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arXiv:1703.02572 · The Journal of Chemical Physics

Dealing with the exponential wall in electronic structure calculations

Peter Fulde, Hermann Stoll

Published May 18, 2017• 20 citations
arXivHTMLPDF

Abstract

An alternative to the density functional theory is the use of wavefunction based electronic structure calculations for solids. In order to perform them, the Exponential Wall (EW) problem has to be resolved. It is caused by an exponential increase of the number of configurations with increasing electron number N. There are different routes one may follow. One is to characterize a many-electron wavefunction by a vector in Liouville space with a cumulant metric rather than in Hilbert space. This removes the EW problem. Another is to model the solid by an impurity or fragment embedded in a bath which is treated at a much lower level than the former. This is the case in the Density Matrix Embedding Theory (DMET) or the Density Embedding Theory (DET). The latter two are closely related to a Schmidt decomposition of a system and to the determination of the associated entanglement. We show here the connection between the two approaches. It turns out that the DMET (or DET) has an identical active space as a previously used Local Ansatz, based on a projection and partitioning approach. Yet, the EW problem is resolved differently in the two cases. By studying a H10 ring, these differences are analyzed with the help of the method of increments.

DOI
10.1063/1.4983207

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An alternative to the density functional theory is the use of wavefunction based electronic structure calculations for solids. In order to perform them, the Exponential Wall (EW) problem has to be resolved. It is caused by an exponential increase of the number of configurations with increasing electron number N. There are different routes one may follow. One is to characterize a many-electron wavefunction by a vector in Liouville space with a cumulant metric rather than in Hilbert space. This removes the EW problem. Another is to model the solid by an impurity or fragment embedded in a bath which is treated at a much lower level than the former. This is the case in the Density Matrix Embedding Theory (DMET) or the Density Embedding Theory (DET). The latter two are closely related to a Schmidt decomposition of a system and to the determination of the associated entanglement. We show here the connection between the two approaches. It turns out that the DMET (or DET) has an identical active space as a previously used Local Ansatz, based on a projection and partitioning approach. Yet, the EW problem is resolved differently in the two cases. By studying a H10 ring, these differences are analyzed with the help of the method of increments.
Wavefunctions for large interacting electron systems lose their meaning due to an exponential growth of the dimensions of Hilbert space with increasing electron number. In order to base electronic-structure calculations for solids on wavefunctions instead on density functional theory one has to resolve this exponential wall problem (EWP). It is shown that the origin of it is the multiplicative character of a wavefunction with respect to independent systems A and B, i.e., ψA/B = ψA ⨂ ψB. The EWP is avoided, if we characterize the system by an additive quantity like an action instead. This can be done by describing the system through the fluctuations of a mean-field state imposed by the interactions. The operators defining these fluctuations span an operator- or Liouville space. In order to obtain additive quantities the metric in Liouville space must be a special one, i.e., based on cumulants. The situation resembles the one of a classical gas, where the interaction contributions to the energy are included by a linked cluster expansion. In this way the EWP is avoided and one obtains a solid basis for wavefunction-based electronic-structure calculations of large systems. Examples are given for the application of the theory.
Wavefunctions for large electron numbers N are plagued by the Exponential Wall Problem (EWP), i.e., an exponential increase in the dimensions of Hilbert space with N. Therefore, they lose their meaning for macroscopic systems, a point stressed, in particular, by Kohn. The EWP has to be resolved in order to provide a solid basis for wavefunction based electronic structure calculations of macroscopic systems, e.g., solids. The origin of the EWP is the multiplicative property of wavefunctions when independent subsystems are considered. Therefore, it can only be avoided when wavefunctions are formulated so that they are additive instead, in particular, when matrix elements involving them are calculated. We describe how this is done for the ground state of a macroscopic electron system. Going over from a multiplicative to an additive quantity requires taking a logarithm. Here it implies going over from Hilbert space to the operator- or Liouville space with a metric based on cumulants. The operators which define the ground-state wavefunction generate fluctuations from a mean-field state. The latter does not suffer from an EWP and therefore may serve as a vacuum state. The fluctuations have to be connected like the ones caused by pair interactions in a classical gas when the free energy is calculated (Meyer's cluster expansion). This fixes the metric in Liouville space. The scheme presented here provides a solid basis for electronic structure calculations for the ground state of solids. In fact, its applicability has already been proven. We discuss also matrix product states, which have been applied to one-dimensional systems with results of high precision. Although these states are formulated in Hilbert space, they are processed by using operators in Liouville space. We show that they fit into the general formalism described above.
