tions involve the vacuum permittivity (now called the electric constant) µ0

and vacuum permeability (now called the magnetic constant) μ0 ; the veloc-

ity of light does not enter explicitly into the equations connecting electric

and magnetic quantities. The SI system is rationalized, meaning that elec-

tric and magnetic potentials, but also energies, ¬elds and forces, are derived

from their sources (charge density ρ, current density j) with a multiplicative

factor 1/(4πµ0 ), resp. μ0 /4π:

1 ρ(r )

¦(r) = dr , (1)

|r ’ r |

4πµ0

μ0 j(r )

A(r) = dr , (2)

|r ’ r |

4π

while in di¬erential form the 4π vanishes:

div E = ’ div grad ¦ = ρ/µ0 , (3)

curl B = curl curl A = μ0 j. (4)

In non-rationalized systems without a multiplicative factor in the integrated

forms (as in the obsolete electrostatic and Gauss systems, but also in atomic

units), an extra factor 4π occurs in the integrated forms:

div E = 4πρ, (5)

curl B = 4πj. (6)

Consistent use of the SI system avoids ambiguities, especially in the use of

electric and magnetic units, but the reader who has been educated with non-

rationalized units (electrostatic and Gauss units) should not fall into one of

the common traps. For example, the magnetic susceptibility χm , which is

the ratio between induced magnetic polarization M (dipole moment per

unit volume) and applied magnetic intensity H, is a dimensionless quantity,

which nevertheless di¬ers by a factor of 4π between rationalized and non-

rationalized systems of units. Another quantity that may cause confusion

is the polarizability ±, which is a tensor de¬ned by the relation μ = ±E

between induced dipole moment and electric ¬eld. Its SI unit is F m2 , but its

non-rationalized unit is a volume. To be able to compare ± with a volume,

the quantity ± = ±/(4πµ0 ) may be de¬ned, the SI unit of which is m3 .

Technical units are often based on the force exerted by standard gravity

(9.806 65 m s’2 ) on a mass of a kilogram or a pound avoirdupois [lb =

0.453 592 37 kg (exact)], yielding a kilogramforce (kgf) = 9.806 65 N, or a

poundforce (lbf) = 4.448 22 N. The US technical unit for pressure psi (pound

Symbols, units and constants xvii

per square inch) amounts to 6894.76 Pa. Such non-SI units are avoided in

this book.

When dealing with electrons, atoms and molecules, SI units are not very

practical. For treating quantum problems with electrons, as in quantum

chemistry, atomic units (a.u.) are often used (see Table 7). In a.u. the

electron mass and charge and Dirac™s constant all have the value 1. For

treating molecules, a very convenient system of units, related to the SI

system, uses nm for length, u (uni¬ed atomic mass unit) for mass, and ps

for time. We call these molecular units (m.u.). Both systems are detailed

below.

SI Units

SI units are de¬ned by the basic units length, mass, time, electric current,

thermodynamic temperature, quantity of matter and intensity of light. Units

for angle and solid angle are the dimensionless radian and steradian. See

Table 5 for the de¬ned SI units. All other units are derived from these basic

units (Table 6).

While the Syst`me International also de¬nes the mole (with unit mol ),

e

being a number of entities (such as molecules) large enough to bring its total

mass into the range of grams, one may express quantities of molecular size

also per mole rather than per molecule. For macroscopic system sizes one

then obtains more convenient numbers closer to unity. In chemical ther-

modynamics molar quantities are commonly used. Molar constants as the

Faraday F (molar elementary charge), the gas constant R (molar Boltzmann

constant) and the molar standard ideal gas volume Vm (273.15 K, 105 Pa)

are speci¬ed in SI units (see Table 9).

Atomic units

Atomic units (a.u.) are based on electron mass me = 1, Dirac™s constant

= 1, elementary charge e = 1 and 4πµ0 = 1. These choices determine the

units of other quantities, such as

4πµ0 2

a.u. of length (Bohr radius) a0 = = , (7)

me e2 ±me c

(4πµ0 )2 3 me a2 0

a.u. of time = = , (8)

4

me e

a.u. of velocity = /(me a0 ) = ±c, (9)

xviii Symbols, units and constants

me e4 ±2 c2 me

a.u. of energy (hartree) Eh = = . (10)

(4πµ0 )2 2 2

Here, ± = e2 /(4πµ0 c) is the dimensionless ¬ne-structure constant. The

system is non-rationalized and in electromagnetic equations µ0 = 1/(4π) and

μ0 = 4π±2 . The latter is equivalent to μ0 = 1/(µ0 c2 ), with both quantities

expressed in a.u. Table 7 lists the values of the basic atomic units in terms

of SI units.

These units employ physical constants, which are not so constant as the

name suggests; they depend on the de¬nition of basic units and on the

improving precision of measurements. The numbers given here refer to con-

stants published in 2002 by CODATA (Mohr and Taylor, 2005). Standard

errors in the last decimals are given between parentheses.

Molecular units

Convenient units for molecular simulations are based on nm for length, u

(uni¬ed atomic mass units) for mass, ps for time, and the elementary charge

e for charge. The uni¬ed atomic mass unit is de¬ned as 1/12 of the mass of a

12 C atom, which makes 1 u equal to 1 gram divided by Avogadro™s number.

The unit of energy now appears to be 1 kJ/mol = 1 u nm2 ps’2 . There is

an electric factor fel = (4πµ0 )’1 = 138.935 4574(14) kJ mol’1 nm e’2 when

calculating energy and forces from charges, as in Vpot = fel q 2 /r. While

these units are convenient, the unit of pressure (kJ mol’1 nm’3 ) becomes a

bit awkward, being equal to 1.666 053 886(28) MPa or 16.66 . . . bar.

Warning: One may not change kJ/mol into kcal/mol and nm into ˚ A

(the usual units for some simulation packages) without punishment. When

√

keeping the u for mass, the unit of time then becomes 0.1/ 4.184 ps =

48.888 821 . . . fs. Keeping the e for charge, the electric factor must be ex-

pressed in kcal mol’1 ˚ e’2 with a value of 332.063 7127(33). The unit of

A

pressure becomes 69 707.6946(12) bar! These units also form a consistent

system, but we do not recommend their use.

Physical constants

In Table 9 some relevant physical constants are given in SI units; the values

are those published by CODATA in 2002.2 The same constants are given

in Table 10 in atomic and molecular units. Note that in the latter table

2 See Mohr and Taylor (2005) and

http://physics.nist.gov/cuu/. A Python module containing a variety of physical constants,

physcon.py, may be downloaded from this book™s or the author™s website.

Symbols, units and constants xix

molar quantities are not listed: It does not make sense to list quantities in

molecular-sized units per mole of material, because values in the order of

1023 would be obtained. The whole purpose of atomic and molecular units

is to obtain “normal” values for atomic and molecular quantities.

xx Symbols, units and constants

Table 1 Typographic conventions and special symbols

Element Example Meaning

c— complex conjugate c— = a ’ bi if c = a + bi