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Local perturbations of periodic systems. Chemisorption and point defects by GoGreenGo.

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Local perturbations of periodic systems. Chemisorption and point defects by GoGreenGo.

Abstract

We present a software package GoGreenGo -- aimed to model local perturbations of periodic systems due to either chemisorption or point defects. The electronic structure of an ideal crystal is obtained by worldwide distributed standard quantum physics/chemistry codes, then processed by various tools performing projection to atomic orbital basis sets. Starting from this, the perturbation is addressed by GoGreenGo with use of the Green's functions formalism, which allows to evaluate its effect on the electronic structure, density matrix and energy of the system. In the present contribution the main accent is made on processes of chemical nature such as chemisorption or doping. We address a general theory and its computational implementation supported by a series of test calculations for benchmark model solids: simple, face-centered and body-centered cubium systems. In addition, more realistic problems of local perturbations in graphene lattice such as lattice substitution, vacancy and "on-top" chemisorption are considered.
SOFTWARE NOTE
Local perturbations of periodic systems. Chemisorption and
point defects by GoGreenGo
Ilya V. Popov | Timofei S. Kushnir | Andrei L. Tchougréeff
A.N. Frumkin Institute of Physical Chemistry
and Electrochemistry RAS, Moscow, Russia
Correspondence
Andrei L. Tchougréeff, A.N. Frumkin Institute
of Physical Chemistry and Electrochemistry
RAS, Moscow, Russia.
Email: andrei.tchougreeff@ac.rwth-aachen.de
Funding information
State Task, Grant/Award Number:
0081-2019-0018
Abstract
We present a software package GOGREENGOan overlay aimed to model local pertur-
bations of periodic systems due to either chemisorption or point defects. The elec-
tronic structure of an ideal crystal is obtained by worldwide-distributed standard
quantum physics/chemistry codes, and then processed by various tools performing
projection to atomic orbital basis sets. Starting from this, the perturbation is
addressed by GOGREENGOwith use of the Green's functions formalism, which allows
evaluating its effect on the electronic structure, density matrix, and energy of the
system. In the present contribution, the main accent is made on processes of chemi-
cal nature, such as chemisorption or doping. We address a general theory and its
computational implementation supported by a series of test calculations of the elec-
tronic structure perturbations for benchmark model solids: simple, face-centered, and
body-centered cubium systems. In addition, more realistic problems of local perturba-
tions in graphene lattice, such as lattice substitution, vacancy, and on-topchemi-
sorption, are considered.
KEYWORDS
chemisorption, electronic structure, Green's functions, point defects in crystals, substitutional
defects
1|INTRODUCTION
A great number of elementary processes of significant importance in
solid-state chemistry are local in nature,
1
although they take place in a
matrixof crystal (periodic) system. Currently, local processes/effects
in solids are addressed by using standard computational chemistry
codes pushing towards the limits of their applicability with a consider-
able loss of efficiency. Indeed, applying molecularcodes to study a
point feature in a crystal requires tricky setting up of boundary condi-
tions in the cluster models of eventually infinite systems.
2
Any way,
one has to consider rather large clusters to compensate boundary
effects. At the same time, the required embedding procedures are
sometimes awkward and not unique, particularly in the case of clus-
ters cut from ionic crystals or metals. A comprehensive review of the
related problems is given in Ref. [3].
An alternative and nowadays predominant approach is to accom-
modate periodic wave function
4
or PAW-DFT
5
based methods to
study local effects. In order to do so, one needs to increase the size of
the unit cell to ensure correct proportion of the point defects and to
minimize lateral interactions between them (see, for example, Refer-
ences [6, 7] and references therein). The unit cell enlargement
increases the computational costs drastically with the scaling up to
ON
3

, where Nis the sizeof the unit cell. The entire study consists
of full calculations of the original (ideal) periodic system and repeating
them for as well periodic models which include intrinsically local fea-
tures. The effect of the local perturbations is then derived by compar-
ing results of these calculations.
Apparently, the state of the locally perturbed crystal cannot differ
too much from that of the ideal one. The mentioned standard
approaches do not consider this fact, although it seems to be profit-
able to avoid repetitive recalculations from the scratch and to make a
better use of the eventually more precise information about the elec-
tronic structure of the ideal crystal. Based on this idea, further embed-
ding schemes had been developed (for example References [811])
Received: 20 June 2021 Revised: 5 September 2021 Accepted: 13 September 2021
DOI: 10.1002/jcc.26766
J Comput Chem. 2021;117. wileyonlinelibrary.com/journal/jcc © 2021 Wiley Periodicals LLC. 1
which employ the information about the ideal crystal for analysis of
the defect ones. In addition, these schemes are not free from numer-
ous ad hoc assumptions leading to an uncontrollable loss of precision
and predictive capacity. Embedding methods based on the optimized
effective potential method
8,9
require density inversion procedures to
determine non-additive kinetic potential term, which can be ill condi-
tioned.
11
Projection-based embedding schemes, employing the idea of
localized molecular orbitals in different manifestations,
10,11
substan-
tially rely on the degree of localization in the system: the quality of
calculations decreases significantly for the systems with intrinsically
delocalized electronic system. Because of that, embedding in metals
presents a considerable challenge for the projection-based methods,
and the problem seems to be conceptual, since it is hard to expect,
that metals will allow any required localization of one-electronic
states. Therefore, the projection-based embedding can hardly be con-
sidered as a universal approach to calculate electronic structure of
local defects in all solids. At the same time, one can expect them to be
a very useful tool to study local defects in insulators.
Remarkably, a detailed theory of local perturbations of periodic
systems using the Green's function (GF) formalism had been devel-
oped yet in 60s
1215
(based on an earlier work
16
) and was applied to
chemisorption in numerous works.
1728
The most characteristic fea-
ture of this approach is that starting from a solution of the ideal peri-
odic problem and from its local perturbation, one obtains the answer
as a correction to the unperturbed solution. By this, (1) the highly inef-
ficient step of solving the perturbed problem from the scratch is
avoided and (2) the result is a pure effect of the perturbation. The the-
ory is highly pedagogically explained in Reference [29], where it is
used to provide pictorial description of the electronic structure pertur-
bations occurring throughout chemisorption. Otherwise more mathe-
matical, but still suitable for a chemistry theory student exposition of
the required apparatus is given in Ref. [30].
Being actively used in 70ies to study local defects in rather simple
model solids, the GF theory was practically abandoned later and rep-
laced by supercell periodic and finite clusters calculations.
31
Undeservedly forgotten, this approach has been getting much less
attention since then, especially in the community of solid-state chem-
istry.* Although, it suits better for discussing chemical problems, it has
never enjoyed any generalization to more realistic systems described
by rigorously defined Hamiltonians. As well, there is no reported soft-
ware implementing the general theory, which would be suitable for a
broad range of solid state and surface chemistry problems (with differ-
ent kinds of defects in crystals and/or chemisorption). The purpose of
the present work, and of the proposed GOGREENGOsoftware package,
is to incorporate the mentioned approach into a context of modern
theoretical methodology used in solid-state chemistry. Since we
mainly focus on the chemical interpretation of electronic structure cal-
culations (described in chemicalterms such as atomic charges or
bond orders), it is implemented for basis of local atomic orbitals. In the
present contribution we are concentrated on perturbations of the
electronic structure caused by the local defects of different kinds,
while structural deformations, which may arise in the defected crystal,
are going to be addressed in future works.
This paper is organized as follows. In the Theory and Implementa-
tion Sections we sketch the generalized theory and implementation
details of the programs included in the GOGREENGOpackage. After that
we provide test results for various benchmarks by considering substi-
tutional and interstitial defects in the model cubic lattices. Finally,
GOGREENGOpackage is applied to tackle more realistic problems
devoted to graphene. One can find more detailed specifications of the
GOGREENGOby following the link.
33
The Supporting information col-
lects details of the theory and analytical results used for the testing.
2|THEORY SKETCH
2.1 |Structure of the electronic problem
Theoretical basis of the proposed software development is the self-
consistent perturbation theory of many-electron systems.
34
It starts
from representing the electronic structure either by a single Slater
determinant formed by one-electron (spin-)orbitals for the wave func-
tion based methods (as exemplified by References [4,35,36]) or by
one-electron density in the DFT context. In the latter case one-
electron orbitals reappear as Kohn-Sham ones, so that either wave-
function or (Kohn-Sham) DFT procedures can be represented as an
iterative eigenvalue-eigenvector problem with some generalized den-
sity dependent Fockian matrix FP½in the functional space spanned
either by exclusively atomic states or by an assembly of plane waves
and atomic local states augmenting (PAW) the former:
εIFP½ðÞλ
ji
¼0:ð1Þ
Solutions of Equation (1) are the eigenvalue-eigenvector pairs ελ,λ
ji
numbered by assemblies of quantum numbers λand satisfying the
well-known relations:
FP½λ
ji
¼ελλ
ji
:ð2Þ
The density Pis determined by the occupied eigenvectors λjiwhose
eigenvalues are subject to the condition ελεF, where εFis the Fermi
energy selected so that the number of occupied one-electron states
equals to that of electrons (Fermi statistics). The eigenvalue-
eigenvector problems Equations (1), (2) are sequentially solved until
the convergence for Pis achieved.
2.2 |Green's function's representation
The eigenvalue-eigenvector problem can be alternatively formulated
in terms of the quantity
:
GzðÞ¼ zIFðÞ
1,ð3Þ
the famous Green's function of a complex argument z. Since the
Fockian F(hereinafter, we omit its Pdependence for brevity) is a
2POPOV ET AL.
Hermitian operator, its eigenvalues ελare always real. Thus, for an
arbitrary complex zunequal to any of ελthe matrix zIFðÞis non-
degenerate and can be inverted producing a z-dependent quantity
Equation (3). Its closest relation to the eigenvalue-eigenvector prob-
lem stems from the spectral representation:
GzðÞ¼X
λ
λ
jiλ
hj
zελ
,ð4Þ
which immediately derives from the expressions for the identity
matrix and the Fockian in the basis of its eigenvectors:
I¼X
λ
λjiλhj;F¼X
λ
ελλjiλhj:
AssoonastheGFisknowninthebasisoftheeigenvaluesofF, where
it is diagonal, it is known in whatever basis. E.g. in the basis of local
atomic spin-orbitals a
ji
,b
ji
,a Greenian matrix is formed by the
elements
Gab zðÞ¼
X
λ
ajλ
hi
λjb
hi
zελ
,ð5Þ
where ajλhiare expansion coefficients of the eigenvector λjiover
atomic basis.
Being defined as a function of complex variable, GF appears in
the expressions for the physical quantities under the integral over the
real axis only. Since GF has many poles on the real axis, it should be
considered there as a distribution (or a generalized function). Such dis-
tribution is defined as a limit:
GεðÞ¼lim
ν!0þGεþiνðÞ,ð6Þ
where zis set to be εþiνwith real εand ν. As described in Supporting
information section 1, evaluation of the limit entering the later equa-
tion leads to the following GF on the real axis
GεðÞ¼GεðÞþiGεðÞ,
GεðÞ¼
X
λ
λjiλhjP 1
εελ

