SNSN-323-63

Hadron Multiplicity and Fragmentation in SIDIS

Nicolas Pierre

CEA Saclay
91000 Gif-sur-Yvette, FRANCE
Institut fuer Kernphysik
Johannes-Gutenberg Universitaet Mainz, 55128 Mainz, GERMANY

> COMPASS final results on multiplicities of charged hadrons and of identified
> pions and kaons produced in the deep inelastic muon scattering off an
> isoscalar target are presented and compared to HERMES results. Measurements
> are done in bins of x, y and z in a wide kinematic range. The hadron and
> pion data show a good agreement with (N)LO QCD expectations. The kaon data
> are long awaited for since they are needed to extract kaon fragmentation
> functions, a crucial ingredient in solving the strange quark polarisation
> puzzle. COMPASS results for kaons differ from the expectations of the old
> NLO DSS fit and they cannot be well described by LO QCD either. In this
> context the importance of $K^{-}$/$K^{+}$ multiplicity ratio at high z is
> discussed.

> PRESENTED AT
>
>
>
>
> CIPANP 2018 - Thirteenth Conference on the Intersection of Particle and
> Nuclear Physics
> Indian Wells, United States of America, May 29 – June 3, 2018

##  1 Introduction : Quark Fragmentation Functions

Quark Fragmentation Functions are non-perturbative objects that are needed to
describe several processes. While light flavoured Quark Fragmentation
Functions like up quark and down quark are well determined and constrained,
the strange quark Fragmentation function is still not well known. Moreover the
largest uncertainty in the $\Delta s$ extraction from polarized SIDIS data
comes from the bad knowledge of strange quark fragmentation function into
kaons. The data available from $e^{+}e^{-}$ and $pp$ collisions are not
sufficient and at too high $Q^{2}$ to allow a good determination of the
strange quark Fragmentation Function.

In order to access the quark fragmentation functions in SIDIS, one measures
the multiplicity of hadrons $lN\rightarrow l^{\prime}hX$ where $h=\pi,K,p$. In
the expression of the multiplicities, the Parton Distribution Functions (PDFs)
and the quark Fragmentation Functions (FFs) are linked :

$\frac{dM^{h}(x,y,z)}{dz}\overset{LO}{=}\frac{\sum_{q}e^{2}_{q}q(x,Q^{2})D^{h}_{q}(z,Q^{2})}{\sum_{q}e^{2}_{q}q(x,Q^{2})},\,\,z=\frac{E_{h}}{\nu}=\frac{E_{h}}{E_{\nu}-E_{\nu^{\prime}}}$
(1)

One can access the quark FF assuming the PDF known. It has to be noted that
while the PDFs depend on $x$, the FFs depend on $z$. When the kaon
multiplicities are measured, one can constrain the product
$s(x,Q^{2}).D^{K}_{s}(z,Q^{2})$.

Thus, pion and kaon multiplicities allow to do global NLO QCD analyses to
extract quark FFs and the strangeness contained in the kaons.

##  2 Multiplicity data

Two experiments have released pion and kaon multiplicity from SIDIS :
HERMES[1] and COMPASS[2]. HERMES released pion and kaon multiplicity data on
both proton and deuteron target while COMPASS released pion and kaon
multiplicity data only on deuteron target (Fig.1 and 2).

Figure 1: From left to right, HERMES multiplicities of $\pi^{+}$, $\pi^{-}$,
$K^{+}$, $K^{-}$ versus $Q^{2}$ in a range going from 1 to 15 GeV2 in bins of
$z$. Proton data are in red dots, deuteron data in blue squares. Figure 2:
From left to right and top to bottom, COMPASS multiplicities of $\pi^{+}$,
$\pi^{-}$, $K^{+}$, $K^{-}$ versus z in bins of $x$ and staggered vertically
with $y$ with different colors. The entire set comprise more than 1200 data
points in total with a strong $z$ dependence. One can note that
$M^{\pi^{+}}\sim M^{\pi^{-}}$ and $M^{K^{+}}>M^{K^{-}}$

##  3 Extraction of Quark Fragmentation Functions from hadron Multiplicities

COMPASS has performed a Leading Order (LO) fits of pion and kaon
multiplicities in order to extract quark fragmentation from it. In the pion
case, the number of independent FFs can be reduced to 2 : $D^{\pi}_{fav}$ and
$D^{\pi}_{unf}$. In Fig.3, $D^{\pi}_{fav}$ is above $D^{\pi}_{unf}$ as
expected. It has also to be noted that COMPASS LO agrees with DSEHS and LSS
LO.

Figure 3: COMPASS LO fits assuming two independent fragmentation functions
$D^{\pi}_{fav}$ and $D^{\pi}_{unf}$. As expected,
$D^{\pi}_{fav}>D^{\pi}_{unf}$ and COMPASS LO fit agrees with the fits of DSEHS
NLO and LSS NLO.

In the kaon case, the number of independent FFs can be reduced to 3 :
$D^{K}_{fav}$, $D^{K}_{str}$ and $D^{K}_{unf}$. The LO fit is not conclusive
due to difficulties fitting high $z$ data even at NLO. These difficulties will
be discussed later. The DSS17 fit includes COMPASS kaon data and finds a
smaller $D^{K}_{str}$ than previously obtained (DSS07 was largely based on
HERMES old data). This is confirmed by a new iterative study of strange PDF
and FF with BSS[3] in order to bring more constrains to the fit. An other way
to constrain FFs is to look at the $K^{+}+K^{-}$ multiplicity sum.

