[Basko]

Does anyone know the energy of a σ1s^2 molecular orbital? I would like to know it in the case of a hydrogen molecule.
The answer I search should be calculated by using LCAO and the variational principle. I really tried to myself but the integral is just too complicated and I didn't even know how to show that we have two separate hydrogen atom in the wavefunctions at the beginning. I believe the energy equals: (α+β) and it is explained here  http://www.chem.qmul.ac.uk/software/download/mo/4.pdf
(I believe the energy ought to be lower than -13,6eV)
Thanks so much!!

[Borek]

Isn't it what Gaussian is for?

[Irlanur]

First of all, let's clarify the myth that He₂ is not a stable molecule. it is: http://aip.scitation.org/doi/10.1063/1.4864002 (0.00112(2) cm^-1)

I never calculated the overlap for H₂ myself, but we could try it together if you want. I am not sure if it's feasible analytically but we could always make it numerically.

How did you start?

Isn't it what Gaussian is for?

That's true, but in the case of H₂ I would suspect (read: I don't know it) the result to be quite different from the "actual" overlap. The  Gaussian basis set expansion is pretty bad for regions close to the nucleus and far away. Also in the case of 2 nuclei it is "easily" possible to calculate the integrals with the actual Slater-Type orbitals.

[Irlanur]

Thinking about it a bit more: The Overlap Integral is not much of a Problem, the Coulomb and Exchange Integrals are quite a big one. I don't think that there is an easy solution to actually solving it.

You can get the Energy of H₂ by looking up the IOnization energy of the Hydrogen Atom and the dissociation energy of H₂.

[Corribus]

I was going to say, photoionization energies are widely available, from which you should be able to easily determine the stabilization energy of the bound state.

[Basko]

Indeed the overlap integral isn't really the problem here since in this approximation we're allowed to neglect it which simplifies the anwswer to (α + β). Alfa is ∫φ1Hφ1 dτ , the Coulomb integral and beta equals ∫φ1Hφ2 dτ , this is the resonance integral. H equals the one electron hamiltonian operator: [tex] \frac {-e^2} {4\pi\epsilon_0r}  + \frac {-\hbar^2} {2m}\nabla^2[/tex] m is the mass of one electron, with the laplacian[tex] \nabla^2=\frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2 \frac{\partial}{\partial r}\right)+\frac{1}{r^2 sin\theta}\frac{\partial}{\partial \theta}\left(sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{r^2 sin^2 \theta}\frac{\partial^2}{\partial \phi^2}\qquad [/tex] and φ equals  [tex] \frac {-1} {\sqrt {\pi a^3}} e^\frac {-r} {a} [/tex] with a the bohr radius and to make sure we combine two separate hydrogen atoms in space I thought of using [tex] \phi_1=\frac {-1} {\sqrt {\pi a^3}} e^\frac {-r} {a}, \phi_2= \frac {-1} {\sqrt {\pi a^3}} e^\frac {-(r-10a)} {a} [/tex]

this is what I make of it

[Corribus]

What makes you think you can neglect the overlap integral? This is a common approximation in the simple Huckel model, but in that case the adjacent (p) orbitals are directed parallel to each other. Here they are face-on, and neglecting the overlap here would in my view lead to an unrealistic approximation.

The coulomb and exchange integrals are typically not analytically solvable for a two electron system and must be approached via approximation methods.

[Basko]

I agree I would be better not to neglect the overlap integral, but for the project I'm doing it's best not to involve the overlap integral . I'm trying to understand and follow what's been said here: http://www.chem.qmul.ac.uk/software/download/mo/4.pdf
In which they do neglect the overlap integral to get to (α+β) which is what I'm looking for I believe. And this is an approximation, right? Using LCAO isn't the exact method of obtaining the wave function I believe.

[Corribus]

I mean, you can neglect anything you want, obviously. The trick is to know what is reasonable to neglect in any given situation, a determination which needs to be contextualized also by what is a reasonable degree of accuracy for your application. And in any case, as Irlanur has already pointed out, the overlap integral is the least of your problems. Certainly the exchange and Coulomb integrals cannot be summarily reduced to nothing, and there's no simple means of calculating them by means of algebraic manipulations. Even in the Huckel treatment, which is probably one of the most complex theoretical treatments of complex molecules that can done by hand, despite its many shortcuts, an experimental value of the beta integral is still used to estimate energy levels. More sophisticated methods necessarily used a wide variety of other approximations and almost always computers - calculations that can still closely approach experimental values for small molecules like H2.

But... the more I go on here it seems like you just want the energy expressed in terms of (unsolved) alpha and beta integrals. In that case, what you've reported here is fine I think, as long as you realize the effect of the approximations you make along the way.

[Basko]

I'll definitely keep in mind that I've made some approximations that will affect my answers. You see I've already done enough math for my project and my teacher doesn't mind if I end with discussing some values I found in literature, so do you know the values of the exchange and coulomb integral in eV for the hydrogen molecule? Whether they're experimental values or nummerically calculated values, (having both values would be best of course) and could you please tell me your source? Where can you find these values (in eV or Joule)?