CHE30324-outline.org

The Classical Foundations

Lecture 0: Introduction

1. Burning lighter
2. Foundations of Physical Chemistry
   1. Quantum mechanics
   2. Statistical mechanics
   3. Thermodynamics, kinetics, spectroscopy
   4. Physical and chemical properties of matter

Lecture 1: Basic statistics

Discrete probability distributions—Coin flip

1. Example of Bernoulli trial, $2^n$ possible outcomes from $n$ flips
2. Number of ways to get $i$ heads in $n$ flips, $_nC_i=n!/i!(n-i)!$
3. Probability of $i$ heads $P_i \propto\ _nC_i$
4. Normalized probability, $˜ P_i = P_i/∑_i P_i =\ _nC_i/2^n$
5. Expectation value $\langle i \rangle = ∑_i i ˜ P_i$

Continuous distributions—temperature

1. Probability density $φ(x)$ has units $1/x$
2. Normalized $˜ φ(x) = φ(x)/∫ φ(x) dx$
3. (Unitless) probability $a < x < b = ∫_a^b ˜ φ(x) dx$
4. Expectation value $\langle f(x) \rangle = ∫ f(x) ˜ φ(x) dx$
5. Mean $= \langle x \rangle$
6. Mean squared $= \langle x^2 \rangle$
7. Variance $σ^2=\langle x^2 \rangle - \langle x \rangle^2$
8. Standard deviation $Δ x = σ$

Temperature example

https://colab.research.google.com/github/wmfschneider/CHE30324/blob/master/Resources/Probability.ipynb

Boltzmann distribution

1. $P(E) \propto e^-E/k_BT$, in some sense the definition of temperature (Figure 1)
2. Energy and its units
3. Absolute temperature and its units
4. $k_BT$ as an energy scale, \SI{0.026}{eV} at \SI{298}{K}
5. Equipartition – energy freely exchanged within and between all degrees of freedom

Boltzmann distribution: Gravity example

1. $E(h)=mgh$, linear, continuous energy spectrum
2. Exponential distribution \[P(h) = \frac{1}{∫_0^∞ exp\left(-mgh/k_BT\right)dh }exp\left(\frac{-mgh}{k_BT}\right ) = \frac{mg}{k_BT}exp\left(\frac{-mgh}{k_BT}\right)\]
3. molecule vs car in a gravitational field (Table ref:carvelectron)
4. Implies exponential decrease in gas density with altitude
5. Barometric law for gases, $P=P_0e^-mgh/k_BT$

Boltzmann distribution: Kinetic energy in 1-D example

1. $KE = \frac{1}{2}m v_x^2$ , $P(v_x) \propto exp \left (-m v_x^2/2 k_B T\right )$
2. Standard Normalized Gaussian distribution of mean $μ$ and variance $σ^2$ \[G(x)=\frac{1}{σ\sqrt{2π}} exp\left ( -\frac{(x-μ)^2}{2σ^2} \right )\]
3. By inspection, $μ=\langle v_x \rangle=0$, $σ^2=\langle v_x^2\rangle =k_BT/m$
4. Normalized velocity distribution \[P_1D(v_x) = \left ( \frac{m}{2π k_B T} \right )^1/2exp\left (-\frac{m|v_x|^2}{2 k_BT} \right ) \]
5. Molecule vs car again (Table ref:carvelectron)

	car	gas molecule
m	\SI{1000}{kg}	\SI{1e-26}{kg}
h	\SI{1}{m}	\SI{1}{m}
mgh	\SI{9800}{J}	\SI{9.8e-26}{J}
	\SI{6.1e22}{eV}	\SI{6.1e-7}{eV}
T	\SI{298}{K}	\SI{298}{K}
\(k_BT\)	\SI{0.026}{eV}	\SI{0.026}{eV}
\(mgh/k_BT\)	\SI{2.4e24}{}	\SI{2.3e-5}{}
\(P(\SI{1}{m})/P(0)\)	\(e-2.4× 10^{-24}\)	0.99998
\(\langle h \rangle\)	\SI{0}{m}	\SI{42}{km}
\(\langle v_x \rangle1/2\)	\SI{2e-12}{m/s}	\SI{640}{m/s}

Lecture 2: Kinetic theory of gases

1. Postulates
   1. Gas is composed of molecules in constant random, thermal motion
   2. Molecules only interact by perfectly elastic collisions
   3. Volume of molecules is $<<$ total volume
2. Maxwell-Boltzmann distribution of molecular speeds (Figure 3)
   1. Speed $v=\sqrt{v_x^2+v_y^2+v_z^2}$, spherical coordinates \begin{eqnarray*} P_{\rm MB}(v) & = &∫\int P_1D(v_x) P_1D(v_y) P_1D(v_z) v^2 sin(θ) dθ dφ
      & = &4π v^2 \left( \frac{m}{2π k_B T}\right)^3/2exp\left(-\frac{m v^2}{2k_B T}\right) \end{eqnarray*}
   2. mean speeds $\langle v \rangle = ∫_0^∞ v P_MB(v)dv \propto \sqrt{T}$
   3. mean kinetic energy $\langle U \rangle = \frac{1}{2} m \langle v^2 \rangle =\frac{3}{2} RT$
   4. heat capacity $C_v= dU/dT = \frac{3}{2} R$
3. Flux and pressure
   1. Velocity flux $j(v_x) dv_x= v_x \frac{N}{V}P(v_x)dv_x$, molecules /area /time /$v_x$
   2. Wall collisions, $J_w = ∫ j(v_x) dv_x$, total collisions /area /time
   3. Momentum change with wall collisions ($Δ$ momentum/area/time): \[ P = ∫_0^∞ 2 m v_x j_x(v_x) dv_x = m (N/V) \langle v_x^2 \rangle = N k_B T/V \]
4. Collisions and mean free path
   1. Collision cross section $σ=π d^2$, area swept by molecule
   2. Molecular collisions per molecule = volume swept * density of targets = $z = σ \langle v \rangle (N/V) \sqrt{2}$
   3. Total collisions per volume = $z_\mathrm{AA} = z (N/V) (1/2)$
   4. Mean free path, $λ = \langle v \rangle/z$ , mean distance between collisions

$\langle v \rangle$	\SI{475}{\meter\per\second} = \SI{1060}{mph}
$J_W$	\SI{0.48}{\mole\per\centi\meter\squared\per\second}
$P$	\SI{1}{bar}
$σ$	\SI{0.43}{nm^2}
$z$	\SI{7e9}{\per\second}
$Z\rm AA$	\SI{8e28}{\per\second\per\centi\meter\cubed}
λ	$\SI{68}{nm} = 183 σ$
$D11$	\SI{1.1e-5}{\meter\squared\per\second}