##  4 Sum of z-Integrated Multiplicities

For an isoscalar deuteron target, the sum of $\pi^{+}$ and $\pi^{-}$
multiplicities integrated over $z$ reads at LO :

$\begin{split}M^{\pi^{+}+\pi^{-}}(x)&=D^{\pi}_{fav}+D^{\pi}_{unf}-\frac{2(s(x)+\overline{s}(x))(D^{\pi}_{fav}-D^{\pi}_{unf})}{5(u(x)+d(x)+\overline{u}(x)+\overline{d}(x))+2(s(x)+\overline{s}(x))}\\\
&\overset{high\,x}{=}D^{\pi}_{fav}+D^{\pi}_{unf}\end{split}$ (2)

Figure 4: From left to right, multiplicity sum of pions and hadrons. COMPASS
pion sum is significantly below HERMES ones. One can also note that there is
no x dependence for COMPASS pion sum as in EMC hadrons sum.

For this quantity, $M^{\pi^{+}+\pi^{-}}$ COMPASS data are significantly below
HERMES ones (Fig.4 left). The absence of x dependence for COMPASS pion data
follows the expectation of Eq.2. As hadrons are mostly pions, the results of
the multiplicity sum for hadrons should be comparable. EMC have done the
measurement of hadrons multiplicities. They concur with measurement of
COMPASS, namely there is no x dependence in hadron multiplicity sum hence in
pion multiplicity sum (Fig.4 right).

For an isoscalar deuteron target, the sum of $K^{+}$ and $K^{-}$
multiplicities integrated over $z$ reads at LO :

$\begin{split}M^{K^{+}+K^{-}}(x)&=\frac{(u+d+\overline{u}+\overline{d})(4D^{K}_{fav}+6D^{K}_{unf})+(s+\overline{s})(D^{K}_{str}+D^{K}_{unf})}{5(u(x)+d(x)+\overline{u}(x)+\overline{d}(x))+2(s(x)+\overline{s}(x))}\\\
&=\frac{QD^{K}_{Q}+SD^{K}_{S}}{5Q+2S}\overset{high\,x,\,S\sim
0}{=}\frac{D^{K}_{Q}}{5}\end{split}$ (3)

Figure 5: Multiplicity sum of kaons. COMPASS kaon sum are significantly above
HERMES results. One can note that the multiplicity sum is steady at low x,
thus pointing to a smaller $D^{K}_{S}$ than previous NLO fits.

COMPASS kaon data are significantly above HERMES ones (Fig.5). At high x, one
can assume that $s(x)$ is close to zero. COMPASS data point to a higher
$D^{K}_{Q}$ (paper value ¡ $D^{K}_{Q}$ ¡ paper value) than previous NLO fits
did ($D^{K}_{Q}\sim 0.43\pm 0.04$). At low x, one can assume that
$D^{K}_{str}>D^{K}_{fav}$ because the sea contribution is larger than the
valence one.

### Comments on radiative corrections

In the COMPASS paper of kaon multiplicities, TERAD was used for the inclusive
corrections viz. the corrections of the lepton line. TERAD without the elastic
and quasi-elastic tails was used for the semi-inclusive corrections viz. the
corrections of the hadron line. Since TERAD was giving z-integrated semi-
inclusive corrections it was used to set an upper limit to the correction to
apply, which was between $1\%$ and $7\%$ depending on kinematics. COMPASS
worked on a way to have a z-dependent semi-inclusive correction and is now
using DJANGOH, which accounts for this z-dependence. Comparison between
DJANGOH and TERAD were conducted. For inclusive corrections, an agreement
between DJANGOH and TERAD was reached within $3\%$, which is a really good
achievment taking into account the fact that the two frameworks use a
different renormalization scheme. Concerning the semi-inclusive corrections
for kaons, the correction is between $0\%$ and $10\%$ depending on the
kinematics with an average at $5\%$. DJANGOH corrections fall well within the
systematic errors in the publication.

##  5 Ratio of z-integrated kaon multiplicities $M^{K^{-}}/M^{K^{+}}$

The kaon multiplicity ratio $M^{K^{-}}/M^{K^{+}}$ is an interesting quantity
to observe as several experimental and theoretical uncertainties cancel
(acceptance, radiative correction etc.).

Figure 6: $R_{K}=K^{-}/K^{+}$ ratio at LO for COMPASS data compared to several
models and LO/NLO calculations.

However, when comparing the obtained $R_{K}$ with predictions[4], there is a
clear disagreement with the models and with the LO/NLO calculations (Fig.6).

In order to understand this discrepancy, one can look at the dependence
$R_{K}$ can have with other kinematic variables. For example, a clear $\nu$
dependence of $R_{K}$ can be observed : as $\nu$ increases, $R_{K}$ gets
closer to pQCD expectations.

Going further in this direction, considering now high-z kaons, the phase space
is reduced for other particles. As there are conservation laws, the missing
mass $M_{X}$ is useful to study this region :

$M_{X}\approx\sqrt{M^{2}_{p}+2M_{p}\nu(1-z)-Q^{2}(1-z)^{2}}$ (4)

The Fig.7 shows that $R_{K}$ and $M_{X}$ are highly correlated while a flat
value of $R_{K}$ is expected from pQCD calculations. It suggests that a
correction within the pQCD formalism is needed to take into account the
available phase space for target remnant hadronization.

Figure 7: Correlation plot between the missing mass $M_{X}$ and the kaon
multiplicity ratio $R_{K}$ for different z bins. The two quantities look to be
highly correlated.

