What is the many-electron wavefunction ?

[erst]

What is the "many-electron wavefunction"?

In the introductory picture, the wavefunction represents the particle's probability amplitude and its modulus squared is probability density. It integrates to 1, representing the fact that, with certainty, there's an electron somewhere.

What, then, is the interpretation of a many-electron wavefunction? What is it a solution of? If it's a five electron wavefunction, does the modulus squared integrate to 5? What does this wavefunction look like? What's the connection to atomic/molecular orbitals as they're commonly visualized? And to band structures?

 

[Simon Bridge]

http://en.wikipedia.org/wiki/Wave_function#Many_spin-0_particles_in_one_spatial_dimension

 

[cgk]

What, then, is the interpretation of a many-electron wavefunction? What is it a solution of?

It's the most complete description of a pure quantum state we have (i.e., it describes the state of the physical system). It's a solution of the Schroedinger equation.

What does this wavefunction look like? What's the connection to atomic/molecular orbitals as they're commonly visualized? And to band structures?

Many-electron wave functions cannot really be visualized in a simple way. There is, however, a clear connection to molecular orbitals and band structure:

1. Both molecular orbitals and band structure are objects which represent a Hartree-Fock (or a similar mean-field) description of the quantum system. That is, they represent the real interacting many-body system in terms of an effective one-particle mean-field system. That representation is *an approximation*, and it is not always applicable.

2. All many-electron eigenstates of such a mean-field description of the quantum system can be written as a Slater determinant. That is, the many electron wave function has the form ψ(x1,x2,x3...) = det[phi_i(x_j)], with x1...xN the combined space-spin coordinates x_i=(r_i,σ_i) of the electrons, and phi_i one-particle wave functions.

3. The phi_i of the last point are called /molecular orbitals/ or /bands/. Only the ones which are occupied in the state are "real". Molecular orbtials and bands are equivalent objects.

4. The phi_i of the last section normally come out delocalized if they are calculated by diagonalizing the mean-field one-particle operator. But this is not essential, as arbitrary non-singular linear combinations of them still lead to the same determinant and consequently the same mean-field state. They can thus normally also be brought into a local form (e.g., Wannier orbitals, localized molecular orbitals), which is equivalent.

5. The molecular orbitals themselves are normally approximated as a linear combination of some basis functions. In the MO case, a few of those basis functions are normally chosen to be atomic orbtials or split atomic orbtials. AOs are just the mean-field orbitals (as described above) when only a many-electron atom is treated as the "full" system.

 

[erst]

Thanks, that clears things up a lot. When does the mean-field approximation break down? Is the many-electron wavefunction useful for real-world calculations?

In applications, I've never encountered anything but single particle descriptions (hence my ignorance).

 

[daveyrocket]

There are many types of mean-field approximations for different types of systems, so they may break down in different ways.

In some cases a mean-field approximation will break down because when one constructs the mean field, it includes the particle that is interacting with it, so this introduces an artificial self-interaction problem. This shows up in DFT calculations when examining systems with tightly bound d and f electrons (so-called strongly correlated systems). Although, this is not necessarily a mean-field problem, because there are various corrections which are still mean-field which attempt to resolve this, such as dynamical mean-field theory.

Other cases might be systems for which there is short-range order but no long-range order, or complicated short-range interactions, or frustration.

 

[cgk]

Thanks, that clears things up a lot. When does the mean-field approximation break down? Is the many-electron wavefunction useful for real-world calculations?

The mean-field approximation is usually fine. For example, in molecules, Hartree-Fock normally provides for >99% of the total energy and gets most properties (e.g., the electron density) more or less right. However, especially in chemistry the accuracy demands are often so high that Hartree-Fock on its own is rather useless. The problem here is often not that the approximation breaks down, but only that it is not accurate to the desired 0.001%. For this reason, actual (i.e., correlated) many-body wave functions are often calculated (e.g., in the coupled cluster method), or methods which employ some ``average'' description of correlation are used (e.g., in Kohn-Sham DFT). In the solid state world, in this case sometimes Green's function perturbation theory based methods are employed (e.g. GW-based methods for band gaps or Bethe-Salpeter-Equation based methods for excitions). These are all based on the notion that the one-particle picture is still valid, just not accurate enough.

