TOMBO Ver.2 Manualfor LDA and GW Calculations

30 October, 2015

TOMBO Group

PrefaceThis Manual is prepared for all users, who want to perform LDA and GW calculations

with TOMBO Ver.2. Chapters 1-5 describe a basic concept of TOMBO, a basic knowl-edge for GW , Input files, Output files, and an example of GW calculation of rutile TiO2,respectively. In Appendix A, usage of the user interface “Materials Studio®” for crystalcalculations is described. TOMBO is a first-principles program using the all-electron mixedbasis approach, in which one-particle (Kohn–Sham or quasiparticle) wave functions are ex-pressed in a linear combination of both atomic orbitals and plane waves. This approachis described in detail in the following reference. Anyone who publishes any result usingTOMBO has to cite this reference:

Shota Ono, Yoshifumi Noguchi, Ryoji Sahara, Yoshiyuki Kawazoe, and Kaoru Ohno,“ TOMBO: All-electron mixed-basis approach to condensed matter physics”,Computer Physics Communications 189, 20-30 (2015). DOI: 10.1016/j.cpc.2014.11.012

TOMBO can handle both isolated systems and crystal (or periodic) systems in a unifiedway. It can perform not only LDA calculations but also GW (+ Bethe–Salpter equation)calculations.

The TOMBO Ver.2 code, manuals, and related information are available from the website of Kaoru Ohno Laboratory at Yokohama National University:

http://www.ohno.ynu.ac.jp/tombo/index.html

Any correspondence can be sent to tombo@ynu.ac.jp.

30 October 2015TOMBO Group

Table of contents

1 Introduction 11.1 Short introduction of TOMBO . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mixed basis formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 All-electron charge density and potential . . . . . . . . . . . . . . . . . . . 61.4 Calculation flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 TDDFT dynamics simulation . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 The GW Approximation 132.1 The quasiparticle band gap . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 The self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Hedin’s equations (GWΓ method) . . . . . . . . . . . . . . . . . . . . . . 162.5 The GW approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 One-shot GW approximation . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Input Files 213.1 COORDINATES.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 rct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.2 nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.3 nv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.4 ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.5 np . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 INPUT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 iApp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 iAlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.3 iexc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.4 iCHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.5 Ulevel, Llevel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Table of contents iv

3.2.6 ippmG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.7 jmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.8 deltaE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.9 npp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.10 nod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.11 nog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.12 ncut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.13 noz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.14 ncs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.15 ncc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.16 nol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.17 nband . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.18 q_eliminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.19 q_min_mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.20 icontinue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.21 isphcut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.22 iTotalE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.23 lpri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.24 nspin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.25 irelative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.26 nion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.27 iExcite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.28 nonadiabatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.29 iMIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.30 smixSCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.31 nSDCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.32 iSblp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.33 iSblp2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.34 mSblp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.35 smixTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.36 iasym . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.37 icalAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.38 iCry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.39 precLDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.40 iFTapp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.41 iCheb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table of contents v

3.2.42 Mcheb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.43 nStep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.44 singlet, triplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 KPOINT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 QPOINT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 SPOINT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Output Files 444.1 OUTPUT.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Status.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 ISYS.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 posit.dat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 eigen.dat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.6 MD_Coordinates.xyz, MD_Coordinates.arc . . . . . . . . . . . . . . . . . 454.7 band.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.8 WaveF_HOMO-1.cube, WaveF_HOMO-1.grd,

WaveF_HOMO-1.vasp, and so on . . . . . . . . . . . . . . . . . . . . . . 464.9 GWA.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.10 PhotoAbsorptionSpectra.out . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Examples 505.1 LDA calculation of isolated dimers . . . . . . . . . . . . . . . . . . . . . . 505.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 GW calculation of rutile TiO2 . . . . . . . . . . . . . . . . . . . . . . . . 555.4 LDA calculation of rutile TiO2 . . . . . . . . . . . . . . . . . . . . . . . . 565.5 Wave function calculation of rutile TiO2 . . . . . . . . . . . . . . . . . . . 59

References 61

Appendix A Materials Studio 63A.1 Build crystal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2 1st Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.3 *.cell file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.3.1 make *.cell file . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.3.2 information of *.cell . . . . . . . . . . . . . . . . . . . . . . . . . 68

Chapter 1

Introduction

TOMBO[1, 2], all-electron mixed-basis code, is short for "TOhoku Mixed Basis Or-bitals ab initio program". In this thesis, electronic structures of transition metal oxides arecalculated by our original all-electron mixed basis code TOMBO developed by Prof. KaoruOhno, et al..

1.1 Short introduction of TOMBO

Density funcitonal theroy (DFT) [3] and local density approximation (LDA) [4] havebeen used in numerous electronic structure calculations and firt-principles molecular dy-namics (MD) simulations. To perform those calculations, Kohn–Sham (KS) equation needsto be solved self-consistently such that the input potential is identical to the output poten-tial [4]. The electronic wave function has to be described by appropriate functions to solvethe KS equation. Among the plane wave (PW) expansion approach has been applied to theab-initio molecular dynamics (MD) simulations with reasonably high accuracy [5].

However, it is difficult to treat phenomena (hyperfine interaction, XPS, XANES, etc.)related to core electrons by this method. One problem in generating good pseudo potentialsis related to the fact that the core contribution to the exchangecorrelation potential is notsimply an additive quantity. Moreover, it is not easy to create efficient pseudopotentials,which require only small number of plane waves.

On the other hand, linear combination of atomic orbitals (LCAO) approaches can treatall electrons. However, these methods have an intrinsic problem of incomplete basis set, andtherefore there is a problem in applying them in perturbation theory or spectral expansion,which requires a description in the complete Hilbert space. It is also difficult to consider anegative affinity problem by these methods. Related but slightly different problem inherent

1.2 Mixed basis formulation 2

to these methods is a basis set superposition error (BSSE) [6, 7]. There is also some troublein the Gaussian basis method [8] to describe cusp in the wave-funcion at the nuclear position.

In these respects, it is highly desirable to develop new method, which combines thePW expansion technique with the LCAO technique to remove pseudopotentials in the PWexpansion methods and to make the basis set complete in the LCAO methods. This is themain idea to introduce the all-electron mixed basis approach. TOMBO [1, 2] is the programpackage using this approach. TOMBO is the all-electron first-principles method, which canbe applicable to both isolated and periodic systems with complete basis set. It is not theovercomplete basis set because only limited number of PWs is used in the computation.

The powerfulness of TOMBO is not only based on these features but also based on thefact that it enables us to perform the state-of-the-art calculations such as GW approximationand Bether–Salpeter equation (BSE). Using these methods, TOMBO can treat the problemsrelated to electron correlations, electronic structure around the band gap, excitation spectra,and so forth. This TOMBO Ver.2 is a new version made by unifying preexiting severalversions: Molecular dynamics code [9], crystal code [10], T -matrix and GW + BSE code[11], and TOMBO Ver.1 [12].

1.2 Mixed basis formulation

The “mixed-basis” indicates the method using both plane waves and Bloch sums madeof atomic orbitals as the basis set.

For an isolated atom, it is possible to solves the Kohn–Sham equation very rigorously,because of the system has a spherical symmetry. In this case, the Kohn-Sham wave functionis expressed as a product of radial function R jnl(r) and spherical harmonics Ylm(r) as

ϕ AOjnlm(r) = R jnl(r)Ylm(r) (1.1)

Here, j, n, l, and m are atomic species, principal quantum number, angular momentumquantum number, and magnetic quantum number.

In the mixed basis code, the KS wave function is expressed as a linear combination ofPWs and AOs (Fig. 1.1),

ψv(r) =1√Ω ∑

GcPW

v (G)ei(k+G)·r +∑j

∑nlm

cAOv ( jnlm)ϕ Bloch

jnlm (k,r) (1.2)

withϕ Bloch

jnlm (k,r) = eik·R j ∑R

eik·Rϕ AOjnlm(r−R j −R). (1.3)

1.2 Mixed basis formulation 3

Here, Ω is the volume of the unit cell, G is the reciprocal lattice vector, c is the expansioncoefficients. Any system must be defined inside the unit cell, which is periodic in space (Fig.1.2). The (valence) radial function R jnl(r) is truncated by subtracting a smooth polynomialfunction that satisfies the matching condition at the surface of atomic sphere, while the restsmooth function (the polynomial function inside and the tail outside atomic sphere) can beexpressed by PWs; see Fig. 1.3.

Fig. 1.1 In the all-electron mixed basis approach, one-electron wave functions are expressedin a linear combination of plane waves (PWs) and atomic orbitals (AOs).

Fig. 1.2 Any system is defined inside the unit cell, which is periodic in space.

Fig. 1.3 Each AO is confined inside the non-overlapping atomic sphere by subtracting asmooth polynomial function.

Usually, AOs are generated at the outset by the atomic LDA code base on a modifiedHerman–Skillman’s algorithm and not updated during the calculation. However, if onewants to update AOs during the self-consistent field (SCF) loop, one can set icalAO = 1 inINPUT.inp. This option is available for cluster calculations only.

1.2 Mixed basis formulation 4

Many-body perturbation theory (such as the GW calculations) or spectral method (suchas the expansion of the wave packet in terms of the eigenstates of the Hamiltonian in theTDDFT dynamics simulations) requires summing over large number of empty states. ThePW basis set can most accurately describe the empty states. In contrast, to describe theelectrons in the core region accurately, the AO basis set works better then the PW basisset. The all-electron mixed basis approach, using both PWs and AOs as a basis set in acombined way, is able to meet the requirements to describe both spatially extended andlocalized states. In our code, AOs are numerically described inside the non-overlappingatomic spheres and the radial part is treated using the logarithmic mesh.

The all-electron mixed basis approach has the following advantages:

1. The number of basis functions can be significantly reduced.

2. In Hamiltonian matrix elements, it is not necessary to store PW-PW part because it isgiven simply by the Fourier components V (G−G′).

3. It is possible to accurately treat core states because we determine AOs by usingHerman–Skillman code with logarithmic radial mesh.

4. There is no complexity to generate and treat pseudoptentials. There is also no problemof transferability.

5. The overlap between AOs and PWs is calculated accurately by first performing angu-lar integral analytically and then performing radial integral of spherical Bessel func-tions numerically in logarithmic radial mesh.

6. Because AOs are confined inside non-overlapping atomic spheres, there is no BSSEproblem, and it is not necessary to calculate overlap integrals between AOs centeredat different atoms, which might produce unnecessary computational errors. Simulta-neously, this reduces the overcompleteness problem.

Since the basis functions (PWs and AOs), fξ (r), are not orthogonal each other, theeigenvalue εν is obtained by solving the following generalized eigenvalue problem:

∑ξ ′

Hξ ξ ′cν ,ξ ′ = εν ∑ξ ′

Sξ ξ ′cν ,ξ ′, (1.4)

where Hξ ξ ′ = ⟨ fξ |H| fξ ′⟩ and Sξ ξ ′ = ⟨ fξ | fξ ′⟩ are, respectively, the Hamiltonian and overlapmatrix elements between the ξ ’th and ξ ′’th basis functions. They looks like

H =

(⟨PW|H|PW⟩ ⟨PW|H|AO⟩⟨AO|H|PW⟩ ⟨AO|H|AO⟩

), S =

(⟨PW|PW⟩ ⟨PW|AO⟩⟨AO|PW⟩ ⟨AO|AO⟩

). (1.5)

1.2 Mixed basis formulation 5

There are Nw PWs (the number of plane waves "nw" in the code), Nao AOs (the number ofatomic orbitals "nao" in the code), and altogether Nbs basis functions (the number of basisset "nbs" in the code). The corresponding column vector is written as

Ψν ≡

cν ,1

cν ,2...

cν ,Nw

cν ,Nw+1...

Cν ,Nbs

1st G2nd G...last G

“nw”

1st AO...last AO

“nao”

(1.6)

Eq.(1.4) is then rewritten as follows:

HΨν = ενSΨν . (1.7)

This generalized eigenvalue problem is transformed to the usual eigenvalue problem byusing Choleski decomposition. The overlap matrix S is expressed as a product of a lowertriangular matrix T and its Hermitian conjugate T †:

S = T T †. (1.8)

Then, Eq. (1.7) becomes

H ′Φν = ενΦν , (1.9)

with the transformed Hamiltonian

H ′ = T−1HT †−1, (1.10)

(H ′ is calculated from H and T−1), and the transformed eigenvectors are give by

Φν = T †Ψν . (1.11)

The resulting ordinary eigenvalue problem can be solved by the standard library programusing the Householder method, if iApp = D is assigned in INPUT.inp.

