THEORY

2-1 The Electric Properties of Matter

2-1-1 Basic relation of electric field, energy and polarizability

There exist both time-dependent and the time-independent electric fields.

Time-dependent field is usually associated with electromagnetic radiation characterized by a frequency of implying dynamic properties, while time-independent field does not vary with accompanying frequency of implying static properties. The present work is focused on the static electric field that is also called the homogeneous electric field here¹⁰.

In the presence of a homogeneous external electric field, the Hamiltonian for the total system (nuclei and electrons) can be written as¹¹

(0) (1) in which the dipole moment operator μ is given by

ˆ

_{i i}

i

µ = ∑ q r

(2.3) with the vector ri indicating the particle i with charge qi

From eq. (2.1) and (2.2) it can be obtained the relation,

ˆ ˆ a x, y, z

_a

where μ^ais the expectation of the a-th component of the dipole moment

From the Hellmann-Feynman theorem it follows

where E⁽⁰⁾ is the energy of the unperturbed term for the system.

Combining eq. (2.7) with (2.8), we can have

In terms of the derivatives in eq. (2.9), the permanent dipole moment, the

the total dipole m om ent (field-dependent) : =- E

the component aa' of the polarizability tensor : = - E =

the component aa'a" of the (first) hyperpolarizability tensor : = - E

In other words, an induced dipole moment, which consists of the linear responses ( α polarizability ) and the nonlinear responses ( β hyperpolarizability) to a homogeneous electric field. The quantities (vectorμ⁰, tensor α and tensorβ) are very important quantities for characterizing the molecule.

A neutral atom is placed in a static external electric field F. If the field is not too large, and the response ( α polarizability ) of the atom is isotropic , and induced dipole moment will be proportional to the electric field, and can be written

ˆ F ˆ

µ = α

(2.15) with α is the constant of proportionality — scalar quantity.

For a molecule with spherical symmetry, the polarizability is well-approximated by a single constant — scalar quantity, the polarizabilty is also isotropic. However, for many molecules being not spherical symmetry, the polarizability cannot be characterized by a single constant. For example, the charge distribution along the internuclear axis is longer than perpendicular to this axis for H2 molecules, and thus it can be expected that the charge separation induced by external electric field is be greater along the internuclear axis than along a perpendicular axis¹². In such case, a single scalar quantity is not sufficient to describe the polarizability. The most general way to transform one vector F to another μ is defined by a second-rank Cartesian tensor as in matrix form

x xx xy xz x and each one has a surface charge density σ. If the medium between plates is a

vacuum, the electric field is derived as ( atomic units), see Fig. 4(a)¹³ _v

0

F = σ

ε (2.18)

But if there existing dielectric medium between two plates, the electric field will be reduced due to the polarization P of the material. The field distorts the atoms and molecules, and orients existing dipoles to create a field opposing the applied field. So the resultant field is reduced¹³. Now P called as polarization is defined as dipole moment per unit volume. Consider a plate of area A, length L. Its total dipole moment due to polarization is

Rearrange eq. (2.20) and combine with eq. (2.18), the basic relation of polarization and electric field is obtained.

0 0 0 0 r 0

0

P= -F =F( - )=F (

σ ε ε ε ε ε

-1)=F ( -1)=F

ε ε ε

_e

ε χ

(2.21)

Electric susceptibility (χe) is defined through eq. (2.21), and is related to the relative permittivity (dielectric constant)

Fig. 4 (a)The field Fv between plates in vacuum.(b) The field in the dielectric medium is polarized, and it is reduced to F.(c) Relation between F and Fv

2-1-3 Lorentz local field and Clausius-Mossotti equation

Let us begin with considering all kinds of electric fields. F0 is applied external field, and F1 is the field of the surface induced charge density on the boundary, and it is due to the uniform polarization of the surface charge density. F1 is also called depolarization field which is opposed to the applied field F0 within the body. F is the addition of F0 and F1, and is called macroscopic electric field. ( see Fig. 5¹⁴ ).

Fig. 5 The applied external field F0. The depolarization field F1 is opposite to P.

