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Local perturbations of periodic systems. Chemisorption and point

defects by GoGreenGo.

Ilya V. Popov1, Timofei S. Kushnir1, Andrei L. Tchougréeﬀ1

June 20, 2021

1 - A.N. Frumkin Institute of Physical Chemistry and Electrochemistry RAS, Moscow, Russia

Abstract

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, 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.

Keywords: Point defects in crystals, electronic structure, chemisorption, substitutional defects, Green’s

functions

1

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 a single particle chemisorption and elementary reactions on catalysts’ surfaces. Theoret-

ical 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 computation-

ally eﬀective quantum chemical calculations of local processes in solids and to provide results in "chemical"

terms.

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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 considerable loss of eﬃciency. Indeed, applying “molecular” codes to study a point

feature in a crystal requires tricky setting up of boundary conditions 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 clusters cut from ionic crystals or metals. A comprehensive review of the related problems is

given in [3].

An alternative and nowadays predominant approach is to accommodate 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, Refs. [6, 7] and references therein). The unit cell enlargement increases the computational

costs drastically with the scaling up to ON3, 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 features. The eﬀect of the local perturbations is then derived by

comparing 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 take this fact into account, although it seems to be proﬁtable

to avoid repetitive recalculations from the scratch and to make a better use of the eventually more precise

information about the electronic structure of the ideal crystal. Based on this idea, further embedding schemes

had been developed (for example Refs. [8, 9, 10]) which employ the information about the ideal crystal for

analysis of the defect ones. Also, these schemes are not free from numerous ad hoc assumptions leading to

an uncontrollable loss of precision and predictive capacity.

Remarkably, a detailed theory of local perturbations of periodic systems using the Green’s function (GF)

formalism had been developed yet in 60-ies [11, 12, 13, 14] (based on an earlier work [15]) and was applied

to chemisorption in numerous works [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The most characteristic

feature of this approach is that starting from a solution of the ideal periodic problem and from its local

perturbation, one obtains the answer as a correction to the unperturbed solution. By this, (i) the highly

ineﬃcient step of solving the perturbed problem from the scratch is avoided and (ii) the result is a pure

eﬀect of the perturbation. The theory is highly pedagogically explained in Ref. [28], where it is used to

provide pictorial description of the electronic structure perturbations occurring throughout chemisorption.

Otherwise more mathematical, but still suitable for a chemistry theory student exposition of the required

apparatus is given in Ref. [29].

Being actively used in 70-ies to study local defects in rather simple model solids, the GF theory was prac-

tically abandoned later and replaced by supercell periodic and ﬁnite clusters calculations [30]. Undeservedly

forgotten, this approach has been getting much less attention since then, especially in the community of

solid state chemistry. 1Although, 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 software implementing the general theory, which would be suitable for a broad range of solid state

and surface chemistry problems (with diﬀerent kinds of defects in crystals and/or chemisorption). The pur-

pose of the present work and of the proposed GoGreenGo software 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 calculations (described in “chemical” terms such

as atomic charges or bond orders), it is implemented for basis of local atomic orbitals.

The paper is organized as follows. In the Theory and Implementation Sections we sketch the generalized

theory and implementation details of the programs included in the GoGreenGo package. After that

we provide test results for various benchmarks by considering substitutional and interstitial defects in the

model cubic lattices. Finally, GoGreenGo package is applied to tackle more realistic problems devoted to

1Relatively modern application of GFs to the problem of point defects in crystals was developed in the framework of Korringa-

Kohn-Rostoker method (KKR-GF) [31], which is based on ideas of multiple-scattering theory 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).

3

graphene. One can ﬁnd more detailed speciﬁcations of the GoGreenGo by following the link [32]. The

Supporting Material collects analytical results used for the testing.

Theory Sketch

Structure of the electronic problem

Theoretical basis of the proposed software development is the self-consistent perturbation theory of many-

electron systems [33]. It starts from representing the electronic structure either by a single Slater determinant

formed by one-electron (spin-)orbitals for the wave function based methods (as exempliﬁed by e.g. [4, 34,

35]) 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 density dependent Fockian matrix F[P]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:

(εI−F[P]) |λi= 0.(1)

Solutions of eq. (1) are the eigenvalue-eigenvector pairs ελ,|λinumbered by assemblies of quantum numbers

λand satisfying the well known relations:

F[P]|λi=ελ|λi.(2)

The density Pis determined by the occupied eigenvectors |λiwhose 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 eqs. (1), (2) are sequentially solved

until the convergence for Pis achieved.

Green’s function’s representation

The eigenvalue-eigenvector problem can be alternatively formulated in terms of the quantity2:

G(z)=(zI−F)−1,(3)

the famous Green’s function of a complex argument z. Since the Fockian F(hereinafter, we omit its P

dependence for brevity) is a Hermitian operator, its eigenvalues ελare always real. Thus, for an arbitrary

complex zunequal to any of ελthe matrix (zI−F)is nondegenerate and can be inverted producing a z-

dependent quantity eq. (3). Its closest relation to the eigenvalue-eigenvector problem stems from the spectral

representation:

G(z) = X

λ

|λihλ|

z−ελ

,(4)

which immediately derives from the expressions for the identity matrix and the Fockian in the basis of its

eigenvectors:

I=X

λ|λihλ|;F=X

λ

ελ|λihλ|.

As soon as the GF is known in the basis of the eigenvalues of F, where it is diagonal, it is known in whatever

basis. E.g. in the basis of local atomic spin-orbitals |ai,|bi, ... a Greenian matrix is formed by the elements

Gab (z) = X

λ

ha|λihλ|bi

z−ελ

,(5)

where ha|λiare expansion coeﬃcients of the eigenvector |λiover atomic basis.3

2Numerous sources are available, in the computational chemistry context Refs. [28, 29] can be recommended.

3The bra-ket notation is used (see, e.g. Ref. [36]).