Lecture 3: Transport

1. Transport of energy, momentum, mass across a gradient.
2. Infinite gradient: effusion and Graham’s law, $\text{effusion rate}\propto MW^-1/2$
3. Finite gradient: Fick’s first law
   1. net flux proportional to concentration gradient
   2. $j_x = -D \frac{d c}{d x}$
   3. Self-diffusion constant, $D=\frac{1}{3}λ \langle v \rangle$
4. Fick’s second law: time evolution of concentration gradient
   1. Continuity with no advection: \(\frac{∂ c}{∂ t} = -∇\cdot \vec{j} + \text{gen}\)
   2. One-dimension, point source: $\frac{d c}{d t} = D \frac{d^2 c}{dx^2}$, $c(x,t=0) =c_0$
   3. Separate variables $c(x,t) = X(x)t(t)$
   4. Diffusion has Gaussian probability distribution: \(c(x,t)/c_0 = [2 \sqrt{π D t}]^-1 exp(-x^2/4Dt)\)
5. Random walk model of diffusion
   1. $N$ steps, $n = n_r - n_l$ net to the right, $P(n) = \left ( \begin{smallmatrix}N \ n_r \end{smallmatrix} \right )2^-N$
   2. Large $N$ and Stirling approximation, $N! ≈ \left (2π N\right)^1/2 N^N e^-N$
   3. Let $x = δ (n_r - n_l)$, $N = t/τ$, Gaussian reappears! \[ P(x,t) = \left ( \frac{2τ}{π t}\right )^1/2 e^{-x^2τ/2tδ^2} \]
   4. Einstein-Smoluchowski relation $D = δ^2/2τ$
6. Knudsen diffusion, $δ = (3/2)l$, $δ/τ = \langle v \rangle$, $D=\frac{1}{3}l \langle v \rangle$
7. Seeing is believing—Brownian motion
   1. Seemingly random motion of large particles (“dust”) due to “kicks” from invisible molecules
   2. Einstein in one of his four 1905 Annus Mirabilis papers shows
      1. Motion of particles suspened in a fluid of molecules must follow same Gaussian diffusion behavior
      2. From steady-state arguments in a field, diffusion constant is Boltzmann energy, $k_B T$, times mobility
      3. Mobility inversely related to viscosity
   3. Stokes-Einstein equation
   4. Allows measurement of Avogadro’s number, final proof of kinetic theory of matter
   5. Similar model for diffusion of liquid molecules, slip boundary

Quantum Mechanics: Blurred Lines Between Particles and Waves

Lecture 4: Duality and demise of classical physics

Properties of waves

1. Characterized by frequency, wavelength, amplitude, \ldots
2. Traveling waves, standing waves
3. Interference, diffraction
4. Characteristic of light, among other thing
5. Expected energy of a classical wave, $\langle ε \rangle _ν = k_B T$ for all $ν$

Blackbody radiation - light emitted by all bodies due to their temperature

1. Blackbody/Hohlraum spectrum (like the sun)
   1. Stefan-Boltzmann law, total irradiance \(I(λ,T)\)
   2. Wien’s displacement law, $λ_\mathrm{text}T = \mathrm{constant}$
2. Rayleigh-Jeans predicts spectrum using classical physics
   1. standing waves + classical wave energy \(→\) ultraviolet catastrophe
   2. \(I(λ, T) = (8π/λ^4) ⋅ k_B T ⋅ c \)
3. Planck model, 1900
   1. Energy spectrum of waves are quantized, $ε_ν=nhν$, $n = 0,1,2, \ldots$
   2. Expected energy of a quantized wave: \[\langle ε \rangle_ν = ∑_n=0^∞ nhν e^-nhν/k_BT = hν/\left ( e^hν/k_BT-1 \right )\]
   3. Intensity: \[I(λ, T) = \frac{8π}{λ^4} ⋅ \langle\epsilon \rangle_ν ⋅ c \]
   4. Correctly reproduces Stefan-Boltzmann and Wien Laws!

Heat capacities of solids

1. Law of DuLong and Pettite, $C_v = 3R$, fails at low $T$
2. Einstein model
   1. Energy of atomic vibrations $ν$ are quantized, $ε_ν=nhν$, $n = 0,1,2, \ldots$
   2. Expected energy of vibration exactly same as Planck’s quantized waves
   3. Heat capacity = derivative of energy wrt temperature goes to zero at low $T$

Photoelectric effect - electrons emitted when light shined on a metal

1. Energy of most weakly bound electrons to a material defined as work function, $W$
2. Shine light on metal, observe kinetic energy of electrons $E_\text{kinetic}=hν -W$
3. Kinetic energy varies with light frequency, number of electrons varies with light intensity
4. Einstein model, 1905 (Nobel prize)
   1. Light is both wave-like and composed of particle-like “photons”
   2. Photon energy related to frequency: \(ε = h ν\ = hc/λ\)
   3. Light intensity related to number of photons

Special theory of relative (Einstein, 1905)

1. speed of light c in a vacuum is a constant for all observes, independent of \(ν\)
2. photons carry momentum $p=h/λ$
3. demonstrated by Compton effect, light scattering off electrons changes $λ$

Rutherford, planetary model of atom

1. Inconsistent with Maxwell’s equations

Bohr model of H atom

1. Bohr model (the old quantum mechanics)
   1. Stable electron “orbits,” quantized angular momentum
   2. Light emission corresponds to orbital jumps, $ν=Δ E/h$
   3. Bohr equations
   4. Comparison with Rydberg formula
   5. Failure for larger atoms
2. Explains discrete H energy spectrum and Rydberg formala

de Broglie relation

1. $λ=h/p$ universally
2. Relation to Bohr orbits
3. Davison and Germer experiment, $e^-$ diffraction off Ni
4. Basis of modern electron diffraction to observe structure of materials

Wave-particle duality

Lecture 5: Postulates of quantum mechanics

Schrödinger equation describes wave-like properties of matter

1. Attempt to mathematically elaborate de’Broglie idea
2. Statement of conservation of energy, kinetic + potential = total
3. One-dimensional, time-independent, single particle Schrödinger equation: \[-\frac{\hbar^2}{2 m} \frac{d^2 ψ(x)}{dx^2} + V(x) ψ(x) = E ψ(x)\]
4. Second-order differential equation, solutions are steady-states of the system, discrete eigenvalues $E$ and eigenvectors $ψ(x)$
5. Applied to H atom by Schrödinger to recover Bohr energies

Born interpretation

1. wavefunction \(ψ(x)\) is a probability amplitude
2. wavefunction squared \(|ψ(x)|^2\) is probability density

Postulates

1. Wavefunction contains all information about a system
2. Operators used to extract that information
   1. QM operators are Hermitian
   2. Have eigenvectors and real eigenvalues, $\hat{O}ψ_i=oψ_i$
   3. Are orthogonal, $\langle ψ_i | ψ_j \rangle = δ_ij$
   4. Always observe an eigenvalue when making an observation
3. Expectation values
4. Energy-invariant wavefunctions given by Schröodinger equation
5. Uncertainty principle

Particle in a box illustrations

Lecture 6: Particle in a box model

Particle between infinite walls, electron confined in a wire

1. Classical solution, either stationary or uniform bouncing back and forth

One-dimesional QM solutions

1. Schrödinder equation and boundary conditions
2. discrete, quantized solutions
3. standing waves, $λ=2 L/n$, $n-1$ nodes, non-uniform probability
4. Ho paper, STM of Pd wire
5. zero point energy and uncertainty
6. correspondence principle
7. superpositions

Multiple dimensions

1. separation of variables, one quantum number for each dimension
2. \(Ψ_lmn(x,y,z) = ψ_l(x) ψ_m(y) ψ_n(z)\), 3dbox notebook
3. \(E_lmn=(l^2+m^2+n^2)π^2\hbar^2/2L^2 \longrightarrow\) degeneracies

Finite walls and tunneling

1. Potential well of finite depth $V_0$
2. Finite number of bound states
3. Classical region, $ψ(x) ~ e^ikx+e^-ikx, k=\sqrt{2mE}/\hbar$
4. “Forbidden” region, $ψ(x) ~ e^{κ x}+e^{-κ x}, κ=\sqrt{2m(V_0-E)}/\hbar$
5. Non-zero probability to “tunnel” into forbidden region
6. Tunneling between two adjacent wells: chemical bonding, STM, nanoelectronics
7. H atom tunneling: NH$_3$ inversion, H transfer, kinetic isotope effect

Pauli principle for fermions

Lecture 7: Harmonic oscillator

Classical harmonic oscillator

1. Hooke’s law, $F=-k(x-x_0)$, $k$ spring constant
2. Continuous sinusoidal motion
3. $x(t)=A sin(\frac{k}{μ})^1/2t, ν=\frac{1}{2π}(\frac{k}{μ})^1/2, E=\frac{1}{2}kA^2$
4. Exchanging kinetic and potential energies

