Derivation of the UHF SCF equations through Lagrange multipliers

Expanding the \(\alpha\)- and \(\beta\)-spinors in the underlying scalar bases requires \(\require{physics} \vb{C}^\alpha\) and \(\vb{C}^\beta\), the \(\alpha\)- and \(\beta\) coefficient matrices. Since the UHF energy only depends on the occupied \(\alpha\)-and \(\beta\)-spinors, whose coefficients are collected in \(\vb{C}_i^\alpha\) and \(\vb{C}_i^\beta\), the UHF energy may be parametrized as: \[\begin{split} \require{physics} E(\vb{C}_i^\alpha, \vb{C}_i^\beta) = &\sum_{\sigma} \sum_i^{N_\sigma} h_{i\sigma, i\sigma}(\vb{C}_i^\alpha, \vb{C}_i^\beta) + \frac{1}{2} \sum_{\sigma \tau} \sum_i^{N_\sigma} \sum_{j}^{N_\tau} (i\sigma i\sigma | j\tau j\tau)(\vb{C}_i^\alpha, \vb{C}_i^\beta) \\ &- \frac{1}{2} \sum_{\sigma} \sum_{ij}^{N_\sigma} (i\sigma j\sigma | j\sigma i\sigma)(\vb{C}_i^\alpha, \vb{C}_i^\beta) \thinspace . \end{split}\]

Since the occupied spinors are required to remain orthonormal, we will construct the Lagrangian \[\begin{equation} \mathscr{L}(\vb{C}_i^\alpha, \vb{C}_i^\beta, \boldsymbol{\epsilon}_\alpha, \boldsymbol{\epsilon}_\beta) = E(\vb{C}_i^\alpha, \vb{C}_i^\beta) - \sum_{ij}^{N_\alpha} \epsilon^\alpha_{ji} \qty[ \vb{C}^{\alpha, \dagger} \vb{S}^{\alpha \alpha} \vb{C}^{\alpha} - \vb{I} ]_{ij} - \sum_{ij}^{N_\beta} \epsilon^\beta_{ji} \qty[ \vb{C}^{\beta, \dagger} \vb{S}^{\beta \beta} \vb{C}^{\beta} - \vb{I} ]_{ij} \thinspace , \end{equation}\] where \(\boldsymbol{\epsilon}_\alpha\) and \(\boldsymbol{\epsilon}_\beta\) are the Lagrange multipliers. The stationary equations are then given by \[\begin{equation} \eval{ \pdv{ \mathscr{L}(\vb{C}_i^\alpha, \vb{C}_i^\beta, \boldsymbol{\epsilon}_\alpha, \boldsymbol{\epsilon}_\beta) }{ C^{\varepsilon *}_{pl} } }_{\vb{C}^{\sigma, \star}_i} = 0 \thinspace , \end{equation}\] where \(\vb{C}^{\sigma, \star}_i\) represents the optimal occupied \(\alpha\)- and \(\beta\) spinor expansion coefficients. Note that this stationary equation must hold for every value of spin \(\varepsilon = \{\alpha, \beta\}\) and indices \(\mu l\), where \(l\) is the index of the \(l\)-th occupied \(\sigma\)-spinor and \(\mu\) is an index of the scalar functions that are used in the expansion of the \(\sigma\) spinor.

