Key equations (need to remember) for Cambridge Part II Theoretical techniques course.
Huckel
The resonance energy is the difference between total Huckel energy and the summed energies of isolated bonds in the structure.
Cyclic polyenes
Energy - this is equivalent to drawing a Frost circle! \[\begin{align} E_s=\alpha + 2\beta \cos(\frac{2s\pi}{N})&&s=0,\pm1,...\pm\frac{N}{2} \end{align}\] Coefficients \[\begin{align} c_s^{(0)}&=\sqrt{\frac{1}{N}}\\ c_s^{(ns)}&=\sqrt{\frac{2}{N}}\sin(\frac{2ns\pi}{N}) && n=1,2,...,\frac{N-1}{2}\\ c_s^{(nc)}&=\sqrt{\frac{2}{N}}\cos(\frac{2ns\pi}{N})\\ c_s^{(N/2)}&=\sqrt{\frac{1}{N}}(-1)^s \end{align}\]
Linear polyenes
Energy \[\begin{align} E_s=\alpha + 2\beta \cos(\frac{s\pi}{N+1})&&s=1,2,...,N. \end{align}\] Coefficients \[\begin{align} c_s^{(n)}&=\sqrt{\frac{2}{N+1}}\sin(\frac{ns\pi}{N+1}) && n=1,2,...,N \end{align}\]
Alternant hydrocarbons
Usually start labelling a starred atom at the end.
- Energies symmetrically placed. \(E=\alpha \pm 2k\beta\).
- Coefficients for pairs of energies: changing the sign of the coefficients on the unstarred atoms.
- In a neutral molecule, population \(q_s=1\).
- In an odd alternant, there is a non-bonding MO - \(E=\alpha\) - where all the unstarred atoms have 0 coefficient, and the sum of starred atoms around an unstarred atom is 0.
Wavefunction
Slater determinant
\[ \Psi=\frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_a(1)&\chi_b(1)&...&\chi_N(1)\\ \chi_a(2)&\chi_b(2)&...&\chi_N(2)\\ ...\\ \chi_a(N)&\chi_b(N)&...&\chi_N(N) \end{vmatrix}. \]
Population and bond order
The population operator is \(\hat{q_s}=|\phi_s\rangle\langle\phi_s|\);
The bond order operator is \(\hat{p_{st}}=\frac{1}{2}(|\phi_s\rangle\langle\phi_t|+|\phi_t\rangle\langle\phi_s|)\).
To calculate from coefficients, use \[\begin{align} q_s=\sum_n f_n|c_s^n|^2\\ p_{st}=\sum_n f_n c_s^nc_t^n \end{align}\]. $$
Physicists’ notation for 2-electron integrals
\[ \langle ab|cd\rangle=\int_1\int_2\psi^*_a(1)\psi^*_b(2)\frac{1}{r_{12}}\psi_c(1)\psi_d(2)\,d\mathbf{r_1}\mathbf{r_2}. \]
Perturbation theory
First order perturbation to the eigenvalues: \[ \Delta \epsilon_i=\langle\psi_i|\Delta \hat{H}|\psi_i\rangle. \]
Normal modes
Mass weighted coordinates
\[ q_i=\sqrt{m_i}x_i \]
Dynamical matrix
\[ K_{ij}=\frac{1}{\sqrt{m_i m_j}}\frac{\partial^2 V}{\partial x_i x_j}\Big{|}_{x=0}. \]
Perturbation of normal mode frequencies
\[ \omega_m^2=(\omega_m^{(0)})^2+\langle Q_m^{(0)}|\hat{K}^{(1)}|Q_m^{(0)}\rangle. \]