Quantum harmonic oscillator

1. Schrödinger equation and boundary conditions
2. Solutions like P-I-A-B + tunneling at boundaries (see Table 10)
3. Zero-point energy and uniform energy ladder
4. Parity operator and even/odd symmetry: \(\langle x \rangle =0 \)
5. Recursion relations: \( \langle x^2 \rangle = α^2 (v+1/2), \langle V(x) \rangle = \frac{1}{2} hν (v+\frac{1}{2})\)
6. Virial theorem: $V(x) \propto x^n → \langle T \rangle = \frac{n}{2}\langle V \rangle$
7. Classical turning point and tunneling
8. Classical limiting behavior: large

HCl example

1. Reduced mass, $\frac{1}{μ}=\frac{1}{m_A}+\frac{1}{m_B}$
2. ZPE, energy spacing in IR, Boltzmann probabilities

Diatomic vibrational spectroscopy

1. Apply harmonic oscillator model
2. Vibrational constant $˜{ν} = (\sqrt{k/μ}/2π)/hc$ cm$^-1$
3. Gross selection rule: dynamic dipole $dμ/dx$ non-zero (heteronuclear, non homonuclear)
4. Specific selection rule: dipole integral $\langle ψ_v|\hat\mu|ψ_v^′ \rangle =0$ unless $Δ v = ± 1$
5. Allowed $Δ ˜{E}_v = ˜{ν}$ cm$^-1$
6. Boltzmann distribution implies $v=0$ states dominate at normal $T$

Polyatomic vibrational spectroscopy

1. Polyatomics, $3n-6$ ($3n-5$ for linear polyatomic) vibrational modes
2. Selection rules and degeneracies affect number of observed features
3. CO$_2$ example

Lecture 8: Rigid Rotor

Classical rigid rotor

1. Compare rotation about an axis vs linear motion
2. Moment of intertia $I=μ r^2$
3. Angular momentum, $\mathbf{l} = I \mathbf{ω} = \mathbf{r}× \mathbf{p}$, $T= l^2/2I$
   1. Angular momentum and energy continuous variables

Quantum rotor in a plane

1. Angular momentum and kinetic energy operators in polar coordinates, $\hat l_z = -i\hbar \frac{d}{dφ}$
2. Eigenfunctions degenerate, cw and ccw rotation
3. No zero point energy
4. Angular momentum eignefunctions, $l_z = m_l \hbar$
5. Energy superpositions and localization

Quantum rotor in 3-D

1. Angular momentum and kinetic energy operators in spherical coordinates
2. Spherical harmonic solutions, $Y_{lm_l}$
3. Azimuthal QN $l=0, 1, \ldots$
4. Magnetic QN $m_l = -l, -l+1, …, l$
5. Energy spectrum, $2 l + 1$ degeneracy
6. Vector model - can only know total total $|L|$ and $L_z$
7. Wavefunctions look like atomic orbitals, $l$ nodes

replace. Tab to end.

Particle angular momentum

1. Fermions, mass, half-integer spin
   1. Electron, $s=1/2, m_s=± 1/2$
2. Bosons, force-carrying, integer spin

Diatomic rotational spectroscopy

1. Apply rigid rotor model
2. Rotational constant $˜{B} = (\hbar^2/2I)/hc = \hbar/4π I c$ cm$^-1$, $I=μ R_\mathrm{eq}^2$
3. Gross selection rule: dynamic dipole moment non-zero (heteronuclear, not homonuclear)
4. Specific selection rule: $Δ l=± 1$, $Δ m_l=0, ±1$
5. $Δ ˜{E_l} = 2˜{B}(l+1)$ cm$^-1$
6. Rotational state populations

Lecture 9: Spectroscopy

Spectroscopy is quantitative measurement of interaction of light with matter

1. Observed $I(ν)/I(ν_0)$
2. Bohr condition, $|E_f-E_i|/h=ν =c˜{ν}=c/λ$
3. Intensities determined by populations of initial and final states (from Boltzmann distribtuion) and transition probabilities (gross and specific selection rules)

Einstein coefficients

1. Stimulated absorption, $dn_1/dt= -n_1 Bρ(ν)$
2. Stimulated emission, $dn_2/dt= -n_2 Bρ(ν)$
3. Spontaneous emission, $dn_2/dt=-n_2 A, A=\left ( \frac{8π h ν^3}{c^3}\right )B$
4. $1/A=$ lifetime

Transition probability

1. Einstein coefficient $B_if=\frac{|μ_if|^2}{6ε_0\hbar^2}$
2. Classical electric dipole, $\overrightarrow{μ}=q ⋅ \overrightarrow{l}$, quantum dipole operator $\hat\mu = e⋅ \overrightarrow{r}$
3. Transition dipole moment, $μ_if = \left( \frac{dμ}{dx}\right ) \langle ψ_i|\hat\mu |ψ_f \rangle$
4. Selection rules—conditions that make $μ_if$ non-zero, “allowed” vs “forbidden” transitions

Lecture 10: Vibrational and rotational spectroscopy

Diatomic rotational spectroscopy

1. Apply rigid rotor model
2. Rotational constant $˜{B} = (\hbar^2/2I)/hc = \hbar/4π I c$ cm$^-1$, $I=μ R_\mathrm{eq}^2$
3. Gross selection rule: dynamic dipole moment non-zero (heteronuclear, not homonuclear)
4. Specific selection rule: $Δ l=± 1$, $Δ m_l=0, ±1$
5. $Δ ˜{E_l} = 2˜{B}(l+1)$ cm$^-1$
6. Rotational state populations

Polyatomic rotational spectroscopy

1. Three distinct moments of intertia ($I_x, I_y, I_z$)
2. Spectra more complex

Diatomic vibrational spectroscopy

1. Apply harmonic oscillator model
2. Vibrational constant $˜{ν} = (\sqrt{k/μ}/2π)/hc$ cm$^-1$
3. Gross selection rule: dynamic dipole $dμ/dx$ non-zero (heteronuclear, non homonuclear)
4. Specific selection rule: dipole integral $\langle ψ_v|\hat\mu|ψ_v^′ \rangle =0$ unless $Δ v = ± 1$
5. Allowed $Δ ˜{E}_v = ˜{ν}$ cm$^-1$
6. Boltzmann distribution implies $v=0$ states dominate at normal $T$

Raman spectroscopy

1. Shine in light of arbitrary frequency $˜{ν_0}$, mostly get out the same
2. Some light comes out at $˜{ν_0}-˜{ν}$ (Stoke’s line)
3. Some light comes out at $˜{ν_0}+˜{ν}$ (anti-Stoke’s line)
4. Gross selection rule: dynamic polarizability non-zero (homonuclear, not heteronuclear)

Anharmonicity, Morse potential

Vibration-rotation spectroscopy

1. Harmonic oscillator + rigid rotor
2. Selection rules: $Δ v = ± 1, Δ l=± 1$
3. $R$ branch: $Δ ˜ E = ˜ ν + 2B(l+1), Δ l = 1$
4. $P$ branch: $Δ ˜ E = ˜ ν - 2B(l), Δ l = -1$

Polyatomic vibrational spectroscopy

1. Polyatomics, $3n-6$ ($3n-5$ for linear polyatomic) vibrational modes
2. Selection rules and degeneracies affect number of observed features
3. CO$_2$ example

Lecture 11: Hydrogen atom

Schrödinger equation

1. Spherical coordinates and separation of variables
2. Coulomb potential $v_\mathrm{Coulomb}(r)=-\frac{e^2}{4π\epsilon_0}\frac{1}{r}$
3. Centripetal potential $v=\hbar^2\frac{l(l+1)}{2μ r^2}$