HERMES and COMPASS results show the same trend in the kaon multiplicity ratio
nevertheless COMPASS data are below HERMES ones. However as seen previously,
there is a strong $\nu$ dependence of $R_{K}$ and this may contribute to
observed disagreement. A direct comparison with the only common kinematic data
point between COMPASS and HERMES data can be made and there an agreement is
reached. In the neighbouring x bins, HERMES $<\nu>$ is smaller than COMPASS
one and HERMES ratio is smaller than COMPASS one, again pointing to the same
kind of dependence described supra. Note also that target mass correction are
suggested by J.V. Guerrero and A. Accardi[5] which could explain part of the
discrepancy in kaon data.

##  6 Conclusion and Open Questions

HERMES and COMPASS offer samples of pion and kaon multiplicities measured in
SIDIS. COMPASS offers the largest sample of kaon multiplicities (over 600 data
points) to constrain Fragmentation Functions. The z-integrated multiplicity
sum shows large discrepancies between COMPASS and HERMES for both pion and
kaon multiplicity sums, going up to $25\%$ in the z-integrated
$M^{\pi^{+}}+M^{\pi^{-}}$ with COMPASS below HERMES and up to $30-40\%$ in the
z-integrated $M^{K^{+}}+M^{K^{-}}$ with COMPASS above HERMES. The kaon
multiplicity ratio $R_{K}=M^{K^{-}}/M^{K^{+}}$, measured for the first time at
high $z$ at COMPASS, shows a large disagreement with pQCD calculations at high
$z$ and low $\nu$. The correlation of $M_{X}$ and $R_{K}$ suggests that a
correction within pQCD formalism is needed to take into account the available
phase space for the target remnant hadronization.

## References

  * [1] A. Airapetian et al. (HERMES Collaboration), Multiplicities of charged pions and kaons from semi-inclusive deep-inelastic scattering by the proton and the deuteron, Phys. Rev. D 87, 074029 (2013).
  * [2] C. Adolph et al. (COMPASS Collaboration), Multiplicities of charged pions and charged hadrons from deep-inelastic scattering of muons off an isoscalar target, PLB 764 (2017) 001 C. Adolph et al. (COMPASS Collaboration), Multiplicities of charged kaons from deep-inelastic muon scattering off an isoscalar target, PLB 767 (2017) 133
  * [3] I. Borsa et al., Probing the sea quark content of the proton with one-particle-inclusive processes, Phys. Rev. D 96, 094020 (2017)
  * [4] R. Akhunzyanov et al. (COMPASS Collaboration), $K^{-}$ over $K^{+}$ multiplicity ratio for kaons produced in DIS with a large fraction of the virtual photon energy, arXiv:1802.00584 (2018) (To be published in PLB)
  * [5] Juan V. Guerrero, Alberto Accardi, Hadron Mass Effects: Kaons at HERMES vs. COMPASS, PoS QCDEV2017 (2017) 055 arXiv:1712.04571


# Self-energy corrected tight binding parameters for few $p$-block
semiconductors in the hybridized atomic orbital basis constructed from first
principles

Manoar Hossain, Joydeep Bhattacharjee  National Institute of Science Education
and Research, HBNI, Jatni, Khurda, Bhubaneswar, 752050, Odisha, India

###### Abstract

We present self-energy corrected tight-binging(TB) parameters in the basis of
the directed hybridised atomic orbitals constructed from first principles, for
nano-diamonds as well as bulk diamond and zinc blende structures made of
elements of group 13, 14 and 15 in the 2$p$, 3$p$ and 4$p$ blocks. With
increasing principal quantum number of frontier orbitals, the lowering of
self-energy corrections(SEC) to the band-gap and consequently to the dominant
inter-atomic TB parameters, is much faster in bulk than in nano-diamonds and
hence not transferable from bulks to nano-structures. However, TB parameters
transfered from smaller nano-diamonds to much larger ones exclusively through
mapping of neighbourhoods of atoms not limited to nearest neighbours, are
found to render HOMO-LUMO gaps of the larger nano-diamonds with few hundreds
of atoms in good agreement with their explicitly computed values at the DFT as
well as DFT+G0W0 levels. TB parameters and their SEC are found to vary
significantly from 2$p$ to 3$p$ block but negligibly from 3$p$ to 4$p$, while
varying rather slowly within each block, implying the possibility of transfer
of SEC across block with increasing principal quantum number. The demonstrated
easy transferability of self-energy corrected TB parameters in the hybrid
orbital basis thus promises computationally inexpensive estimation of quasi-
particle electronic structure of large finite systems with thousands of atoms.

##  I Introduction

Starting with the advent of the quantum theory of solids bloch , tight-
binding(TB) frameworks in the basis of spatially localized Wannier functions
(WF) ashcroft1976solid , have been envisioned as a generic approach to compute
electronic structure of materials. Slater-Koster (SK)
parametrizationslater1954simplified ; koster1954wave  of elements of a TB
Hamiltonian marked the beginning of reasonable band structure calculations of
solids. With the advent of the Kohn-Sham(KS) density functional theory(DFT)
hohenberg1964inhomogeneous ; kohn1965self , the mean field approximation of
KS-DFT lda1 ; gga2  and tight-binding parameters goringe1997tight ;
vogl1983semi ; tb_ordern1  determined by fitting the energy bands at the DFT
level, have been the work horse for studies of ground states properties of
materials with weak to modest localization of valence electrons for about
three decades now.

