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Implicit infinite lattice summations for real space ab initio correlation methods

We suggest a local wave function based ab initio correlation method for infinite periodic systems, which can describe both the near range as well as the long range correlation effects coherently in the same scheme. Specifically, this work introduces a formalism which allows to describe the long range polarization cloud around a quasi particle in a solid explicitly in the formalism of local wave function based ab initio descriptions. To this end we reformulate the infinite lattice summation underlying the quantum chemistry formula to second order in a closed analytic form employing the elliptic theta function of the third kind. All formulas and manipulations are developed explicitely in full detail and a first numeric example demonstrates the principle idea. Good results for the long range polarization effects in LiH and LiF are found in agreement with earlier estimates.


I. INTRODUCTION
The field of many-body theory has been developing at high speed in recent years. A significant amount of effort is directed towards affordable descriptions of electronic correlation in infinite periodic systems or large heterogeneous systems both in the ground state and in excited states. Well established schemes like the density functional theory (DFT) 1 have seen more and more refinements, for example by means of the optimized effective potential method (OEP) 2 , time dependent DFT 3 , screening implementations 4 , the Wigner theory 5 or the reduced density matrix functional theory (RDMFT) 6 . Other correlation schemes have been derived, examples are the dynamical mean field theory (DMFT) 7 or the GW approximation 8 .
Parallel to the density functional approaches wave function based quantum chemical methods have also been trimmed to be applicable to large systems and finally even to polymers and crystals.
To a large extend credit is due to the local ansatz of Fulde, Stollhoff and Horsch 9,10 as well as the systematic development of an incremental scheme by Stoll [11][12][13] . As for ground state properties the ansatz of this local incremental scheme in combination with the coupled cluster (CC) method turned out to be valiant in applications to broad classes of polymers and semiconductors as well as ionic crystals [11][12][13][14][15] . Extensions of this particular approach to band structure calculations were following suit. Gräfenstein and Stoll presented an effective Hamiltonian obtained from cluster approximations for group-IV-semiconductors [16][17][18] , which were subsequently applied successfully to polymers as well [19][20][21] . Further significant advances for the case of excited states have been reported for quantum monte carlo methods (QMC) [22][23][24][25] , algebraic diagrammatic construction (ADC) 26 , or the Green's function approach [27][28][29][30][31][32][33] .
These wave function based methods are straightforwardly applicable to both ground state and excited state calculations alike and are amenable to systematic improvement on the numerical accuracy by their very construction.
The general bottle-neck of steep increase of numerical effort with system size, however, affects all wave function based methods alike. It is precisely this obstacle which was overcome in earlier applications by a formulation of electron correlations in local orbitals and a hierarchy of correlation contributions called the incremental scheme 11,[15][16][17][18] .
Recently we demonstrated that a full Green's function approach with a frequency dependent self energy can in principle also be combined with the incremental scheme. Band structure calculations were performed for LiH and LiF 29-31 and a recent application to a molecule inside a molecular junction also underlines the usefulness of such an approach 32 . The key enabling such calculations was an approach based on local orbitals and a real space formulation of the self energy.
A particular decomposition of energy denominators appearing in quantum chemical expressions led to a considerable speed-up by two orders of magnitude 33 .
However, one problem pertains to all these efforts, namely the presence of a long range polarization cloud around a quasi particle 34 . This effect can shift band structures by some ¢ ¤ £ and has hitherto only been estimated by parametrized continuum approximations 16,17,[29][30][31] in the frame of the aforementioned local orbital based ab initio schemes. While the terms giving rise to the long range polarization of the crystal due to the presence of an extra quasi particle are well known, the underlying infinite lattice summation could not be performed so far, in particular in three dimensional systems. Of course, there are other methods available to describe the long range polarization cloud, like the random phase approximation (RPA) 34,35 , but they cannot be combined with the description of near range correlation effects derived from our local orbital based correlation scheme without double counting. Due to the efficiency in describing correlation effects locally, we would like to stick to the local incremental scheme, while extending it to a description of the long range polarization cloud.
In the present work we device a way to solve this problem. The application of a mathematical identity allows to decompose the two denominators in a perturbative expression so as to prepare the ground for an infinite lattice summation performed analytically. The denominator decomposition has been applied previously in a different context and a slightly different form to energy denominators 33,36 . The ensuing lattice summation bears similarities with the well established Ewald summation technique 37,38 , albeit the mathematical details differ distinctly. The main purpose of this work is to reveal all formulas in detail and to sketch the general idea. Along with this we also present a first numerical application to further exemplify the procedure.
In the following section we describe the theory with the focus on the transformation of the infinite lattice sum. In Sec. III numerical tests are presented together with applications to two realistic ionic systems, LiH and LiF. Sec. IV contains our conclusions.