And the macroscopic electric field F, F=F0+F1

Now an expression of local field Flocal at a general lattice site is discussed. The local field that acts at the site of an atom is different from the macroscopic electric field F. If it is in a sphere, the macroscopic electric field can be seen in eq. (2.21) where the value of depolarization field is from depolarization factor of a sphere¹⁴.

0 1 0

0

F = F + F = F - 1 P

3ε (2.23) The local field at an atom is the sum of the electric field F0 from external sources and of the filed from the dipoles within the specimen. It is written as

0 1 2

local 3

F = F + F + F + F

(2.24) where F0 = applied external field. F1 = depolarization field, from a charge density of the outer surface. F2 = Lorentz cavity field, from polarization charges inside of a spherical cavity cut out of the specimen with the reference atom as center. F3 = field inside cavity. More details in Fig. 6¹⁴

Fig. 6 The internal electric field at an atom in a crystal is the sum of the applied field F0 and of the field due to the other atoms in the crystal. The standard method is summation of the dipole fields of the other atoms. First, it sums individually over a moderate number of neighboring atoms inside an imaginary sphere concentric with the reference atom: this is defined by the field F3. The atoms outside the sphere can be contributed to the field at a reference point is F1 + F2.

If atomic arrangement is cubic symmetry in a crystal or the array of dipoles is random in a liquid or a glass, the inside field F3 is zero. So the field F3 depends on the crystal structure.

Now, let us start to discuss the Lorentz cavity field F2. An annulus on the surface of a cavity is constructed. ( see Fig. 7¹³ )

Fig. 7 charge distributions in a spherical cavity within a dielectric.

From Fig. 7, the charge on the annulus is

cos 2 R sin Rd

P θ π θ

− θ

(2.25)

At the center of the cavity it produces a coulomb field along the annulus. The vertical components cancel off, and the horizontal ones are additive from Fig. 4.

2

2 cavity is spherical, the local field is

loc 0 1 2

0

F =F +F +F =F+ P

3ε (2.28)

eq. (2.28) is the Lorentz relation.

The polarizability α of an atom is defined in term of the local electric field at the atom.

= F

loc

µ α

(2.29)

The polarization of a crystal can be expressed as the product of the polarizability of the atoms multiplied by the local electric field.

loc

where n called as number density is defined as total number of objects per unit volume, and the unit of P is dipole moment per unit volume.

Form eq. (2.30) basic relation between P and F in dielectric is obtained as with which the dielectric constant in eq. (2.22) becomes

e r

( )

⁰

Moreover, a practical form is obtained by replacing the number density by the mass density ρ.

Finally, the Clausius-Mossotti equation can be written as

0 r

The polarizability which is one of atomic properties is related to many important bulk properties of a collection of particles including the dielectric constant εr, the electric susceptibility χe, the refractive index η, etc¹².

If there is the permanent electric dipole moment μ⁰ in the molecule, the polarizability is related to the dielectric constant by the Debye equation.

2

where μ0 is the permanent dipole moment, k is Boltzmann’s constant and T is the absolute temperature. ( more details in 2-1-4 Polarizability )

The index of refraction is related to the polarizability by the Lorentz-Lorenz

This formula is valid for non-polar molecules or at frequency high enough that the permanent dipole moments cannot follow the electric field. The Lorentz-Lorenz equation and Clausius-Mossotti equation are related by the Maxwell relation εr = η^{2 12}.

2-1-4 Orientation and distortion polarization

As is seen from Fig. 4 (b), when a external field in the dielectric medium, the effective electric field between the plates must be reduced. The reduced field due to polarization of the medium is affected by two reasons. The first one is that the molecule of the medium has a permanent dipole moment, and this effect is known as orientation polarization. The orientation polarization is temperature dependent and its value decreases with an increase of temperature because the random thermal collisions oppose the tendency of the permanent dipole moments to orient themselves in the electric field¹⁵. This magnitude of the effect can be calculated from Boltzmann distribution. The energy of a dipole in a local field Flocal along the z-axis is

( )

_{z loc loc}

cos , with 0

E θ = − µ F = − µ F θ ≤ ≤ θ π

(2.39)

The probability dp(θ) that a dipole has an orientation in the range θ to θ+dθ is

( )

^{-E( )}_{( )}^/kT

Finally the average dipole moment along the z-axis is obtained.

