The Scales of Time, Length, Mass, Energy, and Other Fundamental

ARTICLE pubs.acs.org/jchemeduc

The Scales of Time, Length, Mass, Energy, and Other Fundamental Physical Quantities in the Atomic World and the Use of Atomic Units in Quantum Mechanical Calculations Boon K. Teo*,† and Wai-Kee Li‡ †

Department of Chemistry, University of Illinois at Chicago, Chicago, Illinois 60607, United States Department of Chemistry, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, People’s Republic of China

‡

ABSTRACT: This article is divided into two parts. In the first part, the atomic unit (au) system is introduced and the scales of time, space (length), and speed, as well as those of mass and energy, in the atomic world are discussed. In the second part, the utility of atomic units in quantum mechanical and spectroscopic calculations is illustrated with examples. It is shown that the system of atomic units can greatly facilitate calculations and unit conversions. In addition, a number of topics that can be fruitfully treated with the au system, including photoelectric effect, hydrogen atom and different Rydberg constants, positronium atom, atomic mass unit, nuclear binding energy, nuclear reaction energetics, annihilation and massenergy interconversions, femtosecond and attosecond spectroscopies, quantum-size effects, excitons in semiconductors, and so forth are presented and discussed. KEYWORDS: First-Year Undergraduate/General, Second-Year Undergraduate, Upper-Division Undergraduate, Inorganic Chemistry, Physical Chemistry, Textbooks/Reference Books, Nomenclature/Units/Symbols, Quantum Chemistry, Spectroscopy

I

n the atomic or nano world, the scales of physical quantities such as time, length, mass, energy, charge, energy, and so forth are best measured in atomic units. The reason is that, in the microscopic world, the sizes and masses of particles are minute, and they are traveling at an exceedingly fast speed, at least in reference to the macroscopic world. Thus, the time unit of a second in our world is an eternity in the nano realm and the speed limit of 100 km/h for automobiles is but a crawl in the atomic world! There is another compelling reason for adopting the atomic unit system. In chemical, physical, or quantum mechanical calculations, one often encounters unit conversion that could be tedious, time-consuming, or even confusing. In studying the Schr€odinger equation for, say, a hydrogenic atom, one is often confronted with dreary unit conversions involving the mass (me) and charge (e) of an electron, Planck’s constant (h), and the socalled vacuum permittivity ε0. The last-mentioned, now commonly called the electric constant, is needed to describe the electrostatic interaction between the negatively charged electron and the positively charged nucleus. The unit-conversion problem in quantum calculations, as well as in many chemical, physical, or biological calculations, can be alleviated by using the atomic units.1 In fact, all quantum mechanical calculations can be greatly simplified by adopting the atomic unit (au) system. Unfortunately, none of the general chemistry textbooks surveyed2
18 includes a discussion on the atomic units. Neither do most of the inorganic chemistry textbooks surveyed;19
23 only one textbook23 briefly mentioned the au system in the form of a short table. The purpose of this article is to fill this gap, with the hope that this simple system, well-known to the physicists, will be Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.

incorporated into undergraduate chemistry textbooks. In the au system, one measures physical quantities such as the mass, charge, and length in units of the electronic mass (me) and charge (e), and the radius (a0) of the first orbit (1s), of a hydrogen atom, respectively. The angular momentum is measured in units of reduced Planck’s constant p (which is Planck’s constant h divided by 2π). All other physical quantities can be based on, or derived from, these fundamental units. In this way, the often tedious and cumbersome unit conversions can be circumvented by setting these constants to unity (in atomic units). Furthermore, quantum-mechanical expressions can be greatly simplified as well by adopting the au system.24,25 For example, in au, the kinetic energy operator for an electron in Schr€odinger equation,
(p2/2me)r2, becomes simply
1/2r2. This kind of simplification expedites calculations and eliminates tedious unit conversions. This article is divided into two parts. In the first part, we introduce the au system and discuss the scales of time, space (length), and speed, as well as those of mass and energy, in the atomic world. In the second part, we illustrate the utilities of atomic units in quantum-mechanical calculations by worked examples. In particular, a wide range of interesting topics such as photoelectric effect, hydrogen atom and different Rydberg constants, positronium atom, atomic mass unit, nuclear binding energy, nuclear reaction energetics, annihilation and mass-energy interconversions, femtosecond and attosecond spectroscopies, quantum-size effects, excitons in semiconductors, and so forth are presented and Published: April 25, 2011 921

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Table 1. Fundamental Physical Units in Quantum Mechanicsa,b No.

Physical Unit

SI Unitc

Symbol

Atomic Unit

1

electron rest mass

me

9.1093826(16)  10
31 kg

me (au) = 1

2

electron charge

|e|

1.60217653(14)  10
19 C

|e| (au) = 1

3

reduced Planck’s constant

p = h/2π

1.05457168(18)  10
34 J s

p (au) = 1

4

Coulomb’s constant

ke = 1/4πε0

8.9875516  109 J m/C2

ke (au) = 1

5

Bohr radius

a0 = (4πε0)p2/mee2

5.291772108(18)  10
11 m

a0 (au) = 1

= p/mecR

= 0.52917721 Å

6

d

Hartree energy

4.35974417(75)  10
18 J

Eh 4

2

= mee /p (4πε0) = mec2R2

2

= 27.21138386(68) eV

Eh (au) = 1

7

speed of light

c

2.99792458  108 m/s

c (au) = 137.035999068(96)

8

fine-structure constante

R = e2/pc(4πε0)

