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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 GOGREENGO—an 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-top”chemi-

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

“matrix”of 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 “molecular”codes 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 “size”of 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 [8–11])

Received: 20 June 2021 Revised: 5 September 2021 Accepted: 13 September 2021

DOI: 10.1002/jcc.26766

J Comput Chem. 2021;1–17. 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

12–15

(based on an earlier work

16

) and was applied to

chemisorption in numerous works.

17–28

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 “chemical”terms 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 FP½in the functional space spanned

either by exclusively atomic states or by an assembly of plane waves

and atomic local states augmenting (PAW) the former:

εIFP½ðÞλ

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:

FP½λ

ji

¼ελλ

ji

:ð2Þ

The density Pis determined by the occupied eigenvectors λjiwhose

eigenvalues are subject to the condition ελ≤εF, 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

GεðÞ¼ℜGεðÞþiℑGεðÞ,

ℜGεðÞ¼

X

λ

λjiλhjP 1

εελ

,

ℑGεðÞ¼πX

λ

λjiλhjδεελ

ðÞ,ð7Þ

where δεðÞis the Dirac δ-”function”and 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:

P¼π1ðεF

∞

ℑGεðÞdε: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 itself—the

lattice substitution and the vacancy; and (2) interaction of the crystal

with an extra “particle”—dubbed 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

ﬃﬃﬃﬃ

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 “theoretical”GF 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

πKSpℑGεðÞ¼

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

πKℑGaa 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

πKℑGar,br0εðÞ¼

X

αk

ℜakjαk

hi

αkjbk

hi

exp ik r r0

ðÞðÞ½δεεαk

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-”function”and 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 electron–phonon 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 electron–electron 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 “dressing”which 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:

δE¼1

πðεF

∞

εεF

ðÞδℑSpGεðÞdε¼1

πðεF

∞

εδℑSpGεðÞdεΔnεF,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):

δE¼1

πðεF

∞

εεF

ðÞδℑSpGεðÞdε

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 vanishes—outside 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 test—adsorption 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 v¼1; (B) sc v¼1; (C) bcc v¼1; (D) bcc v¼1; (E) fcc v¼1; (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, blue—to 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 one—to 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-

ple—sc, body centered—bcc, and face centered—fcc) 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 [61–64] 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 electron–electron 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) v¼0:5; (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<kκ≤π

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 electron–electron 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 0.4–0.8 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 ℑG0ðÞ

00 is even function of

εand ℜG0ðÞ

00 is odd. In addition, due to the presence of the pole at

ε¼0, ℜG0ðÞ

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 v¼1tðÞcomplemented by a set of further parameters: vari-

ation of the hopping between the impurity and its neighbors

(δt¼0:2), 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 (δγ0¼0:1 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 v¼1 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 v¼1 it shifts downwards

and for v¼1—upwards. 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 orders—the off-diagonal elements of the one-

electron density matrix—are 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 v¼1, 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 cubia—single-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 σ–σinteraction and four others to π–πinteraction. 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]):

εαk¼2tcoskαþκX

β≠α

coskβ

!

ð21Þ

where α,β¼x,y,znumerates bands and projections of k-vector, κis a

ratio between π–πand σ–σhoping parameters and tstands for the

σ–σone. Minus in front of the first cosine in the brackets occurs

because the σ–σoverlap 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 “0”with

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 εF¼1:753t. 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 σ–σhopping 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: v¼0:5,

δt¼0:2, γ0¼0:6, γ1¼0:3, δγ0¼0:1 and δγ1¼0:1.

Perturbed DOS's projected on one of the p-orbitals of the defect

site “0”are 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 v¼0:5

and beyond 4th for v¼0:5. Moreover, charge signs follow a different

pattern: (1) for v¼0:5 the charges are Q0¼0:4152, Q1¼0:0195,

Q2¼0:0102, Q3¼0:0042; (2) for v¼0:5 they are Q0¼0:3039,

Q1¼0:0245, Q2¼0:0044, 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 “0”site (σ-bond

becomes weaker and π-one stronger in all cases). The total changes in

bond orders are PijδBij ¼0:1541 for the negative perturbation and

PijδBij ¼0:2414 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 cluster”in 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

“hydrogen”atom 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 t0—as 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-

ter”become 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 εa¼2.

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 electron–electron 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 εa¼2 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:

εkðÞ¼tﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

two—Dirac—points) 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 “0”th 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 (εl¼3:053) 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 Δn¼1:10). 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 signs—negative

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

C–0.792 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, Q2¼0:050, Q3¼0:045, Q4¼0:032) and be-

come negligible beyond 4th neighbors from the vacancy. On the con-

trary, off-diagonal densities (bond orders) are affected much stronger:

δB01 ¼0:525 (quite obvious since the bond between the vacant

site and its neighbor disappears), δB12 ¼0:085, δB23 ¼0:013,

δB34 ¼0:012, δB45 ¼0:020 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-systems—three π-bonds break and

one new σ-bond appears along with distortion of three C–Cσ-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,

57,74–76

full analysis

of the problem goes beyond the scope of the present paper.

Here, we restrict ourselves by testing the effect of interaction

of “hydrogen”atom 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 [78–83]. Here we employ more advanced self-consistent model

taking into account electron–electron 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 εa¼2:150 and one-center

electron–electron repulsion on the adsorbate s-orbital is γ0

0¼3:952.

Two-center parameters for the C–H 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 one—to 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, 1½with 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, 0½with the maximum at ε¼0:630. 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 Qa¼0:7516 and the

site “0”a positive one of Q0¼0:4607. The charge distribution in

graphene lattice generally follows the same pattern as in the impurity

problem—the 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 ¼0:1672),

six bonds between 1st and 2nd neighbors turn stronger

(δB12 ¼0:0173) and 12 bonds between 2nd and 3rd neighbors again

weaker (δB23 ¼0:01131). 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 ¼1:5080:ð25Þ

At the same time, it loses the energy due to the weakening of three

CC bonds connected with the site “0”:

δECC

b¼6tδB01 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 core–core 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 ε¼3:05 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 “0”interacting 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 core–core repulsion term caused by the defect is not

included.

‡‡

Since in practical calculations we do not have a genuine pole at

ε¼1 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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