Dirac-Hartree-Fock theory
The Hartree-Fock wave function
The Hartree-Fock wave function is the single Slater determinant \[\begin{equation} \require{physics} \ket{\text{HF}} = \prod_{\mathcal{I}}^N \hat{a}^\dagger_{\mathcal{I}} \ket{\text{vac}} \thinspace , \end{equation}\] which means that its one-electron density matrix becomes: \[\begin{equation} D_{\mathcal{P} \mathcal{Q}} \rightarrow \delta_{\mathcal{I} \mathcal{J}} \end{equation}\] and its two-electron density matrix becomes: \[\begin{equation} d_{\mathcal{P} \mathcal{Q} \mathcal{R} \mathcal{S}} \rightarrow \delta_{\mathcal{I} \mathcal{J}} \delta_{\mathcal{K} \mathcal{L}} - \delta_{\mathcal{I} \mathcal{K}} \delta_{\mathcal{J} \mathcal{L}} \thinspace , \end{equation}\] in which \(\mathcal{I}, \mathcal{J}\) are indices of spinors occupied in the Hartree-Fock determinant.
The Dirac-Hartree-Fock equations
Using the form of the Hartree-Fock density matrices, we can find that the Fock operator becomes \[\begin{equation} f_{\mathcal{P} \mathcal{Q}}(\vb{r}) \rightarrow \delta_{\mathcal{I} \mathcal{J}} h^{c, \text{D}}(\vb{r}) + \delta_{\mathcal{I} \mathcal{J}} \sum_{\mathcal{K}}^N \int \dd{\vb{r}'} \phi^\dagger_{\mathcal{K}}(\vb{r}') g^c(\vb{r}, \vb{r}') \phi_\mathcal{K}(\vb{r}') - \int \dd{\vb{r}'} \phi^\dagger_{\mathcal{J}}(\vb{r}') g^c(\vb{r}, \vb{r}') \phi_\mathcal{I}(\vb{r}') \thinspace , \end{equation}\] in which \(g^c(\vb{r}, \vb{r}')\) is the yet unspecified two-electron interaction, i.e. it can still be Coulomb, Gaunt or Breit and \(\mathcal{I} \mathcal{J} \mathcal{K}\) are occupied indices only. Filling this into the general SCF equations leads to the Dirac-Hartree-Fock SCF equations: \[\begin{equation} \qty[ h^{c, \text{D}}(\vb{r}) + \sum_{\mathcal{K}}^N (J^c_\mathcal{K}(\vb{r}) - K^c_\mathcal{K}(\vb{r})) ] \phi_\mathcal{I}(\vb{r}) = \sum_{\mathcal{P} \mathcal{Q}}^K \epsilon_{\mathcal{P} \mathcal{Q}} \phi_{\mathcal{Q}}(\vb{r}) \thinspace , \end{equation}\] in which \(J^c_\mathcal{K}(\vb{r})\) is confusingly called the Coulomb operator: \[\begin{equation} J^c_\mathcal{K}(\vb{r}) \phi_\mathcal{I}(\vb{r}) = \qty(\int \dd{\vb{r}'} \phi^\dagger_{\mathcal{K}}(\vb{r}') g^c(\vb{r}, \vb{r}') \phi_\mathcal{K}(\vb{r}') ) \phi_\mathcal{I}(\vb{r}) \thinspace , \end{equation}\] even though we haven’t yet specified the nature of the two-electron interaction, and \(K^c_\mathcal{K}(\vb{r})\) is called the exchange operator: \[\begin{equation} K^c_\mathcal{K}(\vb{r}) \phi_\mathcal{I}(\vb{r}) = \qty(\int \dd{\vb{r}'} \phi^\dagger_{\mathcal{K}}(\vb{r}') g^c(\vb{r}, \vb{r}') \phi_\mathcal{I}(\vb{r}') ) \phi_\mathcal{K}(\vb{r}) \thinspace . \end{equation}\]
After a unitary transformation of the spinors that diagonalizes the matrix of the Lagrange multipliers, we find the canonical Dirac-Hartree-Fock equations: \[\begin{equation} f^c_{\text{DHF}}(\vb{r}) \phi_\mathcal{I}(\vb{r}) = \epsilon_{\mathcal{I}} \phi_{\mathcal{I}}(\vb{r}) \thinspace , \end{equation}\] in which \(\epsilon_{\mathcal{I}}\) is a diagonal element of the Lagrange multipliers and \(f^c_{\text{DHF}}(\vb{r})\) is called the Dirac-Hartree-Fock operator: \[\begin{equation} f^c_{\text{DHF}}(\vb{r}) = h^{c,\text{D}}(\vb{r}) + \sum_{\mathcal{K}}^N (J^c_\mathcal{K}(\vb{r}) - K^c_\mathcal{K}(\vb{r})) \thinspace . \end{equation}\]
