Equation Sheets
Equation Sheets

CHEM 455 Physical Chemistry

  
Contents
Description Equations
Energy quantization E=nhνE = nh\nu
Average energy of an oscillating dipole ⟨Eosc⟩=hνehν/kBT−1\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}
Spectral radiation density of blackbody (Planck) ρ(ν,T) dν=8πhν3c31ehν/kBT−1 dν\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) ρ(ν,T) dν=8πhν3c3kBTdν\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3} k_B T d\nu
Description Equations
Energy of light E=hνE = h \nu
Photoelectric effect
Kinetic energy of ejected photoelectron		 Ek=hν−ΦE_k = h \nu - \Phi
de Broglie relation p=hλp = \dfrac{h}{\lambda}
Kinetic energy Ek=12mv2=p22mE_k = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}
Description Equations
Hydrogen emission lines
n2>n1n_2 > n_1		 ν~=1λ=RH(1n12−1n22)\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)
Bohr’s radius r=4πε0ℏ2mee2r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}
Energy level in Bohr’s model En=−mee48ε02h2n2E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}
Emission of hydrogen atom
n2>n1n_2 > n_1		 ν=mee48ε02h3(1n12−1n22)\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)
Description Equations
Classical nondispersive wave equation ∂Ψ(x,t)∂x2=1v2∂Ψ(x,t)∂t2\dfrac{\partial \Psi(x, t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial \Psi(x, t)}{\partial t^2}
Wave number k=2πλk = \dfrac{2\pi}{\lambda}
Frequency ν=1T\nu = \dfrac{1}{T}
Angular frequency ω=2πT=2πν\omega = \dfrac{2\pi}{T} = 2\pi\nu
Wave speed v=λνv = \lambda\nu
Euler’s formula eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta
Solution of wave equation Ψ(x,t)=Asin⁡(kx−ωt+ϕ)=Re(Aei(kx−ωt+ϕ′))\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 Ψ(x,t)=A[sin⁡(kx−ωt)+sin⁡(kx+ωt)]=2Asin⁡(kx)cos⁡(ωt)=ψ(x)cos⁡(ωt)\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 −ℏ22md2ψ(x)dx2+V(x)ψ(x)=Eψ(x)-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)
Time-dependent Schrodinger equation −ℏ22m∂2Ψ(x,t)∂x2+V(x,t)Ψ(x,t)=iℏΨ(x,t)-\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 Ψ(x,t)=ψ(x)e−i(E/ℏ)t\Psi(x, t) = \psi(x) e^{-i(E/\hbar)t}
Normalization ∥f(x)∥=∫Df∗f dx=1\Vert f(x) \Vert = \int_D f^* f \ dx = 1
Orthogonality ∫Df∗g dx=0\int_D f^* g \ dx = 0
Use quantum mechanics when … 1. λparticle∼Lproblem\lambda_{\text{particle}} \sim L_{\text{problem}}
2. ΔE≳kbT\Delta E \gtrsim k_bT (discrete energy spectrum)
  1. The state of a quantum-mechanical particle is completely specified by a wave function Ψ(x,t)\Psi(x, t). The probability that the particle will be found at time t0t_0 in a spatial interval of width dxdx centered at x0x_0 is given by Ψ∗(x0,t0)Ψ(x0,t0)dx\Psi^*(x_0, t_0)\Psi(x_0, t_0) dx
  2. For every measurable property of a system, there exists a corresponding operator.
  3. In any single measurement of the observable that corresponds to the operator A^\hat{A}, the only values that will ever be measured are the eigenvalues of that operator.
