CHEM 455 Physical Chemistry
Contents
From Classical to Quantum Mechanics
Blackbody radiation
| Description | Equations |
|---|---|
| Energy quantization | $E = nh\nu$ |
| Average energy of an oscillating dipole | $\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}$ |
| Spectral radiation density of blackbody (Planck) | $\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3}\dfrac{1}{e^{h\nu / k_B T} - 1} \ d\nu$ |
| Spectral radiation density of blackbody (classical) | $\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3} k_B T d\nu$ |
Wave-particle duality
| Description | Equations |
|---|---|
| Energy of light | $E = h \nu$ |
| Photoelectric effect Kinetic energy of ejected photoelectron |
$E_k = h \nu - \Phi$ |
| de Broglie relation | $p = \dfrac{h}{\lambda}$ |
| Kinetic energy | $E_k = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}$ |
Atomic spectra of hydrogen and Bohr’s model
| Description | Equations |
|---|---|
| Hydrogen emission lines $n_2 > n_1$ |
$\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$ |
| Bohr’s radius | $r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}$ |
| Energy level in Bohr’s model | $E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}$ |
| Emission of hydrogen atom $n_2 > n_1$ |
$\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$ |
Waves
| Description | Equations |
|---|---|
| Classical nondispersive wave equation | $\dfrac{\partial \Psi(x, t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial \Psi(x, t)}{\partial t^2}$ |
| Wave number | $k = \dfrac{2\pi}{\lambda}$ |
| Frequency | $\nu = \dfrac{1}{T}$ |
| Angular frequency | $\omega = \dfrac{2\pi}{T} = 2\pi\nu$ |
| Wave speed | $v = \lambda\nu$ |
| Euler’s formula | $e^{i\theta} = \cos\theta + i\sin\theta$ |
| Solution of wave equation | $\begin{aligned}\Psi(x, t) &= A \sin(kx - \omega t + \phi) \\ &= \mathrm{Re}(Ae^{i(kx-\omega t + \phi')})\end{aligned}$ |
| Interfering traveling waves give standing wave | $\begin{aligned}\Psi(x, t) &= A[\sin(kx - \omega t) + \sin(kx + \omega t)] \\ &= 2A \sin(kx)\cos(\omega t) \\ &= \psi(x)\cos(\omega t) \end{aligned}$ |
| Time-independent Schrodinger equation | $-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$ |
| Time-dependent Schrodinger equation | $-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x, t)}{\partial x^2} + V(x, t)\Psi(x, t) = i\hbar\Psi(x, t)$ |
| Stationary states are standing waves | $\Psi(x, t) = \psi(x) e^{-i(E/\hbar)t}$ |
| Normalization | $\Vert f(x) \Vert = \int_D f^* f \ dx = 1$ |
| Orthogonality | $\int_D f^* g \ dx = 0$ |
| Use quantum mechanics when … | 1. $\lambda_{\text{particle}} \sim L_{\text{problem}}$ 2. $\Delta E \gtrsim k_bT$ (discrete energy spectrum) |
Quantum-Mechanical Postulates
- The state of a quantum-mechanical particle is completely specified by a wave function $\Psi(x, t)$. The probability that the particle will be found at time $t_0$ in a spatial interval of width $dx$ centered at $x_0$ is given by $\Psi^*(x_0, t_0)\Psi(x_0, t_0) dx$
- For every measurable property of a system, there exists a corresponding operator.
- In any single measurement of the observable that corresponds to the operator $\hat{A}$, the only values that will ever be measured are the eigenvalues of that operator.
