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If I want to do a very accurate simulation of a molecular system (e.g. 2 hydrogen atoms), then I'll want to use something like diffusion Monte Carlo to determine the energies of these atoms in different configurations.

For example, if I want to create a bonding potential like the harmonic

$$E = K(r - r_0)^2$$

or Morse potential

$$E = D[1-e^{-\alpha(r-r_0)}]^2$$

then I assume what I would do is perform DMC to find the energies for various separation distances, and then perform a regression to fit one of those equations.

However, I was recently thinking perhaps this isn't as accurate as it could be, because you're using quantum mechanics to find energies, and then using classical physics (Newton's laws of motion) to evolve the system.

A more accurate alternative, I assume, would be to evolve a system of two hydrogen atoms using the time-dependent Schrödinger equation and then somehow fit this to some easily computable model.

What I'm curious though is how the resulting simulations would differ? I don't know enough about QM to speculate on what kind of effects the classical system evolution would be missing, or exactly how much (numerically speaking), the two approaches would differ. Does anyone have any idea?

I suppose I could always try both methods and see, but that's going to take quite a bit of work...

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I'm not an expert on the topic, but I know that Jim Mitroy's group at CDU (click through to their publications if interested) are doing a lot of high accuracy few-body (up to about 7 particles) QM calculations. I hear they set up a simulation then let it run on the supercomputer for a month before checkin on the answer. :) But that's to get a ridiculous number of sig. figs. You may be able to get away with much less. –  Michael Brown Feb 15 '13 at 5:18
    
Should mention that the computational complexity goes up exponentially in the number of particles, so if you just have two electrons + two nuclei then there may be cheaper options for you. –  Michael Brown Feb 15 '13 at 5:20
    
Not necessarily. What you are doing first is solving the electronic Schrödinger equation parametrically for each internuclear position. Once you have the potential you can use classical mechanics or quantum mechanics to treat the nuclear (molecular) Hamiltonian dynamics. If you solve this at the quantum level the approximation is in the treatment of the coupling of the electronic and nuclear degrees of freedom. But you can also include these couplings in your study. –  perplexity Feb 15 '13 at 17:09
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2 Answers

It sounds like you are interested in Born-Oppenheimer molecular dynamics, i.e. using classical equations of motion for the nuclei while using quantum mechanics for the electrons. This is a fairly common method implemented in a range of quantum chemistry packages.

A related approximate method is Car-Parinello molecular dynamics, which may be of use to you. This may be your best bet if your system of study is large.

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(This is more like a comment but I don't have enough reputation to comment and is kind of long; however this may be useful for you if I am understanding your question).

If you are only interested in calculating a bonding like potential for a two-electron system such as two hydrogen atoms, that can be done exactly (within the Born-Oppenheimer approx. and for a given basis set, $i.e.$ set of orbitals) using standard quantum chemistry methods. The methods of full configuration interaction (http://en.wikipedia.org/wiki/Full_configuration_interaction), CISD (http://www.gaussian.com/g_tech/g_ur/k_cid.htm) or Coupled Cluster CCSD (http://www.gaussian.com/g_tech/g_ur/k_ccd.htm) would give all exact answers for two electron systems. You can calculate the energy at different bond lengths with these methods using standard quantum chemistry programs. You can then use a program such a gnuplot to fit the obtained point to your harmonic or Morse potential equation.

The methods that I am mentioning above are computationally very expensive, but for your small two electron system it is not a problem to do many calculations at different bond lengths with any modern computer.

Hope this helps, cause I am not sure if I understood perfectly your question. Your question talked about mixing QM with classical methods, but I am not mentioning about anything about the latter because you can get the "exact" quantum answer for your small system (and you said you wanted to do a very accurate calculation).

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Well, ultimately I would like to accurately describe C-C, C-Li, and Li-Li interactions but I think there's too many particles for a fully accurate simulation. –  Nick Feb 15 '13 at 5:36
    
Yes, you cannot do exact calculations for these larger systems (except perhaps Li-Li). I cannot help you much more because I am not very familiar with mixed QM methods. But I can warn you that it will be very hard to get very accurate results for C-C. Even using purely quantum mechanics, many sophisticated methods fail for this system so be very careful on what you do for C-C as you might even get unphysical answers. –  Goku Feb 15 '13 at 5:41
    
@Nick If you have access, you can look at this article jcp.aip.org/resource/1/jcpsa6/v118/i4/p1610_s1?isAuthorized=no. The authors compare dissociation curves of several quantum methods with the "exact" (FCI) solution. The make calculations for potential energy curves of molecules of about the size you are interested in (they show HF, BH and methane). –  Goku Feb 15 '13 at 5:46
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