Units and notation

Units and notation#

Atomic units#

A central problem in molecular physics is to solve the time-independent Schrödinger equation for the electrons in the field of the nuclei. Most often atomic units are then adopted. For the hydrogen atom, we have

\[ \left[ - \frac{\hbar^2}{2 m_\mathrm{e}} \nabla^2 - \frac{e^2}{4\pi\varepsilon_0 r} \right] \psi(\mathbf{r}) = E \psi(\mathbf{r}) \]

where \(\hbar\) is the reduced Planck constant, \(m_\mathrm{e}\) is the electron mass, \(e\) is the elementary charge, and \(\varepsilon_0\) is the electric constant. To cast this equation into dimensionless form, consider a coordinate transformation of the form

\[ \mathbf{r} = (x,y,z) \longrightarrow \lambda \mathbf{r}' = (\lambda x', \lambda y', \lambda z') \]

to arrive at

\[ \left[ - \frac{\hbar^2}{2 m_\mathrm{e} \lambda^2} {\nabla'}^2 - \frac{e^2}{4\pi\varepsilon_0 \lambda r'} \right] \psi(\mathbf{r}') = E \psi(\mathbf{r}') \]

Choose \(\lambda\) so that

\[ \frac{\hbar^2}{m_\mathrm{e} \lambda^2} = \frac{e^2}{4\pi\varepsilon_0 \lambda} \equiv E_h \]

with the solution

\[ \lambda = \frac{\hbar^2 4\pi\varepsilon_0}{m_\mathrm{e} e^2} \equiv a_0; \qquad E_h = \frac{m_\mathrm{e} e^4}{(4\pi\varepsilon_0)^2 \hbar^2} \]

With \(E' = E/E_h\), we get

\[ \left[ - \frac{1}{2} {\nabla'}^2 - \frac{1}{r'} \right] \psi(\mathbf{r}') = E' \psi(\mathbf{r}') \]

with a solution for the ground state energy that is equal to \(E' = -0.5\) a.u. (or Hartree). The defined quantity \(a_0\) is equal to the Bohr radius and the atomic unit of length is therefore also referred to as Bohr.

Four fundamental physical constants are given the value 1 in atomic units, including

Name	Symbol	SI equivalent
reduced Planck constant	\(\hbar\)	1.054 572\(\times 10^{-34}\) J s
elementary charge	\(e\)	1.602 176\(\times 10^{-19}\) C
electron rest mass	\(m_{\mathrm{e}}\)	9.109 383\(\times 10^{-31}\) kg
Coulomb constant	\(k_{\mathrm{e}}\)	8.987 552\(\times 10^{9}\) kg m\(^3\) s\(^{-4}\) A\(^{-2}\)

From these four constants, a system of units can be derived, including

Quantity	Name	Symbol	Expression	SI equivalent
Length	Bohr radius	\(a_0\)	\(\hbar^2/(k_{\mathrm{e}} m_{\mathrm{e}} e^2)\)	5.291 772\(\times 10^{-11}\) m
Energy	Hartree	\(E_{\mathrm{h}}\)	\(\hbar^2/(m_{\mathrm{e}} a_0^2)\)	4.359 744\(\times 10^{-18}\) J
Time		\(t\)	\(\hbar/E_{\mathrm{h}}\)	2.418 884\(\times 10^{-17}\) s
Electric constant		\(\varepsilon_0\)	\(e^2/4 \pi a_0 E_{\mathrm{h}}\)	8.854 188\(\times 10^{-12}\) F m\(^{-1}\)
Fine structure constant		\(\alpha\)	1/137.036 \(e^2 / a_0 E_{\mathrm{h}} 4\pi\varepsilon_0\)	7.297 353\(\times 10^{3}\)
Speed of light		\(c\)	137.036 \(a_0 E_{\mathrm{h}} / \hbar\)	2.997 925\(\times 10^{8}\) m s\(^{-1}\)
Electric field		\(F\)	1 \(E_{\mathrm{h}} / e a_{0}\)	5.142 207\(\times 10^{11}\) V m\(^{-1}\)
Dipole moment		\(\mu\)	1 \(e a_{0}\)	8.478 353\(\times 10^{-30}\) C m