  4. If the system is in a state described by the wave function Ψ(x,t)\Psi(x, t), and the value of the observatle aa is measured once on each of many identically prepared systems, the average value (expectation value) of all of the measurements is given by ⟨a⟩=∫−∞∞Ψ∗A^Ψ dx∫−∞∞Ψ∗Ψ dx \langle a \rangle = \dfrac{\displaystyle\int_{-\infty}^{\infty} \Psi^* \hat{A} \Psi \ dx}{\displaystyle\int_{-\infty}^{\infty} \Psi^*\Psi \ dx}
  5. The evolution in time of a quantum-mechanical system is governed by the time-dependent Schrödinger equation H^Ψ(x,t)=iℏ∂Ψ(x,t)∂t \hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}
Description 1D 3D
Position x^=x\hat{x} = x x^=x\mathbf{\hat{x}} = \mathbf{x}
Linear momentum p^x=−iℏddx\hat{p}_x = -i\hbar \dfrac{d}{dx} p^=−iℏ∇\mathbf{\hat{p}} = -i\hbar\mathbf{\nabla}
Kinetic energy T^=p^x22m=−ℏ22md2dx2\hat{T} = \dfrac{\hat{p}_x^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} T^=−ℏ22m∇2\mathbf{\hat{T}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2
Potential energy V^=V(x)\hat{V} = V(x) V^=V(x)\mathbf{\hat{V}} = V(\mathbf{x})
Total energy Hamiltonian H^=T^+V^=−ℏ22md2dx2+V(x)\hat{H} = \hat{T} + \hat{V} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x) H^=−ℏ22m∇2+V(x)\mathbf{\hat{H}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x})
Description Equations
Time dependent Schrodinger equation H^Ψ(x,t)=iℏ∂Ψ(x,t)∂t\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}
Time independent Schrodinger equation H^ψn(x)=Enψn(x)\hat{H}\psi_n(x) = E_n \psi_n(x)
Stationary state wave function Ψ(x,t)=ψ(x)T(t)\Psi(x, t) = \psi(x) T(t)
Time component of wave function T(t)=eiEt/ℏT(t) = e^{iEt/\hbar}
Probability of finding particle in an interval Prob(x,x+dx)=∣Ψ(x,t)∣2dx=∣ψ(x)∣2dx\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 ψ(x)=∑ncnϕn(x)\psi(x) = \sum\limits_n c_n \phi_n(x)
Expansion coefficients cn=⟨ϕn∣ψ⟩=∫ϕn∗ψ dxc_n = \langle \phi_n \vert \psi \rangle = \int \phi_n^* \psi \ dx
Normalization ∑ncn=1\sum\limits_n c_n = 1
Description Equations
Time independent Schrodinger equation [−ℏ22md2dx2+V(x)]ψ(x)=Eψ(x)\left[ -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x) \right] \psi(x) = E\psi(x)
Wave function
n=0,1,2,…n = 0, 1, 2, …		 ψn(x)=2Lsin⁡(nπxL)\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin\left(\dfrac{n \pi x}{L}\right)
Energy eigenvalues En=h28mL2n2=ℏ2π22mL2n2E_n = \dfrac{h^2}{8mL^2} n^2 = \dfrac{\hbar^2 \pi^2}{2mL^2}n^2
Description Equations
Time independent Schrodinger equation [−ℏ22m∇2+V(x)]ψ(x)=Eψ(x)\left[ -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})
Wave function
nx=0,1,2,…n_x = 0, 1, 2, …
ny=0,1,2,…n_y = 0, 1, 2, …
nz=0,1,2,…n_z = 0, 1, 2, …		 ψnx,ny,nz(x)=ψnx(x)ψny(y)ψnz(z)=2Lx2Ly2Lzsin⁡(nxπxLx)sin⁡(nyπyLy)sin⁡(nzπzLz)\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 En=h28m(nx2Lx2+ny2Ly2+nz2Lz2)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)
