# Transferability of self-energy correction in tight-binding basis
constructed from first principles

Manoar Hossain  Joydeep Bhattacharjee<EMAIL_ADDRESS>National
Institute of Science Education and Research, Homi Bhaba National Institute,
Jatni-752050, Odisha, India

###### Abstract

We demonstrate in this work the transferability of self-energy(“SE”)
correction(“SEC”) of Kohn-Sham(“KS”) single particle states from smaller to
larger systems, when  mapped through localized orbitals constructed from the
KS states. The approach results in a SE corrected TB framework, within which,
the mapping of SEC of TB parameters is found to be transferable from smaller
to larger systems of similar morphology, leading to a computationally
inexpensive approach for estimation of SEC in large systems with reasonably
high accuracy. The scheme has been demonstrated in insulating, semiconducting
and magnetic nanoribbons of graphene and hexagonal boron nitride, where SEC
tends to strengthen the individual $\pi$ bonds, leading to transfer of charge
from edge to bulk. Additionally in magnetic bipartite systems SEC tends to
enhance inter-sublattice spin separation. The proposed scheme thus promises to
enable estimation of SEC of band-gaps of large systems without needing to
explicitly calculate SEC of KS single particle levels which can be
computationally prohibitively expensive.

##  I Introduction

Designing new materials at nanoscale, typically consisting of few tens of
atoms, necessitates increase in accuracy of estimation of electronic structure
preferably without a commensurate increase in computational cost. Particularly
with increasing spatio-temporal resolution of synthesisXia and Halas (2005),
spectroscopic Ding  _et al._ (2016), and transport Lieber and Wang (2007)
measurements of nanostructures, it has become imperative to match measured
values to computed results in order to precisely determine atomic constitution
of samples. Accordingly, computational methodologies have been evolved Elstner
and Seifert (2014) over the years for estimation of electronic structure of
systems typically with a few hundreds of electrons, large enough to be within
the experimentally accessible length-scales, primarily at the level of Kohn-
Sham (KS) density-functional theory (DFT)Kohn and Sham (1965) Hohenberg and
Kohn (1964). As a possible approach to compute self-energy corrected
energetics of electrons in such large systems, in this work we demonstrate
bottom-up transferability of self-energy correction when incorporated in a
suitable tight-binding basis constituted from first principles.

Mean-field approximation of the Kohn-Sham (“KS”) density-functional theory
(“DFT”) Kohn and Sham (1965) Hohenberg and Kohn (1964), has established itself
as a powerful tool for calculation of electronic structures of materials from
first principles, to study ground state properties with reasonable accuracy,
primarily in systems with weak localization of electrons in the valence sub-
shells. Wannier functions Wannier (1937); Marzari and Vanderbilt (1997);
Bhattacharjee and Waghmare (2006), constructed from KS single particle states,
have been used as TB basisQian  _et al._ (2008); Gresch  _et al._ (2018); Carr
_et al._ (2019); Calzolari  _et al._ (2004); Franchini  _et al._ (2012); Jung
and MacDonald (2013) to derive model Hamiltonians to focus only on the
relevant group of orbitals. However, DFT being essentially a ground state
theory, the inherent lack of discontinuity Perdew and Levy (1983) of the
derivative of the static and local Perdew and Zunger (1981) or semi-local
Perdew  _et al._ (1996) mean-field approximations of exchange-correlation
functionals, upon addition or removal of electrons, leads to underestimation
of band-gap compared to their experimentally measured values as the difference
between the ionization potential (“IP”) and electron affinity (“EA”). As a
result, TB parameters computed from DFT often need further tuning parameters
to match experimental data, such as band-gap, particularly in systems with
increased correlation mainly due to localized electrons Cococcioni (2012)
Multitude of efforts to address these inadequacies have been pursued over last
few decades, within and beyond the framework of DFT. Improvement of the
exchange-correlation functionals either by correcting for derivative
discontinuity explicitly Chai and Chen (2013); Armiento and Kümmel (2013) , or
more popularly through incorporation of the inherently non-local nature of
many-electron interactions by deriving non-local functionalsBecke and Roussel
(1989) and partial inclusion of Hartree-Fock exact exchange in hybrid
functionals Heyd  _et al._ (2003),Becke (1993) have been reasonably successful
in addressing particularly the issue of underestimation of band-gap by DFT,
with appropriate choice of relevant parameters.