Unit conversion#

Conversions between units can conveniently be performed using scipy.constants. A number of physical constants are available, including:

import scipy
from scipy.constants import physical_constants

for key in physical_constants.keys():
    print(key)

Wien displacement law constant
atomic unit of 1st hyperpolarizablity
atomic unit of 2nd hyperpolarizablity
atomic unit of electric dipole moment
atomic unit of electric polarizablity
atomic unit of electric quadrupole moment
atomic unit of magn. dipole moment
atomic unit of magn. flux density
deuteron magn. moment
deuteron magn. moment to Bohr magneton ratio
deuteron magn. moment to nuclear magneton ratio
deuteron-electron magn. moment ratio
deuteron-proton magn. moment ratio
deuteron-neutron magn. moment ratio
electron gyromagn. ratio
electron gyromagn. ratio over 2 pi
electron magn. moment
electron magn. moment to Bohr magneton ratio
electron magn. moment to nuclear magneton ratio
electron magn. moment anomaly
electron to shielded proton magn. moment ratio
electron to shielded helion magn. moment ratio
electron-deuteron magn. moment ratio
electron-muon magn. moment ratio
electron-neutron magn. moment ratio
electron-proton magn. moment ratio
magn. constant
magn. flux quantum
muon magn. moment
muon magn. moment to Bohr magneton ratio
muon magn. moment to nuclear magneton ratio
muon-proton magn. moment ratio
neutron gyromagn. ratio
neutron gyromagn. ratio over 2 pi
neutron magn. moment
neutron magn. moment to Bohr magneton ratio
neutron magn. moment to nuclear magneton ratio
neutron to shielded proton magn. moment ratio
neutron-electron magn. moment ratio
neutron-proton magn. moment ratio
proton gyromagn. ratio
proton gyromagn. ratio over 2 pi
proton magn. moment
proton magn. moment to Bohr magneton ratio
proton magn. moment to nuclear magneton ratio
proton magn. shielding correction
proton-neutron magn. moment ratio
shielded helion gyromagn. ratio
shielded helion gyromagn. ratio over 2 pi
shielded helion magn. moment
shielded helion magn. moment to Bohr magneton ratio
shielded helion magn. moment to nuclear magneton ratio
shielded helion to proton magn. moment ratio
shielded helion to shielded proton magn. moment ratio
shielded proton magn. moment
shielded proton magn. moment to Bohr magneton ratio
shielded proton magn. moment to nuclear magneton ratio
{220} lattice spacing of silicon
lattice spacing of silicon
alpha particle-electron mass ratio
alpha particle mass
alpha particle mass energy equivalent
alpha particle mass energy equivalent in MeV
alpha particle mass in u
alpha particle molar mass
alpha particle-proton mass ratio
Angstrom star
atomic mass constant
atomic mass constant energy equivalent
atomic mass constant energy equivalent in MeV
atomic mass unit-electron volt relationship
atomic mass unit-hartree relationship
atomic mass unit-hertz relationship
atomic mass unit-inverse meter relationship
atomic mass unit-joule relationship
atomic mass unit-kelvin relationship
atomic mass unit-kilogram relationship
atomic unit of 1st hyperpolarizability
atomic unit of 2nd hyperpolarizability
atomic unit of action
atomic unit of charge
atomic unit of charge density
atomic unit of current
atomic unit of electric dipole mom.
atomic unit of electric field
atomic unit of electric field gradient
atomic unit of electric polarizability
atomic unit of electric potential
atomic unit of electric quadrupole mom.
