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I am trying to understand the so-called Taylor expansion (or series expansion) of potential energy of a system of $N$-particles. This expanded form is stated without derivation in some molecular dynamics (MD) texts that I've come across. (e.g., this).

This is based on classical mechanics only, as far as I can tell (the source of these potential energies can be quantum mechanical in nature, but the potential energy function itself is used in a classical sense in MD simulations).

In another text, I came across the starting point and the most general form of potential energy of $N$-particles:

$$U = E(\mathrm{all\ atoms}) - \sum_{i=1}^N E_i$$

where $E$(all atoms) is the total energy of the system, and $E_i$ is the total energy of i'th atom when isolated.

After series expanding this, the result is stated as:

$$U = U(\mathbf{r}_1,\mathbf{r}_2,...,\mathbf{r}_N) = \sum_{i} U_1(\mathbf{r}_i) + \sum_{i<j} U_2(\mathbf{r}_i,\mathbf{r}_j) + \sum_{i<j<k} U_3(\mathbf{r}_i,\mathbf{r}_j,\mathbf{r}_k) +\ ...$$

where $\mathbf{r}_i$ is the coordinate of i'th atom, $U_1$ is single potential on an atom (e.g., one resulting from an external force), $U_2$ is pair potential, $U_3$ is triple potential, and so on (it keeps going up to a single $N$-wise potential).

Questions:

  • First equation: isn't potential energy of an isolated atom zero (or undefined)? therefore summing up undefined energies $E_i$ is wrong or misleading? (or in the best case, summing up zero energies will only give zero as a total)
  • How can you go from first equation to the second? Why is this a Taylor expansion? If I recall correctly, Taylor expansion is the expansion of a function around a given value of independent axes. But this expansion appears to be in number of coordinate variables.
  • I understand pair potentials well ($U_2$). But I can't imagine the form or nature of a triple potential ($U_3$), much less quadruple potential and onwards. Could you give me a good textbook example of a triple potential?
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For your first question, $E_i$ would include the electronic ground state energy of the isolated atom. The idea is for $U$ to consist of the potential energy of interaction between all the atoms, so it is a formal way of expressing it as the total energy of the system, minus the energy of the separated atoms, without making any assumptions about what happens when you bring the atoms together. For a metal, for instance, there would be considerable electron rearrangement. $E_i$ might also contain the kinetic energy of the isolated atom, but exactly the same terms would be included in $E$, so they just cancel.

For your second question, I agree with you that it is unusual to refer to this as a Taylor expansion, for exactly the reason you gave. An example of the Taylor expansion can be seen here. Superficially it looks similar but really it is different: it is an expansion in small displacements away from the equilibrium positions. The link you gave is the only place where I've seen the many-body expansion referred to as a Taylor expansion. If you have any more references I should be interested to know.

Your second equation follows from the same picture as the first one. Imagine assembling the system one atom at a time. For simplicity I'll assume $U_1=0$. When you bring the second atom up to the first, $U(\mathbf{r}_1,\mathbf{r}_2)$ may be expressed as a function of the separation between the two atoms. Call this the pair potential. Now bring up the third atom. It makes sense to write $U(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3)$ as a sum of three pair potentials depending on the respective pair separation vectors, but in general there will be a residual amount which cannot be written this way. Call this the three-body term: $$ U_3(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3) = U(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3) - U_2(\mathbf{r}_1,\mathbf{r}_2) - U_2(\mathbf{r}_2,\mathbf{r}_3)- U_2(\mathbf{r}_1,\mathbf{r}_3) . $$ And so the argument may be continued systematically to give your second equation. At each stage, by considering an $n$-atom system, we may define $U_n$ as that part of $U$ which remains after summing up all the previously-defined $2$-body, $3$-body, $\ldots (n-1)$-body functions.

The three-body term arises because the interaction between two atoms is perturbed if we bring a third atom up close to them. A simple example is the Axilrod-Teller potential, due to correlated dipole fluctuations. In the area of intramolecular potentials, bond angle bending is a three-body term, while torsional potentials typically involve four successive atoms in a chain. The Stillinger-Weber potential is an example of a three-body potential designed to model the angle dependence of bonds in materials such as silicon.

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In general, the atomic potential energy as sum of n-body contributions does not come from any Taylor expansion, but, in the case of empirical model interactions it is just a convenient way to add correction terms to a dominant effective two-body interaction. So, this series is quite an empirical approach without any systematic expansion behind.

There is only one relatively well known case where it is possible to obtain in a systematic and well controlled way that sum. It is the case of simple metals, where, if one tries to write the effect of the electrons on the ion-ion interaction as a perturbative series starting from the free electron gas and using the strength of the ion-electron pseudopotential as expansion parameter, one gets, at the lowest order, a pure pair interaction and a position independent contribution, while at the next order a three-body and a four-body interaction appears, plus a term which modifies the two-body term. Since the explicit calculation of such terms would involve the knowledge of electronic non-linear susceptibilities, usually people stopped there, in the past. Nowadays, non-perturbative calculations where electronic degrees of freedom are taken into account explicitly are preferred.

Something which is not always stressed enough, is that many-body potentials are the rule , in condensed matter theory, more that exceptions. The reason is due to the fact that in many cases some kind of adiabatic approximation is justified, thus allowing to eliminate some degrees of freedom. Usually, such procedure results in a many body interaction which is not reducible in a unique way to a sum of 2,3 ... body terms. A simple prototype of a realistic many-body interaction is the case of a system of polarizable ions where ionic dipoles are self-consistently determined by minimizing the energy. The set of values of the induced dipoles depends on all the ionic positions and even if the interactions between two polarizable ions can be easily obtained, there is no simple way to re-use such information to deal with the case of three ions with a unique decomposition.

A final important comment is that in some cases it is even possible to have a direct experimental test about the presence and the quantitative importance of many-body terms. The simplest case being the additional relation between elastic constants of a solid due to the presence of a purely two-body interaction (Cauchy relations).

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