A more general parameter free approach beyond the framework of DFT, is the
many-body perturbation theory(“MBPT”) Onida  _et al._ (2002) Louie and Cohen
(2006)wherein, many-electron effects are treated as perturbation, resulting in
description of interacting electrons as quasi-particles(QP) whose energies
include corrections to the KS single-particle levels due to effective holes
associated with electrons in lieu of their interaction with other electrons.
These corrections, thereby known as self-energy (“SE”) corrections(“SEC”),
computed up to the first order, have been shownAryasetiawan and Gunnarsson
(1998) to be sufficient in accounting for the experimentally accessible SE
corrected band-gap (IP-EA). However, since both the approaches - hybrid
functionals and MBPT, are computationally expensive, MBPT being more so,
scaling typically as $N^{4}$ with system size, it poses a formidable
computational challenge to compute SE corrected band-gap till date even for
nanostructures with dimensions in single digits of nanometers, using standard
computational platforms. Notably though, considerable amount of effort and
progress has been made in recent years in reducing the computational cost by
using specialized basis setsBruneval (2016); Thygesen and Rubio (2007);
Wilhelm  _et al._ (2018); Foerster  _et al._ (2011).

In this work our approach has been to first incorporate the SEC of KS single
particle levels as corrections to TB parameters in a suitable basis and
subsequently see if such corrections derived from a smaller reference systems
can be reasonably applied to TB parameters derived for larger systems for
realistic representation of SEC in such systems without needing to explicitly
compute SEC for KS states which can be computationally prohibitively expensive
with growing system size. Estimation of QP band-gap within a TB framework has
been attempted in recent yearsSawamura  _et al._ (2017); Yu  _et al._ (2018);
Aguilera  _et al._ (2019); Cho and Berkelbach (2019), largely based on tuning
TB parameters to match the relevant QP structure. In the following we first
discuss construction of the atomic orbital basis from the KS single-particle
states, in which SEC is mapped, followed by brief description of the GW
approximation of MBPT used in this work for calculation of SEC. We demonstrate
our approach in graphene nano-ribbons(GNR), where GW approximation of SEC has
been reported in details Yang  _et al._ (2007), and also in hexagonal boron-
nitride ribbons (hBNNR) as example of wide band-gap insulator.

##  II Methodological details

The TB basis used in this work are orthonormal WannierWannier (1937); Marzari
and Vanderbilt (1997); Bhattacharjee and Waghmare (2006) orbitals constructed
as linear combination of KS energy eigen-states with a specific choice of
gauge that maximally retains their character as individual atomic orbtial.
Since in this work we consider only 2$p_{z}$ orbitals, we limit our discussion
here on generation of one orbital per atom. We begin with a template
consisting of one of the 2$p$ orbitals each of B, C, N calculated using norm-
conserving pseudo-potentials. The 2$p$ orbital is chosen to be arbitrarily one
of the lowest three degenerate set of KS states of an isolated B, C or N atom.
The full system is then decorated with such orbitals to associate one single
$2p_{z}$ orbital with each atom, aligned perpendicularly to the local plane
defined by the nearest neighbourhood. These orbitals constitute a non-
orthogonal set of localized basis from which a set of quasi-Bloch states are
constructed as:

$\tilde{\psi}_{\vec{k},j}(\vec{r})=\frac{1}{\sqrt{N}}\sum_{\vec{R}}e^{i\vec{k}\cdot\vec{R}}\phi_{\vec{R},j}(\vec{r}),$
(1)

where $\phi_{\vec{R},j}(\vec{r})$ is the $j$-th member of the non-orthogonal
basis localized in the unit-cell denoted by the lattice vector $\vec{R}$ which
spans over $N$ unit-cells that define the periodicity of the Bloch states.
Next we calculate the projection of the non-orthogonal quasi-Bloch states on
the orthonormal Bloch states constructed from the cell-periodic KS-single
particle states at all allowed $\vec{k}$, as:

$O_{\vec{k},m,j}=\langle\psi^{KS}_{\vec{k},m}\mid\tilde{\psi}_{\vec{k},j}\rangle.$
(2)

Subsequently the overlaps between the representation of the non-orthogonal
quasi-Bloch states within the manifold of of the KS single-particle states,
are calculated as:

$S_{\vec{k},m,n}=\sum_{l}O^{*}_{\vec{k},m,l}O_{\vec{k},n,l}.$ (3)

Finally, we use the Löwdin symmetric orthogonalization Löwdin (1950) scheme to
construct a new set of orthonormal Bloch states from the KS single particle
states as:

$\Psi_{\vec{k},n}(\vec{r})=\sum_{m}S^{-\frac{1}{2}}_{\vec{k},m,n}\sum_{l}O_{\vec{k},l,m}\psi^{KS}_{\vec{k},l}(\vec{r}),$
(4)

using which, a set of localized orthonormal Wannier functions are constructed
as:

$\Phi_{\vec{R^{\prime}},j}(\vec{r})=\frac{1}{\sqrt{N}}\sum_{\vec{k}}e^{-i\vec{k}\cdot\vec{R^{\prime}}}\Psi_{\vec{k},j}(\vec{r}).$
(5)

Löwdin symmetric orthogonalization thus provides a choice of gauge for linear
combination of KS states such that the resultant Wannier functions
$\left\\{\Phi_{\vec{R^{\prime}},j}(\vec{r})\right\\}$ would have minimal
deviation from the non-orthogonal orbtials
$\left\\{\phi_{\vec{R},j}(\vec{r})\right\\}$. Hence we here onwards refer
these Wannier functions as atomic Wannier orbtials (”AWO”).  TB parameters are
computed in the AWO basis as:

$\displaystyle
t_{\vec{R^{\prime}},\vec{R},i,j}=\langle\Phi_{\vec{R^{\prime}},i}\mid
H^{KS}\mid\Phi_{\vec{R},j}\rangle$ (6) $\displaystyle=$
$\displaystyle\sum_{\vec{k}}e^{i\vec{k}.(\vec{R^{\prime}}-\vec{R})}\sum_{l}(OS^{-\frac{1}{2}})^{*}_{li}(OS^{-\frac{1}{2}})_{lj}E^{KS}_{\vec{k},l}$

As obvious, the AWOs used here can in principle be substituted by any
localized atomic orbitals constructed as linear combination of KS single
particle states.

Figure 1: For the three inequivalent atoms marked in (a): (b-f): convergence
of TB parameters ($t_{ij}(r)$) in terms of the number of KS bands considered
for construction AWO(2$p_{z}$); (g-l): self energy correction to TB parameters
($\Delta t_{ij}(r)$) in AGNRs of different families of increasing width.

To estimate self-energy(”SE”) correction(”SEC”) of the KS single particle
levels we followed the GW approximationHybertsen and Louie (1986) of Hedin’s
formulationHedin (1965) Hedin and Lundqvist (1970) of the many-body
perturbation theory (”MBPT”) Onida  _et al._ (2002) Louie and Cohen (2006)to
describe single-particle excitations, wherein, many electron interactions are
represented beyond mean-field by an energy dependent, non-local and non-
Hermitian SE operator $\Sigma$, derived by considering the many-electron
effects as perturbation treated within a self-consistent framework of Dyson’s
equation in terms of the one-particle non-local Green’s functions $G$, as:

$\Sigma(\vec{r},\vec{r}^{\prime},E)=\frac{i}{2\pi}\int dE^{\prime}e^{-i\delta
E^{\prime}}G(\vec{r},\vec{r}^{\prime},E-E^{\prime})W(\vec{r},\vec{r}^{\prime},E^{\prime}),$

where $\delta=0^{+}$ and $W$ is the Coulomb interaction screened by a non-
local dynamic dielectric screening function computed approximately within the
random phase approximation as extension of its static counterpart to finite
frequencies following a generalized plasmon pole modelHybertsen and Louie
(1986). With the underlying assumption that correction to the KS single
particle states are negligible, the quasiparticle energies are approximated
as:

$E^{QP}_{\vec{k},n}=E^{KS}_{\vec{k},n}+\langle\psi^{KS}_{\vec{k},n}\mid\Sigma-V^{KS}_{xc}\mid\psi^{KS}_{\vec{k},n}\rangle$
(7)

where $V^{KS}_{xc}$ is the mean-field exchange-correlation potential derived
from the exchange-correlation functionals used in DFT. Estimation of quasi-
particle energies within GW approximation is computationally expensive
primarily due to the slow convergence of $\epsilon^{-1}$ and $G$, and
therefore of $\Sigma$, with respect to unoccupied single particle KS states.
Substituting $E^{KS}_{\vec{k},n}$ in Eqn.(6) by quasiparticle energies
$E^{QP}_{\vec{k},n}$ we calculate the SE corrected TB parameters
$\left\\{t^{QP}_{\vec{R^{\prime}},\vec{R},i,j}\right\\}$. SEC of the TB
parameters is thus estimated as:

$\Delta
t_{\vec{R^{\prime}},\vec{R},i,j}=t^{QP}_{\vec{R^{\prime}},\vec{R},i,j}-t^{KS}_{\vec{R^{\prime}},\vec{R},i,j},$
(8)

##  III Computational Details

There are two main steps in the proposed approach: (1) computation of the
ground state electronic structure followed by construction of AWOs as per
Eqn.(5) using the KS single particle states transformed as shown in Eqn.(4)
followed further by calculation of TB parameters as per Eqn.(6), (2)
calculation of SEC of KS single particle states using GW approximation and
subsequent estimation of SEC of TB parameters [$\Delta t$] from QP energies.

Ground state electronic structures are calculated using a plane-wave based
implementation of DFT Giannozzi  _et al._ (2009). Unit cells are structurally
optimized with variable cell size. Ground state energies are calculated using
norm conserving pseudo-potentials with Perdew-Zunger (LDA) exchange-
correlation Perdew and Zunger (1981) functional and plane wave energy cutoff
of 60 Rydberg. Grid of $\vec{k}$-points 1x1x15 and 1x1x29 are used for AGNRs
and ZGNRs respectively. Separation of more than 10 Angstrom is used between
periodic images of nano-ribbons. We restrict to the non-self-consistent
($G_{0}W_{0}$) level for estimation of quasi-particle energies using the
BerkeleyGW (BGW) implementation Deslippe  _et al._ (2012). Parameters for
calculation of SEC have been chosen as per Ref.33. Band-gaps have been further
converged with respect to finer $\vec{k}$ grid through interpolation based on
the AWOs. Construction of AWO, calculation of TB parameters in AWO basis, and
estimation of SEC of TB parameters are performed using our implementation
interfaced with the Quantum Espresso code.

Figure 2: With transfer of ($\Delta t$) exemplified in (a) and (b): (c-e):
comparison of band-gaps rendered by DFT,DFT+$G_{0}W_{0}$, and TB+$\Delta t$(of
p=1 & 2) for three families of AGNRs with increasing width; (f) charge density
from TB and TB+$\Delta t$. Figure 3: For the inequivalent atoms marked in (a):
(b-c): TB parameters ($t_{ij}(r)$) in ZBNNR12; (d-e): corresponding self
energy correction $\Delta t_{ij}(r)$ ; (f) comparison of band-gaps rendered by
DFT, DFT+$G_{0}W_{0}$, and TB+$\Delta t$(of ZBNNR12) for wider ZBNNRs.

##  IV Results and Discussion

In the following we primarily demonstrate our approach by accounting for SEC
of KS band-gap in wider ribbons through correction to their TB band-gap
calculated using SEC of TB parameters ($\Delta t$) obtained in a narrower
ribbon. We chose AGNRs, ZGNRs and hBNNRs in order to span a wide range of
band-gaps and magnetism with only 2$p_{z}$ electrons.