Expanding the \(\alpha\)- and \(\beta\) spinors in their underlying scalar bases, we find as an intermediary result: \[\begin{multline} \sum_\mu^{K_\varepsilon} h^{\varepsilon \varepsilon}_{\gamma \mu} C^{\varepsilon \star}_{\mu l} + \sum_\sigma \sum_{\mu}^{K_\varepsilon} \sum_{\nu \rho}^{K_\sigma} P^{\sigma \sigma}_{\nu \rho}(\vb{C}^{\sigma \star}_i) C^{\varepsilon \star}_{\mu l} (\gamma \varepsilon \mu \varepsilon | \nu \sigma \rho \sigma) \\ - \sum_{\mu \nu \rho}^{K_\varepsilon} P^{\varepsilon \varepsilon}_{\nu \mu}(\vb{C}^{\varepsilon \star}_i) C^{\varepsilon \star}_{\rho l} (\gamma \varepsilon \mu \varepsilon | \nu \varepsilon \rho \varepsilon) = \sum_{i}^{N_\varepsilon} \epsilon_{il} \sum_{\mu}^{K_\varepsilon} S^{\varepsilon \varepsilon}_{\gamma \mu} C^{\varepsilon \star}_{\mu j} \thinspace , \end{multline}\] by introducing the scalar basis UHF density matrices \(\vb{P}^{\sigma \sigma}\): \[\begin{equation} P^{\sigma \sigma}_{\mu \nu}(\vb{C}^\sigma_i) = \sum_{i}^{N_\sigma} C^{\sigma *}_{\mu i} C^{\sigma}_{\nu i} \thinspace . \end{equation}\]

Furthermore, we now introduce the direct (or Coulomb) matrices with elements \[\begin{equation} J^{\sigma \sigma}_{\mu \nu}(\vb{C}^\alpha_i, \vb{C}^\beta_i) = \sum_\tau \sum_{\rho \lambda}^{K_\tau} P^{\tau \tau}_{\rho \lambda}(\vb{C}^\tau_i) (\mu \sigma \nu \sigma | \rho \tau \lambda \tau) \end{equation}\] and the exchange matrices with elements \[\begin{equation} K^{\sigma \sigma}_{\mu \nu}(\vb{C}^\sigma_i) = \sum_{\rho \lambda}^{K_\sigma} P^{\sigma \sigma}_{\lambda \rho}(\vb{C}^\sigma_i) (\mu \sigma \rho \sigma | \lambda \sigma \nu \sigma) \thinspace , \end{equation}\] which leaves us with the following stationary equations: \[\begin{equation} \sum_\mu^{K_\varepsilon} h^{\varepsilon \varepsilon}_{\gamma \mu} C^{\varepsilon \star}_{\mu l} + \sum_{\mu}^{K_\varepsilon} J^{\varepsilon \varepsilon}_{\gamma \mu}(\vb{C}^{\sigma \star}_i) C^{\varepsilon \star}_{\mu l} - \sum_{\mu}^{K_\varepsilon} K^{\varepsilon \varepsilon}_{\gamma \mu}(\vb{C}^{\varepsilon \star}_i) C^{\varepsilon \star}_{\mu l} = \sum_{i}^{N_\varepsilon} \epsilon_{il} \sum_{\mu}^{K_\varepsilon} S^{\varepsilon \varepsilon}_{\gamma \mu} C^{\varepsilon \star}_{\mu j} \thinspace . \end{equation}\] Introducing the UHF Fock matrices as \[\begin{equation} \vb{F}^{\sigma \sigma}(\vb{C}^\alpha_i, \vb{C}^\beta_i) = \vb{h}^{\sigma \sigma} + \vb{J}^{\sigma \sigma}(\vb{C}^\alpha_i, \vb{C}^\beta_i) - \vb{K}^{\sigma \sigma}(\vb{C}^\sigma_i) \thinspace , \end{equation}\] we arrive at the following form of the stationary equations: \[\begin{equation} \vb{F}^{\sigma \sigma}(\vb{C}^{\alpha \star}_i, \vb{C}^{\beta \star}_i) \vb{C}^{\sigma \star}_i = \vb{S}^{\sigma \sigma} \vb{C}^{\sigma \star}_i \boldsymbol{\epsilon}^{\sigma} \thinspace , \end{equation}\] which are called the UHF SCF equations. The Fock and overlap matrices are of dimension \((K_\sigma \times K_\sigma)\) and the occupied coefficient matrix \(\vb{C}^\sigma_i\) is of dimension \((K_\sigma \times N_\sigma)\). The occupied spinor coefficients \(\vb{C}^\sigma_i\) are the lowest \(N_\sigma\) generalized eigenvectors of the Fock matrix \(\vb{F}^{\sigma \sigma}\).