With our increasing capability to fabricate devices with ever shrinking size -
now well within double digits in nanometers, interest in accurate computation
of physical and electronic structure of such systems, which typically consist
of thousands of atoms, led to the evolution of tight-binding frameworks for
finite systems with structural inhomogeneities. Efforts to derive TB
parameters for large finite systems, particularly with $sp^{3}$ hybridization
of orbitals, have been reported not only in the orthogonal $sp^{3}$ basis
tserbak1993unified ; sutton1988tight ; goodwin1989generating ; wang1993tight ;
mercer1994tight ; kwon1994transferable  but also more generalizably in the
non-orthogonal basis as well menon1993nonorthogonal ; frauenheim1995density ;
bernstein1997nonorthogonal  and used extensively in molecular dynamics
simulation wang1993tight ; sankey1989ab  and calculation of optical properties
delerue2000excitonic ; proot1992electronic ; delerue1993theoretical ;
niquet2000method ; trani2005tight . Evolution of computational techniques to
construct spatially localized Wannier functions (WF) marzari1997maximally ;
jbwan , opened up the scope for construction of realistic localized orbitals
from KS single particle states, qian2008quasiatomic ; mostofi2008wannier90 ,
leading to explicit computation of realistic TB parameters from first
principles.

With improved spectroscopic measurements enhanced correlation due to spatial
confinement near surfaces and interfaces of nano-structures became apparent
which made it imperative to take the frameworks of computation of electronic
structure beyond the mean field regime leading to formidable increase in
computational complexity and cost. While much effort in this direction has
been underway in terms of improving the exchange-correlation functionals
heyd2003hybrid  to include effects of many-electron interactions within the KS
energy eigen-spectrum, the GW approximation hybertsen1986electron  of the many
body perturbation theory (MBPT) hedin1965new ; hedin1970effects  is so far the
most general abinitio framework to unmissably account for correlations
starting from the KS single particle states. However, both the approaches are
computationally expensive and almost prohibitive for systems with thousands of
atoms. These limitations have led to attempts to incorporate effects of
correlations within the tight-binding(TB) frameworks. Efforts reported in
recent years mostly improve TB parameters qptb1 ; qptb7 ; qptb6 ; qptb5 ;
qptb4  by matching QP structure in specific systems and orbital subspaces.

In this work we calculate self-energy correction(SEC) to TB parameters from
the SEC of the KS energy eigenstates at the DFT+G0W0 level. With tetrahedrally
coordinated atoms exclusively considered in this work except the passivating H
atoms at the surface of the nano-diamonds, TB parameters are calculated in the
orthonormal basis of Wannier functions(WF) constructed following a realistic
template of $sp^{3}$ hybridized atomic orbitals(HAO) obtained from KS single
particle states of isolated atoms. The resultant WFs resemble hybrid atomic
orbitals and orient towards directions of coordination by construction, and
thereby locked to the immediate neighbourhoods of atoms. The aim of this work
is to present a comparative analysis of TB parameters and their SEC in such
directed hybrid orbital basis beyond the nearest neighbourhood for bulk and
finite structures made of a representative variety of 2$p$, 3$p$ and $4p$
block elements which exists in diamond or zinc-blend structures in three
dimensions. We also demonstrate that such TB parameters can be transferred
from smaller nano-diamonds to their much larger iso-structural counterparts
exclusively through mapping of neighbourhoods typically up to 2nd or third
nearest neighbours, to render HOMO-LUMO gaps of the larger systems in good
agreement with their explicitly calculated values at the DFT and DFT+G0W0
level.

Figure 1: Nano-diamonds with projected charge centres of HAOs (pink spheres):
(a) $C_{26}H_{32}$, (b) $C_{281}H_{172}$ and (c) $C_{322}H_{156}$. Charge
centres of H coincides with H.

##  II Methodological details

###  II.1 Construction of hybrid atomic orbitals(HAO) and their transfer to
system of atoms

Hybrid orbital basis (HAO) are constructed for isolated atoms as the
approximate eigenstates of the non-commuting set of three first moment
matrices (FMM) $X,Y$ and $Z$ projected in a finite subspace of orthonormal
basis states. FMMs calculated as:

$(X,Y,Z)_{ij}=\langle\phi_{i}\mid(x,y,z)\mid\phi_{j}\rangle.$ (1)

are maximally joint diagonalized using an iterative scheme based on the Jacobi
method of matrix diagonalization where off-diagonal elements are minimized
through rotation of coordinates by a specific analytic choice of angles as
detailed in ref.hossain2021hybrid . HAOs are naturally oriented towards
direction of coordinations around atom as per the set of basis states
considered for construction of the HAOs. The charge centres of HAOs thus
render coordination polyhedra around an atom for a given sub-shell. For
construction of the $sp^{3}$ HAOs used in this work, the lowest four KS states
of a single atom are used as basis. Although in principle, any localized
atomic orbitals like the pseudo-atomic orbitals or the Slater or Gaussian type
orbitals parametrized for the given atoms can also be used for the purpose, it
is important to have appropriate length-scale of localization of the HAOs for
their correct representation within the KS states of the given system of atoms
where the HAOs are to be used as template to construct WFs. Same pseudo-
potential thus preferably be used for generation of HAOs and also for
calculation of electronic structure of the given system.