Figure 4: For the inequivalent atoms marked in (a) from edge towards bulk: TB
parameters ($t_{ij}(r)$) for spin-1(b-e) and spin-2(f-i) in ZGNRs of
increasing width; corresponding self energy corrections $\Delta t_{ij}(r)$ for
spin-1(j-m) and spin-2(n-q).

###  IV.1 Armchair Graphene Nanoribbons

We consider H-passivated AGNRs and ZGNRs of varying width wherein AGNRs are
specified by a number of the dimer lines and ZGNRs by the number of zigzag
chains. AGNRs are categorized in three different families as per the
number($n$) of dimer lines: $n=3p+0$, $n=3p+1$ and $n=3p+2$, $p$ being an
integer. In this work we consider only the 2$p_{z}$ orbitals since they are
well known to adequately describe the edge of the valence and conduction bands
in GNRs. To maximize the 2$p_{z}$ character of the corresponding AWOs,
sufficient participation of the anti-bonding orbitals are required in order to
match the weightage of bonding orbitals, which in GNRs are represented
increasingly by bands closer to the conduction and valence band edges
respectively. Fig.1(b-f) accordingly shows convergence of $t_{ij}$ for all
inequivalent atoms with respect to the total number of KS bands considered for
construction of the AWOs. The fluctuation of hopping parameters from nearest
to next-nearest and beyond, represent the favorable(unfavorable) nature of
hopping between dissimilar(similar) sub-lattices, as generic in bipartite
systems. As per Fig.1(d), we consider on the average about extra 40 unoccupied
states from the valence band for construction of AWOs.

Fig.1(g-l) shows SEC of TB parameters $\left\\{\Delta t_{ij}\right\\}$ for all
the inequivalent atoms to their neighbours, for a representative set of AGNRs
from all the three families, arranged in increasing order of width.
Fig.1(g,j,k,l) suggests that correction to nearest neighbor(n-n) hopping
reduced marginally after p=1 and convergence beyond p=2. However, Fig.1(g-i or
h-j) indicates that corrections for 3p+1 and 3p+0 are lower than that of 3p+2
for same p, consistent with the fact that AGNRs with n=3p+2 are inherently
metallic in nature with a small gap arising exclusively due to variation in TB
parameters from the edge to the bulk due to relaxation of bond lengths.
Notably, the n-n hopping term, which is between dissimilar sub-lattices, has
the most significant negative correction implying consolidation of the n-n
$\pi$-bond leading to enhanced localization of the $\pi$-bonding orbitals
between atoms due to SEC. Positive correction of further hopping term between
dissimilar sub-lattices also imply the same. Such localization all across the
system, as implied by similar correction to hopping between nearest sites for
all inequivalent atoms, would in effect result into withdrawal of charge from
edge towards bulk, as evident Fig.1(f), due to consolidation of $\pi$ bonds in
the bulk. The resultant overall increase in uniformity of charge distribution
effectively reduces mutual Couloumb repulsion between electrons of opposite
spins, leading to lowering of the on-site term due to SEC. Consistent lowering
of correction to the on-site term with increasing width indicates reduced
levels of SEC in general with increasing value of p. Correction to the hopping
between sites within the same sub-lattice, like the hopping between next
nearest sites, is negligible since it is weak in the DFT level itself.

