TOMBO Ver.2 Manual

TOMBO Ver.2 Manual
for LDA and GW Calculations
30 October, 2015
TOMBO Group

Preface
This 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 knowledge
for GW, Input files, Output files, and an example of GW calculation of rutile TiO 2 ,
respectively. In Appendix A, usage of the user interface “Materials Studio®” for crystal
calculations is described. TOMBO is a first-principles program using the all-electron mixed
basis approach, in which one-particle (Kohn–Sham or quasiparticle) wave functions are expressed
in a linear combination of both atomic orbitals and plane waves. This approach
is described in detail in the following reference. Anyone who publishes any result using
TOMBO 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 unified
way. 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 web
site 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 2015
TOMBO Group

Table of contents
1 Introduction 1
1.1 Short introduction of TOMBO . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mixed basis formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 All-electron charge density and potential . . . . . . . . . . . . . . . . . . . 6
1.4 Calculation flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 TDDFT dynamics simulation . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 The GW Approximation 13
2.1 The quasiparticle band gap . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 The self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Hedin’s equations (GWΓ method) . . . . . . . . . . . . . . . . . . . . . . 16
2.5 The GW approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 One-shot GW approximation . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Input Files 21
3.1 COORDINATES.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 rct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.3 nv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.4 ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.5 np . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 INPUT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 iApp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 iAlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.3 iexc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.4 iCHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.5 Ulevel, Llevel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Table of contents
iv
3.2.6 ippmG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.7 jmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.8 deltaE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.9 npp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.10 nod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.11 nog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.12 ncut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.13 noz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.14 ncs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.15 ncc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.16 nol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.17 nband . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.18 q_eliminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.19 q_min_mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.20 icontinue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.21 isphcut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.22 iTotalE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.23 lpri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.24 nspin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.25 irelative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.26 nion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.27 iExcite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.28 nonadiabatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.29 iMIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.30 smixSCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.31 nSDCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.32 iSblp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.33 iSblp2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.34 mSblp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.35 smixTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.36 iasym . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.37 icalAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.38 iCry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.39 precLDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.40 iFTapp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.41 iCheb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table of contents
v
3.2.42 Mcheb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.43 nStep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.44 singlet, triplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 KPOINT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 QPOINT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 SPOINT.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Output Files 44
4.1 OUTPUT.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Status.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 ISYS.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 posit.dat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 eigen.dat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.6 MD_Coordinates.xyz, MD_Coordinates.arc . . . . . . . . . . . . . . . . . 45
4.7 band.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.8 WaveF_HOMO-1.cube, WaveF_HOMO-1.grd,
WaveF_HOMO-1.vasp, and so on . . . . . . . . . . . . . . . . . . . . . . 46
4.9 GWA.out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.10 PhotoAbsorptionSpectra.out . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Examples 50
5.1 LDA calculation of isolated dimers . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 GW calculation of rutile TiO 2 . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 LDA calculation of rutile TiO 2 . . . . . . . . . . . . . . . . . . . . . . . . 56
5.5 Wave function calculation of rutile TiO 2 . . . . . . . . . . . . . . . . . . . 59
References 61
Appendix A Materials Studio 63
A.1 Build crystal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
A.2 1 st Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.3 *.cell file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.3.1 make *.cell file . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.3.2 information of *.cell . . . . . . . . . . . . . . . . . . . . . . . . . 68

Chapter 1
Introduction
TOMBO[1, 2], all-electron mixed-basis code, is short for "TOhoku Mixed Basis Orbitals
ab initio program". In this thesis, electronic structures of transition metal oxides are
calculated by our original all-electron mixed basis code TOMBO developed by Prof. Kaoru
Ohno, et al..
1.1 Short introduction of TOMBO
Density funcitonal theroy (DFT) [3] and local density approximation (LDA) [4] have
been used in numerous electronic structure calculations and firt-principles molecular dynamics
(MD) simulations. To perform those calculations, Kohn–Sham (KS) equation needs
to be solved self-consistently such that the input potential is identical to the output potential
[4]. The electronic wave function has to be described by appropriate functions to solve
the KS equation. Among the plane wave (PW) expansion approach has been applied to the
ab-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 potentials
is related to the fact that the core contribution to the exchangecorrelation potential is not
simply 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 treat
all electrons. However, these methods have an intrinsic problem of incomplete basis set, and
therefore 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 a
negative 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 trouble
in 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 the
PW expansion technique with the LCAO technique to remove pseudopotentials in the PW
expansion methods and to make the basis set complete in the LCAO methods. This is the
main idea to introduce the all-electron mixed basis approach. TOMBO [1, 2] is the program
package using this approach. TOMBO is the all-electron first-principles method, which can
be applicable to both isolated and periodic systems with complete basis set. It is not the
overcomplete 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 the
fact that it enables us to perform the state-of-the-art calculations such as GW approximation
and Bether–Salpeter equation (BSE). Using these methods, TOMBO can treat the problems
related 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 several
versions: 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 made
of 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 function
is expressed as a product of radial function R jnl (r) and spherical harmonics Y lm (r) as
ϕ AO
jnlm (r) = R jnl(r)Y lm (ˆr) (1.1)
Here, j, n, l, and m are atomic species, principal quantum number, angular momentum
quantum number, and magnetic quantum number.
In the mixed basis code, the KS wave function is expressed as a linear combination of
PWs and AOs (Fig. 1.1),
ψ v (r) = √ 1
Ω
∑
c PW
G
v (G)e i(k+G)·r +∑
j
∑
nlm
c AO
v
( jnlm)ϕ Bloch (k,r) (1.2)
jnlm
with
ϕ Bloch
jnlm (k,r) = eik·R j
∑
R
e ik·R ϕ AO
jnlm (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 expansion
coefficients. 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 polynomial
function that satisfies the matching condition at the surface of atomic sphere, while the rest
smooth function (the polynomial function inside and the tail outside atomic sphere) can be
expressed by PWs; see Fig. 1.3.
Fig. 1.1 In the all-electron mixed basis approach, one-electron wave functions are expressed
in 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 a
smooth polynomial function.
Usually, AOs are generated at the outset by the atomic LDA code base on a modified
Herman–Skillman’s algorithm and not updated during the calculation. However, if one
wants to update AOs during the self-consistent field (SCF) loop, one can set icalAO = 1 in
INPUT.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 (such
as the expansion of the wave packet in terms of the eigenstates of the Hamiltonian in the
TDDFT dynamics simulations) requires summing over large number of empty states. The
PW basis set can most accurately describe the empty states. In contrast, to describe the
electrons in the core region accurately, the AO basis set works better then the PW basis
set. The all-electron mixed basis approach, using both PWs and AOs as a basis set in a
combined way, is able to meet the requirements to describe both spatially extended and
localized states. In our code, AOs are numerically described inside the non-overlapping
atomic 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 is
given simply by the Fourier components V (G − G ′ ).
3. It is possible to accurately treat core states because we determine AOs by using
Herman–Skillman code with logarithmic radial mesh.
4. There is no complexity to generate and treat pseudoptentials. There is also no problem
of transferability.
5. The overlap between AOs and PWs is calculated accurately by first performing angular
integral analytically and then performing radial integral of spherical Bessel functions
numerically in logarithmic radial mesh.
6. Because AOs are confined inside non-overlapping atomic spheres, there is no BSSE
problem, and it is not necessary to calculate overlap integrals between AOs centered
at different atoms, which might produce unnecessary computational errors. Simultaneously,
this reduces the overcompleteness problem.
Since the basis functions (PWs and AOs), f ξ (r), are not orthogonal each other, the
eigenvalue ε ν is obtained by solving the following generalized eigenvalue problem:
∑H ξ ξ ′c ν,ξ ′ = ε ν ∑S ξ ξ ′c ν,ξ ′, (1.4)
ξ ′ ξ ′
where H ξ ξ ′ = ⟨ f ξ |H| f ξ ′⟩ and S ξ ξ ′ = ⟨ f ξ | f ξ ′⟩ are, respectively, the Hamiltonian and overlap
matrix elements between the ξ ’th and ξ ′ ’th basis functions. They looks like
H =
(
⟨PW|H|PW⟩
⟨AO|H|PW⟩
⟨PW|H|AO⟩
⟨AO|H|AO⟩
)
, S =
(
⟨PW|PW⟩
⟨AO|PW⟩
⟨PW|AO⟩
⟨AO|AO⟩
)
. (1.5)