,
GεðÞ¼πX
λ
λjiλhjδεελ
ðÞ,ð7Þ
where δεðÞis the Dirac δ-functionand Pindicates that the integral
of a function fε
ðÞ
, multiplied by εελ
ðÞ
1, with respect to εmust be
understood as the Cauchy principal value. In terms of the GF of a real
argument the general expression for the density operator
(Equation (3) of Supporting information) takes the form:
P¼π1ðεF
GεðÞdε:ð8Þ
2.3 |Perturbations in terms of Green's function
The GF formalism as sketched in Section 2.2 does not add too much
to the usual treatment of the eigenvalue-eigenvector problem. Its
power manifests itself when the perturbations are addressed. For the
GF of the Fockian F, being a sum of an unperturbed one F0ðÞand a
perturbation F0:
F¼F0ðÞ
þF0,
the Dyson equation
GzðÞ¼G0ðÞzðÞþG0ðÞzðÞF0GzðÞ,ð9Þ
holds
38
valid in the whole complex plane. Being solved for GzðÞ,it
gives:
GzðÞ¼ IG0ðÞzðÞF0

1G0ðÞzðÞ,ð10Þ
which generates the perturbation series if one expands the inverse
matrix in the geometric series:
IG0ðÞzðÞF0

1¼IþG0ðÞzðÞF0þG0ðÞzðÞF0G0ðÞzðÞF0þ
GzðÞ¼G0ðÞzðÞþG0ðÞzðÞF0G0ðÞzðÞþG0ðÞzðÞF0G0ðÞzðÞF0G0ðÞzðÞþ
Thus, the general perturbative treatment rewrites in terms of the
Green's functions. Formally, in the case of a point defect in an infinite
crystal the solution of the Dyson equation would require inversion of
a matrix of infinite dimension. However, switching to a local atomic
orbital representation allows to reduce the problem to a finite one,
since in this case a point (local) perturbation acts on a relatively low-
dimensional subspace (P) of the entire space of one-electronic states.
In GOGREENGOwe employ this possibility and consider the perturba-
tion matrices of a form:
F0¼F0
PP 0
00
!
¼V0
00

:ð11Þ
The Greenian matrix is then split in blocks:
G0ðÞ
¼G0ðÞ
PP G0ðÞ
PQ
G0ðÞ
QP G0ðÞ
QQ
0
@1
A,ð12Þ
where Qrefers to the orthogonal complement of the subspace P
(argument zis omitted for brevity). Introducing a dimPdimPmatrix
M
§
:
M¼VI
PP G0ðÞ
PP V

1,ð13Þ
POPOV ET AL.3
and following the procedure given in Supporting information
section 2, one obtains the corrections to the matrix blocks:
GPP G0
ðÞ
PP ¼G0
ðÞ
PP MG 0
ðÞ
PP ,
GQQ G0ðÞ
QQ ¼G0ðÞ
QP MG 0ðÞ
PQ ,
GPQ G0ðÞ
PQ ¼G0ðÞ
PP MG 0ðÞ
PQ ,GQP G0ðÞ
QP ¼G0ðÞ
QP MG 0ðÞ
PP ,ð14Þ
which, respectively, express the effect of the perturbation on the
Green's function in the subspace Pitself, in the subspace where the per-
turbation is absent (Q) and on the coupling between the perturbed and
unperturbed subspaces. Density matrix elements of the perturbed sys-
tem are calculated with Equation (8) once the Dyson equation is solved.
If one takes into account the density dependence of the Fockian, the cal-
culated density serves as an input for a next step of iterative solution.
This setting is referred below as the self-consistent one.
2.4 |Specific of the problems at hand
So far reviewed greenistic representation of the eigenvalue-
eigenvector problem is fairly general. Here, we apply it to the crystal
point defects of different kinds. We consider two types of such
defects: (1) point-wise perturbation of the crystalline matrix itselfthe
lattice substitution and the vacancy; and (2) interaction of the crystal
with an extra particledubbed in different contexts as an interstitial
defect or an adsorbate. In either case we shall employ the generic
term defect. The unperturbed solution described by GF G0ðÞof the
ideal crystal (and of the finite adsorbate in the case (2)) is assumed to
be known. Below we briefly review the specific of the greeninstic for-
malism as applied to infinite periodic systems with point defects.
2.4.1 | Green's functions of crystals
When it comes to solids, the solution of the eigenvalue-eigenvector
problem has specifics described in handbooks on solid-state physics
and chemistry
39
(see chapters devoted to tight binding approximation).
Due to translation invariance of an ideal infinite crystal, its Fockian
accepts the block-diagonal form in the basis of the Bloch sums:
akji¼
1
ffiffiffi
K
pX
r
exp ikrðÞarji
of Aatomic states. Here Ais the number of atomic spin-orbitals
a¼1A,rstands for a unit cell index and K is the number of k-points
involved in the calculation.
k
The blocks are numbered by the wave vec-
tors kfrom the first Brillouin zone and respective eigenvalues form A
functions εαkof k(α¼1A)(electronic) bands with the eigenvector
expansion coefficients forming k-dependent AAmatrices with ele-
ments αkjak
hi
. In this context, the generalized quantum number λ,
labeling the eigenvalues of the Fockian, splits in the pair of the wave
vector kand the band index α:λ¼α,kðÞ. Thus, the GF of the ideal
crystal with Abands reads:
GzðÞ¼
X
αk
αkjiαkhj
zεαk
:ð15Þ
It is as well block-diagonal with AAblocks numbered by k.A
remarkable feature of the theoreticalGF of the crystal is that the
poles coalesce in (quasi-) continuous segments being the allowed
energy bands of electrons (see Reference [39]). Considering the GF
on the real axis according to Equation (7) produces the electronic den-
sity of states (DOS):
DOS εðÞ¼1
πKSpGεðÞ¼
X
αk
δεεαk
ðÞ,ð16Þ
familiar to the workers of the field.
Applying the same trick to the diagonal elements of the Greenian
matrix Equation (5) in the basis of local atomic orbitals
Gaa zðÞ¼
X
αk
akjαk
hi
αkjak
hi
zεαk
,
we arrive to
DOSaεðÞ¼1
πKGaa εðÞ¼
X
αk
akjαk
hi
αkjak
hi
δεεαk
ðÞ
the projection of the DOS upon the atomic state a, as well, familiar
from numerous packages (e.g. LOBSTER
41
or WANNIER90
42
) performing
analysis of numerical data derived from PAW-DFT or whatever com-
puter experiments on solids (see e.g.
41,43
). Treating similarly off-diago-
nal elements:
Gar,br0zðÞ¼
X
αk
akjαk
hi
αkjbk
hi
exp ik r r0