Solutions

1. $ψ(r,θ,φ)=R_nl(r)Y_lm(θ,φ)$
2. Principle quantum number $n=1,2,\ldots$
   1. $K$, $L$, $M$, $N$, … shells
   2. $n-1$ radial nodes
3. Azimuthal quantum number $l=0,1,…,n-1$
   1. $s$, $p$, $d$, … orbital sub-shells
   2. $l$ angular nodes
4. Magnetic quantum number $m_l=-l,-l+1,…,l$
5. Spin quantum number $m_s=± 1/2$
6. Energy spectrum and populations
7. Electronic selection rules
   1. $Δ l=± 1 \quad Δ m_s =0 \quad Δ m_l = 0,± 1$
8. Wavefunctions = “orbitals”, 3d H atom notebook
9. Integrate out angular components to get radial probability function $P_nl(r)=r^2 R_nl^2(r)$
   1. $\langle r\rangle = ∫ r P_nl(r) dr = \left(\frac{3}{2}n^2-l(l+1)\right)a_0$

Variational principle

1. Solutions of Schrödinger equation always form a complete set
2. True wavefunction energy is therefore lower bound on energy of any trial wavefunction \[\langle ψ_\text{trial}^λ | \hat{H} | ψ_\text{trial}^λ\rangle =E_\text{trial}^λ \geq E_0\]
3. Optimize wavefunction with respect to variational parameter \[ \left ( \frac{∂ \langle ψ_\text{trial}^λ | \hat{H} | ψ_\text{trial}^λ\rangle}{∂\lambda} \right ) = 0 → λ_\text{opt} \]

Lecture 12: Many-electron atoms

Many-electron problem, Schrödinger equation not exactly solvable (Sad!)

1. $e^- -e^-$ interaction terms prevent separation of variables
2. Independent electron model basis of all solutions, describes each electron (pair) by its own wavefunction, or “orbital,” $ψ_i$

\[ \left \{ -\frac{\hbar^2}{2m_e}∇^2 - \frac{Z}{r} + v_\text{ee} \right \}ψ_i = ε_i ψ_i \]

Qualitative solutions

1. $ψ_i$ look like H atom orbitals, labeled by same quantum numbers
2. Aufbau principle: “Build-up” electron configuration by adding electrons into H-atom-like orbitals, from bottom up
3. Pauli exclusion principle: Every electron in atom must have a unique set of quantum numbers, so only two per orbital (with opposite spin)
4. Pauli exclusion principle (formally): The wavefunction of a multi-particle system must be anti-symmetric to coordinate exchange if the particles are fermions, and symmetric to coordinate exchange if the particles are bosons
5. Hund’s rule: Electrons in degenerate orbitals prefer to be spin-aligned. Configuration with highest spin multiplicity is the most preferred

$S$	$2S+1$	multiplicity
0	1	singlet
$1/2$	2	doublet
1	3	triplet
$3/2$	4	quartet

Structure of the periodic table

1. Electrons in different subshells experience different effective nuclear charge $Z_\mathrm{eff} = Z - σ_nl$
2. Inner (“core”) shells not shielded well, decrease precipitously in energy with increasing \(Z\)
3. Inner shell electrons “shield” outer electrons well
4. Within a family (column), outmost $n$ increases, further from nucleus, energy goes up
5. Within a period (row), $s$ shielded less than $p$ less than $d$ …, causes degeneracy to break down
6. Electrons in same subshell shield each other poorly, causing ionization energy to increase across the subshell

Quantitative solutions

1. Schrödinger equation \[\hat H Ψ({\bf r}_1, {\bf r}_2,…)=E Ψ({\bf r}_1, {\bf r}_2,…)\] \[\hat H = ∑_i \hat h_i + \frac{e^2}{4 π ε_0}∑_i∑_j>i\frac{1}{|{\bf r}_i-{\bf r}_j|}\] \[\hat h_i = -\frac{\hbar^2}{2m_e}∇^2_i-\frac{Z e^2}{4π\epsilon_0}\frac{1}{|{\bf r}_i|}\]
2. Construct candidate many-electron wavefunction $Ψ$ from one electron wavefunctions (mathematical details vary with exact approach) \[Ψ({\bf r}_1, {\bf r}_2,…)≈ ψ_1({\bf r}_1)ψ_2({\bf r}_2)…ψ_n({\bf r}_n)\]
3. Calculate expectation value of $E$ of approximate model and apply variational principle to find equations that describe “best” (lowest total energy) set of $ψ_i$ \[\frac{∂ E}{∂ ψ_i}=0 \ \ \ ∀ i\] \[\hat fψ=\left\{\hat h + \hat v_\mathrm{Coul}[ψ_i] + \hat v_\mathrm{ex}[ψ_i]+\hat v_\mathrm{corr}[ψ_i] \right\}ψ=ε\psi\] \[E=∑_i ε_i-\frac{1}{2}\langle Ψ |\hat v_\mathrm{Coul}[ψ_i] + \hat v_\mathrm{ex}[ψ_i]+\hat v_\mathrm{corr}[ψ_i]|Ψ \rangle\]
4. Motivate as equation for an electron moving in a “field” of other electrons, adding an electron to a known set of $ψ_i$

Electron-electron interactions

1. Coulomb ($\hat v_\mathrm{Coul}$): classical repulsion between distinguishable electron “clouds”
2. Exchange ($\hat v_\mathrm{ex}$): accounts for electron indistinguishability (Pauli principle for fermions). Decreases Coulomb repulsion because electrons of like spin intrinsically avoid one another
3. Correlation ($\hat v_\mathrm{corr}$): decrease in Coulomb repulsion due to dynamic ability of electrons to avoid one another; “fixes” orbital approximation
4. General form of exchange potential is expensive to calculate; general form of correlation potential is unknown

Popular models

1. Hartree model: Include only classical Coulomb repulsion $\hat v_\mathrm{Coul}$
2. Hartree-Fock model: Include Coulomb and exchange
3. Density-functional theory (DFT): Include Coulomb and approximate expressions for exchange and correlation

Numerical solution

1. All potential terms $\hat v$ depend on the solutions, so equations must be solved iteratively to self-consistency
2. Solved numerically on a grid or by expanding solutions in a basis set

DFT calculations on atoms

1. See README at ../Resources/fda

H Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   1.00      -0.5002      0.5003    1.0005   1.4994

Energy Summary
kinetic energy      =     0.5003
potential energy    =    -1.0005
one-electron energy =    -0.5001
two-electron energy =    -0.0000

total energy =    -0.5002
virial ratio =    -1.9996

He Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   2.00      -0.8998      1.5175    1.7352   0.9133

Energy Summary
kinetic energy      =     3.0349
potential energy    =    -5.8876
one-electron energy =    -3.9058
two-electron energy =     1.0531

total energy =    -2.8527
virial ratio =    -1.9399

Li Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   2.00      -2.2989      3.9238    2.7994   0.5490
2s   1.00      -0.2044      0.2483    0.3695   3.7083

Energy Summary
kinetic energy      =     8.0959
potential energy    =   -15.4017
one-electron energy =    -9.8094
two-electron energy =     2.5036

total energy =    -7.3058
virial ratio =    -1.9024

Na Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   2.00     -39.3997     57.1958   10.6955   0.1417
2s   2.00      -2.4534      7.2764    1.9224   0.7596
2p   6.00      -1.4174      6.5643    1.7927   0.7529
3s   1.00      -0.1925      0.3691    0.3310   3.9570

Energy Summary
kinetic energy      =   168.6993
potential energy    =  -330.3286
one-electron energy =  -230.8553
two-electron energy =    69.2261

total energy =  -161.6293
virial ratio =    -1.9581

B Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   2.00      -7.3382     11.3935    4.7725   0.3195
2s   2.00      -0.4862      1.1651    0.7749   1.8633
2p   1.00      -0.2627      0.8572    0.6432   2.1503

Energy Summary
kinetic energy      =    25.9745
potential energy    =   -50.2880
one-electron energy =   -32.7155
two-electron energy =     8.4020

total energy =   -24.3135
virial ratio =    -1.9361

C Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   2.00     -10.8710     16.5840    5.7583   0.2643
2s   2.00      -0.6769      1.8255    0.9670   1.5010
2p   2.00      -0.3555      1.4282    0.8313   1.6628