4

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 a lot of poles on the real axis, it should be considered

there as a distribution (or a generalized function). Such distribution is deﬁned as a limit :

G(ε) = lim

ν→0+

G(ε+iν),(6)

where zis set to be ε+iν with real εand ν. As described in Appendix A, evaluation of the limit entering

the later equation leads to the following GF on the real axis

G(ε) = <G(ε) + i=G(ε),

<G(ε) = X

λ|λihλ|P1

ε−ελ,(7)

=G(ε) = −πX

λ|λihλ|δ(ε−ελ),

where δ(ε)is the Dirac δ-”function” and Pindicates that the integral of a function f(ε), multiplied by

(ε−ελ)−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 (eq. (29) of Appendix A) takes the form:

P=−π−1ˆεF

−∞ =G(ε)dε. (8)

Perturbations in terms of Green’s function

The GF formalism as sketched in the previous Subsection 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 F(0) and a perturbation F0:

F=F(0) +F0,

the Dyson equation

G(z) = G(0) (z) + G(0) (z)F0G(z)(9)

holds [37]. Being solved for G(z), it gives:

G(z) = I−G(0) (z)F0−1

G(0) (z),(10)

which generates the perturbation series if one expands the inverse matrix in the geometric series:

I−G(0) (z)F0−1=I+G(0) (z)F0+G(0) (z)F0G(0) (z)F0+...

G(z) = G(0) (z) + G(0) (z)F0G(0) (z) + G(0) (z)F0G(0) (z)F0G(0) (z) + ...

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 GoGreenGo we employ this possibility and consider the

perturbation matrices of a form:

F0=F0

P P 0

0 0 =V0

0 0 .(11)

The Greenian matrix is then split in blocks:

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G(0) = G(0)

P P G(0)

P Q

G(0)

QP G(0)

QQ !,(12)

where Qrefers to the orthogonal complement of the subspace P(argument zis omitted for brevity). Intro-

ducing a dim P×dim Pmatrix M:4

M=VIPP −G(0)

P P V−1,(13)

and following the procedure given in Appendix B, one obtains the corrections to the matrix blocks:

GP P −G(0)

P P =G(0)

P P MG(0)

P P ,

GQQ −G(0)

QQ =G(0)

QP MG(0)

P Q,(14)

GP Q −G(0)

P Q =G(0)

P P MG(0)

P Q ,GQP −G(0)

QP =G(0)

QP MG(0)

P P ,

which, respectively, express the eﬀect of the perturbation on the Green’s function in the subspace Pitself,

in the subspace where the perturbation is absent (Q) and on the coupling between the perturbed and

unperturbed subspaces. Density matrix elements of the perturbed system are calculated with eq. (8) once

the Dyson equation is solved. If one takes into account the density dependence of the Fockian, the calculated

density serves as an input for a next step of iterative solution. This setting is referred below as the self-

consistent one.

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: (i) point-wise

perturbation of the crystalline matrix itself – the lattice substitution and the vacancy; and (ii) 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 G(0) of the

ideal crystal (and of the ﬁnite adsorbate in the case (ii)) is assumed to be known. Below we brieﬂy review

the speciﬁc of the greeninstic formalism as applied to inﬁnite periodic systems with point defects.

Green’s functions of crystals

When it comes to solids, the solution of the eigenvalue-eigenvector problem has speciﬁcs described in hand-

books on solid state physics and chemistry [38] (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:

|aki=1

√KX

r

exp (ikr)|ari

of Aatomic states. Here Ais the number of atomic spin-orbitals a= 1÷A,rstands for a unit cell index and

Kis the number of k-points involved in the calculation.5The blocks are numbered by the wave vectors kfrom

the ﬁrst Brillouin zone and respective eigenvalues form Afunctions εαk of k(α= 1 ÷A) - (electronic) bands

with the eigenvector expansion coeﬃcients forming k-dependent A×Amatrices with elements hαk |aki. 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:λ= (α, k). Thus, the GF of the ideal crystal with Abands reads:

G(z) = X

αk

|αkihαk|

z−εαk

.(15)

It is as well block-diagonal with A×Ablocks 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

4sometimes called a mass operator.

5It is equal to the number of unit cells used in a periodic model of the inﬁnite crystal [39].

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electrons (see [38]). Considering the GF on the real axis according to eq. (7) produces the electronic density

of states (DOS):

DOS (ε) = −1

πKSp=G(ε) = X

αk

δ(ε−εαk),(16)

familiar to the workers of the ﬁeld.

Applying the same trick to the diagonal elements of the Greenian matrix eq. (5) in the basis of local

atomic orbitals

Gaa (z) = X

αk

hak |αkihαk |aki

z−εαk

,

we arrive to

DOSa(ε) = −1

πK=Gaa (ε) = X

αk hak |αkihαk |akiδ(ε−εαk )

– the projection of the DOS upon the atomic state a, as well, familiar from numerous packages (e.g. lobster

[40] or wannier90 [41]) performing analysis of numerical data derived from PAW-DFT or whatever computer

experiments on solids (see e.g. [40, 42]). Treating similarly oﬀ-diagonal elements:

Gar,br0(z) = X

αk

hak |αkihαk |bkiexp (ik (r−r0))

z−εαk

,

yields:

−1

πK=Gar,br0(ε) = X

αk <[hak |αkihαk |bkiexp (ik (r−r0))] δ(ε−εαk )(17)

which is a close relative of the crystal orbital overlap and crystal orbital Hamilton populations (respectively,

COOP and COHP [43]) as well widely available in the solid state packages. In these expressions |ari

corresponds 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.

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 eq. (5), which for the real values of argument reduces to:

Gab (ε) = X

λha|λihλ|bi

ε−ελ−iπ ha|λihλ|biδ(ε−ελ),(18)

where the summation goes over discrete levels λ. The imaginary part of the GF consists of discrete signals

located at energies ελand proportional to the Dirac’s δ-”function”. The real part is a continuous function

except simple poles at ε=ελ.

In the general case it is not possible to ﬁnd an analytical solution for the perturbed GF, therefore one has

to treat the initial Greenian matrix of the ﬁnite system numerically. Such treatment requires to approximate

the δ-function in eq. (18) by a Lorentzian of a (small) width νas explained in Appendix A. 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.