In order to calculate the Hamiltonian and Overlap matrix elements, PW-PW, PW-AO,and AO-AO parts are calculated independently. Overlap matrix between two PWs (G,G′)is obviously a unit matrix. Similarly, Hamiltonian matrix between two PWs is the same as

1.3 All-electron charge density and potential 6

the usual PW expansion method:

⟨k+G|H|k+G′⟩= h2(k+G)2

2mδGG′ +V (G−G′). (1.12)

PW-AO matrix elements can be accurately calculated by 1D integration of the product ofthe spherical Bessel function jl′(Gr) and the radial AO function RAO

jnl(r):

⟨k+G|H|ϕ Blochjnlm ⟩ =

e−iG·R j

√Ω

Ylm( ˆk+G)∫

celljl(|k+G|r)

[h2(k+G)2

2m+V (r)

]× R jnl(r)r2dr. (1.13)

AO-AO matrix elements can be accurately calculated by 1D integration of the product oftwo radial AO functions:

⟨ϕ Blochjnlm |H|ϕ Bloch

j′n′l′m′⟩ = δ j j′

∫cell

ϕ AOjnlm(r)

[− h2

2m∇2 +V (r)

]ϕ A0

jn′l′m′(r)d3r. (1.14)

The overlap matrix elements are given by those of (1.13) and (1.14) without [...] inside theintegrands.

1.3 All-electron charge density and potential

In the all-electron mixed basis approach, all-electron charge density ρ(r) is made ofthree contributions: PW-PW, AO-PW,and AO-AO.

ρ(r) = ρPW−PW (r)+∑j

ρAO−PWj (r)+∑

jρAO−AO

j (r). (1.15)

In the all-electron mixed basis approach, the charge density is made of the three contribu-tions,

ρPW−PW (r) =2Ω

occ

∑v

∑G

∑G′

cPW∗v (G′)cPW

v (G)ei(G−G′)·r, (1.16)

ρAO−PWj (r) =

2√Ω

occ

∑v

∑nlm

∑G

cAO∗v ( jnlm)cPW

v (G)×ϕ jnlm(r−R j)ei(G)·r + c.c., (1.17)

ρAO−AOj (r) = 2

occ

∑v

∑n′l′m′

∑nlm

cAO∗v ( jn′l′m′)cAO

v ( jnlm)×ϕ jn′l′m′(r−R j)ϕ jnlm(r−R j). (1.18)

Here the prefactor 2 denotes the spin duplicity, ∑occν means the sum over all occupied states

1.3 All-electron charge density and potential 7

(in a case of non-spin-polarized systems), and c.c. in equation (1.17) means the complexconjugate of the previous term. The first PW-PW contribution can be conveniently treatedin Fourier space. The rest two AO-related contributions are confined only inside the non-overlapping atomic spheres, and can be written together as

ρAOj (r) = ρAO−PW

j (r)ρAO−AOj (r). (1.19)

This can be divided into two parts: one is spherical symmetric part σ j(r j) and the other isasymmetric part. Here we put r j = |r−R j|. The asymmetric part is generally negligible,but if necessary can be taken into account by setting the option parameter iAsym = 1 inINPUT.inp. The treatment of ignoring asymmetric part of the AO-related charge densityis guaranteed when the radii of non-overlapping spheres are set reasonably small enclosingthe core region only.

Spherical potential Vj(r j) inside each non-overlapping atomic sphere is calculated asfollows. First, consider the Hartree potential made by the AO-related spherically symmetriccharge σ j(r j) centered at R j. This potential is easily calculated as the 1D integration inradial direction of the Poisson equation as follows:

vHj (r j) =

4πr j

∫ r j

0σ j(r)r2dr+4π

∫ rc

r j

σ j(r)rdr. (1.20)

According to the Gauss theorem in electrostatics, this Hartree potential behaves as

vHj (r j) =

Q j

r j, for r ≥ rc, (1.21)

where Q j is the symmetric charge defined as

Q j = 4π∫ rc

0σ j(r)r2dr. (1.22)

On the other hand, if we define the screened charge Z∗j as

Z∗j = Z j −Q j, (1.23)

(Z j is atomic number), the sum of the Hartree potential vHj (r j) and nuclear Coulomb poten-

tial −Z j/r j around R j becomes

vHj (r j)−

Z j

r j=−

Z∗j

r j, for r j ≥ rc (1.24)

outside the non-overlapping atomic sphere. Because of this long tail, it is necessary to add

1.3 All-electron charge density and potential 8

all the contributions from surrounding atoms. This summation should be taken not onlyinside the own unit cell, but also surrounding or further apart unit cells. To treat this accu-rately, we use the following Fourier decoupling method. In this method, the simple potentialform −Z∗

j/r j given by Eq.(1.24) connects smoothly to the quadratic function inside the non-overlapping atomic sphere. They should have the same value and the same derivative at theradius of the atomic sphere r = rc:

vinterpoj (r j) =

Z∗j (br2

j +d) for r j < rc,

−Z∗j/r j for : r j ≥ rc.

(1.25)

From the matching condition, we obtain

br2c +d =−1/rc, 2brc = 1/r2

c . (1.26)

From simple calculation, we find that these conditions are identical to

b =1

2r3c, d =− 3

2rc. (1.27)

Thus connected potential is a smooth and analytic function over whole space and is easilytransformed into reciprocal lattice space analytically. We call this potential the interpolatedCoulomb potential and write it as vinterpo

j (r j); see Eq.(1.25). This interpolated Coulomb

potential takes the correct value of the Coulomb potential, vHj (r j)−

Z jr j

, everywhere outsidethe jth atomic sphere; It takes an incorrect value only inside the jth atomic sphere, and itsdifference from the correct value is given by

V truncj (r j) = vH

j (r j)−Z j

r j− vinterpo

j (r j), (1.28)

which we call the truncated Coulomb potential. It has nonzero values only each atomicsphere, and is spherically symmetric. This truncation is schematically illustrated in Fig.1.4.This truncated Coulomb potential is added to the truncated spherical exchange-correlationpotential µxc

j (|r−R j|) and stored as one-dimensional data on the radial logarithmic mesh.Apart from this truncated function, we have to treat separately interpolated Coulomb

potential vinterpoj (r j). This function vinterpo

j (r j) is analytically expressed by Eq.(1.25) in in-finite space, and therefore is able to be analytically transformed into G space. The Fouriercoefficients are explicitly given by

vinterpoj (G) =

4πZ je−iG·R j

Ω

b[(

3(Grc)2 −6

)sinGrc +6Grc cosGrc

]/G5

1.3 All-electron charge density and potential 9

+d sinGrc/G3 + cosGrc/G2

. (1.29)

This analytic Fourier coefficients are added to the rest part of the Fourier coefficients ofboth the Hartree potential ρ rest(G) and the exchange-correlation potential µxcrest(G) to givean additional contribution to the spherical potential in the jth atomic sphere. That is, thisadditional contribution to the spherical potential from the three kinds of Fourier coefficientsare expressed as:

V restj (r j) = ∑

G

sinGr j

Gr jeiG·R j

[∑k

V interpok (G)+

4πΩ

ρ rest(G)

G2 + µxcrest(G)

]. (1.30)

Here we note that vinterpoj (r j) defined by Eq.(1.25) is different from the first term in the l.h.d.

of Eq.(1.30). The latter includes all the tails of the interpolated Coulomb potential centeredat all k = j atoms. This is obvious from the nature of the Fourier transformation.

Thus the total effective potential used in the AO-related matrix elements is given by

Vj(r j) =V truncj (r j)+V rest

j (r j). (1.31)

All the AO-related matrix elements are calculated by using this spherical potential. If iCheb= 1 is set in INPUT.inp, a Chebyshev polynomial fitting to this spherical potential insideatomic sphere is used to accelerate the computational speed. The contribution from theasymmetric part of the potential is treated separately in a large Fourier space when iAsym =1 is set in INPUT.inp.

Fig. 1.4 How to calculate periodic Hartree potential VH(r) inside non-overlapping atomicspheres?

1.4 Calculation flow 10

1.4 Calculation flow

Flowchart of the first-principles molecular dynamics (MD) is shown in Fig. 1.5.

Set initial atomic positions

?Assume initial charge density

?Initialize coefficients Cλ

i

?Calculate charge density

?Mix the charge density with

that of the previous step

Calculate the KS potential

?Calculate matrix elements

?Update Cλ

i according tomatrix diagonalization orsteepest-descent method

?

?

XXXXXXXXX

XXXXXXXXX

If |Enew −Eold|< precLDA Check self-consistencyNO

YES?Calculate forces on atoms

?Update atomic positions

-

Fig. 1.5 Flowchart of the first-principles MD. The convergence of the electronic states ischecked by the difference between the total energies at the present and previous steps. Ifconvergence is achieved, the atomic positions are updated by using the calculated force.

1.5 TDDFT dynamics simulation 11

1.5 TDDFT dynamics simulation

In TOMBO, TDDFT dynamics simulation can be performed by setting “A” in iApp in IN-PUT.inp. If one wants to excite one electron from HOMO to LUMO +n at the outset, onemay define iExcite = 1+ n in INPUT.inp (n can be 0, 1, 2, ...). Default value of iExcite is0, which means no excitation.

According to the time-dependent density functional theory[13], TOMBO can treat thetime-dependent Kohn–Sham (TDKS) equation

i∂∂ t

ψ j(r, t) = Hq(r, t)ψ j(r, t), (1.32)

by setting “A” in iApp in INPUT.inp Here Hq(r, t) is the electronic part of the Hamiltonian.Combining this time-evolution equation with the Newtonian equation of motion for nuclei

MARA =− ∂∂RA

[Eq +V cl], (1.33)

we can perform semi-classical dynamics simulation within the mean-field-type Ehrenfesttheorem. During one simulation, all dynamical variables change in time along one reactionpath according to the choice of the initial atomic coordinates and initial atomic velocities(velocities can be written parallel to the right of the coordinates in COODINATES.inp). Thisway of simulation is called “on the fly” approach. Here Eq is the electronic part of the totalenergy, and, together with the Coulomb potential between nuclei V cl , gives the mean-fieldpotential exerting on nuclei. In these equations, r is the electron coordinate, MA and RA arethe mass and position of the Ath nucleus. This approach is on the fly.

The simulation is basically performed adiabatically, i.e., the atomic motion is assumedslow enough. However, non-adiabatic simulation is possible by considering the coupling tothe atomic velocity, when the input parameter nonadiabatic = 1 is set in INPUT.inp. Belowonly the algorithm of adiabatic simulation is explained, but one may refer to Ref. [14] fornon-adiabatic simulation.

It is necessary to know the exchange-correlation potential to solve Eq.(1.32) step by stepby means of the TDDFT. TOMBO uses a simple LDA exchange-correlation functional thatis local for both space and time. This approximation is called “adiabatic LDA”. In order tointegrate Eq.(1.32) step by step accurately, we use the spectral method[15], in which wavepacket ψ j(r, t) at each instantaneous time is expanded in terms of eigenfunctions ϕk(r, t)(eigenvalues are εk(t))

Hq(r, t)ϕk(r, t) = εk(t)ϕk(r, t). (1.34)

1.5 TDDFT dynamics simulation 12

of the Hamiltonian at that time Hq(r, t). Wave packes are expanded as

ψ j(r, t) = ∑k

c jk(t)ϕk(r, t), (1.35)

where the coefficients c jk(t) are given by the inner product of ϕk(r, t) and ψ j(r, t) as

c jk(t) = ⟨ϕk(r, t)|ψ j(r, t)⟩. (1.36)

Therefore, if we set the time step ∆t smaller than the time scale where the Hamiltonianchanges, the TDKS equation can be integrated as follows:

ψ j(r, t +∆t) = ∑k

exp[−iεk(t)∆t]c jk(t)ϕk(r, t). (1.37)

When the Hamiltonian does not change in time, Eq.(1.37) is exact. Of course, electron wavepackets oscillate 100-1000 times faster than the nuclear motion, but the stability of Eq.(1.37)is excellent. Typically if one set ∆t at 10−2-10−1 fs, the Hamiltonian Hq(r, t) almost doesnot change, Eq.(1.37) becomes good approximation.

This spectral method requires that the basis functions span complete space at least ap-proximately, and in this sense, the all-electron mixed basis method is suitable. In fact, thesummation over the eigenstates in Eq.(1.37) can be restricted to low lying excited states only,although continuum free-electron-like states above energy zero should be also included. Thenumber of levels taken into account in this summation is set as nol in INPUT.inp. Usually,nol = 500 is recommended, but it should be set 1000 or more for larger systems.

Newtonian equation of motion (1.33) involves forces calculated by the Coulomb forcebetween nuclei and the derivative of the electron total energy with respect to the nuclearpositions. These forces are calculated in the same way described in previous section. Ifthe electronic states is on an adiabatic surface, the dynamics using this mean-field potentialbecomes the adiabatic molecular dynamics. If the wave packet ψ j(r, t) is the superpositionof the eigenstates, the forces acting on nuclei becomes the average of the forces calculatedfrom the different eigenstates with the weight of these eigenstates in the wave packet (mixedstate). This is the problem of this mean-field approximation. This mean-filed force becomesunphysical apart from the region where the mean-field potential is valid.