Description Equations
Potential V(x)={0x∈[0,L]V0elsewhereV(x) = \begin{cases}0 & x\in [0, L] \\ V_0 & \mathrm{elsewhere}\end{cases}
Reflection probability R=(E−E−V0)2(E+E−V0)2R = \dfrac{(\sqrt{E} - \sqrt{E - V_0})^2}{(\sqrt{E} + \sqrt{E - V_0})^2}
Transmission probability T=4E(E−V0)(E−E−V0)2T = \dfrac{4\sqrt{E(E - V_0)}}{(\sqrt{E} - \sqrt{E - V_0})^2}
Description Equations
Commutator [A,B]=AB−BA[A, B] = AB - BA
Condition of commutation [A,B]=0[A, B] = 0
Standard deviation (uncertainty) σA=⟨(A−⟨A⟩2⟩)=⟨A2⟩−⟨A⟩2\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) σAσB≥12∣⟨[A^,B^]⟩∣\sigma_A \sigma_B \ge \frac{1}{2} \vert\langle[\hat{A}, \hat{B}]\rangle\vert
Heisenberg uncertainty principle (position-momentum) σxσp≥ℏ2\sigma_x \sigma_p \ge \frac{\hbar}{2}
Description Equations
Hamiltonian of dimer Hdimer=p122m1+p222m2+V(∣x1−x2∣)\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=m1+m2M = m_1 + m_2
Reduced mass μ=m1m2m1+m2\mu = \dfrac{m_1m_2}{m_1 + m_2}
Position in center of mass (COM) coordinate X=m1x1+m2x2M\mathbf{X} = \dfrac{m_1 \mathbf{x}_1 + m_2 \mathbf{x}_2}{M}
Momentum in center of mass (COM) coordinate Π=p1+p2\mathbf{\Pi} = \mathbf{p}_1 + \mathbf{p}_2
Position in relative coordinate x=x1−x2≡r\mathbf{x} = \mathbf{x}_1 - \mathbf{x}_2 \equiv \mathbf{r}
Momentum in relative coordinate p=m1p1+m2p2M\mathbf{p} = \dfrac{m_1 \mathbf{p}_1 + m_2 \mathbf{p}_2}{M}
Hamiltonian of dimer Hdimer=Hfree+Hint=Π22M⏟COM coord+p22μ+V(∣x∣)⏟rel coord\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 Hfree=Π22M\mathbf{H}_{\text{free}} = \dfrac{\mathbf{\Pi}^2}{2M}
Internal Hamiltonian Hint=p22μ+V(∣x∣)\mathbf{H}_{\text{int}} = \dfrac{\mathbf{p}^2}{2\mu} + V(\vert\mathbf{x}\vert)
Dimer wave function Ψdimer=Φ(X)ψ(x)\Psi_{\text{dimer}} = \Phi(\mathbf{X}) \psi(\mathbf{x})
Free particle (COM) wave function Φ(X)=e±iΠX/ℏ\Phi(\mathbf{X}) = e^{\pm i\mathbf{\Pi X}/\hbar}
Internal Hamiltonian Schrodinger equation [−ℏ22μ∇x2+V(r)]ψ(x)=Eψ(x)\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
θ∈[0,π]\theta \in [0, \pi]
ϕ∈[0,2π]\phi \in [0, 2\pi]		 ∇2=1r2∂∂r(r2∂∂r)⏟radial breathing KE, vibration+1r2sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1r2sin⁡2θ∂2∂ϕ2⏟angular breathing KE, rotation\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 H^dimer=H^COM+H^vib+H^rot\hat{H}_{\text{dimer}} = \hat{H}_{\text{COM}} + \hat{H}_{\text{vib}} + \hat{H}_{\text{rot}}
Dimer total energy (see below) E=Π22M+ℏω0(n+12)+ℏ2l(l+1)2IE = \dfrac{\Pi^2}{2M} + \hbar\omega_0 (n+\frac{1}{2}) + \dfrac{\hbar^2 l(l+1)}{2I}
Description Equations
Vibrational Schrodinger equation [−ℏ22μ1r2∂∂r(r2∂∂r)+V(r)]ψ(x)=Eψ(x)\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 ψ(x)=R(r)Y(θ,ϕ)\psi(\mathbf{x}) = R(r)Y(\theta, \phi)
Harmonic approximation V(r)≈12kr2V(r) \approx \frac{1}{2}kr^2