Using the Hartree-Fock density matrices, we can easily write the Dirac-Hartree-Fock energy: \[\begin{equation} E_\text{DHF} = \sum_{\mathcal{I}}^N h^{\text{D}}_{\mathcal{I} \mathcal{I}} + \frac{1}{2} \sum_{\mathcal{I} \mathcal{J}}^N (g_{\mathcal{I} \mathcal{I} \mathcal{J} \mathcal{J}} - g_{\mathcal{I} \mathcal{J} \mathcal{J} \mathcal{I}}) \end{equation}\] In terms of the orthonormal spinors, the matrix elements of the Coulomb operator are \[\begin{align} J_{\mathcal{K}, \mathcal{P} \mathcal{Q}} &= \matrixel{\phi_{\mathcal{P}}}{J^c_{\mathcal{K}}}{\phi_{\mathcal{Q}}} \\ &= g_{\mathcal{P} \mathcal{Q} \mathcal{K} \mathcal{K}} \end{align}\] and those of the exchange operator are \[\begin{align} K_{\mathcal{K}, \mathcal{P} \mathcal{Q}} &= \matrixel{\phi_{\mathcal{P}}}{K^c_{\mathcal{K}}}{\phi_{\mathcal{Q}}} \\ &= g_{\mathcal{P} \mathcal{K} \mathcal{K} \mathcal{Q}} \thinspace , \end{align}\] such that we can rewrite the DHF-energy as \[\begin{equation} E_{\text{DHF}} = \sum_{\mathcal{I}}^N \qty( h^{\text{D}}_{\mathcal{I} \mathcal{I}} + \frac{1}{2} \sum_{\mathcal{J}}^N (J_{\mathcal{J}, \mathcal{I} \mathcal{I}} - K_{\mathcal{J}, \mathcal{I} \mathcal{I}})) \thinspace . \end{equation}\]
Dirac-Hartree-Fock (Coulomb) theory in terms of 2-spinors
If we have a set \(\set{\chi_{\mathcal{P}}^{\text{L}}}\) of \(K_{\text{L}}\) basis \(2\)-spinors available for the large component and a set \(\set{\chi_{\mathcal{P}}^{\text{S}}}\) of \(K_{\text{S}}\) \(2\)-spinors for the small components, we can expand the spinor in terms of these basis functions as \[\begin{equation} \phi_\mathcal{P}(\vb{r}) = \begin{pmatrix} \phi^\text{L}_\mathcal{P}(\vb{r}) \\ \phi^\text{S}_\mathcal{P}(\vb{r}) \end{pmatrix} = \begin{pmatrix} \displaystyle\sum_\mathcal{Q}^{K_\text{L}} C_{\mathcal{Q} \mathcal{P}}^\text{L} \chi^\text{L}_\mathcal{Q}(\vb{r}) \\ \displaystyle\sum_\mathcal{Q}^{K_\text{S}} C_{\mathcal{Q} \mathcal{P}}^\text{S} \chi^\text{S}_\mathcal{Q}(\vb{r}) \end{pmatrix} \thinspace , \end{equation}\] which means that the total number of expansion coefficients that describe a Dirac spinor is equal to \(K = K_{\text{L}} + K_{\text{S}}\). Therefore, we can write down a full coefficient matrix \(\vb{C}\) (with dimensions \((K_{\text{L}} + K_{\text{S}}) \times (K_{\text{L}} + K_{\text{S}})\)) as \[\begin{equation} \vb{C} = \begin{pmatrix} C^{\text{L}}_{1 1} & \cdots & C^{\text{L}}_{1 \mathcal{P}} & \cdots & C^{\text{L}}_{1 (K_{\text{L}} + K_{\text{S}})} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ C^{\text{L}}_{K_{\text{L}} 1} & \cdots & C^{\text{L}}_{K_{\text{L}} \mathcal{P}} & \cdots & C^{\text{L}}_{K_{\text{L}} (K_{\text{L}} + K_{\text{S}})} \\ C^{\text{S}}_{1 1} & \cdots & C^{\text{S}}_{1 \mathcal{P}} & \cdots & C^{\text{S}}_{1 (K_{\text{L}} + K_{\text{S}})} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ C^{\text{S}}_{K_{\text{S}} 1} & \cdots & C^{\text{S}}_{K_{\text{S}} \mathcal{P}} & \cdots & C^{\text{S}}_{K_{\text{S}} (K_{\text{L}} + K_{\text{S}})} \end{pmatrix} \thinspace , \end{equation}\] from which we can clearly see that the expansion coefficients for the \(\mathcal{P}\)-th spinor \(\phi_{\mathcal{P}}\) can be found in the \(\mathcal{P}\)-column of the full coefficient matrix \(\vb{C}\).