zεαk
,
yields:
1
πKGar,br0εðÞ¼
X
αk
akjαk
hi
αkjbk
hi
exp ik r r0
ðÞðÞ½δεεαk
ðÞð17Þ
which is a close relative of the crystal orbital overlap and crystal
orbital Hamilton populations (respectively, COOP and COHP
44
)as
well widely available in the solid-state packages. In these expressions
arjicorresponds to the AO ain the unit cell r, so, the latter formula
allows to calculate the density matrix element for the pair of orbitals
from different unit cells.
2.4.2 | Green's functions of a finite system
The definition of the GF and all related quantities in the case of the
finite system remains the same as sketched in the Subsection devoted
to the GF representation of eigenstates problem. The Greenian matrix
element has a general form Equation (5), which for the real values of
argument reduces to:
4POPOV ET AL.
Gab εðÞ¼
X
λ
ajλhiλjbhi
εελiπajλhiλjbhiδεελ
ðÞ

,ð18Þ
where the summation goes over discrete levels λ. The imaginary part
of the GF consists of discrete signals located at energies ελand pro-
portional to the Dirac's δ-functionand has the dimension of inverse
energy, which becomes apparent from the Lorentzian approximation
of the δ-function. The real part is a continuous function except simple
poles at ε¼ελ.
In the general case it is not possible to find an analytical solu-
tion for the perturbed GF, therefore one has to treat the initial Gre-
enian matrix of the finite system numerically. Such treatment
requires approximating the δ-function in Equation (18) by a
Lorentzian of a (small) width νas explained in Supporting information
section 1. Theoretically, such approximation approaches genuine
result in the limit ν!0. In practical calculations the value of this
parameter has to be finite. It is advised to set νequal to the step of
the energy grid and to choose the latter small enough to guarantee
required accuracy.
3|IMPLEMENTATION AND
COMPUTATIONAL DETAILS
3.1 |General construction
The programmatic implementation of the theory described in the The-
ory Section is provided by GOGREENGOpackage being a set of proce-
dures written in fortran 2010 using the system of high-level objects
provided by the cartesius_fort library.
45
The component procedures
exchange data through intermediate files packed in the hdf5
archive.
46
The flow-chart illustrating the relations between different
programs of the package and the paths of data transfer between them
is shown on the Figure 1. Specifically,
1. program GET_GREEN evaluates Greenian matrix elements for a given
set of local atomic orbitals from the band structure of the ideal
crystal. The program can use eigenvalues derived by VASP,
47
ABINIT.
48
The projections of eigenvectors produced by these
packages to the basis of local orbitals can be obtained by the lob-
ster software
41
whose output format is compatible with get_green.
As well ΘΦ (TetaPhi)
49
produces eigenvalues-eigenvectors in the
format readable by GOGREENGO. This combination allows extracting
Greenian matrix of the pure solid in an atomic basis from the most
popular plane wave DFT codes and using it for impurity calcula-
tions. One can use the band structure of the solid from any other
source, as well, transforming it to the required format. Imaginary
parts of Gab are calculated in get_green for each point of the
energy grid by applying tetrahedron algorithm
50
of integration
over the Brillouin zone. Since the real parts of GF elements are
related with the imaginary parts by the Kramers-Kronig relations
51
they are obtained numerically from the latter as described in Refer-
ence [52]. The user has to define the interval of the energy, where
the Greenian matrix has to be calculated, and the step of the
energy grid. In addition, there is an option to obtain elements of
the Greenian matrix between different unit cells rand r0as in
Equation (17) with no calculations on extended unit cells.
2. HGEN calculates the crystal-defect hopping operator and an array
of two-electron Coulomb integrals (required for the self-consistent
setting). Using hgen is not mandatory: one can also use any exter-
nally prepared hopping operator and Coulomb integrals in the
required format. Hgen calculates the interaction within a semi-
empirical NDDO approximation including different parameteriza-
tions such as MNDO,
53
AM1
54
and PM3
55
which have been
recently shown to be compatible with the PAW-DFT setting.
56
Other options of calculating atomic integrals will be added in the
future releases. To run the hgen utility one has to provide geome-
tries of the defect and crystal, define which atoms interact
(by giving a cut-off distance or listing them explicitly) and type of
FIGURE 1 Flow-chart of the GOGREENGOpackage
POPOV ET AL.5
the AOs basis set to be used (single STO, MAP,
57
Bunge,
58
Koga
59
are available so far). Optionally, one can also change default values
of semi-empirical hopping parameters in order to parameterize the
Hamiltonian for one's needs.
3. Program DYSON reads unperturbed Greenian matrix of the ideal crys-
tal (which comes from GET_GREEN), that of a defect (also comes from
GET_GREEN or can be calculated directly in dyson if one provides
molecular orbital energies and MO LCAO coefficients), initial pertur-
bation operator and a table of two-electronic integrals, if the self-
consistent version is used. The program finds a solution of the
Dyson equation and returns perturbed Greenian matrix together
with the new density matrix and energy correction caused by per-
turbation. In the case of the self-consistent setting it performs itera-
tive solution of the Dyson's equation taking into account an
adjustment of the self-energy at each step. In this case the user has
to specify a convergence threshold. The damping ensuring better
convergence is supported and can be used if necessary.
4. Program GREEN_OPT performs either gradient or simplex optimization
of the defect position and (internal) geometry. Hgen can be used to
generate perturbation matrix on each step. In hgen no gradients are
available so far, so it can only be used for the simplex optimization.
Gradients will be added to hgen in future versions. Regarding geom-
etry optimization we note, that strong local perturbations may cause
considerable deformations of the crystal geometry, having significant
impact on the electronic structure and energetics of the defect for-
mation. Although, in the present paper we only focus on perturba-
tions of the electronic structure, typically, local defects cause local
deformations, which formally translate into additional terms in the
perturbation operator F0. Their impact on the electronic structure
can be treated by the proposed approach as of any other perturba-
tion. Of course, consistent evaluation of the atomic forces and sea-
rch for a deformation, minimizing the total energy, requires special
attention. Authors plan to address this issue in future works by intro-
ducing electronphonon interactions into the model.
4|SOME SPECIFIC FEATURES
4.1 |Self-consistent perturbation theory
As mentioned in the Theory Section, the perturbation of a one-
electron part of the Fockian produces a correction to the density
matrix of the same order as the perturbation itself. Then, due to the
mean field treatment of the electronelectron interaction either in the
wave function or DFT setting, the corresponding Fockian receives fur-
ther corrections proportional to those of the density matrix elements.
They form so called dressingwhich needs to be added to the origi-
nal one-electron (bare) perturbation. The solution of the resulting
Dyson equation yields further changes of density matrix so that one
has to repeat the calculation until the convergence is reached. This
option is implemented in the GOGREENGOpackage.
In metals the effect of perturbation decays with the distance from
the defect as R2κwith some κ> 0 dependent on the form of the
Fermi surface and dimensionality of the crystal structure (κis typically
higher for higher dimension).
28
** This allows one to restrict the range
of the action of the self-consistent (dressed) perturbation by a finite
number of unit cells close to the defect. Of course, the corrections to
the two-electron part of the Fockian involve additional orbitals, and
the dimension of the subspace Phas to be increased. The amount of
this augmentation is system dependent and it is advised to look for an
optimal size by a series of convergence tests.
4.1.1 | Position of the Fermi level
Another important aspect is the position of the Fermi level in the
perturbed system. In general, it does not remain constant although
changes by a small value. The general reason is that the perturbed GF
GεðÞdiffers from unperturbed one G0ðÞεðÞand thus the integral of
SpGεðÞfrom up to εFof the unperturbed crystal not necessarily
yields the same number of electrons as does the integration of
SpG0
ðÞzðÞ. The defect of electron's number Δnis to be eliminated by
shifting the Fermi energy by δεF. As it is explained in Supporting infor-
mation section 3 the value of δεFis determined by the value of Δn=K
and it, evidently, becomes infinitesimally small for the limit K !.In
real calculations K is finite and δεFis non-vanishing in this case. Evalu-
ation of δεFrequired to keep constant the number of electrons in the
system with K unit cells is implemented in the GOGREENGO. Practically,
in most our calculations the value of Δn=K is rather small (we use
3131 31 meshes of k-points for three-dimensional models and
5151 one for graphene) and affects density matrix elements only in
fourth-fifth decimal place. However, in all cases it is advised to thor-
oughly check its impact on the density matrix elements and final
results, especially if the number of k-points K used for the band struc-
ture calculations of the crystal is relatively small as it sometimes hap-
pens in PAW-DFT calculations.
4.1.2 | Energy correction
Even an infinitesimally small shift of the Fermi level causes a finite
correction to the perturbation energy, since a summation over infinite
number of unit cells is implied in calculations. It may be shown
29
that
the perturbation energy is:
δE¼1
πðεF
εεF
ðÞδSpGεðÞdε¼1
πðεF
εδSpGεðÞdεΔnεF,ð19Þ
where Δnis a difference in number of electrons in the perturbed sys-
tem (calculated with the original Fermi level of the unperturbed sys-
tem) and the initial one. In the self-consistent version of the
calculations the modified Equation (25):
δE¼1
πðεF
εεF
ðÞδSpGεðÞdε
1
2X
i,j,k:l
PijPkl ij
kl