To do the TDDFT dynamics simulation, it is necessary to first iterate the SCF loopuntil self-consistency is obtained. Then the TD dynamics loop starts. Typically dTime= ∆t = 0.01 ∼ 0.1 [fs] should be put in COORDINATES.inp. As well as the usual MD,nStep should be also given in INPUT.inp.

Chapter 2

The GW Approximation

The calculated DFT band structures are not improved by the energy-independent nonlocal-density corrections to the LDA, in any case, quasiparticle energies are needed [16]. In orderto improve the accuracy of electronic structure calculation of materials, the GW approxima-tion is introduced on the basis of many-body perturbation theory (MBPT)[17].

2.1 The quasiparticle band gap

In the N-particle many-body neutral system, eigenstates and total energies are denoted|ψ⟩ and EN , respectively. The electron quasiparticle (QP) energies are defined by

εck = EN+1 −EN0 (unoccupied) (2.1)

and the hole QP energies are defined by

εvk = EN0 −EN−1 (occupied) (2.2)

These are excitation energies of the (N ± 1)-particle system relative to the N-particleground-state energy EN

0 and thus correspond to the electron addition and removal energies.It is clear that εck > εF and εvk ⩽ εF , where εF is the Fermi level. The quasiparticle energygap is given by

Egap = εCBM− ε[V BM] = EN+1 +EN−1 −2EN0 (2.3)

where ... and [...] indicate the energetically minimum and maximum k points of ..., and,at each k point, the CBM and VBM refer to the conduction band minimum and the valenceband maximum, respectively.

2.2 The Green function 14

2.2 The Green function

The propagation of one particle through the system is described by the one particleGreen function. The one particle Green function is defined as:

G(x1, t1,x2, t2) = i⟨

ΨN

∣∣∣T [ψ(x1, t1)ψ†(x2, t2)]∣∣∣ΨN

⟩(2.4)

It contains the information on energy and lifetimes of the quasiparticle, and also onthe ground state energy of the system and the momentum distribution. Here, x = (x, t) =(r,σ , t). ΨN is the Heisenberg ground state vector of the interacting N-electron system satis-fying the engenvalue equation H|ΨN⟩= E|ΨN⟩,ψH and ψ†

H are respectively the annihilationand creation field operator and T is the Wick time ordering operator.

ψ(x, t) = eiHtψ(x)e−iHt (2.5)

ψ†(x, t) = e−iHtψ†(x)eiHt (2.6)

and the field operator satisfy the anti-commutation relation:ψ(x),ψ†(x′)

= δ (x−x′) (2.7)

ψ(x),ψ(x′)

=

ψ†(x),ψ†(x′)= 0 (2.8)

and:

T [ψ(x1, t1)ψ†(x2, t2)] =

ψ(x1, t1)ψ†(x2, t2) if t1 > t2ψ(x2, t2)ψ†(x1, t1) if t1 < t2

(2.9)

Therefore, the Green function describes the probability amplitude for the propagationof an electron (hole) from position r2 at t2 to r1 at time t1 for t1 > t2 (t1 < t2) (See Fig.2.1).Insert a complete set of N+1 and N-1 particle states, ∑ j |ψN±1

j ⟩⟨ψN±1j | = 1, we perform a

Fourier transform in energy space we obtain:

G(x1,x2;ω) = ∑s

fs(x1 f ∗s (x2)

ω − εs + iηsgn(εs − εF)(2.10)

2.3 The self-energy 15

(a) add an electron (b) remove an electron

Fig. 2.1 Add/Remove an electron in the N-particle many-body system

where εF is the Fermi level, and the QP energies are given by

εs =

E(N+1)

s −E(N)N for εs ≥ εF

E(N−1)N −E(N−1)

s for εs < εF(2.11)

The subscript s indicate the quantum label of the states of the N±1 system. The amplitudes,i.e., the QP wave functions, fs(x) are defined as:

fs(x) =

⟨ΨN |ψ(x)|ΨN+1,s⟩ for εs ≥ εF

⟨ΨN−1,s|ψ(x)|ΨN⟩ for εs < εF(2.12)

The Green function has the poles at the electron addition (removal) energies and de-scribes quasiparticles. So we can obtain the quasiparticle band gap information from Greenfunction.

2.3 The self-energy

From the definition of Green function (G) and the equation of motion for ψ(r, t) :i ∂

∂ t ψ(r, t) = [ψ(r, t),H]. we can show:

i∂

∂ t1G(1,2) = δ (1,2)+H0(1)G(1,2)− i

∫ν(1+,3)G(1,3,2,3+)d3 (2.13)

2.4 Hedin’s equations (GWΓ method) 16

Here, H0 = T +Vext +VH , the number n stands for (rn, tn), ν(1+,3) is the Coulomb inter-action between electrons. There exists 2-particle Green function, describes the motion of 2particles:

G(1,2,3,4) =−⟨

ΨN0

∣∣∣T [ψ(1)ψ(2)ψ†(4)ψ†(3)]∣∣∣ΨN

0

⟩(2.14)

In frequency Fourier space:

[ω −H0]G(ω)+ i∫

νG2(ω) = 1 (2.15)

Instead of introducing a 2-particles Green function, we introduce the self-energy oper-ator, Σ . The self-energy allows to close formally the hierarchy of equations of motion ofhigher order Green functions. Equation (2.15) is tranformed to 1-particle Green function.

[ω −H0]G(ω)+ i∫

Σ(ω)G(ω) = 1 (2.16)

The self-energy is a non-local and energy dependent operator. From the equation of motion:

[hω −H0(r)]G(r,r′;ω)−∫

Σ(r,r′′;ω)G(r′′,r′;ω)d3r′′ = δ (r− r′) (2.17)

Introducing the Lehmann representation for G. The QP energies and QP wave functionscan be obtained as solutions of a Schrödinger-type QP equation [18]

Hoψnk(r)+∫

dr′Σ(r,r′;εnk)ψnk(r′) = εnkψnk(r) (2.18)

2.4 Hedin’s equations (GWΓ method)

It is possible to calculate energy and lifetimes of quasiparticle excitation solving Eq.(2.18). A formally exact way of calculating the self-energy is given by a set of coupledequations,known as Hedin’s equations [19]:

Self-energy:

Σ(1,2) = i∫

d(34)G(1,3)Γ(3,2,4)W (4,1+) (2.19)

Screened potential:

W (1,2) = ν(1,2)+∫

d(34)ν(1,3)P(3,4)W (4,2) (2.20)

Polarization:P(1,2) =−i

∫d(34)G(1,3)G(4,1+)Γ(3,4,2) (2.21)

2.5 The GW approximation 17

Vertex function:

Γ(1,2,3) = δ (1,2)δ (1,3)+∫

d(4567)∂Σ(1,2)∂G(4,5)

G(4,6)G(7,5)Γ(6,7,3) (2.22)

Dielectric function:ε = 1−νP (2.23)

The screened potential W can also be written as

W (1,2) =∫

d(3)ε−1(1,3)ν(3,2) (2.24)

From Hedin’s equations, we know how to calculate the self-energy. Although the GWΓmethod is implemented in a future version of TOMBO, the following GW approximation isneeded for the moment.

2.5 The GW approximation

A practical approximation to calculate Σ is the GW approximation proposed by Hedin[17]. In this GW approximation, the vertex function Γ is approximated in its zeroth-orderform, Γ(1,2,3) = δ (12)δ (13). So the Hedin’s equations become:

Σ(1,2) = iG(1,2)W (1+,2) (2.25)

G(1,2) = G0(1,2)+∫∫

d(3)d(4)G0(1,3)Σ(3,4)G(4,2) (2.26)

P(1,2) =−iG(1,2)G(2,1) (2.27)

W (1,2) = ν(1,2)+∫∫

d(3)d(4)ν(1,3)P(3,4)W (4,2) (2.28)

This self-consistent GW calculation is possible in TOMBO by setting the first characterof iApp as “G” in INPUT.inp, although crystal calculation and numerical ω integration(which will be explained below) are not available now.

The explicit form of the self-energy Σ in the GW approximation is given by

ΣGW (r,r′;ε) =i

2π

∫dωe−i0+ωG(r,r′;ε −ω)W (r,r′;ω) (2.29)

2.5 The GW approximation 18

The GW self-energy can be separated into two terms ( ΣGW = Σx+Σc ). The exchange partis given by

Σx(r,r′) =i

2π

∫dωeiω0+G(r,r′;ω)ν(r− r′) =−

occ

∑nk

ψnk(r)ψ∗nk(r)

|r− r′|(2.30)

In equation (2.30), the symbol occ in the sum means that the summation is taken over theoccupied states only. The diagonal matrix elements of this exchange part of the self-energybecome

⟨n,k∣∣Σx(r,r′)

∣∣n,k⟩=−occ

∑n′

∑q

∑G

4πΩG2

⟨n,k∣∣∣ei(q+G)·r

∣∣∣n′,k−q⟩⟨

n′k−q∣∣∣e−i(q+G)·r′

∣∣∣n,k⟩(2.31)

The correlation part, Σc can be evaluated by using generalized plasmon-pole (GPP)model [19]. Σc can also be evaluated by using the full ω integration [20]:

Σc(r,r′;ω) =i

2π

∫dω ′e−iω0+G(r,r′;ω −ω ′)

[W (r,r′;ω ′)−ν(r− r′)

](2.32)

In Equation (2.32), it is difficult to perform the ω ′ integral along the real axis, since W andG have a strong structure on this axis. In order to avoid this difficulty, Godby et al.[21–24]restricted the values of ω to small imaginary numbers and changed the contour of the ω ′

integral from real axis to imaginary axis [16]. Then, by analytic continuation, the resultingTaylor series is used to estimate the matrix elements for real values of ω . Ishii and Ohno etal. [20, 25] suggested that this intergration method employed by Godby et al. [16] can beextended rather easily to real number of ω by slightly modifying the contour. The contouralong the real ω ′ axis from −∞ to +∞ for the integral Equation (2.32) can be replaced bythe contour C shown in Fig.2.2.

Here we further use the symmetry W (ω) =W (−ω) to reduce the contour to the positivereal and imaginary parts only. The diagonal matrix elements of the correlation part of theself-energy become⟨

n,k∣∣Σc(r,r′;ω)

∣∣n,k⟩=∑n′

∑q

∑G,G′

⟨n,k∣∣∣ei(q+G)·r

∣∣∣n′,k−q⟩⟨

n′k−q∣∣∣e−i(q+G′)·r′

∣∣∣n,k⟩× i

2π

∫dω ′ [WG,G′(q,ω ′)− (4π/ΩG2)δG,G′

]× (

1ω +ω ′− εk−q,n − iδk−q,n

+1

ω −ω ′− εk−q,n − iδk−q,n)

(2.33)

2.6 One-shot GW approximation 19

Fig. 2.2 Contour C of the ω ′ integration in Equation 2.33

in the case when we only perform the contour integration along the positive (real and imagi-nary) parts of the contour C shown in Fig.2.2 with the help of W (ω)=W (−ω). If we use theoriginal contour C without using W (ω) =W (−ω), the second term in the parentheses of theintegrand in Equation (12) does not appear. This contour is justified because there is no poleinside two quadrants of the complex ω ′ plane, corresponding to (Reω ′ >max(εVBM−ω,0),Imω ′ > 0) and (Reω ′ < min(εCBM −ω ,0), Imω ′ < 0). Here we used the fact that W (ω ′)

has no pole in (Reω ′ > 0, Imω ′ > 0) and (Reω ′ < 0, Imω ′ < 0). This term represents thecontribution related to the electron correlation.

Instead of solving equation (2.18) self-consistently, we adopt here the so-called one shotGW approximation first proposed by Hybertsen and Louie [19]. The Green’s function (G)is replaced by the LDA Green’s function G0 and constructed in a non-self-consistent wayfrom the KS wave functions ψKS

nk (r) and eigenvalues εKSnk (r). It will be discussed in the next

section.

2.6 One-shot GW approximation

In practice, Kohn-Sham orbitals and eigenvalues from a DFT calculation are often used asinput for a GW calculation and the quasiparticle spectrum is evaluated non-selfconsistenlyfrom Eq.(2.33) without updating the Green’s Function or the screened potential, that meansonly one iteration is made. This is known as the “one-shot” GW or G0W0 approximation[16, 19] and has become a standard tool in electronic structure theory. W0 is hereby equal tothe RPA screened potential.

Using the one-shot GW approximation, the quasiparticle equation (Eq. 2.18) is solvedapproximately. The quasiparticle spectrum is calculated with Eq. (2.18) using the first-

2.6 One-shot GW approximation 20

order perturbation theory in ( ΣGW −V LDAxc ), where V LDA

xc is the LDA exchange-correlationpotential. Usually, the KS wave functions ψKS

nk are sufficiently close to the true quasiparticlewave function ψnk(r), so that the first-order estimate of the self-energy correction to theLDA eigenvalues in adequate [19]. The GW quasiparticle energy εGW

nk is then obtained as

εGWnk = εLDA

nk +Znk

⟨ψLDA

nk

∣∣∣ΣGW (r,r′;εLDAnk )−V LDA

xc (r)δ (r− r′)∣∣∣ψLDA

nk

⟩(2.34)

with a renormalization factor as

Znk =

[1− ∂ΣGW (ω)

∂ω

]−1

ω=εLDAnk

(2.35)

The Scheme of the one-shot GW is shown in Fig.(2.3). Firstly, Kohn-Shan equationis solved by DFT and LDA, it obtain Kohn-Sham eigenvalues and wave fuctions. Thenusing one-shot GW approximation, we can get polarization, dielectric function, screenedpotential, et. al.. From Hedin’s equations, we obtain the self-energy. Finally, we calculatethe QP energy by Eq.(2.33).