7.297352570(5)  10
3

R =1/c (au) =

= e2cμ0/4πp

(dimensionless)

7.297352570(5)  10
3

= (4πε0)2p3/mee4 = p/mec2R2

2.418884326505(16)  10
17 s

t (au) = 1

v = a0/t

2.1876912633(73)  106 m/s

v (au) = 1

f (au) = 1

≈ 137.036

= (ε0μ0)1/2 9

10

t = p/Eh

time

velocity

= e2/p(4πε0) = cR 11

force

f = Eh/a0

8.2387225(14)  10
8 N

12

current

j = e Eh/p

6.62361782(57)  10
3 A

j (au) = 1

13

Bohr magnetond

μB = ep /2me

9.27400915(23)  10
24 J/T

μB (au) = 1/2

T = Eh/kB

(or A m2 or C m2/s) 3.1577464(55)  105 K

T (au) = 1

p = Eh/a03

2.9421912(19)  1013 N/m2

p (au) = 1

14

temperature

15

pressure

f

Any four of these fundamental units can be used to fix or derive other physical quantities in quantum mechanics. b Abbreviations of the SI units are kg, kilogram; m, meter; s, second; C, Coulomb; J, Joule; N, Newton; K, Kelvin; A, Ampere; T, Tesla. c International System of Units (see refs 26 and 27). d ε0 is the electric constant or the so-called vacuum permittivity. ε0 = 8.85419  10
12 C2 J
1 m
1. In atomic units, ε0 = 1/4π. e μ0 is the magnetic constant or the so-called vacuum permeability. μ0 = 4π  10
7 N/A2 = 4π  10
7 H/m. In atomic units, μ0 = 4π/c2 = 4πR2. f kB is Boltzmann’s constant: kB = 1.3806510
23 J/K = 8.617310
5 eV/K. a

discussed. We will also derive a few commonly used conversion factors in the process.

atomic unit of energy. In such cases, all quantities related to Eh would need to be modified accordingly. Thus, in atomic units (au), we have

’ ATOMIC UNITS The au is a system of “dimensionless” units wherein the mass, length, charge, momentum, and so forth are measured with respect to that of an electron in the ground state of a hydrogen atom. Thus, the rest mass of an electron, namely, me = 9.1093826(16)  10
31 kg is set to unity, that is, me = 1 au. Similarly, the charge of an electron, |e| = 1.602 17653(14)  10
19 C is 1 au of electrical charge; in other words, |e| = 1 au. Next, the reduced Planck’s constant p = (6.62606896(33)  10
34 J s)/2π = 1.05457168(18)  10
34 J s is 1 au of angular momentum. Also, 1 au of length is the radius of the n = 1 orbit of hydrogen, or the so-called Bohr radius, which is given by a0 = 0.529177 Å = 5.29177  10
11 m = 1 au. This is, in effect, the length scale in the atomic world. Finally, the atomic unit of energy, or hartree, is set equal to twice the ground-state energy (absolute value) of the hydrogen atom: Eh = 2  13.60569 eV = 27.21138 eV = 4.35974417(75)  10
18 J = 1 hartree = 1 au. It should be noted that the subscript h in Eh stands for the unit hartree, not the atomic species hydrogen. Also, some books use the energy (absolute value) of the 1s orbital of hydrogen, or, 13.6057 eV, as the

me ðauÞ ¼ e ðauÞ ¼ p ðauÞ ¼ a0 ðauÞ ¼ Eh ðauÞ ¼ 1 au

ð1Þ

Other physical quantities can be derived from these fundamental constants. A number of important physical constants, in both atomic units (last column) and SI units (International System of Units,26 fourth column), are listed in Table 1. Also included in Table 1 are the expressions of these fundamental physical constants (third column) and the inter-relationships between them for easy comparison and reference. For example, from general chemistry textbooks, we have a0 ¼ ð4πε0 Þp2 =me e2

ð2Þ

Eh ¼ me e4 =p2 ð4πε0 Þ2

ð3Þ

Here ε0 = 8.85419  10
12 C2 J
1 m
1 is the so-called vacuum permittivity. In au, 4πε0 = 1.11265  10
10 C2 J
1 m
1 is also set to unity so that both a0 and Eh are equal to 1 au, because all other terms in eqs 2 and 3 have already been set to unity by definition. In other words, 4πε0 can be ignored in subsequent calculations provided that they are done in atomic 922

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units, 4πε0 ðauÞ ¼ 1 au

The Speed of Light in Atomic Unit: The Speed Limit of Matter

ð4Þ

The speed of light in vacuum, c, is 2.99792458  108 m/s. To convert this to c in au, we make use of the atomic unit of speed described in the previous section:

It can be seen from Table 1 that we can derive all the physical quantities from the set of four fundamental constants, namely, me, e, p, and ε0. In fact, any four of fundamental physical constants listed in Table 1 can be used to fix or derive all other quantities in quantum mechanics. For example, we may as well start with the set of m e, e, p, and a0 , or the set of me, e, p, and Eh, or the set of m e, Eh, p, and a0, and so on. One final note: although the time, length, and mass units in the au system are extremely small, the energy scale is in the range of electronic transitions or chemical bonding interactions, making these units particularly useful in quantum chemical calculations. At the same time, the au system is also commonly used in highenergy and nuclear chemistry and physics, as we shall discuss later with illustrative examples.

