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Something I learned recently which really helped me in understanding exchange is that the case of exchanging two electrons is really a special case of enforcing symmetry (or anti-symmetry) in which the permutations of labels in cycles out to length $N$ are used. $N$ is the number of particles.

Just to write it out, the two-electron exchange integral is defined as, $$K_{ab}=\int d\textbf{r}_1d\textbf{r}_2\psi^*_a(\textbf{r}_1)\psi_b(\textbf{r}_1)r_{12}^{-1}\psi_b^*(\textbf{r}_2)\psi_a(\textbf{r}_2)$$ where everything means what it normally means and I took this integral and notation straight out of Szabo and Ostlund.

Now, this is clearly only a permutation of two electrons, and introduces the only correlation present in Hartree-Fock by allowing for exchange of spin-paired electrons.

Shifting gears, I've been reading Quantum Mechanics and Path Integrals by Richard Feynman and Albert Hibbs. In the chapter on Statistical Mechanics, they talk about liquid helium, which has a peculiar transition around $2K$ in which the heat capacity begins to increase. Feynman was the first to explain that this is due to the exchange interaction, and at this temperature is when permutations of the atom labels larger than just one (i.e. two particle exchange) become important. The following statements from the book are relevant to electrons (as opposed to helium because $\ce{^4He}$ is a boson!).

In a different context, the calculation of partition functions, the authors state:

If we were dealing with Fermi particles, e.g., the isotope of helium which has three nucleons, we would have to include an extra factor of $\pm1$, positive for even permutations and negative for odd permutations. There would also be some extra features which depend upon the spin of the atom in our result.

And more directly about electrons,

Consider the behavior of electrons in a solid metal. The mass of the electron is so much smaller than that of a molecule that the critical temperature is much higher. At room temperature, electrons in a metal are described accurately only by equations which include the exchange effects of these cyclic permutations. From this point of view, room temperature is very cold for electrons.

So, hopefully we have now reached a place where it is clear why I am a bit confused. Namely, to my knowledge, we only ever consider the exchange of two electrons of the same spin in our electronic structure methods. First, is this actually true? That is, are there methods which implement (in principle) $N$-electron integrals to describe all permutations of the labels?

Second, is the comparison I am making even strictly valid? That is, in the case of a partition function, which is what Feynman is talking about, permutations of even length tend to cause the electrons to stay further away from each other. This is the behavior we are used to with exchange. In contrast, for permutations of odd numbers, the contribution to the partition function picks up a minus sign. So, the total partition function for a system of fermions is an alternating sum of positive and negative terms in the path integral formalism. Can I simply substitute partition function for wavefunction and sum for integral and find what would happen in the electronic structure case? That is, will the 3-electron exchange integral (assuming this is real and I'm not mistaken) be a negative number, so that we can picture the wavefunction relaxing a little bit from the so-called "heaps" which form due to exchange?

Finally, if I'm not just delusional about this whole thing, is there evidence that permutations beyond the two labels are important for describing any known chemical processes?


References:

[1]: Szabo, A., & Ostlund, N. S. (2012). Modern quantum chemistry: introduction to advanced electronic structure theory. Courier Corporation.

[2]: Feynman, R. P., Hibbs, A. R., & Styer, D. F. (2010). Quantum mechanics and path integrals. Courier Corporation.