TB Band-gap with TB parameters in AWO basis derived from KS eigen-states of a
given system, would match the KS band-gap of the same system by construction.
Similarly, with SEC of TB parameters, referred here onwards as $\Delta t$, the
correction to TB band-gap (SEC(TB)), would match the SEC of KS band-gap
(SEC(KS)) of a given system, if $\Delta t$ is obtained from SEC of KS single
particle levels of the same system, as evident in Fig.2(c-e) for p=1 and p=2.
Motivated by the overall similarity in SEC of TB parameters within each
families of AGNRs shown in Fig.1(g,j,k,l), we next test if SEC(KS) of a wider
AGNR with p $>$ 1 can be matched by SEC(TB) estimated with TB parameters
calculated from KS states of the same system (p $>$ 1), but using $\Delta t$
obtained for p=1 ($\Delta t(p=1)$) of the same family. As evident in
Fig.2(c-e), within each family, using $\Delta t(p=1)$ and $\Delta t(p=2)$, it
is possible to account for more than 80% of SEC(KS) in wider AGNRs ($p=3,4$),
with no appreciable increase in computational cost beyond computation of TB
parameters for wider ribbons. Owing to the convergence of $\Delta t$ beyond
p=2 [Fig.1(g,j,k,l)], the match between SEC(TB) and SEC(KS) is more accurate
with $\Delta t(p=2)$ than with $\Delta t(p=1)$. The scheme for assignment of
$\Delta t$ from narrower to wider AGNRs is shown in Fig.2(a-b) where the atoms
of matching colors are assigned same corrections. Correction data is collected
for each single atom of the reference system for all its neighbouring pairs
within a cutoff radius chosen to include typically up to 4th or 5th nearest
neighbour beyond which the corrections are practically negligible. The
transfer is done on the basis of two considerations: (1) matching atoms
between the reference and the target systems in terms of their neighbourhood
not limited to nearest neighbours, and (2) by maximally matching distance
between pair of atoms in the reference system to that the target system. In
mapping atoms for the criteria (1) the similarity of average nearest-neighbour
bond-lengths around atoms can be used as a reasonable criteria.

###  IV.2 hBN nanoribbons

Next we demonstrate the scheme in hBNNR, chosen as an example of wide band gap
insulator where the SEC(KS) is substantial. The difference of electro-
negativities of B and N are reflected in the difference in on-site terms in
Fig.3(b,c). $\Delta t$ plotted in Fig.3(d,e) show mild positive and strong
negative SEC for on-site terms for B and N respectively, implying enhanced
polarity of the B-N $\pi$-bond and consolidation of the lone pair of N.
Noticeably, unlike in GNRs, $\Delta t$ in all B and N atoms are very similar
among their own kind irrespective of their proximity to edges, except the ones
exactly at the edges. This is expected to enhance the degree of
transferability of $\Delta t$ across hBN ribbons systems. Fig.3(f) indeed
suggests SEC(TB) to cover more than 90% of SEC(KS) in wider ZBNNRs with
$\Delta t$ calculated in the narrowest of the hBNNRs considered.

Figure 5: (a) Comparison of band-gaps rendered by DFT,DFT+$G_{0}W_{0}$, and
TB+$\Delta t$(of Z12 & Z16) for ZGNRs of larger width (Z20,Z24); (b) charge
density from TB and TB+$\Delta t$; (c) difference of absolute values of spin-
densities($\rho_{spin}=n_{\sigma}-n_{\sigma^{\prime}})$).

###  IV.3 Zigzag edged graphene nano-ribbon

We next demonstrate the scheme in ZGNRs as an example of narrow band-gap
magnetic materials where Coulomb correlation plays a central role in
determining the electronic structure. The difference in on-site energies
[Fig.4(b-i)] of the two spins at the zigzag edges, owes to spin separation
between the two sub-lattices, which leads to localization of 2$p_{z}$
electrons of opposite spins at the two edges characteristic of ZGNRs.
Accordingly, although the C atoms in AGNRs and ZGNRs have the same local
neighbourhood, their TB parameters and their SEC are expected to be
fundamentally different since such spin-separation is completely absent in the
former.  Notably, for the C atoms at the ZGNR edges, the SEC of the on-site
terms Fig.4(j-m) shows higher negative correction for the local majority
spins, compared to those of the C atoms at the interior. This implies enhanced
presence of 2$p_{z}$ electron of one of the spins at an edge and removal of
electron of the other spin from that edge, as a result of SEC. This
enhancement in spin separation across the width of ZGNRs is evident in
Fig.5(c), while Fig.5(b) implies withdrawal of charge from edge to bulk, as
seen in AGNRs as well, due to SEC. Notably, while variation in n-n hopping
itself [Fig.4(b-i)] is small among all inequivalent C atoms in each ZGNR and
similar for all ZGNRs, the magnitude of $\Delta t$ for n-n hopping reduces
from Z12 to Z16 and converged thereafter[Fig.4(j-q)] for both spins.
Accordingly, Fig.5(b) suggests a better accounting of SEC(KS) of wider
ZGNRs(Z20,Z24) using $\Delta t$ of Z16 than that using $\Delta t$ of Z12.

