Dear G. Arivazhagan,

There is NO such thing called dipole moment of a molecule. There is a dipole moment of a functional group. Molecule have dielectric constants. For example water has a dielectric constant of about 78.

For the calculation of dielectric constant using DFT, please read the following publications:

1-Microelectronics Reliability

Volume 39, Issue 2, February 1999, Pages 279–284

Density functional theory applied to the calculation of dielectric constant of low-k materials

A. Courtot-Descharlesa,  F. Piresb, P. Pailleta, J.L. Leraya

doi:10.1016/S0026-2714(98)00237-6

Abstract

The interest of low-k dielectric materials to reduce capacitance in multilevel metal interconnects of integrated circuits is well known in the semiconductor industry. The use of these materials (especially hydrogen silsesquioxane (HSQ) and methyl silsesquioxane (MSQ): intermetal dielectric applications in the back end of line fabrication) leads to a reduction of the dielectric constant from k≅4 in a traditional intermetal dielectric material of silicon dioxide to a value of ≅2.5–3. The physical difference between HSQ or MSQ and a-SiO2 is the presence of Si–H bonds (for HSQ) or Si–CH3 bonds (for MSQ) and the density of the material. A theoretical calculation of bond polarizability (Si–H or Si–CH3) associated to experimental values of electric dipole densities can lead, using the Clausius-Mossotti relationship, to the calculation of the dielectric constant. After validation of the calculation methods both on simulation and experimental values, it is shown that for a constant density, the difference between the materials could be due to the bond polarizability and furthermore that this difference accounts, in part, for the value of dielectric constant. Consequently, even if densification remains the main parameter explaining low-k values, the polarizability of building units of these materials is not negligible.

http://www.sciencedirect.com/science/article/pii/S0026271498002376

2-

Different Methods to Calculate Dielectric Constants

Introduction

Dielectric materials are used to control and store charges and electric energies.They play a key role in modern electronics and electric power system.As the requirements grow for compact,low-cost electronic and electrical power system,as well as for very high energy and power capacitive storage systems,the development of high power and energy density dielectric materials becomes a major promising field. Dielectric constant is an important property for dielectric materials.In literature,there are several methods to calculate dielectric constant,the most popular one at present is density functional perturbation theory (DFPT).In this term paper,two methods will be presented,one is DFPT,the other one is beyond DFT,molecular dynamics.

Theory

In electrostatics,

(1)Di=Ei+4πPi=ϵijEj

This is the fundamental equation from which dielectric constant is calculated.The total polarization may usually be separated into three parts: electronic, ionic and dipolar, which results in the fact that dielectric constant also has three contributions.The electronic contribution arises from the electron cloud shifts from the nucleus,and the ionic contribution is caused by relative displacements between positive and negative ions. Permanent dipoles in polar systems result in the dipolar contribution to dielectric constant.

Density Functional Perturbation Theory (DFPT)

The Hohenberg and Kohn theorem states that all the physical properties of a system of interacting electrons are uniquely determined by its ground-state charge density distribution.This property holds independently of the precise form of the electron-electron interaction.This face was used by Kohn and Sham to map the problem of a system of interacting electrons onto an equivalent noninteracting problem.Since the electrons do not interact with each other, a one-electron Schrodinger equation can be written.

(2)HSCFψn(r)=ϵnψn(r)

(3)HSCF=−ℏ22m∂2∂r2+VSCF(r)

(4)Missing open brace for superscript?

(5)n(r)=2∑n=1N/2|ψn(r)|2

The above equations are the foundation of density funcstional theory (DFT),and once an explicit form for the exchange-correlation energy is available,equation (1) can be solved in a self-consistent way.

In quantum mechanics,perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one.Equation (1) describes the unperturbed system.Now introduce a perturbation to the unperturbed Hamiltonian H0.Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. Let λ be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is

(6)H=H0+λV

Then the Schrodinger equation becomes:

(7)(H0+λV)|ψ⟩=En|ψ⟩

If the perturbation is sufficiently weak, En and |ψ⟩ can be written as power series in λ:

(8)En=E(0)n+λE(1)n+λ2E(2)n+⋯

(9)|ψ⟩=|ψ(0)⟩+λ|ψ(1)⟩+λ2|ψ(2)⟩+⋯

where

(10)E(k)n=1k!dkEndλk

and

(11)|ψ(k)⟩=1k!dk|ψ⟩dλk

When ''λ = 0'', these reduce to the unperturbed values, which are the first term in each series.Substitute equation (7) and (8) into equation (6),thus

(12)(H0+λV)(|ψ(0)⟩+λ|ψ(1)⟩+⋯)=(E(0)n+λE(1)n+λ2E(2)n+⋯)(|ψ(0)⟩+λ|ψ(1)⟩+⋯)

The first order equation is

(13)H0|ψ(1)⟩+V|ψ(0)⟩=E(0)n|ψ(1)⟩+E(1)n|ψ(0)⟩

Apply the above first-order perturbation theory to the unperturbed Kohn-Sham equation,and set |ψ(1)⟩=|△ψn⟩,V=△VSCF,E(1)n=△ϵn

(14)Extra close brace or missing open brace?

