Wannier interpolation of one-particle Green's functions from coupled-cluster singles and doubles (CCSD)
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Wannier interpolation of one-particle Green’s functions fromcoupled-cluster singles and doubles (CCSD)
Taichi Kosugi ∗ Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
Yu-ichiro Matsushita
Laboratory for Materials and Structures, Institute of Innovative Research,Tokyo Institute of Technology, Yokohama 226-8503, Japan (Dated: March 6, 2019)We propose two schemes for interpolation of the one-particle Green’s function (GF) calculatedwithin coupled-cluster singles and doubles (CCSD) method for a periodic system. They use Wannierorbitals for circumventing huge cost for a large number of sampled k points. One of the schemes isthe direct interpolation, which obtains the GF straightforwardly by using Fourier transformation.The other is the self-energy-mediated interpolation, which obtains the GF via the Dyson equation.We apply the schemes to a LiH chain and trans -polyacetylene and examine their validity in detail.It is demonstrated that the direct-interpolated GFs suffer from numerical artifacts stemming fromslow convergence of CCSD GFs in real space, while the self-energy-mediated interpolation providesmore physically appropriate GFs due to the localized nature of CCSD self-energies. Our schemesare also applicable to other correlated methods capable of providing GFs. I. INTRODUCTION
Although electronic-structure calculations based onthe density functional theory (DFT)[1, 2] have been suc-cessful by and large for quantitative explanations andpredictions of the properties of molecules and solids, theyare known to have a tendency to fail in describing thematerial properties even qualitatively for strongly cor-related systems. To remedy such shortcomings of DFT,various approaches have been proposed. There exist suchapproaches based on the Green’s function (GF) theory,including GW method.[3–5] They often use the non-interacting states obtained in DFT calculations as ref-erence states for the construction of interacting GFs. Onthe other hand, many sophisticated approaches based onthe wave function theory have been developed for quan-tum chemistry calculations. The coupled-cluster singlesand doubles (CCSD) method[6] is a widely accepted onesince it achieves moderate balance between its high ac-curacy and high computational cost. Not only is the re-lation between GW and CCSD methods theoretically in-teresting, but also their quantitative comparison is worthexamining[7] from a practical viewpoint.Photoelectron spectroscopy is one of the most activefields in experimental physics of today. Measurements ofthe photoelectric effects in target materials make use ofvarious kinds of techniques such as angle-resolved pho-toemission spectroscopy (ARPES) for clarifying the ma-terial properties. The measured spectra of an interact-ing electronic system are often explained under a cer-tain assumption via the one-particle GF.[8–10] The clearunderstanding of the characteristics of GFs is thus im-portant both for theoretical and practical studies in ma- ∗ [email protected] terial science. Mathematically speaking, the quasiparti-cle and satellite peaks in photoelectron spectra representnothing but the poles of one-particle GF of an interact-ing system. Particularly, the distance between the peaksclosest to zero frequency is the fundamental gap. It hasbeen demonstrated that there exists an analytically solv-able model[11] which helps to obtain transparent insightsinto interacting GFs. Meanwhile, the GFs in the contextof correlated electronic-structure calculations for uniformelectron gases[12] and realistic systems have been draw-ing attention recently[13–16], which we deal with in thepresent study.CCSD[17] and subsequent GF calculations[18–22] aredifficult especially for a periodic system due to their largecomputational cost since a sufficiently large number ofsampled k points is needed. This fact hinders one fromperforming detailed comparison between the band struc-tures obtained by a Hartree–Fock (HF) or DFT calcula-tion and the spectra obtained from CCSD GF, and themeasured spectra. Development of physically appropri-ate interpolation schemes for CCSD GFs is thus desirablefor examining spectral properties of correlated systems,which is nothing but what we do in this study.This paper is organized as follows. In Sect. II, wereview CCSD and GF calculations briefly and explainthe interpolation schemes. In Sect. III, we describe thedetails of our computation. In Sect. IV, we show theresults for the target systems. In Sect. V, our conclusionsare provided. II. METHODA. CCSD and GF for a periodic system
