**Journal of Modern Physics**

Vol.05 No.09(2014), Article ID:47002,17 pages

10.4236/jmp.2014.59083

Cumulant Structure Factors of Jellium

P. Ziesche^{1}, J. Cioslowski^{1,2}

^{1}Max-Planck-Institut für Physik Komplexer Systeme, Dresden, Germany

^{2}Institute of Physics, University of Szczecin, Szczecin, Poland

Email: pz@pks.mpg.de

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 29 January 2014; revised 25 February 2014; accepted 21 March 2014

ABSTRACT

For the ground state of the homogeneous electron gas (jellium), it is shown how the cumulant de- composition of the 2-matrix leads to the cumulant decomposition of the structure factors S_{a,p}(q) for the antiparallel (a) and parallel (p) spin pairs and how it simultaneously allows one to derive the momentum distribution n(k), which is a one-body quantity [Phys. Rev. A 86, 012508 (2012), A 89, 059902 (E)(2014)]. The small-q and large-q behavior of S_{a,p}(q), and their normalizations are de- rived and compared with the results of P. Gori-Giorgi et al. [Physica A 280, 199 (2000) and Phys. Rev. B 61, 7353 (2000)].

**Keywords:**

Jellium, Reduced Density Matrices, Pair Densities, Coulomb Hole, Fermi Hole, Structure Factors, Cumulant Decomposition

1. Introduction

This paper deals with the ground state of an extended spin-unpolarized (paramagnetic) homogeneous electron gas (HEG), which is one of the most widely studied systems with correlated electrons [1] . The gas is assumed to be uniform in space with the electron density. Other characteristic quantities that depend parametrical- ly on r_{s} are the total energy per electron e, from which follows the interaction energy [2] , the momentum distribution, and the static structure factors (SFs) with “a” standing for electron pairs with antiparallel spins and “p” for pairs with parallel spins. For recent papers on the HEG, also known as jellium, see for example [3] - [15] . Recent GWA and QMC calculations for are [3] [4] . (For comparison with experiment see [5] . The value of for r_{s} = 3.9 agrees with the results, obtained in Refs. [6] [7] .)

Overhauser considered the pair densities PDs, which follow from the SFs by Fourier transform, and parameterized them in terms of certain 2-body wave functions (known as geminals) [16] . More general geminals should allow one to calculate not only the 2-body quantities, but also the 1-body quantity. Searching in this direction, one immediately encounters the hierarchy of the reduced density matrices (RDMs) and their cumulant decomposition. (The concept of RDM is of great importance in quantum chemistry, see [17] - [22] .) Here it is assumed that the 2-body RDM (2-matrix) is available from perturbation theory or otherwise (hierarchy truncation etc.) and, with the spin structure as a decisive point, it is elucidated how the contraction has to be performed for an extended system in such a way that the 1-body density is obtained. In view of the thermodynamic limit, this procedure calls for the cumulant decomposition of the 2-matrix into its Hartree- Fock part and its nonreducible remainder, the cumulant 2-matrix, which proves to be the source of both and [10] . Diagonalization of yields cumulant geminals and the corresponding weights (i.e. the spectral resolution).