By collecting the nuclear attraction and the DHF mean-field electron-electron potential in a term \(W^c(\vb{r})\): \[\begin{equation} W^c(\vb{r}) = V^c_\text{nuc}(\vb{r}) + \sum_{\mathcal{K}}^N (J^c_\mathcal{K}(\vb{r}) - K^c_{\mathcal{K}}(\vb{r})) \thinspace , \end{equation}\] we can write the Dirac-Hartree-Fock equations in \(2\)-spinor form as: \[\begin{equation} \begin{pmatrix} W^c(\vb{r}) + q_e \phi^c_\text{ext}(\vb{r}) & c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}) \\ c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}) & W^c(\vb{r}) + q_e \phi^c_\text{ext}(\vb{r}) - 2 m_e c^2 \end{pmatrix} \begin{pmatrix} \phi_\mathcal{I}^L(\vb{r}) \\ \phi_\mathcal{I}^S(\vb{r}) \end{pmatrix} = \epsilon_\mathcal{I} \begin{pmatrix} \phi_\mathcal{I}^L(\vb{r}) \\ \phi_\mathcal{I}^S(\vb{r}) \end{pmatrix} \thinspace , \end{equation}\] which reduces to \[\begin{equation} \label{eq:2-spinor_DHF_equations_no_external} \begin{pmatrix} W^c(\vb{r}) & c \boldsymbol{\sigma} \vdot \vb{p}^c \\ c \boldsymbol{\sigma} \vdot \vb{p}^c & W^c(\vb{r}) - 2 m_e c^2 \end{pmatrix} \begin{pmatrix} \phi_\mathcal{I}^L(\vb{r}) \\ \phi_\mathcal{I}^S(\vb{r}) \end{pmatrix} = \epsilon_\mathcal{I} \begin{pmatrix} \phi_\mathcal{I}^L(\vb{r}) \\ \phi_\mathcal{I}^S(\vb{r}) \end{pmatrix} \end{equation}\] in the absence of any external electromagnetic fields.
We will now integrate the Dirac-Hartree-Fock (Coulomb) equations, i.e. \[\begin{equation} \int \dd{\vb{r}} \chi^\dagger_{\mathcal{P}}(\vb{r}) f^c_{\text{DHF}}(\vb{r}) \phi_\mathcal{I}(\vb{r}) = \epsilon_{\mathcal{I}} \int \dd{\vb{r}} \chi^\dagger_{\mathcal{P}}(\vb{r}) \phi_{\mathcal{I}}(\vb{r}) \thinspace , \end{equation}\] and express the molecular spinors in terms of the introduced basis functions. When working this out, the following definitions arise naturally. We define the Dirac-Hartree-Fock density matrix expressed in atomic \(2\)-spinor basis as \[\begin{equation} D^{\text{XY}}_{\mathcal{P} \mathcal{Q}} = \sum_{\mathcal{I}}^N C^{\text{X}*}_{\mathcal{P} \mathcal{I}} C^{\text{Y}}_{\mathcal{Q} \mathcal{I}} \thinspace , \end{equation}\] the core Hamiltonian as \[\begin{align} h^{\text{D}, \text{LL}}_{\mathcal{P} \mathcal{Q}} &= \matrixel*{\chi^{\text{L}}_{\mathcal{P}}}{V^c_{\text{nuc}} + q_e \phi^c_{\text{ext}}}{\chi^{\text{L}}_{\mathcal{Q}}} \\ h^{\text{D}, \text{LS}}_{\mathcal{P} \mathcal{Q}} &= \matrixel*{\chi^{\text{L}}_{\mathcal{P}}}{c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c}{\chi^{\text{S}}_{\mathcal{Q}}} \\ h^{\text{D}, \text{SL}}_{\mathcal{P} \mathcal{Q}} &= \matrixel*{\chi^{\text{S}}_{\mathcal{P}}}{c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c}{\chi^{\text{S}}_{\mathcal{L}}} \\ h^{\text{D}, \text{SS}}_{\mathcal{P} \mathcal{Q}} &= \matrixel*{\chi^{\text{S}}_{\mathcal{P}}}{V^c_{\text{nuc}} + q_e \phi^c_{\text{ext}} -2 m_e c^2}{\chi^{\text{S}}_{\mathcal{Q}}} \thinspace , \end{align}\] the Coulomb matrix as \[\begin{align} J^{\text{LL}}_{\mathcal{P} \mathcal{Q}} &= \sum_{\mathcal{K}}^N \matrixel*{\chi^{\text{L}}_{\mathcal{P}}}{J^c_{\mathcal{K}}}{\chi^{\text{L}}_{\mathcal{Q}}} \\ &= \sum_{\mathcal{R} \mathcal{S}}^{K_{\text{L}}} D^{\text{LL}}_{\mathcal{R} \mathcal{S}} (\chi^{\text{L}}_{\mathcal{P}} \chi^{\text{L}}_{\mathcal{Q}} | \chi^{\text{L}}_{\mathcal{R}} \chi^{\text{L}}_{\mathcal{S}}) + \sum_{\mathcal{R} \mathcal{S}}^{K_{\text{S}}} D^{\text{SS}}_{\mathcal{R} \mathcal{S}} (\chi^{\text{L}}_{\mathcal{P}} \chi^{\text{L}}_{\mathcal{Q}} | \chi^{\text{S}}_{\mathcal{R}} \chi^{\text{S}}_{\mathcal{S}}) \\ J^{\text{SS}}_{\mathcal{P} \mathcal{Q}} &= \sum_{\mathcal{K}}^N \matrixel*{\chi^{\text{S}}_{\mathcal{P}}}{J^c_{\mathcal{K}}}{\chi^{\text{S}}_{\mathcal{Q}}} \\ &= \sum_{\mathcal{R} \mathcal{S}}^{K_{\text{L}}} D^{\text{LL}}_{\mathcal{R} \mathcal{S}} (\chi^{\text{S}}_{\mathcal{P}} \chi^{\text{S}}_{\mathcal{Q}} | \chi^{\text{L}}_{\mathcal{R}} \chi^{\text{L}}_{\mathcal{S}}) + \sum_{\mathcal{R} \mathcal{S}}^{K_{\text{S}}} D^{\text{SS}}_{\mathcal{R} \mathcal{S}} (\chi^{\text{S}}_{\mathcal{P}} \chi^{\text{S}}_{\mathcal{Q}} | \chi^{\text{S}}_{\mathcal{R}} \chi^{\text{S}}_{\mathcal{S}}) \thinspace , \end{align}\] and the exchange matrix as \[\begin{align} K^{\text{LL}}_{\mathcal{P} \mathcal{S}} &= \sum_{\mathcal{Q} \mathcal{R}}^{K_{\text{L}}} D^{\text{LL}}_{\mathcal{R} \mathcal{Q}} (\chi^{\text{L}}_{\mathcal{P}} \chi^{\text{L}}_{\mathcal{Q}} | \chi^{\text{L}}_{\mathcal{R}} \chi^{\text{L}}_{\mathcal{S}}) \\ K^{\text{LS}}_{\mathcal{P} \mathcal{S}} &= \sum_{\mathcal{Q}}^{K_{\text{S}}} \sum_{\mathcal{R}}^{K_{\text{L}}} D^{\text{LS}}_{\mathcal{R} \mathcal{Q}} (\chi^{\text{S}}_{\mathcal{P}} \chi^{\text{S}}_{\mathcal{Q}} | \chi^{\text{L}}_{\mathcal{R}} \chi^{\text{L}}_{\mathcal{S}}) \\ K^{\text{SL}}_{\mathcal{P} \mathcal{S}} &= \sum_{\mathcal{Q}}^{K_{\text{L}}} \sum_{\mathcal{R}}^{K_{\text{S}}} D^{\text{SL}}_{\mathcal{R} \mathcal{Q}} (\chi^{\text{L}}_{\mathcal{P}} \chi^{\text{L}}_{\mathcal{Q}} | \chi^{\text{S}}_{\mathcal{R}} \chi^{\text{S}}_{\mathcal{S}}) \\ K^{\text{SS}}_{\mathcal{P} \mathcal{S}} &= \sum_{\mathcal{Q} \mathcal{R}}^{K_{\text{L}}} D^{\text{SS}}_{\mathcal{R} \mathcal{Q}} (\chi^{\text{S}}_{\mathcal{P}} \chi^{\text{S}}_{\mathcal{Q}} | \chi^{\text{S}}_{\mathcal{R}} \chi^{\text{S}}_{\mathcal{S}}) \thinspace . \end{align}\] Using these definitions, we can write the Dirac-Hartree-Fock (Coulomb) equations in \(2\)-spinor form as \[\begin{equation} \label{eq:DHF_equations_2-component_expanded} \begin{pmatrix} \vb{f}^{\text{LL}} & \vb{f}^{\text{LS}} \\ \vb{f}^{\text{SL}} & \vb{f}^{\text{SS}} \end{pmatrix} \begin{pmatrix} \vb{C}^{\text{L}} \\ \vb{C}^{\text{S}} \end{pmatrix} = \begin{pmatrix} \vb{S}^{\text{LL}} & \vb{0} \\ \vb{0} & \vb{S}^{\text{SS}} \end{pmatrix} \begin{pmatrix} \vb{C}^{\text{L}} \\ \vb{C}^{\text{S}} \end{pmatrix} \boldsymbol{\epsilon} \thinspace , \end{equation}\] in which the elements of the Fock matrix \(\vb{f}\) are given by \[\begin{align} \vb{f}^{\text{LL}} &= \vb{h}^{\text{D}, \text{LL}} + \vb{J}^{\text{LL}} - \vb{K}^{\text{LL}} \\ \vb{f}^{\text{LS}} &= \vb{h}^{\text{D}, \text{LS}} - \vb{K}^{\text{LS}} \\ \vb{f}^{\text{SL}} &= \vb{h}^{\text{D}, \text{SL}} - \vb{K}^{\text{SL}} \\ \vb{f}^{\text{LL}} &= \vb{h}^{\text{D}, \text{SS}} + \vb{J}^{\text{SS}} - \vb{K}^{\text{SS}} \thinspace . \end{align}\] The coefficient quantity in the DHF-equation \(\eqref{eq:DHF_equations_2-component_expanded}\) is the full coefficient matrix, but written in two blocks to make the L- and S- subblock distinction of the Fock matrix more clear. The L-block of the coefficient matrix is of dimensions \(K_{\text{L}} \times (K_{\text{L}} + K_{\text{S}})\) and the S-block similarly has dimensions \(K_{\text{S}} \times (K_{\text{L}} + K_{\text{S}})\).
Because of the mathematical structure of the Dirac Hamiltonian, there will be \(K_{\text{S}}\) molecular spinors that represent negative-energy states. In order to evade these spinors, we simply remove them in the calculation of the density matrices (Reiher and Wolf 2015), i.e. we remove the first \(K_{\text{S}}\) entries on the diagonal the HF-density matrix expressed in in molecular spinor basis. Therefore, we can calculate the density matrix representation in the expansion basis as \[\begin{equation} D^{\text{XY}}_{\mathcal{P} \mathcal{Q}} = \sum_{\mathcal{I}=K_{\text{S}}}^{K_{\text{S}} + N} C^{\text{X}*}_{\mathcal{P} \mathcal{I}} C^{\text{Y}}_{\mathcal{Q} \mathcal{I}} \thinspace . \end{equation}\]
In terms of the underlying orthonormal \(2\)-spinors, we can write the DHF-energy as \[\begin{equation} \begin{split} E_\text{DHF} = &\sum_{\mathcal{I}}^N \Big( \matrixel{\phi^\text{L}_{\mathcal{I}}}{V^c_{\text{nuc}} + q_e \phi_{\text{ext}}^c}{\phi^\text{L}_{\mathcal{I}}} + c \matrixel{\phi^\text{L}_{\mathcal{I}}}{\boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c}{\phi^\text{S}_{\mathcal{I}}} + c \matrixel{\phi^\text{S}_{\mathcal{I}}}{\boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c}{\phi^\text{L}_{\mathcal{I}}} \notag \\ &+ \matrixel{\phi^\text{S}_{\mathcal{I}}}{V^c_{\text{nuc}} + q_e \phi_{\text{ext}}^c - 2m_e c^2}{\phi^\text{S}_{\mathcal{I}}} \Big) \notag\\ + &\frac{1}{2} \sum_{\mathcal{I} \mathcal{J}}^N \Big( (\phi^\text{L}_{\mathcal{I}} \phi^\text{L}_{\mathcal{I}} | \phi^\text{L}_{\mathcal{J}} \phi^\text{L}_{\mathcal{J}}) + (\phi^\text{L}_{\mathcal{I}} \phi^\text{L}_{\mathcal{I}} | \phi^\text{S}_{\mathcal{J}} \phi^\text{S}_{\mathcal{J}}) + (\phi^\text{S}_{\mathcal{I}} \phi^\text{S}_{\mathcal{I}} | \phi^\text{L}_{\mathcal{J}} \phi^\text{L}_{\mathcal{J}}) + (\phi^\text{S}_{\mathcal{I}} \phi^\text{S}_{\mathcal{I}} | \phi^\text{S}_{\mathcal{J}} \phi^\text{S}_{\mathcal{J}}) \notag \\ &- (\phi^\text{L}_{\mathcal{I}} \phi^\text{L}_{\mathcal{J}} | \phi^\text{L}_{\mathcal{J}} \phi^\text{L}_{\mathcal{I}}) - (\phi^\text{L}_{\mathcal{I}} \phi^\text{L}_{\mathcal{J}} | \phi^\text{S}_{\mathcal{J}} \phi^\text{S}_{\mathcal{I}}) - (\phi^\text{S}_{\mathcal{I}} \phi^\text{S}_{\mathcal{J}} | \phi^\text{L}_{\mathcal{J}} \phi^\text{L}_{\mathcal{I}}) - (\phi^\text{S}_{\mathcal{I}} \phi^\text{S}_{\mathcal{J}} | \phi^\text{S}_{\mathcal{J}} \phi^\text{S}_{\mathcal{I}}) \Big) \thinspace . \end{split} \end{equation}\]
By expanding in a \(2\)-spinor basis, we get \[\begin{equation} \begin{split} E_\text{DHF} = &\sum_{\mathcal{P} \mathcal{Q}}^{K_{\text{L}}} D^{\text{LL}}_{\mathcal{P} \mathcal{Q}} \matrixel{\chi^\text{L}_{\mathcal{P}}}{V^c_{\text{nuc}} + q_e \phi_{\text{ext}}^c}{\chi^\text{L}_{\mathcal{Q}}} + \sum_{\mathcal{P}}^{K_{\text{L}}} \sum_{\mathcal{Q}}^{K_{\text{S}}} D^{\text{LS}}_{\mathcal{P} \mathcal{Q}} c \matrixel{\chi^\text{L}_{\mathcal{P}}}{\boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c}{\chi^\text{S}_{\mathcal{Q}}} \\ &+ \sum_{\mathcal{P}}^{K_{\text{S}}} \sum_{\mathcal{Q}}^{K_{\text{L}}} D^{\text{SL}}_{\mathcal{P} \mathcal{Q}} c \matrixel{\chi^\text{S}_{\mathcal{P}}}{\boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c}{\chi^\text{L}_{\mathcal{Q}}} + \sum_{\mathcal{P} \mathcal{Q}}^{K_{\text{S}}} D^{\text{SS}}_{\mathcal{P} \mathcal{Q}} \matrixel{\chi^\text{S}_{\mathcal{P}}}{V^c_{\text{nuc}} + q_e \phi_{\text{ext}}^c - 2 m_e c^2}{\chi^\text{S}_{\mathcal{Q}}} \\ + &\frac{1}{2} \sum_{\mathcal{P} \mathcal{Q} \mathcal{R} \mathcal{S}}^{K_{\text{L}}} \qty(D^{\text{LL}}_{\mathcal{P} \mathcal{Q}} D^{\text{LL}}_{\mathcal{R} \mathcal{S}} - D^{\text{LL}}_{\mathcal{P} \mathcal{S}} D^{\text{LL}}_{\mathcal{R} \mathcal{Q}}) (\chi^{\text{L}}_{\mathcal{P}} \chi^{\text{L}}_{\mathcal{Q}} | \chi^{\text{L}}_{\mathcal{R}} \chi^{\text{L}}_{\mathcal{S}}) \\ &+ \frac{1}{2} \sum_{\mathcal{P} \mathcal{Q}}^{K_{\text{L}}} \sum_{\mathcal{R} \mathcal{S}}^{K_{\text{S}}} \qty(D^{\text{LL}}_{\mathcal{P} \mathcal{Q}} D^{\text{SS}}_{\mathcal{R} \mathcal{S}} - D^{\text{LS}}_{\mathcal{P} \mathcal{S}} D^{\text{SL}}_{\mathcal{R} \mathcal{Q}}) (\chi^{\text{L}}_{\mathcal{P}} \chi^{\text{L}}_{\mathcal{Q}} | \chi^{\text{S}}_{\mathcal{R}} \chi^{\text{S}}_{\mathcal{S}}) \\ &+ \frac{1}{2} \sum_{\mathcal{P} \mathcal{Q}}^{K_{\text{S}}} \sum_{\mathcal{R} \mathcal{S}}^{K_{\text{L}}} \qty(D^{\text{SS}}_{\mathcal{P} \mathcal{Q}} D^{\text{LL}}_{\mathcal{R} \mathcal{S}} - D^{\text{SL}}_{\mathcal{P} \mathcal{S}} D^{\text{LS}}_{\mathcal{R} \mathcal{Q}}) (\chi^{\text{S}}_{\mathcal{P}} \chi^{\text{S}}_{\mathcal{Q}} | \chi^{\text{L}}_{\mathcal{R}} \chi^{\text{L}}_{\mathcal{S}}) \\ &+ \frac{1}{2} \sum_{\mathcal{P} \mathcal{Q} \mathcal{R} \mathcal{S}}^{K_{\text{S}}} \qty(D^{\text{SS}}_{\mathcal{P} \mathcal{Q}} D^{\text{SS}}_{\mathcal{R} \mathcal{S}} - D^{\text{SS}}_{\mathcal{P} \mathcal{S}} D^{\text{SS}}_{\mathcal{R} \mathcal{Q}}) (\chi^{\text{S}}_{\mathcal{P}} \chi^{\text{S}}_{\mathcal{Q}} | \chi^{\text{S}}_{\mathcal{R}} \chi^{\text{S}}_{\mathcal{S}}) \thinspace , \end{split} \end{equation}\] or by using the Coulomb and exchange matrices, we can simplify to \[\begin{equation} \begin{split} E_\text{DHF} = &\sum_{\mathcal{P} \mathcal{Q}}^{K_{\text{L}}} D^{\text{LL}}_{\mathcal{P} \mathcal{Q}} \qty(h^{\text{D}, \text{LL}}_{\mathcal{P} \mathcal{Q}} + \frac{1}{2} \qty(J^{\text{LL}}_{\mathcal{P} \mathcal{Q}} - K^{\text{LL}}_{\mathcal{P} \mathcal{Q}})) + \sum_{\mathcal{P}}^{K_{\text{L}}} \sum_{\mathcal{Q}}^{K_{\text{S}}} D^{\text{LS}}_{\mathcal{P} \mathcal{Q}} \qty(h^{\text{D}, \text{LS}}_{\mathcal{P} \mathcal{Q}} - \frac{1}{2} K^{\text{SL}}_{\mathcal{P} \mathcal{Q}}) \\ &+ + \sum_{\mathcal{P}}^{K_{\text{S}}} \sum_{\mathcal{Q}}^{K_{\text{L}}} D^{\text{SL}}_{\mathcal{P} \mathcal{Q}} \qty(h^{\text{D}, \text{SL}}_{\mathcal{P} \mathcal{Q}} - \frac{1}{2} K^{\text{LS}}_{\mathcal{P} \mathcal{Q}}) + \sum_{\mathcal{P} \mathcal{Q}}^{K_{\text{S}}} D^{\text{SS}}_{\mathcal{P} \mathcal{Q}} \qty(h^{\text{D}, \text{SS}}_{\mathcal{P} \mathcal{Q}} + \frac{1}{2} \qty(J^{\text{SS}}_{\mathcal{P} \mathcal{Q}} - K^{\text{SS}}_{\mathcal{P} \mathcal{Q}})) \thinspace . \end{split} \end{equation}\]
Furthermore, because of the \(2 \times 2\)-superstructure (with respect to the large and small components of the spinors) of the one-electron Hamiltonian, we find that the small and large component must be related by \[\begin{equation} \phi_\mathcal{I}^\text{S} = \frac{c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}}{2m_e c^2 - W - q_e \phi_\text{ext} + \epsilon_\mathcal{I}} \phi_\mathcal{I}^L \thinspace , \end{equation}\] which for \(2m_e c^2 >> -W + \epsilon_\mathcal{I}\) and in the absence of external fields becomes \[\begin{equation} \phi_\mathcal{I}^\text{S} = \frac{\boldsymbol{\sigma} \vdot \vb{p}}{2m_e c} \phi_\mathcal{I}^L \thinspace , \end{equation}\] which is called the kinetic balance condition. Therefore, in principle, from the small-component basis set should at least be populated by the derivatives of the large-component basis, i.e. the small-component basis will have functions that have angular momentum larger than the large-component basis.