atomic unit of energy
atomic unit of force
atomic unit of length
atomic unit of mag. dipole mom.
atomic unit of mag. flux density
atomic unit of magnetizability
atomic unit of mass
atomic unit of momentum
atomic unit of permittivity
atomic unit of time
atomic unit of velocity
Avogadro constant
Bohr magneton
Bohr magneton in eV/T
Bohr magneton in Hz/T
Bohr magneton in inverse meters per tesla
Bohr magneton in K/T
Bohr radius
Boltzmann constant
Boltzmann constant in eV/K
Boltzmann constant in Hz/K
Boltzmann constant in inverse meters per kelvin
characteristic impedance of vacuum
classical electron radius
Compton wavelength
Compton wavelength over 2 pi
conductance quantum
conventional value of Josephson constant
conventional value of von Klitzing constant
Cu x unit
deuteron-electron mag. mom. ratio
deuteron-electron mass ratio
deuteron g factor
deuteron mag. mom.
deuteron mag. mom. to Bohr magneton ratio
deuteron mag. mom. to nuclear magneton ratio
deuteron mass
deuteron mass energy equivalent
deuteron mass energy equivalent in MeV
deuteron mass in u
deuteron molar mass
deuteron-neutron mag. mom. ratio
deuteron-proton mag. mom. ratio
deuteron-proton mass ratio
deuteron rms charge radius
electric constant
electron charge to mass quotient
electron-deuteron mag. mom. ratio
electron-deuteron mass ratio
electron g factor
electron gyromag. ratio
electron gyromag. ratio over 2 pi
electron mag. mom.
electron mag. mom. anomaly
electron mag. mom. to Bohr magneton ratio
electron mag. mom. to nuclear magneton ratio
electron mass
electron mass energy equivalent
electron mass energy equivalent in MeV
electron mass in u
electron molar mass
electron-muon mag. mom. ratio
electron-muon mass ratio
electron-neutron mag. mom. ratio
electron-neutron mass ratio
electron-proton mag. mom. ratio
electron-proton mass ratio
electron-tau mass ratio
electron to alpha particle mass ratio
electron to shielded helion mag. mom. ratio
electron to shielded proton mag. mom. ratio
electron volt
electron volt-atomic mass unit relationship
electron volt-hartree relationship
electron volt-hertz relationship
electron volt-inverse meter relationship
electron volt-joule relationship
electron volt-kelvin relationship
electron volt-kilogram relationship
elementary charge
elementary charge over h
Faraday constant
Faraday constant for conventional electric current
Fermi coupling constant
fine-structure constant
first radiation constant
first radiation constant for spectral radiance
hartree-atomic mass unit relationship
hartree-electron volt relationship
Hartree energy
Hartree energy in eV
hartree-hertz relationship
hartree-inverse meter relationship
hartree-joule relationship
hartree-kelvin relationship
hartree-kilogram relationship
helion-electron mass ratio
helion mass
helion mass energy equivalent
helion mass energy equivalent in MeV
helion mass in u
helion molar mass
helion-proton mass ratio
hertz-atomic mass unit relationship
hertz-electron volt relationship
hertz-hartree relationship
hertz-inverse meter relationship
hertz-joule relationship
hertz-kelvin relationship
hertz-kilogram relationship
inverse fine-structure constant
inverse meter-atomic mass unit relationship
inverse meter-electron volt relationship
inverse meter-hartree relationship
inverse meter-hertz relationship
inverse meter-joule relationship
inverse meter-kelvin relationship
inverse meter-kilogram relationship
inverse of conductance quantum