A side product of this geminal analysis is the following. Usually the partitioning involves an unperturbed (ideal Fermi gas) term S_{0} and the interaction-induced term arising from the pairwise Cou- lomb repulsion [15] . In contrast to that approach, an alternative partitioning is presented here as a consequence of the cumulant decomposition of the 2-matrix, namely, with the Hartree-Fock term [con- structed from the correlated] and a nonreducible remainder, i.e. the cumulant parallel-spin SF. The advantage of such a partitioning stems from the fact that the cumulant SFs are given within perturbation theory by linked (and therefore size-extensive) diagrams with two open particle-hole lines and arbitrary numbers of closed loops. As usual, the divergent terms are eliminated by a partial summation known as “the random phase approximation” (RPA) with the physical meaning of long-range correlations (screening), which becomes exact at the high-density limit of [23] [24] . RPA is a general method to treat the quantum-mechanical many-body problem, successfully applied not only to the HEG. It is now implemented in electronic structure codes for both solids and molecules [25] . A recent discussion concerns its failure when applied to the dissocia- tion of: it is exact at the dissociation limit, but very inaccurate at intermediate internuclear distances, see Figure 1 in [26] . This RPA has been applied to the HEG quantities e, , n(k), and. There have been recent efforts to obtain n(k) and S(q) beyond RPA [7] [15] . In those publications, analytical constraints and fit- ting procedures to quantum-Monte-Carlo data (as “amends” for experimental data) have been combined for the best possible results. Other attempts to go beyond RPA and quantum-Monte-Carlo calculations are the operator- product expansion (OPE) [27] and the machine learning methods [28] . In the following text, the RPA is com- bined with the cumulant analysis described above. Altogether, the available data on the static SFs of the HEG are compiled and analyzed from a new viewpoint, namely the cumulant decomposition, continuing earlier attempts in that direction [7] [10] [29] - [32] . From a formal point of view, the present study is closely re- lated to the work of P. Gori-Giorgi et al. [15] .

The paper is organized as follows. In Section 2, the 2-matrix and its cumulant decomposition is intro- duced. It is shown how the SFs and follow from. Their relation to the PDs via the Fourier transform is elucidated in Appendix A, where the cusp and curvature theorems are summarized as well. In Section 3, the antiparallel-spin SFs are derived within RPA [the necessary technical details (particle- hole propagator, etc.) being given in Appendix B]. The normalization of yields the short-range correla- tion parameter. In Section 4, the cumulant decomposition of the parallel-spin SFs with a complicated exchange integral is presented. The fourth moment of yields another short-range correlation parameter, namely. Table 1 and Table 2 summarize the small- and large-q behavior and the normalization of the SFs. Section 5 concludes the paper with a summary and an outlook.

2. Basic Equations

The functions mentioned above, i.e. and, determine the total energy via the kinetic energy and the interaction energy,

(2.1)

respectively. In Equation (2.1), is the classical plasma frequency with. The momenta are measured in units of, energies (including) in units of [in a.u. it is]. The GGSB analysis [15] is based upon the decomposition with the term

(2.2)

Table 1. The behavior of the cumulant SFs vs. q with. Note that and. The index “a” means spin-antiparallel, “p” means spin-parallel, d means direct (2-line) diagram, and x means the corresponding exchange (1-line) diagram. For the Löwdin parameter c see Equation (4.1), the other correlation parameters are, c_{1}, and. In addition to the normalization integrals of the last column, it holds that, see Equation (4.12).

Table 2. The behavior of the structure factors (SFs) vs. the momentum transfer q. Note that and . The spin-summed SF is, the magnetic SF is . In addition to the normalization integrals of the last column, it holds that, see Equation (4.12). Comment on: the small-q behavior of is not known so far, the assumption agrees with the Iwamoto expression for the total SF; the normalization of is due to the Pauli principle.

of the ideal Fermi gas with and the term vanishing at the high-density limit, which corresponds to “no Coulomb repulsion”. The linear-cubic combination that appears sever- al times in the following derivations may be regarded as the ideal-Fermi-gas term. has a simple geome- trical meaning: it is the average volume accessible to the occupied momentum when excited by a transfer mo- mentum of amplitude [33] . The interaction energy in the lowest order [i.e. with] equals.

It is interesting to note that the plasmon sum rule [34] requires to possess the small-q asymptotics of

Figure 1. The Feynman diagrams for the Coulomb repulsion in RPA with the partial summation (or screening replacement), the dashed line denotes the bare interaction, the closed loop denotes the particle-hole propagator of Equation (B.3), and the wavy line denotes the effectively screened interaction, where q is the momentum transfer, is the energy or frequency.

, where is the classical plasma frequency (in units of). [From the f-sum rule or Thomas-Reich-Kuhn sum rule of the dynamic SF follows the plasmon sum rule of the static SF.] At first glance, this asymptotic behavior appears incom- patible with the linear dependence of on around. However, these two observations are rea- dily reconciled by writing

(2.3)

where is a function with the asymptotics and.