( ) ( )

^{x -x}

The function L(x) is called Langevin function.

When µFloc << kT corresponding to x<<1, the Langevin function is

Here αo is called orientation polarizability, and the eq. (2.44) exists only at µFloc << kT.

The second version is that it always exists whether the molecule is polar or not.

For the electrons to shift relative to the positive charges, and this is called as electronic polarization, while atoms are shifted relative to each other called as atomic polarization. If the position of a molecule is disturbed by a collision, a new dipole is immediately induced again in the direction of field. However, the

distortion polarization is independent on the temperature¹⁵. The sum of above two polarization effects is called distortion polarization.

2-1-5 A summary of polarization

The electric field gives rise to a dipole moment by the following effects¹⁶: 1. Translation (distortion) effects:

The electrons are shifted relative to the positive charge ( electronic polarization) Atoms or atom groups are displaced relative to each other (atomic polarization) 2. Rotation (orientation) effect:

The electric field trends to direct the permanent dipole moments.

Now let us consider frequency effect toward polarizability. When the frequency of the field is high, the molecule dipole cannot change direction fast enough to follow the field. Therefore permanent dipole moment doesn’t contribute to the polarization at microwave region, orientation polarizability is lost. For higher frequency, because the molecule is bent and stretched in the frequency of Infrared-Ray region by the applied field, the molecule dipole moments change accordingly. The time taken for a molecule to bend is approximately the inverse of the molecular vibrational frequency, so it will lose the contribution of the atomic polarization. At even higher frequency (about visible region), only the electrons are mobile enough to follow the rapidly changing direction of the field, so it only remains electronic polarizability¹⁷. (See Fig. 8)

Fig. 8 The general form of the variation of the polarizability with the frequency.

2-2 Charge population

2-2-1 Mullikan charge

Assuming the MO (φ) can be expanded in a set of basis functions (χ)

The total electrons (Nelec) are equal to integrating and summing over all occupied MOs.

The Mulliken population analysis uses the D‧S matrix for separating the electron density into atomic contributions. A diagonal element DααSαα is the number

electrons shared equally by α AO and β AO. The Mulliken electron population on atom A is defined as in eq. (2.49)

S

The Mulliken net charge on atom A is the sum of the nuclear and electronic contributions.

Q =Z -

_{A A}

ρ

_A (2.50) However, Mulliken population analysis doesn’t offer exact charges of the individual atom, it only provides the trend, because partition of the charge contribution is equal. Moreover, atomic charges calculated from the Mulliken analysis will not converge to a constant value when the size of the basis set is increasing. Larger basis set is usually involving the addition of more polarization basis functions or diffuse basis function, it will give rise to unpredicted change in the atomic charge. So Mulliken population analysis is affected largely by basis functions, and it usually is most useful for comparing trends in charge distributions, when small- or medium-size basis sets are used¹⁰.

2-2-2 Atomic polar tensor charge

The atomic polar tensor V^APT of atom A is defined by the first derivatives of the dipole moment with respect to the nuclear coordinates¹⁸ in eq. (2.52) and it can be used to determine intensities of IR absorptions due to the intensity is given by eq.

(2.51).

2

int ensity IR ⎛∂Rµ ⎞

∝ ⎜⎝∂ ⎟⎠ (2.51)

Here R is the nuclear coordinates.

x x x Because dipole moment derivatives determine IR absorptions, APT charges are directly related to experimentally observable quantities¹⁹. Moreover, APT analysis has the following properties: (1) the atomic charges should be invariant with respect to rotations and translations of the molecule; (2) APT charges sum up to the total electric charges of the molecule; (3) APT charge isn’t directly related to the choice of a particular basis set , its basis set dependent stems only from the fact that the basis set is not complete¹⁸. So the basis-set dependence is modest, although basis-set convergence isn’t reported²⁰. But APT charges are sensitive to the electron correlation in the wave function, it can be seen in Table 1²¹. A measure for the sensitivity of a particular change definition toward electron correlation is provided by the difference q (QCISD) – q (SCF). It is very obvious that APT charge exhibits the largest correlation effect and Mulliken population analysis appears to be relatively in sensitive to the electron correlation²¹. So the observation that APT

appears to be much less sensitive to the basis set than to electron correlation.