c ðauÞ ¼ ð2:99792458  108 m=sÞ= ð2:18769  106 m=sÞ  137:036

According to Einstein’s general relativity theory, this is the maximum speed any object, or matter, can travel. Thus, the speed limit of our universe is 137 au! For comparison, the speed limit of an automobile is on the order of 102 km/h, or 1.27  10
5 au, smaller than c by 7 orders of magnitude! The Fine-Structure Constant

We shall now introduce one of the most important physical constants of all: the fine-structure constant, R, which is the coupling constant characterizing the strength of the electromagnetic interaction. In terms of the fine-structure constant, many of the expressions listed in Table 1 can be greatly simplified. It can be shown that R is given by

The Time Scale in the Atomic World: The Atomic Unit of Time

We live in the macroscopic world where events may last for seconds, minutes, hours, or days. So we choose a second to be the basic time unit, which is one of the seven base units in the SI system.26 What then is the time scale in the atomic world? It turns out that the basic unit of time in au can be derived from Heisenberg’s equation: tðauÞ

R ¼ e2 =pcð4πε0 Þ ¼ e2 cμ0 =4πp ¼ ðε0 μ0 Þ1=2

This is one unit of time in the atomic world! The Scale of Dimension in the Atomic World: The Atomic Unit of Length

In the macroscopic world, we measure our three-dimensional space by meters or fractions thereof. Therefore, meter is also one of the seven base units in the SI system.26 In astronomy, the vast space of the universe is measured in light years, which is the distance traveled by light in one year. What then is the length scale in the atomic world? The answer is, as stated earlier, the Bohr radius a0 = 0.529177 Å = 5.29177  10
11 m = 1 au. Thus, the length scale of the atomic world is very tiny indeed. It is even smaller than 1 nm (10
9 m), the base unit of dimension in nanotechnology, by 20-fold!

RðauÞ ¼ 1=c ðauÞ

ð9Þ

This is an elegant expression relating two extremely important fundamental physical constants. Finally, it is noted that, historically, the fine-structure constant was first introduced by Arnold Sommerfeld in 1916, ten years before the development of quantum mechanics, in connection with his work on the atomic hydrogen spectral line deviation between the Bohr results and the experimental data. Such deviation arises from the neglect of relativistic correction in the Bohr model.

The Speed Unit in the Atomic World

The atomic unit of speed or velocity can be obtained by dividing 1 au of distance, or a0, by 1 au of time (t). Thus,

The Unitless Fine-Structure Constant and Its Properties

One important characteristic of the fine-structure constant, R, is that it is a dimensionless (i.e., unitless) constant. This unique property can easily be shown as follows (note that Coulomb’s constant ke = 1/4πε0 per entry 4 of Table 1):

ð6Þ

That is, v (au) = 2.18769  106 m/s. This is, in effect, the speed unit in the atomic world!

R ¼ ¼ ¼

ð8Þ

Here μ0 is the magnetic constant or the so-called vacuum permeability: μ0 = 4π  10
7 N/A2 = 4π  10
7 H/m (see Table 1, footnotes b and e). In atomic units, μ0 = 4πR2. The other constants have been defined earlier. It follows that, in terms of R and c, the Bohr radius and Hartree energy become a0 = p/mecR and Eh = mec2R2, respectively. Other physical quantities can likewise be simplified. For example, the atomic unit of velocity can be shown to be v = cR (see Table 1, entries 8 and 10). It is interesting to note that, in atomic units, the fine-structure constant, R, is related to the speed of light (in vacuum) in an exceedingly simple manner:

¼ p=Eh ¼ ð1:05457  10
34 J sÞ=4:35974  10
18 J ð5Þ ¼ 2:41888  10
17 s

vðauÞ ¼ a0 ðauÞ=t ðauÞ ¼ 5:29177  10
11 m=2:41888  10
17 s

ð7Þ

e2 =pcð4πε0 Þ ¼ e2 ke =pc ½ð1:602176  10
19 CÞ2  ð8:9875516  109 J m=C2 Þ=ð1:05457  10
34 J s  2:99792  108 m=sÞ 7:29735  10
3

Note that all the units cancel themselves out, and R is thus dimensionless!

We shall now describe a few interesting properties of the finestructure constant R. First, it is the ratio of one atomic unit of 923

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velocity to the speed of light in vacuum (by combining entries 8 and 10 of Table 1): R ¼ v=c

ð10Þ

In atomic units, this reduces to R = 1/137.036 au = 7.29735  10
3 (per eq 9). The fine-structure constant R is also related to the square root of the mass to energy ratio of any particle, as per Einstein’s mass-to-energy conversion equation: E ¼ mc2

ð11Þ

In other words, in atomic units, R ¼ 1=c ¼ ðm=EÞ1=2

’ USING THE ATOMIC UNITS IN QUANTUM MECHANICAL CALCULATIONS: ILLUSTRATIVE EXAMPLES In using the atomic units in quantum mechanical calculations, one sets me = e = p = a0 = 1 au, which is equivalent to simply ignoring these quantities in the quantum mechanical expressions. It greatly simplifies the often tedious or cumbersome unit conversions in the calculations. The final results, in au, can be converted back to the conventional units by multiplying the answer with the appropriate conversion factor. For example, if the result is a length or a distance, it needs to be multiplied by a0 = 0.52918 Å to convert it back to the Å unit, or by a0 = 5.2918  10
11 m to convert to meters. We shall illustrate the use of atomic units in some calculations commonly encountered in chemistry. Photon Energy and Wavelength of Light: The Conversion