The dimension of the Hilbert space needed for the description of an interacting electron system increases exponentially with electron number N. As pointed out by Kohn this exponential wall problem (EWP) limits the concept of many-electron wavefunctions based on solutions of Schrödinger’s equation to [Formula: see text], where [Formula: see text] [Rev. Mod. Phys. 71, 1253 (1999)]. This limitation does not hold when the electronic interactions are neglected or treated in a mean-field approximation. The EWP has directed electronic structure calculations for solids to methods like density-functional theory which avoid dealing with many-electron wavefunctions. We show that the highly unsatisfactory limitation of many-electron wavefunctions to [Formula: see text] can be overcome by generalizing their definition. We show for the ground state of a large system that it is the logarithm of the solution of Schrödinger’s equation which should be used to characterize this system. In this case the wavefunctions of independent subsystems add up rather than multiply. This feature is realized by the action function. It provides a simple physical picture for the resolution of the EWP.
The Hilbert space for an interacting electron system increases exponentially with electron number NNN. This limits the concept of wavefunctions ψψψ based on solutions of the Schrödinger equation to N≤N0N \leq N_0N≤N0​ with N0≃103N_0 \simeq 10^3N0​≃103 \cite{Kohn1999}. It is argued that this exponential wall problem (EWP) is connected with an increasing redundance of information contained, e.g., in the ground-state of the system and it's wavefunction. The EWP as well as redundance of information are avoided when the characterization of the ground state is based on the action function RRR rather than on the solutions ψψψ of the Schödinger equation. Both are related through a logarithm, i.e., R=−iℏ lnψR = -i \hbar \ ln ψR=−iℏ lnψ. Working with the logarithm is made possible by the use of cumulants. It is pointed out the way electronic structure calculations for periodic solids may use this concept.
We developed a periodic version of density matrix embedding theory, DMET, with which it is possible to perform electronic structure calculations on periodic systems and compute the band structure of solid-state materials. Electron correlation can be captured by means of a local impurity model using various wave function methods, such as full configuration interaction, coupled cluster, and multiconfigurational methods. The method is able to describe not only the ground-state energy but also the quasiparticle band picture via the real space-momentum space implementation. We investigate the performance of periodic DMET in describing the ground-state energy as well as the electronic band structure for one-dimensional solids. Our results show that DMET is in good agreement with other many-body techniques at a cheaper computational cost. We anticipate that periodic DMET can be a promising first principle method for strongly correlated materials.
Quantum embedding based on the (one-electron-reduced) density matrix is revisited by means of the unitary Householder transformation. While being exact and equivalent to (but formally simpler than) density matrix embedding theory (DMET) in the noninteracting case, the resulting Householder-transformed density matrix functional embedding theory (Ht-DMFET) preserves, by construction, the single-particle character of the bath when electron correlation is introduced. In Ht-DMFET, the projected ``impurity+bath\mathrm{impurity}+\mathrm{bath}impurity+bath'' cluster's Hamiltonian (from which approximate local properties of the interacting lattice can be extracted) becomes an explicit functional of the density matrix. In the spirit of single-impurity DMET, we consider in this work a closed (two-electron) cluster constructed from the full-size noninteracting density matrix. When the (Householder-transformed) interaction on the bath site is taken into account, per-site energies obtained for the half-filled one-dimensional Hubbard lattice match almost perfectly the exact Bethe ansatz results in all correlation regimes. In the strongly correlated regime, the results deteriorate away from half-filling. This can be related to the electron number fluctuations in the (two-site) cluster which are neither described in Ht-DMFET nor in regular DMET. As expected, the per-site energies dramatically improve when increasing the number of embedded impurities. Formal connections with density/density matrix functional theories have been briefly discussed and should be explored further. Work is currently in progress in this direction.