Table 1: Correlation contribution to different charge definitions at QCISD/cc-pVDZ level

2-3 Dipole moment

¹⁶

The dipole moment of a point charge q relative to a fixed point is defined as qr, in which r is the radius vector from the fixed point to e. Hence, the dipole moment of a system of charges qi, relative to a fixed origin is defined as:

i i i

ˆ = q rˆ

µ

∑

(2.54) If the net charge of the system is zero, the dipole moment is independent of the choice of the origin. The eq. (2.54) can be written in another way by introducing the positive and negative charges in eq. (2.55)

i i i i dipole moment will be equal to charge multiplied by bond length directly.

atoms of a system, we choose the Ａ atom as the origin for (x2, y2) = (0,0), and

2-4 Density Functional Theory

2-4-1 Historical Background

Thomas (1926) and Fermi (1927) are the pioneering scientists to use the electron density rather than wave function for expressing approximate energy for the first time. They applied quantum statistical model (uniform electron gas) to the kinetic energy but treated the electron-nuclear and electron-electron interactions in classical way. The most important part is no exchange and correlation effect in this model. The Thomas-Fermi total energy is expressed in terms of density as²² .

( )

^{2 23 53}

( )

^{1 2} energy can be expressed approximately as

[ ] ( )

⁴³

X X

E ρ ≅C

∫

ρ r^K dr^K (2.60) To improve the quality of exchange energy, semiempirical parameter α was

introduced to CX. It is called Xα method in one of density functional approaches.

Hohenberg and Kohn (1964) brought up two theorems that make big contribution to the present DFT. The first one is that the ground state electronic density is uniquely specified by the given external potential Vext, and it also means electron density ρ(r) defines all terms in the Hamilton operator²³. In other words, the ground state electronic density and the ground state wave function can be used alternatively as full descriptions of the ground state of the system¹¹. The second one

energy by inserting approximate density, see eq. (2.61). E0 represents the minimum value of the system.

E

0

≤ E [ ] [ ] ρ  =T ρ  +E

Ne

[ ] ρ  +E

_ee

[ ] ρ 

(2.61)

Kohn and Sham (1964) considered a fictitious system of non-interacting particles, and in order to ensure that the system has the same density and energy as

the real system, those particles are assumed moving in the external potential

Veff (r)²². In the other hand, the Kohn-Sham system of the electrons, that do not interact with each other at all (as if their charges are equal zero) but interact with the nuclei, they are subject to an external potential Veff (r)¹¹, as described in the following²².

E’xc is called as exchange correlation energy in which exchange part is from Hartree-Fock approximation, and correlation effect is obtained from extra correlation contributions (beyond the Hartree-Fock approximation). Vext is the interaction of nuclei and electrons. Vc is the classical Coulomb operator, then there is a self-interaction of electron cloud with itself.

N

( )

exchange-correlation potential Vxc is to be determined.

^{0 'xc xc} xc

( )

Finally, the total DFT energy of Kohn-Sham theorem can be written of approximations have been carried out for getting the accurate exchange-correlation energy. In the Local Density Approximation (LDA) it is assumed that uniform electron gas is in the system, in order to ensure that the electron density is constant in space, the exchange-correlation energy isn’t related with variations of electron density. It means that we can calculate Vxc for the system as a function of the constant density. In the Generalized Gradient Approximation (GGA) it takes into account a non-uniform electron gas and includes electron density and the derivatives of the density in the expression of the exchange-correlation energy. On the other hand, Exc depends not only on ρat a given point (local), but also at the ρ nearby (non-local). Therefore, it is also called non-local approximations. If it includes higher derivatives of the density, it could be called meta-GGA approximation. At the beginning of the 1990s a new method was called Hybrid method combined the Hartree-Fock theory with Kohn-Sham theorem. Because the exchange energy is given exactly by Hartree-Fock theory and correlation energy is treated only by Kohn-Sham theorem, it will be more accurate to express the exchange-correlation energy. The most popular and famous method B3LYP functional is widely used in computation and it is defined in

depend on the form of exchange and correlation energy. It is the classification of density functional below. (see Table 2)

