LCAO Operators

An LCAO matrix carries two spatial angular indices. A spinful matrix adds a spin-scalar or spin-vector Pauli component. The transformation type follows by identifying these factors in order: first the two AO indices, then the Pauli index, and finally whether spatial and spin rotations are independent or joint.

The Hamiltonian is used as the main example. Overlap and density matrices obey the same angular transformation law under rotations, with the physical qualifications collected near the end of the chapter.

Atomic-Orbital Basis

Let \(\mu=(i,n_1,\ell_1,m_1)\) and \(\nu=(j,n_2,\ell_2,m_2)\) label two spatial atomic orbitals,

\[ \phi_{in\ell m}(\boldsymbol r) = R_{in\ell}\!\left(\lvert\boldsymbol r-\boldsymbol R_i\rvert\right) Y_{\ell m}\!\left(\widehat{\boldsymbol r-\boldsymbol R_i}\right), \]

using the ELFES real spherical-harmonic convention. Write an orthogonal transformation as \(Q=P^\epsilon R\), where \(P=-I\), \(R\in\mathrm{SO}(3)\), and \(\epsilon\in\{0,1\}\). Together with a translation \(\boldsymbol t\), it moves the centers as \(\boldsymbol R'_{\pi(i)}=Q\boldsymbol R_i+\boldsymbol t\).

The actively transformed AO is expressed in the basis attached to the transformed center as

\[ \hat U(Q,\boldsymbol t) |\phi_{in\ell m}\rangle = (-1)^{\epsilon\ell} \sum_{m'=-\ell}^{\ell} |\phi'_{\pi(i)n\ell m'}\rangle \mathcal D^\ell_{m'm}(R). \]

Translation moves the basis center but does not mix its angular components. The finite-dimensional action comes entirely from the two AO angular indices.

Spatial Matrix Blocks

Fix the centers, radial labels, and angular degrees of two AO shells. The corresponding Hamiltonian block has elements

\[ H_{m_1m_2} = \left\langle \phi_{in_1\ell_1m_1} \middle| \hat H \middle| \phi_{jn_2\ell_2m_2} \right\rangle. \]

After transforming the physical system and expressing the result in the transformed AO basis,

\[ {H'}_{m_1'm_2'} = (-1)^{\epsilon(\ell_1+\ell_2)} \sum_{m_1,m_2} \mathcal D^{\ell_1}_{m_1'm_1}(R) \mathcal D^{\ell_2}_{m_2'm_2}(R) H_{m_1m_2}. \]

The block belongs abstractly to \(V_{\ell_1}\otimes V_{\ell_2}^*\) because its second index is an input index. Integer-degree rotation representations are self-dual, and the real Wigner matrices used here are orthogonal, so this space is equivalent to \(V_{\ell_1}\otimes V_{\ell_2}\). Its parity is \((-1)^{\ell_1+\ell_2}\).

The two factors are both spatial and are acted on by the same \(R\). They may therefore be coupled immediately:

\[ H_{LM} = \sum_{m_1,m_2} C^{LM}_{\ell_1m_1,\ell_2m_2} H_{m_1m_2}, \qquad |\ell_1-\ell_2|\leq L\leq\ell_1+\ell_2. \]

For each allowed \(L\), the coupled block transforms as

\[ {H'}_{LM'} = (-1)^{\epsilon(\ell_1+\ell_2)} \sum_M \mathcal D^L_{M'M}(R) H_{LM}. \]

The center pair, radial labels, and parent degrees \(\ell_1,\ell_2\) remain as multiplicity labels. Hermiticity relates blocks with reversed AO indices and centers but does not change this rotation law.

Adding the Pauli Components

Restore the two spin indices and decompose the spinful block as

\[ \mathbf H = \mathsf H^0\otimes\sigma_0 + \sum_{\alpha=x,y,z} \mathsf H^\alpha\otimes\sigma_\alpha, \]

where \(\mathsf H^0=\tfrac12\operatorname{tr}_s\mathbf H\) and \(\mathsf H^\alpha=\tfrac12\operatorname{tr}_s(\sigma_\alpha\mathbf H)\). Each Pauli component is itself a matrix over the two spatial AO indices.

