Quantum Chemistry

1. State Function: The state of a quantum system is described by a wavefunction $\Psi(\mathbf{r}, t)$ containing all information about the system.

2. Observable → Operator: Every measurable observable corresponds to a linear Hermitian operator.

3. Measurement: Measuring an observable $\hat{A}$ yields an eigenvalue $a_n$ of $\hat{A}$:

   $\hat{A}\psi_n = a_n\psi_n$

   The probability of measuring $a_n$ is $|c_n|^2$ where $\Psi = \sum_n c_n\psi_n$.

4. Expectation Value: For a state $\Psi$:

   $\langle A \rangle = \frac{\int \Psi^*\hat{A}\Psi\,d\tau}{\int \Psi^*\Psi\,d\tau}$

5. Time Evolution: $\Psi$ evolves according to the time-dependent Schrödinger equation:

   $i\hbar\frac{\partial \Psi}{\partial t} = \hat{H}\Psi$

For a system with time-independent Hamiltonian:

$\hat{H}\psi = E\psi$

$\left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right]\psi = E\psi$

A particle of mass $m$ confined to $0 \leq x \leq L$ with $V = 0$ inside and $V = \infty$ outside:

$\hat{H}\psi = -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$

Theorem 1 (Particle in a 1D Box):

$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right), \quad E_n = \frac{n^2h^2}{8mL^2}$

where $n = 1, 2, 3, \ldots$

Key features:

• Quantized energy levels; $E_n \propto n^2$.
• Zero-point energy: $E_1 = h^2/(8mL^2) \neq 0$.
• Number of nodes $= n - 1$.

$\psi_{n_x,n_y,n_z}(x,y,z) = \left(\frac{2}{L}\right)^{3/2}\sin\frac{n_x\pi x}{L}\sin\frac{n_y\pi y}{L}\sin\frac{n_z\pi z}{L}$

$E_{n_x,n_y,n_z} = \frac{h^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)$

Definition 1 (Degeneracy): Different sets of quantum numbers that give the same energy are degenerate. For a cubic box, $(1,2,2)$, $(2,1,2)$, and $(2,2,1)$ are triply degenerate.

The probability of finding the particle between $x = a$ and $x = b$:

$P(a \leq x \leq b) = \int_a^b |\psi_n(x)|^2\,dx = \frac{2}{L}\int_a^b \sin^2\frac{n\pi x}{L}\,dx$

Example 1: For a particle in a 1D box of length $L = 1$ nm, find the probability of finding it in the middle third for $n = 1$.

$P\left(\frac{L}{3} \leq x \leq \frac{2L}{3}\right) = \int_{L/3}^{2L/3} \frac{2}{L}\sin^2\frac{\pi x}{L}\,dx$

$= \frac{1}{3} - \frac{\sin(4\pi/3) - \sin(2\pi/3)}{2\pi} = \frac{1}{3} - \frac{-\sqrt{3}/2 - \sqrt{3}/2}{2\pi} = \frac{1}{3} + \frac{\sqrt{3}}{2\pi} \approx 0.61$

$\blacksquare$

Definition 2 (Commutator): $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$.

If $[\hat{A}, \hat{B}] = 0$, the observables can be simultaneously measured with arbitrary precision.

Theorem 2 (Heisenberg Uncertainty Principle):

$\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle|$

$\Delta x \cdot \Delta p_x \geq \frac{\hbar}{2}$

For the hydrogen atom (reduced mass $\mu = m_e m_p/(m_e + m_p) \approx m_e$):

$\left[-\frac{\hbar^2}{2\mu}\nabla^2 - \frac{e^2}{4\pi\varepsilon_0 r}\right]\psi = E\psi$

In spherical coordinates $(r, \theta, \phi)$:

$\psi_{n,\ell,m_\ell}(r,\theta,\phi) = R_{n,\ell}(r)\,Y_\ell^{m_\ell}(\theta,\phi)$

Theorem 3 (Hydrogen Atom Energy):

$E_n = -\frac{\mu e^4}{32\pi^2\varepsilon_0^2\hbar^2}\frac{1}{n^2} = -\frac{13.6 \text{ eV}}{n^2} = -\frac{R_H}{n^2}$

The Rydberg constant $R_H = 2.179 \times 10^{-18}$ J $= 13.6$ eV.