In spectroscopy, it is often necessary to convert the wavelength of light to photon energy and vice versa. The commonly used equation for such conversion is

For instance, for an electron, me = 1 au, E = 0.510997 MeV R

¼ ðm=EÞ1=2 ¼ ½1=ð0:510997 MeV=27:2114 eVÞ1=2 ¼ 7:29735  10
3

E ðeVÞ ¼ 1239:8378=λ ðnmÞ

ð12Þ

For example, a wavelength of 500 nm is equivalent to 2.48 eV. We can derive this conversion equation by combining the particle (photon energy E = hν) and wave (c = νλ) nature of light (i.e., the particle-wave duality):

The same holds for any other particles, be it protons, neutrons, and so forth. Finally, we note that, in atomic units, the combination of ε0 = 1/4π (per 4πε0 = 1) and μ0 = 4πR2 (per μ0 = 4π/c2) gives rise to yet another elegant expression of the relationship between the electric constant ε0 (vacuum permittivity) and magnetic constant μ0 (vacuum permeability) listed in Table 1 (entry 8): R ¼ ðε0 μ0 Þ

1=2

ð14Þ

E ¼ hv ¼ hc=λ ¼ 2πpc=λ

ð15Þ

If we now set p = 1 au and c = 137.036 au and convert nm to au (note that 1 au = 0.0529177 nm), we have E ðeVÞ ¼ ½ð2π  137:036Þ=λ ðnmÞ=0:0529177  27:21138 ¼ 1239:8378=λ ðnmÞ

ð13Þ

It should be emphasized that eqs 14 and 15 apply only to light in vacuum (or air, in good approximation).

Equation 13 can also be derived from the fact that the speed of light, c, in vacuum, is related to ε0 and μ0 by c = 1/(ε0μ0)1/2 (as shown by Maxwell’s laws) along with the relationship R = 1/c (au) in atomic units.

Einstein’s Photoelectric Effect

We can also use the atomic units to calculate the maximum velocity of the outgoing electron ejected from a metal surface when struck by a photon with energy E = hν greater than the threshold energy, the so-called work function, W, of the surface. This is Einstein’s photoelectric effect (for which he was awarded the Nobel Prize in Physics in 1921). The maximum velocity vmax of the outgoing electron is given by

Invariance

It should be noted that one of the advantages in using atomic units in quantum mechanical calculations is that the results are invariant to the exact (i.e., absolute) values of these fundamental physical constants (such as me, e, p, a0, etc.). Hence, even if these physical constants are redefined, revised, or their accuracy somehow improved over the years, the results obtained by using the au system will not be affected. One simply substitutes the new constants in place of the old ones in the final result and there is no need to redo the entire calculation.

1 ðme vmax 2 Þ ¼ E
W ¼ hv
W 2

ð16Þ

For example, when a silver surface, with W = 4.73 eV, is struck with a 150 nm ultraviolet light, the maximum velocity of the ejected electrons is

Scaling

Furthermore, the exact values of certain variables or parameters can be substituted, or modified, at the end of the calculation instead of carrying it through the calculation. For instance, the electron mass me can be replaced by reduced mass μ in the final results. This is particularly useful in dealing with systems of different types of masses such as reduced masses or effective masses of electrons or holes of excitons in semiconductors (vide infra). The actual masses can be substituted for the electron mass in the final results at the conclusion of the calculation. We shall illustrate this scaling property of atomic units in our treatments of a positronium atom and excitons in semiconductors later.

vmax ¼ ½2ðE
WÞ=me 1=2 By converting everything into au, we can easily obtain vmax = 1.12  106 m/s. The maximum kinetic energy of the ejected electrons is thus 1/2(mevmax2) = 3.54 eV, or 5.67  10
19 J. The Hydrogenic Atom

The binding energy of a hydrogenic atom with a nuclear charge Z is given by En ¼
RðZ2 =n2 Þ 924

ð17Þ

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where R is the so-called Rydberg constant given by R ¼ me e4 =2p2 ð4πε0 Þ2 ¼

1 au 2

to obtain Rν = R/h, divide both sides of eq 21a by h: ν ¼
Rν ðnf
2
ni
2 Þ

ð18Þ

Here Rν = 3.289841  10 s is the frequency expression of the Rydberg constant. Finally, to obtain Rw = R/hc, divide both sides of eq 21a by hc:

The simplification in eq 18 using the au system is made possible by the combination of the fact that R = 13.60569 eV = 2.179872  10
18 J (calculated via eq 18 using conventional units) and that the atomic unit of energy, or hartree (Eh), is set equal to twice the ground-state energy (absolute value) of the hydrogen atom (i.e., Eh = 27.21138 eV = 1 au), vide supra. Thus, in atomic units, eq 17 becomes simply En ¼
Z2 =2n2 au ¼
ðZ2 =2n2 Þ  27:21138 eV

1=λ ¼
Rw ðnf
2
ni
2 Þ

¼ ¼

ð19Þ

R ¼

ðn2 =ZÞa0 ¼ ðn2 =ZÞ au ðn2 =ZÞ  0:529177 Å

¼ ðn2 =ZÞ  5:29177  10
11 m

ð20Þ

We can use eq 19 to calculate the energy of the first emission line, from ni = 2 to nf = 1, of the Lyman series of hydrogen (Z = 1), as follows: ΔE = Ef
Ei = [
(1/2  12) þ (1/2  22)] =
3/8 au =
10.2043 eV. This photon energy can be converted, via eq 14, to a wavelength of λ (nm) = 121.50 nm. This is the emission line with the lowest energy, or the longest wavelength, in the Lyman series. This emission line of hydrogen, the most abundant matter in the universe, is of central significance in optical astronomy.