However, we can transform the SCF equations \[\begin{equation} \begin{pmatrix} W & c \boldsymbol{\sigma} \vdot \vb{p} \\ c \boldsymbol{\sigma} \vdot \vb{p} & W - 2 m_e c^2 \end{pmatrix} \begin{pmatrix} \phi_\mathcal{I}^L \\ \phi_\mathcal{I}^S \end{pmatrix} = \epsilon_\mathcal{I} \begin{pmatrix} \phi_\mathcal{I}^L \\ \phi_\mathcal{I}^S \end{pmatrix} \tag{\ref{eq:2-spinor_DHF_equations_no_external}} \end{equation}\] using an approximate ansatz \[\begin{equation} \phi_\mathcal{I}^S \approx \varphi_\mathcal{I}^\text{S} = \frac{\boldsymbol{\sigma} \vdot \vb{p}}{2 m_e c} \varphi_\mathcal{I}^\text{L} \end{equation}\] into \[\begin{equation} \begin{pmatrix} W & \frac{(\boldsymbol{\sigma} \vdot \vb{p})^2}{2 m_e} \\ c \boldsymbol{\sigma} \vdot \vb{p} & (W - 2 m_e c^2) \frac{\boldsymbol{\sigma} \vdot \vb{p}}{2 m_e c} \end{pmatrix} \begin{pmatrix} \phi_\mathcal{I}^L \\ \varphi_\mathcal{I}^\text{L} \end{pmatrix} = \epsilon_\mathcal{I} \begin{pmatrix} \phi_\mathcal{I}^L \\ \frac{\boldsymbol{\sigma} \vdot \vb{p}}{2 m_e c} \varphi_\mathcal{I}^\text{L} \end{pmatrix} \thinspace , \end{equation}\] in which \(\varphi_\mathcal{I}^\text{L}\) can now be expanded in the same basis set as \(\phi_\mathcal{I}^L\).
Dirac-Hartree-Fock theory in terms of 4-spinors
In the quasi-relativistic four-component Hamiltonian from equation, we have given up the real relativistic aspects, as it isn’t Lorentz covariant, and there is an absolute time. Nevertheless, let us explore this Hamiltonian a little further.
We will write every spinor \(\set{\phi_\mathcal{P}(\vb{r})}\) as expanded in a set of known basis functions \(\qty{\chi_\mathcal{P}(\vb{r};\vb{R}_A)}\) centered on the nuclei of the molecule. We can then write \[\begin{equation} \phi_\mathcal{P}(\vb{r}) = \sum_{\mathcal{Q}}^L \begin{pmatrix} C^{(1)}_{\mathcal{Q} \mathcal{P}} \chi_\mathcal{Q}^{(1)}(\vb{r};\vb{R}_A) \\ C^{(2)}_{\mathcal{Q} \mathcal{P}} \chi_\mathcal{Q}^{(2)}(\vb{r};\vb{R}_A) \\ C^{(3)}_{\mathcal{Q} \mathcal{P}} \chi_\mathcal{Q}^{(3)}(\vb{r};\vb{R}_A) \\ C^{(4)}_{\mathcal{Q} \mathcal{P}} \chi_\mathcal{Q}^{(4)}(\vb{r};\vb{R}_A) \\ \end{pmatrix} \thinspace , \end{equation}\] in which \(\qty{\chi_\mathcal{Q}^{(1)}(\vb{r};\vb{R}_A)}\) is used for the first component, and similarly for the other components. The functions \(\qty{\chi_\mathcal{Q}^{(i)}(\vb{r};\vb{R}_A)}\) are usually contracted Gaussian-type orbitals, but the contraction coefficients (and exponents) are usually not transferable between non-relativistic and relativistic calculations.
For the overlap between two \(4\)-spinors, we can then find \[\begin{equation} \braket{\phi_{\mathcal{P}}}{\phi_{\mathcal{Q}}} = \Bigg[ \begin{pmatrix} \vb{C}^\text{L} & \vb{C}^\text{S} \end{pmatrix}^* \begin{pmatrix} \vb{S}^{\text{LL}} & 0 \\ 0 & \vb{S}^{\text{SS}} \end{pmatrix} \begin{pmatrix} \vb{C}^\text{L} \\ \vb{C}^\text{S} \end{pmatrix} \Bigg]_{\mathcal{P} \mathcal{Q}} \thinspace , \end{equation}\] in which \(\vb{S}^{\text{LL}}\) is the overlap matrix for the large components: \[\begin{equation} S^{\text{LL}}_{\mathcal{P} \mathcal{Q}} = \braket{\chi_\mathcal{P}}{\chi_\mathcal{Q}} \end{equation}\] and similarly for the overlap matrix for the small components \(\vb{S}^{\text{SS}}\).