Exact solutions to the quantum many-body problem using the geminal density matrix

• URL: http://arxiv.org/abs/2112.11400v3
• Date: Wed, 20 Apr 2022 20:56:27 GMT
• Title: Exact solutions to the quantum many-body problem using the geminal density matrix
• Authors: Nicholas Cox
• Abstract summary: Two-body reduced density matrix (2-RDM) formalism reduces coordinate dependence to that of four particles. Errors arise in this approach because the 2-RDM cannot practically be constrained to guarantee that it corresponds to a valid wave function. We show how this technique is used to diagonalize atomic Hamiltonians, finding that the problem reduces to the solution of $sim N(N-1)/2$ two-electron eigenstates of the Helium atom.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: It is virtually impossible to directly solve the Schr\"odinger equation for a many-electron wave function due to the exponential growth in degrees of freedom with increasing particle number. The two-body reduced density matrix (2-RDM) formalism reduces this coordinate dependence to that of four particles irrespective of the wave function's dimensionality, providing a promising path to solve the many-body problem. Unfortunately, errors arise in this approach because the 2-RDM cannot practically be constrained to guarantee that it corresponds to a valid wave function. Here we approach this so-called $N$-representability problem by expanding the 2-RDM in a complete basis of two-electron wave functions and studying the matrix formed by the expansion coefficients. This quantity, which we call the geminal density matrix (GDM), is found to evolve in time by a unitary transformation that preserves $N$-representability. This evolution law enables us to calculate eigenstates of strongly correlated systems by a fictitious adiabatic evolution in which the electron-electron interaction is slowly switched on. We show how this technique is used to diagonalize atomic Hamiltonians, finding that the problem reduces to the solution of $\sim N(N-1)/2$ two-electron eigenstates of the Helium atom on a grid of electron-electron interaction scaling factors.

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