- If the system is in a state described by the wave function $\Psi(x, t)$, and the value of the observatle $a$ is measured once on each of many identically prepared systems, the average value (expectation value) of all of the measurements is given by $$ \langle a \rangle = \dfrac{\displaystyle\int_{-\infty}^{\infty} \Psi^* \hat{A} \Psi \ dx}{\displaystyle\int_{-\infty}^{\infty} \Psi^*\Psi \ dx} $$
- The evolution in time of a quantum-mechanical system is governed by the time-dependent Schrödinger equation $$ \hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t} $$
Operators
| Description | 1D | 3D |
|---|---|---|
| Position | $\hat{x} = x$ | $\mathbf{\hat{x}} = \mathbf{x}$ |
| Linear momentum | $\hat{p}_x = -i\hbar \dfrac{d}{dx}$ | $\mathbf{\hat{p}} = -i\hbar\mathbf{\nabla}$ |
| Kinetic energy | $\hat{T} = \dfrac{\hat{p}_x^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}$ | $\mathbf{\hat{T}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2$ |
| Potential energy | $\hat{V} = V(x)$ | $\mathbf{\hat{V}} = V(\mathbf{x})$ |
| Total energy Hamiltonian | $\hat{H} = \hat{T} + \hat{V} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x)$ | $\mathbf{\hat{H}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x})$ |
Simple Quantum Systems
Stationary states
| Description | Equations |
|---|---|
| Time dependent Schrodinger equation | $\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}$ |
| Time independent Schrodinger equation | $\hat{H}\psi_n(x) = E_n \psi_n(x)$ |
| Stationary state wave function | $\Psi(x, t) = \psi(x) T(t)$ |
| Time component of wave function | $T(t) = e^{iEt/\hbar}$ |
| Probability of finding particle in an interval | $\mathrm{Prob}(x, x+dx) = \vert \Psi(x, t) \vert^2 dx = \vert \psi(x) \vert^2 dx$ |
| General solution as linear combination of stationary states | $\psi(x) = \sum\limits_n c_n \phi_n(x)$ |
| Expansion coefficients | $c_n = \langle \phi_n \vert \psi \rangle = \int \phi_n^* \psi \ dx$ |
| Normalization | $\sum\limits_n c_n = 1$ |
Particle in a 1D box
| Description | Equations |
|---|---|
| Time independent Schrodinger equation | $\left[ -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x) \right] \psi(x) = E\psi(x)$ |
| Wave function $n = 0, 1, 2, …$ |
$\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin\left(\dfrac{n \pi x}{L}\right)$ |
| Energy eigenvalues | $E_n = \dfrac{h^2}{8mL^2} n^2 = \dfrac{\hbar^2 \pi^2}{2mL^2}n^2$ |
Particle in a 3D box
| Description | Equations |
|---|---|
| Time independent Schrodinger equation | $\left[ -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})$ |
| Wave function $n_x = 0, 1, 2, …$ $n_y = 0, 1, 2, …$ $n_z = 0, 1, 2, …$ |
$\begin{aligned}&\psi_{n_x, n_y, n_z}(\mathbf{x}) \\ =& \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\ =& \sqrt{\dfrac{2}{L_x}}\sqrt{\dfrac{2}{L_y}}\sqrt{\dfrac{2}{L_z}} \sin\left(\dfrac{n_x \pi x}{L_x}\right)\sin\left(\dfrac{n_y \pi y}{L_y}\right)\sin\left(\dfrac{n_z \pi z}{L_z}\right)\end{aligned}$ |
| Energy eigenvalues | $E_n = \dfrac{h^2}{8m} \left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)$ |
Finite potential well
| Description | Equations |
|---|---|
| Potential | $V(x) = \begin{cases}0 & x\in [0, L] \\ V_0 & \mathrm{elsewhere}\end{cases}$ |
| Reflection probability | $R = \dfrac{(\sqrt{E} - \sqrt{E - V_0})^2}{(\sqrt{E} + \sqrt{E - V_0})^2}$ |
| Transmission probability | $T = \dfrac{4\sqrt{E(E - V_0)}}{(\sqrt{E} - \sqrt{E - V_0})^2}$ |
Commutators and Uncertainty
| Description | Equations |