Pertinently, transfer of charge from edge to bulk as seen in Fig.5(b),
accompanied by enhancement of localization of opposite spins near the edges
seen in Fig.5(c), is also observed within the Hubbard model with increasing
strength of the on-site Coulomb repulsion U, although it is clear from the
contribution of off-diagonal terms of $\Delta t$ in reproducing SEC(KS), that
DFT+U alone will not be sufficient to account for SEC. However, unlike AGNRs,
ZGNRs being magnetic systems, it is reasonable to anticipate that SEC of KS
single particle states will also have impact on the on-site Coulomb repulsion
term U in addition to TB parameters. Therefore while substituting $\Delta t$
of wider ZGNRs by that of Z12 or Z16, we need to account for a possible
underestimation of U. We therefore take recourse to the mean field
approximation of the Hubbard model and self-consistently introduce $\Delta U$
along with $\Delta t$, as

$H=\sum_{i,j,\sigma}(t_{ij}+\Delta
t_{ij})c^{\dagger}_{i\sigma}c_{j\sigma}+\sum_{i,\sigma}\Delta
Un_{i\sigma}\langle n_{i\sigma^{\prime}}\rangle,$ (9)

where $\left\\{t_{ij}\right\\}$ are computed from KS eigen-states of the wider
ZGNRs(Z20,Z24), and $\left\\{\Delta t_{ij}\right\\}$ from Z16. $\Delta U$ is
tuned to match SEC(TB) to SEC(KS) in Z20 and Z24. As evident in Fig.5(a),
indeed with application of a small U in addition to the $\Delta t(Z16)$, it is
possible to match the SE corrected KS band-gap of the wider ribbons with
modest increase in spin density near the edges, implying enhanced inter-
sublattice spin separation due to SEC, which is already hinted in Fig.5(c)
which is with U=0.

##  V Conclusions

In conclusion, we have presented a computationally inexpensive scheme for
estimation of self-energy correction(SEC) of band-gap within a tight-
binding(TB) framework in the basis of atomic Wannier orbitals (AWO)
constructed from KS energy eigen-states. Within the scheme, SEC of TB
parameters are first computed for a smaller reference system from SEC of KS
single particle levels estimated using the GW approximation of MBPT, and then
applied to TB parameters derived for a larger system of similar morphology, in
order to estimate SEC of KS band-gap of the larger system, without needing to
explicitly compute it. The efficacy of the approach, demonstrated in
semiconducting and insulating as well as magnetic and non-magnetic nano-
ribbons of graphene and hexagonal boron-nitride, is found to account for about
90% or more of the SE corrected band-gap for 50% to 100% increase in system
size as assessed in this work, with nominal increase in computational cost.
Notably, the degree of agreement[Fig.(2)(c-e),Fig.(3)(f)] between band-gaps
estimated with mapped SEC in TB basis, and those directly computed
(DFT+$G_{0}W_{0}$), clearly suggests that the scope of transferability should
easily cover further increase in system size compared to that of the reference
systems, particularly with increasing band gap, whereas, the only major
computation beyond DFT for the larger systems in our approach is the
computation of overlap matrices required for Löwding symmetrization and
calculation of TB parameters, which should scale as $\mathcal{O}(N^{2})$,
compared to overall $\mathcal{O}(N^{4})$ scaling of $GW$ approximation.  In
general, the SEC corrected TB framework opens the scope for in-depth analysis
of SEC without having to explicitly generate SE corrected KS states. The
results presented here pave the way for building up a repository of self-
energy corrected TB parameters of different atoms at different chemical
environment for their seamless use in estimation of SEC within a multi-orbital
TB framework.

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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