1.2 Mixed basis formulation 5
There are N w PWs (the number of plane waves "nw" in the code), N ao AOs (the number of
atomic orbitals "nao" in the code), and altogether N bs basis functions (the number of basis
set "nbs" in the code). The corresponding column vector is written as
⎛
Ψ ν ≡
⎜
⎝
Eq.(1.4) is then rewritten as follows:
c ν,1
c ν,2
.
c ν,Nw
c ν,Nw +1
.
C ν,Nbs
⎞
⎟
⎠
⎫
1st G
2nd G
⎪⎬
“nw”
.
last G
⎪⎭
(1.6)
⎫
1st AO ⎪⎬
.
⎪⎭ “nao” last AO
HΨ ν = ε ν SΨ ν . (1.7)
This generalized eigenvalue problem is transformed to the usual eigenvalue problem by
using Choleski decomposition. The overlap matrix S is expressed as a product of a lower
triangular matrix T and its Hermitian conjugate T † :
S = T T † . (1.8)
Then, Eq. (1.7) becomes
H ′ Φ ν = ε ν Φ ν , (1.9)
with the transformed Hamiltonian
H ′ = T −1 HT †−1 , (1.10)
(H ′ is calculated from H and T −1 ), and the transformed eigenvectors are give by
Φ ν = T † Ψ ν . (1.11)
The resulting ordinary eigenvalue problem can be solved by the standard library program
using 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
δ
2m GG ′ +Ṽ (G − G ′ ). (1.12)
PW-AO matrix elements can be accurately calculated by 1D integration of the product of
the spherical Bessel function j l ′(Gr) and the radial AO function R AO
jnl (r):
⟨k + G|H|ϕ Bloch e−iG·R ∫
jnlm ⟩ = j
√ Y lm ( k + ˆ G)
Ω cell
[ ¯h 2 (k + G) 2 ]
j l (|k + G|r)
+V (r)
2m
× R jnl (r)r 2 dr. (1.13)
AO-AO matrix elements can be accurately calculated by 1D integration of the product of
two radial AO functions:
∫ [
⟨ϕ Bloch Bloch
jnlm |H|ϕ j ′ n ′ l ′ m ′⟩ = δ j j ′ ϕ AO
jnlm (r) − ¯h2
cell 2m ∇2 +V (r)
]
ϕ A0
jn ′ l ′ m ′(r)d3 r. (1.14)
The overlap matrix elements are given by those of (1.13) and (1.14) without [...] inside the
integrands.
1.3 All-electron charge density and potential
In the all-electron mixed basis approach, all-electron charge density ρ(r) is made of
three contributions: PW-PW, AO-PW,and AO-AO.
ρ(r) = ρ PW−PW (r) +∑ρ AO−PW
j
j (r) +∑
j
ρ AO−AO
j (r). (1.15)
In the all-electron mixed basis approach, the charge density is made of the three contributions,
ρ AO−PW
j (r) = √ 2 occ
Ω
occ
j (r) = 2
ρ AO−AO
∑
v
ρ PW−PW (r) = 2 Ω
∑ ∑
v
nlm
∑ ∑
n ′ l ′ m ′ nlm
∑
G
occ
∑
v
∑∑
G
c PW∗
v
G ′
Here the prefactor 2 denotes the spin duplicity, ∑ occ
ν
(G ′ )c PW
v (G)e i(G−G′ )·r , (1.16)
c AO∗
v ( jnlm)c PW
v (G) × ϕ jnlm (r − R j )e i(G)·r + c.c., (1.17)
c AO∗
v ( jn ′ l ′ m ′ )c AO
v ( jnlm) × ϕ jn ′ l ′ m ′(r − R j)ϕ jnlm (r − R j ). (1.18)
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 complex
conjugate of the previous term. The first PW-PW contribution can be conveniently treated
in Fourier space. The rest two AO-related contributions are confined only inside the nonoverlapping
atomic spheres, and can be written together as
ρ AO
j
(r) = ρ AO−PW
j
(r)ρ AO−AO
j (r). (1.19)
This can be divided into two parts: one is spherical symmetric part σ j (r j ) and the other is
asymmetric 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 in
INPUT.inp. The treatment of ignoring asymmetric part of the AO-related charge density
is guaranteed when the radii of non-overlapping spheres are set reasonably small enclosing
the core region only.
Spherical potential V j (r j ) inside each non-overlapping atomic sphere is calculated as
follows. First, consider the Hartree potential made by the AO-related spherically symmetric
charge σ j (r j ) centered at R j . This potential is easily calculated as the 1D integration in
radial direction of the Poisson equation as follows:
v H j (r j ) = 4π
r j
∫ r j
0
∫ rc
σ j (r)r 2 dr + 4π σ j (r)rdr. (1.20)
r j
According to the Gauss theorem in electrostatics, this Hartree potential behaves as
v H j (r j ) = Q j
r j
, for r ≥ r c , (1.21)
where Q j is the symmetric charge defined as
∫ rc
Q j = 4π σ j (r)r 2 dr. (1.22)
0
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 v H j (r j) and nuclear Coulomb potential
−Z j /r j around R j becomes
v H j (r j ) − Z j
r j
= − Z∗ j
r j
, for r j ≥ r c (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 only
inside the own unit cell, but also surrounding or further apart unit cells. To treat this accurately,
we use the following Fourier decoupling method. In this method, the simple potential
form −Z ∗ j /r j given by Eq.(1.24) connects smoothly to the quadratic function inside the nonoverlapping
atomic sphere. They should have the same value and the same derivative at the
radius of the atomic sphere r = r c :
⎧
⎨
v interpo Z ∗ j
j
(r j ) =
(br2 j + d) for r j < r c ,
⎩−Z ∗ j /r j for : r j ≥ r c .
From the matching condition, we obtain
(1.25)
br 2 c + d = −1/r c , 2br c = 1/r 2 c. (1.26)
From simple calculation, we find that these conditions are identical to
b = 1
2rc
3 , d = − 3 . (1.27)
2r c
Thus connected potential is a smooth and analytic function over whole space and is easily
transformed into reciprocal lattice space analytically. We call this potential the interpolated
Coulomb potential and write it as v interpo
j
(r j ); see Eq.(1.25). This interpolated Coulomb
potential takes the correct value of the Coulomb potential, v H j (r j) − Z j
r j
, everywhere outside
the jth atomic sphere; It takes an incorrect value only inside the jth atomic sphere, and its
difference from the correct value is given by
Vj
trunc (r j ) = v H j (r j ) − Z j
− v interpo
j
(r j ), (1.28)
r j
which we call the truncated Coulomb potential. It has nonzero values only each atomic
sphere, and is spherically symmetric. This truncation is schematically illustrated in Fig.1.4.
This truncated Coulomb potential is added to the truncated spherical exchange-correlation
potential µ xc
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 v interpo
j
(r j ). This function v interpo
j
(r j ) is analytically expressed by Eq.(1.25) in infinite
space, and therefore is able to be analytically transformed into G space. The Fourier
coefficients are explicitly given by
{
¯v interpo
j
(G) = 4πZ je −iG·R j
b [( 3(Gr c ) 2 − 6 ) ]
sinGr c + 6Gr c cosGr c /G
5
Ω

1.3 All-electron charge density and potential 9
+d sinGr c /G 3 + cosGr c /G 2 }. (1.29)
This analytic Fourier coefficients are added to the rest part of the Fourier coefficients of
both the Hartree potential ¯ρ rest (G) and the exchange-correlation potential ¯µ xcrest (G) to give
an additional contribution to the spherical potential in the jth atomic sphere. That is, this
additional contribution to the spherical potential from the three kinds of Fourier coefficients
are expressed as:
Vj
rest (r j ) = ∑
G
sinGr j
e iG·R j
Gr j
[
∑
k
¯V interpo
k
(G) + 4π Ω
¯ρ rest (G)
G 2
+ ¯µ xcrest (G)
]
. (1.30)
Here we note that v interpo
j
(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 centered
at 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
V j (r j ) = Vj
trunc (r j ) +Vj rest (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 inside
atomic sphere is used to accelerate the computational speed. The contribution from the
asymmetric 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 V H (r) inside non-overlapping atomic
spheres?

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 C λ i
✲
❄
Calculate charge density
❄
Mix the charge density with
that of the previous step
Calculate the KS potential
❄
Calculate matrix ❄ elements
❄
Update Ci
λ according to
matrix diagonalization or
steepest-descent method
❄
NO ✘✘✘✘✘✘✘
✛ ✘❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳
✘ If |E new − E old | < precLDA
✘✘✘✘✘✘✘✘✘
YES
❄
Calculate forces on atoms
❄
Update atomic positions
Check self-consistency
Fig. 1.5 Flowchart of the first-principles MD. The convergence of the electronic states is
checked by the difference between the total energies at the present and previous steps. If
convergence 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, 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.
According to the time-dependent density functional theory[13], TOMBO can treat the
time-dependent Kohn–Sham (TDKS) equation
i ∂ ∂t ψ j(r,t) = H q (r,t)ψ j (r,t), (1.32)
by setting “A” in iApp in INPUT.inp Here H q (r,t) is the electronic part of the Hamiltonian.
Combining this time-evolution equation with the Newtonian equation of motion for nuclei
M A ¨R A = − ∂
∂R A
[
E q +V cl] , (1.33)
we can perform semi-classical dynamics simulation within the mean-field-type Ehrenfest
theorem. During one simulation, all dynamical variables change in time along one reaction
path 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). This
way of simulation is called “on the fly” approach. Here E q is the electronic part of the total
energy, and, together with the Coulomb potential between nuclei V cl , gives the mean-field
potential exerting on nuclei. In these equations, r is the electron coordinate, M A and R A are
the 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 assumed
slow enough. However, non-adiabatic simulation is possible by considering the coupling to
the atomic velocity, when the input parameter nonadiabatic = 1 is set in INPUT.inp. Below
only the algorithm of adiabatic simulation is explained, but one may refer to Ref. [14] for
non-adiabatic simulation.
It is necessary to know the exchange-correlation potential to solve Eq.(1.32) step by step
by means of the TDDFT. TOMBO uses a simple LDA exchange-correlation functional that
is local for both space and time. This approximation is called “adiabatic LDA”. In order to
integrate Eq.(1.32) step by step accurately, we use the spectral method[15], in which wave
packet ψ j (r,t) at each instantaneous time is expanded in terms of eigenfunctions ϕ k (r,t)
(eigenvalues are ε k (t))
H q (r,t)ϕ k (r,t) = ε k (t)ϕ k (r,t). (1.34)

