We demonstrate the accuracy of the hypernetted chain closure and of themean-field approximation for the calculation of the fluid-state properties ofsystems interacting by means of bounded and positive-definite pair potentialswith oscillating Fourier transforms. Subsequently, we prove the validity of abilinear, random-phase density functional for arbitrary inhomogeneous phases ofthe same systems. On the basis of this functional, we calculate analyticallythe freezing parameters of the latter. We demonstrate explicitly that thestable crystals feature a lattice constant that is independent of density andwhose value is dictated by the position of the negative minimum of the Fouriertransform of the pair potential. This property is equivalent with the existenceof clusters, whose population scales proportionally to the density. Weestablish that regardless of the form of the interaction potential and of thelocation on the freezing line, all cluster crystals have a universal Lindemannratio L = 0.189 at freezing. We further make an explicit link between theaforementioned density functional and the harmonic theory of crystals. Thisallows us to establish an equivalence between the emergence of clusters and theexistence of negative Fourier components of the interaction potential. Finally,we make a connection between the class of models at hand and the system ofinfinite-dimensional hard spheres, when the limits of interaction steepness andspace dimension are both taken to infinity in a particularly described fashion.
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机译：我们证明了通过有界和正定对势与振荡傅立叶变换进行相互作用的系统的流体状态计算的超网状链闭合和主题场近似的准确性。随后，我们证明了双线性，随机相密度函数对于同一系统的任意不均匀相的有效性。在此功能的基础上，我们分析地计算了后者的冻结参数。我们清楚地证明，稳定的晶体具有不依赖于密度的晶格常数，其值由对电势的傅立叶变换的负最小值的位置决定。该属性与簇的存在等效，簇的种群与密度成比例地缩放。我们确定，不管相互作用势的形式和在冻结线上的位置如何，所有簇晶体在冻结时的通用Lindemannratio L = 0.189。我们进一步在上述密度泛函和晶体谐波理论之间建立了明确的联系。这使我们能够在簇的出现与相互作用势的负傅立叶分量的存在之间建立等价关系。最后，当相互作用陡度和空间维数的极限都以一种特别描述的方式被推到无穷大时，我们将现有的模型类别与无限维硬球系统联系起来。
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