Energy depends only on $n$; all states with the same $n$ are degenerate (for hydrogen).

The first few radial wavefunctions:

$R_{1,0}(r) = 2\left(\frac{1}{a_0}\right)^{3/2}e^{-r/a_0}$

$R_{2,0}(r) = \frac{1}{2\sqrt{2}}\left(\frac{1}{a_0}\right)^{3/2}\left(2 - \frac{r}{a_0}\right)e^{-r/(2a_0)}$

$R_{2,1}(r) = \frac{1}{2\sqrt{6}}\left(\frac{1}{a_0}\right)^{3/2}\frac{r}{a_0}e^{-r/(2a_0)}$

where $a_0 = 5.292 \times 10^{-11}$ m is the Bohr radius.

Theorem 4 (Spherical Harmonics): The angular part $Y_\ell^{m_\ell}(\theta, \phi)$ are solutions to:

$\hat{L}^2\,Y_\ell^{m_\ell} = \ell(\ell+1)\hbar^2\,Y_\ell^{m_\ell}$

$\hat{L}_z\,Y_\ell^{m_\ell} = m_\ell\hbar\,Y_\ell^{m_\ell}$

Total nodes $= n - 1$ = radial nodes + angular nodes.

Theorem 5 (Angular Momentum Magnitude):

$|\mathbf{L}| = \sqrt{\ell(\ell+1)}\,\hbar$

$L_z = m_\ell\hbar, \quad m_\ell = -\ell, -\ell+1, \ldots, \ell$

The angular momentum vector can never be fully aligned with the $z$-axis (space quantization).

Electrons have intrinsic angular momentum (spin) with $s = 1/2$:

$|S| = \sqrt{s(s+1)}\,\hbar = \frac{\sqrt{3}}{2}\hbar$

$S_z = m_s\hbar, \quad m_s = \pm\frac{1}{2}$

Theorem 6 (Spin-Orbit Coupling): The total angular momentum $\mathbf{J} = \mathbf{L} + \mathbf{S}$:

$|\mathbf{J}| = \sqrt{j(j+1)}\,\hbar, \quad j = |\ell - s|, \ldots, \ell + s$

For an electron with $\ell = 1$, $s = 1/2$: $j = 1/2$ or $3/2$.

Term symbols: ${}^{2S+1}L_J$, e.g., ${}^2P_{3/2}$ for $\ell = 1$, $s = 1/2$, $j = 3/2$.

Theorem 7 (Pauli Exclusion Principle): No two electrons in an atom can have the same set of four quantum numbers $(n, \ell, m_\ell, m_s)$.

Consequence: Each orbital can hold at most 2 electrons (one with $m_s = +1/2$, one with $m_s = -1/2$).

Definition 3 (Aufbau Principle): Electrons fill orbitals in order of increasing energy: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, \ldots$

Definition 4 (Hund”s Rules): For degenerate orbitals:

1. Maximize total spin $S$ (parallel spins first).
2. For a given $S$, maximize $L$.
3. For atoms less than half-filled: minimize $J$; more than half-filled: maximize $J$.

Example 2: Carbon ($1s^2\,2s^2\,2p^2$).

The $2p^2$ configuration: possible microstates lead to terms ${}^3P$, ${}^1D$, ${}^1S$.

By Hund’s rules, the ground state is ${}^3P_0$.

