22.6.3: ii. Exercises
- Page ID
- 85560
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By expanding the molecular orbitals \(\{\phi\kappa\}\) as linear combinations of atomic orbitals \(\{\chi_{\mu}\}\),
\[ \phi_k = \sum\limits_\mu c_{\mu k}\chi_\mu \nonumber \]
show how the canonical Hartree-Fock (HF) equations:
\[ F \phi_i - \epsilon_i\phi_j \nonumber \]
reduce to the matrix eigenvalue-type equation of the form given in the text:
\[ \sum\limits_\nu F_{\mu\nu} C_{\nu i} = \epsilon_i\sum\limits_{\nu} S_{\mu\nu}C_{\nu i} \nonumber \]
where:
\begin{align} F_{\mu\nu} &= \langle \chi_\mu |h|\chi_\nu \rangle + \sum\limits_{\delta \kappa} \left[ \gamma_{\delta \kappa} \langle \chi_\mu \chi_\delta |g| \chi_\nu \chi_\kappa \rangle - \gamma_{\delta \kappa}^{ex} \langle \chi_\mu \chi_\delta |g| \chi_\kappa \chi_\nu \rangle \right], \\ S_{\mu\nu} &= \langle \chi_\mu | \chi_\nu \rangle, \gamma_{\delta \kappa} = \sum\limits_{i=occ} C_{\delta i}C_{\kappa i}, \\ \text{and } \gamma_{\delta \kappa}^{ex} &= \sum\limits_{\substack{\text{occ and}\\\text{same spin}}}C_{\delta i}C_{\kappa i}. \end{align}
Note that the sum over i in \(\gamma_{\delta\kappa} \text{ and } \gamma_{\delta\kappa}\) is a sum over spin orbitals. In addition, show
that this Fock matrix can be further reduced for the closed shell case to:
\[ F_{\mu\nu} = \langle \chi_\mu |h| \chi_\nu \rangle + \sum\limits_{\delta\kappa} P_{\delta\kappa} \left[ \langle \chi_\mu \chi_\delta |g| \chi_\nu \chi_\kappa \rangle - \dfrac{1}{2}\langle \chi_\mu \chi_\delta |g| \chi_\kappa \chi_\nu \rangle \right] , \nonumber \]
where the charge bond order matrix, P, is defined to be:
\[ P_{\delta \kappa} = \sum\limits_{i=occ} 2C_{\delta i}C_{\kappa i}, \nonumber \]
where the sum over i here is a sum over orbitals not spin orbitals.
Q2
Show that the HF total energy for a closed-shell system may be written in terms of integrals over the orthonormal HF orbitals as:
\[ \text{E(SCF) } = 2\sum\limits_{k}^{occ} \langle \phi_k |h| \phi_k \rangle + \sum\limits_{kl}^{occ}\left[ 2\langle k1| gk1 \rangle - \langle k1 |g| 1k \rangle \right] + \sum\limits_{\mu >\nu} \dfrac{Z_\mu Z_\nu}{R_{\mu\nu}}. \nonumber \]
Q3
Show that the HF total energy may alternatively be expressed as:
\[ \text{E(SCF)} = \sum\limits_k^{occ} \left( \epsilon_k + \langle \phi_k |h| \phi_k \rangle \right) + \sum\limits_{\mu > \nu} \dfrac{Z_\mu Z_\nu}{R_{\mu\nu}} \nonumber \]
where the \(\epsilon_k\) refer to the HF orbital energies.