II. THEORY
In earlier works we have designed a formulation of the Green's function correlation method so as to use local HF orbitals as a starting point and then assess the correlation contributions in a full ab initio manner. This correlation method has subsequently been combined with the so called incremental scheme which proved to boost efficiency tremendously as was demonstrated in some solid state band structure calculations [29][30][31][32] .
In earlier works an effective Hamiltonian was set up in a similar way to calculate valence band structures of semiconductors and polymers [16][17][18][19]21 . Analogous schemes were developed even earlier to determine ground state correlation energies of solids as well as polymers [11][12][13]15,20 .
Common to all these efforts is that the correlation hole around a quasi particle is described efficiently in real space by means of a local correlation scheme based on Wannier type HF orbitals as a starting point. While this turned out to be very successful, one shortcoming has never been resolved, namely the problem of the long range polarization cloud. This is a specific type of correlation contribution in infinitely extended periodic systems which is not short range. Quasi particles like extra electrons or holes cause a polarization of the crystal which significantly lowers the overall energy of the system. In band structure calculations this effect was estimated to shift the HF valence and conduction bands in the order of ¥ ¤ ¦ into the fundamental gap. This effect has been estimated from parametrized continuum electrostatics so far [16][17][18][19]21,29,30 .
In this work we demonstrate how the long range polarization cloud can be incorporated into the procedure in a fully ab initio way.
We first briefly repeat the Green's function correlation method and a description of the incremental scheme in Sec. II A and II B.
In the main part Sec. II C a mathematical equality is presented to overcome the numerical obstacle of the infinite lattice summation. The quantum chemical expressions are adapted towards this formula and numerical checks are performed in Sec. III together with a first application to the band structure shifts in the LiH and LiF lattices.

A. The Green's function
The starting point of our approach are localized occupied as well as virtual HF orbitals. While we are interested in solids, the notation is kept very general for the time being. In terms of such orbitals a model space P and excitation space Q are distinguished for the example of virtual states (the case of occupied states being completely analogous) as follows: The model space P -describing the HF level -comprises of the ( § We adopt the index convention that p q 6 s r t 6 s $ u 6 w v and x ¤ 6 G y 6 G 6 w 6 w 6 w represent occupied and virtual HF orbitals, respectively. The idea of locality translates into a restriction of the area the orbital can be chosen from to one or more contiguous spatial parts of the system, as exploited systematically in the incremental scheme to be sketched below. It is important to note that by enlarging the size of the spatial area thus covered, this approximation can be checked in a systematic way for convergence.
With the above notation the Green's function takes the form where T is the time-ordering operator and the brackets denote the average over the exact groundstate. It can be obtained from Dyson's equation as: Here the self energy o R q j i which contains the correlation effects, has been introduced and l represents the unity matrix. The construction of the self energy has been described repeatedly 30? ,31 . Here we just quote the result so as to establish the link to our later formula for the long range polarization cloud.
Only the retarded self energy part is discussed, the case of the advanced part being analogous.
The space of 2-particle 1-hole states (2p1h) is spanned by where is the second order effective Hamiltonian.

B. The incremental scheme
The efficiency of the procedure is derived from the application of an incremental scheme. The task of correlating electrons in a large system is broken down systematically to diagonalizations in smaller subsystems.
As subset of the incremental scheme the unit cell serves as the smallest unit, giving rise to one-cell increments, two-cell increments and so forth.
The main idea of the incremental series (8) is to exploit the mainly local character of correlation corrections to HF results. This feature should manifest itself in a rapid decrease of increments both with the distance between the regions involved and with their number included in the increment.
This means that only a few increments need to be calculated, yet a full account of the near range correlations is achieved this way. It is crucial to emphasize that the cutoff thus introduced in the summation (8) is well controlled, since the decrease of the incremental series can be explicitly monitored.
The shortcoming of this scheme is that the long range polarization cloud cannot be assessed in this way, because it would entail an infinite lattice summation and hence infinitely many very small increments. On the other hand, a straightforward evaluation of the self energy according to the incremental scheme would include to a very vast extend excitations which do not contribute at all to the correlation effects. Only one certain class of excitations needs to be summed up to infinity as described in the next section.

