Equation sheet#
Units#
Constants#
Energy Constant |
Value |
|---|---|
\(h = 6.63 \cdot 10^{-34} \, [J \cdot s]\) |
\(m_u = 1.66 \cdot 10^{-27} \, [kg]\) |
\(k_B = 1.38 \cdot 10^{-23} \, [J \cdot K^{-1}]\) |
\(N_A = 6.02 \cdot 10^{23} \, [mol^{-1}]\) |
\(c = 3.0 \cdot 10^{8} \, [m \cdot s^{-1}]\) |
\(m_e = 9.1093837015 \cdot 10^{-31} \, [kg]\) |
\(R_H = 109680 \, [cm^{-1}]\) |
\(e = 1.602176634 \cdot 10^{-19} \, [C]\) |
Energy Unit Converter#
See here for an interactive option.
Unit |
\(J\) |
\(eV\) |
\(cm^{-1}\) |
\(hartree\) |
\(Hz\) |
|---|---|---|---|---|---|
\(J\) |
\(1\) |
\(6.24181 \cdot 10^{18}\) |
\(5.03445 \cdot 10^{22}\) |
\(2.294 \cdot 10^{17}\) |
\(1.50930 \cdot 10^{33}\) |
\(eV\) |
\(1.60210 \cdot 10^{-19}\) |
\(1\) |
\(8065.73\) |
\(0.0367502\) |
\(2.41804 \cdot 10^{14}\) |
\(cm^{-1}\) |
\(1.98630 \cdot 10^{-23}\) |
\(1.23981 \cdot 10^{-4}\) |
\(1\) |
\(4.55633 \cdot 10^{-6}\) |
\(2.99793 \cdot 10^{10}\) |
\(hartree\) |
\(43.60 \cdot 10^{-19}\) |
\(27.2107\) |
\(219474.63\) |
\(1\) |
\(6.57966 \cdot 10^{15}\) |
\(Hz\) |
\(6.62561 \cdot 10^{-34}\) |
\(4.13558 \cdot 10^{-15}\) |
\(3.33565 \cdot 10^{-11}\) |
\(1.51983 \cdot 10^{-16}\) |
\(1\) |
From classical to Quantum#
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 |
\(E_k = h \nu - \Phi\) |
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 |
\(\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)\) |
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 |
\(\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)\) |
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. \(\lambda_{\text{particle}} \sim L_{\text{problem}}\) |
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 |
\(\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin\left(\dfrac{n \pi x}{L}\right)\) |
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 |
\(\begin{aligned}&\psi_{n_x, n_y, n_z}(\mathbf{x}) \\\ =& \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\\ =& \sqrt{\dfrac{2}{L_x}}\sqrt{\dfrac{2}{L_y}}\sqrt{\dfrac{2}{L_z}} \sin\left(\dfrac{n_x \pi x}{L_x}\right)\sin\left(\dfrac{n_y \pi y}{L_y}\right)\sin\left(\dfrac{n_z \pi z}{L_z}\right)\end{aligned}\) |
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#
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 |
\(\psi(r) = \dfrac{1}{\sqrt{2^2 n!}}\left(\dfrac{\alpha}{\pi}\right)^{1/4} H_n(\sqrt{\alpha} r) e^{-\alpha r^2 / 2}\) |
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 |
\(E_n = (n + \frac{1}{2})\hbar \omega_0\) |
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: quantum 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 |
\(E_l = \dfrac{\hbar^2}{2I}l(l+1)\) |
Angular momentum eigenvalues |
\(L^2 Y = \hbar^2l(l+1) Y\) |
z-component eigenvalues |
\(L_z Y = \hbar m Y\) |
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 |
\(\psi_{nlm}(x) = R_{nl}(r)Y_l^m(\theta, \phi)\) |
Energy eigenvalues |
\(E_n = -\dfrac{e^2}{2a_0}\dfrac{1}{n^2} = -\dfrac{me^4}{2\hbar^2}\dfrac{1}{n^2} - \dfrac{R_H}{n^2}\) |
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 |
\(E = \dfrac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle}\) |