DOI: 10.22661/AAPPSBL.2019.29.1.50
Wavefunctions for Extended Electron Systems
PETER FULDE^{*} MAXPLANCKINSTITUT FÜR PHYSIK KOMPLEXER SYSTEME, NÖTHNITZER STRASSE 38, 01187 DRESDEN, GERMANY (DATED: JANUARY 17, 2019)
^{*} Electronic address: fulde@pks.mpg.de
ABSTRACT
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 electronicstructure 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., ψ_{A/B} = ψ_{A} ⨂ ψ_{B}. 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 meanfield 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 wavefunctionbased electronicstructure calculations of large systems. Examples are given for the application of the theory.
INTRODUCTION
Electronic structure calculations for molecules started almost immediately after the rules for dealing with quantum mechanical systems were formulated by Heisenberg [1] and Schrödinger [2]. The initial goal was to learn more about the origin of chemical binding and the first molecule investigated was, of course, the simplest one, i.e., H_{2}. The work of Heitler and London [3], Hund [4], Mulliken [5] and Hartree [6] is representative of many other pioneers of the rapidly developing new field. In parallel with the development of electronic structure calculations for molecules the corresponding work for periodic solids began. For reviews see, e.g., [710]. Here density functional theory (DFT) [11, 12] initiated a revolution in our understandings of different materials and thus in the field of material sciences. The latter becomes constantly more important in modern technology. Despite all these successes it seems desirable to develop in parallel electronic structure calculations based on wavefunctions since they allow carefully controlled approximations. This is in distinction to DFT based calculations where a proper choice of the density functional depends on the material and on experience. However, wavefunctionbased calculations are affected by what W. Kohn has called the Exponential Wall Problem (EWP). The latter results from mutual electronelectron interactions. It leads to an exponential increase of the dimensions in Hilbert space with increasing electron number N. Indeed, for large molecules and in particular for macroscopic electron systems like solids, the EWP is an obstacle that has to be eliminated in order to create a solid platform for wavefunctionbased calculations. The consequences of the EWP have been described by W. Kohn by the following statement: for a system with N > N_{0} electrons a wavefunction ψ(r_{1} σ_{1}, ..., r_{N} σ_{N}) is no longer a legitimate scientific concept [11]. Here r_{i}, σ_{i} are the position and spin of the ith electron. The number N_{0} depends on a typical error ϵ per dimension in Hilbert space which is unavoidable since an exact determination of an N electron wavefunction is impossible. For ϵ ≃ 10^{2} one finds that N_{0} ≃ 10^{3} [11].
For a legitimate scientific concept a wavefunction must fulfill two conditions: it must be possible to approximate it to a reasonable degree of accuracy and one has to be able to document it properly. Both conditions cannot be met when N > N_{0}. In this case the overlap of any approximation ψ_{app}> to the exact groundstate wavefunction ψ_{0}> is zero for all practical purposes. The latter is <ψ_{app}ψ_{0}> ~ (1ϵ)^{N}. A similar argument holds for the documentation of an exponential number of parameters, which is at least of order 2^{N}. We want to point out that the EWP does not exist for noninteracting electrons or for systems with interactions treated in a selfconsistent field approximation. In this case the groundstate wavefunction is simply given by a Slater determinant. Also DFT does not suffer from an EWP. All electronic degrees of freedom are integrated out, except for those describing the electronic density n(r).
In order to resolve the problem of a proper description of the ground state of an extended electron system we neglect all interactions of the system with its surrounding. In this case we can define stationary states in Hilbert space. It is easy to see that the EWP has its origin in the multiplicative properties of wavefunctions. When ψ_{A}> and ψ_{B}> are the wavefunctions of two separate systems A and B, then the wavefunction of the total system is ψ_{A/B}>=ψ_{A}>⊗ψ_{B}>.
Consider a system of N_{A} separated atoms A with a fixed number of electrons each. Assume, that the correlations among the electrons on a given atomic site can be described with sufficient accuracy by a superposition of M electronic configurations. Then the total number of configurations required for the description of, e.g., the ground state of the total system is M^{NA} and, as expected, is exponentially exploding. Yet, the information we obtain from this wavefunction is all contained in the one for a single atom, i.e., by M configurations. Avoiding the EWP therefore requires finding a representation of the wavefunction in which all redundant information is eliminated and does not appear. This implies giving up a multiplicative representation of a wavefunction and looking instead for an additive representation. Note that Schrödinger went from the additive action or Wirkung function W via W=ℏlnψ to the multiplicative function ψ for which he derived his wave equation [2]. Yet, for a macroscopic electron system we have to return to a description like an action. The associated physical picture is very clear:
Assume a periodic solid with lattice sites I and J related by a lattice translation of the unit cell. An electron at site I will be uncorrelated with one at site J, provided the unit cells are sufficiently apart. Thus, their correlation holes will remain unaffected by the shift. Therefore it will suffice to describe properly only one.
The above considerations suggest splitting the electronic Hamiltonian H for a macroscopic system into two parts H = H_{0}+H_{1}. We choose H_{0} so that its ground state Φ_{0}> is known, e.g., by using the selfconsistent field (SCF) part of H, i.e.,
 (1)