Josephson constant
joule-atomic mass unit relationship
joule-electron volt relationship
joule-hartree relationship
joule-hertz relationship
joule-inverse meter relationship
joule-kelvin relationship
joule-kilogram relationship
kelvin-atomic mass unit relationship
kelvin-electron volt relationship
kelvin-hartree relationship
kelvin-hertz relationship
kelvin-inverse meter relationship
kelvin-joule relationship
kelvin-kilogram relationship
kilogram-atomic mass unit relationship
kilogram-electron volt relationship
kilogram-hartree relationship
kilogram-hertz relationship
kilogram-inverse meter relationship
kilogram-joule relationship
kilogram-kelvin relationship
lattice parameter of silicon
Loschmidt constant (273.15 K, 101.325 kPa)
mag. constant
mag. flux quantum
molar gas constant
molar mass constant
molar mass of carbon-12
molar Planck constant
molar Planck constant times c
molar volume of ideal gas (273.15 K, 100 kPa)
molar volume of ideal gas (273.15 K, 101.325 kPa)
molar volume of silicon
Mo x unit
muon Compton wavelength
muon Compton wavelength over 2 pi
muon-electron mass ratio
muon g factor
muon mag. mom.
muon mag. mom. anomaly
muon mag. mom. to Bohr magneton ratio
muon mag. mom. to nuclear magneton ratio
muon mass
muon mass energy equivalent
muon mass energy equivalent in MeV
muon mass in u
muon molar mass
muon-neutron mass ratio
muon-proton mag. mom. ratio
muon-proton mass ratio
muon-tau mass ratio
natural unit of action
natural unit of action in eV s
natural unit of energy
natural unit of energy in MeV
natural unit of length
natural unit of mass
natural unit of momentum
natural unit of momentum in MeV/c
natural unit of time
natural unit of velocity
neutron Compton wavelength
neutron Compton wavelength over 2 pi
neutron-electron mag. mom. ratio
neutron-electron mass ratio
neutron g factor
neutron gyromag. ratio
neutron gyromag. ratio over 2 pi
neutron mag. mom.
neutron mag. mom. to Bohr magneton ratio
neutron mag. mom. to nuclear magneton ratio
neutron mass
neutron mass energy equivalent
neutron mass energy equivalent in MeV
neutron mass in u
neutron molar mass
neutron-muon mass ratio
neutron-proton mag. mom. ratio
neutron-proton mass ratio
neutron-tau mass ratio
neutron to shielded proton mag. mom. ratio
Newtonian constant of gravitation
Newtonian constant of gravitation over h-bar c
nuclear magneton
nuclear magneton in eV/T
nuclear magneton in inverse meters per tesla
nuclear magneton in K/T
nuclear magneton in MHz/T
Planck constant
Planck constant in eV s
Planck constant over 2 pi
Planck constant over 2 pi in eV s
Planck constant over 2 pi times c in MeV fm
Planck length
Planck mass
Planck mass energy equivalent in GeV
Planck temperature
Planck time
proton charge to mass quotient
proton Compton wavelength
proton Compton wavelength over 2 pi
proton-electron mass ratio
proton g factor
proton gyromag. ratio
proton gyromag. ratio over 2 pi
proton mag. mom.
proton mag. mom. to Bohr magneton ratio
proton mag. mom. to nuclear magneton ratio
proton mag. shielding correction
proton mass
proton mass energy equivalent
proton mass energy equivalent in MeV
proton mass in u
proton molar mass
proton-muon mass ratio
proton-neutron mag. mom. ratio
proton-neutron mass ratio
proton rms charge radius
proton-tau mass ratio
quantum of circulation
quantum of circulation times 2
Rydberg constant
Rydberg constant times c in Hz
Rydberg constant times hc in eV