$\blacksquare$

For helium-like atoms, the Hamiltonian includes electron-electron repulsion:

$\hat{H} = -\frac{\hbar^2}{2m_e}\nabla_1^2 - \frac{\hbar^2}{2m_e}\nabla_2^2 - \frac{Ze^2}{4\pi\varepsilon_0 r_1} - \frac{Ze^2}{4\pi\varepsilon_0 r_2} + \frac{e^2}{4\pi\varepsilon_0 r_{12}}$

The $1/r_{12}$ term makes exact solutions impossible for $N > 1$.

Definition 5 (Slater Determinant): The antisymmetric wavefunction for $N$ electrons:

$\Psi(1,2,\ldots,N) = \frac{1}{\sqrt{N!}}\begin{vmatrix} \chi_1(1) & \chi_2(1) & \cdots & \chi_N(1) \\ \chi_1(2) & \chi_2(2) & \cdots & \chi_N(2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(N) & \chi_2(N) & \cdots & \chi_N(N) \end{vmatrix}$

where $\chi_i$ is a spin-orbital. The determinant ensures antisymmetry under particle exchange, automatically satisfying the Pauli principle.

Theorem 8 (Hartree-Fock Equations): The Hartree-Fock method approximates each electron as moving in the average field of the others:

$\hat{F}\,\phi_i = \varepsilon_i\,\phi_i$

where $\hat{F}$ is the Fock operator and $\varepsilon_i$ are orbital energies. Koopmans’ theorem relates orbital energies to ionization potentials:

$\text{IP} \approx -\varepsilon_i$

Theorem 9 (Born-Oppenheimer Approximation): Since nuclei are much heavier than electrons ($m_p/m_e \approx 1836$), the electronic and nuclear motions can be separated:

$\Psi_{\text{total}} = \psi_{\text{elec}}(\mathbf{r}; \mathbf{R})\,\psi_{\text{nuc}}(\mathbf{R})$

The electronic Schrödinger equation is solved for fixed nuclear positions, giving the potential energy surface (PES).

The PES defines:

• Equilibrium geometry: Minimum on the PES.
• Transition state: Saddle point (first-order saddle point, one imaginary frequency).
• Vibrational frequencies: Second derivatives of the PES at the minimum.

Definition 6 (LCAO-MO): Molecular orbitals are formed as linear combinations of atomic orbitals:

$\psi_i = \sum_\mu c_{\mu i}\,\phi_\mu$

Definition 7 (HOMO and LUMO): The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are frontier orbitals.

The HOMO-LUMO gap $\Delta E = \varepsilon_{\text{LUMO}} - \varepsilon_{\text{HOMO}}$ correlates with:

• Chemical reactivity (smaller gap = more reactive).
• Electronic absorption spectra.
• Electrical conductivity in extended systems.

Homo-nuclear diatomics (second period):

For $\text{O}_2$, $\text{F}_2$ (heavier): $\sigma_{2s} < \sigma_{2s}^* < \sigma_{2p_z} < \pi_{2p_x} = \pi_{2p_y} < \pi_{2p_x}^* = \pi_{2p_y}^* < \sigma_{2p_z}^*$

For $\text{Li}_2$ through $\text{N}_2$ (lighter): $\sigma_{2s} < \sigma_{2s}^* < \pi_{2p_x} = \pi_{2p_y} < \sigma_{2p_z} < \pi_{2p_x}^* = \pi_{2p_y}^* < \sigma_{2p_z}^*$

Bond order:

$\text{BO} = \frac{1}{2}(n_b - n_a)$

where $n_b$ is the number of bonding electrons and $n_a$ is the number of antibonding electrons.

Example 3: $\text{O}_2$ has the configuration $(\sigma_{2s})^2(\sigma_{2s}^*)^2(\sigma_{2p_z})^2(\pi_{2p_x})^2(\pi_{2p_y})^2(\pi_{2p_x}^*)^1(\pi_{2p_y}^*)^1$.

Bond order $= \frac{1}{2}(10 - 6) = 2$.