Energy Summary
kinetic energy      =    39.6755
potential energy    =   -77.0810
one-electron energy =   -51.0043
two-electron energy =    13.5987

total energy =   -37.4055
virial ratio =    -1.9428

N Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   2.00     -15.0801     22.7490    6.7446   0.2254
2s   2.00      -0.8883      2.5980    1.1518   1.2645
2p   3.00      -0.4550      2.1076    1.0101   1.3691

Energy Summary
kinetic energy      =    57.0168
potential energy    =  -111.0407
one-electron energy =   -74.7460
two-electron energy =    20.7221

total energy =   -54.0239
virial ratio =    -1.9475

O Orbital Summary
nl    occ        E           KE       <1/r>     <r>
1s   2.00     -19.9695     29.8903    7.7313   0.1964
2s   2.00      -1.1208      3.4852    1.3328   1.0956
2p   4.00      -0.5609      2.8966    1.1841   1.1696

Energy Summary
kinetic energy      =    78.3376
potential energy    =  -152.8395
one-electron energy =  -104.5798
two-electron energy =    30.0778

total energy =   -74.5019
virial ratio =    -1.9510

Lecture 13: Qualitative models of bonding

Qualtitative bonding

1. What does a molecule (or a solid) have that an atom doesn’t?…more nuclei!
2. Why might those atoms clump together to form molecules or solids?…tunneling! Electrons are happier (lower in energy) when they can wander out of their local potential well
3. Recall particle in a finite well. What matters? Depths of wells and distance between them.

Clamped nucleus (“Born-Oppenheimer”) approximation

1. Write one-electron equations parametrically in terms of positions of all atoms \begin{eqnarray} \hat h & = & -\frac{\hbar^2}{2m_e}∇^2-∑_α \frac{Z_α e^2}{4π\epsilon_0}\frac{1}{|{\bf r}-{\bf R}_α|}
   \hat fψ & = & \left\{\hat h + \hat v_\mathrm{Coul}[ψ_i] + \hat v_\mathrm{ex}[ψ_i]+\hat v_\mathrm{corr}[ψ_i] \right\}ψ=ε\psi \end{eqnarray}
2. Solve as for atoms, using some model for electron-electron interactions
3. Potential energy surface (PES) \[ E({\bf R}_α, {\bf R}_β,…)=E_\mathrm{elec}+\frac{e^2}{4π\epsilon_0}∑_α\sum_β>α\frac{Z_α Z_β}{|{\bf R}_α-{\bf R}_β|} \]

H$_2$ molecule as perturbation on two H atoms brought from infinite distance

1. “Bonding” orbital, \(σ_g({\bf r}) = 1{\rm s_A}+1{\rm s_B}\)
2. “Anti-bonding” orbital, $σ_u({\bf r}) = 1{\rm s_A}-1{\rm s_B}$
3. Interaction scales with “overlap” $S = \langle 1{\rm s_A} | 1{\rm s_B} \rangle$
4. Normalize \begin{displaymath} σ_g = \frac{1}{\sqrt{2(1-S)}}\left ( 1{\rm s_A}+1{\rm s_B} \right) \quad\quad σ_u = \frac{1}{\sqrt{2(1+S)}}\left ( 1{\rm s_A}-1{\rm s_B} \right) \end{displaymath}
5. Energy expectation value \begin{eqnarray*} ε_g = \langle σ_g | \hat{f} | σ_g \rangle & = & \frac{1}{2(1+S)} \left \{ \langle 1{\rm s_A} | \hat{f} | 1{\rm s_A} \rangle + \langle 1{\rm s_B} | \hat{f} | 1{\rm s_B} \rangle + 2 \langle 1{\rm s_A} | \hat{f} |1{\rm s_B} \rangle \right \}
   & = & \frac{1}{1+S} \left ( F_{\rm AA} + F_{\rm AB} \right ) \ ε_u = \langle σ_u | \hat{f} | σ_u \rangle & = & \frac{1}{2(1+S)} \left \{ \langle 1{\rm s_A} | \hat{f} | 1{\rm s_A} \rangle + \langle 1{\rm s_B} | \hat{f} | 1{\rm s_B} \rangle - 2 \langle 1{\rm s_A} | \hat{f} | 1{\rm s_B} \rangle\right \}\ & = & \frac{1}{1-S} \left ( F_{\rm AA} - F_{\rm AB} \right ) \end{eqnarray*}
6. Matrix elements \begin{eqnarray*} F_{\rm AA}=F_{\rm BB}≈ ε_1\mathrm{s}=α
   F_{\rm AB}=F_{\rm BA}=β \ α < β < 0\ \ \mathrm{typically} \end{eqnarray*} \begin{center} \includegraphics[scale=0.5]{./Images/H2-MO} \end{center}
7. From Taylor expansion get picture of atomic orbitals destabilized by electron repulsion $β S$ and split by interaction $β$ \begin{eqnarray*} ε_+≈ α-β S + β
   ε_-≈ α - β S - β \end{eqnarray*}
8. Makes clear that bonding stabilization $&lt;$ anti-bonding destabilization
9. Ground configuration $=σ_g^2$
10. Bond order = $\frac{1}{2}(n-n^*)$
11. Electron-driven bonding in competetition with $1/R$ repulsion between nuclei.

Heteronuclear diatomic: LiH, HF, BH example

1. Only AOs of appropriate symmetry, overlap, and energy match can combine to form MOs \begin{eqnarray*} ε_+≈ α_1- β S - β^2/|α_1-α_2|
   ε_-≈ α_2 - β S + β^2/|α_1-α_2| \end{eqnarray*}
2. LiH: H 1s + Li 2s, bond polarized towards H
3. HF: H 1s + F 2p, bond polarized towards F, lots of non-bonding orbitals
4. BH: H 1s, B 2s and 2p$_z →$ bonding, non-bonding, anti-bonding orbitals

Homonuclear diatomic: O$_2$

1. Assign aos, 1s, 2s, 2p for each atom (10 total)
2. In principle, solve $10× 10$ secular matrix
3. In practice, matrix elements rules mean only a few off-diagonal elements survive
   1. 1s + 1s do nothing
   2. 2s + 2s form $σ$ bond and anti-bond
   3. 2p$_z$ + 2p$_z$ form second bond and anti-bond
   4. 2p$_x,y$ + 2p$_x,y$ form degenerate $π$ bonds and anti-bonds
   5. O$_2$ is a triplet, consistent with experiment!

The Hückel/#+title:

ght binding model: Roberts, Notes on Molecular Orbital Theory

1. $F_ii=α, S_ij=δ_ij, F_ij=β$ iff $i$ adjacent to $j$
2. Ethylene example
3. Butadiene example
4. Benzene example
5. Infinite chain example

Huckel model for pi orbitals of cyclobutadiene

⎡α β 0 β⎤ ⎢ ⎥ ⎢β α β 0⎥ ⎢ ⎥ ⎢0 β α β⎥ ⎢ ⎥ ⎣β 0 β α⎦

Energy state, degeneracy alpha 2

alpha - 2*beta 1

alpha + 2*beta 1

Eigenvectors Eigenvector(s) of state 2 : [Matrix([ [1], [1], [1], [1]])]

Eigenvector(s) of state 1 : [Matrix([ [-1], [ 1], [-1], [ 1]])]

Eigenvector(s) of state 0 : [Matrix([ [-1], [ 0], [ 1], [ 0]]), Matrix([ [ 0], [-1], [ 0], [ 1]])]

./Images/CyclicH.pdf

Band structure of solids

1. Discrete molecular orbitals transform into continuous bands
2. Results in rich range of physical and chemical properties