|---|---|
| Commutator | $[A, B] = AB - BA$ |
| Condition of commutation | $[A, B] = 0$ |
| Standard deviation (uncertainty) | $\begin{aligned}\sigma_A &= \sqrt{\langle (A - \langle A \rangle^2 \rangle)} \\ &= \sqrt{\langle A^2 \rangle - \langle A \rangle^2}\end{aligned}$ |
| Heisenberg uncertainty principle (general) | $\sigma_A \sigma_B \ge \frac{1}{2} \vert\langle[\hat{A}, \hat{B}]\rangle\vert$ |
| Heisenberg uncertainty principle (position-momentum) | $\sigma_x \sigma_p \ge \frac{\hbar}{2}$ |
Spectroscopy
Dimer model
| Description | Equations |
|---|---|
| Hamiltonian of dimer | $\mathbf{H}_{\text{dimer}} = \dfrac{\mathbf{p}_1^2}{2m_1} + \dfrac{\mathbf{p}_2^2}{2m_2} + V(\vert \mathbf{x}_1 - \mathbf{x}_2\vert)$ |
| Total mass | $M = m_1 + m_2$ |
| Reduced mass | $\mu = \dfrac{m_1m_2}{m_1 + m_2}$ |
| Position in center of mass (COM) coordinate | $\mathbf{X} = \dfrac{m_1 \mathbf{x}_1 + m_2 \mathbf{x}_2}{M}$ |
| Momentum in center of mass (COM) coordinate | $\mathbf{\Pi} = \mathbf{p}_1 + \mathbf{p}_2$ |
| Position in relative coordinate | $\mathbf{x} = \mathbf{x}_1 - \mathbf{x}_2 \equiv \mathbf{r}$ |
| Momentum in relative coordinate | $\mathbf{p} = \dfrac{m_1 \mathbf{p}_1 + m_2 \mathbf{p}_2}{M}$ |
| Hamiltonian of dimer | $\begin{aligned}\mathbf{H}_{\text{dimer}} &= \mathbf{H}_{\text{free}} + \mathbf{H}_{\text{int}} \\ &= \underbrace{\dfrac{\mathbf{\Pi}^2}{2M}}_{\text{COM coord}} + \underbrace{\dfrac{\mathbf{p}^2}{2\mu} + V(\vert\mathbf{x}\vert)}_{\text{rel coord}}\end{aligned}$ |
| Free particle Hamiltonian | $\mathbf{H}_{\text{free}} = \dfrac{\mathbf{\Pi}^2}{2M}$ |
| Internal Hamiltonian | $\mathbf{H}_{\text{int}} = \dfrac{\mathbf{p}^2}{2\mu} + V(\vert\mathbf{x}\vert)$ |
| Dimer wave function | $\Psi_{\text{dimer}} = \Phi(\mathbf{X}) \psi(\mathbf{x})$ |
| Free particle (COM) wave function | $\Phi(\mathbf{X}) = e^{\pm i\mathbf{\Pi X}/\hbar}$ |
| Internal Hamiltonian Schrodinger equation | $\left[ -\dfrac{\hbar^2}{2\mu}\nabla_{\mathbf{x}}^2 + V(\mathbf{r}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})$ |
| Laplacian in spherical coordinate $\theta \in [0, \pi]$ $\phi \in [0, 2\pi]$ |
$\nabla^2 = \underbrace{\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right)}_{\text{radial breathing KE, vibration}} + \underbrace{\dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2 \sin^2 \theta}\dfrac{\partial^2}{\partial\phi^2}}_{\text{angular breathing KE, rotation}}$ |
| Dimer Hamiltonian | $\hat{H}_{\text{dimer}} = \hat{H}_{\text{COM}} + \hat{H}_{\text{vib}} + \hat{H}_{\text{rot}}$ |
| Dimer total energy (see below) | $E = \dfrac{\Pi^2}{2M} + \hbar\omega_0 (n+\frac{1}{2}) + \dfrac{\hbar^2 l(l+1)}{2I}$ |
Vibration: quantum harmonic oscillator
| Description | Equations |
|---|---|
| Vibrational Schrodinger equation | $\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + V(r) \right]\psi(\mathbf{x}) = E\psi(\mathbf{x})$ |
| Wave function | $\psi(\mathbf{x}) = R(r)Y(\theta, \phi)$ |
| Harmonic approximation | $V(r) \approx \frac{1}{2}kr^2$ |
| Spring constant | $k = \mu\omega_0^2$ |
| Vibrational Schrodinger equation | $\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{1}{2}kr^2 \right]\psi(r) = E\psi(r)$ |
| Wave function $n = 0, 1, 2, …$ |
$\psi(r) = \dfrac{1}{\sqrt{2^2 n!}}\left(\dfrac{\alpha}{\pi}\right)^{1/4} H_n(\sqrt{\alpha} r) e^{-\alpha r^2 / 2}$ |
| Hermite polynomials | $H_n(r) = (-)^n e^{x^2}\left(\dfrac{d^n}{dx^n}\right)e^{-x^2}$ |
| Constant | $\alpha = \dfrac{m\omega_0}{\hbar}$ |
| Energy eigenvalue $n = 0, 1, 2, …$ |