In this paper, an alternative is presented, namely

(2.4)

Note that what is called in [10] is named here. Equation (2.4) is the so-called cumulant de-

composition into the “Hartree-Fock” component, reducible to the 1-body quantity, which fol- lows from the exact 1-matrix, and the non-reducible “cumulant” component, such that for it is and consequently. In this cumulant decomposition, the Coulomb repulsion enters the SF at two places: 1) the “Fock term” via the correlated momentum distribution and 2) the cumulant SF via linked diagrams, whereas in the GGSB decomposition the Coulomb repulsion is hidden only in that (in the cumulant “language”) equals with [which results from]. For plots of see Figure 1 in paper I of [35] . The Fock term brings in the momentum distribution with its nonanalytical pecu- liarities, caused by the RPA long-range correlation. The complexity of the HEG manifests itself in nonanalytici- ties or singularities of the type or, see e.g. or. This gives rise to the k- or q-dependences in relation to the dependence on r_{s}. For example, according to the perturbation theory, should behave like for small r_{s}. This is true for k values away from the Fermi edge (i.e. and). However, at the Fermi edge involves terms like or it jumps as. This im- plies a contribution that equals, see [29] .

The cumulant decomposition seems to be more complicated than its conventional counterpart but, on the oth- er hand, it may be considered a more natural partitioning. This is so because, within the many-body perturbation theory, the cumulant SF is given by linked diagrams, whereas contains also unlinked diagrams.

As already mentioned above, the spin-summed SF is given by. In terms of the cumu- lant SFs, these quantities read

(2.5)

where, in agreement with Equation (2.4). The negative sign in front of en- sures the positive normalization of the cumulant SFs. Note that (the perfect screening sum rule) implies and, the latter relating the 1-body quantity to the 2-body quantities. The analogs of Equations (2.4) and (2.5) in the direct space, including Fourier transforms and normalizations, are given in Appendix A together with the cusp and curvature theorems for and, respectively, which are short-range correlation parameters of.

The original definitions of the 1- and 2-body RDMs in terms of the anticommuting creation and annihiliation operators and, respectively, are given by

(2.6)

The short hands and are used throughout the text, [instead of].

The hermiticity of is obvious and the permutational antisymmetry means that when 1 and 2 or 1' and 2' are interchanged. is the total particle-number operator. The ground state is an - body state, hence and are -body states, and thus

(2.7)

holds for the normalization. The corresponding contraction sum rule

(2.8)

describes how the 1-matrix for finite results from the 2-matrix. Equation (2.8) does not permit the above mentioned thermodynamic limit.

A way towards a size-extensive contraction is the cumulant decomposition. This means to define the cumulant 2-matrix with

(2.9)

This matrix obviously has the spin structure

(2.10)

with the symmetric and antisymmetric 2-body spin functions

(2.11)

Equations (2.10) and (2.11) define (which does not change sign under the exchange or) and (which changes sign under these exchanges). With , the dimensionless cumulant 2-matrices are introduced. The next step involves deriving (for electron pairs with antiparallel spins) and (for electron pairs with parallel spins) from. With and together with for “a” or for “p”, Equation (2.10) leads to

(2.12)

respectively. Here and in the following, the arguments of these matrices are suppressed for the sake of convenience. Their diagonals (for which and) are denoted by and. They determine the normalizations

(2.13)

where. Their Fourier transforms (with the integral operator) yield the desired cumulant SFs,

(2.14)

see also Appendix A. A consequence of Equation (2.14) and Equations (A.6)-(A.8) is that the short-range cor- relation determines the asymptotic large-q behavior of the SFs and their normalizations.

Where do the cumulant 2-matrices come from? As mentioned above, they are given by linked diagrams. Each direct diagram with two open particle-hole lines (one running from to and another one from to) is associated with an exchange diagram, see for example Figure 2(a) and Figure 3(a). Thus, a series of diagram pairs exists from which the building elements and follow. The next step defines the singlet/triplet components in terms of these diagrams, the relations

(2.15)

following from Equation (2.12). So, the spin-averaged sum can be written differently: The a- and p-compo- nents are equally weighted, whereas the singlet- and triplet-components have the weights equaling 1/4 and 3/4, and the components have the weights equaling 1 and, respectively. The same holds for the PDs. On the other hand, for the cumulant SFs one has:

(2.16)

Note the slightly different definitions of the spin-summed quantities, , and on one hand, and and on the other.

