General spinor bases

A spinor is a suitable non-relativistic one-electron wave function. The single-particle Hilbert space \(\mathscr{H}_1\) is populated by the suitable single-particle state vectors of the problem in question. The \(\qty{\ket{\vb{r}, \sigma}}\)-basis (the eigenfunctions of the complete set of commutating observables \(\qty{\hat{x}, \hat{y}, \hat{z}, \hat{S}^2, \hat{S}_z}\)) is orthonormal: \[\begin{equation} \braket{\vb{r}', \sigma'}{\vb{r}, \sigma} = \delta_{\sigma' \sigma} \delta(\vb{r}' - \vb{r}) \end{equation}\] and admits the closure relation \[\begin{equation} \hat{1} = \sum_\sigma \int \dd{\vb{r}} \dyad{\vb{r}, \sigma} \thinspace . \end{equation}\] Therefore, any of the orthonormal state vectors \[\begin{equation} \require{physics} \mathcal{B} = \set{ \ket{\phi_P}; P=1 \cdots M } \thinspace , \end{equation}\] where \(M\) can be a natural number or \(+ \infty\), can be expressed as \[\begin{equation} \ket{\phi_P} = \sum_\sigma \int \dd{\vb{r}} \phi_P^\sigma(\vb{r}) \ket{\vb{r}, \sigma} \thinspace , \end{equation}\] in which the coordinates of that abstract vector \(\ket{\phi_P}\), i.e. \[\begin{equation} \braket{\vb{r}, \sigma}{\phi_P} = \phi_P^\sigma(\vb{r}) \end{equation}\] depend on four indices: the three continuous indices collected in \(\vb{r}\) and the discrete spin index \(\sigma\).

These functions are then collected in the two-component spinor \(\phi_P(\vb{r})\): \[\begin{equation} \label{eq:Pauli_spinor} \phi_P(\vb{r}) = \begin{pmatrix} \phi_P^\alpha(\vb{r}) \\ \phi_P^\beta(\vb{r}) \end{pmatrix} \thinspace , \end{equation}\] In general, \(\phi(\vb{r})\) is not an eigenfunction of \(\sigma_z\). The adjoint of a spinor is then given as \[\begin{equation} \phi_P^\dagger(\vb{r}) = \begin{pmatrix} \phi_{P \alpha}^*(\vb{r}) & \phi_{P \beta}^*(\vb{r}) \end{pmatrix} \thinspace , \end{equation}\]

The scalar product of two single-particle states is then readily calculated as \[\begin{equation} \braket{\phi_P}{\phi_Q} = \sum_\sigma \int \dd{\vb{r}} \phi_P^{\sigma *}(\vb{r}) \phi_Q^\sigma(\vb{r}) \end{equation}\] by introducing the completeness relation and can alternatively be written as \[\begin{align} \braket{\phi_P}{\phi_Q} & = \int \dd{\vb{r}} \phi_P^\dagger(\vb{r}) \phi_Q(\vb{r}) \\ % & = \int \dd{\vb{r}} \phi_{P \alpha}^*(\vb{r}) \phi_{Q \alpha}(\vb{r}) + \int \dd{\vb{r}} \phi_{P \beta}^*(\vb{r}) \phi_{Q \beta}(\vb{r}) \\ \end{align}\] where the usual matrix product between a row vector and a column vector applies.

We call the quantity \[\begin{equation} \phi_P(\vb{r})^\dagger \phi_Q(\vb{r}) = \phi_{P \alpha}(\vb{r})^* \phi_{Q \alpha}(\vb{r}) + \phi_{P \beta}(\vb{r})^* \phi_{Q \beta}(\vb{r}) \end{equation}\] a Pauli distribution.

If we decompose the complex scalar orbitals \[\begin{equation} \phi_{P \alpha,\beta} : \mathbb{R}^3 \rightarrow \mathbb{C} \end{equation}\] in their real (R) and imaginary (I) parts: \[\begin{equation} ^{\text{R},\text{I}} \phi_{P \alpha,\beta} : \mathbb{R}^3 \rightarrow \mathbb{R} \thinspace , \end{equation}\] we can realize that every general Pauli spinor is actually composed of four independent real scalar functions: \[\begin{equation} \phi_P(\vb{r}) = \begin{pmatrix} \phi_{P \alpha}(\vb{r}) \\ \phi_{P \beta}(\vb{r}) \end{pmatrix} = % \begin{pmatrix} ^{\text{R}}\phi_{P \alpha}(\vb{r}) + i ^{\text{I}}\phi_{P \alpha}(\vb{r}) \\ ^{\text{R}}\phi_{P \beta}(\vb{r}) + i ^{\text{I}}\phi_{P \beta}(\vb{r}) \end{pmatrix} \thinspace . \end{equation}\]