Fig. 2.3 Scheme of the one-shot GW calculation

Chapter 3

Input Files

To perform a LDA or GW calculation of an isolated system (atoms, molecules, or clus-ters), TOMBO needs two input files, COORDINATES.inp and INPUT.inp. These are thecentral input files of TOMBO. The variables related to the unit cell, atomic positions, andalso AOs are written in COORDINATES.inp: the lattice constant, lattice vectors, coordi-nates of the nuclei and the number of orbital-types of AOs for each atom. On the otherhand, INPUT.inp determines "what to do and how to do it", and can contain a relativelylarge number of parameters. In a crystal calculation within a LDA level, two other inputfiles, SPOINT.inp and KPOINT.inp, are required to assign special k points to do special-point sampling within the irreducible wedge of the 1st Brillouin zone (BZ) during the self-consistent field (SCF) loop and output k points to draw band structure diagram, respectively.Usually, the special points are given on the Monkhorst–Pack grid in the irreducible BZ. Fora GW crystal calculation, the assignment of KPOINT.inp is different from a LDA calcula-tion, and another input file QPOINT.inp is also required In order to calculate the polarizationfunction PGG′ for each momentum transfer q (and for each energy transfer ω in the case ofthe full ω integration), k-point sampling is performed for the points on the Gamma gridin the whole BZ assigned as “sum” in KPOINT.inp. Then the correlation part of the self-energy, Σc is calculated by taking q-point summation on the Γ-grid in the irreducible BZ byusing QPOINT.inp. Finally, the quasiparticle energies as well as the expectation values ofthe LDA exchange-correlation potential, and the exchange (Σx) and correlation (Σc) parts ofthe self-energy are evaluated for the k points (typically at symmetry points in the irreducibleBZ) assigned as “out” in KPOINT.inp.

3.1 COORDINATES.inp 22

3.1 COORDINATES.inp

The variables related to both the unit cell and Atomic Orbits (AO) should be written inCOORDINATES.inp: the lattice constant, lattice vectors, coordinates of the nuclei and thenumber of orbit-type AOs for each atom. It can also contain the definitions of “mesh” of theunit cell division (see below) and the basic time step (“dTime” in [fs]) for MD.

A typical form of COORDINATES.inp is shown in Table 3.1.

Table 3.1 Form of COORDINATES.inp

line 1 name of the target system (arbitrary)line 2 length of lattice vector a1 in [Å]line 3 a1x a1y a1z

line 4 a2x a2y a2z

line 5 a3x a3y a3z

line 6 number of atomsline 7 Direct (or Cartesian)line 8 1st atomic species and coordinatesline 9 2nd atomic species and coordinatesline 10 ... ...line 11 nth atomic species and coordinatesline 12 XX_rct (radius of atomic sphere of XX atom)line 13 XX_ nc (number of kinds of core states)line 14 XX_nv (number of kinds of valence states)line 15 XX_ns (number of s-type Atomic Orbits for XX atom)line 16 XX_np (number of p-type Atomic Orbits for XX atom)line 17 XX_nd (number of d-type Atomic Orbits for XX atom))line 18 ... ...line 19 mesh (mesh division of unit cell (same for all directions))line 20 END_PARA

Note:

1. Line 1 is arbitrary and can be used for user’s memo of this COORDINATES.inp file.

2. Line 2 is the lattice constant, i.e. the length of the lattice vector a1 in [Å].

3.1 COORDINATES.inp 23

3. a1x, a1y, a1z are given in line 3, where a1 should be normalized to 1, i.e., |a1|= 1 or

a21x +a2

1y +a21z = 1. (3.1)

4. a2x, a2y, a2z and a3x, a3y, a3z are given in line 4 and line 5, respectively, in the sameunit of a1x, a1y, a1z.

5. Line 7:"Direct" gives fractional coordinates of atomic positions (coefficients for a1,a2, a3) "Cartesian" gives Cartesian coordinates of atomic positions in [Å] unit.

6. From line 8 to line 11, if "Direct" is chosen, atomic positions should be put aroundthe center of the unit cell for symmetry. (Note that any coefficients must be between0 and 1, and cannot be either 0 or 1. Instead of putting 0, you have to put 0.0000001.)"Cartesian" does it automatically.

7. In order to fix the position of certain atom during the dynamics, F or f should bewritten at the right of the atomic coordinate as

Si 0.500000000 0.500000000 0.500000000 F

User can continue MD as a subsequent job by copying the two coordinates at thelast time step and at the time step one before the last step in MD_Coordinates.xyz toCOORDINATES.inp as

C 7.8734002 8.2943019 9.5324198 7.8728956 8.2894257 9.5373841

line by line for all atoms. The coordinates are given in Cartesian in [Å] unit. In thiscase, ivelocity = 0 [default] should be set somewhere below line 12.

8. User can set initial velocity of atoms in a similar way:

C 7.8734002 8.2943019 9.5324198 0.010000 0.000000 0.000000

In this case, ivelocity should be set at either 1, 2, or 3 somewhere below line 12. Here,ivelocity = 1, ivelocity = 2, and ivelocity = 3 mean that the velocity is given in unitsof [angstrome/fs], [m/s], and [a.u.], respectively.

9. In order to treat an antiferromagnetic system having zero total magnetic moment, oneshould write nspin = -1 in INOUT.inp and simultaneously, one may write the spinmagnetic moment of each atom as m 1 or m -1 at the right of the atomic coordinate as

Mn 0.250000000 0.250000000 0.250000000 m 1

Mn 0.750000000 0.750000000 0.750000000 m -1

3.1 COORDINATES.inp 24

10. The Atomic Orbitals implemented in TOMBO code are listed in Fig. 3.1: s-, p-, andd-orbitals are implemented, while f -orbital is still under implementation. Neverthe-less, in principle, TOMBO code can handle all atoms by increasing the cutoff energyfor Plane Waves.

11. Lines 12 to 18 are related to the radius of non-overlapping atomic sphere and thechoice of AOs for each atom.

12. In line 19, mesh of the unit cell division is defined if necessary. Otherwise defaultvalue is used for mesh. For accurate calculation, large mesh such as 128 or 192 isrecommended as well as noz, which should be defined in INPUT.inp.

An example of COORDINATES.inp for rutile TiO2 crystal [26] is shown in Table 3.2.

Table 3.2 COORDINATES.inp of rutile TiO2

line 1 system=Rutile_TiO2line 2 4.59line 3 1.000000000 0.000000000 0.000000000line 4 0.000000000 1.000000000 0.000000000line 5 0.000000000 0.000000000 0.643883326line 6 6line 7 Directline 8 O 0.553188499 0.553188499 0.250000000line 9 O 0.946811501 0.946811501 0.250000000line 10 O 0.446811501 0.053188499 0.750000000line 11 O 0.053188499 0.446811501 0.750000000line 12 Ti 0.250000000 0.250000000 0.250000000line 13 Ti 0.750000000 0.750000000 0.750000000line 14 Ti_rct=0.8line 15 Ti_nc=3line 16 Ti_nv=4line 17 Ti_ns=3line 18 Ti_np=2line 19 mesh=64line 19 END_PARA

3.1 COORDINATES.inp 25

Fig.

3.1

Ato

mic

orbi

tals

(AO

s)av

aila

ble

inTO

MB

Oco

de:s

-,p-

,and

d-or

bita

lsar

eim

plem

ente

d

3.2 INPUT.inp 26

3.1.1 rct

"XX_rct" is the radius of the non-overlapping atomic sphere (“rct”) of XX atom in unitsof [Å]. Here, note that user should write S _rct = 0.8 to define “rct” of S atom, for example,because the atomic symbols H, B, C, N, O, F, S, K, V, Y, and I are one character only,one blank is necessary between the atomic symbol and _. The sum of two “rct” values ofadjacent atoms should not exceed the interatomic distance. The value of “rct” may stronglyaffect the one-shot GW calculation results. In such a case, “rct” should be chosen as smallas possible, although the necessary cut-off energy for plane waves is increased.

3.1.2 nc

"XX_nc" is the number of core AOs for which no truncation is performed. One shouldwrite N _nc = 1 [default] to define “nc” of N atom (1s only), for example. For titanium, thecore orbitals of Ti are 1s22s22p6, so that there are 3 core AOs: 1s, 2s and 2p; so Ti_nc = 3.

3.1.3 nv

"XX_nv" is the number of valence AOs for which truncation (to confine inside the atomicsphere) is performed. One should write N _nv = 2 [default] to define “nv” of N atom (2sand 2p), for example. For titanium, the valene orbitals of Ti are 3s23p63d24s2, so that thereare 4 valence AOs: 3s, 3p, 3d and 4s; so Ti_nv = 4.

3.1.4 ns

"XX_ns" is the number of s-type AOs for XX atom. One should write N _ns = 2 [default]to define “ns” of N atom (1s and 2s), for example. For titanium, 1s22s22p63s23p6 AOs areused because 4s is quite extended. Therefore there are 3 s AOs: 1s, 2s and 3s; so Ti_ns = 3.

3.1.5 np

"XX_np" is the number of p-type AOs for XX atom. One should write N _np =1 [default] to define “np” of N atom (2p), for example. For titanium, AOs of Ti are1s22s22p63s23p6, so that there are 2 kinds of p orbits: 2p and 3p; so Ti_np = 2.

3.2 INPUT.inp

A typical form of INPUT.inp is shown in Table 3.3. # and mean comments.

3.2 INPUT.inp 27

Table 3.3 Form of INPUT.inp

line 1 name of the target system (arbitrary)line 2 iAppline 3 iAlgline 4 ippmGline 5 jmaxline 6 deltaEline 7 nppline 8 nodline 9 nogline 10 ncutline 11 nozline 12 ncsline 13 nolline 14 nbandline 15 q_eliminateline 16 icontinueline 17 isphcutline 18 iTotalEline 19 lpriline 20 nspinline 21 iMIXline 22 smixSCFline 23 nSDCGline 24 iSblpline 25 iSblp2line 26 smixTDline 27 iasymline 28 icalAOline 29 iCryline 30 precLDAline 31 iFTappline 32 iChebline 33 nstepline 34 END_PARA

3.2 INPUT.inp 28

An example of INPUT.inp for one-shot GW calc. of rutile TiO2 [26] is shown in Table 3.4.

Table 3.4 INPUT.inp of rutile TiO2

line 1 system=Rutile_TiO2line 2 iApp =LGNNline 3 iAlg =Dline 4 ippmG =4line 5 jmax =200line 6 deltaE =2.0d0line 7 npp =100line 8 nod =8line 9 nog =18line 10 ncut =5line 11 noz =18line 12 ncs =22line 13 nol =400line 14 nband =10line 15 q_eliminate =0.005d0line 16 icontinue =0line 17 isphcut =0line 18 iTotalE =0line 19 lpri =1line 20 nspin =0line 21 iMIX =0line 22 smixSCF =0.80d0line 23 nSDCG =8line 24 iSblp =80line 25 iSblp2 =15line 26 smixTD =0.80d0line 27 iasym =0line 28 icalAO =0line 29 iCry =1line 30 precLDA =0.0001d0line 31 iFTapp =0line 32 iCheb =0line 33 nstep =1line 34 END_PARA

3.2 INPUT.inp 29

3.2.1 iApp

G for GW, B for Bethe–Salpeter, D for both of them.1st character indicates the method of the SCF Loop:

L or N (None): the default, LDA.H: Hartree–Fock (available only for Γ-point calculations).G: Self-consistent GW (available only for Γ-point calculations).

2nd to 4th characters indicate the method after the SCF Loop:G: one-shot GW calculation.B: Bethe–Salpeter calculation.D: one-shot GW + Bethe-Salpeter calculation.T: T-matrix calculation.S: Second-order Møller–Plesset calculation.M: molecular dynamics (MD) using Newton’s equation of motion for nuclei available

for LDA only.X: structural relaxation using Broyden algorithm available for LDA only.A: Adiabaic LDA, electron dynamics calculation by solving the TD Kohn–Sham

equation.W: Wave functions (WaveF_HOMO, WaveF_LUMO, ...) output.

3.2.2 iAlg

D: Matrix Diagonalization (recommended for small target systems)S: Steepest DescentC: Conjugate Gradient + RMM-DIIS + Davidson.

3.2.3 iexc

Type of the exchange-correlation functional.iexc =3 [default] means that Perdew–Zunger’s interpolation formula [27] is used for the

exchange-correlation functional of the DFT. Usually, this is not written in INPUT.inp.