P0ðÞ
ij

0P0ðÞ
kl

0ij
kl

0

,ð20Þ
taking into account changes in the self-energy needs to be used. The
summation in the last term goes over all orbitals involved in the
6POPOV ET AL.
perturbed subspace P. Due to the decaying effect of the perturbation,
it is a finite set as explained above. Two-center two-electronic inte-
grals in this equation change in the perturbed system only if the
geometry gets distorted.
4.1.3 | Local and virtual states
As it was stressed yet in works,
12,20
the defects may produce addi-
tional poles of the perturbed GF of two types. Either so-called local or
virtual states may arise depending on the strength of the perturbation
as related to the energy spectrum of the unperturbed system. The
local states correspond to the poles on the real axis and, therefore,
appear as narrow peaks of the perturbed DOS in the energy ranges,
where the unperturbed DOS vanishesoutside the allowed energy
bands of the ideal crystal. By contrast, the virtual states are related
with the poles in the complex plane
19
and manifest themselves on the
real axis as wide Lorentzian peaks of the perturbed DOS inside the
allowed energy band (see Supporting information section 4). Both
features are perfectly reproduced by the GOGREENGOpackage as dem-
onstrated in Figures 2 and 3. For more details about handling the pole
structure of the perturbed GF in GOGREENGOsee Supporting informa-
tion section 4.
5|TEST RESULTS AND DISCUSSION
Since the described approach, although, well established theoretically
(analytically), did not so far enjoy full scale program implementation, it
requires a thorough testing against analytically solvable models, even
looking out oversimplified. Below, we present such tests and round
up with an intermediate testadsorption on graphene, which on one
hand can be traced analytically far enough to provide necessary refer-
ence and on the other hand provides, although a simple, but realistic,
example eventually suitable for experimental check. In the main text
we mostly concentrate on the numerical results obtained by
GOGREENGO, while analytical solutions, used for control, are collected
in the Supporting information. Even in the case of simple models not
FIGURE 2 Perturbed densities of states DOS0εðÞ(blue lines) in comparison with the initial ones (red lines) for the cubia lattices and schematic
representations of atomic charges distributions induced by the substitutional defects. Dashed vertical lines show the Fermi level. Individual panels
correspond to the following systems: (A) sc v¼1; (B) sc v¼1; (C) bcc v¼1; (D) bcc v¼1; (E) fcc v¼1; (F) fcc v¼1. Radii of the spheres are
proportional to the charge of the atom (notice the different scales of Q0for different lattices in Table 2). Red spheres correspond to the positive
charge, blueto the negative. The biggest sphere in all cases is located on the defect site (0). On the plots red filling between the curves
corresponds to the increase of the diagonal electronic density on the site and blue oneto decrease
POPOV ET AL.7
all functionality of the package can be tested against analytical solu-
tions, since the later are available only for the simplest local perturba-
tions and are inaccessible in the frame of the self-consistent
approach. Therefore, we test GOGREENGOagainst very simple bench-
marks and then demonstrate its capabilities for more realistic and
comprehensive setting.
5.1 |GoGreenGo for perturbation of cubia
Cubia (see e.g. Reference [60]) are simplest thinkable models of 3D
metals. They are formed by s-orbitals centered at the vertices of (sim-
plesc, body centeredbcc, and face centeredfcc) cubic lattices
with one-electron hopping tbetween the nearest neighbors of a given
node of a lattice. The dispersion laws of electronic bands for such
models
61
allowing for analytic solutions as given in Table 1. Corre-
spondingly, the eigenvectors (Bloch states) related to these eigen-
values are known and, consequently, the Green's functions.
Specifically, diagonal GFs for the systems listed in Table 1 are known
from References [6164] and their plots are presented in Supporting
information section 6. One can also find the corresponding graphs of
the electronic DOS at the website.
65
The off-diagonal elements of the
respective Greenian matrices are as well accessible through
GOGREENGO.
It is as well possible to find analytical solutions for the ideal
cubia in the framework of the extended Hubbard model taking into
account electronelectron interactions as described in Supporting
information section 5. In our subsequent consideration we will use
the extended Hubbard model as a starting point for the self-
consistent calculations. All energy parameters are given in units of t,
that is t¼1 everywhere below. The diagonal matrix elements of the
unperturbed Fockian are set to zero being by this the energy refer-
ence. In further Subsections we present as coherence tests the results
of numerical treatment of various local perturbations of cubia with
use of GOGREENGOpackage.
5.1.1 | Lattice substitutions in cubia
First, we consider lattice substitution defects in cubia, where one
atom of the crystal (denoted as 0) is replaced by a different atom. In
general, the substitute can be a many-electron atom, but here we
restrict our tests by single-electron impurities. Chemically this corre-
sponds to substitution defects in alkali metals.
FIGURE 3 Perturbed densities of states DOS0εðÞ(blue lines) compared with the ideal ones (red lines) for the substitution defect in p-cubium:
(A) v¼0:5; (B) v¼0:5. Color scheme is the same as in Figure 2
TABLE 1 Dispersion relations, Fermi level and the nearest neighbor Coulson bond-order for cubia with single (one electron per site)
occupation. Indices kicorrespond to projection of wave vector kto orthogonal basis vectors of reciprocal space chosen so that in all cases the
cubic Brillouin zone is defined as π<kκπ
εkWεFB
Sc 2tcoskxþcoskyþcoskz
ðÞ 12t00:3324
Bcc 8tcoskxcoskycoskz16t00:2605
Fcc 4tcoskxcoskyþcoskxcoskzþcoskycoskz
ðÞ16t0:915t0:2184
8POPOV ET AL.
Complexity of the model can be gradually increased by including
different terms into the perturbation operator so that different pro-
gram features are tested independently. In the simplest possible set-
ting we neglect electronelectron interactions and only take into
account one-center perturbation: the difference vbetween diagonal
matrix elements of the Fockian over impurity AO and the AO's of the
unperturbed crystal. In this case an analytical solution is available
(Supporting information section 8). Comparing numerical results to
this solution shows that GOGREENGOproduces the perturbed Greenian
matrix identical to the analytical one up to eighth decimal sign. In
addition, this setting allows making qualitative sketch of the features
of the perturbed system, which remain valid for more involved crys-
tals and perturbations of this kind. As one can conclude from the ana-
lytical form of the mass operator (see Supporting information
section 4) and of the cubia GFs G0ðÞ
00 (Supporting information
section 6), the additional poles in the perturbed system appear for:
(1) sc vjj>4:01; (2) bcc vjj>6:10; (3) fcc v<9:38 in the case of
occupied local state and v>0:83 in the case of vacant local state. In all
these cases except for the local state above the band in the fcc lattice,
the required value of vis unreasonably high and it is difficult to
expect any local state to appear if, say, indeed, one alkali atom is
exchanged by another (the typical difference in core attractions lies
in the range of 0.40.8 eV
66
much smaller than the typical band-
width). The only option is an appearance of the vacant local state
inthefcclattice,butitpresents a minor interest since it does
not contribute to the electronic density and the energy of the sys-
tem. Another conclusion, which can be drawn from the analytical
solution, is the behavior of the perturbed function in the vicinity
of the pole of the initial GF in bcc. For bcc G0ðÞ
00 is even function of
εand G0ðÞ
00 is odd. In addition, due to the presence of the pole at
ε¼0, G0ðÞ
00 has a discontinuity there. Therefore, one would observe a
discontinuity of the perturbed function DOS0εðÞat ε¼0, which is
indeed observed in our numerical results described below (see
Figure 4).
FIGURE 4 Perturbed diagonal densities of states DOS0εðÞin comparison with the initial ones and schematic representation of the charge
distributions in graphene with the substitution defects: (A) boron, (B) nitrogen. Color code and other legend is the same as in Figure 2. Narrow
peaks on the plots above the band in (A) and below the band in (B) correspond to the local states formed predominantly of the defect orbital
POPOV ET AL.9
Although described model provides a simple and pictorial solution,
it is rather far from realistic description of the substitution defects,
requiring more involved interaction operator and the self-energy cor-
rections. GOGREENGOsupports such description and to test this, we
included corrections to one-center two-electron and two-center hop-
ping integrals, Coulomb interactions, and applied the self-consistent
procedure. We performed calculations within this setting for cubia lat-
tices for v¼1tðÞcomplemented by a set of further parameters: vari-
ation of the hopping between the impurity and its neighbors
(δt¼0:2), one-center Coulomb repulsion in the unperturbed system
(γ0¼0:6), two-center (nearest neighbor) Coulomb repulsion in the
unperturbed system (γ1¼0:3) and a variation of the Coulomb inte-
grals in the defect (δγ0¼0:1 and δγ1¼0:1). The resulting perturbed
DOS0εðÞtogether with obtained charge distributions are depicted in
Figure 4; numerical values of charges and the electronic
††
energy vari-
ations due to the defect formation are collected in Table 2.
In all cases the charge distributions induced by the defect have a
similar oscillating-decaying behavior. The defect site (0) carries the
highest (by absolute value) net-charge, negative (electron density
accumulation) for v¼1 and positive (electron density depletion) for
v¼1. Absolute values of the net-charges on other sites decay with
the separation from the defect and have alternating signs so that for
any site its charge has the opposite sign to the charges of its neigh-
bors. The decay rate is rather high as expected for 3D metals.
28
The
fastest one is observed for fcc, where the induced charges are insig-
nificant (<0.01) beyond the 2nd neighbors. For bcc the corrections
become negligible beyond 3rd neighbors and for sc beyond the 4th
ones. In all cases the Fermi level shifts as described in the Implemen-
tation Section and Supporting information section 3, but the absolute
values of δεFdo not exceed 7106. For v¼1 it shifts downwards
and for v¼1upwards. Such a small shift insignificantly affects indi-
vidual density matrix elements. Maximal correction of the diagonal
density matrix elements due to the Fermi level shift equals to Δn=K.