Separate sets of HAOs are constructed for each types of atoms, and transfered
to the corresponding atoms in the given system through mapping of directions
of coordination from the atoms to their nearest neighbours on to the
directions of charge centres of HAOs from the isolated atom for which they are
explicitly constructed. Through such mapping a distribution of projected
charge centres of HAOs are generated for the entire system prior to the
transfer of HAOs, as seen in Fig.1. For perfect tetrahedral coordination, as
seen in the bulk diamond and zinc blende structures, the transfered HAOs would
thus retain intra-atomic orthonormality, whereas for the nano-diamonds such
intra-atomic orthogonality will be marginally compromised owing to deviations
of perfect tetrahedral coordination more towards the surface. However this is
not a problem as such not only because the deviations are minimal but because
all transfered HAOs are symmetrically orthonormalized during the construction
of the WFs.

Figure 2: HAWOs in bulk (a)C (b)Si and (c)Ge. Figure 3: TB and self-energy
correction to TB parameters in (a) carbon, (b) silicon and (c) germanium nano-
diamonds from adamantane to pentamantane. Figure 4: TB and self-energy
correction to TB parameters of hybrid SiC and SiGe nano-diamonds(pentamantane)
along with those of C, Si and Ge nano-diamonds(pentamantane).

###  II.2 Wannier functions based on HAOs

HAOs transferred from isolated atoms to the system of atoms constitute a non-
orthogonal basis primarily due to their inter-atomic non-orthogonality. A set
of quasi-Bloch states are subsequently constructed as:

$\tilde{\psi}_{\vec{k},j}(\vec{r})=\frac{1}{\sqrt{N}}\sum_{\vec{R}}e^{i\vec{k}\cdot\vec{R}}\phi_{\vec{R},j}(\vec{r}),$
(2)

where $\phi_{\vec{R},j}(\vec{r})$ is the $j$-th HAO localized in the unit-cell
denoted by the lattice vector $\vec{R}$ which spans over $N$ unit-cells as per
the the Born-von Karman periodicity. The quasi-Bloch states are projected on
the orthonormal Bloch states constructed from the KS single-particle states at
all allowed $\vec{k}$:

$O_{\vec{k},m,j}=\langle\psi^{KS}_{\vec{k},m}\mid\tilde{\psi}_{\vec{k},j}\rangle.$
(3)

The subspace of KS states is of size same as the number of total number of
HAOs transfered to the system, which is twice the number of occupied states in
our case. Overlaps between the non-orthogonal quasi-Bloch states are
calculated within the manifold of the considered KS states as:

$S_{\vec{k},m,n}=\sum_{l}O^{*}_{\vec{k},l,m}O_{\vec{k},l,n}.$ (4)

Figure 5: Variation of HOMO-LUMO gap computed with TB and SEC-TB parameters
mapped to un-relaxed structures from relaxed pentamantane structure, and the
same computed from first principles at the DFT and DFT+G0W0 level with relaxed
structure, as function of size of carbon nano-diamonds increasing in (a)
pyramidal and (b) bi-pyramidal structures. Comparison of DOS of C68H56
calculated from (c) TB mapped to un-relaxed structures and DFT of relaxed
structure, and (d) SEC-TB mapped to un-relaxed structure and DFT+G0W0 of
relaxed structure . Figure 6: Variation of HOMO-LUMO gap of nanodiamonds from
adamantane to pentamantane made of (a) carbon, (b) silicon and (c) germanium.
Bulk band gaps are shown in dotted lines.

Representability of an HAO within the set of KS states considered, is ensured
by setting a lower cutoff on individual $S_{\vec{k},n,n}$ values for all HAO
index $n$, which we typically set at 0.85. In all the systems considered in
this work, such a cutoff is found to be well satisfied for all HAOs within the
set of KS states whose count is set to the total number of HAOs of all the
atoms in the given system. Using the Löwdin symmetric orthogonalization
lowdin1950non  scheme a new set of orthonormal Bloch states are constructed
from the KS single particle states as:

$\Psi_{\vec{k},n}(\vec{r})=\sum_{m}S^{-\frac{1}{2}}_{\vec{k},m,n}\sum_{l}O_{\vec{k},l,m}\psi^{KS}_{\vec{k},l}(\vec{r}),$
(5)

where $l$ spans over the KS states and the sum over $m$ ensures
orthonormalization. Finally a orthonormal set of localized Wannier functions
referred here onwards in this paper as the hybrid atomic Wannier orbitals
(HAWO), are constructed as:

$\Phi_{\vec{R^{\prime}},j}(\vec{r})=\frac{1}{\sqrt{N}}\sum_{\vec{k}}e^{-i\vec{k}\cdot\vec{R^{\prime}}}\Psi_{\vec{k},j}(\vec{r}).$
(6)

The resultant HAWO resemble the corresponding HAOs
$[\left\\{\phi_{\vec{R},j}(\vec{r})\right\\}]$ as seen in Fig.2. Notably, the
number of KS states which can be considered for construction of HAWOs are not
limited by the total number of HAOs transfered to the system. In fact the
localization of the HAWOs increase with increasing size of the KS subspace
used. However in such a case the matrix $(OS^{-\frac{1}{2}})$ would become
semi-unitary. TB parameters are computed in the HAWO basis from energetics of
KS single particle states as:

$\displaystyle
t_{\vec{R^{\prime}},\vec{R},i,j}=\langle\Phi_{\vec{R^{\prime}},i}\mid
H^{KS}\mid\Phi_{\vec{R},j}\rangle$ (7) $\displaystyle=$
$\displaystyle\sum_{\vec{k}}e^{i\vec{k}.(\vec{R^{\prime}}-\vec{R})}\sum_{l}(OS^{-\frac{1}{2}})^{*}_{li}(OS^{-\frac{1}{2}})_{lj}E^{KS}_{\vec{k},l}.$