ð24Þ

Rν ¼ R=h ¼ ð1=4πÞ au
1

ð25Þ

Rw ¼ R=hc ¼ ð1=4πcÞ au
1

ð26Þ

Unit Conversions

Unit conversion is an important process of any scientific calculation. It ensures the physical or chemical significance of the parameters or variables involved as well as the results obtained. In the final analysis, if the units are incorrect, the results must be in error. As an example, undergraduate students often find the conversion factors 1 eV = 1.60218  10
19 J = 96.485 kJ/mol mysterious and difficult to comprehend, not realizing that the first conversion is between quantities for single particles (electrons, atoms, etc.), whereas the second conversion requires the multiplication by Avogadro’s number NA = 6.02214  1023. In other words, 1.60218  10
19 J is the energy required to drag one electron across an electric field of 1 V, whereas 96.485 kJ/mol is the one mole equivalent of the energy (i.e., must be multiplied by the Avogadro’s number): 1.60218  10
19 J  6.02214  1023/mol = 96.485 kJ/mol. Even more bewildering to the students is the unit called wavenumber, cm
1, commonly used in vibrational spectroscopy; the conversion factor here is 1 eV = 8065.532 cm
1. This conversion factor can be obtained by taking the ratio of the corresponding Rydberg constants, Rw/R, discussed earlier, as follows:

In fact, the emission spectrum of atomic hydrogen consists of a number of series, one in the ultraviolet region, called Lyman series, one in the visible region, called the Balmer series, and several others in the infrared region, including the Paschen, Brackett series, and so forth. We now know that the Lyman, Balmer, Paschen, and Brackett series correspond to transitions from higher energy levels with ni to a lower energy level with nf = 1, 2, 3, and 4, respectively. The photon energies, Ephoton, or equivalently the frequencies, ν, or the wavelengths, λ, of these characteristic emission lines can be correlated by eqs 21, 22, and 23, respectively, to be described below. These equivalent equations have different but interrelated Rydberg constants R, Rν, and Rw, which are the energy, frequency, and wavenumber expressions, respectively. (Note that wavenumber is simply the inverse of wavelength λ.) We can use the atomic unit system to simplify the derivation, as well as the interconversion, of these Rydberg constants. The energy of an emitted photon that occurs in an electronic transition of the hydrogen atom (Z = 1) when the electron falls from a higher (with an initial quantum number ni) to a lower (with a final quantum number nf) level (i.e., ni > nf) is given by the energy difference of the two electronic states which can be calculated by using eq 17 (Z = 1 for hydrogen):

Rw =R ¼ 1:0973721  105 cm
1 =13:6057 eV ¼ 8065:532 cm
1 =eV In other words, the wavenumber equivalent of 1 eV is 8065.532 cm
1. The Positronium Atom

ð21Þ

The positronium atom28 is an “exotic atom” that resembles a hydrogen atom but with a positron, instead of a proton, as its nucleus. In other words, it has a positively charged positron at the center with the negatively charged electron revolving around it (or vice versa since the two have the same mass). A positron is the antimatter counterpart of electron: the two are identical in all aspects (mass, spin, etc.) but with opposite charges. For a

Here R = 13.60569 eV or 2.179872  10
18 J is the energy expression of the Rydberg constant. By making use of the relationships ΔE = hν and ν = c/λ, we can rewrite eq 21 as hν ¼ hc=λ ¼
Rðnf
2
ni
2 Þ

1 au 2

These constants can be converted back to the conventional units by the following conversion factors: 1 au (energy) = 27.21138 eV for R, 1 au (time) = 2.418884  10
17 s for Rν, and 1 au (length) = 5.29177  10
9 cm for R w. Note that for Rν and Rw, the conversion factor appears in the denominator.

The Rydberg Constants

Ephoton ¼ ΔE ¼ Ef
Ei ¼
Rðnf
2
ni
2 Þ

ð23Þ

Here Rw = 1.09737  105 cm
1 is the wavenumber expression of the Rydberg constant. In terms of au system, the three Rydberg constants can be greatly simplified to:

Similarly, the radius of the nth Bohr orbit can be simplified to rn

ð22Þ

15
1

ð21aÞ 925

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Again here NA is Avogadro’s constant. To convert amu into au, we divide 1 amu in kg by the mass of an electron 1 amu ¼ 1:6605387  10
27 =9:1093826  10
31 au ¼ 1822:888 au The mass of 1 amu can be converted to energy via Einstein’s mass
energy equation: E ¼ mc2 ¼ 1822:888  ð137:036Þ2 au ¼ 931:491 MeV

ð31Þ

Thus, the mass of 1 amu can be converted to a tremendous energy of 931.491 MeV. Nuclear Binding Energies: Mass
Energy Interconversions

We can now use the result derived in the previous subsection to calculate nuclear binding energy of a nucleus or the energy released from a nuclear reaction. Take deuterium, an isotope of hydrogen, as an example. The nuclear binding energy of a deuterium nucleus (which has a proton and a neutron) can be calculated as follows (given the masses of deuterium, mD, proton, mp, and neutron, mn): Δm

Figure 1. A comparison of the energy levels (E) and the radii (r) of the first three orbits (n = 1, 2, and 3) of the hydrogen atom and the positronium atom.

