)^2 \Big] e^{-R/a_0}$$
where ##R## is the distance between the atoms.
Relevant Equations
None.
Hi! Some help with this problem would be much appreciated.

The overlap integral is defined as ##S = \int \phi_A (\mathbf{r}_A) \phi_B (\mathbf{r}_B) \,d\mathbf{r}##. For the two orbitals, I have that
$$\phi_A = \frac{1}{\sqrt{\pi}} \Big( \frac{1}{a_0} \Big)^{3/2} e^{-r_A / a_0}$$
for the 1s orbital and
$$\phi_B = \frac{1}{4\sqrt{2\pi}} \Big( \frac{1}{a_0} \Big)^{5/2} \cos \theta r_B e^{-r_B / 2a_0}$$
for the 2p orbital. This would give an overlap integral of
$$S = \frac{1}{4\sqrt{2}\pi} \Big( \frac{1}{a_0} \Big)^4 \int \cos \theta r_B e^{-(r_A+0.5r_B)/a_0} \,d\mathbf{r}$$
Now, the textbook I'm using suggests using spheroidal coordinates ##u=\frac{r_A + r_B}{R}## and ##v=\frac{r_A - r_B}{R}##. Since ##\theta## is the angle between ##r_B## and the z-axis, this gives me ##\cos \theta = \frac{z - R/2}{r_B}## and since ##z=\frac{R}{2}uv## I get ##\cos \theta = \frac{R(uv-1)}{2 r_B}##. The overlap integral is now
$$S = \frac{R}{8\sqrt{2}\pi a_0^4} \int (uv-1) e^{-R(3u-v)/(4a_0)} \,d\mathbf{r}$$
The volume element is ##\frac{R^3}{8}(u^2-v^2)## and the variables ##(u, v, \theta')## have the intervals ##1 \leq u \leq \infty##, ##-1 \leq v \leq 1## and ##0 \leq \theta' \leq 2\pi##.

But I can't get the same solution for this integral as stated in the exercise. Already from the beginning I have a factor of ##1 / \sqrt{2}## that will follow through and not disappear. I've tried solving the integral, but it's a really tedious and long calculation. Do I at least have the right integral?

I've done a similar version of this exercise for a hydrogen molecule with two 1s orbitals, and got the right solution for that. That integral was MUCH easier to solve than this.

PhDeezNutz
No one seems to have answered this but I think your waveforms are not correct.

## What is the overlap integral of a hydrogen molecule?

The overlap integral of a hydrogen molecule is a measure of the overlap between the two atomic orbitals of the hydrogen atoms in the molecule. It is used to determine the strength of the bond between the two atoms.

## How is the overlap integral calculated?

The overlap integral is calculated by integrating the product of the two atomic orbitals over all space. This can be done analytically or numerically using computational methods.

## What does a high overlap integral indicate?

A high overlap integral indicates a strong bond between the two atoms in the molecule. This means that the electrons are more likely to be shared between the two atoms, resulting in a stable molecule.

## What is the significance of the overlap integral in molecular bonding?

The overlap integral is a crucial factor in determining the strength and stability of a molecule. It helps to explain the formation of chemical bonds and the properties of different molecules.

## How does the overlap integral change with different molecular geometries?

The value of the overlap integral can vary depending on the orientation and distance between the two atoms in the molecule. As the molecular geometry changes, the overlap integral may increase or decrease, affecting the strength of the bond and the overall stability of the molecule.

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