When trying to find the form of Pauli spinors in optimization problems, we usually do not vary the actual underlying scalar functions themselves, but we expand them as a linear combination of known scalar functions. If we have a set of \(K_\alpha\) known scalar basis functions \(\{ \chi^\alpha_\mu(\vb{r}) \}\) that we use to expand the \(\alpha\)-component of the spinors in and we also have a, possibly different, set of \(K_\beta\) known scalar basis functions \(\{ \chi^\beta_\mu(\vb{r}) \}\) for the \(\beta\)-components, we can write the expansion of the components a general spinor with index \(P\) as \[\begin{equation} \phi_P(\vb{r}) = % \begin{pmatrix} \phi_{P \alpha}(\vb{r}) \\ \phi_{P \beta}(\vb{r}) \end{pmatrix} = % \begin{pmatrix} \displaystyle \sum_\mu^{K_\alpha} \chi^\alpha_\mu(\vb{r}) C^{\alpha}_{\mu P} \\ \displaystyle \sum_\mu^{K_\beta} \chi^\beta_\mu(\vb{r}) C^{\beta}_{\mu P} \end{pmatrix} \thinspace . \end{equation}\] This means that we use \(M = K_\alpha + K_\beta\) expansion coefficients to characterize one single spinor. The total coefficient matrix is then of dimension \((M \times M)\), in which every column describes the expansion of a single spinor: \[\begin{equation} \label{eq:coefficient_matrix_spinor_expansion} \vb{C} = \begin{pmatrix} \vb{C}^\alpha \\ \vb{C}^\beta \end{pmatrix} = \begin{pmatrix} C^{\alpha}_{11} & \cdots & C^{\alpha}_{1P} & \cdots & C^{\alpha}_{1, M} \\ C^{\alpha}_{21} & \cdots & C^{\alpha}_{2P} & \cdots & C^{\alpha}_{2M} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ C^{\alpha}_{K_\alpha 1} & \cdots & C^{\alpha}_{K_\alpha P} & \cdots & C^{\alpha}_{K_\alpha M} \\ C^{\beta}_{11} & \cdots & C^{\beta}_{1P} & \cdots & C^{\beta}_{1M} \\ C^{\beta}_{21} & \cdots & C^{\beta}_{2P} & \cdots & C^{\beta}_{2M} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ C^{\beta}_{K_\beta 1} & \cdots & C^{\beta}_{K_\beta P} & \cdots & C^{\beta}_{K_\beta M} \\ \end{pmatrix} \thinspace . \end{equation}\] The total coefficient matrix \(\vb{C}\) is then divided into two parts: the top \(K_\alpha\) rows describe the expansion of the \(\alpha\)-components of the spinors in terms of the \(\alpha\)-scalar basis and the bottom \(K_\beta\) rows describe the expansion of the \(\beta\)-components. We should remark that the total coefficient matrix \(\vb{C}\) is still a square matrix, but the top sub-block \(\vb{C}^\alpha\) is rectangular with dimensions \((K_\alpha \times M)\) and the bottom sub-block \(\vb{C}^\beta\) is also rectangular with dimensions \((K_\beta \times M)\).

In order to describe the resulting \(M = K_\alpha + K_\beta\) spinors, we thus require \(M^2\) complex expansion coefficients \(\{ C^{\alpha}_{\mu P}, C^{\beta}_{\mu P} \}\) and thus \(2M^2\) real numbers.

We should note that, when we have used scalar \(\alpha\)- and \(\beta\) scalar bases, an initial coefficient matrix of \[\begin{equation} \vb{C} = \vb{I} \end{equation}\] means that the non-orthogonal spinors are spin-separated. For example, the first spinor we could make is \[\begin{equation} \phi_1(\vb{r}) = \begin{pmatrix} \chi_1^\alpha(\vb{r}) \\ 0 \end{pmatrix} \end{equation}\] and the last spinor would then be \[\begin{equation} \phi_M(\vb{r}) = \begin{pmatrix} 0 \\ \chi_{K_\beta}^\beta(\vb{r}) \end{pmatrix} \thinspace . \end{equation}\]