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80 Charged Particle and Photon Interactions with Matter

1995b). The decomposition of the strength into occupied orbitals is shown in Figure 4.10 below

25 eV. The calculated energy of HOMO orbital is 12.0 eV, which is slightly larger than the mea-

sured ionization potential, 11.4 eV. For this molecule, a detailed analysis has previously been

reported in Nakatsukasa and Yabana (2001).

4.4.2.2 ethylene

We show the TDDFT calculation for the oscillator strength distribution of the ethylen; C

, mol-

ecule below 40eV in Figure 4.11, in comparison with measurement (Holland et al., 1997; Cooper

etal., 1995a). The decomposition of the strength into occupied orbitals is shown in Figure 4.12

below 25eV. The calculated energy of HOMO orbital is 11.67eV, which is slightly larger than the

measured ionization potential, 11.0eV. For this molecule, a detailed analysis has previously been

reported in Nakatsukasa and Yabana (2001).

0.8

0.6

0.4

Total

1π

3σ

2σ

0.2

0 5 10 15

Energy (eV)

Oscillator strength distribution (eV

–1

)

20 25

Figure 4.10 Decomposition of the calculated oscillator strength distribution of C

molecule into

orbitals.

0.8

0.6

0.4

0.2

0 5 10 15 20

Energy (eV)

Oscillator strength distribution (eV

–1

)

Cal.

(e,e) exp.

Photon exp.

30 35 40

Figure 4.11 Oscillator strength distribution of ethylene, C

, molecule. Calculation by TDDFT is com-

pared with measurement. (Data from Holland, D.M.P. et al. Chem. Phys., 219, 91, 1997; Cooper, G. et al.,

Chem. Phys.,

194, 175, 1995a.)

Time-Dependent Density-Functional Theory for Oscillator Strength Distribution 81

4.4.2.3 benzene C

We show the TDDFT calculation for the oscillator strength distribution of the benzene, C

, mole-

cule below 60eV in Figure 4.13, in comparison with measurement (Feng et al. 2002). The decompo-

sition of the strength into occupied orbitals is shown in Figure 4.14 for the energy region of 5–13eV.

The calculated energy of HOMO orbital is 11.18eV, which is larger than the measured ionization

potential, 9.24eV. As seen from Figure 4.13, the overall shape of the oscillator strength distribution

is very nicely reproduced by the calculation. Although several structures appear in the TDDFT

calculation, only a few shoulders appear in the measurement above the ionization threshold. In the

spectrum, the intense transition of the 1e

orbital appears at 6.9eV. There appear several transitions

around 10–12 eV region, which are seen as a single shoulder in the measurement. In the TDDFT

calculation, they come from several occupied orbitals. Among them, the orbitals of 3e

, 3e

, and

contribute substantially in this energy region.

0.8

0.6

0.4

0.2

0 5 10 15 20

Energy (eV)

Oscillator strength distribution (eV

–1

)

Total

Figure 4.12 Decomposition of the calculated oscillator strength distribution of C

molecule into orbitals.

1.5

0.5

Cal.

Exp. (

J. Electron Spectrosc.

Relat. Phenom.

123 (2002) 199)

0 10 20 30

Energy (eV)

Oscillator strength distribution (eV

–1

)

40 50 60

Figure 4.13 Oscillator strength distribution of benzene, C

, molecule. Calculation by TDDFT is com-

pared

with measurement. (Data from Feng, R. et al. J. Electron Spectrosc. Relat. Phenom., 123, 199, 2002.)

82 Charged Particle and Photon Interactions with Matter

Recently, Gokhberg et al. (2009) reported the oscillator strength distribution of benzene mol-

ecule

by the Stieltjes–Chebyshev moment theory.

4.4.2.4

naphthalene

We show the TDDFT calculation for the oscillator strength distribution of the naphthalene, C

molecule below 50eV in Figure 4.15, in comparison with measurement (de Souza et al., 2002).

The spectrum of naphthalene resembles that of benzene. The calculated spectrum of naphthalene

is somewhat smoother than that of benzene, probably because of the higher density of states in the

excited state. In the naphthalene spectrum, a bump is seen around 23eV, which is not seen in benzene.

However, it is not clear in the measured spectrum. The absolute value of the measured spectrum is

somewhat

lower than the calculation.

1.4

1.2

0.8

0.6

0.4

0.2

5 6 7 8 9

Energy (eV)

Oscillator strength distribution (eV

–1

)

10 11 12 13

Total

Figure 4.14 Decomposition of the calculated oscillator strength distribution of C

molecule into orbitals.

2.5

Cal.

Exp. (

J. Electron Spectrosc.

Relat. Phenom.