The CC state for a reference state | Ψ i is constructedby performing an exponentially parametrized transformas | Ψ CC i = e ˆ T | Ψ i , where ˆ T is a so-called cluster op-erator. The normalization of our CCSD wave functionsobeys the bi-variational formulation,[23–25] with whichwe calculate the CCSD one-particle GFs[18–20] in therecently proposed procedure[21, 22] as well as in our pre-vious studies.[13–15]Here we review briefly the calculation of CCSD GF fora periodic system. The GF in frequency domain is givenby G ( k , ω ) = G (h) ( k , ω ) + G (e) ( k , ω ) , (1)where G (h) pp ′ ( k , ω ) = h Ψ | (1 + ˆΛ) a † k p ω + H a k p ′ | Ψ i (2)and G (e) pp ′ ( k , ω ) = h Ψ | (1 + ˆΛ) a k p ω − H a † k p ′ | Ψ i (3)are the partial GFs from the hole and electron excita-tions, respectively. k is a wave vector and ω is a com-plex frequency. p is the composite index of a spatialorbital and a spin direction for an occupied or unoccu-pied single-electron state. For the original Hamiltonianˆ H , we defined the similarity transformed Hamiltonian H ≡ e − ˆ T ˆ He ˆ T − E measured from the CCSD total energy E . We also defined the transformed creation and anni-hilation operators a † k p = e − ˆ T ˆ a † k p e ˆ T and ¯ a k p = e − ˆ T ˆ a k p e ˆ T , respectively. ˆΛ is the parametrized de-excitation opera-tor determined in the Λ-CCSD calculation,[21, 22] whichhas to be introduced since the CCSD operator e ˆ T is notunitary.In order to avoid the computational difficulty in treat-ing the inverse matrix ( ω ± H ) − in eqs. (2) and (3),the parametrized operators ˆ X k p ( ω ) and ˆ Y k p ( ω ) are in-troduced so that[21, 22]( ω + H ) ˆ X k p ( ω ) | Ψ i = a k p | Ψ i (4)and ( ω − H ) ˆ Y k p ( ω ) | Ψ i = a † k p | Ψ i . (5)The linear equation for the non-Hermitian matrix in eq.(4) is called the ionization potential (IP) equation-of-motion (EOM) CCSD equation, while that in eq. (5) iscalled the electron affinity (EA) EOM-CCSD equation.After obtaining the parametrized operators, we use themin eqs (2) and (3) to get G (h) pp ′ ( k , ω ) = h Ψ | (1 + ˆΛ) a † k p ˆ X k p ′ ( ω ) | Ψ i (6)and G (e) pp ′ ( k , ω ) = h Ψ | (1 + ˆΛ) a k p ˆ Y k p ′ ( ω ) | Ψ i . (7) The k -resolved spectral function is defined via the GFas A ( k , ω ) = − π ImTr G ( k , ω + iδ ) (8)for a real ω with a small positive constant δ ensuringcausality. The spectral function calculated in this wayreflects our correlated approach, to be compared with theband structures obtained in mean-field-like approachessuch as HF and DFT.Before moving on to the description of our interpo-lation schemes, it is noted here that there exists an al-ternative to obtain correlated spectra or band structurefor arbitrary k points without resorting to interpolation.Specifically, usage of a large series of shifted regular k meshes enables one to perform EOM-CCSD calculationsto get the excitation energies for an arbitrarily fine k mesh, as adopted by McClain et al.[17] This approach re-quires large computational cost for the accuracy ensuredby the EOM-CCSD framework itself. B. Wannier interpolation
1. Wannier orbitals
Wannier orbitals (WOs)[26] and their variants in solidsare analogues of Foster–Boys orbitals[27, 28] in molec-ular systems. In particular, maximally localized WOs(MLWOs)[29] are widely used not only for analyses ofchemical bonds but also for accurate calculations ofanomalous Hall conductivity and transport properties.The generic expression of a WO is w R n ( r ) = 1 N k X k ,p e − i k · R ψ k p ( r ) U ( k ) pn . (9) R is the lattice point where the unit cell containing the n th WO is located. U ( k ) is a unitary matrix at k forthe construction of localized orbitals from the extendingBloch orbitals ψ k p ( r ). When the transformation matrix U ( k ) is identity at each k , the normal WOs (NWOs)[26]are obtained. When the matrices are determined so thatthe spread functional[30, 31] is minimized, on the otherhand, the MLWOs are obtained.
