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{{ :public:projects:mol_0000.gif?nolink |Quantum-nuclei-simulation isomorphic to P coupled classical systems}}

The path integral formalism represents an isomorphism between a quantum system and an equivalent classical model system. In the latter, each original quantum particle is represented as an ensemble of <latex>P \in \mathbb{N}</latex> classical particles (beads), which are connected among themselves with springs in a ring topology, and interact with their corresponding beads of the other (“quantum”) particles via their physical interaction potential. Due to this analogy, a particle in the isomorphic classical system is often referred to as a “ring polymer” or “ring necklace”, and the system of corresponding beads is called replica.

The conventional path integral formulation is based on the real space representation of the hightemperature density matrix. In neutron scattering experiments, however, the momentum space density <latex>$n(\mathbf{k}) = |\Psi(\mathbf{k})|^2$</latex> is probed, which cannot be obtained directly from the regular density. Instead, the standard scheme has to be modified in such a way that the polymer is opened.

Path integrals were made popular by R.Feynman, implemented (in combination with classical potentials) and applied to superfluid helium by D.Ceperley and more recently used to investigate the quantum nature of protons in complex systems (in particular liquid water) within a density functional theory description by D.Marx.

With the help of the Trotter expansion, which corresponds to transform the density matrix to a higher temperature, it is possible to write the partition function as

<latex> $\begin{align*}
Z = \text{Tr} \left[ \left(e^{-\frac{\beta}{P}\hat H}\right)^P\right] = \int d^{3N}R \ \ \langle\mathbf{R}|e^{-\frac{\beta}{P} \hat H} \dots e^{-\frac{\beta}{P} \hat H} |\mathbf{R} \rangle \ \ \text{with}\  P \in \mathbb{N} .
\end{align*} $ </latex>

The new aspect is that with a modification of the conventional path integral scheme, it is possible to express not only quantities in real space (**R**-space), but also momentum densities. A complete derivation of this modified path integral formalism would exceed the space available here, but it can be shown that the momentum density of a nucleus can be expressed as: 

<latex> $\begin{align*}
n(\mathbf{k}) = & \int d^3d_2 \dots d^3k_N\ | \Psi(\mathbf{k_1}=\mathbf{k}, \dots \mathbf{k_N})|^2\\
= & \frac{1}{(2\pi)^3} \int d^3 R_1\, d^3R'_1 \ e^{-i\mathbf{k}(\mathbf{R}_1 - \mathbf{R}'_1)/\hbar}\, n(\mathbf{R}_1, \mathbf{R}'_1)
\end{align*}$ </latex>

using the off-diagonal direct-space density matrix:

<latex> $\begin{align*}
n(\mathbf{R}_1, \mathbf{R}'_1) = \int d^3R_2^{(1)}\dots d^3R_N^{(1)}\dots d^{3N}R_N^{(P)} \rho_{1_{R_1}2} \rho_{23}\dots \rho_{N1_{R'_1}}
\end{align*}$ </latex>

While the conventional classical isomorphism corresponds to a ring polymer, this new modification describes a linear polymer, in which one real-space point (**R1**) is duplicated (yielding **R1** and **R´1**), with new harmonic potential between these new coordinates.

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