Overlap integral

[Prometheus]

I have just come across the overlap rule used to calculate the probability of finding any particular energy eigenvalue, but little about it was explained. Is an overlap integral the same as a convolution of two functions?

[MigL]

I thought it was an indication/metric of orbital overlap, which is itself, an explanation for some of the 'non-standard' molecular angles.

That being my understanding, but I'm no chemist.

[studiot]

Look up the Linear Combination of Atomic Orbitals (LCAO method) It is the simplest calculational one.

 

https://www.google.co.uk/search?hl=en-GB&source=hp&biw=&bih=&q=lcao+method+pdf&gbv=2&oq=LCAO+method&gs_l=heirloom-hp.1.2.0l4j0i22i30l6.2532.8313.0.14110.11.10.0.1.1.0.203.1155.2j6j1.9.0....0...1ac.1.34.heirloom-hp..1.10.1202.iMTmLjEJPCs

[Prometheus]

Thanks, but there was quite a lot to choose from and i was only confusing myself further. Maybe if i give some details someone could help narrow it down.

 

In my text the overlap rule was defined as:

 

[latex]p_i = |\int_{-\infty}^{\infty} \psi_i^*(x) \Psi(x,t) dx|^2[/latex]

 

Where [latex] \psi_i^*(x) [/latex] is the conjugate of the energy eigenfunction, [latex] \Psi(x,t)[/latex] is the wave function of the system and [latex] p_i [/latex] the probability of obtaining the ith energy eigenvalue. No explanation as to how this was derived is given but the text does describe it as being the overlap between the energy eigenfunction and the wave function.

 

This got me wondering whether this has any relation then to the general form of a convolution:

 

[latex]f*g(t)= \int_{-\infty}^{\infty} f(\tau)g(t-\tau) d\tau[/latex]

 

which i understand has a similar interpretation as a sum of the degree of overlap between the function f and g?

 

[studiot]

Err what overlaps then?

 

I think this is the one that intests you

 

From Atkins : Molecular Quantum Mechanics

 

Edited November 5, 2016 by studiot

[Mordred]

Its a slightly different form in the first equation. However you will probably find this link helpful.

 

http://web.mit.edu/~emin/www.old/writings/quantum/quantum.html

 

If I'm not mistaken this relates to the first equation. I believe the first equation relates to the Heisenburg uncertanty. However my higher QM is rusty lol

Edited November 5, 2016 by Mordred

[Prometheus]

Cheers, this is the sort of stuff i was looking for. I'll pick through the details later. I also vaguely remember that convolutions involving Gaussian integrals had some special properties that may be pertinent - i'll have to dig up some old notes too and let you guys know.

[studiot]

You don't need to get so complicated however if you want a physical interpretation rather than a description of how to do the maths.

 

I asked what overlaps?

 

This is because the integral arises when atoms combine to create a molecule by way of overlap of their atomic orbitals.

 

The simplest situation is the two protons forming the hydrogen molecule.

 

The lowest energy wave function in a hydrogen atom correspond to the 1s orbital and have the Schrodinger equation has a solution

 

[math]{\Psi _{1s}} = \frac{1}{{\sqrt \pi }}{\left( {\frac{1}{{{a_0}}}} \right)^{\frac{3}{2}}}\exp \left( {\frac{{ - r}}{{{a_0}}}} \right)[/math]

If we now consider two hydrogen nuclei, A and B, influencing one electron as in the diagram

The distances of the electron from the respective nuclei are r_a and r_b respectively.

So the atomic wave functions are

[math]{\Psi _{1sa}} = \frac{1}{{\sqrt \pi }}{\left( {\frac{1}{{{a_0}}}} \right)^{\frac{3}{2}}}\exp \left( {\frac{{ - {r_a}}}{{{a_0}}}} \right)[/math]

and

[math]{\Psi _{1sb}} = \frac{1}{{\sqrt \pi }}{\left( {\frac{1}{{{a_0}}}} \right)^{\frac{3}{2}}}\exp \left( {\frac{{ - {r_b}}}{{{a_0}}}} \right)[/math]

Now consider a trial molecular wave function

[math]{\Psi _{molecule}} = \frac{1}{{\sqrt 2 }}\left( {{\Psi _{1sa}} + {\Psi _{1sb}}} \right)[/math]

Integrate this over all space

[math]\int\limits_{allspace} {{\Psi ^*}\Psi dr} = \frac{1}{2}\int {{{\left( {{\Psi _{1sa}} + {\Psi _{1sb}}} \right)}^2}d\tau } [/math]

[math] = \frac{1}{2}\int {\left[ {{{\left( {{\Psi _{1sa}}} \right)}^2} + 2\left( {{\Psi _{1sa}}{\Psi _{1sb}}} \right) + {{\left( {{\Psi _{1sb}}} \right)}^2}} \right]d\tau } [/math]

which divides into the sum of three integrals.

[math] = \frac{1}{2}\int {{{\left( {{\Psi _{1sa}}} \right)}^2}d\tau + \frac{1}{2}\int {{{\left( {{\Psi _{1sb}}} \right)}^2}d\tau + \int {\left( {{\Psi _{1sa}}{\Psi _{1sb}}} \right)d\tau } } } [/math]

The first two are theoriginal atomic wavefunctions integrated over all space with suitable constants and equal 1

The third is called the overlap integral for obvious reasons.

Edited November 11, 2016 by studiot

[Mordred]

nice post Studiot +1

[Prometheus]

Hmm...

 

This is all hurting my head a little. The text i am following had only been considering single particle systems when it introduced the overlap rule, so i am a little confused by your question as to what overlaps.

 

I think i will put this on the back-burner until the text moves onto systems with more particles - that may clear up some of my confusion, and then i can revisit this thread. The help is appreciated.

[Prometheus]

Still haven't got onto looking at problems with more than one particle but something just struck me about the question of what overlaps.

 

So i've been dealing with single particle problems and the functions in the overlap integral given in my text is the wave function of the particle and one of the energy eigenfunctions of that system. Now, i understand the wave function is itself a linear combination of all possible energy eigenfunctions - so it makes sense that the magnitude to which any one energy eigenfunction and the wave function overlap is proportional to the probability of measuring the system with the corresponding eigenvalue.

 

So there's nothing physically 'overlapping', just the energy eigenfunctions and the wave function. I wonder if my text is using none-standard terminology?

[Prometheus]

The text i'm following, in preparation for using Dirac notation, is now arguing that the overlap integral is analogous to a dot product in some kind of generalised vector space called a function space. Not looked at generalised vector spaces before so this will be interesting. Hopefully it will shed light on the OP.

[Mordred]

It will definetely help. Dirac notation is a handy tool. One example of dot product is the inner dot product of the Minkowskii metric.

 

[latex]U\bullet V=V\bullet U [/latex]

 

Which essentially gives the vector symmetry under change in sign

Edited December 21, 2016 by Mordred