With and given from the Feynman diagrams and with and following from them, the spin-resolved components of the interaction energy are

(2.17)

and

(2.18)

which can be written as. Thus, ,. In the following, RPA is used (index “r”) and the terms and mean all the (hopefully small) contributions beyond RPA: and.

3. Antiparallel Spins: The Coulomb Hole

We move through the chain, where “a” means antiparallel spins and “d” means “direct” diagrams (of the type depicted in Figure 2(a)). In its Cartesian space representation, is given by (using the definitions of Appendix B)

(3.1)

with the abbreviations and. [Remember to read instead of.] The integral operator generalizes the particle-hole propagator, see Equation (B.4). The diagonal elements define the cumulant PD [see Equation (B.5)]

(3.2)

When going from Equation (3.1) to Equation (3.2), the contour integration, which replaces the fre- quency integration by the velocity integration, leads to the real function for the particle- hole propagator, see e.g. Equation (B.1) in [11] . This leads to Equation (B.5) taking the form , where is referred to as the Macke function [23] [36] [37] . The effective interaction

is the RPA screened Coulomb interaction with a Yukawa-like cut-off,. The reverse replacement can- cels out with, thus restoring the original perturbation expression in the lowest order. The

Figure 2. 2a denotes the lowest-order renormalized cumulant 2-matrix, 2b denotes the non-divergent cumulant PD or the cumulant SF (Ya- suhara) [46] , (Kimball) [38] , and 2c denotes the non-divergent interaction energy (Macke, Gell-Mann/Brueckner) [23] [24] .

Figure 3. The RPA exchange terms corresponding to Figure 2 with 3a corresponding to (following from 2a corresponding to through the exchange replacement, 3b corresponding to or (non- divergent), and 3c corresponding to. As shown by Onsager et al. [54] , already does not diverge (needs no RPA renormalization). For the im- portance of “x” see [40] .

qualitative behavior of is depicted in Figure 11 of [7] ; it begins at as a positive number and then decreases, becoming negative, and finally approaches zero from below, ensuring that its normalization vanishes, see Equation (A.6).

The Fourier transform of is the cumulant SF within RPA:

(3.3)

[38] . Note that parametrically depends on, approaching the Macke function for [with the properties (B.8), see also Figure 4]; may be referred to as the “generalized Macke function”. determines via, which yields the interaction energy of the antiparallel spin pairs

(3.4)

with at the high-density limit. The short-range correlation parameter

(3.5)

also follows from in RPA with

(3.6)

at the high-density limit [38] - [40] . Actually, it is exactly this high-density behavior of and for that results from the ladder theory, which constitutes the best method to treat the short-range correlation [41] - [46] (for the recent developments see [27] ). The correctness of Equations (3.3)-(3.6) confirms the starting point (3.1).

Figure 4. The functions and.

How does behave for small?

(3.7)

in agreement with an ansatz of Iwamoto [47] . The coefficients are due to multipair and quasiparticle-qua- sihole excitations, respectively. They follow from as and [48] . Note that the region shrinks with decreasing and finally vanishes for. As contains (in the paramagnetic gas) half of the plasmon term, the other half has to come from, which also has to compensate the (-independent) ideal-Fermi-gas term.

In the transitional region between the small and large values of there is a peculiar point. Namely, upon approaching from below the transition momenta of, the topology changes from two overlapping to two non-overlapping Fermi spheres. This topology change causes a jump discontinuity in and at [36] , namely

(3.8)

the ellipsis representing the terms beyond RPA. In the Cartesian space these jump discontinuities cause the Friedel oscillations in. They make interpolations between and with Padé approximants [49] or the robust interpolation scheme [50] questionable.