Implementation and Computational Details

General construction

The programmatic implementation of the theory described in the Theory Section is provided by GoGreenGo

package being a set of procedures written in fortran 2010 using the system of high-level objects provided

7

by the cartesius_fort library [44]. The component procedures exchange data through intermediate ﬁles

packed in the hdf5 archive [45]. The ﬂow-chart illustrating the relations between diﬀerent programs of the

package and the paths of data transfer between them is shown on the Fig.1. Speciﬁcally,

•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 [46], ABINIT

[47]. The projections of eigenvectors produced by these packages to the basis of local orbitals can be

obtained by the lobster software [40] whose output format is compatible with get_green. As well

ΘΦ (TetaPhi) [48] produces eigenvalues-eigenvectors in the format readable by GoGreenGo. This

combination allows to extract Greenian matrix of the pure solid in an atomic basis from the most

popular plane wave DFT codes and use it for impurity calculations. 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

[49] 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 Ref. [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 eq.(17) with no calculations on extended

unit cells.

•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 parameterizations 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 re-

leases. To run the hgen utility one has to provide geometries of the defect and crystal, deﬁne which

atoms interact (by giving a cut-oﬀ distance or listing them explicitly) and type of 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.

•program dyson reads unperturbed Greenian matrix of the ideal crystal (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 perturbation 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 perturbation. In the case of the self-consistent setting it performs iterative

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.

•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.

Some speciﬁc features

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 further corrections proportional to those of the density matrix elements. They form so called

8

Figure 1: Flow-chart of the GoGreenGo package.

“dressing” which needs to be added to the original 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 GoGreenGo package.

In metals the eﬀect of perturbation decays with the distance from the defect as R−2κwith some κ > 0

dependent on the from of the Fermi surface and dimensionality of the crystal structure (κis typically higher

for higher dimension) [27].6This 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.

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 G(ε)

diﬀers from unperturbed one G(0) (ε)and thus the integral of SpG(ε)from −∞ up to εFof the unperturbed

crystal not necessarily yields the same number of electrons as does the integration of SpG(0) (z). The defect

of electron’s number ∆nis to be eliminated by shifting the Fermi energy by δεF. As it is explained in

Appendix C the value of δεFis determined by the value of ∆n

/Kand it, evidently, becomes inﬁnitesimally

small for the limit K→ ∞. In real calculations Kis ﬁnite and δεFis non-vanishing in this case. Evaluation

of δεFrequired to keep constant the number of electrons in the system with Kunit cells is implemented in

the GoGreenGo. Practically, in most our calculations the value of ∆n

/Kis rather small (we use 31×31 ×31

meshes of k-points for three-dimensional models and 51 ×51 one for graphene) and aﬀects density matrix

elements only in fourth-ﬁfth decimal place. However, in all cases it is advised to thoroughly check its impact

on the density matrix elements and ﬁnal results, especially if the number of k-points Kused for the band

structure calculations of the crystal is relatively small as it sometimes happens in PAW-DFT calculations.

6This is the most unfavorable situation – the decay in insulators is even faster.

9

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 [28] that

the perturbation energy is:

δE =−1

πˆεF

−∞

(ε−εF)δ=SpG(ε)dε =−1

πˆεF

−∞

εδ=SpG(ε)dε −∆nεF,(19)

where ∆nis a diﬀerence in number of electrons in the perturbed system (calculated with the original Fermi

level of the unperturbed system) and the initial one. In the self-consistent version of the calculations the

modiﬁed equation [24]:

δE =−1

πˆεF

−∞

(ε−εF)δ=SpG(ε)dε−

−1

2X

i,j,k.l Pij Pkl (ij||kl)−P(0)

ij 0P(0)

kl 0(ij| |kl)0(20)

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 perturbed subspace P. Due to the decaying eﬀect of the perturbation, it is a

ﬁnite set as explained above. Two-center two-electronic integrals in this equation change in the perturbed

system only if the geometry gets distorted.

Local and virtual states

As it was stressed yet in works [11, 19], the defects may produce additional 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 [18] and manifest themselves on the real axis as wide

Lorentzian peaks of the perturbed DOS inside the allowed energy band (see Appendix D). Both features are

perfectly reproduced by the GoGreenGo package as demonstrated in Figs. 4, 6. For more details about

handling the pole structure of the perturbed GF in GoGreenGo see Appendix D.

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 reference 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 Materials. Even in the case of simple models

not all functionality of the package can be tested against analytical solutions, since the later are available

only for the simplest local perturbations and are inaccessible in the frame of the self-consistent approach.

Therefore, we test GoGreenGo against very simple benchmarks and then demonstrate its capabilities for

more realistic and comprehensive setting.

GoGreenGo for perturbation of cubia

Cubia (see e.g. Ref.[60]) are simplest thinkable models of 3D metals. They are formed by s-orbitals centered

at the vertices of (simple - 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. Correspondingly, the eigenvectors

10

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.

εkW εFB

sc −2t(cos kx+ cos ky+ cos kz) 12t0 0.3324

bcc −8tcos kxcos kycos kz16t0 0.2605

fcc −4t(cos kxcos ky+ cos kxcos kz+ cos kycos kz) 16t0.915t0.2184

(Bloch states) related to these eigenvalues are known and, consequently, the Green’s functions. Speciﬁcally,

diagonal GFs for the systems listed in Table 1 are known from Refs. [61, 62, 63, 64] and their plots are

presented in Supporting Material Sec. 1. 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 Appendix E. In our sub-

sequent 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 t= 1 everywhere below. The diagonal

matrix elements of the unperturbed Fockian are set to zero being by this the energy reference. In further

Subsections we present as coherence tests the results of numerical treatment of various local perturbations

of cubia with use of GoGreenGo package.

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 corresponds to substitution defects in alkali metals.

Complexity of the model can be gradually increased by including diﬀerent terms into the perturbation

operator so that diﬀerent program features are tested independently. In the simplest possible setting we

neglect electron-electron interactions and only take into account one-center perturbation: the diﬀerence v

between 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 Material Sec. 3). Comparing numerical results to

this solution shows that GoGreenGo produces the perturbed Greenian matrix identical to the analytical

one up to eighth decimal sign. In addition, this setting allows to make qualitative sketch of the features of

the perturbed system, which remain valid for more involved crystals and perturbations of this kind. As one

can conclude from the analytical form of the mass operator (see Appendix D) and of the cubia GFs G(0)

00

(Supporting Material Sec. 1), the additional poles in the perturbed system appear for: (i) sc |v|>4.01; (ii)

bcc |v|>6.10; (iii) 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 v

is 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 bandwidth). The only option is an appearance of the vacant local state in the fcc

lattice, but it presents a minor interest since it does not contribute to the electronic density and the energy

of the system. 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 =G(0)

00 is even function of ε

and <G(0)

00 is odd. In addition, due to the presence of the pole at ε= 0,<G(0)

00 has a discontinuity there.