Spring constant k=μω02k = \mu\omega_0^2
Vibrational Schrodinger equation [−ℏ22μ1r2∂∂r(r2∂∂r)+12kr2]ψ(r)=Eψ(r)\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,…n = 0, 1, 2, …		 ψ(r)=122n!(απ)1/4Hn(αr)e−αr2/2\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 Hn(r)=(−)nex2(dndxn)e−x2H_n(r) = (-)^n e^{x^2}\left(\dfrac{d^n}{dx^n}\right)e^{-x^2}
Constant α=mω0ℏ\alpha = \dfrac{m\omega_0}{\hbar}
Energy eigenvalue
n=0,1,2,…n = 0, 1, 2, …		 En=(n+12)ℏω0E_n = (n + \frac{1}{2})\hbar \omega_0
Transition dipole moment μ⃗fi=dμ⃗(x0)dx⟨ψf∣x^∣ψi⟩\vec{\mu}_{fi} = \dfrac{d\vec{\mu}(x_0)}{dx} \langle \psi_f \vert \hat{x} \vert \psi_i \rangle
Vibrational selection rule Δn=±1\Delta n = \pm 1
Description Equations
Angular momentum L=x×p=Iω⃗\mathbf{L} = \mathbf{x} \times \mathbf{p} = I\vec{\omega}
Linear velocity v⃗=R0ω⃗\vec{v} = R_0 \vec{\omega}
Moment of inertia I=mR02I = mR_0^2
Rotational kinetic energy E=12Iω2=L22IE = \dfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}
Description Equations
Angular momentum operator L^=x^×p^\hat{\mathbf{L}} = \hat{\mathbf{x}} \times \hat{\mathbf{p}}
z-component of angular momentum operator Lx=ℏi∂∂ϕL_x = \dfrac{\hbar}{i}\dfrac{\partial}{\partial\phi}
Magnitude of angular momentum operator L^2=L2=−ℏ2[1sin⁡θ∂∂θ(sin⁡θ∂∂θ+1sin⁡2θ∂2∂ϕ2)]\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 L^\hat{\mathbf{L}} does not commute [L^i,L^j]=iℏL^k[\hat{L}_i, \hat{L}_j] = i\hbar \hat{L}_k
Components of L^\hat{\mathbf{L}} commute with its magnitude [L^i,L2]=0[\hat{L}_i, L^2] = 0
Description Equations
Rotational Schrodinger equation −ℏ22μR02[1r2sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1r2sin⁡2θ∂2∂ϕ2]Y(θ,ϕ)=EY(θ,ϕ)-\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 Ylm(θ,ϕ)=(−)m(2l+1)4π(l−m)!(l+m)!Plm(cos⁡θ)eimϕY_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 Plm(x)=12ll!(1−x2)m/2d(l+m)dx(l+m)(x2−1)lP_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,…l = 0, 1, 2, …		 El=ℏ22Il(l+1)E_l = \dfrac{\hbar^2}{2I}l(l+1)
Angular momentum eigenvalues
l=0,1,2,…l = 0, 1, 2, …		 L2Y=ℏ2l(l+1)YL^2 Y = \hbar^2l(l+1) Y
z-component eigenvalues
m=−l,…,0,…,lm = -l, …, 0, …, l		 LzY=ℏmYL_z Y = \hbar m Y
Transition dipole moment μfi=⟨ψf∣μzcos⁡θ∣ψi⟩\mu_{fi} = \langle \psi_f \vert \mu_z \cos\theta \vert \psi_i \rangle
Rotational selection rule Δl=±1,Δm=0\Delta l = \pm 1, \Delta m = 0
Description Equations
Hydrogen atom Schrodinger equation [−ℏ2me1r2∂∂r(r2∂∂r)+L⃗2mer2−e2r]ψ(x)=Eψ(x)\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 Veff=ℏl(l+1)2mr2−e2rV_{\text{eff}} = \dfrac{\hbar l(l+1)}{2mr^2} - \dfrac{e^2}{r}
Wave function
n=1,2,…n = 1, 2, …		 ψnlm(x)=Rnl(r)Ylm(θ,ϕ)\psi_{nlm}(x) = R_{nl}(r)Y_l^m(\theta, \phi)
Energy eigenvalues
n=1,2,…n = 1, 2, …		 En=−e22a01n2=−me42ℏ21n2−RHn2E_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 RH=2.179×10−18J=13.6 eVR_H = 2.179 \times 10^{-18} \mathrm{J} = 13.6 \ \mathrm{eV}