$E_n = (n + \frac{1}{2})\hbar \omega_0$ |
| Transition dipole moment | $\vec{\mu}_{fi} = \dfrac{d\vec{\mu}(x_0)}{dx} \langle \psi_f \vert \hat{x} \vert \psi_i \rangle$ |
| Vibrational selection rule | $\Delta n = \pm 1$ |
Rotation: rigid rotor
Classical rigid rotor
| Description | Equations |
|---|---|
| Angular momentum | $\mathbf{L} = \mathbf{x} \times \mathbf{p} = I\vec{\omega}$ |
| Linear velocity | $\vec{v} = R_0 \vec{\omega}$ |
| Moment of inertia | $I = mR_0^2$ |
| Rotational kinetic energy | $E = \dfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}$ |
Quantum rigid rotor
| Description | Equations |
|---|---|
| Angular momentum operator | $\hat{\mathbf{L}} = \hat{\mathbf{x}} \times \hat{\mathbf{p}}$ |
| z-component of angular momentum operator | $L_x = \dfrac{\hbar}{i}\dfrac{\partial}{\partial\phi}$ |
| Magnitude of angular momentum operator | $\hat{\mathbf{L}}^2 = L^2 = -\hbar^2 \left[ \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta} \left( \sin\theta\dfrac{\partial}{\partial\theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2}{\partial\phi^2} \right) \right]$ |
| Components of $\hat{\mathbf{L}}$ does not commute | $[\hat{L}_i, \hat{L}_j] = i\hbar \hat{L}_k$ |
| Components of $\hat{\mathbf{L}}$ commute with its magnitude | $[\hat{L}_i, L^2] = 0$ |
| Description | Equations |
|---|---|
| Rotational Schrodinger equation | $-\dfrac{\hbar^2}{2\mu R_0^2}\left[ \dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2 \sin^2 \theta}\dfrac{\partial^2}{\partial\phi^2} \right]Y(\theta, \phi) = EY(\theta, \phi)$ |
| Spherical harmonics | $Y_l^m(\theta, \phi) = (-)^m \sqrt{\dfrac{(2l+1)}{4\pi}\dfrac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}$ |
| Legendre polynomial | $P_l^m(x) = \dfrac{1}{2^l l!}(1-x^2)^{m/2} \dfrac{d^{(l+m)}}{dx^{(l+m)}}(x^2-1)^l$ |
| Energy eigenvalues $l = 0, 1, 2, …$ |
$E_l = \dfrac{\hbar^2}{2I}l(l+1)$ |
| Angular momentum eigenvalues $l = 0, 1, 2, …$ |
$L^2 Y = \hbar^2l(l+1) Y$ |
| z-component eigenvalues $m = -l, …, 0, …, l$ |
$L_z Y = \hbar m Y$ |
| Transition dipole moment | $\mu_{fi} = \langle \psi_f \vert \mu_z \cos\theta \vert \psi_i \rangle$ |
| Rotational selection rule | $\Delta l = \pm 1, \Delta m = 0$ |
Hydrogen atom
| Description | Equations |
|---|---|
| Hydrogen atom Schrodinger equation | $\left[ -\dfrac{\hbar}{2m_e}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{\vec{L}}{2m_er^2} - \dfrac{e^2}{r} \right]\psi(x) = E\psi(x)$ |
| Effective potential | $V_{\text{eff}} = \dfrac{\hbar l(l+1)}{2mr^2} - \dfrac{e^2}{r}$ |
| Wave function $n = 1, 2, …$ |
$\psi_{nlm}(x) = R_{nl}(r)Y_l^m(\theta, \phi)$ |
| Energy eigenvalues $n = 1, 2, …$ |
$E_n = -\dfrac{e^2}{2a_0}\dfrac{1}{n^2} = -\dfrac{me^4}{2\hbar^2}\dfrac{1}{n^2} - \dfrac{R_H}{n^2}$ |
| Rydberg’s constant | $R_H = 2.179 \times 10^{-18} \mathrm{J} = 13.6 \ \mathrm{eV}$ |
| Bohr’s radius | $a_0 = \dfrac{\hbar^2}{me^2}$ |
| Radial probability distribution | $P_{nl}(r) dr = r^2 R^2_{nl}(r) dr$ |
Many Electron and Proton System
Many electron atom
| Description | Equations |
|---|---|
| Helium Schrodinger equation | $\left[ \underbrace{-\dfrac{\hbar^2}{2m}\nabla_1^2}_{\text{KE of }e^-_1} \overbrace{-\dfrac{\hbar^2}{2m}\nabla_2^2}^{\text{KE of }e^-_2} \underbrace{-\dfrac{2e^2}{\vert\mathbf{x}_1\vert}}_{e^-_1\text{-N attraction}} \overbrace{-\dfrac{2e^2}{\vert\mathbf{x}_2\vert}}^{e^-_2\text{-N attraction}} + \underbrace{\dfrac{e^2}{\vert\mathbf{x}_1 - \mathbf{x}_2\vert}}_{e^-_1 \text{-} e^-_2 \text{repulsion}} \right] \psi(\mathbf{x}_1, \mathbf{x}_2) = E \psi(\mathbf{x}_1, \mathbf{x}_2)$ |