3.2.4 iCHG

Flag to output total Charge Density (ChargeDensity files) after the SCF loop is converged.Either 1 [on] or 0 [off] should be set if necessary. Default value of iCHG is 0.

3.2 INPUT.inp 30

3.2.5 Ulevel, Llevel

Levels to draw wave functions.When “W” is assigned in iApp to output wave functions, user can set "Ulevel = ***"

and "Llevel = ***" to control the number of wave functions to be generated. "Ulevel" and"Llevel" indicate the topmost and lowermost levels, and all the wave functions between (orincluding) these levels will be generated. Default is that Llevel = noc-1 (HOMO-1) andUlevel = noc+1 (LUMO+1). If the system is spin polarized, wave functions are generatedfor up and down spins separately.

3.2.6 ippmG

Selection of plasmon pole model or full ω integration.ippmG = 0 [default] (or 3) means the use of the generalized plasmon-pole (GPP) model

in the evaluation of the correlation part of the self-energy, Σc, while ippmG = 1 and ippmG= 2 mean the use of the plasmon pole model by Engel–Farid and von der Linden–Horsch.Note: ippmG = 1 (Engel–Farid) is not available for crystal system. If ippmG = 4 is set, thecorrelation part of self-energy, Σc, is evaluated by using the full ω integration along the realaxis and then turned 90 parallel to the imaginary axis (only positive part). If ippmG = 5 isset, the full ω integration along positive real axis is performed.

3.2.7 jmax

Number of mesh points used for the full ω integration.Note: Typically "jmax = 200" is used for ippmG = 4 and "jmax = 1000" for ippmG =

5, but "jmax" should be set as 0 [default] for ippmG = 0-3 when plasmon pole models areused.

If jmax = 200 is used with ippmG = 4, 201 points are set at 0.1+0.2n [eV] and 20.1+(1+ 2n)i [eV] for n = 0, 100 along the positive real axis and then rotated 90 parallel tothe positive imaginary axis. If jmax = 1000 is used with ippmG = 5, 1001 points are set at0.1+9.2n [eV].

3.2.8 deltaE

Finite difference ∆E in the numerical derivative of the renormalization factor Z.For the renormalization factor Z in Eq. (2.6), it is necessary to evaluate the derivative of

the self-energy Σ(ε) with respect to ε . "deltaE" is the fine difference (in units of [eV]) of∆ε of this numerical derivative ∂Σ(ε)/∂ε |εLDA

nk. Default value is deltaE = 0.5d0 [eV].

3.2 INPUT.inp 31

Note: When numerical ω integration is performed (ippmG=4,5), deltaE = 2.0 is recom-mended instead of the default value.

3.2.9 npp

Number of plasmon poles. Default value for npp is 999.Note: Typically 100 is chosen in the case of ippmG = 1 and 2. This parameter is ignored

in the case of ippmG = 0, 3, 4, 5.

3.2.10 nod

Maximum number of nodes in PWs in each direction (necessary item for any type ofcalculation).

Max number of |n1|, |n2|, |n3| of reciprocal lattice vector G = n1b1 + n2b2 + n3b3 forPWs. Default value for "nod" is 10.

The cut-off energy can be obtained by:

Ecut−o f f =

(2π ×nod

A1

)2

(3.2)

in the unit of [Ry]. Here A1 is the lattice constant (length of lattice vector a1) in the unit ofBohr radius.

Note: The cut-off energy can also be found in the "OUPUT.out" file.Note: Instead of giving nod, one may give Ecutoff, which is the cutoff energy of PWs,

corresponding the maximum kinetic energy of PWs, in units of [Ry]. (For example, one mayset Ecutoff = 24.d0 in INPUT.inp.) For isolated systems, 18 Ry is required for hydrogenatoms and 24 Ry required for transition metal, but 7 Ry is enough for carbon atoms and 3Ry is enough for sodium and lithium atoms. In principle, one has to check the convergenceof the total energy or level energies with increasing the cut-off energy (or "nod"). For crystalsystems, because one has to reduce the radii of non-overlapping atomic spheres to, e.g., 0.8[Å], "nod" (or "Ecutoff") should be chosen larger.

3.2.11 nog

Maximum number of nodes in G waves in the Fock exchange (necessary item for HFand GW calculations only.)

Max number of |n1|, |n2|, |n3| of reciprocal lattice vector G = n1b1 + n2b2 + n3b3 forthe exchange part of the self-energy Eq. (2.31). “nog” is also used for the Fourier space

3.2 INPUT.inp 32

calculation of the asymmetric potential inside the non-overlapping atomic sphere if iasym =1 is set. Default value for "nog" is 20.

Note: Typically set double of nod, because the spatially localized core contribution isquite important in the Fock exchange term.

Note: The cut-off energy can also be found in the "OUPUT.out" file.

3.2.12 ncut

Maximum number of nodes in G,G’ waves in the polarization function and the correla-tion part of the self-energy (necessary item for GW calculation only).

Max number of |n1|, |n2|, |n3| of reciprocal lattice vector G = n1b1+n2b2+n3b3 for thepolarization function and the correlation part of the self-energy Eq. (2.33). Default valuefor "ncut" is 8.

Note: Typically set the same number as nod or a little less than nod. (necessary item forGW calculations).

Note: The cut-off energy can also be found in the "OUPUT.out" file.

3.2.13 noz

Number of Fourier mesh for nuclear Coulomb tail.A smooth polynomial function inside the non-overlapping sphere connected to the 1/r

long tail of nuclear Coulomb potential (see Fig. 1.4) is analytically Fourier transformed asEq. (1.29) and summed over all atoms. "noz" is the maximum number of nodes in G wavesin this Fourier transformation. It is then numerically Fourier back transformed along theradial direction in each atomic sphere to make spherical potential felt by AOs as Eq. (1.30).

Default is noz = 24. For accurate calculation, large mesh such as 128 or 192 is recom-mended as well as mesh, which should be defined in COORDINATES.inp.

3.2.14 ncs

Number of core states."ncs" is the number of core states, which is excluded in the calculation of the correla-

tion part of the self-energy in the GW calculation or fixed in the TDDFT dynamics in theadiabatic LDA calculation. The default value is ncs = 0.

For example, in rutile TiO2, there are 2 Ti atoms and 4 O atoms in a unit cell:Ti: 1s22s22p63s23p6 1 + 1 + 3 +1 +3 = 9

Note: p orbital contains px, py and pz, so the number is 3 for p orbital.

3.2 INPUT.inp 33

O : 1s2 There is only one s orbital, so the number is 1 for s orbital.In rutile TiO2, 9×2+1×4 = 22, sothat ncs = 22.

3.2.15 ncc

Number of core constituents."ncc" is the number of core constituents to be excluded from the Fock exchange in HF

and GW calculations. For the intermediate states in the calculation of the exchange term,the states from “number of core constituents” (ncc) + 1 to “number of occupied states” (noc)are used in the code. Deep core states also contribute largely to the exchange term, so that"ncc" should be usually set at 0 [default].

3.2.16 nol

Number of levels."nol" is the number of levels (states) to be included in the GW calculations of the corre-

lation parts (the polarization function and the self-energy). Default value for iSblp is 300.Note: If "nol" is not enough, it leads band gap become large.Note: "nol" must be less than the number of basis set (functions) "nbs". The value of

"nbs" can be found in "OUTPUT.out" file.

3.2.17 nband

Number of empty states to display in GWA.out."nband" is the number of output empty levels (band), i.e., the number of empty states in

conduction bands to be displayed in GW (GWA.out). Default is nband = 0.

3.2.18 q_eliminate

Criterion to remove AO.AO is eliminated, if truncated AO has norm less that "q_eliminate". For example, if

one sets q_eliminate = 0.001 as small enough, no AO will be removed from the basis set.The default value of q_eliminate is 0.3, if the unit cell size is less than 4.0 Å, and 1×10−2

otherwise.

3.2.19 q_min_mod

Criterion to truncate AOs.

3.2 INPUT.inp 34

Whether the subtraction of a polynomial from AO is performed or not depends on thenorm of the original AO outside the atomic sphere. If this norm exceeds q_min_mod,the subtraction is performed. q_min_mod can be set in INPUT.inp. The default value ofq_min_mod is 1×10−4.

3.2.20 icontinue

continue job to TDDFT dynamics or to one-shot GW.Frag to continue to the previous calculation by skipping the LDA SCF Loop. Set 0

[default] to perform a new calculation or 1 to skip the LDA SCF Loop.

3.2.21 isphcut

Coulomb spherical cut (available for isolated systems only).-1 : Flag to truncate Coulomb potential tail at half of the lattice constant;0 : (default) not to truncate.

3.2.22 iTotalE

Total Energy calculation.To calculate SCF (LDA, HF, or self-consistent GW) Total Energy, set iTotalE = 1, but

otherwise set iTotalE = 0 [default].

3.2.23 lpri

Output detailed information of calculations in ISYS.out.Either 1 [on] or 0 [off] should be set. Default value of lpri is 0.

3.2.24 nspin

Number up spin electrons minus number of down spin electrons.For the spin polarized system, one can set nspin = 1 for spin doublet. nspin = 2 for spin

triplet, ... nspin = # for spin (# + 1) multiplet in INPUT.inp. Its default value is nspin = 0. Inorder to treat an antiferromagnetic system having zero total magnetic moment, one shouldwrite nspin = -1 in INOUT.inp and simultaneously, one may write the spin magnetic momentof each atom as m 1 or m -1 at the right of the atomic coordinate in COORDINATES.inp.(nspin = -1 can be used together with iExcite = 1, 2, ...)

3.2 INPUT.inp 35

3.2.25 irelative

Semi-relativistic effect.Flag to include the semi-relativistic effect into the LDA calculation (1 for ON, 0 for

OFF). The Darwin and mass-velocity terms are included as a relativistic effect in the LDAcalculation, when irelative = 1 is set.

3.2.26 nion

Number of excess electrons for ionic system.For isolated systems using spherical Coulomb cut-off (isphcut = -1), user can define

“nion” to calculate ionic systems. Since nion means the excess number of electrons, so thatnion > 0 corresponds to minus ion, and nion < 0 corresponds to plus ion. For example, nion= +2 corresponds to -2 ion. Note that plus and minus signs are different from the usualconvention. Of course, nion = 0 (defual) means a neutral system.

3.2.27 iExcite

Number of levels to excite an up-spin electron.If one wants to excite one electron from HOMO to LUMO +n at the outset, one may

define iExcite = 1+ n in INPUT.inp (n can be 0, 1, 2, ...). Default value of iExcite is 0,which means no excitation. Since only up-spin electron is excited, so that nspin should beset either -1 or some positive numbers.

3.2.28 nonadiabatic

Nonadiabatic treatment in TDDFT dynamics simulations.Nonadiabatic correction proportional to the velocity of atoms can be treated in the update

of wave packet if nonadiabatic = 1 is set. Default value is nonadiabatic = 0.

3.2.29 iMIX

Method of SCF Charge Density Mixing.0: Linear, 1: Optimized Linear, 2: Pulay (DIIS), 3: Broyden Charge Mixing. Default

setting is iMIX = 2. But, usually, iMIX = 3 is recommended in usual cases. Usually iMIX =3 (or 2) is recommended, but, if not converged, set iMIX = 0 with smixSCF = 0.90 or 0.95.

3.2 INPUT.inp 36

3.2.30 smixSCF

Charge Mixing Rate in SCF loop for LoopTD = 1.Rate of mixing (MAX) previous Charge Density for the 1st MD step (LoopTD = 1).

Default value is smixTD = 0.8d0.

3.2.31 nSDCG

Number of Steepest Descent Loops.Number of Steepest-Descent (SD) or Conjugate-Gradient (CG) iterations without up-

dating the Hamiltonian. Default value is nSDCG = 8.

3.2.32 iSblp

Max SCF Loop for LoopTD = 1.Number of max SCF Loops for the 1st step in the starting SCF Loop (LoopTD = 1).

Default value for iSblp is 999.

3.2.33 iSblp2

Max SCF Loop with only Gamma point. Default value for "iSblp2" is 1.LoopSCF > iSblp2 : special point loop using SPOINT. Default value for "iSblp" is 1.

3.2.34 mSblp

Maximum number of SCF loops during MD or relaxation.Number of max SCF Loops in the dynamics Loop (LoopTD > 1). During this loop,

Hamiltonian is updated with fixed atomic positions. Default value for "mSblp" is 1.In order to perform a simulation with conserved total energy, it is recommended to set

mSblp = 20.

3.2.35 smixTD

Charge Mixing Rate (in or outside SCF loop) for LoopTD > 1.Rate of mixing (MAX) previous Charge Density after the 1st MD step (LoopTD > 1).

Default value is smixTD = 0.5d0.

3.2 INPUT.inp 37

3.2.36 iasym

Asymmetric potential inside atomic sphere.Frag to treat asymmetric AO charge inside atomic spheres (1 for ON, 0 for OFF). Default

value of "iasym" is 0.