Therefore, one would observe a discontinuity of the perturbed function DOS0(ε)at ε= 0, which is indeed

observed in our numerical results described below (see Fig. 2).

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 corrections.

GoGreenGo supports such description and to test this, we included corrections to one-center two-electron

and two-center hopping integrals, Coulomb interactions, and applied the self-consistent procedure. We

performed calculations within this setting for cubia lattices for v=±1(t)complemented by a set of further

11

Table 2: Parameters of charge distribution in cubia with the substitutional defect and electronic energy of

the defect formation. Qi= 1 −nicorresponds to the atomic charge of i-th neighbor of the defect site "0"

with nibeing the electronic population of the i-th site in the perturbed system.

v Q0Q1Q2Q3Pi∼jδBij ∆n δE

sc

−1–0.3478 0.1216 –0.0712 0.0555 –0.1620 0.0882 –1.9964

10.3478 –0.1217 0.0711 –0.0555 –0.1594 –0.0899 –0.0029

bcc

−1–0.4067 0.1058 –0.0835 –0.0389 –0.0600 0.1488 –2.0900

10.4043 –0.1088 0.0819 0.0379 –0.0592 –0.0121 –0.0866

fcc

−1–0.1984 0.0243 –0.0115 –0.0025 –0.0532 0.0294 –2.1583

10.2903 -0.0293 0.0116 0.0025 –0.0413 –0.0766 –0.2671

parameters: variation of the hopping between the impurity and its neighbors (δt =−0.2), one-center Coulomb

repulsion in the unperturbed system (γ0= 0.6), two-center (nearest neighbor) Coulomb repulsion in the

unperturbed system (γ1= 0.3) and a variation of the Coulomb integrals in the defect (δγ0=−0.1and

δγ1= 0.1). The resulting perturbed DOS0(ε)together with obtained charge distributions are depicted in

Fig.2; numerical values of charges and the electronic7energy variations 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 v=−1and positive (electron density depletion) for v= 1. 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 neighbors. The decay rate is rather high as expected for 3D metals

[27]. The fastest one is observed for fcc, where the induced charges are insigniﬁcant (<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 Implementation Section and Appendix C, but the absolute

values of δεFdo not exceed 7·10−6. For v=−1it shifts downwards and for v= 1 – upwards. Such a small

shift insigniﬁcantly aﬀects individual density matrix elements. Maximal correction of the diagonal density

matrix elements due to the Fermi level shift equals to ∆n

/K.For 31 ×31 ×31 k-mesh used in our calculations

and values of ∆nfrom Table 2 they never exceed 5·10−5for 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 Pi∼jδBij 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 eq. (20). An alternative expression for it

is given in Appendix E. 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 v=−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 distribution, and

the correction to the bonding energy, which is always positive as we saw above.

Substitution defects in p-cubium

In the previous Subsection, we have tested the main functionality of GoGreenGo for 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

correspond to σ-σinteraction and four others to π-πinteraction. In addition we assume the following for

7Correction to core-core repulsion term caused by the defect is not included.

12

Figure 2: Perturbed densities of states DOS0(ε)(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 v=−1; b) sc v= 1; c) bcc v=−1; d) bcc v= 1; e) fcc v=−1; f) fcc v= 1. 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.

13

the two-electron Coulomb integrals (ab|cd)=(aa|cc)δabδcd, which allows to keep the same number of two-

electron parameters as previously. The analytical band-structure for the ideal pc contains three degenerate

bands (see e.g. Ref. [60]):

εαk =−2t

−cos kα+κX

β6=α

cos kβ

(21)

where α, β =x, y, z numerates 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 consideration we set κ= 0.40 and t= 1. The elements of the Greenian matrix for pc are

accessible through GoGreenGo and can be found in Supporting Material Sec. 1. We consider pc with one

electron per unit cell, yielding the Fermi level to be εF=−1.753t. Coulson bond-order Bσfor the σ-bond

is −0.2753 and for the π-bond is Bπ= 0.1474. Note, that Bσis negative and being combined with positive

σ-σhopping produces a negative contribution to the energy. In our further discussion of overall change in

the bond orders it is convenient, therefore, to use values of |Bσ|.

To test the whole functionality of the program in the case of multi-band crystal we consider substitutional

defects in the most rigorous setting by taking into account the self-energy correction and changes of one- and

two-center integrals as it was done in the previous Subsection. The set of parameters used for calculations:

v=±0.5,δt =−0.2,γ0= 0.6,γ1= 0.3,δγ0=−0.1and δγ1= 0.1.

Perturbed DOS’s projected on one of the p-orbitals of the defect site "0" are plotted on Fig. 3 for the

negative and positive perturbation. 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 v=−0.5and beyond 4th for v= 0.5. Moreover, charge signs follow a diﬀerent pattern: (i)

for v=−0.5the charges are Q0=−0.4152,Q1= 0.0195,Q2= 0.0102,Q3=−0.0042; (ii) for v= 0.5they

are Q0= 0.3039,Q1=−0.0245,Q2=−0.0044,Q3= 0.0217. 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 nonsymmetric location of the Fermi

level in the band (like in fcc cubium). Values of ∆nare 0.1523 for the negative perturbation and −0.2524

for the positive one. As previously, such small values do not cause signiﬁcant shift of the Fermi level for our

31 ×31 ×31 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

Pi∼jδBij =−0.1541 for the negative perturbation and Pi∼jδBij =−0.2414 for the positive perturbation,

so the system loses bonding energy, upon formation of the defect.

One-orbital interstitial impurity in simple cubium

Further class of problems accessible for GoGreenGo is that of interactions 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 illustrating

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 augmented 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 lattice (eight neighbors). These nine orbitals form the perturbation subspace (the P-subspace of eq.