123 (2002) 315)

1.5

0.5

0 10 20

Energy (eV)

Oscillator strength distribution (eV

–1

)

30 40 50

Figure 4.15 Oscillator strength distribution of naphthalene, C

, molecule. Calculation by TDDFT is

compared with measurement. (Data from de Souza, G.G.B. et al., J. Electron Spectrosc. Relat. Phenom.,

123, 315, 2002.)

Time-Dependent Density-Functional Theory for Oscillator Strength Distribution 83

4.4.3 fullerene c

As a nal example of our calculations, we show the TDDFT calculation for the fullerene molecule, C

(Kawashita et al., 2009). Here, we employ the LDA for exchange-correlation potential. The C

mol-

ecule has a quite high spatial symmetry of icosahedron. Reecting the symmetry, the density of states

of dipole-allowed transitions is rather low in comparison with the other molecules of similar size. We

show in Figure 4.16 the calculated oscillator strength distribution of C

. We take a sufciently large

spatial area in the calculation so that the spectrum converges with respect to computational param-

eters such as the box size and mesh spacing. Below the ionization threshold, four transitions are seen

in the calculation, which show a reasonable agreement with measurements where three transitions are

observed. One of the interesting features of the spectrum of C

is the presence of a number of sharp

transitions in the spectrum, even up to 35eV. In the measurements, however, such ne structures are

smeared out, probably by the electron-vibration coupling in the molecule. In Figure 4.17, we show a

0 10 20 30

Energy (eV)

Oscillator strength distribution (eV

–1

)

40 50 60

Figure 4.16 Oscillator strength distribution of fullerene, C

, molecule. (Taken from Kawashita, Y. et al.,

J. Mol. Struct. THEOCHEM,

914, 130, 2009.)

1600

1400

1200

1000

800

600

400

200

0 20 40 60

Energy (eV)

Cross section (Mb)

TDDFT

Exp. (gas)

Exp. (film)

80 100 120

Figure 4.17 Oscillator strength distribution of C

molecule. Calculated result by TDDFT is shown, in com-

parison with measurements in gas phase and in lm. (Data from Kae, B.P. et al., J. Phys. Soc. Jpn., 77, 014302,

2008; Yagi, H. et al., Carbon 47, 1152, 2009; and Kawashita, Y. et al., J. Mol. Struct. THEOCHEM, 914, 130, 2009.)

84 Charged Particle and Photon Interactions with Matter

comparison of the TDDFT oscillator strength distribution with recent measurement (Kae et al., 2008;

Yagi et al., 2009). For the TDDFT calculation, we made a smoothing using a Gaussian function with

a width of 0.27eV

. As is seen in Figure 4.17, the overall features of the oscillator strength distribution

is well reproduced by the calculation. There are several shoulders and plateaus in the measurements,

at 10, 13, 18, 29–34, and 40–50eV. At corresponding positions, structures are also seen in the TDDFT

calculation. However, the detailed assignment of these structures has not been made yet.

4.5 other optiCal Quantities studied with tddFt

The applications of the TDDFT are not limited to the oscillator strength distributions, which we

have shown in the previous section, but are extended to a lot of optical observables. In this section,

outline several other applications of the TDDFT briey.

4.5.1 nonlinear polarizabilitieS

As the intensity of the optical field increases, materials come to show the nonlinear optical

responses, which are related to technologically important nonlinear optical phenomena such

as second-harmonic generation, degenerate four-wave mixing, two-photon absorption, and

so on (Brédas et al., 1994). The nonlinear responses of molecules are characterized by the

nonlinear polarizabilities, which are obtained by a straightforward extension of the linear

response theory to the nonlinear response theory (higher-order perturbation theory). The first

nonlinear response calculation based on the TDDFT was performed for rare gas atoms in 1987

(Senatore and Subbaswamy, 1987). The TDDFT calculation for molecular hyperpolarizabili-

ties has been performed in 1998 based on a quantum chemical approach (Van Gisbergen et al.,

1998a,b, 1999), and in 2001 the same calculation has been performed based on the real-space

approach (Iwata etal., 2001).

4.5.2 dielectric functionS of bulk MaterialS

The TDDFT has also been applied for innitely periodic systems as well as the isolated systems

such as molecules. The linear optical response of bulk materials is characterized by the dielectric

function ε

μν

(ω). The real-space and real-time TDDFT calculation on the dielectric functions was

performed

by us, as detailed in Bertsch et al. (2000).

The

TDDFT describes well the overall features of the dielectric function for a wide energy range.

However the TDDFT gives poor results for the dielectric response of semiconductors and insulators

in the energy range near the bandgap. It is well known that the LDA systematically underestimates

the bandgap energies. The failure of the LDA cannot be recovered by the TDDFT with the adiabatic

LDA. Thus, several attempts have been achieved to incorporate correlation effects that are lacked in

the TDDFT. The Bethe–Salpeter theory based on the GW approximation (Onida et al., 2002) is one

of the successful approaches for the dielectric function, but the computational cost is much higher

than

that of the TDDFT.

