Hartree-Fock correction to e-e interaction

[CAF123]

The corrections to the energy per electron in a jellium model (uniform distribution of positive ion charge approximation to the regulated long range order ionic array) is given by (in units of Ry) $$E/N = \frac{2.21}{(r_s/a_o)^2} - \frac{0.916}{(r_s/a_o)} + 0.0622 \ln (r_s/a_o) - 0.096 + O(r_s/a_o) $$ (##r_s## is the radius of a sphere that holds the same volume as one conduction electron and ##a_o## is the bohr radius, but I don't think is so important to my questions). See e.g the derivation and this same result in Ashcroft and Mermin P.336

My question is why does the last three terms contain contributions to the kinetic and potential energy? It seems to me that it should only be potential energy since the source of the extra terms comes from the perturbation which is precisely due to the e-e interactions, but I have been told otherwise.

My other question is this whole procedure was implemented to better account for the interaction between the electrons in a metal. This expansion above is only valid for ##r_s/a_o \ll 1 ## and for most metals this value is from 2 to 6. So what is the usefulness of this result then if it does not apply to the very thing it sought out to describe?

Thanks for any replies.

 

[Astronuc]

The last (right) three terms are the energy correlation (or correlation energy, or electron correlation energy) which describes the electron correlation.

See Equations 13, 14 and 15 herein:
http://www.phys.ufl.edu/~pjh/teaching/phz7427/7427notes/ch2.pdf

Other notes:
http://en.wikipedia.org/wiki/Electronic_correlation
http://vergil.chemistry.gatech.edu/notes/ci/node1.html
http://vergil.chemistry.gatech.edu/courses/chem6485/pdf/intro-e-correlation.pdf
http://dasher.wustl.edu/chem478/lectures/lecture-04.pdf

http://www.tcm.phy.cam.ac.uk/~mdt26/tmo/scm_talk.html

 

[CAF123]

Hi Astronuc,

So the last three terms are corrections to both the result obtained via first order perturbation theory (the second term in the expansion) and corrections to the free electron energy. I think that is right. I am not really sure if there is a way to argue this is the case or should I just accept that this is true? I mean Hartree-Fock imposes anti symmetry of the wave function and the PEP in place means that electrons with same spins can't get close to each other thereby altering the free electron kinetic energy.

As for the expansion, I still do not really understand why such an expansion is useful it is only valid for ##r_s/a_o \ll 1## and for most metals this value is about 2-6. Could you explain this part? I also see that the first two terms in the expansion graphically look like a typical Leonard jones potential with a repulsive and attractive component, with minimum ##r_s/a_o \approx 4.8## in the range of the typical values of this quantity for metals. What does this imply?

Thanks.

 

[Astronuc]

I'm not an expert in this area, but it is an area of interest. I'm also trying to learn the material.

Here is what seems a reasonable comment on HF - http://en.wikipedia.org/wiki/Hartree–Fock_method#ApproximationsI'll try to find the answers to the other questions.See exercise 18 in this - http://www.uio.no/studier/emner/mat...ningsmateriale/Weekly exercises/exercises.pdf

 

[DrDu]

I think you found already an answer to the first part of your question. The electron-electron interaction is a perturbation to the whole hamiltonian and introduces e.g. the correlation hole, i.e. a zero of the wavefunction where two electron positions coincide. This evidently has an impact on the kinetic energy, too.
You may also argue that a system hold together by Coulombic forces has to obey the virial theorem, i.e. ##\langle T\rangle=-1/2 \langle V \rangle##, so that altering the strength of the Coulomb interaction will also alter the mean kinetic energy.

The second question about the validity of the high density expansion of the energy for real metals is justified.
The problem is that for metallic densities, there is no parameter which is small so that a perturbational analysis can be based on it. On the other hand, there are also low density expansions and you can hope that the behaviour for intermediate densities somehow interpolates between the two limits.
This has first been attempted by Wigner.
Finally, there are corrections which take into account some effects which are thought to be important at realistic densities.
The book by Gerald D. Mahan, "Many-Particle Physics" discusses this more in depth.