For 3131 31 k-mesh used in our calculations and values of Δn
from Table 2 they never exceed 5 105for cubia, which is fairly
negligible.
The (Coulson) bond ordersthe off-diagonal elements of the one-
electron density matrixare affected much less than the diagonal
matrix elements of the density. Again, these corrections decay rapidly
with the distance from the defect. In all cases, the bonds formed by
the defect site are weaker than those of the innate atom, but even
this correction occurs only at the third decimal place. Nevertheless,
the sum of all corrections to the bond orders PijδBij can be fairly
noticeable due to the large number of bonds in the cubia lattices.
These values are presented in Table 2. As one can see, in all cases the
total variation of the bond orders is negative, meaning that the system
loses a part of the bonding energy because of the defect.
Electronic energy of the defect formation can be evaluated by
Equation (20). An alternative expression for it is given in Supporting
information section 5. As it can be seen from Table 2, in all cases the
total energies are negative and the absolute values are much higher
for the systems with v¼1, which is obviously explained by the sign
of the one-center contribution of the impurity AO. Further significant
contributions are the Coulomb attraction of the atomic charges, which
is always negative due to the oscillatory behavior of the charge distri-
bution, and the correction to the bonding energy, which is always pos-
itive as we saw above.
5.1.2 | Substitution defects in p-cubium
In the previous Subsection, we have tested the main functionality of
GOGREENGOfor cubiasingle-band crystals. To demonstrate package's
capabilities in treating multi-band solids we use a p-cubium
(pc) model, which is formed by three p-orbitals located in the vertices
of the simple cubic lattice. Due to symmetry, each orbital overlaps
only with its own nearest translation images in the frame of the tight-
binding approximation (e.g., pxorbital overlaps only with neighboring
pxorbitals and orthogonal pyand pz). Two of such overlaps corre-
spond to σσinteraction and four others to ππinteraction. In addi-
tion we assume the following for the two-electron Coulomb integrals
ab jcdÞ¼ aa jccÞδabδcd ,ðð , which allows keeping the same number of
two-electron parameters as previously. The analytical band-structure
for the ideal pc contains three degenerate bands (see
e.g. Reference [60]):
εαk¼2tcoskαþκX
βα
coskβ
!
ð21Þ
where α,β¼x,y,znumerates bands and projections of k-vector, κis a
ratio between ππand σσhoping parameters and tstands for the
σσone. Minus in front of the first cosine in the brackets occurs
because the σσoverlap of two p-orbitals is negative (and hopping is
positive), if they are aligned in the same direction. In our further con-
sideration we set κ¼0:40 and t¼1. The elements of the Greenian
matrix for pc are accessible through GOGREENGOand can be found in
Supporting information section 1. We consider pc with one electron
TABLE 2 Parameters of charge
distribution in cubia with the
substitutional defect and electronic
energy of the defect formation. Qi¼
1nicorresponds to the atomic charge
of ith neighbor of the defect site 0with
nibeing the electronic population of the
ith site in the perturbed system
vQ
0Q1Q2Q3PijδBij ΔnδE
sc 10.3478 0.1216 0.0712 0.0555 0.1620 0.0882 1.9964
1 0.3478 0.1217 0.0711 0.0555 0.1594 0.0899 0.0029
bcc 10.4067 0.1058 0.0835 0.0389 0.0600 0.1488 2.0900
1 0.4043 0.1088 0.0819 0.0379 0.0592 0.0121 0.0866
fcc 10.1984 0.0243 0.0115 0.0025 0.0532 0.0294 2.1583
1 0.2903 0.0293 0.0116 0.0025 0.0413 0.0766 0.2671
10 POPOV ET AL.
per unit cell, yielding the Fermi level to be εF¼1:753t. Coulson
bond-order Bσfor the σ-bond is 0:2753 and for the π-bond is
Bπ¼0:1474. Note, that Bσis negative and being combined with posi-
tive σσhopping produces a negative contribution to the energy. In
our further discussion of overall change in the bond orders it is conve-
nient, therefore, to use values of Bσ
jj
.
To test the whole functionality of the program in the case of
multi-band crystal we consider substitutional defects in the most rig-
orous setting by taking into account the self-energy correction and
changes of one- and two-center integrals as it was done in Sec-
tion 5.1.1. The set of parameters used for calculations: v¼0:5,
δt¼0:2, γ0¼0:6, γ1¼0:3, δγ0¼0:1 and δγ1¼0:1.
Perturbed DOS's projected on one of the p-orbitals of the defect
site 0are plotted on Figure 5 for the negative and positive perturba-
tion. Resulting charge distributions exhibit similar behavior as in sc
with some specific variations. First of all, the charges decay faster
than in sc: they become negligible beyond 3rd neighbors for v¼0:5
and beyond 4th for v¼0:5. Moreover, charge signs follow a different
pattern: (1) for v¼0:5 the charges are Q0¼0:4152, Q1¼0:0195,
Q2¼0:0102, Q3¼0:0042; (2) for v¼0:5 they are Q0¼0:3039,
Q1¼0:0245, Q2¼0:0044, Q3¼0:0217. That is, the charges
change their signs every two neighbors from the defect. Also, it can
be seen that the negative perturbation causes a larger charge than the
positive one of the same magnitude. This is a consequence of a non-
symmetric location of the Fermi level in the band (like in fcc cubium).
Values of Δnare 0:1523 for the negative perturbation and 0:2524
for the positive one. As previously, such small values do not cause sig-
nificant shift of the Fermi level for our 31 3131 k-mesh.
As for corrections to the bond orders, we see that they change
significantly only for the bonds incident to the 0site (σ-bond
becomes weaker and π-one stronger in all cases). The total changes in
bond orders are PijδBij ¼0:1541 for the negative perturbation and
PijδBij ¼0:2414 for the positive perturbation, so the system loses
bonding energy, upon formation of the defect.
5.1.3 | One-orbital interstitial impurity in simple
cubium
Further class of problems accessible for GOGREENGOis that of interac-
tions of solids with some extra additive termed as an interstitial
defect. The simplest example of such defect is an interstitial (impurity)
atom represented by a single s-orbital, which may be considered as a
model of a hydrogen atom. We considered the effect of such impurity
placed in a cubic void of the sc lattice being a nontrivial example illus-
trating general features of the methodology. In the cubic void, the
defect is surrounded by eight identical atoms of the crystal system.
The unperturbed Fockian of such a system is one of the crystal aug-
mented by an extra row and column having the energy of the impurity
orbital on their intersection and filled by off-diagonal zeroes signifying
no interaction between the impurity and the crystal. The perturbation
consists, in the first approximation, of one-electron hopping between
the defect (s-)orbital and those of its immediate neighbors in the lat-
tice (eight neighbors). These nine orbitals form the perturbation sub-
space (the P-subspace of Equation (11)) hereinafter addressed as the
impurity cluster.
The position of the defect s-orbital relative to the cubium Fermi
level is given by a difference between ionization potentials of the
interstitial atom and crystal atom of the lattice. For all metals, atomic
ionization potentials are smaller than for hydrogen, therefore we
assume that the diagonal matrix element over the impurity orbital is
negative. In the cubia models the nearest neighbor hopping parameter
tof the crystal provides the natural energy scale. We, at first, consid-
ered numerically the effect of interaction on the atomic charges and
bond orders in the impurity clusterin the frame of the Anderson
model (see Supporting information section 8.2). Analytical solutions
for the Anderson model used for testing are given in Supporting infor-
mation section 8.2.
Perturbed GFs obtained numerically with GOGREENGOfor a
hydrogenatom in a cubic void (for the first step of the self-
FIGURE 5 Perturbed diagonal DOS for the site nearest to the vacancy (left) and schematic illustration of the charge distribution caused by
the vacancy defect in graphene. Legend for the plot and color scheme are the same as in previous figures
POPOV ET AL.11
consistent procedure) were compared with corresponding analyti-
cal equations from Supporting information section 8.2. The values
coincide up to 6th decimal place. Corrections to density matrix
elements are not accessible analytically due to a very involved
functional form of G0ðÞeven for the sc cubium. Numerical values
are collected in the Table 3. As one can conclude, if the adsor-
bate level is positioned below the Fermi level the impurity atom
acquires a negative charge, while for εa¼εF¼0 the charge is posi-
tive. The absolute value of charge depends on the impurity-matrix
hopping parameter t0as it increases, the charge decays which is a
result of changing in coupling patterns (see the four types of
perturbed adsorbate levels in Supporting information section 8.2).
Atoms of the crystal matrix in all cases acquire a small positive
charge.
Bond orders Bbetween the crystal atoms of the impurity clus-
terbecome smaller (the bonds become weaker) and this energetically
unfavorable change is compensated by formation of new bonds with
the order B0between impurity atom and lattice atoms. With increasing
hopping parameter t0the orders of new bonds increase and the orders
of original bonds decrease. For εa¼0 the bond orders B0are larger
than for εa¼2.
In a more extended setting, we turn on the Coulomb interactions
in the crystal. In this case, non-vanishing matrix elements between the
sites outside the impurity cluster appear, which makes an analytical
treatment unfeasible. The point to be checked here, is the extension
of the number of atoms in the impurity cluster for accounting of the
perturbation dressing through electronelectron interactions.
According to the tests, it is sufficient to include first and second
neighbors of the lattice atoms directly interacting with impurity.
As it can be concluded from Table 4 including two-electron Cou-
lomb terms and the self-energy correction does not have major influ-
ence on the resulting density matrices and energies; all trends remain
fairly the same with slight numerical modifications. The only exception
is the case of εa¼2 and t¼0:83, where one observes more signifi-
cant differences in density matrix elements na,Band B0between two
settings, which translates into quite considerable difference in
energies.
5.2 |GoGreenGo for perturbations of graphene
Two-dimensional graphene sheets are widely studied experimentally
and theoretically being nature models for two dimensional solids and,
respectively, surfaces. Either lattice impurities or adsorbates in/on
graphene are of considerable interest from the experimental and the-
oretical points of view.
67,68
The honeycomb lattice of graphene has a primitive cell containing
two carbon atoms. The π-system is formed by pπ-orbitals, one for