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  • $\begingroup$ The electron-electron interaction is a 2 particle operator ($\frac{1}{r_{ij}}$), which means only terms with 2 electrons ($i$ and $j$) show up. There us no need to consider 3-electron integrals (or higher). $\endgroup$ – Feodoran Aug 9 '17 at 6:28
  • $\begingroup$ Right. But what I'm saying is in principle there should be a 3-particle exchange and 4-particle exchange, etc. Or at least there is when considering fermionic systems in calculating a partition function. It could be that all higher order permutations are zero but I'd have to be shown why this should be the case. $\endgroup$ – jheindel Aug 9 '17 at 6:54
  • $\begingroup$ You might be confusing partition and wave function. The partition function is about the distribution of states, which in turn are defined by wave functions. Those are two different layers of abstraction. Considering permutations about which particle occupies which state is a different thing than the Exchange Integral in HF. $\endgroup$ – Feodoran Aug 9 '17 at 7:11
  • $\begingroup$ No. I was just drawing an analogy. In looking a bit deeper I think that explicitly correlated (F12) methods actually deal with precisely these integrals. There are"cyclic" operators of the $\frac{1}{r_{12}}\frac{1}{r_{13}}\frac{1}{r_{23}}$. I think this is what I'm asking about but these papers are very heavy on math and very light in physical interpretation. Haha. I'll add some references tomorrow. $\endgroup$ – jheindel Aug 9 '17 at 7:47
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    $\begingroup$ I would be careful with an analogy between permutations of particles and the HF exchange integrals. This exchange integrals is only named like this because in its mathematical equation you exchange two electronic coordinates with respect to the coulomb integral. Since there is no classic analog to this, it is difficult to interpret this physically. So yes, analogies due to the mathematical form of the equations may show up. But with analogies in the physical interpretations I would be careful. $\endgroup$ – Feodoran Aug 9 '17 at 8:16
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The Hartree–Fock wave function in fact has the correct permutational symmetry for any $n$-particle exchange, because the determinant has it. This follows directly from the definition of a determinant as a sum over all permutations of rows (columns). But as @Feodoran says in the comment, the Coulomb potential is two-body, and it's expectation value, $\langle\psi|\hat V|\psi\rangle$, picks only the two-electron exchange.

So, to directly answer your question: First, we actually do consider all $n$-particle exchanges, but they simply do not show up in the relevant integrals. Second, I'm no expert on statistical physics, but I don't see why there should be any strong analogy between the wave function and the partition function in classical statistical physics, which is a thermodynamic quantity and does not consider the microscopic dynamics of a system, that involves correlations and state transitions. (There is also a partition function in QFT, and there perhaps may be some analogies between that and the wave function, but I know even less about QFT.)

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  • $\begingroup$ Maybe an analogy to CI, where you consider all possible permutations of $N$ electrons in $M$ orbitals, will work better here? $\endgroup$ – Feodoran Aug 9 '17 at 8:30
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I am posting this answer not as the definitively correct answer, but as an example of the type of thing I am talking about.

azago's answer is correct in pointing out that the Slater determinant contains within it the correct permutational symmetry of the wavefunction required by electrons. Namely, it is antisymmetric upon interchange of two orbitals. Additionally, it symmetric upon interchange of three (or any odd number of) orbitals. This is the symmetry which must be present in higher cycles of exchanges.

What I have found upon further reading is that I could have been more precise in the wording of my question. Namely, these higher-order exchange interactions are not surprising if I had made it clear that these are necessarily three-center, four-center,... integrals. Then it is more clear that HF, while having the correct symmetry to adhere to Pauli exclusion, does not include these integrals as it only deals with 2-electron integrals at the most.

So, I will discuss three contexts I have found in which the types of integrals I am thinking of show up.

Hamiltonian for a Three-Atom System:

Following along with ref. [1], the Hamiltonian for the three atom system, $\ce{H3}$ is, $$ H=\sum_i\left(-\frac{1}{2}\nabla^2_i-\sum_{\alpha}\frac{1}{r_{\alpha i}}+\sum_{j\gt i}\frac{1}{r_{ij}} \right)+\sum_k\frac{1}{R_k} $$ where $i$ and $j$ are labels for the three electrons, and $\alpha$ is the label for each Hydrogen atom with locations $R_1$, $R_2$, and $R_3$.

It is clear from this Hamiltonian that we have the usual coulomb terms and the usual two-electron integrals. Something weird is going to happen with the exchange terms though. There are actually three possible configurations of the atoms, but the only point I'm making here is that the matrix elements for one of the atomic configurations can be written as, $$ \langle\psi_1|H|\psi_1\rangle=4J+4K_1-2K_2-2K_3-4K_{123} $$ I have deviated from their notation here for consistency with current convention. These integrals are written explicitly as: $$ \begin{align} J&=\langle abc|H|abc\rangle \\ K_1&=\langle abc|H|bac\rangle \\ K_2&=\langle abc|H|acb\rangle \\ K_3&=\langle abc|H|cba\rangle \\ K_{123}&=\langle abc|H|cab\rangle \\ \end{align} $$ where now it is quite clear that $K_{123}$ is a double exchange. That is, two electron labels are changed, and it is clear this is distinct from the other exchange integrals which are the usual exchanges we are used to.