3.2.37 icalAO

AO redetermination at each SCF loop.Frag to update Atomic Orbitals (AOs) at each SCF step (1 for ON, 0 for OFF). Default

value of "icalAO" is 0.

3.2.38 iCry

Cluster/Crystal Calculation.Frag to calculate crystal calculation (0 for cluster calculation, 1 for crystal calculation).

Default set is iCry = 0, i.e., cluster calculation.

3.2.39 precLDA

Convergence criterion for Total Energy in units of [eV].Criterion of Total Energy convergence (SCF loop ends when ∆E becomes less than this).

Default value is precLDA = 0.0001d0.

3.2.40 iFTapp

If iFTapp = 1 is set, an approximate treatment of this 1D (in radial coordinate insideatomic sphere) Fourier transformation is used to accelerate the computational speed. Defaultvalue of iFTapp is 0.

3.2.41 iCheb

Chebyshev fitting for atomic potential to accelerate the computational speed.Frag to fit AO potential in atomic spheres to Chebyshev polynomial (1 for ON, 0 for

OFF), in order to reduce the time required for the computation of < PW |V |AO > matrixelements. Default value of iCheb is 1, i.e., Chebyshev fitting ON.

3.3 KPOINT.inp 38

3.2.42 Mcheb

Number of Chebyshev polynomials for fitting AO potential in atomic spheres. Defaultvalue of MCheb is 30.

3.2.43 nStep

Maximum number of MD (or Relaxation or TDDFT) steps. Default value of nStep is 1.

3.2.44 singlet, triplet

Initial guess for the singlet and triplet optical gaps.In solving the Bethe–Salpeter equation to obtain photoabsorption spectra, an initial

guess for optical gap ("singlet" and "triplet") must be given in units of [eV]. "singlet" isa value for the singlet exciton, and "triplet" is a value for the triplet exciton. Default valuesare singlet = 0.d0 and triplet = 0.d0.

3.3 KPOINT.inp

In a crystal calculation (iCry = 1), user has to define k points (usually on symmetrylines) for the energy-band outputs. For each band, a list of energies of all the k points aregiven by "band_0001.out", "band_0002.out", .... The total list including all bands is givenby band.out. Form of KPOINT.inp is given in Table 3.5.

Table 3.5 Form of KPOINT.inp file

line 1 Direct/Cartesianline 2 numbers of k points for "out+sum", "out", and "sum"line 3 1st k point and weightline 4 2nd k point and weightline 5 3rd k point and weightline 6 ... ...line 7 nth k point and weight

For a three-dimensional lattice defined by its primitive vectors (a1, a2, a3), the reciprocallattice is generated by its three reciprocal primitive defined by the formulae

b1 = 2πa2 ×a3

a1 · (a2 ×a3), b2 = 2π

a3 ×a1a2 · (a3 ×a1)

, b3 = 2πa1 ×a2

a3 · (a1 ×a2). (3.3)

3.3 KPOINT.inp 39

Note: line 1Direct: k points in the fractional coordinates (coefficients for b1, b2, b3.);Cartesian: coefficients for 2π/A1 in Cartesian coordinates; (A1 is the lattice constant.)

Note: line 2For the LDA band calculation, “out + sum” should be equal to the number of k points

listed below; “sum” and “out” should be set equal to 0. For the one-shot GW calculation,“out” is the number of k-points for output and “sum” is the number of k-points for thek-point sampling in the calculation of the polarization function for the correlation part ofthe self-energy (k points in “out” are inside the irreducible Brillouin Zone (BZ), but “sum”should be given in the whole BZ. Usually, “sum” should be given on Γ grid.); “out + sum”is the number of k-points for both “out” and “sum”.Note: line 3-7

weights for “out+sum” and “sum” are meaningful to calculate the polarization function.

Table 3.6 KPOINT.inp for LDA of Si

line 1 Directline 2 16 0 0line 3 0.500000000000 0.250000000000 0.750000000000 0.11764705882line 4 0.500000000000 0.333333333333 0.666666666667 0.05882352941line 5 0.500000000000 0.416666666667 0.583333333333 0.05882352941line 6 0.500000000000 0.500000000000 0.500000000000 0.05882352941line 7 0.375000000000 0.375000000000 0.375000000000 0.05882352941line 8 0.250000000000 0.250000000000 0.250000000000 0.05882352941line 9 0.125000000000 0.125000000000 0.125000000000 0.05882352941line 10 0.000000000000 0.000000000000 0.000000000000 0.05882352941line 11 0.100000000000 0.000000000000 0.100000000000 0.05882352941line 12 0.200000000000 0.000000000000 0.200000000000 0.05882352941line 13 0.300000000000 0.000000000000 0.300000000000 0.05882352941line 14 0.400000000000 0.000000000000 0.400000000000 0.05882352941line 15 0.500000000000 0.000000000000 0.500000000000 0.05882352941line 16 0.500000000000 0.125000000000 0.625000000000 0.05882352941line 17 0.437500000000 0.312500000000 0.750000000000 0.05882352941line 18 0.375000000000 0.375000000000 0.750000000000 0.05882352941

Table 3.6 is KPOINT.inp for a LDA calculation of Si. The fourth column, i.e., theweight, is no meaning in the LDA calculation. The k points listed in Line 3, line 6, line

3.4 QPOINT.inp 40

10, line 15, and line 18 corresponds to the W, L, Γ, X, and K points, respectively. The 1stBrillouin zone of bcc and fcc crystals is shown in Fig. 3.2.

Fig. 3.2 The first Brillouin zone of bcc and fcc crystals.

Table 3.7 is KPOINT.inp for a one-shot GW calculation of rutile TiO2 [26].

Table 3.7 KPOINT.inp for one-shot GW of rutile TiO2

line 1 Directline 2 6 0 2line 3 0.00000 0.00000 0.00000 1.00 #Γline 4 0.00000 0.00000 0.50000 1.00 #Zline 5 0.00000 0.50000 0.00000 1.00 #Xline 6 0.00000 0.50000 0.50000 1.00 #Rline 7 0.50000 0.50000 0.00000 1.00 #Mline 8 0.50000 0.50000 0.50000 1.00 #Aline 9 0.50000 0.00000 0.50000 1.00line 10 0.50000 0.00000 0.00000 1.00

3.4 QPOINT.inp

q points for the momentum transfer on Γ grid inside the irreducible BZ. This file isrequired for the one-shot GW crystal calculation only. Form of QPOINT.inp file is given byTable 3.8.

3.4 QPOINT.inp 41

Table 3.8 Form of QPOINT.inp file

line 1 Direct/Cartesianline 2 The number of q points in the irreducible zone, and the number of all grid points (see Appendix B of [? ]

in the whole zoneline 3 1st q point and weightline 4 2nd q point and weightline 5 3rd q point and weightline 6 ... ...line 7 nth q point and weight

Note:The 3rd line should be "0.00000 0.00000 0.00000".

Note:Increasing the number of q-points, it improves the accuracy of GW results, but it will

take too much time.Note: line 1

This line is the same as KPOINT.inp.Note: line 2

Rutile TiO2 has the symmetry group of P42/MNM, and 16 symmetry operations.Here, 6 q-points are chosen as shown in Fig. (3.3).

Fig. 3.3 q-points and wights in b1, b2 plane

3.5 SPOINT.inp 42

Table 3.9 QPOINT.inp of rutile TiO2

line 1 Directline 2 6 27line 3 0.00000 0.00000 0.00000 1.00line 4 0.00000 0.00000 0.50000 1.00line 5 0.50000 0.00000 0.00000 2.00line 6 0.50000 0.00000 0.50000 2.00line 7 0.50000 0.50000 0.00000 1.00line 8 0.50000 0.50000 0.50000 1.00

3.5 SPOINT.inp

In SPOINT.inp file, give special k points used for the SCF iteration loop in a crystalcalculation. Usually, special points are on the Monkhorst–Pack grid inside the irreducibleBrillouin zone. Form of SPOINT.inp file is given by Table 3.10.

Table 3.10 Form of SPOINT.inp file

line 1 Direct/Cartesianline 2 number of k pointsline 3 1st k point and weightline 4 2nd k point and weightline 5 3rd k point and weightline 6 ... ...line 7 nth k point and weight

Note: line 1This line is the same as KPOINT.inp.

Table 3.11 is SPOINT.inp for a LDA calculation of Si. Here the fourth column, i.e., theweigh, has important meaning when the special point sampling is performed to make thetotal electron density.

3.5 SPOINT.inp 43

Table 3.11 SPOINT.inp for LDA of Si

line 1 Directline 2 2line 3 0.2500000 0.2500000 0.2500000 1.00line 4 0.2500000 0.2500000 -0.2500000 3.00

Table 3.12 is SPOINT.inp for a one-shot GW calculation

Table 3.12 SPOINT.inp for one-shot GW of rutile TiO2

line 1 Directline 2 24line 3 0.40000000 0.40000000 0.43750000 0.04000000line 4 0.40000000 0.40000000 0.31250000 0.04000000line 5 0.40000000 0.40000000 0.18750000 0.04000000line 6 0.40000000 0.40000000 0.06250000 0.04000000line 7 0.40000000 0.20000000 0.43750000 0.08000000line 8 0.40000000 0.20000000 0.31250000 0.08000000line 9 0.40000000 0.20000000 0.18750000 0.08000000line 10 0.40000000 0.20000000 0.06250000 0.08000000line 11 0.40000000 0.00000000 0.43750000 0.04000000line 12 0.40000000 0.00000000 0.31250000 0.04000000line 13 0.40000000 0.00000000 0.18750000 0.04000000line 14 0.40000000 0.00000000 0.06250000 0.04000000line 15 0.20000000 0.20000000 0.43750000 0.04000000line 16 0.20000000 0.20000000 0.31250000 0.04000000line 17 0.20000000 0.20000000 0.18750000 0.04000000line 18 0.20000000 0.20000000 0.06250000 0.04000000line 19 0.20000000 0.00000000 0.43750000 0.04000000line 20 0.20000000 0.00000000 0.31250000 0.04000000line 21 0.20000000 0.00000000 0.18750000 0.04000000line 22 0.20000000 0.00000000 0.06250000 0.04000000line 23 0.00000000 0.00000000 0.43750000 0.01000000line 24 0.00000000 0.00000000 0.31250000 0.01000000line 25 0.00000000 0.00000000 0.18750000 0.01000000line 26 0.00000000 0.00000000 0.06250000 0.01000000

Chapter 4

Output Files

Note:

1. In every calculation, output files includes OUTPUT.out, Status.out, and ISYS.out.There appear also CHARGE.bin for the next job when icontinue = 1 is set.

2. In the dynamics calculation when “M”, “X” or “A” is assigned in iApp of INPUT.inp,output files include also: posit.dat and eigen.dat as well as MD_Coordinates.xyz andMD_Coordinates.arc.

3. In the crystal calculation, output files includes also: band.out, and so on.

4. If “W” is assigned in iApp of INPUT.inp, output files includes also: WaveF_HOMO-1.cube, WaveF_HOMO-1.grd, WaveF_HOMO-1.vasp, and so on.

5. If iCHG = 1 is set in INPUT.inp, output files includes also: ChargeDensity.cube,ChargeDensity.grd, and ChargeDensity.vasp.

6. In the GW calculation, output files include also: GWA.out.

7. In the Bethe–Salpeter equation (BSE) calculation, output files include also: PhotoAb-sorptionSpectra.out.

4.1 OUTPUT.out

In OUTPUT.out file, you can find the number of occupied states "noc", the number ofplane waves "nw", the number of atomic orbitals "nao", and the cut-off energies for planewaves, Fock exchange and correlation, and so on. This file appears just after the executionstarts, so you can check these information quickly.

4.2 Status.out 45

4.2 Status.out

In Status.out, the job process is displayed during execution. One can display it by typing"tail -f Status.out".

4.3 ISYS.out

In ISYS.out file, you can find the detailed output during execution. First, the informationof the unit cell, the result of Herman-Skillman’s atomic LDA calculation for generatingAOs, the symmetry operations, the information of the truncation of AOs are written in thisISYS.out. After the mixed-basis calculation starts, for example, the LDA eigenvalues ateach SCF step and the force contributions at each TD step are written in this ISYS.out. Inthe GW calculations, some information of the polarization and dielectric functions are alsowritten.

4.4 posit.dat

When MD or other dynamics calculation is performed, atomic positions and forces ateach time step are written in position.dat.

4.5 eigen.dat

When MD or other dynamics calculation is performed, eigenvalues at each TD step arewritten in eigen.dat.

4.6 MD_Coordinates.xyz, MD_Coordinates.arc

When MD or other dynamics calculation is performed, atomic trajectories are writtenin these files for visualization. MD_Coordinates.xyz is for VMD, MD_Coordinates.arc forMaterials Studio.