But what most severely restricts the analysis of the uniform electron gas in terms of density is the non-uniformity of the background which is formed by the electric point charges of the nuclei and not by a homogeneous positive background in real metals.
In density functional theory this is taken into account using e.g. gradient expansions and corrections obtained from isolated atoms.

 

[CAF123]

But what most severely restricts the analysis of the uniform electron gas in terms of density is the non-uniformity of the background which is formed by the electric point charges of the nuclei and not by a homogeneous positive background in real metals.
In density functional theory this is taken into account using e.g. gradient expansions and corrections obtained from isolated atoms.

I have read somewhere online that ##r_s## is the typical distance between two electrons (although I am not sure why this is the case). So given that the minimum of the graph I mentioned before happens at ##r_s/a_o \approx 4.8##, then if ##r_s## is such that ##r_s/a_o < 4.8## the configuration is unstable and there is vast repulsion. I think it is right to say that for metals such an expansion can get more accurate by including more terms but I am still trying to understand what this graph (in particular the min at ##r_s/a_o \approx 4.8##) means with regard to the fact that the expansion only holds for ##r_s/a_o \ll 1##. I mean take a metal such that ##r_s/a_o \approx 4## is the expansion valid? The fact this number is above 1 and that it lies in he repulsive component of the graph seems to suggest no, but I am not sure how to interpolate the results when r_s/a_o comes closer to the min and therefore closer to the attractive component. Thanks!

 

[cgk]

CAF123, the validity of the expansion is indeed a serious problem, and in practice it is not used for small densities. For small densities numerically calculated correlation energies (typically with some form of QMC/DMC) are employed in order to obtain reasonable fits. The high-density expansion (=weak correlation expansion) and the low density expansion (=strong correlation expansion) are then combined to form a curve which spans the entire range of [itex]r_s[/itex]. If you want to see how this is done, I recommend reading one of the original papers which does this: http://dx.doi.org/10.1103/PhysRevB.45.13244 . It is quite readable. It is this functionally fitted combined curve which is then used in practice, normally not the perturbative ones (and certainly not the HF ones, which are only responsible for the leading Dirac-exchange term---there are no correlation contributions in HF and it is only valid in the high density limit). Further corrections are then included by using the gradient expansions DrDu mentioned (e.g.: http://dx.doi.org/10.1103/PhysRevLett.77.3865 , also quite readable and useful to know from a general physics standpoint... it forms the practical backbone of much of contemporary condensed matter and molecular physics).

Now, that being said, it must be stated that treating the uniform electron gas at low densities is still a serious problem to this day. It is a *very* difficult strong correlation problem. For example, serious defects are known in Ceperly's QMC numerical data which was used in the PW92 paper above, and around [itex]r_s\approx 1[/itex], which is arguably the single most important region, both the QMC (low density) and the perturbative (high density) expansions are quite bad. We will likely see some better parameterizations in the near future.

 

[DrDu]

After cgk's comment who knows the actual scientific frontier much better than I, I just wanted to make two comments on terminology:
The first two terms give the Hartree Fock energy for all densities by definition. All the following terms make up the correlation energy, also by definition.

Edit: Thinking about it, I don't see why the minimum in the HF energy should lead to an instability. Formally, r_s is a fixed parameter and the system would only become instable (phase separation) for a maximum at some given r_s.
A more physical question is probably to ask for the cohesive energy of a metal which is related to the density dependence of of the energy. This is also discussed in Mahan.

 

[cgk]

DrDu, I am also not sure about this. I think there could be some instability of the HF solution with regard to symmetry breaking of either spin or space symmetries (similarly to the Coulson-Fisher-Point in H2, where a broken-symmetry UHF solution appears. There the H2-distance is also only a fixed parameter). But I would also expect a restricted, spin-adapted HF solution to exist over the entire range of r_s. For the RPA or other perturbation theory on top of it this might not necessarily hold---I guess this could explode at any point. However, I am not a electron gas person and I am not that deep into this sort of thing. I am interested in this solely from a general interest and application point of view, and sometimes speak to people who are more into this.

@CAF: Just a notice: r_s is not exactly the average distance of the electrons, it is the radius of a sphere whose volume is equal to the average volume/electron: http://en.wikipedia.org/wiki/Wigner–Seitz_radius . It is mainly used because it arises as a natural quantity in equations.