By making use of eq 31, we obtain the total nuclear binding energy of

hydrogen atom, the reduced mass μ is for all practical purposes the same as me, since a proton is nearly 2000 times heavier than an electron. The binding energy and the radius of the electron in the nth shell are given by eqs 19 and 20, respectively, with Z = 1. For the positronium atom, the reduced mass is μ ¼ me meþ =ðme þ meþÞ

ΔE

Nuclear Reactions: Energy Matters

Likewise, the energy released or absorbed during a nuclear reaction can be calculated easily with the au system. For instance, the energy released during the deuterium (2H)
tritium (3H) fusion reaction (eq 32) can be calculated as follows:

ð28Þ

2

In other words, the binding energy in a positronium atom is half of that in a hydrogen atom. By the same token, substituting μ in place of me in a0 (eq 2) gives rise to the radius of the nth orbit of a positronium atom (from eq 20): rn ¼ ðn2 =μÞ au ¼ ð2n2 Þ au ¼ ð2n2 Þ  0:52918 Å

Δm ¼ ¼ ¼ ΔE ¼ ¼

ð29Þ

In other words, the radii of the orbits (orbitals) in a positronium atom are twice as large as those of a hydrogen atom. The energy levels and the radii of the first three orbits of the hydrogen atom (left) and the positronium atom (right) are compared in Figure 1.

H þ 3 H f 4 He þ n

ð32Þ

mHe þ mn
mD
mT 4:00150 þ 1:00866
2:01355
3:01550
0:01889 amu Δm  931:491 MeV
17:59586 MeV

Here mHe, mD, mT are the masses of the nuclei of helium, deuterium (D), and tritium (T), respectively. Thus, the energy released for each nuclear fusion reaction (eq 32) during a hydrogen bomb explosion is 17.59586 MeV, a tremendous amount of energy! Of course an activation barrier needs to be overcome before the nuclear chain reaction can happen.

Atomic Mass Unit (amu) vs Atomic Unit (au)

In atomic units, the mass m of a particle is expressed in terms of the multiples of the rest mass of an electron, me. This should not be confused with the atomic mass unit (amu) commonly used in chemistry. In chemistry, the amu unit is defined as 1 amu ¼ 1 g=NA ¼ 1 g=6:022142  1023 ¼ 1:6605387  10
27 kg

¼ Δm  931:491 MeV ¼
2:22626 MeV

for deuterium. Or, half of this value, 2.22626 MeV/2 = 1.11313 MeV is the binding energy per nucleon for deuterium.

ð27Þ

Here me and meþ are the masses of electron and positron, respectively. Since me = meþ = 1 au, we have μ = 1/2 au. By substituting μ in place of me in eq 18, eq 19 becomes En ¼
μ=2n2 au ¼
1=4n2 au ¼
ð1=4n2 Þ  27:21138 eV

¼ ðmD
mp
mn Þ ¼ 2:01355
1:00728
1:00866 ¼
0:00239 amu

Annihilation of Matter and Antimatter: Matter Matters

There are two kinds of matters in nature: matter and antimatter. A particle and its antimatter counterpart (called antiparticle) have the same mass but opposite charges and they are otherwise identical in all other aspects. When a particle meets its

ð30Þ 926

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antiparticle, they annihilate each other and two photons of tremendous energy are released. We can also use Einstein’s E = mc2 equation to calculate the quantity of energy released. Once again, the au system greatly simplifies the calculations. In fact, in high-energy physics, as well as in astrophysics and nuclear chemistry, the masses of particles are often expressed in terms of their energies as per Einstein’s mass
energy equation. Thus, for an electron, me = 1 au, E ¼ mc2 ¼ 1  ð137:036Þ2 au ¼ 0:510997 MeV

fundamentally changed our view of chemical reactions; now we can actually “see” them like a time-lapsed “movie!” It is of interest to note that Svante Arrhenius received the Nobel Prize in Chemistry in 1903, for proposing the famous equation that bears his name, which relates reaction rate with temperature. At that time, Arrhenius considered the reactants macroscopically (i.e., not on the molecular level) and he assumed that reactions take a relatively long time to proceed to completion. Nowadays chemists can study detailed mechanisms and dynamics of reactions using femto- and attosecond spectroscopies.

ð33Þ

Quantum-Size Effects: Size Matters

Therefore, the mass of an electron may be expressed as 0.510997 MeV/c2. For a proton, m = 1836.15me = 1836.15 au, we have, E ¼ mc2 ¼ 1836:15  ð137:036Þ2 au ¼ 938:272 MeV

Quantum-size or quantum confinement effects refer to the changes of properties of a material when its size is reduced to below a characteristic value in the nanometer regime. The resulting properties of these nanomaterials (nanodots 0D, nanowires 1D, or quantum wells 2D) are significantly different from that of the bulk and are often “tailorable” and tunable with size. Take silicon as an example. It is a semiconductor with a bulk band gap of 1.11 eV. In the quantum-size regime, the band gap of silicon nanowires or nanodots can be continuously tuned from 1.1 to 3.5 eV as the size of the nanowires or nanodots decreases from 7 to 1.3 nm.30 We shall use silicon as an example to show that size matters in the atomic, as well as the nano, world! In a semiconductor, the filled valence band (VB) is separated from the energetically higher conduction band (CB) by a band gap. When an electron is excited from the VB to the CB by, say, absorption of a photon (photoexcitation), a hole is created in the VB. The electron in the CB is attracted to the localized, positively charged hole in the VB by the Coulombic electrostatic force, thereby forming an electron
hole pair called exciton. The exciton can be treated as a quasi hydrogen atom with energies and radii different from material to material. In fact, we can use the equations developed above for hydrogen atom, properly scaled with the dielectric constant and the effective masses of electron and hole, to estimate the energies and Bohr radius of the exciton of a semiconductor. Referring to eqs 2 and 20 (for n = 1 and Z = 1), the excitonic Bohr radius of a semiconductor is given by