Let \(R\) act on the spatial system and let \(S\in\mathrm{SO}(3)_{\mathrm{spin}}\) be a global spin rotation. The spin-scalar block transforms only through its AO indices. The Pauli-vector block transforms as

\[ {H'}_{m_1'm_2'}^{\alpha'} = (-1)^{\epsilon(\ell_1+\ell_2)} \sum_{m_1,m_2,\alpha} \mathcal D^{\ell_1}_{m_1'm_1}(R) \mathcal D^{\ell_2}_{m_2'm_2}(R) S^{\alpha'}{}_{\alpha} H_{m_1m_2}^{\alpha}. \]

Coupling the two AO indices first gives \(H^0_{LM}\) and \(H^\alpha_{LM}\). Under independent proper rotations, their types are respectively

\[ (L,0) = V_L^{\mathrm{space}}\otimes V_0^{\mathrm{spin}}, \qquad (L,1) = V_L^{\mathrm{space}}\otimes V_1^{\mathrm{spin}}. \]

This first coupling is always valid because both AO indices belong to spatial space. What can be done with the remaining Pauli-vector index depends on the relation between \(R\) and \(S\).

Independent Rotations Without Spin–Orbit Coupling

Without spin–orbit coupling or another spin–lattice coupling, \(R\) and \(S\) are independent. The Pauli-vector block \(H^\alpha_{LM}\) is already an irreducible product-group feature of type \((L,1)\). There is no further Clebsch–Gordan decomposition between \(M\) and \(\alpha\).

A pure spatial rotation changes the AO angular components while leaving the Pauli vector fixed. A pure global spin rotation mixes \(H^x,H^y,H^z\) while leaving the AO components fixed. A correct equivariant description must allow both actions.

Joint Rotations With Spin–Orbit Coupling

With spin–orbit coupling, proper physical rotations act jointly, so \(S=R\). Convert the Cartesian spin index to the real degree-one spherical basis described in Real-Space Fields:

\[ H_{LM_Lm_s} = \sum_\alpha \left(T^{1\leftarrow\mathrm{cart}}\right)_{m_s\alpha} H^\alpha_{LM_L}, \qquad m_s=-1,0,1. \]

Now \(M_L\) and \(m_s\) are acted on by the same rotation. Their product can be coupled to total angular degree \(J\):

\[ H_{JM}^{(L)} = \sum_{M_L,m_s} C^{JM}_{LM_L,1m_s} H_{LM_Lm_s}, \qquad |L-1|\leq J\leq L+1. \]

The resulting components obey

\[ {H'}_{JM'}^{(L)} = (-1)^{\epsilon(\ell_1+\ell_2)} \sum_M \mathcal D^J_{M'M}(R) H_{JM}^{(L)}. \]

The intermediate \(L\) is retained because different AO coupling paths can produce the same final \(J\). Under the joint \(\mathrm{O}(3)\) action, the output parity remains \((-1)^{\ell_1+\ell_2}\): the Pauli vector is axial and contributes parity \(+1\).

The Pauli-scalar block does not participate in this second coupling and remains a degree-\(L\) spatial feature. The complete joint decomposition therefore contains the scalar path \(L\) together with all allowed vector paths \(L\otimes1\to J\).

Hamiltonian, Overlap, and Density Matrices

The angular construction applies componentwise to Hamiltonian, overlap, and density matrices. For a spin-independent AO basis, the overlap has only a Pauli-scalar component, \(\mathbf S=\mathsf S\otimes\sigma_0\). Hamiltonian and density matrices may carry both scalar and vector components; the Pauli decomposition alone does not assign a particular exchange or spin–orbit origin to any matrix entry.

In a nonorthogonal AO basis, the density coefficient matrix is not the covariant matrix of the density operator: the latter is \(\mathbf S\mathbf D\mathbf S\). Their meanings and general changes of LCAO coordinates differ. Under the orthogonal angular rotations used here, however, their AO component arrays obey the same displayed rotation law.

Rotational equivariance also does not determine whether the component matrices are real or complex. Hermiticity, orbital real structures, time reversal, and the distinction between local fields and two-index kernels control those questions separately.

Representation Flow

The spatial AO indices are always coupled first:

\[ V_{\ell_1}^{\mathrm{space}} \otimes V_{\ell_2}^{\mathrm{space}*} \longrightarrow V_L^{\mathrm{space}}. \]

The Pauli scalar then has type \((L,0)\). The Pauli vector has type \((L,1)\) under independent rotations and decomposes as \(\bigoplus_{J=|L-1|}^{L+1}V_J\) only after restriction to joint rotations. This separation is the organizing principle for spinful LCAO equivariance.