1.5 TDDFT dynamics simulation 12
of the Hamiltonian at that time H q (r,t). Wave packes are expanded as
ψ j (r,t) = ∑c jk (t)ϕ k (r,t), (1.35)
k
where the coefficients c jk (t) are given by the inner product of ϕ k (r,t) and ψ j (r,t) as
c jk (t) = ⟨ϕ k (r,t)|ψ j (r,t)⟩. (1.36)
Therefore, if we set the time step ∆t smaller than the time scale where the Hamiltonian
changes, the TDKS equation can be integrated as follows:
ψ j (r,t + ∆t) = ∑exp[−iε k (t)∆t]c jk (t)ϕ k (r,t). (1.37)
k
When the Hamiltonian does not change in time, Eq.(1.37) is exact. Of course, electron wave
packets 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 H q (r,t) almost does
not change, Eq.(1.37) becomes good approximation.
This spectral method requires that the basis functions span complete space at least approximately,
and in this sense, the all-electron mixed basis method is suitable. In fact, the
summation 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. The
number 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 force
between nuclei and the derivative of the electron total energy with respect to the nuclear
positions. These forces are calculated in the same way described in previous section. If
the electronic states is on an adiabatic surface, the dynamics using this mean-field potential
becomes the adiabatic molecular dynamics. If the wave packet ψ j (r,t) is the superposition
of the eigenstates, the forces acting on nuclei becomes the average of the forces calculated
from the different eigenstates with the weight of these eigenstates in the wave packet (mixed
state). This is the problem of this mean-field approximation. This mean-filed force becomes
unphysical 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 loop
until 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 nonlocaldensity
corrections to the LDA, in any case, quasiparticle energies are needed [16]. In order
to improve the accuracy of electronic structure calculation of materials, the GW approximation
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 E N , respectively. The electron quasiparticle (QP) energies are defined by
ε ck = E N+1 − E0 N (unoccupied) (2.1)
and the hole QP energies are defined by
ε vk = E0 N − EN−1 (occupied) (2.2)
These are excitation energies of the (N ± 1)-particle system relative to the N-particle
ground-state energy E0 N and thus correspond to the electron addition and removal energies.
It is clear that ε ck > ε F and ε vk ⩽ ε F , where ε F is the Fermi level. The quasiparticle energy
gap is given by
E gap = ε {CBM} − ε [V BM] = E N+1 + E N−1 − 2E0 N (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 valence
band 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 particle
Green function. The one particle Green function is defined as:
∣ [
]∣ ⟩
∣∣T
G(x 1 ,t 1 ,x 2 ,t 2 ) = i
⟨Ψ N ψ(x 1 ,t 1 )ψ † ∣∣ΨN
(x 2 ,t 2 )
(2.4)
It contains the information on energy and lifetimes of the quasiparticle, and also on
the ground state energy of the system and the momentum distribution. Here, x = (x,t) =
(r,σ,t). Ψ N is the Heisenberg ground state vector of the interacting N-electron system satisfying
the engenvalue equation H|Ψ N ⟩ = E|Ψ N ⟩,ψ H and ψ † H
are respectively the annihilation
and creation field operator and T is the Wick time ordering operator.
ψ(x,t) = e iHt ψ(x)e −iHt (2.5)
and the field operator satisfy the anti-commutation relation:
ψ † (x,t) = e −iHt ψ † (x)e iHt (2.6)
{
}
ψ(x),ψ † (x ′ ) = δ(x − x ′ ) (2.7)
and:
{
ψ(x),ψ(x ′ ) } {
}
= ψ † (x),ψ † (x ′ ) = 0 (2.8)
T [ψ(x 1 ,t 1 )ψ † (x 2 ,t 2 )] =
{
ψ(x 1 ,t 1 )ψ † (x 2 ,t 2 ) if t 1 > t 2
ψ(x 2 ,t 2 )ψ † (x 1 ,t 1 ) if t 1 < t 2
(2.9)
Therefore, the Green function describes the probability amplitude for the propagation
of an electron (hole) from position r 2 at t 2 to r 1 at time t 1 for t 1 > t 2 (t 1 < t 2 ) (See Fig.2.1).
Insert a complete set of N+1 and N-1 particle states, ∑ j |ψ N±1 | = 1, we perform a
Fourier transform in energy space we obtain:
j
⟩⟨ψ N±1
j
G(x 1 ,x 2 ;ω) = ∑
s
f s (x 1 fs ∗ (x 2 )
ω − ε s + iηsgn(ε s − ε F )
(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 ε F is the Fermi level, and the QP energies are given by
ε s =
{
E s
(N+1) − E (N)
N
− E (N−1)
E (N−1)
N
for ε s ≥ ε F
(2.11)
s for ε s < ε F
The subscript s indicate the quantum label of the states of the N ±1 system. The amplitudes,
i.e., the QP wave functions, f s (x) are defined as:
f s (x) =
{
⟨Ψ N |ψ(x)|Ψ N+1,s ⟩ for ε s ≥ ε F
⟨Ψ N−1,s |ψ(x)|Ψ N ⟩ for ε s < ε F
(2.12)
The Green function has the poles at the electron addition (removal) energies and describes
quasiparticles. So we can obtain the quasiparticle band gap information from Green
function.
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 ∂
∫
G(1,2) = δ(1,2) + H 0 (1)G(1,2) − i
∂t 1
ν(1 + ,3)G(1,3,2,3 + )d3 (2.13)

2.4 Hedin’s equations (GWΓ method) 16
Here, H 0 = T +V ext +V H , the number n stands for (r n ,t n ), ν(1 + ,3) is the Coulomb interaction
between electrons. There exists 2-particle Green function, describes the motion of 2
particles:
⟨
[
]∣ ⟩
G(1,2,3,4) = − Ψ N ∣
0 ∣T ψ(1)ψ(2)ψ † (4)ψ † ∣∣Ψ N
(3) 0
In frequency Fourier space:
(2.14)
∫
[ω − H 0 ]G(ω) + i
νG 2 (ω) = 1 (2.15)
Instead of introducing a 2-particles Green function, we introduce the self-energy operator,
Σ . The self-energy allows to close formally the hierarchy of equations of motion of
higher order Green functions. Equation (2.15) is tranformed to 1-particle Green function.
∫
[ω − H 0 ]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 − H 0 (r)]G(r,r ′ ;ω) −
Σ(r,r ′′ ;ω)G(r ′′ ,r ′ ;ω)d 3 r ′′ = δ(r − r ′ ) (2.17)
Introducing the Lehmann representation for G. The QP energies and QP wave functions
can be obtained as solutions of a Schrödinger-type QP equation [18]
∫
H o ψ nk (r) +
dr ′ Σ(r,r ′ ;ε nk )ψ nk (r ′ ) = ε nk ψ nk (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 coupled
equations,known as Hedin’s equations [19]:
Self-energy:
Screened potential:
∫
Σ(1,2) = i
d(34)G(1,3)Γ(3,2,4)W(4,1 + ) (2.19)
∫
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) +
Dielectric function:
d(4567) ∂Σ(1,2) G(4,6)G(7,5)Γ(6,7,3) (2.22)
∂G(4,5)
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 GWΓ
method is implemented in a future version of TOMBO, the following GW approximation is
needed 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-order
form, 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) = G 0 (1,2) +
d(3)d(4)G 0 (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 character
of 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
Σ GW (r,r ′ ;ε) = i ∫
2π
dωe −i0+ω G(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 ( Σ GW = Σx + Σ c ). The exchange part
is given by
Σ x (r,r ′ ) = i ∫
2π
dωe iω0+ G(r,r ′ ;ω)ν(r − r ′ ) = −
occ
∑
nk
ψ nk (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 the
occupied states only. The diagonal matrix elements of this exchange part of the self-energy
become
⟨
n,k
∣ ∣Σx (r,r ′ ) ∣ ∣ n,k
⟩
= −
occ
∑
∑
n ′ q
∑
G
4π
⟨
ΩG 2
∣
n,k∣e i(q+G)·r∣ ⟩⟨
∣ ∣n ′ ,k − q n ′ ∣
k − q∣e −i(q+G)·r′∣ ⟩
∣ ∣n,k
(2.31)
The correlation part, Σ c can be evaluated by using generalized plasmon-pole (GPP)
model [19]. Σ c can also be evaluated by using the full ω integration [20]:
Σ c (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 and
G 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 resulting
Taylor series is used to estimate the matrix elements for real values of ω. Ishii and Ohno et
al. [20, 25] suggested that this intergration method employed by Godby et al. [16] can be
extended rather easily to real number of ω by slightly modifying the contour. The contour
along the real ω ′ axis from −∞ to +∞ for the integral Equation (2.32) can be replaced by
the contour C shown in Fig.2.2.
Here we further use the symmetry W(ω) = W(−ω) to reduce the contour to the positive
real and imaginary parts only. The diagonal matrix elements of the correlation part of the
self-energy become
⟨
n,k
∣ ∣Σc (r,r ′ ;ω) ∣ ∣ n,k
⟩
= ∑
∑
n ′ q
× i
2π
⟨
∑
G,G ′
∫
∣
n,k∣e i(q+G)·r∣ ⟩⟨
∣ ∣n ′ ,k − q n ′ ∣
k − q∣e −i(q+G′ )·r ′∣ ⟩
∣ ∣n,k
dω ′ [ W G,G ′(q,ω ′ ) − (4π/ΩG 2 ]
)δ G,G ′
1
× (
ω + ω ′ +
− ε k−q,n − iδ k−q,n
1
ω − ω ′ − ε k−q,n − iδ k−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 imaginary)
parts of the contour C shown in Fig.2.2 with the help of W(ω) =W(−ω). If we use the
original contour C without using W(ω) = W(−ω), the second term in the parentheses of the
integrand in Equation (12) does not appear. This contour is justified because there is no pole
inside two quadrants of the complex ω ′ plane, corresponding to (Reω ′ > max(ε VBM −ω,0),
Imω ′ > 0) and (Reω ′ < min(ε CBM − 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 the
contribution related to the electron correlation.
Instead of solving equation (2.18) self-consistently, we adopt here the so-called one shot
GW approximation first proposed by Hybertsen and Louie [19]. The Green’s function (G)
is replaced by the LDA Green’s function G 0 and constructed in a non-self-consistent way
from the KS wave functions ψnk KS (r) and eigenvalues εKS
nk
(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 as
input for a GW calculation and the quasiparticle spectrum is evaluated non-selfconsistenly
from Eq.(2.33) without updating the Green’s Function or the screened potential, that means
only one iteration is made. This is known as the “one-shot” GW or G 0 W 0 approximation
[16, 19] and has become a standard tool in electronic structure theory. W 0 is hereby equal to
the RPA screened potential.
Using the one-shot GW approximation, the quasiparticle equation (Eq. 2.18) is solved
approximately. The quasiparticle spectrum is calculated with Eq. (2.18) using the first-

