Abstract
Due to advances in computer hardware and new algorithms, it is now possible to perform highly accurate many-body simulations of realistic materials with all their intrinsic complications. The success of these simulations leaves us with a conundrum: how do we extract useful physical models and insight from these simulations? In this article, we present a formal theory of downfolding--extracting an effective Hamiltonian from first-principles calculations. The theory maps the downfolding problem into fitting information derived from wave functions sampled from a low-energy subspace of the full Hilbert space. Since this fitting process most commonly uses reduced density matrices, we term it density matrix downfolding (DMD).
Highlights
TO DOWNFOLDING THE MANY ELECTRON PROBLEMIn multiscale modeling of many-particle systems, the effective Hamiltonian is one of the most core concepts
Due to advances in computer hardware and new algorithms, it is possible to perform highly accurate many-body simulations of realistic materials with all their intrinsic complications. The success of these simulations leaves us with a conundrum: how do we extract useful physical models and insight from these simulations? In this article, we present a formal theory of downfolding–extracting an effective Hamiltonian from first-principles calculations
We develop a general theory for generating effective quantum models that is exact when the wave functions are sampled from the manifold of low-energy states
Summary
In multiscale modeling of many-particle systems, the effective Hamiltonian (or Lagrangian) is one of the most core concepts. One might think to reconcile the fitting approach used in classical force fields with quantum models by matching eigenstates between a quantum model and ab initio systems, varying the model parameters until the eigenstates match [14] This strategy does not work well in practice because it is often not possible to obtain exact eigenstates for either the model or the ab initio system. We develop a general theory for generating effective quantum models that is exact when the wave functions are sampled from the manifold of low-energy states Because this method is based on fitting the energy functional, we will show the practical application of this theory using both exact solutions and ab initio quantum Monte Carlo (QMC) to derive several different quantum models. £ In section 4, we discuss future prospects of applications of the DMD method, ongoing challenges and clear avenues for methodological improvements
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