2. Direct interpolation
The Bloch sum of the localized orbital in eq. (9) for awave vector k is defined as w k n ( r ) = P R e i k · R w R n ( r ) , which extends over the whole crystal. The Bloch sumsof the target bands allows one to transform the CCSDGF in the band representation, which is also said to bein the Bloch gauge, to the new one in the Wannier gaugeas G nn ′ ( k , ω ) = X p,p ′ ( U ( k ) † ) np G pp ′ ( k , ω ) U ( k ) p ′ n ′ . (10)For the calculated GF at N k sampled k points in theBrillouin zone (BZ), we perform Fourier transformationas e G nn ′ ( R , ω ) = 1 N k sampled X k e − i k · R G nn ′ ( k , ω ) , (11)which is ideally equal to the exact Fourier transform G nn ′ ( R , ω ) in the limit of an infinite number of sampled k points. The real-space representation defined above en-ables us to obtain the GF for an arbitrary wave vectorvia inverse Fourier transformation as e G d nn ′ ( k , ω ) = X R e i k · R e G nn ′ ( R , ω ) , (12)which we call the direct interpolation hereafter.It is clear from eq. (8) that the spectral function e A d ( k , ω ) calculated from direct interpolation does notdepend on the matrices U ( k ) since they are unitary. Itis also clear from eq. (12) that the interpolated spectralfunction integrated over an arbitrarily fine k mesh is iden-tical to the original spectra integrated over the sampled k points: e A d ( ω ) = A ( ω ).
3. Self-energy-mediated interpolation
We cannot avoid being concerned about the reliabil-ity of e G nn ′ ( R , ω ) defined in eq. (11) since the numberof sampled k points has to be small in general due tothe large computational cost of CCSD and subsequentGF calculations. To circumvent the difficulty in increas-ing the number of sampled k points, we propose anotherinterpolation scheme for GFs here.The self-energy Σ is obtained via the Dyson equation G − ( k , ω ) = G − ( k , ω ) − Σ( k , ω ) , (13)where G is the HF GF. Substituting the CCSD GF in eq.(1) into the matrix equation above, we get the CCSD self-energy. It is noted here that the CCSD self-energy doesnot contain the contributions from the HF self-energydiagrams, which are already contained in G .[32] TheHF GF in the Bloch gauge is diagonal in reciprocal space,whose component is given by( G − ) pp ′ ( k , ω ) = ( ω − ε k p ) δ pp ′ , (14)where ε k p is the HF orbital energy.The interpolation procedure is as follows. We first cal-culate the CCSD self-energy in the Bloch gauge via eq.(13), which is then transformed into the Wannier gaugeas well as in eq. (10). We apply Fourier transformationto it using the sampled k points to get e Σ nn ′ ( R , ω ) simi-larly to eq. (11). From this real-space representation, wecan interpolate the self-energy e Σ nn ′ ( k , ω ) for an arbitrary wave vector via inverse Fourier transformation, which weplug into the Dyson equation to get the interpolated GF e G sem ( k , ω ) = [ e G − ( k , ω ) − e Σ( k , ω )] − . (15)We call this scheme the self-energy-mediated interpola-tion hereafter. Since this scheme includes inversion ofmatrices, the resultant spectral function depends on theconstruction of WOs since the unitary matrices U ( k ) de-pend on k in general.There exists an attempt for interpolating GW quasi-particle band structure using MLWOs done by Hamannand Vanderbilt.[33] Their scheme uses the GW quasi-particle wave functions and their orbital energies to getthe GW Hamiltonian in real space by adopting a man-ner computationally similar to our direct interpolation.Their formalism for efficient interpolation of correlatedband structure stems from the localized shapes of ML-WOs. The self-energy-mediated interpolation, on theother hand, relies on the localized nature of self-energies,as will be demonstrated later. It will be interesting toexamine the interpolation using the GW self-energy inthe future. III. COMPUTATIONAL DETAILS
We adopt STO-3G basis set for the CartesianGaussian-type basis functions[6] of all the elements inthe present study. The Coulomb integrals between AOsare calculated efficiently.[34] By transforming them usingthe results of the HF calculations for periodic systems,we obtain the integrals between the Bloch orbitals[35],with which we perform the CCSD calculations by suc-cessive substitution. We solve the IP-EOM-CCSD andEA-EOM-CCSD equations in eqs. (4) and (5), respec-tively, by using the shifted BiCG method.[36–38] We set δ = 0 .
