$$ \newcommand{\bra}[1]{\left\langle#1\right|} \newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\ketbra}[2]{\left|#1\right\rangle\left\langle#2\right|} \newcommand{\bracket}[2]{\left\langle #1 \middle| #2 \right\rangle} \newcommand{\matrixel}[3]{\left\langle #1 \middle| #2 \middle| #3 \right\rangle} \newcommand{\avg}[1]{\left\langle #1 \right\rangle} \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\bh}[1]{\boldsymbol{\hat{#1}}} \newcommand{\bv}[1]{\boldsymbol{\vec{#1}}} $$

Constants

Planck constant \(h = 6.62607004 \times 10^{-34} J \cdot s\)
Boltzmann constant \(k_B = 1.38064852 \times 10^{-23} J \cdot K^{-1}\)
Avogadro's number \(N_A = 6.0221409 \times 10^{23} mol^{-1} \)
Speed of light \(c = 2.99792458 \times 10^{8} m \cdot s^{-1}\)
Bohr radius \(a_0 = 5.29177211 \times 10^{-11} m \)
Charge of electron \(e = 1.60217662 \times 10^{-19} C \)
Mass of electron \(m_e = 9.10938356 \times 10^{-31} kg \)
Mass of proton \(m_p = 1.6726219 \times 10^{-27} kg \)

Unit Conversion

Energy

$$ \begin{array}{|c|c|c|c|c|c|} \hline Energy & au[Hartree] & eV & J & cal & K & cm^{-1} \\ \hline 1au[Hartree] & 1 & 2.72114 \times 10^{1} & 4.35974 \times 10^{-18} & 1.04200 \times 10^{-18} & 3.15775 \times 10^{5} & 2.19475 \times 10^{5} \\ 1eV & 3.67493 \times 10^{-2} & 1 & 1.60218 \times 10^{-19} & 3.82929 \times 10^{-20} & 1.16045 \times 10^{4} & 8.06554 \times 10^{3} \\ 1J & 2.29371 \times 10^{17} & 6.24151 \times 10^{18} & 1 & 2.39006 \times 10^{-1} & 7.24297 \times 10^{22} & 5.03412 \times 10^{22} \\ 1cal & 9.59689 \times 10^{17} & 2.61145 \times 10^{19} & 4.18400 \times 10^{0} & 1 & 3.03046 \times 10^{23} & 2.10627 \times 10^{23} \\ 1K & 3.16681 \times 10^{-6} & 8.61733 \times 10^{-5} & 1.38065 \times 10^{-23} & 3.29983 \times 10^{-24} & 1 & 6.95035 \times 10^{-1} \\ 1cm^{-1} & 4.55634 \times 10^{-6} & 1.23984 \times 10^{-4} & 1.98645 \times 10^{-23} & 4.74772 \times 10^{-24} & 1.43878 \times 10^{0} & 1 \\ \hline \end{array} $$

Mass

$$ \begin{array}{|c|c|c|} \hline Mass & au[m_e] & kg & m_p \\ \hline 1au[m_e] & 1 & 9.10938 \times 10^{-31} & 5.44617 \times 10^{-4} \\ 1kg & 1.09777 \times 10^{30} & 1 & 5.97864 \times 10^{26} \\ 1m_p & 1.83615 \times 10^{3} & 1.67262 \times 10^{-27} & 1 \\ \hline \end{array} $$

Length

$$ \begin{array}{|c|c|c|} \hline Length & au[Bohr] & m & \unicode{xC5} \\ \hline 1au[Bohr] & 1 & 5.29177 \times 10^{-11} & 5.29177 \times 10^{-1} \\ 1m & 1.88973 \times 10^{10} & 1 & 1.00000 \times 10^{10} \\ 1\unicode{xC5} & 1.88973 \times 10^{0} & 1.00000 \times 10^{-10} & 1 \\ \hline \end{array} $$

Time

$$ \begin{array}{|c|c|c|c|} \hline Time & au & s & ps & fs \\ \hline 1au & 1 & 2.41888 \times 10^{-17} & 2.41888 \times 10^{-5} & 2.41888 \times 10^{-2} \\ 1s & 4.13414 \times 10^{16} & 1 & 1.00000 \times 10^{12} & 1.00000 \times 10^{15} \\ 1ps & 4.13414 \times 10^{4} & 1.00000 \times 10^{-12} & 1 & 1.00000 \times 10^{3} \\ 1fs & 4.13414 \times 10^{1} & 1.00000 \times 10^{-15} & 1.00000 \times 10^{-3} & 1 \\ \hline \end{array} $$

