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Time-Dependent Density-Functional TheoryConcepts and Applications$

Carsten A. Ullrich

Print publication date: 2011

Print ISBN-13: 9780199563029

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199563029.001.0001

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(p.416) Appendix A Atomic units

(p.416) Appendix A Atomic units

Source:
Time-Dependent Density-Functional Theory
Publisher:
Oxford University Press

A.1 Atomic units in vacuum

In this book, we have consistently employed Hartree atomic units (a.u.). This convenient system of units is obtained by assigning a value of 1 to the free-electron mass and charge and to the reduced Planck's constant, i.e., m = e = ħ = 1.

Table A.1 lists various physical quantities, their fundamental atomic units, and the associated numerical values. Note that all references to the first Bohr orbit and the ionization energy of the hydrogen atom assume an infinite mass of the proton.

The numerical values in Table A.1 are expressed in the familiar SI units. Conversion factors to Gaussian units and more significant figures of the fundamental constants can be found in Drake (2006).

In SI units, the speed of light is given by

(A.1)
c = 1 0 μ 0 = 2.99792 × 10 8 m/s,
where 0 = 8.85419 × 10–12 C2/N m2 and μ 0 = 4π × 10–7 N/A2 are the permittivity and the permeability, respectively, of free space. Note that the fine structure constant α is a dimensionless quantity and is thus independent of the system of units adopted:
(A.2)
α = e 2 4 π 0 c = 1 137.036 .

Together with the Bohr radius a 0 = 1 a.u., this implies that in a.u. we have

(A.3)
c = α 1 = 137.036 ,

(A.4)
0 = 1 / 4 π ,

(A.5)
μ 0 = 4 π α 2 .

The atomic unit of the magnetic field can be defined in several different ways. Since we work in SI units, we choose here the definition in which one unit of magnetic field, B 0, causes a unit Lorentz force on an electron moving with unit velocity. The value of B 0 is then 2.35052 × 105 T.

An often-used alternative definition of the magnetic-field unit posits that the electric and magnetic fields in a plane wave have the same magnitude. This implies that Maxwell's equations in atomic units and in Gaussian units have the same appearance, and it causes the magnetic-field unit to be (1/c)B 0 = 1.715 × 103 T. (p.417)

Table A.1 Fundamental atomic units.

Quantity

Physical meaning

Unit

Value in SI units

Mass

Free-electron mass

m

9.10938 × 10–31 kg

Charge

Absolute value of free-electron charge

e

1.60218 × 10–19 C

Angular momentum

Reduced Planck's constant

ħ

1.05457 × 10–34 Js

Length

Bohr radius of H atom

a 0 = 4 π 0 2 m e 2

5.29177 × 10–11 m

Velocity

Electron speed in the first Bohr orbit

υ 0 = e 2 4 π 0

2.18769 × 106 m/s

Time

Time it takes an electron in the first Bohr orbit to travel one Bohr radius

τ 0 = a 0 υ 0

2.41888 × 10–17 s

Energy

Twice the ionization energy of the H atom

E H = e 2 4 π 0 a 0

4.35974 × 10–18 J

Electric field

Electric field in the first Bohr orbit

ε 0 = e 4 π 0 a 0 2

5.14221 × 1011 V/m

Intensity

Intensity of plane electromagnetic wave with electric-field amplitude ξ 0

I 0 = 1 2 0 μ 0 ε 0 2

3.50945 × 1020 W/m2

Magnetic field

Causes unit Lorentz force on electron with speed υ 0

B 0 = e a 0 2

2.35052 × 105 T

A.2 Atomic units in the effective-mass approximation

Next, we consider the case of charge carriers in semiconductors, which are treated in the effective-mass approximation (see Appendix K). Working with rescaled atomic units is convenient for describing the electronic structure and excitations in semiconductor nanostructures.

In the effective-mass approximation, the free-electron mass is replaced by an effective mass

(A.6)
m * = m r m ,
where the relative effective mass m r is a material-dependent, dimensionless number which accounts for the fact that the electrons do not move in free space but in a periodic crystal. Furthermore, the permittivity of free space is replaced by the static permittivity of the material, = r 0, where r is the dielectric constant. To account for the dielectric screening, we define the effective charge as
(A.7)
e * = e / r .

The magnetic permeability μ r of typical semiconductor materials, on the other hand, remains close to 1 and will be ignored in the following.

(p.418)

Table A.2 Atomic units for a semiconductor with effective mass m* and effective charge e*. The GaAs results use mr = 0.067 and r = 12.4.

Quantity

Unit

a.u.*/a.u. for GaAs

Mass

m* = mr m

0.067

Charge

e * = e / r

0.284

Angular momentum

ħ* = ħ

1

Length

a 0 * = 4 π 0 2 m * e * 2 = r m r a 0

185

Velocity

υ 0 * = e * 2 4 π 0 = υ 0 r

0.0833

Time

τ 0 * = a 0 * υ 0 * = r 2 m r τ 0

2.29 × 103

Energy

E H * = e * 2 4 π 0 a 0 * = m r r 2 E H

4.36 × 10–4

Electric field

ε 0 * = e * 4 π 0 r a 0 * 2 = m r 2 r 3 ε 0

2.35 × 10–6

Intensity

I 0 * = 1 2 r 0 μ 0 ε 0 * 2 = m r 4 r 11 / 2 I 0

19.5 × 10–12

Magnetic field

B 0 * = e * r a 0 * 2 = m r 2 r 2 B 0

1.03 × 10–4

The atomic units for such a semiconductor are given in Table A.2. We also give numerical results for the common semiconductor GaAs, where r = 12.4 and m r = 0.067 for conduction-band electrons. It is found that the effective Bohr radius a 0 * is larger than a 0 by about a factor of 200, and the effective Hartree energy E H * is smaller than E H by about a factor of 2000.

The reader may wonder why we use the static dielectric constant of the material in the definitions of effective atomic units. This seems appropriate if only static phenom-ena are to be studied. However, the situation is less clear if one wants to study carrier dynamics and excitations, which would require the use of the frequency-dependent di-electric function. To avoid inconsistencies, the regime of validity of the effective-mass approximation is restricted to frequencies much below the band gap energy, which is typically much less than 1 eV. The use of this approximation is therefore limited to studying dynamical phenomena on low-frequency scales where it is justified to work with the static dielectric constant.