02 Ht in eq. (8) throughout this study. For the con-struction of MLWOs, we calculate the overlaps betweenthe cell-periodic parts of the Bloch orbitals as input towannier90.[39]
IV. RESULTS AND DISCUSSIONA. LiH chain
1. Band structure and CCSD GF
For a LiH chain composed of equidistant atoms, wefirst optimized the lattice constant via HF calculationsusing N k = 12 × × k points. We obtainedthe optimized lattice constant a = 3 .
28 ˚A, in reasonableagreement with previous studies.[40, 41] We obtained arestricted HF (RHF) solution for this lattice constant andadopted it as the reference state for the CCSD calcula-tion. - π / a π / a ω ( e V ) k x HFMLWOs
LowHigh A ( k , ω ) FIG. 1. HF band structure of a LiH chain as circles and thatobtained with the MLWOs as curves. The spectral function A ( k , ω ) calculated from the CCSD GF at 12 sampled k pointsare also shown. The chain extends in the x direction. We constructed the MLWOs from all the 6 bands. TheMLWOs can be used for interpolation of the originalbands.[30, 31] The HF bands and their Wannier inter-polation are plotted in Fig. 1, where the original bandsare accurately reproduced. The flat valence band at ω = −
10 eV comes from the H 1 s orbital, while theconduction bands are dispersive. The CCSD spectralfunction A ( k , ω ) is also shown in the figure. We findclear correspondence between the HF band energies andthe quasiparticle peaks in the CCSD spectra. In addi-tion, low intensities exist in the CCSD spectra, knownas the satellite peaks.[13] They are direct consequencesof many-body effects taken into account by the corre-lated approach. The locations of quasiparticle peaks be-low (above) the Fermi level are closer to ω = 0 thanthose of the valence (conduction) HF band energies are,as generic characteristics of correlation effects. Since thesystem is spin unpolarized, the spectral intensities arethe same at an arbitrary k and − k due to time reversalsymmetry.
2. Direct interpolation
The spectral function e A d ( k , ω ) calculated from directinterpolation is shown in Fig. 2 (a). One finds soonthat three obviously unfavorable features exist in the in-terpolated spectra. First, the quasiparticle peaks for thehighest conduction band consist of spots separated by thedistance ∆ k x between the neighboring sampled k points.Second, there exist trains of specks at ω = 10 and 5 eV,where each speck is separated by ∆ k x again. The spec-tral intensities for some of the specks are, even worse,unphysically negative. Third, the time reversal symme-try is not preserved in the spectra, particularly for thetrains of specks.For the sampled frequencies in a range −
40 eV < - (cid:1) / a (cid:0) / a (cid:2) ( e V ) k x LowHigh A d ( k , (cid:3) ) ~ (a)(b) | R |/ a | G nn ( R , (cid:4) ) | ( a . u . ) ~ near Fermi level with NWOs × outside with NWOsnear Fermi level with MLWOsoutside with MLWOs × FIG. 2. (a) Spectral function e A d ( k , ω ) calculated from thedirect interpolation of CCSD GF for a LiH chain. (b) Theabsolute values | e G nn ( R , ω ) | of diagonal components of theGFs as functions of | R | . Those obtained using the NWOsand MLWOs for the energy region near the Fermi level ( − < ω <
22 eV) and the outside region are plotted. ω <
40 eV, the absolute values of diagonal componentsof e G ( R , ω ) in the region near the Fermi level ( −
12 eV < ω <
22 eV) and the outside region are plotted in Fig.2 (b). Although the decreasing tendencies of those val-ues for the frequencies near the Fermi level are seen forboth kinds of WOs, their convergence is slow for the in-crease in | R | . In contrast, the diagonal components forthe other frequencies decrease rapidly enough already at | R | /a = 2. These observations indicate that the sam-pled k points are too few for the direct interpolationnear the Fermi level despite the fact that the HF bandsare sufficiently convergent with respect to the k points.The unfavorable features of the direct-interpolated spec-tra enumerated above are numerical artifacts due to theinsufficient number of sampled k points.
