Hatano Y., Katsumura Y., Mozumder A. (Eds.) Charged Particle and Photon Interactions with Matter - Recent Advances, Applications, and Interfaces

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60 Charged Particle and Photon Interactions with Matter

3.5 ConClusions

To understand the details of radiation interaction with matter, we need information about the elemen-

tary collision processes between electrons and molecules. The study of electron–molecule collisions

has a long history, but recently it deals with (large) biomolecules. In this chapter, therecent advances

in experimental and theoretical studies of electron collisions with molecules are reviewed in con-

junction with a view on radiation science. The collision processes considered are elastic scattering;

momentum transfer; excitations of rotational, vibrational, and electronic states; ionization; disso-

ciation; and electron attachment to the molecule. Total scattering and stopping cross sections are

also presented in relation to their importance in the study of radiation effects. Sample cross-section

data are graphically shown, mainly for two representative molecules, H

O and CH

. Cross-section

databases currently available are briey mentioned. The present status of the study of electron–mol-

ecule collisions is not completely satisfactory. Available cross-section data are not comprehensive.

Further

efforts are needed to obtain more complete and more accurate cross sections.

aCknowledgments

One of the authors, Hiroshi Tanaka, is grateful to Profs. S. J. Buckman, M. J. Brunger, and H. Cho

for their kind international collaboration among Japan, Australia, and Korea with the experimental

studies. Also, we would like to thank Drs. M. Hoshino and H. Kato for their kind assistance in

preparing

the gures.

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Time-Dependent

Density-Functional

Theory

for Oscillator

Strength

Distribution

Kazuhiro Yabana

University of Tsukuba

Tsukuba,

Japan

Yosuke Kawashita

University of Tsukuba

Tsukuba,

Japan

Takashi Nakatsukasa

RIKEN Nishina Center

Wako,

Japan

Jun-Ichi Iwata

University of Tsukuba

Tsukuba,

Japan

Contents

4.1 Introduction ............................................................................................................................66

4.2 Formalism............................................................................................................................... 67

4.2.1 Denition

of the Oscillator Strength Distribution......................................................67

4.2.2

Density–Density

Response Function and the Polarizability ......................................68

4.2.3

Time-Dependent

Density-Functional Theory for Polarizability ................................69

4.2.4

Exchange-Correlation Potential.................................................................................. 71

4.3 Computational

Details............................................................................................................72

4.3.1

Real-Space

Grid Representation................................................................................. 72

4.3.2

Real-Time

Method......................................................................................................72

4.3.3

Treatment

of Outgoing Boundary Condition for Photoionization Process................. 74

4.4

Oscillator Strength Distributions of Molecules...................................................................... 75

4.4.1 Diatomic

and Triatomic Molecules ............................................................................ 75

4.4.1.1

Molecule ................................................................................................. 75

4.4.1.2 O

Molecule .................................................................................................76

4.4.1.3 H

O Molecule ..............................................................................................77

4.4.1.4 CO

Molecule...............................................................................................78

4.4.2 Hydrocarbon

Molecules..............................................................................................79

4.4.2.1

Acetylene

............................................................................................79

66 Charged Particle and Photon Interactions with Matter

4.1 introduCtion

The oscillator strength distribution is a basic physical quantity that characterizes the interaction of

photons with atoms and molecules. Most of the oscillator strength of atoms and molecules distributes

in the excitation energy region above the ionization threshold, in which the oscillator strength is a

continuous function of energy. Below the ionization threshold, the oscillator strength distribution is

composed of discrete transitions characterized by the values of the oscillator strength and the excita-

tion energy.

In the last decade, the theoretical description of the oscillator strength distribution has shown

remarkable progress by virtue of the development of the time-dependent density-functional

theory (TDDFT). Nowadays, the density-functional theory (DFT) is recognized as a standard

computational method for the various properties of matters including atoms, molecules, nano-

materials, solids, and so on. The DFT allows us to describe and predict the various properties of

these matters from the rst principles. However, the application of the ordinary DFT is limited to

the electronic ground state. The TDDFT is an extension of the DFT so as to describe the electron

dynamics induced by a time-dependent external potential. Since the oscillator strength distribu-

tion is related to a density change induced by a spatially uniform electric eld, the TDDFT is

applicable for it.

The rst application of the TDDFT for the oscillator strength distribution appeared in 1980 for rare

gas atoms (Zangwill and Soven, 1980). It was found that the TDDFT provides an accurate description

of the oscillator strength distribution of rare gas atoms for a wide energy region. The application was

then extended to the oscillator strength distributions of small molecules (Levine and Soven, 1984),

Mie plasmon (a surface plasma oscillation of metallic clusters) (Ekardt, 1984), nonlinear polarizabili-

ties (Senatore and Subbaswamy, 1987), dielectric functions (Bertsch et al., 2000), and so on. The

TDDFT is nowadays implemented in many computational chemistry programs as a standard tool to

calculate electronic excitations (Dreuw and Head-Gordon, 2005). However, they are limited to dis-

crete excitations.