(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, considered numerically the eﬀect of interaction on the atomic

14

Figure 3: Perturbed densities of states DOS0(ε)(blue lines) compared with the ideal ones (red lines) for the

substitution defect in p-cubium : a) v=−0.5; b) v= 0.5. Color scheme is the same as in Figure 2.

Table 3: GoGreenGo results obtained for the hydrogen impurity atom in the cubic void with the Anderson

Hamiltonian (Supporting Material Sec. 3.2). Parameter γ= 0.40 for all cases.

εa−2 0

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

δE –1.8278 –2.2008 –2.3433 –2.0104 –0.2086 –2.0233 –2.6101 –3.0670

charges and bond orders in the "impurity cluster" in the frame of the Anderson model (see Supporting

Material Sec. 3.2). Analytical solutions for the Anderson model used for testing are given in Supporting

Material Sec. 3.2.

Perturbed GFs obtained numerically with GoGreenGo for a “hydrogen” atom in a cubic void (for

the ﬁrst step of the self-consistent procedure) were compared with corresponding analytical equations from

Supporting Material Sec. 3.2. 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 G(0) even for the sc cubium.

Numerical values are collected in the Table 3. As one can conclude, if the adsorbate level is positioned below

the Fermi level the impurity atom acquires a negative charge, while for εa=εF= 0 the charge is positive.

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 Material Sec. 3.2.). Atoms of the crystal matrix in all cases acquire a small positive

charge.

Bond orders Bbetween the crystal atoms of the "impurity cluster" 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 εa= 0 the bond orders B0are larger than for εa=−2.

In a more extended setting, we turn on the Coulomb interactions in the crystal. In this case, non-vanishing

15

Table 4: GoGreenGo results obtained for the hydrogen impurity atom in the cubic void with taking into

account inter-electronic interactions in the crystal. Parameters γ=γ0= 0.4,γ1=γ0= 0.2for all cases (see

Supporting Material Sec. 3.2 for the equations).

εa−2 0

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

δE –1.8720 –2.2704 –4.5067 –2.0715 –0.2247 –2.0645 –2.6345 –3.3018

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 Coulomb terms and the self-energy correction

does not have major inﬂuence on the resulting density matrices and energies; all trends remain fairly the

same with slight numerical modiﬁcations. The only exception is the case of εa=−2and t= 0.83, where one

observes more signiﬁcant diﬀerences in density matrix elements na,Band B0between two settings, which

translates into quite considerable diﬀerence in energies.

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 theoretical 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:

ε(k) = ±tq3 + 2 coskx+ 2 cos kx+ 2 cos (kx−ky)(22)

where "−" corresponds to the ﬁlled band and "+" to the empty one. Eq. (22) allows analytical evaluation

of the diagonal DOS [69] (see also Supporting Material Sec.2 ). The Fermi level εF= 0 and the Coulson

bond order B= 0.525 between nearest neighbors.

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.

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 Material Sec. 4. The

parameters used to study boron and nitrogen impurities 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 perturbation 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 formed by 7×7original cells, whose unperturbed Greenian

16

Table 5: Atomic parameters used in this work to describe boron and nitrogen impurities in graphene lattice.

All values are in units of t.v β γ0γ1

C - 0.792 2.100 0.792

B 2.488 0.914 1.228 0.699

N -1.251 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. Qicorresponds to the atomic charge of i-th neighbor of the defect site "0".

Q0Q1Q2Q3Q4Q5Q6Pi∼jδBij ∆n δE

B 0.74 –0.39 0.24 –0.22 –0.16 0.14 –0.09 –0.76 –1.10 1.04

N –0.42 0.27 –0.17 0.15 0.12 –0.10 0.07 –0.42 0.91 –1.42

matrix was derived from the band structure of the unit cell with two atoms eq. (22). As previously we

denote the defect site as "0".

Diagonal DOS on the "0"-th site together with schematic representation of the charge distribution in

the perturbed systems is given in Fig. 4. The charge distribution parameters and electronic energies of the

defect formation are collected in Table 6. As one can see from the plots of DOS0(ε)the local states outside

the band appear, respectively, below the band for nitrogen (εl=−3.053) and above it for boron (εl= 3.773).

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 contribute. The Fermi level in both

cases shifts since nitrogen brings one more electron to the system (hence ∆n= 0.91) and boron withdraws

one electron (hence ∆n=−1.10). However, the value of ∆n

/Kis again very small and does not aﬀect the

individual density matrix elements signiﬁcantly (we used 51 ×51 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 perturbation induced by substitutions. Spectacular alternation of signs of the charges induced by

substitution seen in Figure 4 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 benzene 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 manifests 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 Pi∼jδBij 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.

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 forming an extra bond with some unsaturated particle,

which acquiring the sp3hybridization, leaves the π-system. In the case of not self-consistent setting analytical

GFs can be found, which are given in Supporting Materials Sec. 4. Comparison of GoGreenGo numerical

GFs to analytical ones showed they coincide up to eight decimal place. To study vacancy in a more realistic

setting we again apply the 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

17

Figure 4: Perturbed diagonal densities of states DOS0(ε)in 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 Fig. 2. Narrow peaks on the plots above the band in panel a)

and below the band in panel b) correspond to the local states formed predominantly of the defect orbital.

18

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.

nearest neighbors. The perturbation itself reduces to nullifying the hopping matrix elements between the

excluded site and its neighbors. The resulting charge distribution as well as AO-projected DOS for the site

nearest to the vacancy are shown in Fig.5. Remarkably, the situation is quite diﬀerent from that of the

previous Subsection. For the predominantly oﬀ-diagonal perturbation, corrections to the diagonal densities

are not that large (Q1= 0.096,Q2=−0.050,Q3= 0.045,Q4=−0.032) and become negligible beyond

4th neighbors from the vacancy. On the contrary, oﬀ-diagonal densities (bond orders) are aﬀected much

stronger: δB01 =−0.525 (quite obvious since the bond between the vacant site and its neighbor disappears),

δB12 = 0.085,δB23 =−0.013,δB34 = 0.012,δB45 =−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

i∼j

δBij ≈3δB01 + 6δB12,(23)

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.