ð34Þ

The mass of a proton can thus be expressed as 938.272 MeV/c2. Therefore, the annihilation of an electron and a positron produces two high-energy photons, each with the energy of 0.510997 MeV, as per eq 33. By the same token, the annihilation of a proton and an antiproton (the antimatter of proton with the same mass but opposite charge) also produces two high-energy photons, each with the energy of 938.272 MeV, as calculated by eq 34. Femtosecond and Attosecond Spectroscopies: Time Matters

In the previous sections, we discussed the interconversion of mass and energy via Einstein’s E = mc2 equation. In this section, we shall consider the importance of time scale in the studies of the kinetics, dynamics, and mechanisms of chemical reactions. As described earlier, one unit of time in the atomic world is 2.41888  10
17 s, somewhere between a femtosecond (10
15 s) and an attosecond (10
18 s). At the first glance, one unit of time in the atomic world is such a fleeting moment that it is not likely to be useful, even in the laboratory. But this is definitely untrue. In laser technology, now about 50 years of age, scientists can generate pulses with durations of just a few femtoseconds. Although this time interval is 41 times longer than one au of time, it has already found many applications in chemistry and the allied sciences. Using these ultrafast pulses as “cameras”, chemists can now study (and actually observe) very swift processes, including the progression of a reaction, the existence of an intermediate or a transition state, and so forth. More recently, pulses in the attosecond (10
18 s) range, which is 24 times faster than one au of time, have become available for the study of even faster reactions or processes. This should open up new frontiers in science and technology. In 1999, Ahmed H. Zewail was awarded the Nobel Prize in Chemistry for his pioneering work in the field of femtochemistry. Examples of the experiments carried out by Zewail and his coworkers29 are briefly described here. In one of their first experiments, Zewail’s group studied the dissociation of iodocyanide: ICN f I þ CN. They were able to confirm the existence of a transition state at which the I
C bond was about to break and the whole reaction took about 200 fs to complete. In another experiment, Zewail’s team studied the reaction H þ CO2 f CO þ OH, a reaction that is known to occur in the atmosphere and in combustion. Their results showed that the reaction proceeds through a transition state (HOCO) that lasts about 1,000 fs (i.e., 1 ps). Femtochemistry has thus

aB ¼ ðε=μÞa0

ð35Þ

where a0 = 0.529 Å = 1 au is the Bohr radius of hydrogen atom, μ is the reduced effective mass, and ε is the dielectric constant of the semiconductor. Here μ ¼ me mh =ðme  þ mh Þ ð36Þ where me* and mh* are the effective masses of the electron and the hole, respectively (in a way similar to eq 27). For Si, ε = 11.8, me* = 0.97, mh* = 0.16, μ = 0.137 (via eq 36), aB can be calculated as follows: aB = (11.8/0.137)a0 = 86.1 au = 45.6 Å = 4.56 nm. Similarly, the binding energy of the ground-state exciton of a semiconductor can be scaled (per eqs 17
19 for n = 1 and Z = 1) as EB ¼
ðμ=2ε2 ÞEh

ð37Þ

where Eh = 27.21 eV = 1 au. For Si, ε = 11.8, μ = 0.137 (per eq 36), EB can be calculated as EB =
(0.137/2(11.8)2) au =
4.91  10
4 au =
0.0134 eV. Once again, the use of au system greatly simplifies the calculation. Notice that the Bohr radius of the exciton is quite large and the energy rather small in comparison with the corresponding values of hydrogen atom. This is due to the screening of the electrostatic attraction by other electrons in the semiconductor (dielectric 927

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constant) and the small effective masses of the electron and the hole. Furthermore, when the size of the nanomaterial is reduced to a value less than, or on the order of, the excitonic Bohr radius, it exhibits quantum-size or quantum confinement effects. For Si, this occurs at around 5 nm. Below this size, the band gap widens (vide supra) so that when the electron and the hole eventually recombine to emit a photon (the so-called photoluminescence), the wavelength of the emitted light is blue-shifted (i.e., gets shorter in wavelength or of higher energy) as the size gets smaller. Indeed, the color of the photoluminescence of “porous silicon” (which contains both silicon nanowires and nanodots) changes from red to yellow to green to blue as the nanowires and nanodots get smaller in this quantum size range.