2.6 One-shot GW approximation 20
order perturbation theory in ( Σ GW −Vxc
LDA ), where Vxc
LDA is the LDA exchange-correlation
potential. Usually, the KS wave functions ψnk KS are sufficiently close to the true quasiparticle
wave function ψ nk (r), so that the first-order estimate of the self-energy correction to the
LDA eigenvalues in adequate [19]. The GW quasiparticle energy εnk
GW is then obtained as
⟨
⟩
= εLDA nk + Z nk ψnk
LDA ∣
∣Σ GW (r,r ′ ;εnk
LDA LDA
) −Vxc (r)δ(r − r ′ ∣
) ∣ψnk
LDA (2.34)
ε GW
nk
with a renormalization factor as
Z nk =
[
1 − ∂Σ ]
GW (ω) −1
∂ω ω=εnk
LDA
(2.35)
The Scheme of the one-shot GW is shown in Fig.(2.3). Firstly, Kohn-Shan equation
is solved by DFT and LDA, it obtain Kohn-Sham eigenvalues and wave fuctions. Then
using one-shot GW approximation, we can get polarization, dielectric function, screened
potential, et. al.. From Hedin’s equations, we obtain the self-energy. Finally, we calculate
the 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 clusters),
TOMBO needs two input files, COORDINATES.inp and INPUT.inp. These are the
central input files of TOMBO. The variables related to the unit cell, atomic positions, and
also AOs are written in COORDINATES.inp: the lattice constant, lattice vectors, coordinates
of the nuclei and the number of orbital-types of AOs for each atom. On the other
hand, INPUT.inp determines "what to do and how to do it", and can contain a relatively
large number of parameters. In a crystal calculation within a LDA level, two other input
files, SPOINT.inp and KPOINT.inp, are required to assign special k points to do specialpoint
sampling within the irreducible wedge of the 1st Brillouin zone (BZ) during the selfconsistent
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. For
a GW crystal calculation, the assignment of KPOINT.inp is different from a LDA calculation,
and another input file QPOINT.inp is also required In order to calculate the polarization
function P GG ′ for each momentum transfer q (and for each energy transfer ω in the case of
the full ω integration), k-point sampling is performed for the points on the Gamma grid
in the whole BZ assigned as “sum” in KPOINT.inp. Then the correlation part of the selfenergy,
Σ c is calculated by taking q-point summation on the Γ-grid in the irreducible BZ by
using QPOINT.inp. Finally, the quasiparticle energies as well as the expectation values of
the LDA exchange-correlation potential, and the exchange (Σ x ) and correlation (Σ c ) parts of
the self-energy are evaluated for the k points (typically at symmetry points in the irreducible
BZ) 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 in
COORDINATES.inp: the lattice constant, lattice vectors, coordinates of the nuclei and the
number of orbit-type AOs for each atom. It can also contain the definitions of “mesh” of the
unit 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
line 2
name of the target system (arbitrary)
length of lattice vector a 1 in [Å]
line 3 a 1x a 1y a 1z
line 4 a 2x a 2y a 2z
line 5 a 3x a 3y a 3z
line 6 number of atoms
line 7
line 8
line 9
Direct (or Cartesian)
1st atomic species and coordinates
2nd atomic species and coordinates
line 10 ... ...
line 11
line 12
line 13
line 14
line 15
line 16
line 17
nth atomic species and coordinates
XX_rct (radius of atomic sphere of XX atom)
XX_ nc (number of kinds of core states)
XX_nv (number of kinds of valence states)
XX_ns (number of s-type Atomic Orbits for XX atom)
XX_np (number of p-type Atomic Orbits for XX atom)
XX_nd (number of d-type Atomic Orbits for XX atom))
line 18 ... ...
line 19
line 20
mesh (mesh division of unit cell (same for all directions))
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 a 1 in [Å].

3.1 COORDINATES.inp 23
3. a 1x , a 1y , a 1z are given in line 3, where a 1 should be normalized to 1, i.e., |a 1 | = 1 or
a 2 1x + a 2 1y + a 2 1z = 1. (3.1)
4. a 2x , a 2y , a 2z and a 3x , a 3y , a 3z are given in line 4 and line 5, respectively, in the same
unit of a 1x , a 1y , a 1z .
5. Line 7:"Direct" gives fractional coordinates of atomic positions (coefficients for a 1 ,
a 2 , a 3 ) "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 around
the center of the unit cell for symmetry. (Note that any coefficients must be between
0 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 be
written 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 the
last time step and at the time step one before the last step in MD_Coordinates.xyz to
COORDINATES.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 this
case, 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 units
of [angstrome/fs], [m/s], and [a.u.], respectively.
9. In order to treat an antiferromagnetic system having zero total magnetic moment, one
should write nspin = -1 in INOUT.inp and simultaneously, one may write the spin
magnetic 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-, and
d-orbitals are implemented, while f -orbital is still under implementation. Nevertheless,
in principle, TOMBO code can handle all atoms by increasing the cutoff energy
for Plane Waves.
11. Lines 12 to 18 are related to the radius of non-overlapping atomic sphere and the
choice of AOs for each atom.
12. In line 19, mesh of the unit cell division is defined if necessary. Otherwise default
value is used for mesh. For accurate calculation, large mesh such as 128 or 192 is
recommended as well as noz, which should be defined in INPUT.inp.
An example of COORDINATES.inp for rutile TiO 2 crystal [26] is shown in Table 3.2.
Table 3.2 COORDINATES.inp of rutile TiO 2
line 1 system=Rutile_TiO2
line 2 4.59
line 3 1.000000000 0.000000000 0.000000000
line 4 0.000000000 1.000000000 0.000000000
line 5 0.000000000 0.000000000 0.643883326
line 6 6
line 7 Direct
line 8 O 0.553188499 0.553188499 0.250000000
line 9 O 0.946811501 0.946811501 0.250000000
line 10 O 0.446811501 0.053188499 0.750000000
line 11 O 0.053188499 0.446811501 0.750000000
line 12 Ti 0.250000000 0.250000000 0.250000000
line 13 Ti 0.750000000 0.750000000 0.750000000
line 14 Ti_rct=0.8
line 15 Ti_nc=3
line 16 Ti_nv=4
line 17 Ti_ns=3
line 18 Ti_np=2
line 19 mesh=64
line 19 END_PARA