3. Self-energy-mediated interpolation
To circumvent the direct interpolation, let us next trythe self-energy-mediated interpolation. We impose thetime reversal symmetry condition on the spectral func-tion from the self-energy-mediated interpolation as e A semTR ( k , ω ) ≡ e A sem ( k , ω ) + e A sem ( − k , ω )2 . (16)The spectral functions calculated in this way by usingthe NWOs and MLWOs are shown in Fig. 3 (a), wherethe unfavorable features for the direct interpolation donot appear. The spectra for the two kinds of NWOs arealmost indistinguishable from each other. The absolutevalues of diagonal components of e Σ( R , ω ) in the sameregions as in Fig. 2 (b) are plotted in Fig. 3 (b). Thosevalues decrease rapidly enough already at | R | /a = 1 forall the frequencies. This means that the number of sam-pled k points is sufficient for the description of the vari-ation in CCSD self-energy in reciprocal space, and hencethe self-energy-mediated interpolation of GF is reliablewithin the accuracy ensured by our preceding procedureof CCSD GF calculations.The spectral functions integrated over k points, orequivalently the densities of states, for the original CCSDGF and the interpolated GFs using the WOs are shownin Fig. 4 (a). Those for the two kinds of WOs look indis-tinguishable, in addition to which they almost coincidewith the original spectra.To see whether the self-energy-mediated interpolationusing a smaller number of sampled k points reproducesthe original spectra, we calculated the interpolated spec-tra for N k = 6 and plotted them in Fig. 4 (b). The in-terpolated spectra from N k = 12 and those from N k = 6look quite similar to each other, implying the usefulnessof our scheme for k -integrated spectra. B. trans -polyacetylene
1. Band structure and CCSD GF
For trans -polyacetylene, we adopted the structural pa-rameters provided by Teramae[42] to construct the unitcell consisting of two C atoms and two H atoms, wherethe bond alternation has occurred.[43, 44] We obtainedan RHF solution for this geometry using N k = 8 × × k points and adopted it as the reference state forthe CCSD calculations. Although it has been shown[45]that the band picture on this system is dubious by re-sorting to DFT calculations incorporating the zero-pointvibrations of atoms, we keep to the band picture sincethe main purpose of present study is to propose the in-terpolation schemes.We constructed the MLWOs from the 10 bands nearthe Fermi level. The HF bands and their Wannier inter-polation are plotted in Fig. 5, where the original bandsare accurately reproduced. The calculated band gap of8.9 eV at X ( k x = ± π/a ) is in reasonable agreement ob-tained by Teramae[46] using the same basis set. Thesecalculated gaps are much larger than the experimentalones[44, 47] of 1 - 2 eV, as is often the case with HF cal-culations. The CCSD spectral function is also shown in ω ( e V ) - π / a π / a k x (a)(b) | R |/ a | (cid:5) nn ( R , ω ) | ( a . u . ) ~ with NWOswith MLWOs - π / a π / a k x (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14) ω (cid:15)(cid:16)(cid:17) Low
H(cid:18)(cid:19)(cid:20) A T(cid:21) ( k , ω ) ~ s e m near Fermi level with NWOs × outside with NWOsnear Fermi level with MLWOsoutside with MLWOs × FIG. 3. (a) Spectral functions e A semTR ( k , ω ) calculated from theself-energy-mediated interpolation for a LiH chain by usingthe NWOs and MLWOs are shown in the upper and lowerpanels, respectively. (b) The absolute values | e Σ nn ( R , ω ) | ofdiagonal components of the self-energies as functions of | R | . the figure, where the satellite peaks for Γ ( k x = 0) havestronger intensities than for k x = 0.
