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How would one calculate the dynamic time evolution of the shape of an electron wave function in a chemical reaction?

  • I'll first start with my understanding of the theory and so if I'm wrong you can correct me. Take a simplistic model where you have an hydrogen atom. Then you apply an electric field nearby strong enough to ionize the hydrogen atom. I would think that when there is no external electric field, the electron wave function is the spherical 1s orbital. Then when the electric field is turned on, the wave-function would distort itself somewhat like a pear and at the time where the atom is being ionized *something* would happen (collapse of the wave function? bubbling out? function is undefined?). How would one calculate this pear modification of the electron wave-function. How would it work with more complex systems? Are there already software that exist to visualize this? I've read Elements of Quantum Mechanics but I am still working to fully understand it. This is what I've got so far. Looking for pointers, ideas and explanations. Thanks

  • Answer:

    There are two separate questions that you are asking here, I think. So let me look at the first one, first. I promise this will not be a long or a complete answer. To find out how the electron wave functions change during the process of forming a bond between two atoms, you need to solve the time-dependent Schrödinger equation for a many body system involving, in the simplest case, four bodies. The simplest case is two hydrogen atoms bonding to form H2. (That's not quite true, you can also think about the case of H2+, the ionized state of molecular hydrogen … but that is still a three body problem.) The problem is already far too hard to be solvable in closed form: but it can be treated using numerical methods. These methods are highly developed these days thanks to the increase in computing power that has happened over time, but the answers are complicated, and perhaps, also, not particularly illuminating. You might well be able to learn most of what is important about the binding of the H2 molecule, for most purposes, from the study of the static equations in various approximations … it's clear that the molecule forms in some time-dependent way, but the question is, what does more detailed knowledge of the wave function during the time when that is happening give you? Here's a paper that describes the result of some simulations of the solution for the way the electron orbitals change while a chemical bond is being formed. http://www.mpi-halle.mpg.de/mpi/publi/br/gross_persoenlich/CBMG07.pdf Your other question relates to the Stark effect at strong electric fields. I want to think a little before answering that one.

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You calculate using the adiabatic approximation, you can assume the electrons stay in the ground state, and the nuclei are classical. Then you allow the nucleus to move around, and the wavefunction to change to the instantaneous ground state of the potential given the position of the nuclei. This is the Born Oppenheimer approximation, and it's exact for normal temperature systems, basically excluding liquid helium and nothing else. The result is easy for Hydrogen with one electron, but you can't do this type of simulation in real time with more than 2 or 3 electrons, because the wavefunction is high-dimensional. In this case, you want to change to density functional theory. The existing codes for this will give you the ground state property of complex molecules, using a core potential and valence wavefunctions which are approximately correct in shape, for any deformation of the nuclei. The best method is the Car-Parrinello method, which integrates the electronic structure in real time.

Ron Maimon

I can't add much to David's answer. Basically, when there is no "external field," as you say, you solve the time-dependent Schrödinger equation for a single electron. It is not necessarily true that it will be in the 1s state. The probability is highest for it to be in this state, but that's no guarantee. At any rate, when you're talking about bonds you're now really talking about molecular orbitals. There's a nice discussion of the hydrogen molecular ion in Slater's book Quantum Theory of Matter (which might be out of print?). Molecular orbitals can generally be calculated for molecules with few electrons using the Hartree-Fock method. But it gets messy quickly as David pointed out.

Ian T. Durham

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