| Orbital approximation | $\psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) = \phi(\mathbf{x}_1)\phi(\mathbf{x}_2)\dots\phi(\mathbf{x}_n)$ |
| Hartree orbital equations | $\left[ -\dfrac{\hbar^2 \nabla_i^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} + \sum\limits^N_{j=1, j\not= i} \displaystyle\int \dfrac{e^2 \phi_j^*(\mathbf{x}')\phi_j(\mathbf{x}')}{\vert\mathbf{x} - \mathbf{x}' \vert} d^3\mathbf{x}' \right] \phi_i(\mathbf{x}) = \varepsilon_{ii} \phi_i(\mathbf{x})$ |
Spin
| Description | Equations |
|---|---|
| Components of $\hat{\mathbf{S}}$ does not commute | $[\hat{S}_i, \hat{S}_j] = i\hbar \hat{S}_k$ |
| Components of $\hat{\mathbf{S}}$ commute with its magnitude | $[\hat{S}_i, S^2] = 0$ |
| Eigenvalue of $\hat{\mathbf{S}}^2$ | $\hat{\mathbf{S}}^2 \leftrightarrow \hbar^2 s(s+1)$ |
| Eigenvalue of $\hat{S}_z$ | $\hat{S}_z \leftrightarrow \hbar m_s$ |
Electron spin
| Description | Equations |
|---|---|
| Electron spin | $s = \frac{1}{2}$ |
| Spin up function | $\alpha(m_s) = \begin{cases} 1 & m_s = +\frac{1}{2} \\ 0 & m_s = -\frac{1}{2} \end{cases}$ |
| Spin down function | $\beta(m_s) = \begin{cases} 0 & m_s = +\frac{1}{2} \\ 1 & m_s = -\frac{1}{2} \end{cases}$ |
| $\alpha$ is eigenfunction of $\hat{S_z}$ | $\hat{S_z} \alpha = +\frac{1}{2}\hbar \alpha$ |
| $\beta$ is eigenfunction of $\hat{S_z}$ | $\hat{S_z} \beta = -\frac{1}{2}\hbar \beta$ |
| $\alpha, \beta$ are eigenfunctions of $\hat{S^2}$ | $\hat{S^2} \alpha = \hbar^2 s(s+1) \alpha = \frac{3}{4} \hbar^2 \alpha \newline \hat{S^2} \beta = \hbar^2 s(s+1) \beta = \frac{3}{4} \hbar^2 \beta$ |
| Normalization | $\sum\limits_{m_s} \alpha^*\alpha = \sum\limits_{m_s} \beta^*\beta = 1$ |
| Orthogonality | $\sum\limits_{m_s} \alpha^*\beta = \sum\limits_{m_s} \beta^*\alpha = 0$ |
Identical particles
| Description | Equations |
|---|---|
| Spin-spin permutation operator | $P_{ij} \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_j, \dots, \mathbf{r}_N) = \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_j, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N)$ |
| Doing nothing | $P_{ij}P_{ij} = 1$ |
| Symmetric eigenvalue | $\lambda = 1$ |
| Anti-symmetric eigenvalue | $\lambda = -1$ |
| Fermions (e.g. electron) | $\frac{1}{2}$-integer spin, anti-symmetric |
| Bosons | integer spin, symmetric |
| Pauli exclusion principle | $\psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = - \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = 0$ |
| Slater determinant | $\Psi(\mathrm{x}_1, \mathrm{x}_2, \cdots, \mathrm{x}_N) = \dfrac{1}{\sqrt{N!}} \begin{vmatrix}\chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix}$ |
| Hartree-Fock orbital equations | $\left[ -\dfrac{\hbar^2\nabla^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} \right] \phi_i(\mathbf{r}) + \displaystyle\sum_{j=1}^N \left[ \phi_i(\mathbf{r}) \int \dfrac{e^2 \phi^*_j(\mathbf{r}) \phi_j(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' - \phi_j(\mathbf{r}) \int \dfrac{e^2 \phi^*_j(\mathbf{r}) \phi_i(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' \right] = \varepsilon_i \phi_i(\mathbf{r})$ |
| Molecular orbital by linear combination of atomic orbitals (MO-LCAO) | $\psi(\mathbf{x}) = c_1 \phi_1 (\mathbf{x}) + c_2 \phi_2 (\mathbf{x}) \newline \mathrm{MO} = c_1 (\mathrm{AO}) + c_2 (\mathrm{AO})$ |
| Variational principle | $E = \dfrac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle}$ |