each site of the lattice. Dispersion law in the approximation of the
nearest neighbor hopping is:
εkðÞ¼tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3þ2coskxþ2coskxþ2cos kxky
ðÞ
q,ð22Þ
where corresponds to the filled band and +to the empty one.
Equation (22) allows analytical evaluation of the diagonal DOS
69
(see
also Supporting information section 7). The Fermi level εF¼0 and the
Coulson bond order B¼0:525 between nearest neighbors.
TABLE 3 GOGREENGOresults obtained for the hydrogen impurity atom in the cubic void with the Anderson Hamiltonian (supporting
information ssection 8.2). Parameter γ¼0:40 for all cases
εa20
t00:20 0:70 0:83 0:95 0:20 0:70 0:83 0:95
na1.9910 1.6823 1.4164 1.2164 0.6872 0.6023 0.6319 0.6557
ni0.9987 0.9608 0.9347 0.9130 0.9887 0.9534 0.9474 0.9418
B0.3311 0.2932 0.2672 0.2454 0.3211 0.2858 0.2798 0.2742
B00.0108 0.1286 0.2072 0.2446 0.1381 0.2772 0.2909 0.2995
δE1.8278 2.2008 2.3433 2.0104 0.2086 2.0233 2.6101 3.0670
TABLE 4 GOGREENGOresults obtained for the hydrogen impurity atom in the cubic void with taking into account inter-electronic interactions
in the crystal. Parameters γ¼γ0¼0:4, γ1¼γ0¼0:2 for all cases (see supporting information section 8.2 for the equations)
εa20
t00.20 0.70 0.83 0.95 0.20 0.70 0.83 0.95
na1.9914 1.6814 1.4592 1.2321 0.6242 0.6079 0.6387 0.6654
ni0.9470 0.9306 0.9330 0.8935 1.0098 0.9816 0.9748 0.9701
B0.3296 0.2887 0.2779 0.2313 0.3198 0.2833 0.2770 0.2728
B00.0116 0.1336 0.2417 0.2511 0.1450 0.2791 0.2923 0.3027
δE1.8720 2.2704 4.5067 2.0715 0.2247 2.0645 2.6345 3.3018
12 POPOV ET AL.
In the present Section the Greenian matrix of pure graphene
serves as a starting point for the study of several types of local
defects. We concentrate on the effect of the local perturbations on
the π-system of graphene and for the time being ignore the σ-system.
5.2.1 | Substitution defects in graphene
Substitution defects in graphene can be described in the same manner
as in cubia. Here we aim to study defects, which are closer to the
experimental situation. Specifically, we consider the graphene layer,
where one of the carbon atoms is replaced by nitrogen or boron. This
is modeled by an appropriate choice of parameters describing one-
and two-center interactions as described in Supporting information
section 9. The parameters used to study boron and nitrogen impuri-
ties in graphene lattice are collected in Table 5.
The bare perturbed subspace is spanned by four π-orbitals, one of
boron/nitrogen and three more of their nearest neighbors. The pertur-
bation itself touches the diagonal matrix element of the substituted
atom and the hopping integrals with its nearest neighbors.
For 2D graphene being a poor metal (Fermi surface degenerates to
twoDiracpoints) one can expect a slower decay of perturbation
effect with the separation from the defect. To ensure we do not miss
any significant corrections to the density matrix elements we performed
self-consistent calculations of substitutional defects using an impurity
cluster for med by 7 7 original cells, whose unperturbed Greenian
matrix was derived from the band structure of the unit cell with two
atoms Equation (22). As previously we denote the defect site as 0.
Diagonal DOS on the 0th site together with schematic repre-
sentation of the charge distribution in the perturbed systems is given
in Figure 2. The charge distribution parameters and electronic ener-
gies of the defect formation are collected in Table 6. As one can see
from the plots of DOS0ε
ðÞthe local states outside the band appear,
respectively, below the band for nitrogen (εl¼3:053) and above it
for boron (εl¼3:773). For nitrogen the local state contributes 0.38 to
the diagonal density matrix element which is 26.95% of the total
value. In the case of boron the local state is vacant and does not con-
tribute. The Fermi level in both cases shifts since nitrogen brings one
more electron to the system (hence Δn¼0:91) and boron withdraws
one electron (hence Δn¼1:10). However, the value of Δn=K is again
very small and does not affect the individual density matrix elements
significantly (we used 5151 k-mesh).
The qualitative behavior of the atomic charge distribution is fairly
similar to that observed in cubia with the only exception that values
of the charges decay much slower here, as anticipated. In both cases
the corrections to density become negligible (<0.01) beyond 6th
neighbor. This is rather reasonable response to very strong perturba-
tion induced by substitutions. Spectacular alternation of signs of the
charges induced by substitution seen in Figure 2 is nothing, but the
manifestation of the Coulson's law of alternating polarity,
70
which
had been known already to Hückel
71
and served to explain the rules
of ortho-meta-para orientation in the electrophilic substitution in ben-
zene derivatives known in organic chemistry
72
.
In complete analogy with cubia, the bond orders are much less
affected by the predominantly diagonal perturbation. This effect man-
ifests only for the bonds closest to the defect and become negligible
as from the bonds between 3rd and 4th neighbors. The order of the
bond closest to the defect decreases by 0.048 for nitrogen and 0.165
for boron. The total change in bond orders PijδBij is, however, quite
significant due to the number of bonds involved (see Table 6) and
comprises the contributions of the bond orders up to those between
3rd and 4th neighbors, therefore being more local than the variation
of diagonal densities (charges).
Energies of the defect formation have different signsnegative
for nitrogen and positive for boron. In the latter case this is the result
of the huge positive one-center contribution.
5.2.2 | Vacancy in graphene
Vacancies in graphene lattice are interesting from two viewpoints.
First, they can appear in the material during synthesis and, therefore,
can have an impact on the properties of available samples. Second,
vacancies in the π-system are tentative models of a carbon atom for-
ming an extra bond with some unsaturated particle, which acquiring
the sp3 hybridization, leaves the π-system. In the case of not self-
consistent setting analytical GFs can be found, which are given in
Supporting information section 9. Comparison of GOGREENGOnumeri-
cal GFs to analytical ones showed they coincide up to eight decimal
place. To study vacancy in a more realistic setting we again apply the
TABLE 5 Atomic parameters used in this work to describe boron
and nitrogen impurities in graphene lattice. All values are in units of t
vβγ
0γ1
C0.792 2.100 0.792
B2.488 0.914 1.228 0.699
N1.251 0.904 3.573 0.862
TABLE 6 Parameters of charge distribution in the graphene with the substitutional defects and electronic energy of the defect formation. Qi
corresponds to the atomic charge of ith neighbor of the defect site 0
Q0Q1Q2Q3Q4Q5Q6PijδBij ΔnδE
B0.74 0.39 0.24 0.22 0.16 0.14 0.09 0.76 1.10 1.04
N0.42 0.27 0.17 0.15 0.12 0.10 0.07 0.42 0.91 1.42
POPOV ET AL.13
self-consistent option of the package with one- and two-center
parameters of carbon being the same as in the previous Subection.
As before the perturbation cluster involves four orbitals one
excluded from the π-system and three its nearest neighbors. The per-
turbation itself reduces to nullifying the hopping matrix elements
between the excluded site and its neighbors. The resulting charge dis-
tribution as well as AO-projected DOS for the site nearest to the
vacancy are shown in Figure 6. Remarkably, the situation is quite differ-
ent from that of the previous Subsection. For the predominantly off-
diagonal perturbation, corrections to the diagonal densities are not that
large (Q1¼0:096, Q2¼0:050, Q3¼0:045, Q4¼0:032) and be-
come negligible beyond 4th neighbors from the vacancy. On the con-
trary, off-diagonal densities (bond orders) are affected much stronger:
δB01 ¼0:525 (quite obvious since the bond between the vacant
site and its neighbor disappears), δB12 ¼0:085, δB23 ¼0:013,
δB34 ¼0:012, δB45 ¼0:020 and all other changes occur only at a
third decimal place. With a good accuracy (up to third decimal place)
the following holds:
X
ij
δBij 3δB01 þ6δB12,ð23Þ
so that, all other oscillations of bond orders compensate each other
and the total change of the bonding energy is a sum of two local terms
energy of three broken bonds and energies of six nearest bonds,
which become stronger.
5.2.3 | Adsorption of hydrogen/alkali metal atom
on graphene
On-top adsorption of atomic hydrogen on graphene is a process of
considerable interest in material science because it leads to a
formation of sp
3
defects, which usually present in synthetic graphene
and affect its properties. It is known
73
that chemisorption of
hydrogen forces carbon atom to rise above the plane by 0.4 Å and to
form a σ-bond with the adsorbate. This causes a change of hybridiza-
tion state of the carbon atom (from sp
2
to sp
3
) and consequent
reorganization of both π- and σ-systemsthree π-bonds break and
one new σ-bond appears along with distortion of three CCσ-bonds.
Complete treatment of such process can be performed in σπ
approximation by considering π-bonds breaking as a vacancy
forms (as described above) and a rigorous evaluation of the σ-system
reorganization energy. Although theoretical basis for that has
been already established in our previous works,
57,7476
full analysis
of the problem goes beyond the scope of the present paper.
Here, we restrict ourselves by testing the effect of interaction
of hydrogenatom only with the graphene π-system, neglecting
possible distortion of graphene geometry and not touching the
σ-core. Treatment of the on-top adsorption on graphene in the frame-
work of the standard Anderson impurity model was given in Refer-
ences [7883]. Here we employ more advanced self-consistent model
taking into account electronelectron interactions at different atomic
sites.
Parameters required to describe interactions are calculated
within the MNDO setting which has shown its validity for description
of carbon allotropes.
57
We take RCH
ðÞ
¼1:1 Å The adsorbate
level lies below the Fermi level by εa¼2:150 and one-center
electronelectron repulsion on the adsorbate s-orbital is γ0
0¼3:952.
Two-center parameters for the CH pair have the following values
β0¼1:095 and γ0
1¼1:41. All energies are in the units of t, which is
known to be 2.4 eV in graphene.
77
We denote the graphene site,
interacting with the H atom, as 0. The bare perturbation acts in the
two-dimensional space (0th π-AO of graphene and adsorbate s-
orbital):
FIGURE 6 Diagonal AO-projected DOS for the atomic orbital of adsorbate (A) and for the orbital of graphene's site interacting with the H
atom (B). On the right plot (B) red curve corresponds to the initial (unperturbed) DOS and the blue oneto the perturbed DOS. Narrow peaks on
the plots above the band corresponds to the local state. On the plot a) one can also observe a virtual state being wide peak centered at the
point ca. 0.5
14 POPOV ET AL.
δF¼0β0
β00