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 sp3defects, 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 hybridization state of the carbon

atom (from sp2to sp3) 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–C σ-bonds. Complete treatment of such process

can be performed in σ−πapproximation 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 [56, 74, 75, 76], full analysis of the problem goes beyond

19

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 framework of the standard Anderson

impurity model was given in Refs. [78, 79, 80, 81, 82, 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 [56]. We take R(C−H) = 1.1Å. The adsorbate level lies

below the Fermi level by εa=−2.150 and one-center electron-electron repulsion on the adsorbate s-orbital

is γ0

0= 3.952. Two-center parameters for the C-H pair have the following values β0= 1.095 and γ0

1= 1.41.

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 (0-th π-AO

of graphene and adsorbate s-orbital):

δF =0−β0

−β00,(24)

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 7×7original graphene cells, whose unperturbed Greenian matrix was

derived from the band structure calculated with the original unit cell of two atoms.

Perturbed AO-projected DOS for adsorbate:DOSa(ε)and for the 0-th π-AO of graphene: DOS0(ε)are

plotted in Fig. 6. The adsorbate DOS below the Fermi level consists of two broad peaks. The ﬁrst one lies

in the interval [−3,−1] with the maximum at ε=εa. It is obviously the adsorbate level broadened due to

interaction with the graphene π-band. The second peak is much smaller than the ﬁrst one and lies in the

interval [−1,0] with the maximum at ε=−0.630. At the point ε=−1the DOS drops to zero 8due to the

logarithmic singularity of the pure graphene DOS (GF). There is a local state above the band at ε= 3.090,

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 Qa=−0.7516 and the site "0" a positive one of Q0= 0.4607.

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. The

net-charges of the neighbors from 1st to 6th are Q1=−0.3137,Q2= 0.1942,Q3=−0.1355,Q4= 0.1174,

Q5=−0.0775 and Q6= 0.0538. Due to the alternation of atomic charges, the contribution of two-center

Coulomb interactions is negative, favoring the perturbed state.

Corrections to the bond orders decay much faster with the distance than atomic charges. Coulson bond

order for the C-H bond formed upon adsorption is Ba0= 0.5802. Three bonds of the adsorption site with

its neighbors in the layer turn weaker (δB01 =−0.1672), six bonds between 1st and 2nd neighbors turn

stronger (δB12 = 0.0173) and twelve bonds between 2nd and 3rd neighbors again weaker (δB23 =−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:

δEC−H

b=−2β0+γ0

1

2Ba0Ba0=−1.5080.(25)

At the same time it loses the energy due to the weakening of three C-C bonds connected with the site "0":

δEC−C

b=−6tδB01 = 1.0032.(26)

In total, the change in bonding energy is negative.

Electronic energy of chemisorption can be calculated by eq. (20) or alternatively one can adapt eq. (57),

which in both cases gives the value of δE =−1.392 and favors the adsorption in agreement with DFT

calculations [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

8Since in practical calculations we do not have a genuine pole at ε=−1and 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.

20

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.

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

/4= 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 eq. (24), the following fast-decaying charge distribution

Qa=−0.8372,Q0= 0.1103,Q1=−0.0187,Q2= 0.0081 is obtained (other charges are negligible). The

local state below the band at ε=−3.05 appears, which is not there in the self-consistent calculations. This

example demonstrates that in some cases the self-consistency is important, when considering point defects,

not only quantitatively perspective, but also qualitatively.

Conclusion

We present the GoGreenGo package 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 containing defects. It builds the Green’s function of

the unperturbed crystal 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 various local defects in model cubia crystals, in more

realistic graphene and simple multiband model of p-cubium. In all cases, when the analytic solutions were

available, the package manifested perfect agreement with the former.

21

Data Availability Statement

The data that support the ﬁndings of this study are available either in Supplementary Materials or from the

corresponding author upon reasonable request.

Acknowledgments

This work is supported by the State Task 0081-2019-0018 «Fundamental physicochemical laws of adsorp-

tion, adsorptive separation, adsorptive electrochemical ion exchange processes in nanoporous materials and

fundamentals of targeted synthesis of new adsorbents».

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A Some elements of the general theory of GF

Somewhat more reﬁned explanation of relations between the GF and corresponding eigenvector-eigenvalue

problem along with the separation of the GF into real and imaginary parts can be built upon the theory of

residues of the functions of complex variable [84]. Since G(z)given by eqs. (4), (5) is continuous everywhere

in the complex plane except for the simple, ﬁrst-order poles located on the real axis at the points z=ελ,

one can express the solution of the eigenvector-eigenvalue problem in terms of the GF. Its poles, obviously,

correspond to the eigenvalues, while residues:

Res

ελ

G(z) = 1

2πi ˛

Cελ

G(z)dz = lim

z→ελ

(z−ελ)G(z) = |λihλ|(27)

are the projection operators to the eigenvectors of the Fockian F. In eq. (27) the ﬁrst equality is the

deﬁnition of the residue, the second one is the theorem expressing the integral over a contour encircling

a simple pole, and the last one is the result of the calculation for the ﬁrst-order poles. Correspondingly,

the density (matrix) of the system in its ground state which is the sum of the operators projecting to the

occupied |λi’s expresses through the integral over the contour Cencircling all poles ελ≤εFFig.7:

P=X

λ

ελ≤εF

|λihλ|=1

2πi ˛

C

G(z)dz. (28)

The density matrix elements in the basis of atomic functions|ai,|bi, ... can be found by integration of

the matrix elements Gab of the GF over the contour Cencircling all poles ελ≤εF:

Pab =1

2πi ˛

C

Gab (z)dz. (29)

In physics books (and in the main text) these results are frequently represented with use of the intuitively

less clear recipe involving the δ-”functions” and Kramers-Kronig relations as applied to the GF’s of the real

argument. It tacitly employs the crucial fact that in the expressions for the physical quantities the GF

appears under the integral over the real axis, where it has a plenty of poles. Thus, the Green’s functions

25

Figure 7: A form of the contour C[29] used to calculate integrals of the GF – the elements of the perturbed

density matrix. Black crosses are the poles of the unperturbed GF referring to the allowed energy band of

the crystal. Blue cross on the abscissa to the left of the allowed band is the additional pole referring to the

local state outside the allowed band which appeared due to perturbation – due to absence of intrinsic width

it generates extremely narrow peaks in the DOS which can be seen in Figs. 4, 5. The cyan cross represents

an additional pole within the allowed band. The corresponding virtual state provides a contribution to the

perturbed DOS which has an energy spread proportional to the unperturbed DOS occurred at the energy

corresponding real part of the pole. Similarly, the red cross represent a pole within the allowed energy band,

but close to the Fermi level.

