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.

Table: Atomic unit conversion factors.

Quantity

Symbol

Atomic unit

SI equivalent

Energy

\(E\)

1 \(E_\mathrm{h}\)

4.359 744\(\times 10^{-18}\) J

Reduced Planck constant

\(h = 2\pi\hbar\)

1 \(\hbar\)

1.054 572\(\times 10^{-34}\) J s

Time

\(t\)

1 \(\hbar E_\mathrm{h}^{-1}\)

2.418 884\(\times 10^{-17}\) s

Length

\(l\)

1 \(a_0\)

5.291 772\(\times 10^{-11}\) m

Speed of light

\(c\)

137.036 \(a_0 E_h \hbar^{-1}\)

2.997 925\(\times 10^{8}\) m s\(^{-1}\)

Electric constant

\(\varepsilon_0\)

1 \(4\pi\varepsilon_0\)

8.854 188\(\times 10^{-12}\) F m\(^{-1}\)

Fine structure constant

\(\alpha\)

1/137.036 \(e^2( a_0 E_h 4\pi\varepsilon_0)^{-1}\)

7.297 353\(\times 10^{3}\)

Charge

\(q\)

1 \(e\)

1.602 176\(\times 10^{-19}\) C

Electric field

\(F\)

1 \(E_h (e a_{0})^{-1}\)

5.142 207\(\times 10^{11}\) V m\(^{-1}\)

Dipole moment

\(\mu\)

1 \(e a_{0}\)

8.478 353\(\times 10^{-30}\) C m

Mass

\(m\)

1 \(m_e\)

9.109 383\(\times 10^{-31}\) kg

Unit conversion#

Conversions between units can conveniently be performed using scipy.constants.

Notation#

Orbitals and wave functions#

symbol

meaning

\(\Psi\)

multi-electron wave function

\(\psi\)

spin orbital

\(\phi\)

molecular orbital

\(\chi\)

atomic orbital

Indices#

indices

meaning

\(ij\ldots\)

occupied orbitals or electrons

\(ab\ldots\)

unoccupied orbitals

\(pq\ldots\)

general orbitals

\(tu\ldots\)

active space orbitals

\(AB\ldots\)

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