How does of Equation (3.3) behave for? As the large-q asymptotics does not require the RPA, the perturbation theory holds in the lowest order, hence in Equation (3.3) the “descreening” replacement is possible. The integration over yields the Macke function with the large-q asymp- totics (B.8) starting with. Thus, for one has

(3.9)

Are there perhaps corrections due to terms beyond RPA? Indeed, the electron-electron coalescence cusp theo- rem together with the Kimball trick [51] makes the first term decorated with an additional factor. One may expect the next term to be modified in a similar way, thus

(3.10)

with unknown correlation parameters and vanishing for and with the beyond-RPA term different from. The meaning of Equation (3.10) is as follows: the short-range correlation parameter determines the large-wavenumber asymptotics of [41] [51] [52] . Because depends on the parameter, one may read Equation (3.5) as an integral equation to be solved iteratively. One may also ask how to modify the RPA-result of Equation (3.3) in such a way that it absorbs the correct large-q behavior (3.10).

4. Parallel Spins: The Fermi Hole

For the parallel-spin, SF it is appropriate to write it as with, , and. Thus besides one also needs the quantities and, which arise from the same source, namely. and do not contribute to the normalization of, but they do contri- bute to the interaction energy [Equation (2.13)] and to the on-top Fermi hole curvature [Equation (4.12)]. Whereas describes the smooth transition from 0 for small to 1 for as it is known from the ideal- Fermi gas, the term describes the interaction induced corrections.

is known to some extent (at least sufficient for a qualitative discussion). As it can be seen from Figure 1 of [35] , it is for and for with defining. The Fock function defined in Equation (2.4) contributes to. It has the properties (4.1)-(4.4):

(4.1)

The quantity is referred to as the Löwdin parameter or the index of non-idempotency. [P.-O. Löwdin was first to address the meaning of the trace of the squared 1-matrix.] It is a correlation parameter that measures the correlation-induced non-idempotency of the momentum distribution and vanishes for. For the dependence of on see [7] . For the kinetic part of the correlation energy it holds

(4.2)

The small- and large-q asymptotics of are

(4.3)

respectively. The small behavior has been demonstrated with the decomposition, where is continuous (jump-free), see Equation (3.8) in [7] or Equation (5.1) in [35] . The quantity is the quasiparticle weight with within RPA. Note the non- analytical behavior of with terms and note that here the prefactor of the ideal-Fermi-gas term depends on. The Löwdin parameter and the term quadratic in have to be canceled out by and replaced by the other half of the plasmon term. The large-q behavior arises from [52] [53] , which is influenced by the short-range correlation parameter, in analogy to in Equations (3.10) and (4.12), respectively. At, there is a singular (topologically caused) jump

(4.4)

Together with that arises from, it contributes to, the source of the Friedel oscillations in, whereas Equation (3.8) leads to the Friedel oscillations of.

Next, is to be considered. It follows from the Feynman diagrams that, where “p” means parallel spins and “x” stands for exchange. According to Figure 3(a), the exchange counterpart to is given by

(4.5)

Its diagonal elements define the cumulant PD (within RPA)

(4.6)

and (after a tedious derivation) its Fourier transform turns out to be equal

(4.7)

Unfortunately this complicated integral is not known so far as an explicit function of and. If it is approximately substituted by, then the integration over yields the well-known energy denominator for the exchange term:

(4.8)

resulting in

(4.9)

with, in analogy to Equation (3.3). is referred to as the Gutlé function [33] ; for its definition and properties see Equations (B.9), (B.10), and Figure 4. The quantity introduced above in Equation (4.7) can be regarded as its generalization. It is worth mentioning that, whereas in the screened interaction is necessary to remove the long-range divergences of the bare interaction, this is not the case for as it follows from

(4.10)

This is the famous result of Onsager, Mittag, and Stephen [54] [55] . The total interaction energy equals with, , and.

Within the approximation, the exchange analog of Equation (3.5) reads

(4.11)

which agrees with Equation (3.5) in the corresponding approximation thanks to, see Equations (B.8) and (B.9).

