2 volume derivative in representability problems are widespread in coarse- grained descriptions of soft-matter systems[9]. They may also be important important in other coarse- grained simulations. For example, we recently used the structural route to derive radially symmetric potentials for water from a more sophisticated underlying model [7]. We explicitly constructed v$_{g}$ ( r ) and v$_{U}$ ( r ) and they look very different, as anticipated by Eq. 5. At some of the state-points we studied, using v$_{g}$ ( r ) to calculate the virial pressure resulted in a dimensionless compressibility fac- tor Z = βP/ρ that was almost two orders of magnitude larger than that of the original multi-site water model used to parameterise v$_{g}$ ( r ). Similar deviations from the underlying water model were also seen in ref. [10]. Admittedly, it may not be surprising that an isotropic potential should perform so poorly when the underly- ing fluid has complex orientational correlations. In fact treating water as a pair potential is probably not such a good idea One way of improving on the thermodynamic perfor- mance of the structurally derived potential v$_{g}$ ( r ) is to use constraints to simultaneously fit to properties like the virial pressure or the internal energy. Due to the uniqueness of v$_{g}$ ( r ), this potential would no longer cor- rectly reproduce the pair correlations, but since the ther- modynamic properties are scalar quantities, it might not come at too large a cost for the structure. A nice exam- ple of applying this to water can be found in this paper by the Mainz group [10] who showed that the structural properties were affected (and by extension the isothermal compressibility which can be written as an integral over g ( r )), but not too badly. something first pointed out in 1969 by Ascarelli and Harrison[11] in the context of density dependent pair po- tentials used for modelling liquid metals. C. more general problems with effective potentials should also act on v ( r ; ρ ), resulting in a virial equation with an extra ∂v ( r ; ρ ) /∂ρ term: βP = ( ∂ log Q ( N, V, T ) ∂V )$_{N,T}$ (7) Z ρ vir = βP ρ (8) = 1 − 2 3 βπρ ∫ ∞ 0 r 2 { r ∂v ( r ; ρ ) ∂r − 3 ρ ∂v ( r ; ρ ) ∂ρ } g ( r ) dr, Now on the surface this all looks very kosher – I took my potential, plugged it into my partition function, turned the statistical mechanical crank, and out popped Eq. 8 with a correction to the pressure due to the state- dependence of the effective potential. However, when we tried this with the v$_{g}$ ( r ) derived for the water model [7], the correction actually made the agreement far worse. In ref. [9] I give other examples were this correction has the wrong sign (and magnitude). Although the analysis leading to Eq. 8 has the veneer of statistical mechanical respectability, it is in fact a disreputable result. Taking a step back, it is not at all surprising that this derivation cannot be right [9]. The correction term in Eq. 8 only takes into account a local dependence on den- sity. If you took the same potential v ( r ; ρ ), and used it in a grand-canonical ensemble, then you would need infor- mation from all densities ρ , not just the local derivative. So the two ensembles would not be equivalent, a good sign that there are more problems afoot. Warnings about such difficulties abound, for example, about 40 years ago Barker et al. [12] wrote: We record our opinion that the use of density-dependent effective pair potentials can be misleading unless it is recognised that these are mathematical constructs to be used in specified equations rather than physical quan- tities and similar reservations were sounded by other distin- guished investigators [8, 13, 14]. The point of this exercise is not to say that effective po- tentials are useless. I’ve happily used them myself [15]. Sometimes a fine solution to a dubious Hamiltonian is better than a dubious solution to a very fine Hamilto- nian. It is just that one must be careful when using coarse-graining procedures not to automatically treat the coarse-grained potentials as if they are real bona-fide po- tentials of the type one uses in a Hamiltonian. Also, in practice it is also good to remember that these potentials are always compromises and that fitting too strongly to one quantity may generate larger errors in other quanti- ties that one also wants to measure [7, 9]. where Λ is the usual thermal de Broglie wavelength. The Q ( N, V, T ) = Λ − 3 N N ! ∫ d r N exp   $^{−}$$^{β}$ ∑ i<j v ( r$_{ij}$ ; ρ )   $^{.}$ (6) Here is an example of how things can go horribly wrong. Consider a homogeneous fluid in a volume V , with N particles interacting with a spherically symmet- ric pair potential v ( r ; ρ ), that depends on the state point (that could be density, temperature, etc...). For simplic- ity, here we consider only the density ρ = N/V . The standard way to derive the virial equation is directly through the canonical partition function In ref. [9] some more general issues with effective po- tentials are reviewed. For example, most of the time our effective simplified potentials are really state depen- dent, that is if you were to derive them at different state points they would be different. I argue that you shouldn’t naively treat them as if they are real two-body potentials of a Hamiltonian system.