4.7 band.out

In a crystal calculation when iCry = 1 set in INPUT.inp, band.out is the result of energyeigenvalues (at sequential k points) in all bands. This file can be read, for example, by

4.8 WaveF_HOMO-1.cube, WaveF_HOMO-1.grd,WaveF_HOMO-1.vasp, and so on 46

Microsoft EXCEL to draw the energy band diagram as a line graph.

band_0001.out is the result of energy eigenvalues (at sequential k points) in the 1st band;band_0002.out is the result of energy eigenvalues (at sequential k points) in the 2nd band;band_0003.out is the result of energy eigenvalues (at sequential k points) in the 3rd band;... ...

4.8 WaveF_HOMO-1.cube, WaveF_HOMO-1.grd,WaveF_HOMO-1.vasp, and so on

If “W” is assigned in iApp of INPUT.inp, wave output files are generated for all wavefunctions between Llevel and Ulevel assigned in INPUT.inp. Files with an extension “cube”is for GaussView, files with an extension “grd” is for Materials Studio, and files with anextension “vasp” is for VESTA.

4.9 GWA.out

"GWA.out" file saves the GW calculation results.In a GW cluster calculation, GWA.out looks like Table 4.1.In a GW crystal calculation, please find the sentence The Result of q-point sum in

GWA.out file. Information above this sentence is for intermediate calculations and not thefinal result. The final result is below this sentence, and looks like Table 4.2 for rutile TiO2.Note: Loopk_k = n (n=1, 2, ...)

n means nth k point, corresponding to KPOINT.inp file.Note: level with * symbol

It means this level is the highest occupied level, and level *+1 is the lowest empty level.Note: exc(LDA) is the LDA exchange-correlation potential, V LDA

xc , in [eV].Note: eps(LDA) is LDA eigenvalue in [eV].Note: xg(Fock) is exchange part of self-energy, Σx, in [eV].Note: slf(GWA) is correlation part of self-energy, Σx, in [eV].Note: QP(GWA) is quasi-particle energy ε0

n,k in [eV].

ε0n,k = εLDA

n,k +∫

drdr′ψLDAn,k

∗(r)[Σ(r,r′;εLDA

n,k )−V LDAxc δ (r− r′)]ψLDA

n,k (r′) (4.1)

4.9 GWA.out 47

Note: RQP(GWA) is renormalized quasi-particle energy εGWn,k in [eV],

εGWn,k = ε0

n,k +(ε0n,k − εLDA

n,k )∂ΣGW

n,k (ε)∂ε

∣∣∣∣∣ε=εLDA

n,k

Zn,k, (4.2)

which is equal to Eq. (2.6). Here Zn,k is a renormalization factor defined by

Zn,k =

[1−

∂ΣGWn,k (ε)∂ε

]−1

ε=εLDAn,k

. (4.3)

Table 4.1 GWA.out of Li2 GW cluster calculation

Loopk_q = 0 Loopk_k = 1Na_ 2: mesh= 48 size= 2.117 nod=6 nog=12 nol= 300 ncut= 6

q=0 only, iDiag 0 Cor RenorLDA -relation -malized

<μ xc >　 eigenvalue <Σ x > <Σ c > QP energy QP energyNa 2 exc(LDA) eps(LDA) xg(Fock) slf(GWA) QP(GWA) RQP(GWA)

* 3 -5.5682 -3.5408 -7.3972 -0.8465 -6.2163 -5.60384 -3.5836 -1.7990 -1.3020 -0.9539 -0.4713 -0.70525 -3.8425 -1.5722 -1.2141 -1.0514 0.0048 -0.29006 -3.8425 -1.5722 -1.2141 -1.0513 0.0049 -0.28997 -3.0842 -1.0258 -0.9242 -1.0423 0.0920 -0.10848 -1.7537 -0.0763 -0.2596 -0.8076 0.6102 0.47849 -2.0835 -0.0312 -0.2372 -1.0481 0.7669 0.5806

10 -2.0835 -0.0312 -0.2372 -1.0482 0.7669 0.580611 -1.6420 0.4796 -0.1403 -0.8559 1.1254 0.991912 -1.6420 0.4796 -0.1403 -0.8559 1.1254 0.991913 -1.0370 1.0156 -0.0488 -0.5180 1.4857 1.4327

Eg(LDA): 1.742 Eg(GWA): 5.745IP(LDA): 3.541 IP(GWA): 6.216

4.9 GWA.out 48

Table 4.2 GWA.out of TiO2 GW crystal calculation

The Result of q-point sumLoopk_k = 1Level exc(LDA) eps(LDA) xg(Fock) slf(GWA) QP(GWA) RQP(GWA)

23 -20.6370 -8.7741 -28.3616 7.9457 -8.5530 -8.615724 -21.8867 -7.5498 -29.4496 8.0295 -7.0832 -7.231625 -22.5537 -6.8436 -29.9669 7.2448 -7.0120 -6.995126 -22.5484 -6.7963 -29.9329 7.2063 -6.9746 -6.958027 -18.7654 3.1389 -20.0824 3.4084 5.2303 4.750028 -20.1376 3.4787 -20.8328 3.3223 6.1058 5.446829 -16.8424 3.7836 -18.2006 3.6425 6.0679 5.598730 -20.8289 4.9844 -21.5778 3.3271 7.5625 7.036931 -20.8219 5.0327 -21.5454 3.2865 7.5957 7.085932 -19.8849 6.2660 -21.1415 3.2785 8.2879 8.006033 -19.9192 6.2689 -21.1534 3.2696 8.3042 8.044834 -20.8307 6.8520 -22.0483 3.4441 9.0785 8.578235 -21.9237 8.1268 -22.6500 2.7348 10.1353 9.822536 -21.7953 8.3549 -22.3521 2.7431 10.5412 10.117037 -21.8323 8.3580 -22.3833 2.7383 10.5453 10.1039

* 38 -21.5760 8.7647 -22.2072 2.9078 11.0413 10.475639 -22.3485 10.4435 -13.0584 -4.9269 14.8067 14.122840 -21.9881 10.7806 -13.3086 -4.4607 14.9993 14.170641 -22.7476 10.9637 -13.6819 -4.7350 15.2944 14.162342 -23.2351 11.0461 -13.7080 -5.0459 15.5273 14.529843 -23.2203 11.0883 -13.6749 -4.9537 15.6800 14.653244 -25.2912 13.0470 -16.3438 -5.1293 16.8650 16.249345 -24.9397 13.1399 -15.1763 -5.7544 17.1488 16.174146 -26.0406 15.1183 -17.3744 -4.6174 19.1671 18.162347 -26.0336 15.1694 -17.3581 -4.5614 19.2835 18.262948 -26.7895 16.2643 -17.1368 -5.6359 20.2812 19.1626

Loopk_k = 2Level exc(LDA) eps(LDA) xg(Fock) slf(GWA) QP(GWA) RQP(GWA)

23 -22.0740 -7.3505 -29.6158 7.7835 -7.1088 -7.142724 -22.1741 -7.3054 -29.6361 7.3740 -7.3934 -7.389925 -22.0695 -7.3024 -29.5826 7.7010 -7.1144 -7.134026 -22.1688 -7.2586 -29.6010 7.2227 -7.4681 -7.4680... ...

4.10 PhotoAbsorptionSpectra.out 49

4.10 PhotoAbsorptionSpectra.out

Photoabsorption spectra calculated by solving the Bethe–Salpeter equation (BSE) iswritten in PhotoAbsorptionSpectra.out as Table 4.3.

BSE routine is available only for a cluster calculation.Note: line 1: singlet and triplet is the values assigned in INPUT.inp. If not assigned,

0.000000000 (eV) is written.Note: line 2: multiplicity 0 corresponds to singlet and multiplicity 1 (must be 3 in usual

definition) corresponds to triplet.Note: lines 3-: First column is the photoabsorption energy in units of [eV], second and

third columns are the absorption coefficients for singlet and triplet excitons, respectively.

Table 4.3 OUTPUT of TiO2 GW calculation

singlet = 0.000000000 (eV), triplet = 0.000000000 (eV)multiplicity = 0　　　 1　　　0.000000000 0.002468937 0.1851386210.005440000 0.002480351 0.1926297770.010880000 0.002491845 0.2005845220.016320000 0.002503421 0.2090418620.021760000 0.002515078 0.2180449850.027200000 0.002526818 0.2276418120.032640000 0.002538641 0.2378856330.038080000 0.002550548 0.2488358380.043520000 0.002562540 0.2605587780.048960000 0.002574617 0.2731287610.054400000 0.002586781 0.286629224energy (eV) singlet exciton triplet exciton

Chapter 5

Examples

5.1 LDA calculation of isolated dimers

Figs. 5.1(a) and (b) represent the force and total energy of nitrogen dimer (N2) in the LDAcalculations with and without the spherical cut of Coulomb potential. In both calculations,unit cell is simple cubic with the lattice constant of 12 [Å], "nspin=0", "N _rct=0.50", and"nod=12" corresponding to the PW cut-off energy of 11.0 [Ry]. The two results with andwithout the spherical cut are almost the same. There is a minimum of the total energy atdistance 1.15 [Å], where the Force crosses zero.

(a) with spherical cut (b) without spherical cut

Fig. 5.1 Force and total energy of nitrogen dimer (N2) versus bond length in a calculationwith and without spherical cut.

5.1 LDA calculation of isolated dimers 51

Figs. 5.2(a) and (b) represent the force and total energy of iron dimer (Fe2) in the LSDAcalculations with and without the spherical cut of Coulomb potential. In both calculations,unit cell is simple cubic, "nspin=6", and "Fe_rct=1.00". The lattice constant is 12 [Å] (6.8[Å]) and "nod=17" ("nod=10") corresponding to the PW cut-off energy of 22.2 [Ry] (23.9[Ry]) for the calculation with (without) the spherical cut. The two results with and withoutthe spherical cut are slightly different. There is a minimum of the total energy at distance2.00 [Å] (1.97 [Å]), where the Force crosses zero.

(a) with spherical cut (b) without spherical cut

Fig. 5.2 Force and total energy of iron dimer (Fe2) versus bond length in a calculation withand without spherical cut.

Figs. 5.3(a) and (b) represent the force and total energy of iron dimer (Ni2) in the LSDAcalculations with and without the spherical cut of Coulomb potential. In both calculations,unit cell is simple cubic, "nspin=2", and "Ni_rct=1.00". The lattice constant is 12 [Å] (6.8[Å]) and "nod=17" ("nod=10") corresponding to the PW cut-off energy of 22.2 [Ry] (23.9[Ry]) for the calculation with (without) the spherical cut. The two results with and withoutthe spherical cut are slightly different. There is a minimum of the total energy at distance2.03 [Å] (2.01 [Å]), where the Force crosses zero.

5.1 LDA calculation of isolated dimers 52

(a) with spherical cut (b) without spherical cut

Fig. 5.3 Force and total energy of nickel dimer (Ni2) versus bond length in a calculationwith and without spherical cut.

The COORDINATES.inp and INPUT.inp files for iron dimer (Fe2) calculation is shownin Tables 5.1 and 5.2.

Table 5.1 COORDINATES.inp of Fe2

line 1 SYSTEM=Fe2line 2 12.000000000000line 3 1.000000000 0.000000000 0.000000000line 4 0.000000000 1.000000000 0.000000000line 5 0.000000000 0.000000000 1.000000000line 6 2line 7 Cartesianline 8 Fe 0.0000000000 0.0000000000 0.0000000000line 9 Fe 1.1547005384 1.1547005384 1.1547005384line 10 Fe_rct=1.00line 11 Fe_ns=3line 12 Fe_nc=3line 13 Fe_nv=4line 14 mesh=64line 14 END_PARA

5.2 Molecular Dynamics 53

Table 5.2 INPUT.inp of Fe2

line 1 SYSTEM=Fe2line 2 iApp =LMNNline 3 iAlg =Dline 4 nod =14line 5 ncs =0line 6 noz =64line 15 icontinue =0line 16 isphcut =-1line 17 iTotalE =1line 18 lpri =1line 19 nspin =6line 20 iMIX =0line 21 smixSCF =0.90line 23 iSblp =300line 26 iasym =0line 27 icalAO =0line 28 iCry =0line 29 iExcite =0line 30 precLDA =0.000001d0line 31 iFTapp =0line 32 iCheb =0line 33 nstep =1line 34 END_PARA

5.2 Molecular Dynamics

As an example of a first-principles molecular dynamics simulation, Fig. 5.4 shows severalsnapshots of a chemical reaction producing a formic acid (HCOOH) molecule from a carbondioxide (CO2) molecule and two hydrogen atoms (2H) [2].

In this MD simulation, COORDINATES.inp file given in Table 5.3 and INPUT.inp filesgiven in Table 5.4 are used.

5.2 Molecular Dynamics 54

Fig. 5.4 Snap shots of a molecular dynamics (MD) simulation producing HCOOH fromCO2 + 2H.