$\text{O}_2$ is paramagnetic (two unpaired electrons in $\pi^*$ orbitals).

$\blacksquare$

For heteronuclear diatomics like CO or HF, the MOs are weighted combinations where the more electronegative atom contributes more to bonding orbitals. Electronegativity differences shift the energy levels.

Definition 8 (Symmetry Adapted Linear Combinations): For polyatomic molecules, symmetry-adapted linear combinations (SALCs) of atomic orbitals are constructed using group theory.

Huckel theory makes three approximations for $\pi$-electron systems:

1. Only $\pi$ electrons are considered explicitly.
2. $\langle \phi_\mu | \hat{H} | \phi_\mu \rangle = \alpha$ (Coulomb integral, same for all $p$ orbitals).
3. $\langle \phi_\mu | \hat{H} | \phi_\nu \rangle = \beta$ (resonance integral, nonzero only for bonded neighbors).
4. Overlap integrals: $\langle \phi_\mu | \phi_\nu \rangle = \delta_{\mu\nu}$.

For ethylene (2 $\pi$ centers):

$\begin{vmatrix} \alpha - E & \beta \\ \beta & \alpha - E \end{vmatrix} = 0$

Setting $x = (\alpha - E)/\beta$:

$x^2 - 1 = 0 \implies x = \pm 1 \implies E = \alpha \pm \beta$

The bonding orbital has $E = \alpha + \beta$ and the antibonding orbital has $E = \alpha - \beta$.

For benzene, the secular determinant gives $x^6 - 6x^4 + 9x^2 - 4 = 0$ with roots $x = \pm 2, \pm 1, \pm 1$.

Energy levels: $E = \alpha + 2\beta$, $\alpha + \beta$ (doubly degenerate), $\alpha - \beta$ (doubly degenerate), $\alpha - 2\beta$.

Definition 9 (Huckel Rule): A planar monocyclic system with $(4n + 2)$ $\pi$ electrons is aromatic.

Benzene ($n = 1$, 6 $\pi$ electrons) satisfies this rule.

Definition 10 (Delocalization Energy): The energy lowering due to electron delocalization:

$E_{\text{deloc}} = E_\pi(\text{delocalized}) - E_\pi(\text{localized})$

For benzene: $E_\pi = 2(\alpha + 2\beta) + 4(\alpha + \beta) = 6\alpha + 8\beta$. Three isolated double bonds: $3 \times 2(\alpha + \beta) = 6\alpha + 6\beta$. Delocalization energy: $2\beta$.

• Minimal basis: STO-3G — each orbital represented by 3 Gaussian functions.
• Split-valence: 3-21G, 6-31G — valence orbitals split into multiple functions.
• Polarization: 6-31G*, 6-31G** — add $d$ functions on heavy atoms, $p$ on H.
• Diffuse: 6-31+G* — add diffuse functions for anions and excited states.

• Moller-Plesset perturbation theory (MP2, MP4): Includes electron correlation.
• Configuration Interaction (CI): Expands the wavefunction in excited configurations.
• Coupled Cluster (CCSD(T)): Gold standard for single-reference systems.
• Density Functional Theory (DFT): Uses electron density instead of wavefunction; B3LYP is a popular functional.

Definition 11 (BSSE): In calculating interaction energies, each monomer artificially borrows basis functions from the other. Corrected using the counterpoise method.

Theorem 10 (First-Order Correction): For $\hat{H} = \hat{H}_0 + \hat{H}'$:

$E_n^{(1)} = \langle \psi_n^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle$

$\psi_n^{(1)} = \sum_{m \neq n} \frac{\langle \psi_m^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}}\,\psi_m^{(0)}$

Theorem 11 (Second-Order Energy Correction):

$E_n^{(2)} = \sum_{m \neq n} \frac{|\langle \psi_m^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}}$

Theorem 12 (Variational Principle): For any trial wavefunction $\tilde{\Psi}$:

$\langle \tilde{\Psi} | \hat{H} | \tilde{\Psi} \rangle \geq E_0$

where $E_0$ is the true ground state energy. This underpins the Hartree-Fock and DFT methods.

