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 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 N. This limits the concept of wavefunctions ψ based on solutions of the Schrödinger equation to N≤N0 with N0≃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 R rather than on the solutions ψ of the Schödinger equation. Both are related through a logarithm, i.e., R=−iℏlnψ. 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.
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.
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.
Electronic structure calculations for solids based on many-electron wavefunctions have been hampered by the argument that for large electron numbers wavefunctions are not a legitimate scientific concept, because they face an exponential wall problem. We show that this problem can be avoided by using cumulant techniques in formulating wavefunctions. Therefore calculations for solids based on many-electron functions are possible and useful. This includes also systems with strongly correlated electrons.
The calculation of a wavefunction took about two afternoons, and five wavefunctions were calculated in the whole …. Wigner and Seitz, 1933 Summary In order to explain many important properties of materials and phenomena, it is necessary to go beyond independent-particle approximations and directly account for many-body effects that result from electronic interaction. The many-body problem is a major scientific challenge, but there has been great progress resulting from theoretical developments and advances in computation. This chapter is a short introduction to the interacting-electron problem, with some of the history that has led up to the concepts and methods described in this book. The many-body interacting-electron problem ranks among the most fascinating and fruitful areas of research in physics and chemistry. It has a rich history, starting from the early days of quantum mechanics and continuing with new intellectual challenges and opportunities. The vitality of electronic structure theory arises in large part from the close connection with experiment and applications. It is spurred on by new discoveries and advances in techniques that probe the behavior of electrons in depth. In turn, theoretical concepts and calculations can now make predictions that suggest new experiments, as well as provide quantitative information that is difficult or not yet possible to measure experimentally. This book is concerned with the effects of interactions between electrons beyond independent-particle approximations. Some phenomena cannot be explained by any independent-electron method, such as broadening and lifetime of excited states and two-particle bound states (excitons) that are crucial for optical properties of materials. There are many other examples, such as the van derWaals interaction between neutral molecules that arises from the dipole–induced dipole interaction. This force, which is entirely due to correlation between electrons, is an essential mechanism determining the functions of biological systems. Other properties, such as thermodynamically stable magnetic phases, would not exist if there were no interactions between electrons; even though mean-field approximations can describe average effects, they do not account for fluctuations around the average. Ground-state properties, such as the equilibrium structures of molecules and solids, can be described by density functional theory (DFT) and the Kohn–Sham independent-particle equations. However, present approximations are often not sufficient, and for many properties the equations, when used in a straightforward way, do not give a proper description, even in principle.
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.