We have been developing a real-space computational method to solve the time-dependent Kohn–

Sham (TDKS) equation, which is the basic equation of the TDDFT (Yabana and Bertsch, 1996,

1999; Nakatsukasa and Yabana, 2001; Yabana et al., 2006). In this method, we express single-

electron orbitals on uniform grid points in the three-dimensional Cartesian coordinate. One of the

advantages of the real-space method is the exibility to impose the boundary condition, both for

isolated and innitely periodic systems. To describe photoionization processes, one must impose an

outgoing boundary condition for photoemitted electrons. We have developed the real-space method

imposing the outgoing boundary condition for electrons, and have applied the method for the oscil-

lator strength distribution of small molecules, metallic clusters (Nakatsukasa and Yabana, 2001),

and fullerenes (Kawashita et al., 2009). We have also developed a similar method for innitely

periodic systems and applied the method to describe the dielectric functions of crystalline solids as

well

(Bertsch et al., 2000).

4.4.2.2

Ethylene

..............................................................................................80

4.4.2.3 Benzene

............................................................................................... 81

4.4.2.4 Naphthalene

.......................................................................................82

4.4.3 Fullerene

...............................................................................................................83

4.5 Other

Optical Quantities Studied with TDDFT .....................................................................84

4.5.1

Nonlinear

Polarizabilities...........................................................................................84

4.5.2

Dielectric

Functions of Bulk Materials ......................................................................84

4.5.3

Optical

Activities of Chiral Molecules.......................................................................84

4.6

Summary ................................................................................................................................85

References........................................................................................................................................85

Time-Dependent Density-Functional Theory for Oscillator Strength Distribution 67

In this chapter, we would like to present theoretical results for the oscillator strength distribution

of molecules based on the TDDFT, taking several molecules as example. We concentrate on the

theoretical descriptions of electronic excitations, freezing the positions of the atomic nuclei. We will

show the calculations of small molecules, N

, O

, H

O, and CO

; several hydrocarbon molecules

of small and medium size, acetylene, ethylene, benzene, and naphthalene; and a fullerene C

as an

example of a large molecule. These results will show that the TDDFT works well to reproduce the

measured oscillator strength distributions accurately, in both aspects of the absolute value and the

individual

structures that reect bound-to-bound and bound-to-continuum excitations.

The

organization of this chapter is as follows: In Section 4.2, we present a basic formalism of the

TDDFT for the oscillator strength distribution. In Section 4.3, we present the computational details

of our approach. In Section 4.4, we present the results of the calculations of oscillator strength dis-

tributions. In Section 4.5, we discuss the extensions of our framework for other optical observables

including nonlinear polarizabilities, optical activities of chiral molecules, and dielectric functions

innitely periodic systems. In Section 4.6, a summary will be presented.

4.2 Formalism

4.2.1 definition of the oScillator Strength diStribution

We take ħ = 1 throughout this chapter. We consider a molecule in which the positions and the charge

numbers of atomic nuclei are denoted as R

(a = 1 − N

) and Z

, respectively, and the positions of

electrons are denoted as r

(i = 1 − N

). We freeze the positions of the atomic nuclei R

throughout

the present analyses, ignoring the coupling of the electrons with molecular vibrations. The electronic

Hamiltonian of a molecule is given as follows:

Z e e

i a

i j

ee e

= −

−

=== <

∑∑∑ ∑

2 2

111

2 r R

r r

. (4.1)

For bound states, we denote the eigenvalues and eigenfunctions of the Hamiltonian as E

< 0)

and Φ

, respectively, which satisfy HΦ

= E

. We denote the ground state as Φ

. Above the ioniza-

tion threshold, we denote the eigenvalues and eigenfunctions as E(E > 0) and Φ

, respectively, which

satisfy HΦ

= EΦ

. The wave function Φ

satises the appropriate scattering boundary condition.

The eigenfunctions Φ

and Φ

satisfy the orthonormal relationship, 〈Φ

n′

〉 = δ

nn′

for bound states

and

〈Φ

E′

〉 = δ(E − E′) for scattering states, respectively.

The linear optical properties of molecules are characterized by the frequency-dependent dipole

polarizability. Consider that an external electric eld of a xed frequency ω is applied to a molecule.