2. Direct interpolation
The spectral function e A d ( k , ω ) calculated from directinterpolation is shown in Fig. 6 (a), where one finds unfa- (cid:22)(cid:23)(cid:24) (cid:25)(cid:26)(cid:27) (cid:28)(cid:29)(cid:30) ω (cid:31)!" ) with NWOs A sem ( ω ) ~ with MLWOs A sem ( ω ) ~ A ( ω ) f89 :; k pts00.40.81.2 <=> ?@A A sem ( ω ) ~ BCDE FG k pts A sem ( ω ) ~ IJKL M k pts A ( ω ) NOP QR k pts SUVWXYZ[\]^_‘abcdeghi ) ω jklm (a)(b) FIG. 4. (a) k -integrated spectral functions of a LiH chainfor the original CCSD GF at 12 sampled k points and theinterpolated GFs using the WOs. (b) The original spectra andthe self-energy-mediated interpolated ones using the NWOsfor 12 sampled k points. The latter for 6 sampled k pointsare also shown. nopqrtvwx Low yz{| ω ( }~ ) - π / a π / a k x A ( k , ω ) (cid:127)(cid:128) MLWOs
FIG. 5. HF band structure of trans -polyacetylene as circlesand that obtained with the MLWOs as curves. The spectralfunction A ( k , ω ) calculated from the CCSD GF at 8 sampled k points are also shown. The polymer extends in the x direction. a is the lattice constant. (cid:129)(cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136)(cid:137) Low (cid:138)(cid:139)(cid:140)(cid:141) ω (cid:142)(cid:143)(cid:144)(cid:145) - π / a π / a k x A d ( k , ω ) ~ (a)(b) near Fermi level with NWOs × outside with NWOsnear Fermi level with MLWOsoutside with MLWOs × | G nn ( R , ω ) | ( a . u . ) ~ | R |/ a near Fermi level with NWOs × outside with NWOsnear Fermi level with MLWOsoutside with MLWOs × FIG. 6. (a) Spectral function e A d ( k , ω ) calculated from thedirect interpolation of CCSD GF for trans -polyacetylene. (b)The absolute values | e G nn ( R , ω ) | of diagonal components ofthe GFs as functions of | R | . Those obtained using the NWOsand MLWOs for the energy region near the Fermi level ( − < ω <
33 eV) and the outside region are plotted. vorable features similarly to the case of a LiH chain. Forthe sampled frequencies in a range −
60 eV < ω <
50 eV,the absolute values of diagonal components of e G ( R , ω )in the region near the Fermi level ( −
33 eV < ω < ω < −
25 eV can be unphysically negative, as seen inFig. 6 (a).
3. Self-energy-mediated interpolation
The spectral functions e A semTR ( k , ω ) calculated via self-energy-mediated interpolation by using the NWOs andMLWOs are shown in Fig. 7 (a). Unphysical inten-sity does not appear in the interpolated spectra near theFermi level. The absolute values of diagonal componentsof e Σ( R , ω ) in the same frequency regions as in Fig. 6 (b)are plotted in Fig. 7 (b). The diagonal components nearthe Fermi level for the NWOs are large for | R | = 0 com-pared to | R | 6 = 0. This is also the case for the MLWOs. LowHigh ω (cid:146)(cid:147)(cid:148)(cid:149) - π / a π / a k x (a)(b) | R |/ a ω (cid:150)(cid:151)(cid:152)(cid:153) A T R ( k , ω ) ~ s e m with NWOswith MLWOs | (cid:154) nn ( R , ω ) | ( a . u . ) ~ near Fermi level with NWOs × outside with NWOsnear Fermi level with MLWOsoutside with MLWOs × - π / a π / a k x FIG. 7. (a) Spectral functions e A semTR ( k , ω ) calculated from theself-energy-mediated interpolation for trans -polyacetylene byusing the NWOs and MLWOs are shown in the upperand lower panels, respectively. (b) The absolute values | e Σ nn ( R , ω ) | of diagonal components of the self-energies asfunctions of | R | . On the other hand, there exist significant contributionsfrom | R | 6 = 0 for the frequencies far from the Fermi levelin contrast to the case of a LiH chain. The unphysicalintensities are thus seen for −
25 eV < ω <
60 eV at Γ,where the two kinds of WOs give slightly different spec-tra. [See Fig. 7 (a)] The spectral functions integrated over k points for theoriginal CCSD GF and the interpolated GFs using theWOs are shown in Fig. 8 (a). Those for the two kindsof WOs look indistinguishable even for ω < −