Notably, although TB parameters have been derived from first principles based
on the maximally localized Wannier function marzari2012maximally ; lee2005band
; atomicorb2 ; calzolari2004ab ; franchini2012maximally ; jung2013tight  or
atomic orbitals atomicorb1 ; qian2010calculating ; d2016accurate ;
agapito2016accurate  constructed from KS states over the years, attempts to
calculate TB parameters in the basis of directed hybrid orbital has been
primarily limited so far to analytical modelsyue2017thermal ;
popov2019deductive . Self-energy corrected TB parameters (SEC-TB)
$\left\\{t^{QP}_{\vec{R^{\prime}},\vec{R},i,j}\right\\}$ in the HAWO basis are
calculated by substituting $E^{KS}_{\vec{k},n}$ in Eqn.(7) by quasi-particle
energies $E^{QP}_{\vec{k},n}$ obtained at the $G_{0}W_{0}$ level of the GW
approximation of MBPThedin1965new ; hybertsen1986electron . Efforts have been
reported in recent years on incorporating SEC in TB parameters computed in
terms of the MLWFsqptb3 ; qptb5 ; qptb6 .

Figure 7: (a-p) TB parameters and their self-energy corrections, and (q)
summary of onsite and dominant inter-atomic terms, for 2$p$, 3$p$ and 4$p$
block elements in diamond and zinc blende structures.

###  II.3 Bottom-up mapping of TB parameters

The orientation of HAWOs being locked to the local atomic neighbourhood of
each atom, the multi-orbital TB parameters derive in HAWO basis are easily
mappable from one system to another with matching atomic neighbourhoods.
Transferability of the TB parameters in terms of reproducibility of band-gaps
and band-width within acceptable ranges of deviation from their direct
estimates from first principles, is demonstrate by transferring TB parameters
from smaller reference systems to structurally similar larger target systems.
Transfer of TB parameters is done in two steps.
First all individual pairs of atoms of the target system beyond the nearest
neighbourhood, are mapped on to matching pairs of atoms in the reference
system through comparison of a collection of structural parameters like
spatial separation, angles and dihedral angles subtended by the nearest
neighbours and a measure ($\zeta$) of proximity of atoms with respect to
surfaces, interfaces and inhomogeneities defined as:

$\zeta_{i}=\sum^{N_{i}}_{j}Z_{j}w(r_{i,j})$ (8)

$N_{i}$ being the number of neighbours of the $i$-th atom within a chosen
cutoff radius, and $w$ being a weight factor which is a function of the
distance $r_{i,j}$ of the $j$-th neighbour of the $i$-th atom, and $Z_{i}$
being a characteristic number to be associated with each type of atom, for
example, the atomic weight.

For mapping of orbital pairs from reference to target, a projected map
[Fig.1(b-c)] of locations of charge centres of HAOs for the target system is
generated first and then among the mapped pair of atoms, pair of HAOs of
system are mapped to pair of HAOs of the reference through mapping of their
respective charge centres much like the mapping of atom pairs. The weight
factor $w$ is typically chosen to be 1.0 within half of the cutoff radius and
smoothly reduced to zero at the cutoff radius using a cosine function. Cutoff
radius is chosen based on the size of the reference system, since it should
neither be too large for variations in $Z_{i}$ to average out, nor should it
be too small such that $\zeta$ varies abruptly only at the structural
inhomogeneities. Cutoff radius and $Z_{i}$ values should thus be so chosen
that $\zeta$ values can effectively differentiate between atoms at the
interior of a finite system from those at the surfaces in both reference and
target systems such that relaxed structures of target systems are not required
to adequately transfer effects of variation in bond lengths near the surfaces
from reference to target systems.

##  III Computational details

All the ground state geometries as well as ground state electronic structures
are calculated using the QUANTUM ESPRESSO (QE) code giannozzi2009quantum
which is a plane wave based implementation of density functional theory (DFT).
BFGS scheme has been used to obtain the relaxed structures. For bulk systems
variable cell relaxation has been performed to optimize lattice parameters and
ionic positions. A separation of at least 10 Å between periodic images has
been ensured for all the finite systems. The KS ground state properties are
calculated within the Perdew-Zunger approximation of the exchange-correlation
functional implemented in a norm-conserving pseudo-potential. Plane wave basis
with kinetic energy cutoff in excess of 60 Rydberg along with a 15x15x15
Monkhorst-Pack grid of k-points have been used for all the bulk systems. The
self-energy correction to KS energy eigenvalues are calculated at the G0W0
level of GW approximation of the many-body perturbation theory (MBPT)
implemented in the BerkeleyGW (BGW) codedeslippe2012berkeleygw . To calculate
the static dielectric matrix and extend to the finite frequencies the
generalized plasmon-pole model hybertsen1986electron  is used. For convergence
of dielectric function as well as the self-energy term we used in excess of
250(5000) bands for bulks(nano-diamonds). To get a well converged band-gap for
bulk structures we used a finner k-mesh of 30x30x30 grid. Finally, in-house
implementation interfaced with the QE code is used for generation of HAOs,
HAWOs from KS states, calculation of TB parameters in the HAWO basis, and
mapping of TB parameters from smaller reference systems to larger target
systems.