Formula

Vector and Matrix

Vector calculus $$ \bm{\nabla} \cdot (\bm{\nabla} \times \bm{v}) = 0 \\ \bm{\nabla} \times (\bm{\nabla} \phi) = \bm{0} \\ \bm{\nabla} \times (\bm{\nabla} \times \bm{v}) = \bm{\nabla}(\bm{\nabla} \cdot \bm{v}) - \bm{\nabla}^2\bm{v} \\ \bm{\nabla} \cdot (\bm{\nabla} \phi) = \bm{\nabla}^2 \phi \\ \\ \bm{\nabla} \cdot (\bm{v} \times \bm{w}) = (\bm{\nabla} \times \bm{v}) \cdot \bm{w} - \bm{v} \cdot(\bm{\nabla} \times \bm{w}) \\ \bm{\nabla} \times (\bm{v} \times \bm{w}) = \bm{v} (\bm{\nabla} \cdot \bm{w}) - \bm{w} (\bm{\nabla} \cdot \bm{v}) + (\bm{w} \cdot \bm{\nabla}) \bm{v} - (\bm{v} \cdot \bm{\nabla}) \bm{w} \\ \bm{u} \cdot (\bm{v} \times \bm{w}) = \bm{v} \cdot (\bm{w} \times \bm{u}) = \bm{w} \cdot (\bm{u} \times \bm{v}) \\ \bm{u} \times (\bm{v} \times \bm{w}) = (\bm{u} \cdot \bm{w}) \bm{v} - (\bm{u} \cdot \bm{v}) \bm{w} \\ \bm{u} \times (\bm{v} \times \bm{w}) + \bm{v} \times (\bm{w} \times \bm{u}) + \bm{w} \times (\bm{u} \times \bm{v}) = 0 \\ $$ Matrix identity $$ e^{A+B} = e^A e^B e^{-\frac{1}{2} [A,B]} \\ $$

Function and Integration

Genearal function $$ e^{\lambda\frac{\partial}{\partial x}} f(x) = f(x + \lambda) \\ $$ Delta function $$ \delta(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk \\ $$ Gaussian function $$ \int_{-\infty}^{\infty} e^{-Ax^2+Bx}dx = \sqrt{\frac{\pi}{A}}e^{B^2/4A} \text{ , } A \gt 0 \\ $$

Derivative Coupling

$$ \bm{d}_{jk} \equiv \matrixel{j}{\bm{\nabla}}{k} = \frac{\matrixel{j}{\bm{\nabla}H}{k}}{E_k - E_j} \\ $$

Maxwell's Equations

$$ \bm{\nabla} \cdot \bm{E} = \frac{\rho}{\varepsilon_0} \\ \bm{\nabla} \cdot \bm{B} = 0 \\ \bm{\nabla} \times \bm{E} = -\frac{\partial \bm{B}}{\partial t} \\ \bm{\nabla} \times \bm{B} = \mu_0 \left(\bm{J} + \varepsilon_0 \frac{\partial \bm{E}}{\partial t}\right) \\ $$

Second Quantization

$$ \hat{H} = \frac{\hat{p}^2}{2m} + \frac{m\omega^2}{2}\hat{x}^2 = \hbar\omega\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\ \hat{a} = \frac{1}{\sqrt{2}}\left[\sqrt{\frac{m\omega}{\hbar}}\hat{x} + i\frac{\hat{p}}{\sqrt{\hbar m\omega}}\right] \text{ , } \hat{a}^{\dagger} = \frac{1}{\sqrt{2}}\left[\sqrt{\frac{m\omega}{\hbar}}\hat{x} - i\frac{\hat{p}}{\sqrt{\hbar m\omega}}\right] \\ \hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\left(\hat{a} + \hat{a}^{\dagger}\right) \text{ , } \hat{p} = -i\sqrt{\frac{\hbar m\omega}{2}}\left(\hat{a} - \hat{a}^{\dagger}\right) \\ [\hat{x}, \hat{p}] = i\hbar \text{ , } [\hat{a}, \hat{a}^{\dagger}] = 1 \\ \hat{a}\ket{n} = \sqrt{n} \ket{n-1} \text{ , } \hat{a}^{\dagger} \ket{n} = \sqrt{n+1} \ket{n+1} \\ \hat{a}(t) = \hat{a}e^{-i\omega t} \text{ , } \hat{a}^{\dagger}(t) = \hat{a}^{\dagger}e^{i\omega t} \\ $$

Marcus Rate

$$ k_{1 \leftarrow 2} = \frac{|V_{12}|^2}{\hbar} \sqrt{\frac{\pi}{k_BTE_r}} e^{-(E_{21} - E_r)^2 / 4k_BTE_r} \\ $$

Order of Magnitude

Rates

$$ \text{ benzophenone intersystem crossing } \frac{1}{k_{ISC}} \approx 10ps \text{ (Tamai, N.; Asahi, T.; Masuhara, H. Chem. Phys. Lett. 1992, 198, 413−418) } \\ \text{ benzophenone phosphorescence } \frac{1}{k_{phos}} \approx 0.712 ms \text{ (Biron, M.; Longin, P. Chem. Phys. Lett. 1985, 116 (2−3), 250−253) } $$