Furthermore, the on-top curvature of the Fermi hole follows from and, vice versa, it de- termines the large-q asymptotics of,

(4.12)

does not contribute to the -term because its expansion begins with. The proof of Equation (4.12) employs the curvature theorem (A.11) and the Kimball trick. With and Equation (4.2) it is

(4.13)

Note, that begins at the second order with and, see Equation (B.8). The terms proportional to, which would cause the integral to diverge, cancel each other. At the high-density limit one has [56] [57]

(4.14)

with.

Next the small-q behavior of is studied. Since the integral that enters Equation (4.7) is not known in a closed form, understanding the small-q behavior of is not as straightforward as that of. For with Equation (3.3) for and Equation (4.7) for it holds

(4.15)

where the first ellipsis stands for higher-order terms [of the order or] of Equation (3.3) and the second ellipsis means the beyond-RPA terms beginning with. With introduced above, for it follows (within RPA) that

(4.16)

What contributes to the second term as a function of and remains to be studied. If we assume that its power expansion begins with then, in analogy with Equation (3.7), one obtains

(4.17)

and thus (again for)

(4.18)

in agreement with the plasmon sum rule and [15] . Note that and [from which follows] are indirectly related due to the fact that they have their common origin in the same Feynman diagrams. Namely, the starting point is the 4-point function (see Figure 3(a) and also Figure 3(a) in [10] ). Upon carrying out the contraction procedure, followed by integration over and the Fourier transform for, the source quantity for the momentum distribution emerges, see [10] . On the other hand, upon taking the diagonal elements and and carrying out the Fourier transform for one obtains. Certainly a further study of and within RPA and the beyond-RPA terms is warranted.

Next, the large-q behavior of is studied. For, the RPA quantity approaches. Thus, with the large-q asymptotics of [see Equation (B.9)] and the beyond RPA-corrections analogue Equation (3.9), one obtains

(4.19)

perhaps with an additional correlation parameter. In comparison with Equation (3.10), the tail is strongly enhanced (i.e. four times larger). Note that the difference of the prefactors is just. The large-q asymptotics of begins with the same expression as, thus begins with a term proportional to:

(4.20)

which makes and even integrable, see Equation (A.14). Comparison with Equation (A.13), that follows from the curvature theorem [Equation (A.11)], shows that as a sum rule, hence the asymptotics (4.12) is recovered.

The asymptotics of the SFs are summarized in Table 1 and Table 2; for the high-density limit with and, the decoration drops off and the RPA limiting values reemerge. They show once more how the short-range correlation parameters determine the large-q asymptotics of the SFs, what is effected by the cusp and curvature theorems. As a side result, the on-top curvature of the Fermi hole is rederived in terms of, see Equation (4.12). As it follows from Table 2 the function, defined at the beginning, see Equation (2.3), has the asymptotics and. Besides, for the inflexion point of is,. For this means. The inflexion-point trajectory for, namely, is in agreement with the slope of the ideal-Fermi-gas term at.

5. Summary and Outlook

The focus of this paper is on the mathematics of the HEG, which on one hand is only a marginal point in the broad realm of correlated systems lacking rigorous solutions, but on the other constitutes an archetype of an extended many-body Fermi system. The small-q and large-q behaviors of the cumulant structure factors and the structure factors as well as their normalizations are summarized in Table 1 and Table 2, respectively. In these tables, “a” stands for the electron pairs with antiparallel spins and “p” stands for the pairs with parallel spins. This analysis is based upon rigorous constraints such as the perfect screening sum rule (or the charge neutrality condition), the plasmon sum rule with its inflexion-point trajectory, the on-top (the zero electron-electron distance) theorems for the pair densities (cusp for “a” and curvature for “p”) and the “old” RPA (with the Feynman diagrams like those depicted in Figure 2(a) and Figure 3(a) giving exact results for). They are all summarized in a way similar to the GGSB paper [15] (where has been pa- rameterized by combining analytical constraints with fitting of numerical data), but modified here under the novel cumulant point of view. This is done under the premise that the cumulant decompositions (2.4), (2.5), and (A.2) are more fundamental than the seemingly simpler decomposition into the ideal-Fermi- gas component and the interaction-induced remainder. A formal analysis is presented with the spin structure (2.10) as a decisive point, leaving the task to parameterize analytically as functions of and in a manner similar to that carried out in the GGSB paper [15] . The agreement of Equation (3.5) with the results of Macke [23] and Gell-Mann/Brueckner [24] and of Equation (4.10) with the calculations of Onsager et al. [54] also confirms the expressions (3.1) for “dr” and (4.5) for “xr”, where “r” means RPA and “d” and “x” mean direct and exchange Feynman diagrams, respectively, see Figure 2(a) and Figure 3(a). A complete and self-consistent RPA description needs both, which is available from Equation (3.3), and, which is actually given by a complicated exchange integral in Equation (4.7). This is needed for in the combination, where the Fock term appears, which brings in the correlated momentum distribution with such terms as (i.e. the Löwdin or idempotency parameter) and (i.e. the jump of at), the quantity c to be compensated by. With and, the final expression for is. The combination makes the term to have the correct prefactor, such that it disappears in, leaving twice the half plasmon term. One may ask how the Fock term, respectively the combination influences the PD?