(15)△n(r)=4∑n=1N/2ψn∗(r)△ψn(r)

(16)|△ψn⟩=∑m≠n|ψm⟩⟨ψm|△VSCF|ψn⟩ϵn−ϵm

In equation (13),△VSCF is a linear functional of △n(r),then equation (13) and (14) form a set of self-consistent equations for the perturbed system completely analogous to the Kohn-Sham equations in the unperturbed case,and this becomes the foundation of the density functional perturbation theory.

From equation (15),the wave function response to a given perturbation depends only on the off-diagonal matrix elements of the perturbing potential between eigenfunctions of the unperturbed Hamiltonian.When the perturbation is a macroscopic electric field, then such matrix elements are indeed well defined,as can be seen by rewriting them in terms of the commutator between r and the unperturbed Hamiltonian,which is a lattice periodic operator:

(17)⟨ψm|r|ψn⟩=⟨ψm|HSCF,r|ψn⟩ϵm−ϵn,∀m≠n

When calculating the response of a crystal to an applied electric field E0,one must consider that the screened field acting on the electrons is

(18)E=E0−4πP

Where P is the electronic polarization linearly induced by the screened field E:

(19)P=−eV∫Vr△En(r)dr

Combine equation (14) and (18),

(20)Pα=−4eV∑n=1N/2⟨ψn|rα|△Eψn⟩=−4eV∑n=1N/2∑m=N/2+1∞⟨ψn|[HSCF,rα]|ψm⟩ϵn−ϵm×⟨ψm|△Eψn⟩

Where the subscript α indicates Cartesian components.Then introduce the wave function ψ¯¯¯nα(r) defined as

(21)ψ¯¯¯nα(r)=∑m≠nψm(r)⟨ψm|[HSCF,rα]|ψn⟩ϵm−ϵn

Thus,

(22)Extra close brace or missing open brace?

The electronic contribution to the dielectric tensor,ϵ∞αβ,can be derived from simple electrostatics

(23)E0α=(Eα+4πPα)=∑βϵ∞αβEβ

Using equation (21) to calculated the polarization induced in the α direction when a field is applied in the β direction,thus

(24)Extra close brace or missing open brace?

The total dielectric constant has two parts,one is the electronic contribution which has been derived,and the other one is ionic contribution.It can be defined as:

(25)ϵ=ϵ∞αβ+4πe2V∑λSmωλ2

Where Sm is the mode oscillator strength, it is proportional to the born effective charge and the displacement of each atom.And ωλ is the frequency of the λth IR-active phonon normal mode.In DFPT,the ionic contribution to dielectric constant can also be calculated,the detailed procedure is presented here,as it is similar to that of electronic contribution.

Molecular Dynamics

In the Frohlich's theory,the dielectric constant is related to the total dipole moment fluctuation.To derive this relation,there are usually two steps.First,with the help of statistical mechanics,establish a relation between the mean total dipole moment in the presence of a homogeneous external field and the equilibrium fluctuations in the absence of any external perturbations.Second,classical electrostatics is used to calculated the macroscopic polarization induced by electric field.After eliminating electric field from both equations one obtains a relation between the fluctuation and the dielectric constant.

For a system of N particles with generalized coordinates qN=(q1,…,qN and interaction energy U(qN),enclosed in a volume V at temperature T, the mean value of the total dipole moment in the direction of a homogeneous external field E0,

(26)⟨M⟩E0=∫dqNexp(−[U(qN)−M(qN)E0]/kBT)M(qN)∫dqNexp(−[U(qN)−M(qN)E0]/kBT)

to first order in E0 is given by

(27)⟨M⟩E0=E0⟨△M2⟩3kBT

In classical electrostatics,

(28)Extra close brace or missing open brace?

The polarization is the dipole moment per unit volume,so,

(29)ϵ−1=4π⟨△M2⟩3VkBT

(30)⟨△M2⟩=⟨M2⟩−⟨M⟩2

According to this definition of dielectric constant in molecular dynamics,it may provide the ionic and dipolar contributions to dielectric constant.

Results and Discussion

To compare this two methods,the dielectric constants of crystal polyethylene and ice are calculated.

Polyethylene has –CH2-CH2- as its repeating unit. It has an orthorhombic unit cell in which there are two chains and each chain has one –CH2-CH2- unit.16 Figure 1 shows the crystal structure of Polyethylene.For ice,here ice XI is considered due to its simple structure.Ice XI is an orthorhombic, low-temperature equilibrium form of hexagonal ice. It is ferroelectric. It is considered the most stable configuration of ice Ih.Study indicated that the temperature below which ice XI forms is −36 °C (240K).The crystal structure of ice XI is shown in Figure 2.