4.5.3 optical activitieS of chiral MoleculeS

For chiral molecules, linear optical properties are characterized by the optical activity as well as

the polarizability. The TDDFT is straightforwardly applicable to them, and the calculations of

circular dichroism have been extensively discussed in the literature (Yabana and Bertsch, 1999).

Time-Dependent Density-Functional Theory for Oscillator Strength Distribution 85

4.6 summary

We have presented a theoretical description for the oscillator strength distribution of molecules

for a full spectral region of valence electrons below 100 eV. We rely upon the TDDFT, which is an

extension of the DFT, so as to describe electron dynamics in the rst-principles calculation. We

rst describe a theoretical basis of the TDDFT to calculate the oscillator strength distribution.

We then show the calculated results of the oscillator strength distributions for molecules, taking

diatomic and triatomic molecules—N

, O

, H

O, and CO

; several hydrocarbon molecules of

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

as an

example of a large molecule. It has been shown that the TDDFT provides accurate and quantita-

tive descriptions for the oscillator strength distributions of these molecules. Finally, we mention

briey the applications of the TDDFT for other optical quantities, including hyperpolarizabilities,

the optical activities of chiral molecules, and the dielectric function for bulk materials.

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Generalized Oscillator

Strength

Distribution

Liquid Water

Hisashi Hayashi

Japan Women’s University

Tokyo,

Japan

Yasuo Udagawa

Tohoku University

Sendai,

Japan

5.1 introduCtion

Water is the most common liquid substance on the earth’s surface. Many fundamental processes,

in particular those related to the development of life, take place in aqueous solutions. Accordingly,

water is the most extensively studied liquid from theoretical as well as experimental points of view;

several chapters of this book have the word “liquid water” in their title. Nevertheless, our knowledge

about liquid water is far from complete; even now there is a debate on such a fundamental issue as

to whether a water molecule is hydrogen-bonded with two other molecules or four in the liquid state

(Smith

etal., 2004; Wernet etal., 2004).

The

optical properties of liquid water in the vacuum ultraviolet (VUV) region, like extinction

coefcient (κ), refractive index (n), and reectivity (R), are some of those which are not fully explored

in spite of their importance. They all depend on photon energy, h

ω, and are connected with each

other and also to the complex dielectric response function (ε(ω) = ε

(ω) + iε

(ω)) by the following

equations: ε(ω)

1/2

= n + iκ and R = [(n − 1)

+ κ

]/[(n + 1)

+ κ

]. It is the dielectric response function

that governs interactions between matter and photons or charged particles, which is the theme of this

chapter. If one of those optical or dielectric properties is known for a wide energy range, not only

can other optical properties be calculated, but also various other physical properties can be evaluated

Contents

5.1 Introduction ............................................................................................................................87

5.2 Photoabsorption

and Optical Oscillator Strength...................................................................88

5.3

Inelastic

X-Ray Scattering and Generalized Oscillator Strength........................................... 91

5.4

Inelastic

Electron Scattering and Generalized Oscillator Strength........................................93

5.5

GOS

Measurements by IXS ...................................................................................................94

5.6

Optical Limit ..........................................................................................................................96

5.7 Comparison

of GOS from EELS and IXS .............................................................................99

5.8

Bethe

Surface of Liquid Water .............................................................................................100

5.9

Concluding

Remarks ............................................................................................................ 101

Acknowledgments.......................................................................................................................... 102

References...................................................................................................................................... 102

88 Charged Particle and Photon Interactions with Matter

(Inokuti, 1983; Williams etal., 1991). For example, the mean excitation energy of a material, the most

important quantity required to describe the interaction with ionizing radiations, can be calculated

on absolute scale. Unfortunately, such studies have been hampered by the lack of optical data in the

VUV. To the best of our knowledge, optical data available in this energy region are a series of reec-

tance measurements at Oak Ridge (Painter etal., 1969; Kerr etal., 1971, 1972; Heller etal., 1974),

but they are not wide enough, only up to 26eV, and it has been estimated that the measurements are

accurate to perhaps 30% (Kutcher and Green, 1976).

The dielectric response function ε is in fact a function of momentum transfer q also, and should be

expressed as ε(q,ω). Not only are the available optical data not wide enough, but also what reectance

measurements provide are merely ε(0,ω), because the momentum of a VUV photon is very small. If

ε(q,ω) of a medium is known over a wide energy and momentum range, the interaction of a charged

particle with the medium can be fully understood. Such quantities as the mean free path of charged

particles, cross sections for various interactions, and energy deposition, all of which are crucial in radia-

tion chemistry/biology in vivo but are difcult to determine experimentally, can be evaluated. They

ultimately lead to a track-structure analysis, which is one of the main subjects of radiation research as

well as that of the previous version of this book (Nikjoo and Uehara, 2003; Nikjoo etal., 2006). Hence,

many efforts have been made to estimate ω and q-dependence of ε(q,ω) of liquid water. In most of such

studies, the Tennessee group’s ε(0,ω) are extrapolated to a higher energy region and combined with

some assumptions to extend them to nite momentum transfers (Dingfelder etal., 1998). Lacking exper-

imental results to compare with, it has not been possible to evaluate how accurate those calculations are.