The polarizability α

μν

(ω) relates the polarization in μ-direction, p

(t), with the electric eld in

ν-direction,

(t) = E

−iωt

p t E t

µ µν ν

α ω( ) ( ) ( ).=

(4.2)

The explicit expression of the polarizability α

μν

(ω) is given by

α ω

ω δ ω δ

µν µ ν

( ) ,=

− −

+ +













∑ ∑ ∑

E i E i

r r

n n

n n i

0 0

1 1

Φ Φ Φ Φ (4.3)

where we dene E

= E

− E

, and the sum over excited states n extends both bound and scattering

states.

68 Charged Particle and Photon Interactions with Matter

We dene the oscillator strength for the bound states n as

n i

∑

Φ Φr .

(4.4)

Similarly, for the scattering states, we define the oscillator strength distribution at energy

E(E > 0) as

E i

( )

∑

Φ Φr (4.5)

where ω = E − E

holds. Combining them, we introduce the strength function S(ω) that is dened

for

the whole energy region,

S E f

( ) ( )

( )

.ω δ ω

θ ω= − + +

( )

∑

0 0

(4.6)

The

Thomas–Reiche–Kuhn (TRK) sum rule for the strength function may be given as follows:

d S f d

ω ω ω

( )

∫

∑

∫

= + =

∞

(4.7)

The

photoabsorption cross section σ(ω) is related to the strength function S(ω) by

σ ω

ω( ) ( ).=

2 2

S (4.8)

The

imaginary part of the polarizability is related to the strength function by

( ) Im ( ).ω

α ω

µµ

∑

2 1

(4.9)

4.2.2 denSity–denSity reSponSe function and the polarizability

We introduce a density operator of electrons dened by

( ) ( ).n

r r r= −

∑

(4.10)

The ground state density n

(r) is then expressed as n

(r) = 〈Φ

n(r)

〉. When a weak and time-dependent

external potential, V

ext

(r,t), is applied to a molecule, it induces a change of the electron density, which

is a linear functional of the external potential. We denote the electron density at time t as n(r, t).

The density–density response function, χ(r,r′,t − t′), relates the density change δn(r,t) = n(r,t) − n

(r)

with the external potential,

Time-Dependent Density-Functional Theory for Oscillator Strength Distribution 69

δ χn t dt d t t V t

( , ) ( , , ) ( , ),r r r r r=

′ ′ ′

−

′ ′ ′

∫∫

−∞

ext

(4.11)

where

we assume that the system is independent of time in the absence of the external potential.

One

may derive an explicit expression of the density–density response function in quantum

mechanics

employing an elementary time-dependent perturbation theory. The result is as follows:

χ θ( , , ) ( ) [ ( , ), ( , )] ,r r r r

′

−

′

= −

′ ′ ′

t t

t t n t n t

0 0

〈

| |

〉Φ Φ

(4.12)

where

θ(t) is the step function. n

(r,

is dened by

( , )

( ) .n t e n e

iHt iHt

r r=

−

(4.13)

For a perturbation of a xed frequency ω, the relevant response function is given by the Fourier

transform of Equation 4.12. We write the explicit expression of the response function as follows:

χ ω χ

ω δ

( , , ) ( , , )

( ) ( )

r r r r

′

+ −

−∞

∞

∫

dt e t

n n

i E

i t

n n

Φ Φ Φ Φ

0 0

ˆ ˆ

∑∑

−

′

+ +

Φ Φ Φ Φ

0 0

( )

n n

i E

n n

r r

ω δ

(4.14)

where the sum over n extends all the electronic excited states of the system including scattering

states. δ is a small positive number. The density–density response function in the frequency repre-

sentation thus includes information on the electronic excited states: excitation energies as poles and

the

transition densities as residues.

consider a dipole eld in ν-direction with frequency ω. The potential is given by V

ext

(r,t) =

−iωt

, where r

is one of the Cartesian coordinates, x, y, z. Since the polarizability α

μν

(ω) relates

the polarization in μ-direction, p

(t), with the electric eld in ν-direction, E

(t) = E

−iωt

p t d e r n t E t

µ µ µν ν

δ α ω( ) ( ) ( , ) ( ) ( ).= − =

∫

r r

(4.15)

The polarizability may be expressed in terms of the density–density response function as

α ω χ ω

µν µ

( ) ( , , ).= −

′ ′ ′

∫

e d d r r

r r r r

(4.16)

4.2.3 tiMe-dependent denSity-functional theory for polarizability

In the TDDFT, one expects that the density change δn(r,t) induced by an external potential V

ext

(r,t)

accurately described by the TDKS equation,

t h n t V t t

i i

∂

= +

{ }

ψ ψ( , ) [ ( )] ( , ) ( , ),r r r

KS ext

(4.17)