We call Φ_{0}> the vacuum state, i.e., Φ_{0}> ↔ vac> and we will use both expressions interchangeable. Then the interactions H_{1} generate vacuum fluctuations. The latter are used to specify the ground state of the system. As an example consider a configurational interaction representation of a wavefunction ψ_{0}>, i.e.,
 (2)

Thus we can specify ψ_{0}> by the sum of products of electronic creation and annihilation operator where Greek indices are used for occupied orbitals in vac> and Latin indices for unoccupied ones with the spin index included. These operator products span an operator or Liouville space. However, when the expansion (2) is terminated, an unavoidable step for a macroscopic electron system, the approximated form of the wavefunction results in an energy which is generally not size extensive. This is changed when we apply a cumulant metric in Liouville space, in analogy to the computation of the energy for a classical imperfect gas (see below). We denote in the following by Ω) the point in Liouville space that specifies the ground state of the electronic system. The rounded ket indicates that the metric in Liouville space is a special one. It is based on cumulants.
This review is structured as follows. First we recall some facts about cumulants. Next the form of Ω) and the Schrödinger equation is given. This is followed by a documentation of Ω), an important step for numerical applications. Finally, we list some applications of the additive form of the wavefunction. A summary and an outlook into the future completes this review.
USE OF CUMULANTS
Many years ago Kubo [13] pointed out the usefulness of cumulants. They are an important tool when multiplicative functions occur in relation with additive functions. As an example consider the multiplicative partition function Z of a classical gas of N particles with pair interaction 𝜙_{ij}, i.e., a potential energy . It is Z = Z_{0} · Z_{U} where Z_{0} is the partition function for 𝜙_{ij} = 0. The corresponding free energy F(T) = F_{0} + F_{U} is an additive function.
We define
 (3)

and write
 (4)

Therefore, the interaction part Z_{U} of the partition function is
 (5)

where <...> is the average over all configurations of the gas. Consequently
 (6)

Working with the logarithm is avoided by using cumulants. One definition of cumulants often used is
 (7)

where A is an arbitrary operator or function and c indicates taking the cumulant. The latter is defined by assuring that both sides are identical when they are expanded in powers of λ. For example it is
 (8)

When applied to Eq. (6) we find
 (9)

It demonstrates that only linked pair interactions contribute to the free energy, an observation also termed Mayer's cluster expansion [14]. The close relation of these findings with the EWP in the quantum case will become clear below. More generally cumulants are defined through
 (10)

where Φ_{1}> and Φ_{2}> are arbitrary state vectors in Hilbert space with the condition that <Φ_{1}Φ_{2}>≠0 [15]. The one of the number 1 is
 (11)

while for the unit operator 1_{op} we find when Φ_{1}>=Φ_{2}>:
 (12)

Note the independence from the norm of the vectors Φ_{1}> and Φ_{2}>:
 (13)