./Images/band.pdf

Non-bonding interactions

1. Chemical covalent bonds have energies on the order of several eV
2. Even things that are not “bonded” still attract one another
   1. permanent dipoles (~0.1 eV)
   2. induced dipoles (dispersion)—scales with number of electrons
3. Results in physical properties, eg trends in boiling point (He < Ne < Kr < Xe; \ce{CH4} < \ce{C2H6} < \ce{C3H8} )

Lecture 14: Quantitative Models of Bonding

Numerical Schrödinger equation solvers for discrete (molecule) and periodic (solids/liquids/interfaces) readily available today

Have to specify:

1. Identity of atoms
2. Positions of atoms (distances, angles, $\ldots$)
3. (spin multiplicity)
4. exact theoretical model (how are Coulomb, exchange, and correlation described?)
   1. Hartree, Hartree-Fock, DFT (various flavors), $\ldots$
5. basis set to express wavefunctions in terms of
6. initial guess of wavefunction coefficients (often guessed for you)

Secular equations solved iteratively until input coefficients = output coefficients

1. “self-consistent field”
2. Output
   1. energies of molecular orbitals
   2. occupancies of molecular orbitals
   3. coefficients describing molecular orbitals
   4. total electron wavefunction, total electron density, dipole moment, $\ldots$
   5. total molecular energy
   6. derivatives (“gradients”) of total energy w.r.t. atom positions
3. Plot total energy vs internal coordinates: potential energy surface (PES)
4. Search iteratively for minimum point on PES (by hand or using gradient-driven search): equilibrium geometry
5. Find second derivative of energy at minimum point on PES: harmonic vibrational frequency
6. Find energy at minimum relative to atoms (or other molecules): reaction energy

H$_2$ example

1. Choose “B3LYP” model for Coulomb, exchange, and correlation potentials
2. Choose “6-31G(d)” basis set
3. Compute total energy vs distance
4. Fit energies to quadratic near minimum
5. Predict minimum from fit
6. Extract harmonic force constant \(k\) from second derivative of fit
7. Compute harmonic frequency from force constant
8. Compute zero point vibrational energy from frequency, ZPE \(=0.5 hν\).

-----------------------------------
                  B3LYP       EXPT
-----------------------------------
H-H     (Ang):    0.747       0.742
nu~    (cm-1):    4768        4401

E H2     (eV):   -31.81                
ZPE H2   (eV):     0.29
2*E H    (eV):   -27.04                
                  -----                  
E Dissoc (eV):     4.47        4.48
-----------------------------------

Polyatomic molecules

1. Gradient-driven optimizations, $3n-6$ degrees of freedom
2. Hessian matrix for frequencies
3. Computational Chemistry Comparison and Benchmark Database

Solids

1. Materials project
2. OQMD

Statistical Mechanics: The Bridge from the Tiny to the Many

Lecture 17: Statistical mechanics

Need machinary to average QM information over macroscopic systems

Equal a priori probabilities

1. Any way to distribute energy amongst elements of a system are as likely as any other

Two-state model

1. Box of particles, each of which can have energy 0 or $ε$
2. Thermodynamic state defined by number of elements $N$, and number of quanta $q$, $U=qε$
3. Degeneracy of given $N$ and $q$ given by binomial distribution: \begin{displaymath} Ω(N,q)=\frac{N!}{q!(N-q)!} \end{displaymath}
4. Allow energy (heat!) to exchange between two such systems
   1. Energy of composite system is sum of individual systems (first law, \(q_1+q_2=q\))
   2. Degeneracy of composite system is always $\geq$ degeneracy of the starting parts! \[Ω(N_1+N_2,q_1+q_2) > Ω(N_1,q_1)⋅ Ω(N_2,q_2) \]
   3. Boltzmann’s tombstone, $S = k_B ln Ω$
   4. Second Law:

Die Energie der Welt ist constant. Die Entropie der Welt strebt einem Maximum zu. - Clausius

Large two-state system

1. Stirling’s approximation: \[Ω(N,q) ≈ N^N/(N-q)^(N-q)\]
2. Composite system \[Ω(N,q) = ∑_{i≤ q} Ω(N_1,i)⋅ Ω(N_2,q-i) \]
3. For large $N$, one term overwhelmingly dominates sum

Consequences of energy flow between two large systems

1. Each subsystem has energy $U_i$ and degeneracy $Ω_i(U_i)$
2. Bring in thermal contact, $U=U_1+U_2$, $Ω=∑_{U_1}Ω_1(U_1)Ω_2(U-U_1)$
3. If systems are very large, one combination of $U_1$, $U_2$ will dominate Ω sum. Find largest term. \begin{displaymath} \left ( \frac{∂ Ω}{∂ U_1} \right )_N = 0 \end{displaymath} \begin{displaymath} \left ( \frac{∂ ln Ω_1}{∂ U_1} \right )_N = \left ( \frac{∂ ln Ω_2}{∂ U_2} \right )_N \end{displaymath} \begin{displaymath} \left ( \frac{∂ S_1}{∂ U_1} \right )_N = \left ( \frac{∂ S_2}{∂ U_2} \right )_N \end{displaymath}
4. Thermal equilibrium is determined by equal temperature! \begin{displaymath} \frac{1}{T}=\left ( \frac{∂ S}{∂ U} \right )_N \end{displaymath}
5. Equal temperatures → most probable distribution of energy between subsystems.
6. (Same arguments lead to requirement that equal pressures ($P_i$) and equal chemical potentials ($μ_i$) maximize entropy when volumes or particles are exchanged)

Two-state model in limit of large $N$

1. Large $N$ and Stirling’s approximation
2. Fundamental thermodynamic equation of two-state system: \begin{displaymath} S(U)=k_Bln Ω(N,q) = \ldots = -k_B \left ( x ln x + (1-x) ln (1-x) \right ), \mathrm{where}\ x = q/N = U/Nε \end{displaymath}
3. Temperature is derivative of entropy wrt energy, yields \begin{displaymath} \left( \frac{∂ S}{∂ U} \right )_N = T → U(T) = \frac{Nε e^-ε/k_BT}{1+e^-ε/k_BT} \end{displaymath}
4. $T → 0, U → 0, S → 0$, minimum degeneracy, only 1 possible state
5. $T → ∞, U → Nε/2, S → k_B ln 2$, maximum degeneracy, $_N C_N/2 = 2^N$ possible states
6. Differentiate again to get heat capacity \begin{displaymath} C_N = \left ( \frac{∂ U}{∂ T} \right )_N = \frac{(ε/k_B T)^2 e^-ε/k_BT}{(1+e^{-ε/k_B T})^2} \end{displaymath}

Example of microcanonical (“$NVE$”) ensemble

1. Direct evaluation of $S(U)$ is generally intractable, so seek simpler approach

Lecture 18: Canonical ($NVT$) ensemble

Partition function

1. Imagine a system brought into thermal equilibrium with a much larger “reservoir” of constant $T$, such that the aggregate has a total energy $U$
2. Degeneracy of a given system microstate $j$ with energy $U_j$ is $Ω_res(U-U_j)$ \begin{eqnarray*} T = \frac{dU_res}{k_Bdln\Omega_res}
   Ω_res(U-U_j) \propto e^{-U_j/k_B T} \end{eqnarray*}
3. Probability for system to be in a microstate with energy $U_j$ given by Boltzmann distribution! \begin{displaymath} P(U_j) \propto e^{-U_j/k_B T} = e^{-U_j β} \end{displaymath}
4. Partition function “normalizes” distribution, $Q(T,V) = ∑_j e^{-U_j β}$
5. Partition function counts the number of states accessible to a system at a given $V$ and in equilibrium with a reservoir at $T$

Energy factoring (sidebar)

1. If system is large, how to determine it’s energy states $U_j$? There would be many, many of them!
2. One simplification is if we can write energy as sum of energies of individual elements (atoms, molecules, degrees of freedom) of system: \begin{align} U_j&=ε_j(1)+ε_j(2) + … + ε_j(N)
   Q(N,V,T) &= ∑_j e^-U_jβ \ &=∑_je^{-(ε_j(1)+ε_j(2) + … + ε_j(N))β} \end{align}
3. If molecules/elements of system can be distinguished from each other (like atoms in a fixed lattice), expression can be factored: \begin{align} Q(N,V,T)&=\left ( ∑_j e^-ε_j(1)β\right )\cdots \left ( ∑_j e^-ε_j(N)β\right )
   &= q(1)\cdots q(N) \ \text{Assuming all the elements are the same:}\ &= q^N \ q&=∑_j e^{-ε_j β}: \mathrm{molecular\ partition\ function} \end{align}
4. If not distinguishable (like molecules in a liquid or gas, or electrons in a solid), problem is difficult, because identical arrangements of energy amongst elements should only be counted once.
5. Approximate solution, good almost all the time: \begin{equation} Q(N,V,T)=q^N/N! \end{equation}
6. Sidebar: “Correct” factoring depends on whether individual elements are fermions or bosons, leads to funny things like superconductivity and superfluidity.