3.1 COORDINATES.inp 25
Fig. 3.1 Atomic orbitals (AOs) available in TOMBO code: s-, p-, and d-orbitals are implemented

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 units
of [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 of
adjacent atoms should not exceed the interatomic distance. The value of “rct” may strongly
affect the one-shot GW calculation results. In such a case, “rct” should be chosen as small
as 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 should
write N _nc = 1 [default] to define “nc” of N atom (1s only), for example. For titanium, the
core orbitals of Ti are 1s 2 2s 2 2p 6 , 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 atomic
sphere) is performed. One should write N _nv = 2 [default] to define “nv” of N atom (2s
and 2p), for example. For titanium, the valene orbitals of Ti are 3s 2 3p 6 3d 2 4s 2 , so that there
are 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, 1s 2 2s 2 2p 6 3s 2 3p 6 AOs are
used 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 are
1s 2 2s 2 2p 6 3s 2 3p 6 , 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
line 2
line 3
line 4
line 5
line 6
line 7
line 8
line 9
line 10
line 11
line 12
line 13
line 14
line 15
line 16
line 17
line 18
line 19
line 20
line 21
line 22
line 23
line 24
line 25
line 26
line 27
line 28
line 29
line 30
line 31
line 32
line 33
line 34
name of the target system (arbitrary)
iApp
iAlg
ippmG
jmax
deltaE
npp
nod
nog
ncut
noz
ncs
nol
nband
q_eliminate
icontinue
isphcut
iTotalE
lpri
nspin
iMIX
smixSCF
nSDCG
iSblp
iSblp2
smixTD
iasym
icalAO
iCry
precLDA
iFTapp
iCheb
nstep
END_PARA

3.2 INPUT.inp 28
An example of INPUT.inp for one-shot GW calc. of rutile TiO 2 [26] is shown in Table 3.4.
Table 3.4 INPUT.inp of rutile TiO 2
line 1 system=Rutile_TiO2
line 2 iApp =LGNN
line 3 iAlg =D
line 4 ippmG =4
line 5 jmax =200
line 6 deltaE =2.0d0
line 7 npp =100
line 8 nod =8
line 9 nog =18
line 10 ncut =5
line 11 noz =18
line 12 ncs =22
line 13 nol =400
line 14 nband =10
line 15 q_eliminate =0.005d0
line 16 icontinue =0
line 17 isphcut =0
line 18 iTotalE =0
line 19 lpri =1
line 20 nspin =0
line 21 iMIX =0
line 22 smixSCF =0.80d0
line 23 nSDCG =8
line 24 iSblp =80
line 25 iSblp2 =15
line 26 smixTD =0.80d0
line 27 iasym =0
line 28 icalAO =0
line 29 iCry =1
line 30 precLDA =0.0001d0
line 31 iFTapp =0
line 32 iCheb =0
line 33 nstep =1
line 34 END_PARA

3.2 INPUT.inp 29
3.2.1 iApp
G for GW, B for Bethe–Salpeter, D for both of them.
1 st 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).
2 nd to 4 th 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 Descent
C: 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 (or
including) these levels will be generated. Default is that Llevel = noc-1 (HOMO-1) and
Ulevel = noc+1 (LUMO+1). If the system is spin polarized, wave functions are generated
for 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, Σ c , 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, the
correlation part of self-energy, Σ c , is evaluated by using the full ω integration along the real
axis and then turned 90 ◦ parallel to the imaginary axis (only positive part). If ippmG = 5 is
set, 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 are
used.
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 to
the positive imaginary axis. If jmax = 1000 is used with ippmG = 5, 1001 points are set at
0.1 + 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| ε
LDA. Default value is deltaE = 0.5d0 [eV].
nk

3.2 INPUT.inp 31
Note: When numerical ω integration is performed (ippmG=4,5), deltaE = 2.0 is recommended
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 of
calculation).
Max number of |n 1 |, |n 2 |, |n 3 | of reciprocal lattice vector G = n 1 b 1 + n 2 b 2 + n 3 b 3 for
PWs. Default value for "nod" is 10.
The cut-off energy can be obtained by:
E cut−o f f =
( 2π × nod
A 1
) 2
(3.2)
in the unit of [Ry]. Here A 1 is the lattice constant (length of lattice vector a 1 ) in the unit of
Bohr 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 may
set Ecutoff = 24.d0 in INPUT.inp.) For isolated systems, 18 Ry is required for hydrogen
atoms and 24 Ry required for transition metal, but 7 Ry is enough for carbon atoms and 3
Ry is enough for sodium and lithium atoms. In principle, one has to check the convergence
of the total energy or level energies with increasing the cut-off energy (or "nod"). For crystal
systems, 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 HF
and GW calculations only.)
Max number of |n 1 |, |n 2 |, |n 3 | of reciprocal lattice vector G = n 1 b 1 + n 2 b 2 + n 3 b 3 for
the 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 is
quite 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 correlation
part of the self-energy (necessary item for GW calculation only).
Max number of |n 1 |, |n 2 |, |n 3 | of reciprocal lattice vector G = n 1 b 1 +n 2 b 2 +n 3 b 3 for the
polarization function and the correlation part of the self-energy Eq. (2.33). Default value
for "ncut" is 8.
Note: Typically set the same number as nod or a little less than nod. (necessary item for
GW 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 as
Eq. (1.29) and summed over all atoms. "noz" is the maximum number of nodes in G waves
in this Fourier transformation. It is then numerically Fourier back transformed along the
radial 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 recommended
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 correlation
part of the self-energy in the GW calculation or fixed in the TDDFT dynamics in the
adiabatic LDA calculation. The default value is ncs = 0.
For example, in rutile TiO 2 , there are 2 Ti atoms and 4 O atoms in a unit cell:
Ti: 1s 2 2s 2 2p 6 3s 2 3p 6 1 + 1 + 3 +1 +3 = 9
Note: p orbital contains p x , p y and p z , so the number is 3 for p orbital.

3.2 INPUT.inp 33
O : 1s 2
There is only one s orbital, so the number is 1 for s orbital.
In rutile TiO 2 , 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 correlation
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 the
norm 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 of
q_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. In
order to treat an antiferromagnetic system having zero total magnetic moment, one should
write nspin = -1 in INOUT.inp and simultaneously, one may write the spin magnetic moment
of 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 LDA
calculation, 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 that
nion > 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 usual
convention. 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 be
set 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 1 st 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 updating
the Hamiltonian. Default value is nSDCG = 8.
3.2.32 iSblp
Max SCF Loop for LoopTD = 1.
Number of max SCF Loops for the 1 st 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 1 st 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 inside
atomic sphere) Fourier transformation is used to accelerate the computational speed. Default
value 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 > matrix
elements. 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. Default
value 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" is
a value for the singlet exciton, and "triplet" is a value for the triplet exciton. Default values
are 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 symmetry
lines) for the energy-band outputs. For each band, a list of energies of all the k points are
given by "band_0001.out", "band_0002.out", .... The total list including all bands is given
by band.out. Form of KPOINT.inp is given in Table 3.5.
Table 3.5 Form of KPOINT.inp file
line 1 Direct/Cartesian
line 2 numbers of k points for "out+sum", "out", and "sum"
line 3 1 st k point and weight
line 4 2 nd k point and weight
line 5 3 rd k point and weight
line 6 ... ...
line 7 n th k point and weight
For a three-dimensional lattice defined by its primitive vectors (a 1 , a 2 , a 3 ), the reciprocal
lattice is generated by its three reciprocal primitive defined by the formulae
a 2 × a 3
b 1 = 2π
a 1 · (a 2 × a 3 ) ,
b a 3 × a 1
2 = 2π
a 2 · (a 3 × a 1 ) ,
b a 1 × a 2
3 = 2π
a 3 · (a 1 × a 2 ) . (3.3)

3.3 KPOINT.inp 39
Note: line 1
Direct: k points in the fractional coordinates (coefficients for b 1 , b 2 , b 3 .);
Cartesian: coefficients for 2π/A 1 in Cartesian coordinates; (A 1 is the lattice constant.)
Note: line 2
For 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 the
k-point sampling in the calculation of the polarization function for the correlation part of
the 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 Direct
line 2 16 0 0
line 3 0.500000000000 0.250000000000 0.750000000000 0.11764705882
line 4 0.500000000000 0.333333333333 0.666666666667 0.05882352941
line 5 0.500000000000 0.416666666667 0.583333333333 0.05882352941
line 6 0.500000000000 0.500000000000 0.500000000000 0.05882352941
line 7 0.375000000000 0.375000000000 0.375000000000 0.05882352941
line 8 0.250000000000 0.250000000000 0.250000000000 0.05882352941
line 9 0.125000000000 0.125000000000 0.125000000000 0.05882352941
line 10 0.000000000000 0.000000000000 0.000000000000 0.05882352941
line 11 0.100000000000 0.000000000000 0.100000000000 0.05882352941
line 12 0.200000000000 0.000000000000 0.200000000000 0.05882352941
line 13 0.300000000000 0.000000000000 0.300000000000 0.05882352941
line 14 0.400000000000 0.000000000000 0.400000000000 0.05882352941
line 15 0.500000000000 0.000000000000 0.500000000000 0.05882352941
line 16 0.500000000000 0.125000000000 0.625000000000 0.05882352941
line 17 0.437500000000 0.312500000000 0.750000000000 0.05882352941
line 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., the
weight, 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 1st
Brillouin 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 TiO 2 [26].
Table 3.7 KPOINT.inp for one-shot GW of rutile TiO 2
line 1 Direct
line 2 6 0 2
line 3 0.00000 0.00000 0.00000 1.00 #Γ
line 4 0.00000 0.00000 0.50000 1.00 #Z
line 5 0.00000 0.50000 0.00000 1.00 #X
line 6 0.00000 0.50000 0.50000 1.00 #R
line 7 0.50000 0.50000 0.00000 1.00 #M
line 8 0.50000 0.50000 0.50000 1.00 #A
line 9 0.50000 0.00000 0.50000 1.00
line 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 is
required for the one-shot GW crystal calculation only. Form of QPOINT.inp file is given by
Table 3.8.