It is useful to consider the behaviour of the cumulant when we transform the vector Φ_{2}> in Eq. (10) into another vector ψ> in Hilbert space. For this purpose we apply a sequence of infinitesimal transformations e^{δS} in Hilbert space taking us on a path from Φ_{2}> to ψ> [15]. We subdivide this path into L steps. After the first step we obtain for the cumulant of any operator A, but now taken with respect to the vectors Φ_{1}> and e^{δS}Φ_{2}>
 (14)

After L steps this results in
 (15)

with
 (16)

We draw attention that Ω is not unique since many different paths can be chosen in order to go over from Φ_{2}> to
ψ>. Until now ψ> has been any vector not equal to Φ_{2}>. Later we shall choose for it the ground state ψ_{0}> of H and for Φ_{2}> the ground state Φ_{0}> = vac> of H_{0}. In this case the operator Ω transforms the ground state of uncorrelated electrons into the ground state of the correlated electron system [1618]. Note that when ψ> is any eigenstate of H and Φ> is any vector in Hilbert space with <Φψ> ≠0, then for any operator A (not a cnumber!) the following equation holds [17]
 (17)

The matrix element factorizes and therefore the cumulant vanishes. In general the expectation values of operators A in the ground state of the system are obtained from A_{exp} = (ΩAΩ).
CLUSTERS OF VACUUM FLUCTUATIONS
In order to describe the ground state of the electron system in a form that is additive, all vacuum fluctuations that enter the description of the ground state must not factorize. We include them by the following vector in Liouville space
 (18)

i.e., whenever a matrix element involving ψ_{0}>^{c} is calculated the cumulant of this matrix element must be taken. We have here adopted the notation of Eqs. (15,16) and identified Φ_{1}> with vac> and ψ> with the ground state ψ_{0}> of H. In the following we will always assume that <vacψ_{0}>≠0, although this overlap becomes exponentially small with increasing electron number N in Eq. (2). Equation (18) suggests introducing the following metric in Liouville space
 (19)

where A and B are arbitrary operators. The groundstate energy E_{0} is obtained from
 (20)

With the help of Eq. (17) this expression is rewritten in condensed form as
 (21)

We call Ω) the cumulant wave operator in analogy to Møller's wave operator . The latter relates ψ_{0}> and vac> in Hilbert space through
 (22)

As seen from Eq. (15) Ω) is of the generic form Ω) = 1+S) and therefore S) is called a cumulant scattering operator. It describes connected vacuum fluctuations. Thus the energy E_{0} decomposes into E_{0} = E_{SCF} + E_{corr} with
 (23)

One notices that with Eqs. (18, 21) we have gone from a wavefunction ψ_{0}> in Hilbert space, which is of a multiplicative form to a characterization of the ground state by vacuum fluctuations in Liouville space, i.e., Ω) which is additive. There is no EWP in the latter case. Any approximation to Ω) leads just to a small change δS) in the cumulant scattering operator and a corresponding change in the correlation energy δE_{corr} = (HδS). Note that Eq. (21) corresponds to the time independent Schrödinger equation for the ground state formulated in Liouville space. For small electronic systems both forms, i.e. the one in Hilbert or Liouville space may be used, which ever is more convenient. Next we shall derive some relations that are very useful for practical calculations of Ω).
Starting from the identity
 (24)

where the ψ_{n}> are a complete set of orthonormal eigenfunctions of H, it follows from Eqs. (13,18) that
 (25)

The right hand side remains finite in the limit λ → ∞. This remaining part can be obtained from a Laplace transform and can be written as [19]:
 (26)

We have used the fact that ... ... H_{0}) = 0 since Φ_{0}> is an eigenstate of H_{0} and therefore any cumulant vanishes. When Eq. (25) is substituated into Eq. (23) we obtain the energy contributions of the linked fluctuations in the form of a perturbation expansion. Note the connections to Kato's expansions [20] and to Goldstone's diagrammatic energy expansion [21].
While Eq. (25) allows for an evaluation of Ω) in the form of a perturbation expansion, one may also adopt a quite different approach based on projections. In the case where one has a clear physical picture about the most important vacuum fluctuations one may limit oneself to these and thus to a relevant subspace 𝕽_{0} of the full Liouville space 𝕽 that they span [22]. Let us assume that the orthonormal operators A_{v} span this subspace 𝕽_{0}. Then an ansatz of the form of
 (27)

is suggested. The parameters η_{v} can be determined from Eq. (17), i.e.,
 (28)

with the solution
 (29)

and
 (30)