(10) Tro, N. J. Chemistry: A Molecular Approach; Prentice-Hall: Upper Saddle River, NJ, 2008. (11) Whitten, K. W.; Davis, R. E.; Peck, L. M.; Stanley, G. G. General Chemistry, 7th ed.; Thompson Brooks/Cole: Belmont, CA, 2007. (12) Zumdahl, S. S.; Zumdahl, S. A. Chemistry, 7th ed.; Houghton Mifflin: Boston, MA, 2007. 13. (13) Petrucci, R. H.; Harwood, W. S.: Herring, F. G. General Chemistry: Principles and Modern Applications, 9th ed.; Prentice-Hall: Upper Saddle River, NJ, 2007. (14) Averill, B. A.; Eldredge, P. A. Chemistry: Principles, Patterns, and Applications; Benjamin Cummings: San Francisco, CA, 2006. (15) Spencer, J. N.; Bodner, G. M.; Rickard, L. H. Chemistry: Structure and Dynamics, 3rd ed.; Wiley: Hoboken, NJ, 2006. (16) Chang, R. General Chemistry: The Essential Concepts, 4th ed.; McGraw-Hill: Boston, MA, 2006. (17) Oxtoby, D. W.; Freeman, W. A.; Block, T. F. Chemistry: Science of Change, 4th ed.; Thompson Brooks/Cole: Pacific Grove, CA, 2003. (18) Radel, S. R.; Navidi, M. H. Chemistry, 2nd ed.,Kendall/Hunt: Dubuque, IA, 2001. (19) Shriver, D. F.; Atkins, P. Inorganic Chemistry, 4rd ed.; Freeman: New York, 2006. (20) Housecroft, C. E. Sharpe, A. G. Inorganic Chemistry, 3rd ed.; Pearson Prentice-Hall: Harlow, U.K., 2008. (21) Huheey, J. E.; Keiter, E. A.; Keiter, R. L. Inorganic Chemistry: Principles of Structure and Reactivity, 4th ed.; HarperCollins, Hinsdale, IL, 1993. (22) Cotton, F. A. Wilkinson, G. Advanced Inorganic Chemistry, 6th ed.; Wiley: NewYork, 1999. (23) Li, W.-K.; Zhou, G.-D.; Mak, T. C. W. Advanced Structural Inorganic Chemistry; Oxford University Press: Oxford, 2008; p 43. (24) Griffiths, D. J. Introduction to Quantum Mechanics, PrenticeHall: Upper Saddle River, NJ, 1995. (25) Pilar, F. L. Elementary Quantum Chemistry, 2nd ed., McGrawHill: New York, 1990. (26) The International System of Units (SI) is called the SI units, an abbreviation from the French term Systeme International d’Unite0 s. There are seven base units in the SI system, as per BIPM 2006.27 They are meter (m), second (s), kilogram (kg), mole (mol), Kelvin (K), ampere (A), and candela (cd), with the standard symbols in parentheses. All other physical units can be derived from these base units. (27) BIPM stands for Bureau International des Poids et Mesures, in French, or “International Bureau of Weights and Measures,” in English. For the US version of the BIPM 2006, see Taylor, B. N.; Thompson, A., Eds.; NIST Special Publication 330; National Institute of Standards and Technology: Gaithersburg, MD, 2008. (28) See, for example , Mogensen, O. E. Positron Annihilation in Chemistry; Springer-Verlag: Berlin, 1995. (29) Zewail, A. H. Sci. Am. 1990, December, 40
46. (30) See, for example Teo, B. K.; Sun, X. H. Chem. Rev. 2007, 107, 1454–1532 and references cited therein.

’ CONCLUSION This article has two parts. In the first part, we introduce the atomic unit (au) system and discuss the scales of time, space (length), and speed, as well as those of mass and energy, in the atomic world. The fundamental physical constants in quantum mechanics are expressed in terms of the SI and the au system and their inter-relationships and interconversions are discussed in detail. The invariance and scaling properties of the au system are stressed. In the second part, we use worked examples to demonstrate the utility of atomic units in quantum mechanical calculations as well as unit conversions in a number of chemical, physical, or spectroscopic calculations. In particular, a wide range of topics such as photoelectric effect, hydrogen atom, Rydberg constants, positronium atom, atomic mass unit, nuclear binding energy, energetics of nuclear reactions, annihilation of matter and antimatter, mass
energy interconversions, femtosecond and attosecond spectroscopies, quantum-size effects, excitons in semiconductors, and so forth are presented and discussed. The last sections illustrate that energy, mass, time, and size matter a great deal in the atomic or nano world! Finally, in view of the efficacy illustrated in these examples, we urge the authors of general chemistry and inorganic chemistry texts to promote the use of atomic units in their books.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ REFERENCES (1) Shull, H.; Hall, G. G. Nature 1959, 184, 1559. (2) LeMay, H. E. ; Bursten, B. E.; Murphy, C. J.; Brown, T. L.; Woodward, P. Chemistry: The Central Science, 11th ed.; Prentice- Hall: Upper Saddle River, NJ, 2009. (3) Ebbing, D.; Gammon, S. D. General Chemistry, 9th ed.; Houghton Mifflin: Boston, MA, 2009. (4) Kelter, P.; Mosher, M.; Scott, A. Chemistry: The Practical Science; Houghton Mifflin: Boston, MA, 2009. (5) Silberberg, M. S. Chemistry: The Molecular Nature of Matter and Change, 5th ed.; McGraw-Hill: Boston, MA, 2009. (6) Burdge, J. Chemistry; McGraw-Hill: Boston, MA, 2009. (7) Lotz, J. C.; Treichel, P. M.; Townsend, J.Chemistry & Chemical Reactivity, 7th ed.; Thompson Brooks/Cole: Belmont, CA, 2008; Volumes 1 and 2. (8) McMurry, J. E.; Fay, R. C. Chemistry, 5th ed.; Pearson PrenticeHall: Upper Saddle River, NJ, 2008. (9) Moore, J. W.; Stanitski, C. L.; Jurs, P. C. Chemistry: The Molecular Science, 3rd ed.; Thompson Brooks/Cole: Belmont, CA, 2008. 928

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