But there are also important cases where the mean-field description breaks down. This normally happens when orbitals with some covalent bonding component have very small overlap with the orbitals they are bonding to (e.g., 3d metals or during bond forming/dissociation processes). In this case there will be a large number of quite different competing states with similar energy, and mean-field methods like HF or Kohn-Sham cannot handle them properly. Then different methods which employ actualy many-body wave functions are used (e.g., multi-configuration self-consistent field (MCSCF) or multi-reference configuration interaction (MRCI)).
In the solid state world, this also applies to various ``strange'' phenomena, like heavy fermions, correlated metals, super-conductivity etc. The mentioned DMFT also falls in this application region (although it is really not a mean-field method in the usual sense of the word).

In applications, I've never encountered anything but single particle descriptions (hence my ignorance).

That's okay. The single-particle picture is normally applicable. There are just cases where it is not.

 

[DrDu]

or methods which employ some ``average'' description of correlation are used (e.g., in Kohn-Sham DFT).

I think Kohn-Sham DFT can be proven to be basically exact, independently of correlation being high or low (the corresponding theorems refer to "N-representability" or "V-representability"). The point is that DFT, does not consider the many body wavefunction at all but only tries to represent the density or spin density in terms of one-particle orbitals. However the potentials to describe these one particle orbitals are unknown and approximate schemes, like LDA, break down when the underlying system is strongly correlated.

 

[DrDu]

Thanks, that clears things up a lot. When does the mean-field approximation break down? Is the many-electron wavefunction useful for real-world calculations?

In applications, I've never encountered anything but single particle descriptions (hence my ignorance).

Explicitly correlated functions have been used in the high precision calculations of the ground state of H_2 by James and Coolidge or of Helium by Hylleraas.
see e.g.
http://jcp.aip.org/resource/1/jcpsa6/v1/i12/p825_s1
Also in valence bond theory you explicitly construct a many particle wavefunction.
The resonating valence bond approach is investigated as a possibly good approximation for the ground state of high temperature superconductors.

There are quite a lot of methods to do calculations with explicitly correlated wavefunctions today which don't start from an effective one-particle picture. I remember e.g. the expansion into hyperspherical harmonics.

 

[Einstein Mcfly]

How close to more advanced methods of QC get you to the exact many-electron wavefunction? In cases where you go beyond single determinants and use post HF methods (CASSCF, MCSCF, CI, MP2, CCSD(T) etc), the single particle “orbitals” that you get out of HF and DFT no longer have clear meanings. What is it that you have at the end of these calculations that you compare to experiment? For example, if you’re going to do a multi-reference calculation to try to model the excited state spectra of a particular system, in the end you must have some notion of a “ground state” and some “excited states” that give you the proposed energy difference corresponding to a photon that would excite the system from the ground to excited states. So what is it that you have? Also, do these mathematical objects better preserve the rules of QM (unlike the usual orbitals with defined n,l,m and s for each electron)?

Thanks for your help. My experience is in DFT calcs, not post-HF stuff.

 

[DrDu]

In cases where you go beyond single determinants and use post HF methods (CASSCF, MCSCF, CI, MP2, CCSD(T) etc), the single particle “orbitals” that you get out of HF and DFT no longer have clear meanings.

You still can define so called "natural orbitals", i.e. those orbitals which diagonalize the 1-density matrix. However, these orbitals do not characterize completely the wavefunction (in contrast to HF).

 

[Einstein Mcfly]

So what do they mean? Are they the usual "solutions to the SE that, when taken as a product with all of the other occupied solutions give the total wavefunction"? If so, what do they tell you? Are they analogous to HF orbitals but include some correlation?

Also, if the solution with post-HF methods aren't written as products of one-particle functions, what are they comprised of? Is there any smaller unit of interpretation in these methods that allows one to make more sense of the results?

Thanks for your previous response.

 

[DrDu]

The natural orbitals and their occupation numbers allow you to calculate all expectation values of one particle operators. To describe two particle operators (namely energy), you need geminals (eigenstates of the two particle density matrix). Only for HF, geminals can be expressed simply in terms of one particle eigenstates.
E.g. the onset of superconductivity, in which electronic correlation is essential, can be ascribed to one eigenvalue of the two-particle density matrix becoming large.

You can also introduce some orbitals related to the poles of the 1-Greensfunction, which allow you to calculate the intensities of electronic transitions.
The interpretation of many-particle wavefunctions is a huge topic and lot has been written about it.