I do not think this has anything to do with the fact there are three atoms present, however. Only that there are three electrons. Thus, I believe that even a complete description of Lithium would include these three-electron exchanges. Thus, the reason we do not see these integrals in HF is that they are left out of the Hamiltonian.

Explicitly-Correlated Methods:

I'll be brief here because explicitly-correlated methods are pretty complicated and I haven't spent enough time studying them to really know what's going on, but one thing which does happen is that you get matrix elements of the following form[2]: $$ Z_{\nu w,p}^{xy,q}=\langle w_{\nu w,p}|(1-\hat{p}_{13})r_{13}^{-1}+(1-\hat{p}_{23})r_{23}^{-1}|w_{xy,q}\rangle $$ The only important in this horrible integral is that the exchange operator $\hat{p}_{ij}$ appears with the labels of three-distinct electrons. Now, I have no idea if it's valid to interpret this as the type of integral I'm talking about, but these kinds of integrals appear all over the place in explicitly correlated methods.

Three-Body Symmetry-Adapted Perturbation Theory

Now, two-body symmetry-adapted perturbation theory (SAPT) is quite a common method, but there is a 3-body version of SAPT which can be quite a useful tool at times.

In Ref. [3], the authors present the relevant equations for 3-body SAPT, but they note that they approximate the exchange term out to the fourth power of orbital overlap. That is, for the exchange operator they do the following: $$ P\approx 1+P_2+P_3+P_4 $$ They describe this as follows:

In this expression, $P_2$ represents the operator which interchanges one electron from $X$ with one electron from $Y$ and therefore produces terms proportional to $S^2$. The operator $P_3$ includes double exchanges between three monomers involving three electrons. This operator, absent for a dimer, generates terms proportional to $S^3$. The operator $P_4$ is a double exchange operator which interchanges four electrons among the monomers and generates terms proportional to $S^4$. The double exchanges produced by $P_4$ are of the $(XY)(XY)$ or $(XY)(XZ)$ types and can be thought of as two exchanges of the type produced by $P_2$ occurring simultaneously. In contrast, double exchanges of the $(XY)(XYZ)$ type are included in $P_5$ and double exchanges of the $(XYZ)(XYZ)$ type are included in $P_6$, leading to terms proportional to $S^5$ and $S^6$, respectively.

The whole purpose of three-body SAPT is to be able to describe the 3-body non-additivities which arise from intramolecular interactions. It has already been pointed out that the electrostatic interaction is additive up to 2-body because Coulomb's law follows the superposition principle. This then makes it clear that these higher-order exchanges involving more than two electrons generate some of the non-additive forms of coulomb correlation.

Finally, the authors of ref. [3] provide the form of these three-electron cycles I was trying to describe which can be written as: $$ P_{111}=\sum{i\in A}\sum{i\in B}\sum{i\in C}[P_{ij}P_{jk}+P_{ik}P_{jk}] $$ which interchanges electrons in a cycle on atoms $A$, $B$, and $C$. This operator generates all possible three-electron exchanges between three atoms. There are no intra-monomer exchanges however.


[1] Porter, R. N., & Karplus, M. (1964). Potential energy surface for H3. The Journal of Chemical Physics, 40(4), 1105-1115.

[2] Hättig, C., Klopper, W., Köhn, A., & Tew, D. P. (2011). Explicitly correlated electrons in molecules. Chemical reviews, 112(1), 4-74.

[3] Lotrich, V. F., & Szalewicz, K. (1997). Symmetry-adapted perturbation theory of three-body nonadditivity of intermolecular interaction energy. The Journal of chemical physics, 106(23), 9668-9687.

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