At this stage a general comment is appropriate. Since Ω) is size extensive, different quantum chemical methods like the Coupled Electron Pair Approximation (CEPA0) and variations of it [2325] or the Coupled Cluster (CC) method [2629] etc, can be incorporated in it. For CEPA0 and CEPA2 this was done in Ref. [30] and for the CC theory in Ref. [31]. The different methods vary in the way the different vacuum fluctuations are selected and summed up. Which form is preferable depends on the given situation. A discussion of the advantages and disadvantages of the different methods, however, is not the subject of this review. In passing we note that the effects of an unavoidable coupling of a nearly isolated system with its surroundings on Ω) can be neglected, since Ω) is an additive quantity [32, 33].
DOCUMENTATION OF THE WAVE OPERATOR
After having shown that the EWP does not appear if wavefunctions are formulated via vacuum fluctuations in Liouville space, i.e., by the fluctuations of a meanfield state Φ_{0}> defined as vacuum, we indicate how the wave operator can be documented and therefore applied for realistic calculations of the ground state of solids. The formulation of Ω) in terms of linked vacuum fluctuations puts us in a position to reduce the treatment of electronic correlations to a small number of electrons. This is done as follows. The starting point is a set of L basis function f_{i} (r) centered at different lattice sites I, J etc. In terms of them the field operators ψ_{σ} (r) are expressed as
 (31)

For the basis functions orthogonalized sets of Gausstype orbitals are usually chosen. In this case the corresponding creation and annihilation operators , a_{iσ} fulfill the usual anticommutation relations. The Hamiltonian expressed in terms of these operators is
 (32)

We split the Hamiltonian into H = H_{0} + H_{1} where H_{0} is the selfconsistent field (SCF) Hamiltonian and H_{1} is the remaining residual interaction part. The SCF groundstate is defined as the vacuum state and is of the form where the create electrons in the canonical SCF or Bloch spin orbitals μσ. The index μ includes the momentum k and a subband index while 0> is the empty state. The vacuum fluctuations generated by H_{1} are rather local and generate the correlation hole of an electron. Therefore we replace the occupied Bloch orbitals by Wannier orbitals. The latter are obtained by a unitary transformation U in the space spanned by the occupied canonical spinorbitals
 (33)

so that . The unitary transformation is chosen so that the Wannier orbitals are as localized as possible [34]. The unoccupied or virtual SCF spin orbitals are best expressed in terms of modified basis functions
_{i}(r) that are the f_{i}(r) orbitals but orthogonalized to the occupied space, i.e., to the Wannier orbitals. The index I indicates the site (or bond) at which the Wannier or virtual orbitals are centered. With these definitions the residual interactions can be decomposed in the form
 (34)

The brackets refer to pairs, triplets and quadrupoles of sites or bonds. The residual interaction part of H has one or two destruction and creation operators and the subscrips I, IJ etc specify where these two or four operators are centered. For example, H_{I} tells us that they are all centered at site (or bond) I, while IJ implies they are centered at sites (or bond) I and J and so on.
Equation (26) suggests the introduction of operators
 (35)

with α running over all contributions to H_{1}, i.e., H_{I}, H_{IJ}, H_{IJK}, H_{IJKL}. Thus from the expansion (26) we obtain
 (36)

This form is very suitable for the determination of the most important vacuum fluctuations contributing to S) and the correlation energy E_{corr} = (H_{1}S). In general correlationenergy contributions from H_{I}, i.e., from fluctuations at a given site I will be more important than from fluctuations involving several sites, e.g., H_{IJ}. Thus the following ordering of the various terms in (36) suggests itself
 (37)

