Polymer density functional approach to efficient evaluation of path integrals

A polymer density functional theory (P-DFT) has been extended to the case of quantum statistics within the framework of Feynman path integrals. We start with the exact P-DFT formalism for an ideal open chain and adapt its efficient numerical solution to the case of a ring. We show that, similarly, the path integral problem can, in principle, be solved exactly by making use of the two-particle pair correlation function (2p-PCF) for the ends of an open polymer, half of the original. This way the exact data for one-dimensional quantum harmonic oscillator are reproduced in a wide range of temperatures. The exact solution is not, though, reachable in three dimensions (3D) because of a vast amount of storage required for 2p-PCF. In order to treat closed paths in 3D, we introduce a so-called “open ring” approximation which proves to be rather accurate in the limit of long chains. We also employ a simple self-consistent iteration so as to correctly account for the interparticle interactions. The algorithm is speeded up by taking convolutions with the aid of fast Fourier transforms. We apply this approximate path integral DFT (PI-DFT) method to systems within spherical symmetry: 3D harmonic oscillator, atoms of hydrogen and helium, and ions of He and Li. Our results compare rather well to the known data, while the computational effort (some seconds or minutes) is about 100 times less than that with Monte Carlo simulations. Moreover, the well-known “sign problem” is expected to be considerably reduced within the reported PI-DFT, since it allows for a direct estimate of the corresponding partition functions.

Figure 1

The normalized harmonic spring distributions, $g_1^{\left(0\right)}\left(\Delta r\right)\sim \exp [-{\beta }_n{K}_f{\left(\Delta r\right)}^2]$ (the same as $h\left(\Delta r\right)$ except for normalization), at ${\beta }^{*}=1$ are presented both in 1D (top) and 3D (bottom). The effect of increasing the number of monomers/vertices, $n$, is illustrated.

Figure 2

The electron distributions in the hydrogen atom and in the artificial ${\mathrm{H}}^{-}$ system (proton and two electrons). In both cases $n=2048$ and $T=10000\mathrm{K}(R_{\text{cut}}=20.48\mathrm{\AA },N_{\text{grid}}=1024)$. The known exact solution of the Schrödinger equation is also shown for comparison, and the first Bohr radius, $R_B=0.529\mathrm{\AA }$, is marked by the arrow.