3.4 QPOINT.inp 41
Table 3.8 Form of QPOINT.inp file
line 1 Direct/Cartesian
line 2 The number of q points in the irreducible zone, and the number of all grid points (see Appendix B
in the whole zone
line 3 1 st q point and weight
line 4 2 nd q point and weight
line 5 3 rd q point and weight
line 6 ... ...
line 7 n th 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 TiO 2 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 b 1 , b 2 plane

3.5 SPOINT.inp 42
Table 3.9 QPOINT.inp of rutile TiO 2
line 1 Direct
line 2 6 27
line 3 0.00000 0.00000 0.00000 1.00
line 4 0.00000 0.00000 0.50000 1.00
line 5 0.50000 0.00000 0.00000 2.00
line 6 0.50000 0.00000 0.50000 2.00
line 7 0.50000 0.50000 0.00000 1.00
line 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 crystal
calculation. Usually, special points are on the Monkhorst–Pack grid inside the irreducible
Brillouin zone. Form of SPOINT.inp file is given by Table 3.10.
Table 3.10 Form of SPOINT.inp file
line 1 Direct/Cartesian
line 2 number of k points
line 3 1 st k point and weight
line 4 2 nd k point and weight
line 5 3 rd k point and weight
line 6 ... ...
line 7 n th k point and weight
Note: line 1
This 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., the
weigh, has important meaning when the special point sampling is performed to make the
total electron density.

3.5 SPOINT.inp 43
Table 3.11 SPOINT.inp for LDA of Si
line 1 Direct
line 2 2
line 3 0.2500000 0.2500000 0.2500000 1.00
line 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 TiO 2
line 1 Direct
line 2 24
line 3 0.40000000 0.40000000 0.43750000 0.04000000
line 4 0.40000000 0.40000000 0.31250000 0.04000000
line 5 0.40000000 0.40000000 0.18750000 0.04000000
line 6 0.40000000 0.40000000 0.06250000 0.04000000
line 7 0.40000000 0.20000000 0.43750000 0.08000000
line 8 0.40000000 0.20000000 0.31250000 0.08000000
line 9 0.40000000 0.20000000 0.18750000 0.08000000
line 10 0.40000000 0.20000000 0.06250000 0.08000000
line 11 0.40000000 0.00000000 0.43750000 0.04000000
line 12 0.40000000 0.00000000 0.31250000 0.04000000
line 13 0.40000000 0.00000000 0.18750000 0.04000000
line 14 0.40000000 0.00000000 0.06250000 0.04000000
line 15 0.20000000 0.20000000 0.43750000 0.04000000
line 16 0.20000000 0.20000000 0.31250000 0.04000000
line 17 0.20000000 0.20000000 0.18750000 0.04000000
line 18 0.20000000 0.20000000 0.06250000 0.04000000
line 19 0.20000000 0.00000000 0.43750000 0.04000000
line 20 0.20000000 0.00000000 0.31250000 0.04000000
line 21 0.20000000 0.00000000 0.18750000 0.04000000
line 22 0.20000000 0.00000000 0.06250000 0.04000000
line 23 0.00000000 0.00000000 0.43750000 0.01000000
line 24 0.00000000 0.00000000 0.31250000 0.01000000
line 25 0.00000000 0.00000000 0.18750000 0.01000000
line 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 and
MD_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: PhotoAbsorptionSpectra.out.
4.1 OUTPUT.out
In OUTPUT.out file, you can find the number of occupied states "noc", the number of
plane waves "nw", the number of atomic orbitals "nao", and the cut-off energies for plane
waves, Fock exchange and correlation, and so on. This file appears just after the execution
starts, 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 information
of the unit cell, the result of Herman-Skillman’s atomic LDA calculation for generating
AOs, the symmetry operations, the information of the truncation of AOs are written in this
ISYS.out. After the mixed-basis calculation starts, for example, the LDA eigenvalues at
each SCF step and the force contributions at each TD step are written in this ISYS.out. In
the GW calculations, some information of the polarization and dielectric functions are also
written.
4.4 posit.dat
When MD or other dynamics calculation is performed, atomic positions and forces at
each time step are written in position.dat.
4.5 eigen.dat
When MD or other dynamics calculation is performed, eigenvalues at each TD step are
written in eigen.dat.
4.6 MD_Coordinates.xyz, MD_Coordinates.arc
When MD or other dynamics calculation is performed, atomic trajectories are written
in these files for visualization. MD_Coordinates.xyz is for VMD, MD_Coordinates.arc for
Materials Studio.
4.7 band.out
In a crystal calculation when iCry = 1 set in INPUT.inp, band.out is the result of energy
eigenvalues (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 1 st band;
band_0002.out is the result of energy eigenvalues (at sequential k points) in the 2 nd band;
band_0003.out is the result of energy eigenvalues (at sequential k points) in the 3 rd 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 wave
functions 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 an
extension “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 the
final result. The final result is below this sentence, and looks like Table 4.2 for rutile TiO 2 .
Note: Loopk_k = n (n=1, 2, ...)
n means n th 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, Vxc
LDA , in [eV].
Note: eps(LDA) is LDA eigenvalue in [eV].
Note: xg(Fock) is exchange part of self-energy, Σ x , in [eV].
Note: slf(GWA) is correlation part of self-energy, Σ x , in [eV].
Note: QP(GWA) is quasi-particle energy εn,k 0 in [eV].
∫
εn,k 0 = εLDA n,k +
drdr ′ ψ LDA
n,k
∗
(r)[Σ(r,r ′ ;ε LDA
n,k
LDA
) −Vxc
δ(r − r ′ )]ψn,k LDA (r′ ) (4.1)

4.9 GWA.out 47
Note: RQP(GWA) is renormalized quasi-particle energy ε GW
n,k
ε GW
n,k
= ε0 n,k + (ε0 n,k − εLDA
n,k )∂ΣGW n,k (ε)
∂ε
∣
∣
ε=ε
LDA
n,k
in [eV],
which is equal to Eq. (2.6). Here Z n,k is a renormalization factor defined by
Z n,k =
[
1 − ∂ΣGW
∂ε
n,k (ε)
] −1
ε=ε LDA
n,k
Z n,k , (4.2)
. (4.3)
Table 4.1 GWA.out of Li 2 GW cluster calculation
Loopk_q = 0 Loopk_k = 1
Na_ 2: mesh= 48 size= 2.117 nod=6 nog=12 nol= 300 ncut= 6
q=0 only, iDiag 0 Cor Renor
LDA -relation -malized
< xc > eigenvalue < x > < c > QP energy QP energy
Na 2 exc(LDA) eps(LDA) xg(Fock) slf(GWA) QP(GWA) RQP(GWA)
* 3 -5.5682 -3.5408 -7.3972 -0.8465 -6.2163 -5.6038
4 -3.5836 -1.7990 -1.3020 -0.9539 -0.4713 -0.7052
5 -3.8425 -1.5722 -1.2141 -1.0514 0.0048 -0.2900
6 -3.8425 -1.5722 -1.2141 -1.0513 0.0049 -0.2899
7 -3.0842 -1.0258 -0.9242 -1.0423 0.0920 -0.1084
8 -1.7537 -0.0763 -0.2596 -0.8076 0.6102 0.4784
9 -2.0835 -0.0312 -0.2372 -1.0481 0.7669 0.5806
10 -2.0835 -0.0312 -0.2372 -1.0482 0.7669 0.5806
11 -1.6420 0.4796 -0.1403 -0.8559 1.1254 0.9919
12 -1.6420 0.4796 -0.1403 -0.8559 1.1254 0.9919
13 -1.0370 1.0156 -0.0488 -0.5180 1.4857 1.4327
Eg(LDA): 1.742 Eg(GWA): 5.745
IP(LDA): 3.541 IP(GWA): 6.216