Table 5.3 COORDINATES.inp of 2H + CO2

line 1 CO2+H2=>HCOOHline 2 8.000000000000line 3 1.000000000 0.000000000 0.000000000line 4 0.000000000 1.000000000 0.000000000line 5 0.000000000 0.000000000 1.000000000line 6 2line 7 Cartesianline 8 C 1.2000000000 0.0000000000 0.0000000000line 9 O 0.0000000000 0.0000000000 0.0000000000line 10 O 2.4000000000 0.0000000000 0.0000000000line 11 H 1.2000000000 1.7000000000 0.0000000000line 12 H 2.4000000000 -1.7000000000 0.0000000000line 13 C _rct=0.60line 14 C _ns=2line 15 C _np=1line 16 O _rct=0.50line 17 O _ns=2line 18 O _np=1line 19 mesh=48line 20 dTime=0.1d0line 21 END_PARA

5.3 GW calculation of rutile TiO2 55

Table 5.4 INPUT.inp of 2H + CO2

line 1 CO2+H2=>HCOOHline 2 iApp =LMNNline 3 iAlg =Sline 4 nod =12line 5 ncs =0line 6 noz =48line 15 icontinue =0line 16 isphcut =0line 17 iTotalE =1line 18 lpri =1line 19 nspin =0line 20 iMIX =3line 21 smixSCF =0.80line 23 iSblp =100line 26 iasym =0line 27 icalAO =0line 28 iCry =0line 29 iExcite =0line 30 precLDA =0.00001d0line 31 iFTapp =0line 32 iCheb =0line 33 nstep =1000line 34 END_PARA

5.3 GW calculation of rutile TiO2

Five input files are needed for the GW crystal calculationCOORDINATES.inp (Table 3.2)INPUT.inp (Table 3.4)KPOINT.inp (Table 3.6)QPOINT.inp (Table 3.8)SPOINT.inp (Table 3.10)

in addition to a sh file for submitting job. Table 5.5 is an example of sh file.

5.4 LDA calculation of rutile TiO2 56

Table 5.5 A sh file for submitting job (run.sh)

mkdir tmpdate./home/program/maindateunsetenv LDR_CNTRL

Note: "/home/program/main" is the file path of TOMBO code.

When the job is finished, the GW result is list in "GWA.out" file.

5.4 LDA calculation of rutile TiO2

Method: Self-consistent field (SCF) loop with special-point sampling (SPOINT.inp) andband structure calculations at the k-points (KPOINT.inp) on symmetry lines are performedwithin standard local density approximation (LDA) of density functional theory (DFT).

In the LDA calculation, it needs 4 input files:

COORDINATES.inpINPUT.inpKPOINT.inpSPOINT.inp

The COORDINATES.inp and SPOINT.inp for LDA calculation are the same with thosefor GW calculation. In INPUT.inp file, some parameters should be modified:

—————————————-iApp = LGNN → LNNN—————————————-For example, KPOINT.inp for LDA calculation is list as follow:

line 1 Directline 2 51 0 0line 3 0.00000000000 0.00000000000 0.0000000000 1.00 #Γline 4 0.00000000000 0.00000000000 0.0500000000 1.00line 5 0.00000000000 0.00000000000 0.1000000000 1.00line 6 0.00000000000 0.00000000000 0.1500000000 1.00

5.4 LDA calculation of rutile TiO2 57

line 7 0.00000000000 0.00000000000 0.2000000000 1.00line 8 0.00000000000 0.00000000000 0.2500000000 1.00line 9 0.00000000000 0.00000000000 0.3000000000 1.00line 10 0.00000000000 0.00000000000 0.3500000000 1.00line 11 0.00000000000 0.00000000000 0.4000000000 1.00line 12 0.00000000000 0.00000000000 0.4500000000 1.00line 13 0.00000000000 0.00000000000 0.5000000000 1.00 #Zline 14 0.00000000000 0.05000000000 0.4500000000 1.00line 15 0.00000000000 0.10000000000 0.4000000000 1.00line 16 0.00000000000 0.15000000000 0.3500000000 1.00line 17 0.00000000000 0.20000000000 0.3000000000 1.00line 18 0.00000000000 0.25000000000 0.2500000000 1.00line 19 0.00000000000 0.30000000000 0.2000000000 1.00line 20 0.00000000000 0.35000000000 0.1500000000 1.00line 21 0.00000000000 0.40000000000 0.1000000000 1.00line 22 0.00000000000 0.45000000000 0.0500000000 1.00line 23 0.00000000000 0.50000000000 0.0000000000 1.00 #Xline 24 0.00000000000 0.50000000000 0.0500000000 1.00line 25 0.00000000000 0.50000000000 0.1000000000 1.00line 26 0.00000000000 0.50000000000 0.1500000000 1.00line 27 0.00000000000 0.50000000000 0.2000000000 1.00line 28 0.00000000000 0.50000000000 0.2500000000 1.00line 29 0.00000000000 0.50000000000 0.3000000000 1.00line 30 0.00000000000 0.50000000000 0.3500000000 1.00line 31 0.00000000000 0.50000000000 0.4000000000 1.00line 32 0.00000000000 0.50000000000 0.4500000000 1.00line 33 0.00000000000 0.50000000000 0.5000000000 1.00 #Rline 34 0.05000000000 0.50000000000 0.4500000000 1.00line 35 0.10000000000 0.50000000000 0.4000000000 1.00line 36 0.15000000000 0.50000000000 0.3500000000 1.00line 37 0.20000000000 0.50000000000 0.3000000000 1.00line 38 0.25000000000 0.50000000000 0.2500000000 1.00line 39 0.30000000000 0.50000000000 0.2000000000 1.00line 40 0.35000000000 0.50000000000 0.1500000000 1.00line 41 0.40000000000 0.50000000000 0.1000000000 1.00line 42 0.45000000000 0.50000000000 0.0500000000 1.00

5.4 LDA calculation of rutile TiO2 58

line 43 0.50000000000 0.50000000000 0.0000000000 1.00 #Mline 44 0.50000000000 0.50000000000 0.0500000000 1.00line 45 0.50000000000 0.50000000000 0.1000000000 1.00line 46 0.50000000000 0.50000000000 0.1500000000 1.00line 47 0.50000000000 0.50000000000 0.2000000000 1.00line 48 0.50000000000 0.50000000000 0.2500000000 1.00line 49 0.50000000000 0.50000000000 0.3000000000 1.00line 50 0.50000000000 0.50000000000 0.3500000000 1.00line 51 0.50000000000 0.50000000000 0.4000000000 1.00line 52 0.50000000000 0.50000000000 0.4500000000 1.00line 53 0.50000000000 0.50000000000 0.5000000000 1.00 #A

Note:1st k point, 2nd k point, ..., and 51st k point are listed from line 3 to line 53, and the

energy-band outputs of these k points are "band_0001.out", "band_0002.out", ..., "band_0051.out".Note:In line 2, numbers of k points (the first number) is meaningful within LDA band structure

calculation, but the second and third numbers( , 0, 0 ) is anyway required.Note:weight is meaningless in the LDA calculation and “out”-only k points, but should be

given.)

How to look at output data:band_0001.out energy eigenvalues (at sequential k points) in the 1st bandband_0002.out energy eigenvalues (at sequential k points) in the 2nd band

... ...band.out energy eigenvalues (at sequential k points) in all bands.

With the GWA.out and band.out files, we obtain GW and LDA calculation results. Wemake the figure of band structure, shown in Fig. 5.5

5.5 Wave function calculation of rutile TiO2 59

Fig. 5.5 Band structure of the pure rutile TiO2. (Lines are the LDA results and dots are theGW results. The zero of energy is placed at the top of the valence band ([VBM] at the Γpoint).[26]

5.5 Wave function calculation of rutile TiO2

After GW calculation, we can continue Wave Function calculation. It needs to modifyINPUT.inp file:

—————————————-iApp = LGNN → LWNNicontinue = 0 → 1—————————————-and we make a new KPOINT.inp:

Table 5.7 KPOINT.inp for Wave Function calculation

line 1 Directline 2 1 0 0line 3 0.00000 0.00000 0.00000 1.00 #Γ

Note:line 3: the k pint of wave function calculation.Note:line 3: weight is meaningless.

5.5 Wave function calculation of rutile TiO2 60

When wave function calculation is finished, the below output files are obtained:

WaveF_HOMO-1.cubeWaveF_HOMO-1.vaspWaveF_HOMO-1.grdWaveF_HOMO.cubeWaveF_HOMO.vaspWaveF_HOMO.grdWaveF_LUMO.cubeWaveF_LUMO.vaspWaveF_LUMO.grdWaveF_LUMO+1.cubeWaveF_LUMO+1.vaspWaveF_LUMO+1.grd

Note: "HOMO" is the highest occupied state, and "LUMO" is the lowest empty state.

Here, wave functions are calculated at Γ point (0, 0.0, 0.0) and R point (0, 0.5, 0.5).From the results of *.cube files, contour plots of the partial charge densities on (0 0 1) planeare made with same magnification using VESTA software, shown in Fig. 5.6.

(a) The highest occupied state at the Γ poin (b) The lowest empty state at the R point

Fig. 5.6 Contour plot of the partial charge density of the highest occupied state and thelowest empty state of the pure rutile TiO2 on (0 0 1) plane.[26]

References

[1] K. Ohno, S. Ono, R. Kuwahara, Y. Noguchi, R. Sahara, Y. Kawazoe, and M. H. F.Sluiter. TOMBO Group. http://www.ohno.ynu.ac.jp/tombo/index.html.

[2] S. Ono, Y. Noguchi, R. Sahara, Y. Kawazoe, and K. Ohno. Tombo: All-electronmixed-basis approach to condensed matter physics. Computer Phys. Comm., 189:20–30, 2015.

[3] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev., 136(3B):B864–B871, 1964.

[4] W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlationeffects. Phys. Rev., 140:A1133–A1138, 1965.

[5] K. Ohno, F. Mauri, and S. G. Louie. Magnetic susceptibility of semiconductors by anall-electron first-principles approach. Phys. Rev. B, 56(3):1009–1012, 1997.

[6] S.F. Boys and F. Bernardi. The calculation of small molecular interactions by the dif-ferences of separate total energies. some procedures with reduced errors. Mol. Phys.,19:553, 1970.

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Appendix A

Materials Studio

Materials Studio is software for simulating and modeling materials. It is developedand distributed by Accelrys. Now, I introduce how to use Materials Studio to build crystalmodel, 1st Brillouin zone, and make SPOINT.inp file.

Fig. A.1 Materials Studio

A.1 Build crystal model 64

A.1 Build crystal model

Step 1File → New... → 3D Atomistic

Step 2Build → Crystals → Build crystal...

Step 3Set "Space Group" and "Lattice Parameters".

Fig. A.2 Build Crystal

The default is space group P1, i.e. no symmetry. Rutile TiO2 has the space groupP42/MNM. By telling Materials Studio this symmetry it will automatically apply it to theatoms, thus generating atoms at the symmetry points.

Add the lattice constant — click on the “Lattice” tab near the top of the “Build Crystal”window. Set Lengths and Angles.Step 4

Build → Add Atoms

A.2 1st Brillouin zone 65

Fig. A.3 Add atoms

Add atoms at the origin, by changing the “element” from its default and clicking “Add”.By default the co-ordinates are in fractionals, but you can change this on the “Options” tab.

Note:For some common crystal model, it can be imported:File → Import... → Structures/Examples

A.2 1st Brillouin zone

Step 1Modules → CASTEP → Calculation;

Step 2Properties → Band structure → More...;

Step 3choose "Path...";

Step 4In "Brillouin Zone Path", click "Create", and choose "Display reciprocal lattice".

A.3 *.cell file 66

Fig. A.4 1st Brillouin zone

A.3 *.cell file

A.3.1 make *.cell file

Step 1Modules → CASTEP → Calculation;

Step 2In "CASTEP Calculation", choose " k-points", and click "More...";

A.3 *.cell file 67

Fig. A.5 CASTEP Calculation

Step 3In "CASTEP Electronic Options", choose " Electronic", click "Custom grid parameters",

and set a, b and c of k-points in "Grid parameters" using Monkhorst-Pack grid.

Fig. A.6 CASTEP Electronic Options

A.3 *.cell file 68

Step 4In "CASTEP Calculation", click "Files...", and click "Save Files". Then we can obtain

the input files of Materials Studio.Step 5

In "Project", right click, choose "Open Containing Folder".

Fig. A.7 Input file 1

Step 6Input files: (take ZnO for example) some files are hidden files.

Fig. A.8 Input file 2

A.3.2 information of *.cell

Take wurtzite ZnO for example, Open *.cell file, shown in Fig. A.9Note:

A.3 *.cell file 69

Fig. A.9 ZnO.cell file

From "%BLOCK LATTICE_CART" to "%ENDBLOCK LATTICE_CART", these arelattice paremeters in the Cartesian coordinates, can be used to make "CORRDINATE.inp"file;Note:

From "%BLOCK KPOINTS_LIST" to "%ENDBLOCK KPOINTS_LIST", these arek-points for LDA/GGA calculation, can be used to make "SPOINT.inp" file.