Entanglement density functional theory

It is worth noting that although we obtained the total energy (4.123) in the Bohmian mechanics context, it showcases a clear electronic density dependency, not under a density functional (as DFT would require) but merely as a spatial function, which is a direct reflection of the entanglement behavior of Bohmian theory through the involvement of a quantum potential. However, in most cases, and especially for atomic systems, Eq. (4.123) will yield numerical values under custom density function realizations. 

The statistical theory is remarkable in that it enables the macroscopic deformation behaviour of an elastomer to be predicted from considerations of how the molecular structure responds to an applied strain. However, it is important to realize that it is only an approximation to the actual behaviour and has significant limitations. Perhaps the most obvious problem is with the assumption that end-to-end distances of the chains can be described by the Gaussian distribution. This problem has been highlighted earlier in connection with solution properties (Section 3.3) where it was shown that the distribution cannot be applied when the chains become extended. It can be overcome to a certain extent with the use of more sophisticated distribution functions, but the use of such functions is beyond the scope of this present discussion. Another problem concerns the value of N. This will be governed by the number of junction points in the polymer network which can be either chemical (crosslinks) or physical (entanglements) in nature. The structure of the chain network in an elastomer has been discussed earlier (Section 4.5). There will be chain ends and loops which do not contribute to the strength of the network, but if their presence is ignored it follows that if all network chains are anchored at two crosslinks then the density, p, of the polymer can be expressed as... 

The sedimentation coefficient s(c) is related to D by equation (96) and can thus also be written as a universal function of c/c. The gel shear modulus is related to the density of entanglements or to the number of two-body contacts in the solution ( c ). The theory of rubber elasticity predicts. 

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