As is well-known, the so-called averaged energy of two-electrons Coulomb repulsion U has been introduced both in quantum theory of atoms / molecules and of condensed matter, which is typically defined, as is shown in the attached figure (a), - where the integral written there is taken over the whole 6-dimensional configurational space (r1=(x1, y1, z1) and r2 = (x2, y2, z2)).
For instance, the on-site two-electrons repulsion energy U appears in superexchange theory, where the antiferromagnetic contribution to exchange integral is obtained as: Jaf ~ b2/U (where b is a hopping integral), it appears in LDA+U approach intended to reproduce the band structure of strongly correlated crystalline systems correctly, and so on.
As far, as I can judge, the U energy is introduced as was shown (or in equivalent way) in manifold textbooks and papers.
But it is absolutely evident, that the integral defined so diverges, in other words, is equal to infinity, except the trivial and physically insignificant case, if at least one of one-electron orbitals is identically zero at the whole space. Actually, most probably no other physically reasonable form of one-electron orbitals can be proposed to eliminate the singularity in the denominator at ANY point of "line" r1 = r2 in 6-dimensional space (see also attached figure (b), where this point is symbolically depicted for the case of "one-dimensional" electrons).
Note, that the approach like "Let`s deviate from "line" r1 = r2, next, take the integral over the whole space except the deviation vicinity (see also figure (b)), and finally calculate the limit of the result approaching the measure of deviation to zero".. so, something like that evidently is not valid - because it also does not eliminate the singularity, actually, the integral over the whole space except the deviation vicinity might be arbitrarily large (keeping its sign to be invariable), as depends on the deviation measure value.
Sorry for a long text, but it is related to my question directly. On the one hand - I cannot find the logical errors in argumentation given above, as well, as cannot find the explanation in textbooks and publications I have ever seen. On the other hand, manifold sources deal with the definitely FINITE values of U (typically, some eV).
Can someone explain me, how this contradiction could be solved?
I would be VERY grateful.
The integrand is singular as you say. But the integral is finite in 2 or 3 dimensions due to phase space considerations. If you change coordinates to center of mass R=(r1+r2)/2 and difference r=r2-r1, then the integrand has 1/|r| singularity. But the measure of integration gives r^2dr (3D) or rdr (2D) that cancels that singularity. So in 2D or higher the integral is finite and convergent (assuming the orbitals fall off to zero as r1,r2 go to infinity --- but that is a separate problem).
Most models involving two or more electrons in the solid state are
too hard to solve, so there is a simplified model, called the Hubbard
model for this repulsion that contains this U, when two electrons of opposite
spin occupy the same basic site.
If this potential, as you properly say diverges, it means in QM that there is
zero probability density of finding the two particles very near each other.
This justifies the use of this finite U.
Sometimes an additional potential is used for nearest neighbours , and so on.
It is most usefull in the so called tight binding method, basically a discretized
form of Schrodinger in the form of a mesh, but is further backed up by the use
of the so called Wigner functions.
Ithink the key point of your question has been answered: while the argument of the integral diverges, it can still yield a finite value under suitable conditions.
Let me just add that quantum mechanically you have to compute the expectation value of the Coulomb interaction in a 2 (or more, if appropriate) electron state. In the example you gave, the 2-electron-function is chosen to be a product function. A more physical choice for two electrons would be to consider a singlet or a triplet state. The difference of Coulomb interaction between the two cases is also known as the exchange integral. In case of a local interaction, the triplet (spin 1) state has the lower Coulomb energy, since the probability of the two electrons to be found at the same location is identically zero.
In atomic physics books you will find this treated for the He atom, which has two level schemes, one for the singlets, and one for the triplets.
U is the screened potential. So, you should pick RPA or Thomas-fermi to described the dressed quasi-particle and then, the integral will not diverge. There is no sense for the integral if the potential energy does not get screened within a few lattice sites, or is in the case of the Hubbard model, screening within a distance slightly larger than the Bohr radius. Thus, when you see U, understand that it is really Vexternal/dielectric function.
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