Rydberg constant times hc in J
Sackur-Tetrode constant (1 K, 100 kPa)
Sackur-Tetrode constant (1 K, 101.325 kPa)
second radiation constant
shielded helion gyromag. ratio
shielded helion gyromag. ratio over 2 pi
shielded helion mag. mom.
shielded helion mag. mom. to Bohr magneton ratio
shielded helion mag. mom. to nuclear magneton ratio
shielded helion to proton mag. mom. ratio
shielded helion to shielded proton mag. mom. ratio
shielded proton gyromag. ratio
shielded proton gyromag. ratio over 2 pi
shielded proton mag. mom.
shielded proton mag. mom. to Bohr magneton ratio
shielded proton mag. mom. to nuclear magneton ratio
speed of light in vacuum
standard acceleration of gravity
standard atmosphere
Stefan-Boltzmann constant
tau Compton wavelength
tau Compton wavelength over 2 pi
tau-electron mass ratio
tau mass
tau mass energy equivalent
tau mass energy equivalent in MeV
tau mass in u
tau molar mass
tau-muon mass ratio
tau-neutron mass ratio
tau-proton mass ratio
Thomson cross section
triton-electron mag. mom. ratio
triton-electron mass ratio
triton g factor
triton mag. mom.
triton mag. mom. to Bohr magneton ratio
triton mag. mom. to nuclear magneton ratio
triton mass
triton mass energy equivalent
triton mass energy equivalent in MeV
triton mass in u
triton molar mass
triton-neutron mag. mom. ratio
triton-proton mag. mom. ratio
triton-proton mass ratio
unified atomic mass unit
von Klitzing constant
weak mixing angle
Wien frequency displacement law constant
Wien wavelength displacement law constant
atomic unit of mom.um
electron-helion mass ratio
electron-triton mass ratio
helion g factor
helion mag. mom.
helion mag. mom. to Bohr magneton ratio
helion mag. mom. to nuclear magneton ratio
Loschmidt constant (273.15 K, 100 kPa)
natural unit of mom.um
natural unit of mom.um in MeV/c
neutron-proton mass difference
neutron-proton mass difference energy equivalent
neutron-proton mass difference energy equivalent in MeV
neutron-proton mass difference in u
standard-state pressure
alpha particle relative atomic mass
Bohr magneton in inverse meter per tesla
Boltzmann constant in inverse meter per kelvin
conventional value of ampere-90
conventional value of coulomb-90
conventional value of farad-90
conventional value of henry-90
conventional value of ohm-90
conventional value of volt-90
conventional value of watt-90
deuteron relative atomic mass
electron gyromag. ratio in MHz/T
electron relative atomic mass
elementary charge over h-bar
helion relative atomic mass
helion shielding shift
hyperfine transition frequency of Cs-133
lattice spacing of ideal Si (220)
luminous efficacy
neutron gyromag. ratio in MHz/T
neutron relative atomic mass
nuclear magneton in inverse meter per tesla
Planck constant in eV/Hz
proton gyromag. ratio in MHz/T
proton relative atomic mass
reduced Compton wavelength
reduced muon Compton wavelength
reduced neutron Compton wavelength
reduced Planck constant
reduced Planck constant in eV s
reduced Planck constant times c in MeV fm
reduced proton Compton wavelength
reduced tau Compton wavelength
shielded helion gyromag. ratio in MHz/T
shielded proton gyromag. ratio in MHz/T
shielding difference of d and p in HD
shielding difference of t and p in HT
tau energy equivalent
triton relative atomic mass
triton to proton mag. mom. ratio
vacuum electric permittivity
vacuum mag. permeability
W to Z mass ratio
alpha particle rms charge radius
Copper x unit
Molybdenum x unit