The calculated dielectric constants are shown in the table below.The molecular dynamics calculation was done at 65 K.

The experiment value of polyethylene dielectric constant is around 2, so DFPT result shows good agreement with experiment value.From the structure of polyethylene,it does not have spontaneous polarization,and thus the dipolar contribution is 0,which is in agreement with the result of MD for polyethylene.For ice XI,it has permanent dipoles along the z direction.This can be verified by the dielectric constant calculated from MD.In x and y direction,there are no spontaneous polarization and the dielectric constant are due to ionic contribution.Compare the result of ionic contribution from DFPT with that of MD,they are very close to each other.While for z direction,MD has larger value than DFPT,which results from the dipolar contribution.

Summary

Based on the theory and results of DFPT and MD,a brief comparison of these two methods can be made:

DFPT can give relatively accurate dielectric constants for both polar and nonpolar systems, while MD is suitable for polar system

MD can calculate the dielectric constants of amorphous and fluid systems, but for DFPT it is difficult.

DFPT gives the dielectric constant at 0 K, while MD can calculate dielectric constant at different temperatures.

They may be compensation to each other. For polar system: ϵtol= Electronic (DFPT) + Ionic (DFPT) + Dipolar (MD)

References

(1) C. Kittel, Introduction to Solid State Physics

(2) A. Messiah, Quantum Mechanics

(3) S. Baroni, Stefano de Gironcoli, and Andrea Dal Corso. Rev. Mod. Phys. 73 (2001)

(4) X. Gonze and C. Lee. Phys.Rev.B 55 10355-10368 (1996)

(5) X. Gonze. Phys.Rev.B 55 10337-10354 (1996)

(6) M. Neumann, Molecular Physics 50 841-858 (1983)

(7) D. J. Price and C. L. Brooks III. J.Chem.Phys 121 (2004)

(8) R. Howe and R. W. Whitworth. J.Chem.Phys 90 (1989)

http://electronicstructure.wikidot.com/chenchen

Hoping this will be helpful,

Rafik

Thank you very much.Is it possible to do DFPT calculations using Gaussian program? Can we calculate the complex permittivity?

I understood that the term polar molecule means a molecule that has a dipole moment. The polar nature of the molecule is due to the dipole moment of the constituent functional group. I used the phrase in that sense. The dipole moment derivative is required to investigate the nature of H-bonds (blue and red shifting).

Dear G. Arivazhagan,

Dipole moment is a measure of the functional group in the molecule. The sume of all the dipole moments in the group gives the term called dielectric constant.

To my knoweledge, there is no way to run DFPT in Gaussian.

For DFPT  calculations, see the following link:

http://pubs.rsc.org/en/content/articlehtml/2014/fd/c4fd00168k

For calculating complex permititivity, use the attached file.

Hoping this time my answer is to the point.

Rafik

Thank you Sir. I have downloaded the paper that you have recommended. If the sum of all the dipole moments in a functional group gives the term called dielectric constant, then the dipole moment should equal the dielectric constant. But they are found to assume different values for a given molecule. Why?

Simple answer is a single molecule does NOT have a dielectric constant, but it definitely possess a quantity called dipole moment. A dielectric constant is in contrast to the dipole moment a macroscopic quantity that starts to make sense as soon as you have an ensemble of molecules which also includes interaction between the molecules. Indeed liquid water is the best example as it is the hydrogen bonding between molecules that is responsible for the high dielectric constant in liquid water (and the rotation of the permanent dipole in an electric field of comparably low frequenccy). Ice on the other hand has a comparably low dielectric constant (as the rotation is frozen) - but it is the same molecule!

Hello Dr. G.Arivazhagan,

    In the frequency calculation with Gaussian, dipole derivatives are calculated to determine infrared intensity.  If you look at the last part of the output file of Gaussian, you can find the following session: in the case of H2O at the RHF/6-31G level,

GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad

 1|1|UNPC-E1400BBQ|Freq|RHF|6-31G|H2O1|FITO|18-Feb-2013|0||#N Geom=AllC

 heck Guess=TCheck SCRF=Check GenChk RHF/6-31G Freq||H2O test||0,1|O,-0

 .0188774341,-0.0244713386,0.|H,0.9290620083,-0.0810959382,0.|H,-0.3143

 476107,0.8780212779,0.||Version=IA32W-G09RevA.02|State=1-A'|HF=-75.985

 3592|RMSD=3.326e-010|RMSF=1.696e-006|ZeroPoint=0.0224877|Thermal=0.025

 3221|Dipole=0.6008698,0.778974,0.|DipoleDeriv=-0.5105623,0.0508084,0.,

If you would like to obtain full tables for the dipole derivatives, use the FORMCHK command to get formatted chekpoint file *fch from binary chekpoint file *.chk.  As far as I remember, dipole derivatives are punched in the *.fch file in the atomic unit with input orientation.

I hope this would be of some help for you. 

Thank you very much Sir. I will try to do that.