25 eV incontrast to the k -resolved spectra. [See Fig. 7 (a)] Fur-thermore, negative intensities do not appear for those fre-quencies in the k -integrated spectra. These observationsimply that accurate interpolation of k -resolved spectrarequires more sampled k points than k -integrated spec-tra do.To see whether the self-energy-mediated interpolationusing a small number of sampled k points allows one toaccess the k -integrated spectra which would be obtainedfor a larger number of k points, we calculated the inter-polated spectra for N k = 6 and plotted them in Fig. 8(b). The interpolated spectra from N k = 8 and thosefrom N k = 6 look quite similar to each other, indicativeof well converged self-energy with respect to N k . On theother hand, the peak locations of the original spectra for −
10 eV < ω <
15 eV differ slightly from those of theinterpolated spectra, implying slow convergence of theoriginal GF. These results corroborate the usefulness ofthe self-energy-mediated interpolation scheme as well asin the LiH chain case.It has been demonstrated that the self-energy-mediated interpolation is successful for our two systemsat least near the Fermi level. Our results are consistentwith the often adopted assumption that the self-energy ofan electronic system is more localized than the GF. Thedynamical mean-field theory (DMFT)[48] and its appli-cation in electronic-structure calculations[49] are basedon this assumption and have been used successfully.
V. CONCLUSIONS
We proposed two schemes for interpolation of the one-particle GF calculated within CCSD method for a peri-odic system. These schemes employ transformation ofrepresentation from reciprocal to real spaces by usingWOs for circumventing huge cost for a large numberof sampled k points. One of the schemes is the directinterpolation, which obtains the GF straightforwardlyby using Fourier transformation. The other is the self-energy-mediated interpolation, which obtains the GF viathe Dyson equation. We applied the schemes to two in-sulating systems, a LiH chain and trans -polyacetylene,and examined their validity in detail. We found thatthe direct-interpolated GFs suffered from numerical ar-tifacts stemming from slow convergence of CCSD GFsin real space. The self-energy-mediated interpolation, onthe other hand, was found to provide more physicallyappropriate GFs due to the localized nature of CCSDself-energies. We should keep in mind that in a metallicsystem, whose density matrix[50, 51] and GF[52] decayonly algebraically at a zero temperature, a large num-ber of sampled k points would be required for sufficientlyconvergent results. Remembering the widely accepted (cid:155)(cid:156)(cid:157) (cid:158)(cid:159)(cid:160) ¡¢£ ⁄¥ƒ §¤'“«‹›fifl(cid:176)–†‡·(cid:181)¶•‚„”» ) ω (eV) with NWOs A sem ( ω ) ~ with MLWOs A sem ( ω ) ~ A ( ω ) …‰(cid:190) ¿ k pts (cid:192)`´ ˆ˜¯ ˘˙¨ (cid:201)˚ A sem ( ω ) ~ ¸(cid:204)˝˛ ˇ k pts A sem ( ω ) ~ —(cid:209)(cid:210)(cid:211) (cid:212) k pts A ( ω ) (cid:213)(cid:214)(cid:215) (cid:216) k pts00.4 ω (eV) (cid:217)(cid:218)(cid:219)(cid:220)(cid:221)(cid:222)(cid:223)(cid:224)Æ(cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)ŁØŒº(cid:236)(cid:237) ) (a) (cid:238)(cid:239)(cid:240) FIG. 8. (a) k -integrated spectral functions of trans -polyacetylene for the original CCSD GF at 8 sampled k pointsand the interpolated GFs using the WOs. (b) The originalspectra and the self-energy-mediated interpolated ones usingthe NWOs for 8 sampled k points. The latter for 6 sampled k points are also shown. assumption that the self-energy of an interacting systemis more localized than the GF, the self-energy-mediatedinterpolation is expected to be more suitable for genericsystems than the direct interpolation.Since our interpolation schemes are not restricted toCCSD method, they are applicable to any correlatedmethods in quantum chemistry as long as it provides away to obtain one-particle GFs. Development of variouscorrelated methods with GFs in solids is thus importantfor reliable explanations and predictions of their spectralshapes and excitation energies. ACKNOWLEDGMENTS
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