(on the real axis) are not functions in the usual sense anymore (although they are in the complex plane),

rather they are distributions – linear functionals on the space of the true functions and the item of interest

are the values of these functionals as such and the limits they may converge to. The reasoning is based on

a high-school formula for the inverse of a complex number:

z=x+iy ⇒1

z=x−iy

x2+y2,

where zis set equal to ε+iν with an energy εand (half-)width ν. Then GF separates into real and imaginary

parts:

G(z) = X

λ|λihλ| ε−ελ

(ε−ελ)2+ν2−iν

(ε−ελ)2+ν2!=<G(z) + i=G(z).(30)

The second term in the brackets above is nothing but the Lorentzian of half-width νmultiplied by iπ –

one of the well-known probability (Cauchy) distributions. Making ν→0+nulliﬁes it everywhere except for

ε=ελwhere it diverges: very unnatural and peculiar behavior for a true function. Only, bearing in mind

the intentional use of the above expression as of a multiplier in an integrand over εand not as a function of

εper se, and then, taking the limit while ν→0+, supplies meaning to the formal expression:

lim

ν→0+

1

ε+iν −ελ

=P1

ε−ελ−iπδ (ε−ελ)(31)

abundant in physics books. It can be shown that if multiplied by a function f(ε)then put under the integral

and integrated ﬁrst and making the ν→0+afterwards produces exactly the value of fat ελ:f(ελ)-

that what the Dirac δ-function is expected to do. Analogously, treating the ﬁrst term in the brackets as a

multiplier to an integrand we arrive at the Hilbert transform of the function f(ε).

26

B Matrix solution of the Dyson equation for local perturbation

For the local perturbation F0which aﬀects only a subspace Pof the entire space of the one-electronic states

it is natural to divide the latter into Pand its orthogonal complement Q. The matrix to be inverted in order

to solve the Dyson equation then acquires a special form:

I−G(0)F0= IP P −G(0)

P P V0

−G(0)

QP V IQQ !(32)

where subscripts refer to the subspaces where the respective matrices act and the local perturbation Vacts

only in P(is a dim P×dim Pmatrix) . Although the matrix to be inverted has a very large (or even inﬁnite)

dimension of the entire orbital space of the crystal, its special structure permits to ﬁnd its inverse relatively

easily [85]:

I−G(0)F0−1=

IP P −G(0)

P P V−10

G(0)

QP VIP P −G(0)

P P V−1

IQQ

,(33)

that is, only a dim P×dim Pmatrix needs to be actually inverted (the rest is accessed through matrix

multiplication). The correction to the unperturbed Greenian matrix caused by perturbation is then:

G−G(0) =

IPP −G(0)

P P V−1−IPP 0

G(0)

QP VIPP −G(0)

P P V−10

G(0)

P P G(0)

P Q

G(0)

QP G(0)

QQ !(34)

From the latter, one obtains equations given in the main text.

C Shift of the Fermi level

The trace of the Greenian matrix after the perturbation changes as

δSpG(ε) = −∂

∂ε ln det G(ε) + ∂

∂ε ln det G0(ε) = ∂

∂ε ln det G0(ε)

det G(ε).(35)

The ﬁrst equality comes from the fact, that determinant equals to the product of eigenvalues gλ(ε)of

the Greenian matrix:

−∂

∂ε ln det G(ε) = −∂

∂ε ln Y

λ

gλ(ε) = −lim

ν→0+X

λ

∂

∂ε ln (ε+iν −ελ)−1=

= lim

ν→0+X

λ

(ε+iν −ελ)−1=X

λ

gλ(ε) = SpG(ε)(36)

From the Dyson’s equation we get:

det G(ε) = det I−G(0) (ε)F0−1det G(0) (ε).(37)

The latter allows to express the trace correction as

δSpG(ε) = ∂

∂ε ln det I−G(0) (ε)F0(38)

If the Fermi level had had not changed, the conservation of number of electrons would require that

−1

πˆεF

−∞

∂

∂ε =ln det I−G(0) (ε)F0dε = 0 (39)

27

which is, generally, not satisﬁed. Let us denote the integral on the left of the last equation as ∆n. The

condition of conservation of number of electrons reads then

KˆεF+δεF

εF

DOS0(ε)dε =−∆n(40)

where DOS0(ε)is a sum of diagonal projected DOS for a single unit cell in the initial (unperturbed) system.

From this equation one can see that for K→ ∞ the shift of the Fermi level approaches zero. However, in

the ﬁnite dimension models this shift needs to be taken into account especially in the case of small K(not

very dense k-mesh). Note, that for negative ∆nthe Fermi level shifts up, while for positive – down.

D Pole structure of the perturbed GF

Due to a speciﬁc role which poles of the GF play throughout the integration prescribed by eq. (29), the

qualitative eﬀect of perturbation crucially depends on whether it causes any additional poles to appear on

the right hand side of eq. (14). A sketch of evolving events comes from considering simplest perturbation

v, acting on the one-dimensional subspace Pspanned by the state |0i. The 1×1matrix to be inverted to

solve the Dyson equation (see Appendix B) is

1−vG(0)

00 ,

and the only nonvanishing matrix element of the mass operator eq. (13) takes the form:

M00 =v

1−vG(0)

00

.(41)

The tentative additional poles are zeroes of the denominator in the latter equation. Separating G(0)

00 into

real and imaginary parts we obtain an equation for the poles:

1−v<G(0)

00 (εl)−iv=G(0)

00 (εl)=0 (42)

For the complex quantity to vanish its real and imaginary parts have to vanish separately. The imaginary

part =G(0)

00 (ε)is zero everywhere in the forbidden energy bands of a crystal (where the DOS vanishes). In

this case only one condition needs to be satisﬁed:

<G(0)

00 (εl) = 1

v(43)

at a point εloutside the allowed energy band. Thisdeﬁnes position of the local state and is referred to in

the main text throughout the discussion of additional poles in the case of substitutional defect in cubia.