,ð24Þ
but the size of the cluster used for the self-consistent calculations is,
of course, larger due to Coulomb interactions. To ensure we do not
miss any significant changes in the density matrix elements, we, as
previously, use the impurity cluster of 77 original graphene cells,
whose unperturbed Greenian matrix was derived from the band struc-
ture calculated with the original unit cell of two atoms.
Perturbed AO-projected DOS for adsorbate: DOSaεðÞand for the
0th π-AO of graphene: DOS0ε
ðÞare plotted in Figure 3. The adsorbate
DOS below the Fermi level consists of two broad peaks. The first one
lies in the interval 3, 1½with the maximum at ε¼εa. It is obviously
the adsorbate level broadened due to interaction with the graphene
p-band. The second peak is much smaller than the first one and lies in
the interval 1, 0½with the maximum at ε¼0:630. At the point ε¼
1 the DOS drops to zero
‡‡
due to the logarithmic singularity of the
pure graphene DOS (GF). There is a local state above the band at
ε¼3:090, which does not contribute to the electronic density. No
local states show up below the band in this case.
Adsorbate s-AO acquires a negative charge Qa¼0:7516 and the
site 0a positive one of Q0¼0:4607. The charge distribution in
graphene lattice generally follows the same pattern as in the impurity
problemthe values of the net-charges decay with the distance from
the defect with alternating signs. The decay is rather slow and the
charges become negligible (<0.01) only beyond 6th neighbors of the
adsorption site (values are collected in Table 7). Due to the alternation
of atomic charges, the contribution of two-center Coulomb interac-
tions is negative, favoring the perturbed state.
Corrections to the bond orders decay much faster with the dis-
tance than atomic charges. Coulson bond order for the C-H bond
formed upon adsorption is Ba0 ¼0:5802. Three bonds of the adsorp-
tion site with its neighbors in the layer turn weaker (δB01 ¼0:1672),
six bonds between 1st and 2nd neighbors turn stronger
(δB12 ¼0:0173) and 12 bonds between 2nd and 3rd neighbors again
weaker (δB23 ¼0:01131). All further corrections to the bond orders
are negligible (<0.01) and do not contribute significantly to the
adsorption energy. Due to the formation of the C-H bond the system
gains the energy:
δECH
b¼ 2β0þγ0
1
2Ba0