Bohr’s radius a0=ℏ2me2a_0 = \dfrac{\hbar^2}{me^2}
Radial probability distribution Pnl(r)dr=r2Rnl2(r)drP_{nl}(r) dr = r^2 R^2_{nl}(r) dr
Description Equations
Helium Schrodinger equation [−ℏ22m∇12⏟KE of e1−−ℏ22m∇22⏞KE of e2−−2e2∣x1∣⏟e1−-N attraction−2e2∣x2∣⏞e2−-N attraction+e2∣x1−x2∣⏟e1−-e2−repulsion]ψ(x1,x2)=Eψ(x1,x2)\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 ψ(x1,x2,…,xn)=ϕ(x1)ϕ(x2)…ϕ(xn)\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 [−ℏ2∇i22m−Ze2∣x∣+∑j=1,j≠iN∫e2ϕj∗(x′)ϕj(x′)∣x−x′∣d3x′]ϕi(x)=εiiϕi(x)\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})
Description Equations
Components of S^\hat{\mathbf{S}} does not commute [S^i,S^j]=iℏS^k[\hat{S}_i, \hat{S}_j] = i\hbar \hat{S}_k
Components of S^\hat{\mathbf{S}} commute with its magnitude [S^i,S2]=0[\hat{S}_i, S^2] = 0
Eigenvalue of S^2\hat{\mathbf{S}}^2 S^2↔ℏ2s(s+1)\hat{\mathbf{S}}^2 \leftrightarrow \hbar^2 s(s+1)
Eigenvalue of S^z\hat{S}_z S^z↔ℏms\hat{S}_z \leftrightarrow \hbar m_s
Description Equations
Electron spin s=12s = \frac{1}{2}
Spin up function α(ms)={1ms=+120ms=−12\alpha(m_s) = \begin{cases} 1 & m_s = +\frac{1}{2} \\ 0 & m_s = -\frac{1}{2} \end{cases}
Spin down function β(ms)={0ms=+121ms=−12\beta(m_s) = \begin{cases} 0 & m_s = +\frac{1}{2} \\ 1 & m_s = -\frac{1}{2} \end{cases}
α\alpha is eigenfunction of Sz^\hat{S_z} Sz^α=+12ℏα\hat{S_z} \alpha = +\frac{1}{2}\hbar \alpha
β\beta is eigenfunction of Sz^\hat{S_z} Sz^β=−12ℏβ\hat{S_z} \beta = -\frac{1}{2}\hbar \beta
α,β\alpha, \beta are eigenfunctions of S2^\hat{S^2} S2^α=ℏ2s(s+1)α=34ℏ2αS2^β=ℏ2s(s+1)β=34ℏ2β\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 ∑msα∗α=∑msβ∗β=1\sum\limits_{m_s} \alpha^*\alpha = \sum\limits_{m_s} \beta^*\beta = 1
Orthogonality ∑msα∗β=∑msβ∗α=0\sum\limits_{m_s} \alpha^*\beta = \sum\limits_{m_s} \beta^*\alpha = 0
Description Equations
Spin-spin permutation operator Pijψ(r1,r2,…,ri,…,rj,…,rN)=ψ(r1,r2,…,rj,…,ri,…,rN)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 PijPij=1P_{ij}P_{ij} = 1
Symmetric eigenvalue λ=1\lambda = 1
Anti-symmetric eigenvalue λ=−1\lambda = -1
Fermions (e.g. electron) 12\frac{1}{2}-integer spin, anti-symmetric
Bosons integer spin, symmetric
Pauli exclusion principle ψ(r1,r2,…,ri,…,ri,…,rN)=−ψ(r1,r2,…,ri,…,ri,…,rN)=0\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 Ψ(x1,x2,⋯ ,xN)=1N!∣χ1(x1)χ2(x1)⋯χN(x1)χ1(x2)χ2(x2)⋯χN(x2)⋮⋮⋱⋮χ1(xN)χ2(xN)⋯χN(xN)∣\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 [−ℏ2∇22m−Ze2∣x∣]ϕi(r)+∑j=1N[ϕi(r)∫e2ϕj∗(r)ϕj(r)∣x−x′∣d3r′−ϕj(r)∫e2ϕj∗(r)ϕi(r)∣x−x′∣d3r′]=εiϕi(r)\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) ψ(x)=c1ϕ1(x)+c2ϕ2(x)MO=c1(AO)+c2(AO)\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=⟨ψ∣H∣ψ⟩⟨ψ∣ψ⟩E = \dfrac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle}
Updated on 2021-12-23
Back | Home