This dictionary provide the numerical value of the constants, including unit and uncertainty. For example, \(\hbar\), \(e\), \(m\), \(a_0\), \(E_{\mathrm{h}}\), and \(t\) from above can be obtained as:

print("hbar   :", physical_constants["reduced Planck constant"])
print("e      :", physical_constants["atomic unit of charge"])
print("m      :", physical_constants["atomic unit of mass"])
print("a_0    :", physical_constants["atomic unit of length"])
print("E_h    :", physical_constants["atomic unit of energy"])
print("t      :", physical_constants["atomic unit of time"])

hbar   : (1.0545718176461565e-34, 'J s', 0.0)
e      : (1.602176634e-19, 'C', 0.0)
m      : (9.1093837139e-31, 'kg', 2.8e-40)
a_0    : (5.29177210544e-11, 'm', 8.2e-21)
E_h    : (4.359744722206e-18, 'J', 4.8e-30)
t      : (2.4188843265864e-17, 's', 2.6e-29)

VeloxChem returns energies in Hartree, and conversion to other units is conveniently done as:

E = 1.0

au2eV = physical_constants["Hartree energy in eV"][0]
au2J  = physical_constants["Hartree energy"][0]
h     = physical_constants["Planck constant"][0]
c     = physical_constants["speed of light in vacuum"][0]
N_A   = physical_constants["Avogadro constant"][0]
cal   = scipy.constants.calorie
au2cm = au2J / (100. * h * c)
au2nm = 1.0e9 * h * c / au2J
au2kjmol = au2J * N_A / 1000.
au2kcalmol = au2kjmol / cal 

print("Hartree     eV        cm-1       nm      kJ/mol   kcal/mol")
print(58 * "-")
print(
    f"{E:4.3f} {E*au2eV:12.5f} {E*au2cm:9.1f} {au2nm/E:9.5f} {E*au2kjmol:9.3f} {au2kcalmol*E:9.4f}"
)

Hartree     eV        cm-1       nm      kJ/mol   kcal/mol
----------------------------------------------------------
1.000     27.21139  219474.6  45.56335  2625.500  627.5095

Note

Included above is the energy expressed in nm. This is often used in spectroscopy, with the interpretation that it is the associated photon wavelength.

Notation#

This section describe the notation adopted in this book:

Orbitals and wave functions#

symbol	meaning
\(\Psi\)	multi-electron wave function
\(\psi\)	spin orbital
\(\phi\)	molecular orbital
\(\chi\)	atomic orbital
\(\mid \psi_1,\ldots,\psi_N\rangle\)	Slater determinant
\(\mid k_1 \ldots k_p \ldots k_M\rangle\)	occupation number vector

Indices#

indices	meaning
\(ij\ldots\)	occupied orbitals or electrons
\(ab\ldots\)	unoccupied orbitals
\(pq\ldots\)	general orbitals
\(tu\ldots\)	active space orbitals
\(\alpha\beta\ldots\)	atomic orbitals or Cartesian tensor component
\(AB\ldots\)	nuclei

Matrices and vectors#

symbol	meaning
\(\mathbf{D}\)	matrix
\(D_{\alpha \beta}\)	matrix elements
\(\mathbf{r}\)	vector
\(r_{\alpha}\)	vector elements

Common matrices#

symbol	meaning
\(\mathbf{D}\)	density matrix
\(\mathbf{C}\)	MO coefficient matrix
\(\mathbf{S}\)	overlap matrix
\(\mathbf{F}\)	Fock matrix

Two-electron integrals#

Integrals with respect to spin orbitals are expressed in the physicist’s notation

\[ \omega_{abcd} = \langle \psi_a \psi_b | \hat{\omega} | \psi_c \psi_d \rangle = \int \psi_a^\dagger(\mathbf{r}_1) \psi_b^\dagger(\mathbf{r}_2) \, \hat{\omega} \, \psi_c(\mathbf{r}_1) \psi_d(\mathbf{r}_2) \, d^3\mathbf{r}_1 d^3\mathbf{r}_2 \]

Integrals with respect to spatial (or molecular) orbitals are expressed in the chemist’s notation

\[ \omega_{abcd} = ( \phi_a \phi_b | \hat{\omega} | \phi_c \phi_d ) = \int \phi_a^\ast(\mathbf{r}_1) \phi_b(\mathbf{r}_1) \, \hat{\omega} \, \phi_c^\ast(\mathbf{r}_2) \phi_d(\mathbf{r}_2) \, d^3\mathbf{r}_1 d^3\mathbf{r}_2 \]

Consistency of notation#

This book is written with a consistent notation in mind. However, on account of the large number of authors and the large number of areas covered, full consistency is not always possible. We hope that any deviations are made clear, but please contact us if you see any issues.

The same symbol is sometimes used for several properties, e.g.

\(E\) typically designates energies, but is also used for electric fields

Short-hand notation#

Implied indices are occasionally used for brevity, such as

\(F\) in molecular properties collectively denotes electric field amplitudes at all different perturbation frequencies and along the three Cartesian molecular axes