Table 2: Perdew classification of exchange-correlation functions¹⁰

Level Name Variables Examples

1 Local density ρ L(S)DA , Xα

2 GGA ρ, ▽ρ BLYP , OLYP , PW86 , PW91 , PBE

3 m-GGA ρ, ▽ρ,▽²ρ or more B95 , VSXC , PKZB , TPSS

4 Hybid

ρ, ▽ρ,▽²ρ or more and HF exchange

H+H , B3LYP , O3LYP , PBE0 , TPSSh

2-4-2 M05 and M05-2X

Now still there is one problem called as self-correlation error in some DFT methods. It stemmed from electron interacting with itself. Table 3 is about self-correlation error for one electron of Hydrogen atom. We can see the HF theory is free of self-correlation error for one-electron system, but other DFT methods clearly have self-correlation error.

Table 3: Energy components [Eh] of various functionals for the hydrogen atom²³.

M05²⁴ (for Minnesota 2005) and M05-2X²⁵ (for Minnesota 2005 with double the amount of nonlocal exchange) are the newly developed functionals. They both can be called hybrid methods, because they incorporate electron spin density, density gradient, kinetic energy density and Hartree-Fock exchange energy. The two new functionals have three advantages; one is incorporating kinetic energy density in a balanced way in the exchange and correlation functionals, the other is to satisfy the uniform electron gas limit, the third is free of self-correlation²⁵. The M05 functional was parametrized by including both metals and nonmetals and is broadly applicable to organometallic, inorganometallic and nonmetallic bonding, thermochemistry, thermochemical kinetics, and noncovalent interaction²⁴. So M05 performs well not only for main-group thermochemistry and radical reaction barrier but also for transition-metal to transition-metal interaction. The M05-2X functional was parametrized only for nonmetals and performed well for thermochemical kinetics and noncovalent interactions (weak interaction like hydrogen bond …etc), excluding metals²⁵.

2-5 Thermoeletric Phenomena

2-5-1 Historical Background

In 1822, Thomas Johann Seebeck published a paper describing that a compass needle was deflected when it was placed in a closed loop of two dissimilar metals that had one junction heated*. That was the first observation of the phenomenon of thermoelectricity. But he tried to relate this phenomenon to the earth’s magnetic field. Of course, it didn’t work because the phenomenon wasn’t from the magnetic effect at all. Although Seebeck didn’t understand it at that time, that effect was known as the Seebeck effect. Later in 1835, Jean Charles Athanase Peltier discovered that the temperature of a junction between two dissimilar metals changes when current flows between them²⁶. At that time, it was not realized that this was related to the Seebeck effect. In 1838, Lenz discovered that heat is liberated or absorbed depending on the direction of the current flow across a junction of two

In 1822, Thomas Johann Seebeck published a paper describing that a compass needle was deflected when it was placed in a closed loop of two dissimilar metals that had one junction heated*. That was the first observation of the phenomenon of thermoelectricity. But he tried to relate this phenomenon to the earth’s magnetic field. Of course, it didn’t work because the phenomenon wasn’t from the magnetic effect at all. Although Seebeck didn’t understand it at that time, that effect was known as the Seebeck effect. Later in 1835, Jean Charles Athanase Peltier discovered that the temperature of a junction between two dissimilar metals changes when current flows between them²⁶. At that time, it was not realized that this was related to the Seebeck effect. In 1838, Lenz discovered that heat is liberated or absorbed depending on the direction of the current flow across a junction of two

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