##  IV Results and discussion

Nano-diamonds starting with adamantane followed by diamantane, triamantane,
tetramantane with increasing number of atoms up to pentamantane have been
considered for explicit calculation of TB parameters at the DFT level and
their further refinement due to SEC of KS energy eigenvalues at the G0W0
level. As evident in Fig.3 the dominant nearest neighbour(nn) TB parameters as
well as those beyond the nearest neighbourhood, remains effectively invariant
with respect to system size, implying transferability of TB parameters across
length-scales of nano-diamonds at the DFT level as we demonstrate in this
paper. Magnitudes of TB parameters in Si and Ge are similar but vary
significantly from that of C, which is consistent with variation in their
inter-atomic separations. Notably however, the SEC to the dominant TB
parameters reduces slowly with increasing system size in a similar rate for C,
Si and Ge based nano-diamonds, although the magnitude of correction itself
reduces significantly with increasing principal quantum number owing to
increased de-localization. The slow variation of SEC across system size also
suggests possibility of SEC corrected TB parameters to be transferable as well
to a workable degree.

Fig.4 clearly suggests that TB parameters of either Si or Ge can be used for
SiGe nano-diamond as well, whereas for the SiC nano-diamonds the dominant
interatomic TB parameters are effectively the average of that of the C and Si
based nano-diamonds. These observations hint at increased transferability of
SEC across elements in a block with increasing quantum number of frontier
orbitals. The reorganization of the on-site terms of SiC nano-diamond compared
to those made of Si or C can be attributed to the hetero-polarity of the Si-C
nn bond due to difference of electro-negativities of Si and C, whereas
electro-negativities of Si and Ge are similar and less than that of C.

To study transferability of TB parameters from smaller nano-diamonds to their
larger counterparts we have mapped TB parameters from pentamentane to larger
pyramidal and bi-pyramidal carbon based nano-diamonds with close to 1000
atoms. We have considered un-relaxed coordinates of the larger nano-diamonds
and mapped the TB parameters calculated for relaxed structure of pentamentane,
using $\zeta$ values calculated with a cutoff radius considered up to the next
nearest neighbour. As evident in Fig.5(a,b), the match of the DFT HOMO-LUMO
gap with that calculated from the transferred TB parameters from pentamentane
is found to be quite exact for both pyramidal as well as bi-pyramidal nano-
diamonds up to about 500 atoms, which is almost about ten times escalation of
size of the target system compared to that of the reference system. Notably,
for all the larger nano-diamonds the DFT HOMO-LUMO gaps were calculated
explicitly with well relaxed structures, whereas the structures used to
transfer TB parameters from pentamentane were completely un-relaxed. Therefore
the match of the exact DFT HOMO-LUMO gaps with that evaluated with TB
parameters transferred to an un-relaxed structure point to the efficacy of
$\zeta$ in differentiating atoms as per their proximity to the surface in both
reference and target systems so that the transferred TB parameters have the
correct variation from surface to interior. The match of the quasi-particle
(QP) HOMO-LUMO gap is also found quite satisfactory up to more than two time
escalation of system size beyond pentamentane, till which G0W0 calculations
could be reliably performed with computational facilities at our end. QP HOMO-
LUMO gap data available in the literature raty2005optical ; raty2003quantum ;
drummond2005electron ; sasagawa2008route  allows us to compare for a bit
further. These results thus confirms transferability of TB parameters with and
without SEC over substantial escalation of system size.

Figure 8: Comparison of band-structures computed from first principles with
that obtained with TB parameters derived from first principles at the DFT
(upper panel) and DFT+G0W0(lower panel) levels for 2$p$, 3$p$ and 4$p$ block
elements.

Interatomic TB parameters for bulk Si and Ge in diamond structures are
consistent with those in nano-diamonds notwithstanding the substantial drop of
band-gap in bulk to about 0.5 eV(1 eV) from HOMO-LUMO gap around 4 eV(7 eV) in
nano-diamonds at the DFT(DFT+G0W0 ) level as shown in Fig.6(b,c). However for
bulk C in diamond structure the dominant nn hopping term strengthens
substantially to -10 eV (Fig.7(a)) from -6.5 eV (Fig.3(a) red plots) in nano-
diamonds, although the band-gap in bulk is comparable to that of HOMO-LUMO gap
of nano-diamonds, which are about 4.2 eV and 5 eV respectively at the DFT
level. The reason of this contrasting trend for C compared to those for Si and
Ge can be traced to one obvious fact that $sp^{3}$ HAWOs in Si and Ge are more
de-localized than that of C owing to increase in principal quantum number, but
in addition to that, the C-C nn separation clearly decreases in bulk from
nano-diamonds by about 2% to 3 %, as has also been reported
saani2007comparison  earlier, whereas the Si-Si and Ge-Ge nn separations
remain largely same in bulk and nano-diamonds. Furthermore, as evident in
Fig.2, the major lobe (positive one in this case) of the $sp^{3}$ HAWOs of C
is stretched towards the nearest C compared to the HAWOs of Si and Ge. Overlap
of such $sp^{3}$ HAWOs of C would thus increase appreciably with reduction in
C-C nn separation from nano-diamonds to bulk, leading to the increase in
magnitude of nn hopping term. Thus the TB parameters are not transferable from
bulk to nano-diamond for C even at the DFT level. However it may still be
feasible for Si and Ge.

Consistent with the localized nature of HAWOs, SEC is higher for C based bulk
and nano-diamonds compared to their Si and Ge based counterparts. However the
relative variation of localization of HAWOs of C, Si and Ge are much dominated
over by the localization imposed by the finiteness of nano-diamonds, leading
to comparable values of SEC of band-gap of C, Si and Ge based nano-diamonds
ranging from about 4 eV to 3 eV leading to comparable SEC correction to nn
hopping term ranging from -1.75 eV to -1.3 eV. From nano-diamonds to bulk the
SEC to band gap and thereby to the nn hopping term reduces consistently on
account of de-localization. Thus the self-energy corrected TB parameters are
not transferable from bulk to nano-diamonds even for Si and Ge.