Experimentally, q-dependence of dielectric functions ε(q,ω) can be determined on gases and solids

with inelastic electron scattering or electron energy loss spectroscopy (EELS) (Bonham and Fink,

1974; Egerton, 1996). For example, Lassettre etal. and Takahashi etal. have made extensive studies

about dielectric functions of gaseous water (Lassettre and Skerbele, 1974; Lassettre and White, 1974;

Takahashi etal., 2000) and Daniels of solid water and ice (Daniels, 1971). However, because of experi-

mental difculties, EELS measurements have never been carried out on volatile liquids like water.

We have pointed out that inelastic x-ray scattering (IXS) is free from difculties inherent to

optical as well as EELS measurements, the most serious of which is the need of vacuum, and can

provide ε(q,ω) of volatile liquids for a wide energy and momentum-transfer range (Watanabe etal.,

1997, 2000; Hayashi etal., 1998; Hayashi etal., 2000). IXS studies do not require target materials

to be placed in vacuum and hence volatile liquids can be studied. Since momentum of a hard x-ray

photon is large, q-dependence can easily be determined by changing the scattering angle. In the

following, the manner in which photoabsorption is related to the interaction between charged particles

and materials is described rst, then the principles of IXS and EELS are reviewed, and nally IXS

studies

on liquid water are detailed.

5.2 photoabsorption and optiCal osCillator strength

It is well known that there is a close connection between the interaction of fast moving charged particles

with a material and photoabsorption (Ritchie, 1982; Inokuti, 1986). In order to understand the relation

heuristically, let us consider the following model depicted in Figure 5.1, where a particle with charge ze

and velocity v is traveling along near an atom or a molecule with an impact parameter b. The particle

exerts an electric eld upon the target. Although the electric eld has two components, as illustrated in

the gure, only E

⊥

is important because E

∥

changes its sign at the distance of closest approach (t = 0),

and as a result, the effect of E

∥

almost vanishes if integrated over the whole period. The electric eld E

⊥

has a bell-shaped distribution in time and a simple calculation shows that its FWHM is about 2.6b/v.

Suppose the particle is an electron with a kinetic energy of 10keV and b= 1nm; the velocity v is calcu-

lated to be 6 × 10

m/s and hence the FWHM of the bell is 4 × 10

−17

s, short enough to approximate the

electric eld as a δ-function in time. From the Fourier principle, a δ-function-like electric eld in the time

domain is equivalent to white light in the frequency domain. That is, the effect of the fast charged

particle

traveling

nearby is almost the same as that of white light illuminated on the target.

Generalized Oscillator Strength Distribution of Liquid Water 89

The interaction between white light and a target results in an absorption of a part of the white

light, accompanied by excitation of the target. From the time-dependent perturbation theory, the

absorption cross section, σ

abs

, for transition from the ground state |0〉 with energy E

to an excited

state |n〉 with energy E

, dened as energy absorbed per unit time by the atom, divided by energy

ux

of the radiation eld, is given by (Sakurai, 1994)

δ ω

abs













⋅ − −

⋅



n e E E

k r

pε ( ),

(5.1)

where

k = (ω/c)n is the wave number vector of the monochromatic light propagating toward n with

polarization

vector ε

p the electron momentum operator

r the coordinate of the electron involved in the transition

the use of dipole approximation combined with the commutation relation [x,H

] = ih¯ p

/m,

Equation

5.1 is transformed to

σ π αω δ ω ω

abs

= −4 0

0n n

n x| | ( ). (5.2)

Here, the radiation eld is assumed to propagate in the z direction with the polarization vector along

the x-axis, α = e

c is the dimensionless ne-structure constant, and h

= E

− E

. Atoms and

molecules have many excited states, and hence their response to white light is represented by simply

integrating

the above over all the excited states:

σ ω π αω

abs

∫

∑

=d n x

4 0

| | . (5.3)

In atomic and molecular physics it is common to use a dimensionless quantity called optical

oscillator strength, or simply oscillator strength, dened by

Target (atom

or molecule)

ze/b

~2.6

b/v

Figure 5.1 A particle with charge ze and velocity v passes by a target at an impact parameter b, and

exerts the electric eld E that depends on time t. The two components of E are given by E

⊥

= zeb/(v

+ b

)

3/2

and E

∥

= zevt/(v

+ b

)

3/2