Here we only mention that for cubia <G(0)

00 is positive above the Fermi level and negative below it (see

Supporting Materials Sec.1), therefore a positive perturbation (v > 0) can only lead to a local state above

the allowed band, while a negative one – only below it. Values of vrequired to satisfy the condition for cubia

are discussed in the main text.

If eq. (43) is satisﬁed for the complex point εv+iνvwith its real part εvlying inside the band (=G(0)

00 (εv)6=

0), then a virtual state appears – a pole in the complex plane. It manifests itself on the real axis as a maximum

of the DOS in the allowed energy band [11]. Behavior of DOS0(ε)in the vicinity of the virtual state εv

derives from the following equation for the perturbed GF with mass operator eq. (41):

G00 (ε) = G(0)

00 (ε) + G(0)

00 (ε)MG(0)

00 (ε) = G(0)

00 (ε)

1−vG(0)

00 (ε).

Evaluating imaginary part of this, one obtains

DOS0(ε) = −π−1=G(0)

00 (ε)

1−v<G(0)

00 (ε)2+v2=G(0)

00 (ε)2.(44)

28

If =G(0)

00 (ε)changes slowly in the vicinity of εv, the perturbed density of states is proportional to a Lorentzian:

DOS0(ε)∼π−1η

(ε−εv)2+η2(45)

with a half-width

η==G(0)

00 (εv)

∂<G(0)

00 (ε)/∂εε=εv

,(46)

which follows from the expansion

1−v<G(0)

00 (ε) = −v(ε−εv)∂<G(0)

00 (ε)

∂ε ε=εv

.(47)

If the half-width νεv, one would observe a sharp maximum of DOS at εv, otherwise, there would not

be a deﬁnite peak. In addition, from this analysis one can explicitly observe that for the pole outside the

band the half-width vanishes (η= 0) yielding a discrete peak in DOS, which corresponds to the local state

discussed above.

It should be noted that the condition eq. (42) may not be possible to satisfy for a given value of

perturbation v. In this case there are no local or virtual states inﬂuencing the resulting DOS, and its form

is simply given by eq. (44).

In the general case a similar analysis of the pole structure can be performed if the mass operator Mis

expressed according to Kramer’s rule through the adjugate matrix and determinant D= det IPP −G(0)

P P V:

M=

Vadj IPP −G(0)

P P V

D.(48)

The sought additional poles are the roots of D, which is a polynomial of the power dim P. If a root of

determinant z0=εlis real, then corresponding pole describes a local state. If it is complex then it refers to

a virtual state. In the impurity problems local states appear outside the band where =D= 0 for all points

and position of the local state is therefore determined by roots of <D. Positions of the virtual states are

determined by zeroes of <Das well, but located inside the band [11].

While virtual states do not provide any diﬃculties in evaluating integrals eqs. (19) and (16), the local

states do. In their presence those integrals are understood in terms of Cauchy principal value. Contribution

of the local state outside the band to the density matrix elements can be evaluated by considering residue

at z0, which for the root of the ﬁrst order has a form

Res

z0

Gab (z) = lim

z→z0

(z−z0)Gab (z) = ha|lihl|bi(49)

From eq. (48) it follows that the perturbed function Gab may be represented in a form f(z)/D (z). If both

Dand fare holomorphic and D0(z0)6= 0 then the residue can be evaluated as

Res

z0

Gσ

ab (z) = f(z0)

D0(z0)(50)

This allows to express contributions of localized states to the density matrix elements through the derivatives

of D. With the same approach it may be shown that local state outside the band contributes to the energy

correction δε as εl.

E Model periodic systems

In order to test our approach we consider impurity atoms in several simple model metals: simple cubium,

face centered cubium and body centered cubium each bearing one s-orbital per lattice node. Besides that

29

we consider perturbations of π-bands of graphene. All these lattices have one electron per site. We consider

them in the framework of HF approach with extended Hubbard Hamiltonian:

H=−UX

i,σ

c+

iσciσ −mγ1X

i,σ

c+

iσciσ −βX

i∼j,σ c+

iσcj σ +h.c.+

+γ0X

i

c+

iαciα c+

iβ ciβ +γ1X

i∼j,σ,κ

c+

iσciσ c+

jκ cjκ ,(51)

featuring following energy contributions: one-center core attraction (term proportional to U), one-center

Coulomb repulsion (γ0), nearest neighbor electron hopping (β) and Coulomb repulsion (γ1); the term pro-

portional to mthe number of neighbors of each atom in respective lattice models the core attraction to the

nearest neighbors which is necessary for overall energy balance. For spin and charge symmetric states we set

c+

iαciα =Dc+

iβ ciβ E=1

2,(52)

c+

iαcj α=Dc+

iβ cjβ E=B

2,(53)

where the last equation holds for neighbor sites iand j. In the Hartree-Fock approximation the following

Fockian appears:

F=αX

i

c+

iσciσ −tX

i∼jc+

iσcj σ +h.c.,(54)

where αand tare:

α=−U−mγ1+γ0c+

iαciα +γ1X

j∼iX

σc+

jσ cjσ =−U+γ0

2,(55)

t=β+γB

2.(56)

In all further discussions we set the energy reference at α= 0. In principle these equations are to be

solved iteratively for the only nontrivial density matrix element B. However, in cubia and graphene the bond

order Bdoes not depend on Hamiltonian parameters due to the symmetry and no self-consistent procedure

is required to obtain analytical solutions in these cases.

In some cases to get anlytical estimates we will be interested in a simpler model without two-center

Coulomb terms in Hamiltonian (simple Hubbard Hamiltonian). Then tdoes not depend on the density

matrix elements.

The energy of the defect formation can be expressed through the parameters of the Hamiltonian and the

density matrix elements corrections as

δE =v+δγ0

4δn0n0−2δt X

i∼0

Bi0+δγ1X

i∼0

(δn0δni−1) −2tX

i∼j

δBij +

+γ1

2BX

i∼j

δBij +γ0

4X

i

δn2

i+γ1X

i∼j

δniδnj−∆nεF.(57)

This equation has an advantage that it allows to analyze contributions of diﬀerent terms and it is being

referred to in the main text in the discussions of the defects’ energies.

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