Obviously the operator S_{α}) is the cumulant scattering operator of a Hamiltonian H_{0} + H_{α}. The remaining part in Eq. (36) consists of operators T_{αβ}> involving more than a single H_{α}. A discussion of the T_{αβ}> is found in Ref. [15].
When only fluctuations at a single site I are included (onecenter approximation) then S_{I}) is the cumulant scattering matrix of H_{c}(I) = H_{0} + H_{I}. Diagonalizing H_{I} is a manybody problem involving a small electron number only, i.e., electrons at site I. Strong onsite correlations imply strong vacuum fluctuations S_{I}). With Ω) = 1+Σ_{I}S_{I}) they are taken into account at all sites. In Hilbert space such a generalization is not possible, as this would require dealing with an exponential number of configurations.
In an improved approximation twosite scattering matrices are also treated. This implies including not only H_{I} but also H_{IJ} when the cumulant scattering matrix S) is determined (twocenter approximation). Since the index α runs now over all interaction matrix elements involving sites I, J, and IJ we note that S_{α}) contains contributions of the form S_{IJ}).
For example, the operator
 (38)

is then the cumulant scattering operator of the Hamiltonian H_{c} (I, J, IJ) = H_{0} + H_{I} + H_{J} + H_{IJ}, i.e.,
 (39)