It may be that in the present paper a deeply lying confrontation emerges, namely between 1) perturbation theory, which is linear in the Feynman diagrams, and 2) the cumulant decomposition, for which higher-order reduced density matrices in terms of products of lower-order ones are characteristic.

Going back to the starting point, which rests upon the belief that the geminals and the corresponding weights that diagonalize the cumulant 2-matrix, are the most “natural” 2-body quantities to describe an extended many- body Fermi system, establishes the direction of the future studies. If these quantities are known, then they de- termine the structure factors, the interaction energy and?according to [10] ?also the momentum dis- tribution in terms of closed-form expressions. The next task would be to derive an effective approximate 2-body scheme (a modified 2-body Schrödinger equation or the Bethe-Salpeter equation?) from the hierarchy of contracted Schrödinger equations such that it yields the cumulant geminals and their weights.

Note added in proof: Equations (4.7) and (4.16), Figure 3, and Ref. [25] at the middle of Section 1 are related to attempts around RPA+SOSEX, see [58] - [61] .

Acknowledgements

The authors thank P. Gori-Giorgi in particular for drawing their attention to the Kimball trick, and are grateful to C. Gutlé, and M. Holzmann for valuable discussions and to U. Saalmann for critically reading the manuscript. They acknowledge P. Fulde and the Max Planck Institute for the Physics of Complex Systems Dresden for supporting this work. One of the authors (J. C.) also acknowledges funding from NCN (Poland) under grant DEC-2012/07/B/ST4/00553.

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Appendix A: The Cumulant Pair Densities and Their Cusp and Curvature Theorems

The analogue of Equation (2.4) in the direct space is

(A.1)

with being the (dimensionless) 1-matrix [and of Equation (2.4) being the Fourier transform of]. Equation (2.5) corresponds to the Coulomb and Fermi holes,

(A.2)

with the spin-summed PDs and. For the qualitative behavior of see Figure 11 in [7] or Figure 2 and Figure 3 in [31] . The PDs are the Fourier transformed SFs,

(A.3)

(A.4)

and

(A.5)

The definition of the spin-summed SFs, namely and, brings about the factors 1/2 in Equations (A.5)-(A.7). Note also the asymmetry with respect to “a” (Coulomb hole) and “p” (Fermi hole). The conditions fix the normalizations of, known as the perfect screening sum rule, whereas the coalescing (or the on-top) values and (as a conse- quence of the Pauli principle) fix the normalizations of and, respectively, and fixes the fourth moment of. In the following, the identities [obtained from Equations (A.3)-(A.5), res- pectively]

(A.6)

(A.7)

and

(A.8)

are also used. For plots of with various values of the parameter see [7] , Figure 11. For the on-top (the zero electron-electron distance) curvature of the Fermi hole, it holds according to Equation (A.2) that

(A.9)

as the consequence of. Note that and.

Cusp and curvature theorems: The large-q behavior of is related to the electron-electron coal- escence theorems (a.k.a. the cusp theorems, “a” denoting antiparallel spin pairs) [31] [51] [52] [57]

(A.10)

and the curvature theorems (with “p” standing for parallel spin pairs) [57]

(A.11)

respectively. Therefore, the short-range parameters and determine their derivatives and, respectively. These theorems allow one to correct the large-q behavior of the SFs and of and beyond RPA.