As evident in Fig.7(a,c,e,g,i,k), the nn hopping parameter is almost exactly
same for diamond structures of group 14 (C,Si,Ge) and zinc blende structures
made of groups 13 and 15 (BN,AlP,GaAs) within each block, whereas the SEC to
the nn (Fig.7(b,d,f,h,j,l)) hopping are consistently more for zinc blend
structures than the diamond structures in each block owing likely to the
enhanced localization of the bonding orbitals in the heteropolar B-N, Al-P and
Ga-As nn bonds than in the C-C, Si-Si, and Ge-Ge bonds. Fig.7(q) summarizes
the variation of the onsite terms and the major inter-atomic hopping
parameters at the DFT and DFT+G0W0 level. As evident in Fig.7(m,o,n,p,q), for
zinc blend structure of SiC(SiGe) the nn hopping and it’s SEC both are average
of their counterparts in diamond structures of Si and C(Ge). Fig.7(q) clearly
suggests the stepped evolution of TB parameters and SEC from 2$p$ to 3$p$ and
4$p$ blocks and the similarity in the later two. Band structures obtained with
TB parameters shown in Fig.8 are plotted along with DFT band structure.
Notably, for the exact reproduction of band structure the TB parameters have
to be obtained not only in a dense enough grid of $\vec{k}$ but also using a
KS band subspace of size exactly same as the number of HAOs transfered for the
entire unit-cell in order to maintain the unitary nature of the matrix
$(OS^{-\frac{1}{2}})$. If we use a larger KS subspace then the reproduction
energy bands will become increasingly inexact with increasing energy.

##  V Conclusions

For a representative variety of $p$ block elements, we have presented TB
parameters calculated at the level of DFT and DFT+G0W0 in the orthonormal
hybridized atomic Wannier orbitals(HAWO) basis constituted of the KS single
particles and are directed towards coordination by construction. We present TB
parameters for nano-diamonds made of C, Si, Ge, SiC and SiGe and their bulks
in diamond or zinc blende structures, and also of bulk BN, AlP and GaAs in the
three consecutive $p$ blocks. Transferability of inter-atomic TB parameters
between bulk structures to nano-diamonds is generally poor and worsens with
self-energy correction(SEC). However among nano-diamonds, the inter-atomic TB
parameters at the DFT level are found to remains effectively unaltered with
increasing size, implying robust transferability of the TB parameters from
smaller to larger nano-diamonds as demonstrated. Slow reduction of SEC to TB
parameters with increasing system size also implies good transferability of
self-energy corrected TB parameters across nano-diamonds. Transfer of TB
parameters are performed exclusively by mapping up to second or third nearest
neighbourhoods between the reference and the target systems, and thus does not
necessitates relaxed geometries of the target systems as long as all
neighbourhoods could be mapped. Similarity of TB parameters and their self-
energy corrections across 3$p$ and 4$p$ blocks compared to that in the 2$p$
block hints at the possible transferability of SEC across blocks with
increasing principal quantum number.

##  VI Acknowledgments

Computations have been performed in computing clusters supported by the
Nanomission of the Dept. of Sci.& Tech. and Dept. of Atomic Energy of the
Govt. of India.

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code programmed in high-level logic, will be compiled into byte code and
executed in the virtual machine of miners. Miners will charge additional
crytocurrency payments called gas costs, because the extra computational tasks
incurred by smart contracts will be broadcast throughout the blockchain. Smart
contracts are implemented in a high-level programming language, such as
Solidity (Language, 2020).

It is worth noting that Bitcoin and Ethereum were only supposed to enable
decentralization, but do not ensure privacy. In fact, the transaction
histories of many crytocurrencies are visible to the public. There are certain
high-profiled prosecution of darknet operators based on the evidence of
Bitcoin transactions. Supporting privacy in blockchain is a crucial on-going
research topic.

###  E.2. Smart Contract Implementation

We next explain how Multi-Signature smart contract can achieve the step (5)
and (6) of the stage Cost-sharing Payment by the following methods:

  1. (1)

submitTransaction(). This method allows each user to submit the ${\tt
zkpSum}[{\tt Cost}_{\tt ess},(P_{i})_{i=1}^{N}]$ that they have agreed upon
off the chain. The method will compare whether users have submitted the same
${\tt zkpSum}$.

  2. (2)

confirmTransaction(). On one hand, this method allows each user to confirm
that the stored ${\tt zkpSum}$ in the smart contract is the one that they have
agreed upon off the chain. On the other, each user is required to submit a
${\tt nzkpNN}[{\tt Bal}({\tt ad}_{i})-P_{i}]_{i=1}^{N}$, which will be
validated to prove that there is sufficient balance in his account to pay for
the energy cost.

  3. (3)

executeTransaction(). This method can only be executed by the operator unless
all the users have already confirmed the transaction. The method will validate
the ${\tt zkpSum}$ before calling ESToken smart contract to credit ${\tt
Cost}_{\tt ess}$ to the operator’s account and debit the corresponding payment
from each user’s account.

  4. (4)

secretlyJointTransfer(). This method, defined within the EStoken smart
contract is invoked by executeTransaction(), which actually performs the real
transfer between multiple accounts.