The expansion of S) can be continued so that the cumulant scattering matrix involves an increasing number of sites with vacuum fluctuations, i.e., δS_{IJK}) etc. When a solid is periodic all S_{I}) with I changed by a lattice translation are equivalent. For nonperiodic solids the S_{I}) are site dependent. A similar argument holds true for the other terms.
Fig. 1: Examples of different linked vacuum fluctuations S_{I}, S_{KL}, S_{MNT} contributing to S). On the right hand side: vacuum fluctuations contributing to the correlation hole around site I. Different colours refer to vacuum fluctuations involving electrons on different numbers of sites.
In Fig.1 we show examples of different terms in S). They represent one, two and three center fluctuations. The associated correlation energy improves rapidly with increasing number of increments [35, 36].
The decomposition of S) and with it the computation of the correlation energy in the form of increments has reduced the computations for a periodic macroscopic system to one of a few electrons namely those at sites (or bonds) I, IJ, IJK etc. The corresponding fluctuations can be determined by using any of the size extensive quantum chemical methods. Which of the different methods is the most economical one to treat these electrons depends on the specific system. Often a CC or a CEPA calculation will be the method of choice. A special comment with respect to metals is in order. Here we deal with the difficulty that in a metal the Wannier functions fall off only algebraically in contrast to the exponential drop in systems with an energy gap [37]. One way to improve localization here is to define from the occupied canonical or Bloch orbitals only as many localized orbitals as can be doubly occupied. In the case of Li metal this implies that the localized orbitals 𝜙_{i}, determined, e.g., by the method of Pipek and Mezey [38], are set up with respect to Li_{2} units. For more details we refer to Ref. [39].
APPLICATIONS
The main purpose of this review has been to provide a solid basis for groundstate calculations based on wavefunctions when the electron numbers are macroscopic. Yet, it is reassuring to see that the theory can and has been successfully applied to solids. When one consults the original literature for the given examples, one will notice that the calculations described are often using a somewhat different language. This is not surprising since the condensed form presented here of resolving the EWP problem has been developing slowly over the years. However, the essence of the applied computational schemes in the given examples is precisely the same as described here.
Groundstate calculations have been performed for semiconductors of group IV [40], III  V [40, 41], II  VI [42] compounds, on oxides MgO [43], and CaO [44] to name a few. Also the rareearth compound GaN [45] has been treated with the 4f electrons kept in the core. The accuracy of the findings, e.g., for the cohesive energy or the bulk modulus has been analysed in detail for some of these systems with good results [7, 46].
The overall impression is that connected vacuum fluctuations are of rather small spatial extent! For example, for gapped systems the correlation energy due to twobody increments S_{IJ}) falls off asymptotically as van der Waals interactions do, i.e., as r^{6}. An analysis shows that one and twocenter correlations are usually sufficient to obtain satisfactory results for quantities like the cohesive energy, bulk modulus or bond length. This assumes that reasonably sized basis sets of Gaussiantype orbitals (GTO) are used. The influence of the size of the basis sets on the quality of the calculated physical quantities has also been discussed, e.g., in Refs. [7, 46]. A general finding is that large energy gaps lead to spatially reduced correlation holes. Raregas solids are special, since binding is not obtained at a SCF level. In this case H_{0} is chosen so that it describes a collection of free atoms that are considered as the vacuum. The Hamiltonian H_{1} and with it the vacuum fluctuations take care of the interactions between them. The decomposition of S) starts therefore with the contributions S_{IJ}) where the indices refer to different atoms. They lead to binding and are at large distances of van der Waals type. By including threebody corrections of the form S_{IJK}) the accuracy of the calculated cohesive energy can be improved to a satisfactory degree [47].
SUMMARY AND PERSPECTIVES
The aim of this review has been to address and resolve the EWP that one is facing in Hilbert space for wavefunctions of large interacting electron systems. The exponential increase of the dimensions in Hilbert space with electron number renders the concept of wavefunctions obsolete in this case. This problem must be resolved in order to have a solid basis for wavefunctionbased electronic structure calculations for solids. As was demonstrated, it is the multiplicative property of a wavefunction with respect to independent subsystems that is causing the EWP. Therefore it is avoided when we characterize the ground state of macroscopic systems by quantities that are additive instead of multiplicative. This is possible by choosing the ground state in the meanfield approximation as a vacuum state and by characterizing the groundstate of H by Ω), i.e., the operators that generate vacuum fluctuations through H_{1}. These fluctuations define a vector in operator or Liouville space. However, for Ω) to be additive a cumulant metric in Liouville space is required. The logarithm of a multiplicative function changes it into an additive one and cumulants avoid dealing with the logarithm. A good example is a classical interacting gas where the logarithm of the multiplicative partition function changes it into an additive function proportional to the free energy and where working with the logarithm is avoided by a cumulant expansion (Mayer's cluster expansion [24]) of the pair interactions. Thus for a macroscopic electron system we may start from a mean field, e.g., HartreeFock ground state and use the cumulant scattering operator S) to define the ground state in Liouville space through Ω) = 1+S). As explained in the text Ω) does not suffer from the EWP. Examples for the application of wavefunctionbased electronic structure calculations for solids do exist and were pointed out.
Which are the perspectives that may result from the present work? As pointed out before, density functional theory [11, 12, 48] has revolutionized electronic structure calculations for molecules and solids. Yet, in practice it has to rely on uncontrolled approximations for the functional which may lead to unsatisfactory results, in particular when electronic correlations are strong. Therefore it seems highly desirable to develop in parallel methods that apply controlled quantum chemical methods based on wavefunctions. For solids, calculations of this type had always to struggle with the criticism that one is dealing here with objects, i.e., wavefunctions that due to the EWP have lost their meaning for large systems. The counter argument has been, of course, that, nevertheless, meaningful results for the energy and other physical quantities can be obtained from wavefunctions, the simplest example being manybody perturbation theory. As we have shown, defining states via cumulant wave operators in Liouville space makes this discussion obsolete. Having provided a solid framework for wavefunctionbased structure calculations it becomes possible to make substantial progress in dealing with periodic solids with weakly as well as strongly correlated electrons. Also one would like to generalize the above concept in an efficient way to energy band calculations [49]. It seems that a big, barely tapped field of research is lying in front of us. It should be mentioned that this review is based in parts on Ref. [50].
Acknowledgements: I would like to thank Hermann Stoll for helpful discussions.
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Peter Fulde is a professor emeritus and former director at the Max Planck Institute for the Physics of Complex Systems (MPIPKS) in Dresden (Germany). He received his diploma in physics from the University of Hamburg (Germany) and his PHD in 1963 from the University of Maryland. From 1968  1971 he held a chair in theoretical physics at the University of Frankfurt/Main. Since 1972 he has been a scientific member of the MaxPlanckGesellschaft, until 1993 as a director at the MPI for Solid State Research in Stuttgart and thereafter as the founding director of the MPIPKS. From 2007  2013 he served as a president of the APCTP in Pohang. His research focussed on condensed matter theory. 