Proof of Equation (4.12): The Kimball trick is used [51] thanks to P. Gori-Giorgi. Namely, a function with the asymptotics of for is defined first through. Then, from the Fourier transform (A.4) and with the help of the Mathematica software [48] , one obtains

(A.12)

from which the derivatives of [including] follow. With the above curvature theorem (A.11), it finally results in Equation (A.9), which is equivalent with

(A.13)

Comparing this result with Equation (3.10), allows one to conclude that the sum rule holds. A side product of this analysis is

(A.14)

Proof: As an intermediate step towards the above expression for, one derives (with the help of the Mathematica software [48] ) the equations

(A.15)

which lead to the first equation of Equation (A.14). From this result, the second equation follows by means of

and Equation (A.9). Remark on the sign of: According to Equation (A.7), the integral has to vanish. This is because the negative asymptotics (4.16) of is compensated by the positive branch (4.17) for small. An additional factor suppressing the latter and enhancing the former. This makes the integral negative, implying. For the function, see for example [7] .

Appendix B: The Particle-Hole Propagator and the Special Functions I(q) and I_{x}(q)

The building elements of the RPA Feynman diagrams (cf. Figure 1, Figure 2(a), Figure 3(a)) are the Coulomb repulsion with the coupling constant and the one-body Green’s function of free electrons with,

(B.1)

From, it follows that the particle-hole propagator within RPA, given by

(B.2)

results in

(B.3)

The denominators contain the excitation energy to create a hole with the momentum inside the Fermi sphere and a particle with the momentum outside the Fermi sphere. A complicated step function, defined by for and, and 0 otherwise, arises from the Pauli principle. A ge- neralization of Equation (B.3) reads

(B.4)

defining the integral operator,.

In calculations of the cumulant SF (see Figure 2(b) with the bare Coulomb repulsion), the Macke function appears via

(B.5)

When dealing with the corresponding exchange term (see Figure 3(b) with the bare Coulomb repulsion), one obtains by means of contour integration

(B.6)

where

(B.7)

defines the exchange counterpart of the Macke function. The functions and have the following properties

(B.8)

and

(B.9)

see Figure 4. is known from the work of Macke [23] ; it is explicitly presented e.g. in [36] , where also is given, see Equation (C.2). is known from Gutlé [33] . The original expression for can be markedly simplified for, in which case

(B.10)

The small-q behavior causes the “Heisenberg divergence”. The asymptotics ensures finiteness of the integral, which has been congenially calculated by Onsager et al. [54] , see Equation (4.10). The flattening in the small-q region, observed upon going from to, together with the “area conservation”, is partly compensated by the heavy enhancement of the tail (from 2/5 to 8/5), see Figure 4. Both and consist of two branches,which encounter each other at the Fermi edge with transition momenta. They have a simple analytical behavior for and, but “in between”, namely at it is

(B.11)

with, for and, for. For it similarly holds

(B.12)

with (for)

(B.13)

The analog case is not known so far. The peculiar behavior at reminds of the behavior of near, see [7] , Equations (6) and (7).

Remark on the types: Note that indices like pl, a, p, d, x, dr, xr are to be represented by upright types, not by italics as this is falsely partly the case e.g. in Table 1 and Table 2.

NOTES

^{1}Another approach with the same result and coining of the term “RPA” is due to D. Bohm and D. Pines (1953).

^{2}In Equation (27) the correct prefactor is 2αr_{s}/(5π).

^{3}The last sentence of the Abstract should be deleted.

^{4}It is shown, that the pair-density geminals, see [16] , do not satisfy the plasmon sum rule.

^{5}In Appendix C the Macke function I(q) is explicitly given. The extension to the spinpolarized electron gas is in [37] .

^{6}Here the exchange integral of the 3-dimensional HEG has been evaluated analytically. This rather herculean work was extended to the d-dimensional electron gas by M. L. Glasser [55] .