C. Derivation of the implicit infinite lattice summation formulas
The kind of excitations responsible for the long range polarization are identifiable, and most recently we have found that such contributions lead to closed forms described by elliptic theta func- With the local schemes developed so far this cannot be achieved and has hence been taken into account by some continuum approximation in previous applications.
In diagrammatic notation the polarization of the crystal due to some extra charge in orbital When switching to spin-free orbital notation and focussing on a closed shell system, the dia- for the matrix element ó õ ô E ô of an effective Hamiltonian or for the matrix element of the self energy. The two formulas are related by as was discussed in previous studies 30,31 . In the following it does not matter which case is pursued and we stick to the effective Hamiltonian expression (9).
We now explicitely perform the sum over all unit cells to describe local excitations from orbital © to orbital in unit cell . Thus in the following the orbital summations only refer to the central unit cell , while the unit cell is indicated explicitely for each orbital index: Due to the degeneracy with respect to the unit cell the energy denominator can be taken out of the lattice summation which then only runs over the two-electron integrals: The index ' is a reminder that the lattice sums above are to be performed over the entire lattice and will be dropped henceforth. Inserting the representation of each local HF orbital in the basis set ( 0 ) 2 1 ú ü 0 3 5 4 3þ ô ( 7 6 8 1 (14) will transform the lattice sum as: With the two-electron integrals written out the lattice sum (16) takes the form:

Denominator Decomposition
The basic idea leading to a solution of the integral including the infinite lattice summation is to decompose the denominators into Gaussian lobes. Since the basis functions b d c are ultimately also decomposed into Gaussian lobes, the integrals can be performed analytically leaving us with a lattice sum over Gaussian type functions, which we can finally transform into special functions.
We start by representing the basis functions b e c in terms of Gaussian lobes and where c is a normalization constant and the sum in (20) The two-electron integrals are commonly broken down to the primitive Gaussian lobe level in quantum chemistry program packages such as WANNIER 39 , involving another transformation to: This leaves us with the task of evaluating and summing the product of two two-electron integrals.
This part of Eq. (25) takes the form: We now turn to the decomposition of the denominators appearing in Eq. (27,28) according to a formula analyzed by Hackbusch and Khoromskij 40 : where is a suitable integration width and can be taken from ref. 40 , where extensive studies concerning this decomposition are available, and have the explicit forms: A similar decomposition has been studied extensively by us in a previous work 33 . Further studies to the numerical accurateness of truncating expansion (29) will be presented in Sec. III.
With the abbreviations » ¼ · X ½ ¿ 3 p À ¼ ·¨½ (32) the integral over one typical expression originating from (27) together with the expansion (20) takes the form of an integral over five Gaussian lobes: is the summation index of the original infinite lattice summation. This double integral over five Gaussians is straightforward to solve. Applying formula (62) from appendix A yields a single Gaussian with respect to Ú and some lengthy prefactors as: where we used the abbreviations was used.
Checking the symmetry helps to make this formula plausible. . The result is stated in Eq.

Lattice summation
We now turn to the lattice summation contained in Eq. (38). For convenience we first confine the discussion to the one-dimensional case along the To the best of our knowledge this type of summation cannot be identified with a special function.
The problem is the shift r in the exponent, which is added to the summation index r before the square is performed. But with one special twist the summation can be transformed as follows.
The latter step is decisive, since now the summation is in a form matching a special function. In fact the elliptic theta function of the third kind r is defined as 41 : The sum in reciprocal space can thus be expressed as and we finally resolve the lattice summation (38) as: We also state the straightforward extension to simple cubic three-dimensional lattices for later use in an obvious notation: f or band index ± due to the long range polarization cloud is obtained to second order from subtracting the respective part from Eq. (12): where in the first sum of the last line the lattice summation runs over the infinite lattice and is performed as described in this work, and the second sum runs over the near range treated by the incremental scheme as in previous works. The latter sum is evaluated explicitely.
As a result the overall shift due to the long range polarization effects is isolated and can be compared with previous estimates as is done in the next section.

III. RESULTS AND DISCUSSION
While the main purpose of this work is to put forth the mathematical idea with all formulas worked out, we would also like to further exemplify our ansatz with a first numerical application.
Full scale quantum chemistry routines, however, have not been implemented for this work so far due to the tremendous programming efforts typical of ab initio codes. The approximations presented here, however, yield very encouraging results and focus on demonstrating the feasibility of the suggested implicit lattice summation in Sec. III B. First some numerical checks are presented in Sec. III A.