4.9 GWA.out 48
Table 4.2 GWA.out of TiO 2 GW crystal calculation
The Result of q-point sum
Loopk_k = 1
Level exc(LDA) eps(LDA) xg(Fock) slf(GWA) QP(GWA) RQP(GWA)
23 -20.6370 -8.7741 -28.3616 7.9457 -8.5530 -8.6157
24 -21.8867 -7.5498 -29.4496 8.0295 -7.0832 -7.2316
25 -22.5537 -6.8436 -29.9669 7.2448 -7.0120 -6.9951
26 -22.5484 -6.7963 -29.9329 7.2063 -6.9746 -6.9580
27 -18.7654 3.1389 -20.0824 3.4084 5.2303 4.7500
28 -20.1376 3.4787 -20.8328 3.3223 6.1058 5.4468
29 -16.8424 3.7836 -18.2006 3.6425 6.0679 5.5987
30 -20.8289 4.9844 -21.5778 3.3271 7.5625 7.0369
31 -20.8219 5.0327 -21.5454 3.2865 7.5957 7.0859
32 -19.8849 6.2660 -21.1415 3.2785 8.2879 8.0060
33 -19.9192 6.2689 -21.1534 3.2696 8.3042 8.0448
34 -20.8307 6.8520 -22.0483 3.4441 9.0785 8.5782
35 -21.9237 8.1268 -22.6500 2.7348 10.1353 9.8225
36 -21.7953 8.3549 -22.3521 2.7431 10.5412 10.1170
37 -21.8323 8.3580 -22.3833 2.7383 10.5453 10.1039
* 38 -21.5760 8.7647 -22.2072 2.9078 11.0413 10.4756
39 -22.3485 10.4435 -13.0584 -4.9269 14.8067 14.1228
40 -21.9881 10.7806 -13.3086 -4.4607 14.9993 14.1706
41 -22.7476 10.9637 -13.6819 -4.7350 15.2944 14.1623
42 -23.2351 11.0461 -13.7080 -5.0459 15.5273 14.5298
43 -23.2203 11.0883 -13.6749 -4.9537 15.6800 14.6532
44 -25.2912 13.0470 -16.3438 -5.1293 16.8650 16.2493
45 -24.9397 13.1399 -15.1763 -5.7544 17.1488 16.1741
46 -26.0406 15.1183 -17.3744 -4.6174 19.1671 18.1623
47 -26.0336 15.1694 -17.3581 -4.5614 19.2835 18.2629
48 -26.7895 16.2643 -17.1368 -5.6359 20.2812 19.1626
Loopk_k = 2
Level exc(LDA) eps(LDA) xg(Fock) slf(GWA) QP(GWA) RQP(GWA)
23 -22.0740 -7.3505 -29.6158 7.7835 -7.1088 -7.1427
24 -22.1741 -7.3054 -29.6361 7.3740 -7.3934 -7.3899
25 -22.0695 -7.3024 -29.5826 7.7010 -7.1144 -7.1340
26 -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) is
written 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 TiO 2 GW calculation
singlet = 0.000000000 (eV), triplet = 0.000000000 (eV)
multiplicity = 0 1
0.000000000 0.002468937 0.185138621
0.005440000 0.002480351 0.192629777
0.010880000 0.002491845 0.200584522
0.016320000 0.002503421 0.209041862
0.021760000 0.002515078 0.218044985
0.027200000 0.002526818 0.227641812
0.032640000 0.002538641 0.237885633
0.038080000 0.002550548 0.248835838
0.043520000 0.002562540 0.260558778
0.048960000 0.002574617 0.273128761
0.054400000 0.002586781 0.286629224
energy (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 (N 2 ) in the LDA
calculations 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 and
without the spherical cut are almost the same. There is a minimum of the total energy at
distance 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 (N 2 ) versus bond length in a calculation
with 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 (Fe 2 ) in the LSDA
calculations 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 without
the spherical cut are slightly different. There is a minimum of the total energy at distance
2.00 [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 (Fe 2 ) versus bond length in a calculation with
and without spherical cut.
Figs. 5.3(a) and (b) represent the force and total energy of iron dimer (Ni 2 ) in the LSDA
calculations 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 without
the spherical cut are slightly different. There is a minimum of the total energy at distance
2.03 [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 (Ni 2 ) versus bond length in a calculation
with and without spherical cut.
The COORDINATES.inp and INPUT.inp files for iron dimer (Fe 2 ) calculation is shown
in Tables 5.1 and 5.2.
Table 5.1 COORDINATES.inp of Fe 2
line 1 SYSTEM=Fe2
line 2 12.000000000000
line 3 1.000000000 0.000000000 0.000000000
line 4 0.000000000 1.000000000 0.000000000
line 5 0.000000000 0.000000000 1.000000000
line 6 2
line 7 Cartesian
line 8 Fe 0.0000000000 0.0000000000 0.0000000000
line 9 Fe 1.1547005384 1.1547005384 1.1547005384
line 10 Fe_rct=1.00
line 11 Fe_ns=3
line 12 Fe_nc=3
line 13 Fe_nv=4
line 14 mesh=64
line 14 END_PARA

5.2 Molecular Dynamics 53
Table 5.2 INPUT.inp of Fe 2
line 1 SYSTEM=Fe2
line 2 iApp =LMNN
line 3 iAlg =D
line 4 nod =14
line 5 ncs =0
line 6 noz =64
line 15 icontinue =0
line 16 isphcut =-1
line 17 iTotalE =1
line 18 lpri =1
line 19 nspin =6
line 20 iMIX =0
line 21 smixSCF =0.90
line 23 iSblp =300
line 26 iasym =0
line 27 icalAO =0
line 28 iCry =0
line 29 iExcite =0
line 30 precLDA =0.000001d0
line 31 iFTapp =0
line 32 iCheb =0
line 33 nstep =1
line 34 END_PARA
5.2 Molecular Dynamics
As an example of a first-principles molecular dynamics simulation, Fig. 5.4 shows several
snapshots of a chemical reaction producing a formic acid (HCOOH) molecule from a carbon
dioxide (CO 2 ) molecule and two hydrogen atoms (2H) [2].
In this MD simulation, COORDINATES.inp file given in Table 5.3 and INPUT.inp files
given in Table 5.4 are used.

5.2 Molecular Dynamics 54
Fig. 5.4 Snap shots of a molecular dynamics (MD) simulation producing HCOOH from
CO 2 + 2H.
Table 5.3 COORDINATES.inp of 2H + CO 2
line 1 CO2+H2=>HCOOH
line 2 8.000000000000
line 3 1.000000000 0.000000000 0.000000000
line 4 0.000000000 1.000000000 0.000000000
line 5 0.000000000 0.000000000 1.000000000
line 6 2
line 7 Cartesian
line 8 C 1.2000000000 0.0000000000 0.0000000000
line 9 O 0.0000000000 0.0000000000 0.0000000000
line 10 O 2.4000000000 0.0000000000 0.0000000000
line 11 H 1.2000000000 1.7000000000 0.0000000000
line 12 H 2.4000000000 -1.7000000000 0.0000000000
line 13 C _rct=0.60
line 14 C _ns=2
line 15 C _np=1
line 16 O _rct=0.50
line 17 O _ns=2
line 18 O _np=1
line 19 mesh=48
line 20 dTime=0.1d0
line 21 END_PARA

5.3 GW calculation of rutile TiO 2 55
Table 5.4 INPUT.inp of 2H + CO 2
line 1 CO2+H2=>HCOOH
line 2 iApp =LMNN
line 3 iAlg =S
line 4 nod =12
line 5 ncs =0
line 6 noz =48
line 15 icontinue =0
line 16 isphcut =0
line 17 iTotalE =1
line 18 lpri =1
line 19 nspin =0
line 20 iMIX =3
line 21 smixSCF =0.80
line 23 iSblp =100
line 26 iasym =0
line 27 icalAO =0
line 28 iCry =0
line 29 iExcite =0
line 30 precLDA =0.00001d0
line 31 iFTapp =0
line 32 iCheb =0
line 33 nstep =1000
line 34 END_PARA
5.3 GW calculation of rutile TiO 2
Five input files are needed for the GW crystal calculation
COORDINATES.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 TiO 2 56
Table 5.5 A sh file for submitting job (run.sh)
mkdir tmp
date
./home/program/main
date
unsetenv 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 TiO 2
Method: Self-consistent field (SCF) loop with special-point sampling (SPOINT.inp) and
band structure calculations at the k-points (KPOINT.inp) on symmetry lines are performed
within standard local density approximation (LDA) of density functional theory (DFT).
In the LDA calculation, it needs 4 input files:
COORDINATES.inp
INPUT.inp
KPOINT.inp
SPOINT.inp
The COORDINATES.inp and SPOINT.inp for LDA calculation are the same with those
for 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 Direct
line 2 51 0 0
line 3 0.00000000000 0.00000000000 0.0000000000 1.00 #Γ
line 4 0.00000000000 0.00000000000 0.0500000000 1.00
line 5 0.00000000000 0.00000000000 0.1000000000 1.00
line 6 0.00000000000 0.00000000000 0.1500000000 1.00

