The necessity of time-dependence in making approximations of the H2+ molecule

[IHateMayonnaise]

I am currently undergoing a project for my undergraduate course in Quantum Physics on electron tunneling in the $$H_2^+$$ ion. Essentially, I am trying to model this situation in Fortran based on the variables that the user supplies (For example, the energy levels of the atoms, the separation distance, etc..). An illustration of my goal is examined at http://www.chem1.com/acad/webtut/bonding/TunnelBond.html" . At first glance it seems this situation boils down to nothing more than a fancy double well potential problem - however upon beginning to draw out the wavefunctions for the various regions involved I find that I may have been mistaken.

For normal potential-well situations, we have nice and neat wavefunctions that take the form (for instance):

$$\Psi_(x)=Ae^{Kx}+B_Ie^{-Kx}$$

where

$$K^2=\frac{2mE}{\hbar^2}$$

HOWEVER, since I am concerning myself with a Hydrogenic molecule - the wave function takes the form:

$$\Psi_{nlm}(r,\theta,\phi)=\left(\sqrt{\left(\frac{2}{na_o}\right)^3\frac{(n-l-1)!}{2n[(n+l)!]^3}}\right)e^{\frac{-r}{na_0}}\left(\frac{2r}{na_0}\right)^lL_{n-l-1}^{2l+1}\left(\frac{2r}{na_0}\right)Y_l^{m_l}(\theta,\phi)$$

So, before I can get anywhere I need to modify this wavefunction to properly describe my system. But, as you will notice (as I surly did), that the hydrogen wavefunction does not explicitly include a value for the energy, as the previous wavefunction does. This leads me to believe that I may have to delve into various time-dependent perturbation methods of the Schrodinger equation - which my introductory coursebook offers little insight into. Since this seems like it would be a fairly common computational endevour, and the community here are extremely experienced, I was wondering if anybody could offer some insight as to how to get me started on this project (literature, specific topics of relevance, etc..). Any help would be greatly appreciated. Thanks all

IHateMayonnaise

 

[genneth]

What do you get if you act the Hamiltonian on that wavefunction? Is it an eigenvector? What's the eigenvalue?

 

[JustinLevy]

At first glance it seems this situation boils down to nothing more than a fancy double well potential problem - however upon beginning to draw out the wavefunctions for the various regions involved I find that I may have been mistaken.

If your program expects a fixed distance between the protons as an input, then yes, this basically is just a double well problem. More over, the site you linked to, and the phrasing used there (tunnelling between the wells) would suggest you can best make this comparision by viewing the problem as a double well.

For normal potential-well situations, we have nice and neat wavefunctions that take the form (for instance):

$$\Psi_(x)=Ae^{Kx}+B_Ie^{-Kx}$$

where

$$K^2=\frac{2mE}{\hbar^2}$$

That wave function describes a free particle. So you are either misunderstanding something here, or you need to explain it a bit more for us to understand what you mean here.

HOWEVER, since I am concerning myself with a Hydrogenic molecule - the wave function takes the form:

$$\Psi_{nlm}(r,\theta,\phi)=\left(\sqrt{\left(\frac{2}{na_o}\right)^3\frac{(n-l-1)!}{2n[(n+l)!]^3}}\right)e^{\frac{-r}{na_0}}\left(\frac{2r}{na_0}\right)^lL_{n-l-1}^{2l+1}\left(\frac{2r}{na_0}\right)Y_l^{m_l}(\theta,\phi)$$

That's the wavefunction for energy levels of a hydrogen atom. Not the molecule.

So, before I can get anywhere I need to modify this wavefunction to properly describe my system. But, as you will notice (as I surly did), that the hydrogen wavefunction does not explicitly include a value for the energy, as the previous wavefunction does.

The energy is related to the quantum number n you used in that equation. Your book should derive this relation when it derived the wavefunction above (or at least presents these relations somewhere).

This leads me to believe that I may have to delve into various time-dependent perturbation methods of the Schrodinger equation - which my introductory coursebook offers little insight into.

If the external potential is constant (your program takes as a parameter a constant distance between the protons), then no time-dependant analysis is necessary to get the wave function of the various energy levels.

Since this seems like it would be a fairly common computational endevour, and the community here are extremely experienced, I was wondering if anybody could offer some insight as to how to get me started on this project (literature, specific topics of relevance, etc..). Any help would be greatly appreciated. Thanks all

There are several ways to do this.

If the protons are far appart so that the electron has to actually tunnel to get across the "barrier" between the protons, then as a first approximation, let's look at the tunelling as a "perturbation".

I will use as a (truncated) basis, the ground state wavefunction of the electron if proton 1 was the only proton (so the wavefunction is centered on proton 1) and the ground state wavefunction of the electron if only proton 2 existed (so the wavefunction is centered on proton 2).

In the limit the protons are really far appart, the energy should be the same as if it was just a hydrogen atom. So the Hamiltonian in this basis would be:
$$H = \left( \begin{array}{cc} E_0 & 0 \\ 0 & E_0 \end{array} \right)$$
Where E_0 is the ground state energy of the hydrogen.

Now bring them closer, and we'll start to have a tunnelling term:
$$H = \left( \begin{array}{cc} E_0 & t \\ t & E_0 \end{array} \right)$$

You can diagonalize this to get the new energy states, and eigen vectors.
The matrix element t of course can be calculated by hand by doing the integral $$t = \langle \Psi_1 | H | \Psi_2 \rangle = \int \Psi_1^*(\vec{r}) \hat{H} \Psi_2(\vec{r}) dx \ dy \ dz$$ Where $\Psi_1(\vec{r})$ is the first basis function (the hydrogen ground state centered on proton one), and similarly for the second basis function.

As the protons get closer, this gross first approximation will get worse. Another method uses the fact that no wavefunction can have a lower energy expectation value than the ground state. So we can consider a wavefunction built of many linearly independent pieces, and vary the coefficients in front of each piece in order to minimize the energy. This is called the variational method.

In the limit that we consider a complete basis set of linearly independent pieces, this method is exact. In practice we usually make some good guesses and use a finite basis set. For example you could use the first couple of hydrogen wavefunctions centered on each proton as a basis set. In fact, people have already wrote programs that will do this calculation for you, and there are nice programs that can display the results as graphs of the electron density or other plots of information regarding the wavefunction.

If you want to go that way, the program I recommend since it is free (you need to click some forms and email someone though), is PC GAMESS.
http://classic.chem.msu.su/gran/gamess/index.html

The input files for that look much more complicated that you really need to know, so if you decide to go that route, let me know and I'll make a simple input file for you, and explain what it means, for you to get started.