Distinguishable vs. indistinguishable particles

1. $q(V,T)$ counts states available to a single element of a system, like a molecule in a gas or in a solid
2. Distinguishable (e.g., in a solid): $Q(N,V,T) = q(V,T)^N$
3. Indistinguishable (e.g., a gas): $Q(N,V,T)≈ q(V,T)^N/N!$

Two-state system again

1. Partition function, $q(T)=1+e^-ε\beta$
2. State probabilities
3. Internal energy $U(T)$ \begin{equation} U(T)=-N \left ( \frac{∂ ln(1+e^-ε\beta)}{∂\beta} \right)=\frac{Nε e^-ε\beta}{1+e^-ε\beta} \end{equation}
4. Heat capacity $C_v$
   1. Minimum when change in states with $T$ is small
   2. Maximize when chagne in states with $T$ is large
5. Helmholtz energy, $A= -ln q/β$, decreasing function of $T$
6. Entropy

Thermodynamic functions in canonical ensemble

Lecture 19: Molecular Partition Functions

Ideal gas of molecules

\begin{displaymath} Q_ig(N,V,T) = \frac{(q_\mathrm{trans}q_\mathrm{rot}q_\mathrm{vib})^N}{N!} \end{displaymath}

Particle-in-a-box (translational states of a gas)

1. Energy states $ε_n=n^2ε_0, n=1,2, \ldots$, $ε_0$ tiny for macroscopic $V$
2. $Θ_\mathrm{trans} = ε_0/k_B$ translational temperature
3. $Θ_\mathrm{trans} &lt;&lt; T →$ many states contribute to $q_\mathrm{trans}→$ integral approximation \begin{eqnarray*} q_\mathrm{trans,1D} ≈ ∫_0^∞ e^{-x^2β\epsilon_0}dx = L/Λ
   Λ = \left ( \frac{h^2β}{2π m} \right )^1/2\ \mathrm{thermal\ wavelength} \ q_\mathrm{trans,3D} = V/Λ^3 \end{eqnarray*}
4. Internal energy
5. Heat capacity
6. Equation of state (!)
7. Entropy: Sackur-Tetrode equation

Rigid rotor (rotational states of a gas)

1. sum over rigid energy states and degeneracies of rigid rotor
2. $Θ_\mathrm{rot} = \hbar^2/2 I k_B$
3. “High” T $q_\mathrm{rot}(T) ≈ σ Θ_\mathrm{rot}/T$, most often true

Harmonic oscillator (vibrational states of a gas)

1. sum over harmonic oscillator energy states
2. $Θ_\mathrm{vib}=hν/k_B$, typically 100’s to 1000’s K
3. introduce strong non-linear $T$ dependence to thermodynamic properties

Electronic partition functions $→$ spin multiplicity

Many-particle molecule

1. partition function is a product of all degrees of freedom \begin{displaymath} q(T,V) = q_\text{trans} \left ( ∏_i=1^3 q_\text{rot}⁽ⁱ⁾\right ) \left ( ∏_{i = 1}^3N-6 q_\text{vib}⁽ⁱ⁾\right ) q_\text{elec} \end{displaymath}
2. thermodynamic quantities are sums of all degrees of freedom

Non-ideality

1. Real molecules interact through vdW interactions
2. Particle-in-a-box model is a start, have to elaborate to get at properties of liquids, solutions, ….
3. See Hill, J. Chem. Ed. 1948, 25, p. 347 http://dx.doi.org/10.1021/ed025p347

	Characteristic	Characteristic	States @ RT
	Energy (cm-1)	Temperature (K)
translational	$\hbar^2/2 m L^2 ≈ 10-21$	$10-21$	$1030$	classical limit
rotational	$≈ 1$	$≈ 1$	100’s	semi-classical
vibrational	$≈ 1000$	$≈ 1000$	1	non-classical
electronic	$≈ 10,000$	$≈ 10,000$	1	non-classical

Lecture 20: Chemical reactions and equilibria

Isothermal, isbaric separation for ideal gas mixture

\[ \text{A/B} (N_A,N_B,V,T) → \text{A}(N_A,x_AV,T) + \text{B}(N_B,x_B,V,T) \]

1. Apply ideal gas expressions to all parts and compute a difference!
2. Internal energy, $Δ U(T) = 0$
3. Entropy, $Δ S(T)/(N_A+N_B) = k_B(x_Aln(x_A) + x_B ln(x_B))$
4. Minimum work of separation, $Δ A(T) = Δ U - TΔ S &gt; 0$
5. Entropy favors mixing

Chemical reaction thermodynamics

1. Transformation that conserves atoms
2. Example: vinyl alcohol to acetaldehyde, \ce{H2C=CH(OH) -> CH3CH(O)}
3. Differences between well defined initial and final states \[ \ce{H2C=CH(OH)} (\SI{1}{mol},\SI{1}{bar},\SI{298}{K}) \ce{-> CH3CH(O)} (\SI{1}{mol},\SI{1}{bar},\SI{298}{K}) \]
4. Reaction entropy captures contributions of all degrees of freedom \[Δ S^ˆ(T) = Δ S^ˆ_\text{trans}(T)+ Δ S_\text{rot}(T) +Δ S_\text{vib}(T)\]
5. Reaction energy (internal, Helmholtz, …) must also capture difference in \SI{0}{K} electronic energy \[Δ U^ˆ(T) = Δ U^ˆ_\text{trans}(T)+ Δ U_\text{rot}(T) +Δ U_\text{vib}(T) + Δ E_\text{elec}(0) + Δ ZPE\]
6. “Standard state”
   1. derives from concentration dependence of entropy
   2. corresponds to some standard choice, $(N/V)^ˆ = c^ˆ$, e.g. \SI{1}{mol/l} (T-independent), or $(N/V)^ˆ = P^ˆ/RT$, e.g. \SI{1}{bar} (T-dependent)
7. Permits functions to be easily computed at other concentrations, e.g. \begin{displaymath} A(T,N/V) = A^ˆ(T) + k T ln\left ( (N/V)/(N/V)^ˆ \right ) =A^ˆ(T) + k T ln \left ( c/c^ˆ \right ) \ G(T,P) = G^ˆ(T) + k T ln\left ( P/P^ˆ \right ) \end{displaymath}