Ba0 ¼1:5080:ð25Þ
At the same time, it loses the energy due to the weakening of three
CC bonds connected with the site 0:
δECC
b¼6tδB01 ¼1:0032:ð26Þ
In total, the change in bonding energy is negative.
Electronic energy of chemisorption can be calculated by Equa-
tion (20) or alternatively one can adapt Equation (31) from Supporting
information section 5, which in both cases gives the value of δE¼
1:392 and favors the adsorption in agreement with DFT calcula-
tions.
73
However, numerical comparison of binding energies is not
possible at the moment, since the calculated electronic energy does
not include significant positive terms of corecore repulsion, which
will be added in the next release. We, also, did not take into account
rehybridization effects. The most significant negative contributions to
the binding energy come from one-center term (1:62), two-center
Coulomb and bonding terms mentioned above. The most significant
positive contribution comes from one-center repulsion of adsorbate
electrons with different spin-projections which equals to γ0
0n2
a=4¼
1:7306 (note, that we employ a non-magnetic solution for the
perturbed system, but take into account that unperturbed hydrogen
atom bears one unpaired electron).
In conclusion of this discussion, we mention that omitting the
self-consistency procedure in the above model calculations leads to
the charge distribution drastically differing from that obtained above.
In the non-self-consistent setting with the bare perturbation Equa-
tion (24) the fast-decaying charge distribution is obtained as shown in
Table 7. The local state below the band at ε¼3:05 appears, which is
not there in the self-consistent calculations. This example demon-
strates that in some cases the self-consistency is important, when
considering point defects, not only quantitatively perspective, but also
qualitatively.
6|CONCLUSION
We present the GOGREENGOpackage intended for description of point
defects in crystals as well as for analysis of adsorption processes on
surfaces. The package employs the Green's functions formalism in
order to obtain the densities of states of the perturbed crystal con-
taining defects. It builds the Green's function of the unperturbed crys-
tal from the output produced by major ab initio solid-state quantum
chemistry codes and solves the Dyson equation for the perturbed
GF. Results of this calculation are processed so that the perturbed
densities of states, charge distributions and the off-diagonal matrix
elements of the density are available in the atomic basis. The package
is tested for the electronic structure perturbations caused by various
local defects in model cubia crystals, in more realistic graphene and
TABLE 7 Atomic charges induced by the on-top chemisorption of hydrogen atom on graphene calculated within the self-consistent and non-
self-consistent settings. Qicorresponds to the atomic charge of ith neighbor of the carbon atom 0interacting with adsorbate. Qastands for the
charge of the hydrogen atom. corresponds to a negligibly small charge
QaQ0Q1Q2Q3Q4Q5Q6
self 0.7516 0.4607 0.3137 0.1942 0.1355 0.1174 0.0775 0.0538
non self 0.8372 0.1103 0.0187 0.0081 –– ––
POPOV ET AL.15
simple multiband model of p-cubium. In all cases, when the analytic
solutions were available, the package manifested perfect agreement
with the former.
ACKNOWLEDGMENTS
This work is supported by the State Task 0081-2019-0018 Funda-
mental physicochemical laws of adsorption, adsorptive separation,
and adsorptive electrochemical ion exchange processes in nanoporous
materials and fundamentals of targeted synthesis of new adsorbents.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available either in
Supplementary Materials or from the corresponding author upon rea-
sonable request.
ORCID
Andrei L. Tchougréeff https://orcid.org/0000-0002-8077-1168
ENDNOTES
* Relatively modern application of GFs to the problem of point defects in
crystals was developed in the framework of Korringa-Kohn-Rostoker
method (KKR-GF),
32
which is based on ideas of multiple-scattering the-
ory and, consequently, relies on rather different formulation of the GF
theory compared to the one used in above cited references and in the
present paper (see below).
Numerous sources are available, in the computational chemistry context
Refs. [29,30] can be recommended.
The bra-ket notation is used (see, e.g. Ref. [37]).
§
Sometimes called a mass operator.
k
It is equal to the number of unit cells used in a periodic model of the infi-
nite crystal.
40
** This is the most unfavorable situation the decay in insulators is even
faster.
††
Correction to corecore repulsion term caused by the defect is not
included.
‡‡
Since in practical calculations we do not have a genuine pole at
ε¼1 and it is approximated by a peak with small yet finite width, the
numerical DOS does not drop to zero exactly. However, this fact does
not influence an integral of DOS.
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Article
We present a software package GoGreenGo—an overlay aimed to model local perturbations of periodic systems due to either chemisorption or point defects. The electronic structure of an ideal crystal is obtained by worldwide‐distributed standard quantum physics/chemistry codes, and then processed by various tools performing projection to atomic orbital basis sets. Starting from this, the perturbation is addressed by GoGreenGo with use of the Green's functions formalism, which allows evaluating its effect on the electronic structure, density matrix, and energy of the system. In the present contribution, the main accent is made on processes of chemical nature, such as chemisorption or doping. We address a general theory and its computational implementation supported by a series of test calculations of the electronic structure perturbations for benchmark model solids: simple, face‐centered, and body‐centered cubium systems. In addition, more realistic problems of local perturbations in graphene lattice, such as lattice substitution, vacancy, and “on‐top” chemisorption, are considered. Point defects in crystals form a wide class of processes being of great importance in solid‐state chemistry. Only by considering surface chemistry one can propose a numerous examples ‐ from formation of isolated surface defects to single particle chemisorption and elementary reactions on catalysts' surfaces. Theoretical investigation of these processes, aiming to understand their mechanisms from the electronic structure perspective, presents one of many important branches of solid‐state chemistry deserving close attention. In this work we present a new software package GoGreenGo specifically designed to perform computationally effective quantum chemical calculations of local processes in solids and to provide results in “chemical” terms.
Book
I. The Born-Oppenheimer Hamiltonian. 1. Separating the center of mass motion in quantum mechanics. 1.1. Reducing the two-body problem to two one-body ones. 1.2. The center of mass in quantum mechanics 1.3. Free atoms and atomcules. 2. The Born--Oppenheimer approximation. 2.1. Introductory remarks. 2.2. The Born-Oppenheimer separation. 2.3. Why the Born-Oppenheimer separation is not exact? 2.4. Approximate decoupling. 2.5. A note on the Born-Oppenheimer separation. II. General Theorems And Principles. 1. The variation principle. 1.1. The Rayleigh quotient. 1.2. The variation principle for the ground state. 1.3. The variation principle as an equivalent of the Schrodinger equation: a useful formulation of the variation principle. 1.4. Eckart's inequality. 1.5. Excited states. 2. The Hellmann - Feynman theorem. 2.1. The differential Hellmann - Feynman theorem. 2.2. The integral Hellmann -Feynman theorem. 3. The virial theorem in quantum mechanics. 3.1. Time dependence of a physical quantity. 3.2. The virial theorem. 3.3. Scaling - a connection with the variation principle. 3.4. The virial theorem in the Born-Oppenheimer approximation. 3.5. The virial theorem and the chemical bonding. III. The Linear Variational Method And Lowdin's Orthogonalization Schemes. 1. The linear variational method (Ritz -method) 2. Lowdin's symmetric orthogonalization. 2.1. Matrix SAND-1/2. 2.2. The S∧-1/2 transformation. 2.3. The Lowdin basis. 2.4. The stationary property of Lowdin's symmetric orthogonalization scheme. 2.5. Lowdin-orthogonalization: a two-dimensional example. 3. Linear independence of the basis and Lowdin's canonic orthogonalization. 3.1. Eigenvalues of the overlap matrix: a measure for the linear. 3.2. Lowdin's canonic orthogonalization. IV. Perturbational Methods. 1. Non-degenerate Rayleigh-Schrodinger perturbation theory. 1.1. The problem. 1.2. 'Algebraic' expansion. 1.3. The use of the reduced resolvent in the Rayleigh-Schrodinger perturbation theory. 1.4. Wigner's 2n+1 theorem. 2. Variational-perturbational method: the Hylleraas-functional.3. Degenerate Rayleigh-Schrodinger perturbation theory. 4. Brillouin-Wigner perturbation theory. 4.1. The size-consistency problem. 5. Size consistency of the Rayleigh-Schrodinger perturbation. 5.1. Formal considerations based on the properties of power series. 5.2. Size consistency of the perturbational expansions. 6. Lowdin's partitioning method. V. Determinant Wave Functions. 1. Spin-orbitals. 2. Many-electron spin states. 3. Slater determinants. 3.1. Two-electron examples. 4. The antisymmetrizing operator. 4.1. The projection character of the antisymmetrizing operator. 4.2. Commutation properties of the antisymmetrizing operator. 5. Invariance of the determinant wave function with respect of. 6. Matrix elements between determinant wave functions. 6.1. Overlap. 6.2. One-electron operators. 6.3.
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