A. Numerical checks
In this section we check the denominator decomposition (29)  . Applying our method of denominator decomposition, subsequent transformation to reciprocal space and identification with the elliptic theta function of the third kind yields: On the other hand we know from Riemann's Zeta function 42 Instead of an infinite lattice summation one is thus left with a double sum over only a very finite number of terms in Eq. (44). This is the merit of the optimized denominator decomposition described in Eq. (29)(30)(31). Symmetry considerations and numerical cutoff criteria will eventually reduce the summation in Eq. (44) further.

B. Results for the LiH Crystal
In earlier studies we have presented ab initio studies on the band structure of LiH both with the effective Hamiltonian method 31 as well as with the Green's function approach 29  -irrespective of the basis set, i. e. assuming a complete basis -was found.
In the following we will apply the mechanism developed here to obtain this shift from first principles. LiH is chosen for various reasons. First, it is both experimentally and theoretically well studied. Secondly, the HF gap of 13 is severely reduced by correlation effects to 5 thus placing LiH between the two prototype semiconductors diamond and silicon. Last of all, it is well suited for pioneering ab initio studies due to the small size of the unit cell with 4 electrons.
We take the lattice constant H from experiment to be 4.06Å 43 .
In order to apply our scheme of infinite lattice summation, the crystal is alternatively described as simple cubic with 8 atoms at each Bravais lattice sight, rather than a fcc lattice. The unit cell thus consists of the following ions at the respective positions in units of The situation is sketched in Fig. 3. Starting from an ionic picture, the 1s orbitals both at the Li and at the H are doubly occupied. The lowest lying conduction band at the fundamental gap then arises from the 2s orbital of the Li cation. In the following we identify the index from the formulas above with this band. The presence of this extra electron at the site of U causes a polarization of the crystal which is basically due to polarizations of the extended, doubly occupied 1s cloud of the H anions. Thus the state in the notation of Fig. 2 is identified with the Li 2s orbital at the origin, and the pair describes a local excitation at a hydrogen site from the H 1s orbital ( ) to higher lying polarization functions like 2p, 3p, 3d, etc. for orbital . In order to arrive at first numerical results so as to demonstrate the proposed method, the set-up of a full quantum chemistry ab initio code is avoided here. Rather, we approximate the HF orbitals with a simple representation in terms of Gaussian lobes. Apart from the aspect of numerical savings, this might also be feasible for future studies on the ground that at long distances such an approximation might be justified. At short ranges, the contributions had been assessed with the full ab initio machinery, and the simple Gaussian lobe approximation in the short range neighborhood is subtracted from the infinite lattice summation by hand anyway.
To proceed, we performed a solid state HF calculation with a valence quadruple-zeta (VQZ) basis using the aforementioned solid state program package WANNIER 39 . This gives us the energies used in the denominator of Eq. (9). Rather then using the HF Wannier-type orbitals themselves in the transformations (15,24), we use simple lobes to mimic these orbitals as discussed above, thus saving the expensive transformations. We again stress that our goal here is to obtain first numerical results for the infinite lattice summation rather than presenting full quantum chemical ab initio calculations.
With this in mind we evaluate the polarization diagram of Fig. 2   The last column repeats the result of the continuum approximation 29  tion we thus also recover the fact that the polarization effect is much less pronounced in LiF as it is in LiH. The reason is the unusually large polarizability of the H anion.

IV. CONCLUSIONS
In conclusion we have presented a formalism which allows to include a description of the long range polarization cloud into an overall local wave function based ab initio correlation scheme for infinitely extended periodic systems. It should be emphasized that with this progress all parts of the correlation effects can be assessed within the same local orbital scheme, so that all contributions are cleanly evaluated on the same footing with double counting avoided. While the previously well established local orbital based incremental scheme was very efficient in describing the near range polarization cloud, it can now be extended to also account for the long range polarization effects for the first time.
The feasibility of the infinite lattice summation formulas were demonstrated in a first numerical application for the LiH and LiF crystals and good agreement with previous estimations was found.
We believe that this constitutes a major step ahead in the design of a general local orbital based ab initio scheme for solids.

V. ACKNOWLEDGMENTS
The author is grateful for an inspiring discussion with Boris N. Khoromskij from the Max-Planck-Institute for Mathematics in the Sciences (Leipzig/Germany). Support from the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) in the frame of the priority program SPP 1145 is also appreciated.

VI. APPENDIX A: DERIVATION OF EQ. (34,35)
We now prove that where the exponent¯takes the form We have again used the shorthand (36) for the°Å First the integral in Eq. (33) is solved with respect to Ö . Omitting ² by defining (for convenience of writing), this part of the integral takes the form: Rearranging the exponent yieldş The above abbreviations are defined as: where the definitions The definitions (56,63) were used and the result is indeed v 9 i defined in (35