Chemical equilibrium

1. Reaction advancement \(ξ\) describes progress from reactants to products
   1. “ICE”: \(n_i = n_i0 -ν_i ξ \)
2. Free energy of a mixture of reactants and products \[G(T,ξ) = ξ (Δ G^ˆ + k T ∑_i ν_i ln P_i/P^ˆ) \]
3. Equilibrium condition—minimize \(G\) with respect to \(ξ\)
4. Equilibrium condition—equate chemical potentials \begin{eqnarray*} μ_A(N,V,T) & = & μ_B(N,V,T)
   E_A(0) - k T ln (q_A/N_A) & = & E_B(0) - k T ln (q_B/N_B) \ \frac{N_B}{N_A} = \frac{N_B/V}{N_A/V} & = &\frac{q_B(T,V)/V}{q_A(T,V)/V} e^{-Δ U(0)/kT} \end{eqnarray*}
5. \(q/V = 1/Λ^3\) has units of number/volume, or concentration
6. Equilibrium constant—convert units to some standard concentration \(c^ˆ\) or pressure \(P^ˆ\) \begin{eqnarray*} q_A^ˆ(T) & = & (q_A(T,V)/V) (1/c^ˆ)
   q_A^ˆ(T) & = & (q_A(T,V)/V)(k_B T/P^ˆ) \ K_eq(T) & = &\frac{q_B^ˆ(T)}{q_A^ˆ(T)} e^{-Δ U(0)/kT} = e^{-Δ G^ˆ(T)/kT} \end{eqnarray*}
7. ICE/equilibrium calculation for \ce{H2C=CH(OH) -> CH3CH(O)}
8. Free energy convolutes energy and entropy effects
   1. \(Δ H\), \(Δ S\) weakly \(T\)-dependent
   2. \(Δ G = Δ H - TΔ S\) can be strongly \(T\)-dependent
9. Gibbs-Helmholtz relation \[ \left ( \frac{∂ G/T}{∂ T}\right )= -\frac{H}{T^2}\] \[ \left ( \frac{∂ Δ G^ˆ/T}{∂ T}\right )= -\frac{Δ H^ˆ}{T^2}\] \[ \left ( \frac{∂ ln K(T)}{∂ 1/T}\right )= -\frac{Δ H^ˆ}{R}\]
10. van’t Hoff relationship, when $T$ dependence of \(Δ H\) is small \[ ln\left ( \frac{K(T_2)}{K(T_1)}\right )= -\frac{Δ H^ˆ}{R}\left ( \frac{1}{T_2}-\frac{1}{T_1}\right ) \]
11. ICE/equilibrium calculation for ethane dehydrogenation, \ce{C2H6 -> C2H4 + H2}, 1 bar standard state

Le’Chatlier’s principle

1. Example: \ce{H2C=CH(OH) -> CH3CH(O)}, endothermic
2. Response to temperature: Boltzmann distribution favors higher energy things as $T$ increases
3. Example: ethane dehydrogenation, \ce{C2H6 -> C2H4 + H2}, positive entropy
4. Equilibrium composition starting from \ce{C2H6}, at constant pressure \[ K_p(T) = \frac{q_\ce{C2H4}^ˆ(T) q_\ce{H2}^ˆ(T)}{q_\ce{C2H6}^ˆ(T)} e^{-Δ E(0)/k_BT} = \frac{P_\ce{C2H4}P_\ce{H2}}{P_\ce{C2H6}}\frac{1}{P^ˆ} = \frac{P}{P^ˆ}\frac{x^2}{(1-x)(1+x)} \]
5. Response to pressure change: translational DOFs increasingly favor side with fewer molecules as volume decreases/pressure increases

Thermodynamic tables

1. General chemical reaction \(∑_i ν_i A_i = 0\), \(ν_i\) stoichiometric coefficients
2. Thermodynamic change \(Δ W^ˆ(T) = ∑_i ν_i W^ˆ_i(T)\), where \(W = A, U, S, G, \ldots \)
3. Tabulations a common source of standard state H and S, eg http://webbook.nist.gov
   1. \(S^ˆ(T)\) referenced to \SI{0}{K}, because \(S(0) = 0\) (Third law) \[ S^ˆ(T^′) = S^ˆ(T) + ∫_T^{T^′} \frac{C^ˆ_p(T)}{T} dT\]
   2. Enthalpies of elements in their most stable form at \(T=\SI{298}{K}\), \(P=\SI{1}{bar}\) defined to be zero
   3. Enthalpies of substances tabulated as formation enthalpies relative to constiuent elements \[ Δ H^ˆ(T) = ∑_i ν_i Δ H^ˆ_f,i(T) \] \[ Δ H^ˆ(T^′) = Δ H^ˆ(T) + ∫_T^{T^′} Δ C^ˆ_p(T) dT\]

Lecture 21: Chemical kinetics

Kinetics and reaction rates

1. Rate: number per unit time per unit something

Empirical chemical kinetics

1. Rate laws, rate orders, and rate constants
2. Functions of $T$, $P$, composition $C_i$
3. differential vs integrated rate laws
4. Arrhenius expression, $k=A e^-E_a/k_BT$
   1. Arrhenius plot, \(ln k\) vs \(1/T\)

	differential rate	integrated rate	half-life
First order	$r = kC_A$	$C_A = CA0 e-k τ$	$ln 2/k$
Second order	$r = kC_A^2$	$1/C_A = 1/CA0 + k τ$	$1/kCA0$

Reaction mechanisms

1. Elementary steps and molecularity
2. Ozone decomposition, rate second-order at high \(P_\ce{O2}\), first-order at low \(P_\ce{O2}\)

   \ce{2 O3 -> 3 O2}
   \ce{O3 ->[k_1] O2 + O}
   \ce{O2 + O ->[k_-1] O3}
   \ce{O + O3 ->[k_2] 2 O2}

3. Collision theory
   1. A + B → products
   2. rate proportional to A/B collision frequency $z_AB$ weighted by fraction of collisions with energy $&gt; E_a$ \begin{displaymath} r = k C_A C_B , k = \left ( \frac{8 k_B T}{π μ} \right )^1/2 σ_AB N_av e^-E_a/k_BT \end{displaymath}
   3. upper bound on real rates

Transition state theory (TST)

1. Assumptions
   1. Existence of reaction coordinate (PES)
   2. Existence of dividing surface
   3. Equilibrium between reactants and “transition state”
   4. Harmonic approximation for transition state
2. rate proportional to concentration of “activated complex” over reactants times crossing frequency \begin{eqnarray*} r & = & k C_AC_B
   & = & k^\ddagger C_AB^\ddagger \ & = & ν^\ddagger K^\ddagger C_A C_ B \ & = & ν^\ddagger \frac{k_BT}{hν^\ddagger}\bar{K}^\ddagger(T) C_A C_B \ & = & \frac{k_B T}{h} \frac{q^\ddagger(T)}{q_A(T) q_B(T)} e^{-{Δ E(0)/k_BT}} C_A C_B \end{eqnarray*}
3. application to atom - atom collision
4. application to two molecules - vinyl alcohol to acetaldehyde
5. microscopic reversibility
6. equilibrium requirement \(K_eq(T) = k_f(T)/k_r(T)\)

Locating transition states computationally

Thermodynamic connection

1. Relate activated complex equilibrium constant to activation free energy \[ \(\bar{K}^\ddagger(T) = e^{-Δ G^{ˆ \ddagger}(T)/kT} = e^{-Δ H^{ˆ \ddagger}(T)/k_BT}e^{Δ S^{ˆ \ddagger}(T)/k_B} \]
2. Compare to Arrhenius expression \[E_a = Δ H^{ˆ \ddagger}(T) + kT, A = \frac{k_B T}{h}e^1e^{Δ S^{ˆ \ddagger}(T)/k_B}\]

Vinyl alcohol to TS  216 kJ/mol

Application: gas-phase reactions

1. Ethane pyrolysis, \ce{C2H6 -> C2H4 + H2}, doi:10.1021/jp206503d

Heterogeneous reactions and catalysis

1. molecule-surface collisions
2. surface reactions
3. Ammonia oxidation, \ce{NH3 + O2 -> NO, N2}, doi:10.1021/acscatal.8b04251

Diffusion-controlled reactions

1. Intermediate complex
2. Steady-state approximation
3. Diffusion-controlled limit ($k_D = 4π (r_A + r_B) D_AB$)
4. Reaction-controlled limit ($k_app=(k_D/k_-D)k_r$)

Lecture 22: Conclusion

1. Do you think about the burning lighter any differently now?