5.4 LDA calculation of rutile TiO 2 57
line 7 0.00000000000 0.00000000000 0.2000000000 1.00
line 8 0.00000000000 0.00000000000 0.2500000000 1.00
line 9 0.00000000000 0.00000000000 0.3000000000 1.00
line 10 0.00000000000 0.00000000000 0.3500000000 1.00
line 11 0.00000000000 0.00000000000 0.4000000000 1.00
line 12 0.00000000000 0.00000000000 0.4500000000 1.00
line 13 0.00000000000 0.00000000000 0.5000000000 1.00 #Z
line 14 0.00000000000 0.05000000000 0.4500000000 1.00
line 15 0.00000000000 0.10000000000 0.4000000000 1.00
line 16 0.00000000000 0.15000000000 0.3500000000 1.00
line 17 0.00000000000 0.20000000000 0.3000000000 1.00
line 18 0.00000000000 0.25000000000 0.2500000000 1.00
line 19 0.00000000000 0.30000000000 0.2000000000 1.00
line 20 0.00000000000 0.35000000000 0.1500000000 1.00
line 21 0.00000000000 0.40000000000 0.1000000000 1.00
line 22 0.00000000000 0.45000000000 0.0500000000 1.00
line 23 0.00000000000 0.50000000000 0.0000000000 1.00 #X
line 24 0.00000000000 0.50000000000 0.0500000000 1.00
line 25 0.00000000000 0.50000000000 0.1000000000 1.00
line 26 0.00000000000 0.50000000000 0.1500000000 1.00
line 27 0.00000000000 0.50000000000 0.2000000000 1.00
line 28 0.00000000000 0.50000000000 0.2500000000 1.00
line 29 0.00000000000 0.50000000000 0.3000000000 1.00
line 30 0.00000000000 0.50000000000 0.3500000000 1.00
line 31 0.00000000000 0.50000000000 0.4000000000 1.00
line 32 0.00000000000 0.50000000000 0.4500000000 1.00
line 33 0.00000000000 0.50000000000 0.5000000000 1.00 #R
line 34 0.05000000000 0.50000000000 0.4500000000 1.00
line 35 0.10000000000 0.50000000000 0.4000000000 1.00
line 36 0.15000000000 0.50000000000 0.3500000000 1.00
line 37 0.20000000000 0.50000000000 0.3000000000 1.00
line 38 0.25000000000 0.50000000000 0.2500000000 1.00
line 39 0.30000000000 0.50000000000 0.2000000000 1.00
line 40 0.35000000000 0.50000000000 0.1500000000 1.00
line 41 0.40000000000 0.50000000000 0.1000000000 1.00
line 42 0.45000000000 0.50000000000 0.0500000000 1.00

5.4 LDA calculation of rutile TiO 2 58
line 43 0.50000000000 0.50000000000 0.0000000000 1.00 #M
line 44 0.50000000000 0.50000000000 0.0500000000 1.00
line 45 0.50000000000 0.50000000000 0.1000000000 1.00
line 46 0.50000000000 0.50000000000 0.1500000000 1.00
line 47 0.50000000000 0.50000000000 0.2000000000 1.00
line 48 0.50000000000 0.50000000000 0.2500000000 1.00
line 49 0.50000000000 0.50000000000 0.3000000000 1.00
line 50 0.50000000000 0.50000000000 0.3500000000 1.00
line 51 0.50000000000 0.50000000000 0.4000000000 1.00
line 52 0.50000000000 0.50000000000 0.4500000000 1.00
line 53 0.50000000000 0.50000000000 0.5000000000 1.00 #A
Note:
1 st k point, 2 nd k point, ..., and 51 st 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 1 st band
band_0002.out energy eigenvalues (at sequential k points) in the 2 nd 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. We
make the figure of band structure, shown in Fig. 5.5

5.5 Wave function calculation of rutile TiO 2 59
Fig. 5.5 Band structure of the pure rutile TiO 2 . (Lines are the LDA results and dots are the
GW results. The zero of energy is placed at the top of the valence band ([VBM] at the Γ
point).[26]
5.5 Wave function calculation of rutile TiO 2
After GW calculation, we can continue Wave Function calculation. It needs to modify
INPUT.inp file:
—————————————-
iApp = LGNN → LWNN
icontinue = 0 → 1
—————————————-
and we make a new KPOINT.inp:
Table 5.7 KPOINT.inp for Wave Function calculation
line 1 Direct
line 2 1 0 0
line 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 TiO 2 60
When wave function calculation is finished, the below output files are obtained:
WaveF_HOMO-1.cube
WaveF_HOMO-1.vasp
WaveF_HOMO-1.grd
WaveF_HOMO.cube
WaveF_HOMO.vasp
WaveF_HOMO.grd
WaveF_LUMO.cube
WaveF_LUMO.vasp
WaveF_LUMO.grd
WaveF_LUMO+1.cube
WaveF_LUMO+1.vasp
WaveF_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) plane
are 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 the
lowest empty state of the pure rutile TiO 2 on (0 0 1) plane.[26]

References
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[3] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev., 136(3B):B864–
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[4] W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlation
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[5] K. Ohno, F. Mauri, and S. G. Louie. Magnetic susceptibility of semiconductors by an
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[7] B. Liu and A.D. McLean. Accurate calculation of the attractive interaction of two
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[9] T. Ohtsuki, K. Ohno, K. Shiga, Y. Kawazoe, Y. Maruyama, and K. Masumoto. Insertion
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[10] K. Ohno, F. Mauri, and S. G. Louie. Magnetic susceptibility of semiconductors by an
all-electron first-principles approach. Phys. Rev. B, 56:1009–1012, Jul 1997.
[11] Y. Noguchi, S. Ishii, and K. Ohno. Two-electron distribution functions and short-range
electron correlations of atoms and molecules by first principles T-matrix calculations.
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References 62
[12] M. S. Bahrany, M. H. F.. Sluiter, and Y. Kawazoe. First-principles calculations of hyperfine
parameters with the all-electron mixed-basis method. Phys. Rev. B, 73:045111,
2006.
[13] E. Runge and E. K. U. Gross. Density-functional theory for time-dependent systems.
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[14] U. Saalmann and R. Schmidt. Non-adiabatic quantum molecular dynamics: basic
formalism and case study. Z. Phys. D, 38:153, 1996.
[15] T. Sawada and K. Ohno. Time-dependent density functional approach to chemical
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[16] R. W. Godby, M. Schlüter, and L. J. Sham. Self-energy operators and exchangecorrelation
potentials in semiconductors. Phys. Rev. B, 37:10159–10175, 1988.
[17] L. Hedin. New method for calculating the one-particle Green’s function with application
to the electron-gas problem. Phys. Rev., 139(3A):A796–A823, 1965.
[18] L. Hedin and S. Lundqvist. Effects of electron-electron and electron-phonon interactions
on the one-electron states of solids. volume 23 of Solid State Physics, pages 1 –
181. Academic Press, 1970.
[19] M. S. Hybertsen and S. G. Louie. Electron correlation in semiconductors and insulators:
Band gaps and quasiparticle energies. Phys. Rev. B, 34(8):5390–5413, 1986.
[20] K. Ohno, K. Esfarjani, and Y. Kawazoe. Computational materials science: from ab
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[21] M. S. Hybertsen and S. G. Louie. Theory of quasiparticle surface states in semiconductor
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[22] R. W. Godby, M. Schlüter, and L. J. Sham. Accurate exchange-correlation potential for
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Appendix A
Materials Studio
Materials Studio is software for simulating and modeling materials. It is developed
and distributed by Accelrys. Now, I introduce how to use Materials Studio to build crystal
model, 1 st Brillouin zone, and make SPOINT.inp file.
Fig. A.1 Materials Studio

A.1 Build crystal model 64
A.1 Build crystal model
Step 1
File → New... → 3D Atomistic
Step 2
Build → Crystals → Build crystal...
Step 3
Set "Space Group" and "Lattice Parameters".
Fig. A.2 Build Crystal
The default is space group P1, i.e. no symmetry. Rutile TiO 2 has the space group
P42/MNM. By telling Materials Studio this symmetry it will automatically apply it to the
atoms, 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 1 st 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 1 st Brillouin zone
Step 1
Modules → CASTEP → Calculation;
Step 2
Properties → Band structure → More...;
Step 3
choose "Path...";
Step 4
In "Brillouin Zone Path", click "Create", and choose "Display reciprocal lattice".

A.3 *.cell file 66
Fig. A.4 1 st Brillouin zone
A.3 *.cell file
A.3.1
make *.cell file
Step 1
Modules → CASTEP → Calculation;
Step 2
In "CASTEP Calculation", choose " k-points", and click "More...";

A.3 *.cell file 67
Fig. A.5 CASTEP Calculation
Step 3
In "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 4
In "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 6
Input 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.9
Note:

A.3 *.cell file 69
Fig. A.9 ZnO.cell file
From "%BLOCK LATTICE_CART" to "%ENDBLOCK LATTICE_CART", these are
lattice paremeters in the Cartesian coordinates, can be used to make "CORRDINATE.inp"
file;
Note:
From "%BLOCK KPOINTS_LIST" to "%ENDBLOCK KPOINTS_LIST", these are
k-points for LDA/GGA calculation, can be used to make "SPOINT.inp" file.