# 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