1. Confusing the time-dependent and time-independent Schrödinger equations. The TDSE governs time evolution; the TISE gives stationary states and energy eigenvalues. Fix: Use the TISE for bound-state problems and the TDSE for time-dependent phenomena.
2. Using the wrong angular momentum formula. The magnitude is $|L| = \sqrt{\ell(\ell+1)}\hbar$, not $\ell\hbar$. Fix: This is a quantum correction; $L$ can never equal $n\hbar$ exactly.
3. Applying the simple hydrogen energy formula to multi-electron atoms. $E_n = -13.6/n^2$ only works for hydrogen-like atoms. Fix: For multi-electron atoms, use effective nuclear charge or empirical data.
4. Ignoring the Pauli principle when writing configurations. Each orbital holds at most 2 electrons. Fix: Always check that no more than 2 electrons occupy any orbital and that spin assignments are antisymmetric.
5. Confusing Huckel $\alpha$ and $\beta$ signs. $\beta < 0$ (bonding), so $E = \alpha + \beta$ is lower than $\alpha$. Fix: Remember that $\alpha$ is the reference and bonding lowers energy.
6. Wrong orbital ordering for light vs heavy diatomics. $\text{N}_2$ and earlier have $\pi_{2p} < \sigma_{2p}$; $\text{O}_2$ and later have $\sigma_{2p} < \pi_{2p}$. Fix: Check the $s$-$p$ mixing for $\text{Li}_2$ through $\text{N}_2$.
7. Misinterpreting Koopmans’ theorem. $-\varepsilon_i$ equals the ionization energy only at the Hartree-Fock level with frozen orbitals. Fix: For DFT, the HOMO energy approximates IP but not exactly (Janak’s theorem).

• Schrödinger equation: $\hat{H}\psi = E\psi$; foundation of quantum chemistry.
• Particle in a box: $E_n = n^2h^2/(8mL^2)$; introduces quantization and zero-point energy.
• Hydrogen atom: $E_n = -13.6/n^2$ eV; quantum numbers $n, \ell, m_\ell, m_s$.
• Angular momentum: $|L| = \sqrt{\ell(\ell+1)}\hbar$; $L_z = m_\ell\hbar$; spin $s = 1/2$.
• Pauli exclusion: No two electrons share all four quantum numbers.
• MO theory (LCAO): $\psi_i = \sum c_{\mu i}\phi_\mu$; bonding vs antibonding; bond order.
• Huckel theory: $\pi$-electron approximation; aromaticity ($4n + 2$ rule).
• Born-Oppenheimer: Separates electronic and nuclear motion; defines the PES.
• Variational principle: Any trial energy $\geq E_0$; basis for computational methods.

Problem: Calculate the energy of the n=3 level of a hydrogen atom and the wavelength of the photon emitted in the transition n=3 to n=2. Solution: E_n = -13.6/n^2 eV. E_3 = -13.6/9 = -1.51 eV. E_2 = -13.6/4 = -3.40 eV. Delta E = E_3 - E_2 = -1.51 - (-3.40) = 1.89 eV. lambda = hc/Delta E = 1240 eV nm / 1.89 eV = 656 nm (in the visible range, H-alpha line).

Problem: For O2, the molecular orbital ordering has pi_2p below sigma_2p. What is the bond order, and is O2 paramagnetic? Solution: Electron configuration of O2 (12 electrons): sigma_2s^2 sigma_2s*^2 sigma_2p_z^2 pi_2p_x^2 pi_2p_y^2 pi_2p_x*^1 pi_2p_y*^1. Bond order = (1/2)(bonding - antibonding) = (1/2)(8 - 4) = 2. Since there are two unpaired electrons in the pi_2p* orbitals, O2 is paramagnetic.