Tight-Binding Model

The tight-binding (TB) model represents the real-space Hamiltonian as a linear combination of geometry-dependent matrix elements and scalar model parameters.

The model assumes that each Hamiltonian block associated with a lattice translation vector $\mathbf{R}$ can be written as

\[H^{\mathbf{R}} = \sum_v V_v^0 \, h_v^{\mathbf{R}},\]

where $V_v^0$ are the adjustable TB parameters and $h_v^{\mathbf{R}}$ contains the geometry-dependent contribution for interaction channel $v$.

In reciprocal space, the Hamiltonian is obtained by Fourier transformation,

\[H(\mathbf{k}) = \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} H^{\mathbf{R}}.\]

Localized Basis

The TB model is formulated in a localized orbital basis. Each orbital centered on atom $i$ can be written as a linear combination of atomic orbitals expressed in real spherical harmonics,

\[\psi_i(\mathbf{r}-\mathbf{R}) = \frac{1}{A_i} \sum_{lm} f_{lm}(\boldsymbol{\omega}_i) \phi_{lm}(\mathbf{r}-\boldsymbol{\tau}_i-\mathbf{R}),\]

where $\boldsymbol{\omega}_i$ defines the orbital orientation, $A_i$ is a normalization constant, and $f_{lm}$ are expansion coefficients.

Angular Dependence

The angular dependence of hopping matrix elements is handled using the Slater-Koster idea. For a pair of atoms $i$ and $j$, a rotated coordinate frame is constructed such that the local $z'$ axis is aligned with the connecting vector between the two atoms.

If $U$ denotes the rotation into this local frame, the orbital orientation transforms as

\[\boldsymbol{\omega}_i^r = U \boldsymbol{\omega}_i.\]

In this rotated frame, matrix elements between orbitals can be written as

\[\langle \chi_i | \hat{H} | \chi_j \rangle = \frac{1}{A_i A_j} \sum_{ll'm} f^*_{lm}(\boldsymbol{\omega}_i^r) f_{l'm}(\boldsymbol{\omega}_j^r) \langle \phi_{lm} | \hat{H} | \phi_{l'm} \rangle.\]

Due to symmetry, only matrix elements with matching magnetic quantum number $m$ contribute in the rotated frame.

The orientation-dependent prefactor is collected into a tensor

\[C_{vij}^{\mathbf{R}} = f^*_{lm}(\boldsymbol{\omega}_i^r) f_{l'm}(\boldsymbol{\omega}_j^r),\]

where $v$ is a composite index labeling the interaction channel, for example $ss\sigma$, $sp\sigma$, $pp\sigma$, or $pp\pi$.

Distance Dependence

Within the two-center approximation, the remaining atomic-orbital matrix element is written as a product of an adjustable TB parameter and a distance-dependent factor,

\[\langle \phi_{lm} | \hat{H} | \phi_{l'm} \rangle \approx V_{ll'm}^0 \, V_{ll'm}(\Delta r),\]

where $\Delta r$ is the distance between the two atomic centers.

Instead of introducing a separate flexible parametrization for each distance dependence, the model computes

\[V_{ll'm}(\Delta r) = \langle \phi_{lm}(\mathbf{r}) | \phi_{l'm}(\mathbf{r}-\Delta r\,\hat{\mathbf{e}}_z) \rangle.\]

The radial part is based on hydrogen-like orbitals with an effective core charge $Z_{\mathrm{eff}}$. This reduces the complexity of the distance dependence to a small number of physically interpretable hyperparameters.

The required overlap integrals are precomputed and interpolated using cubic splines, making Hamiltonian construction efficient during training and evaluation.

Final Matrix Element

Combining angular and distance dependence gives the real-space TB matrix element

\[H_{ij}^{\mathbf{k}} = \frac{1}{A_i A_j} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \sum_v C_{vij}^{\mathbf{R}} V_v(\Delta r) V_v^0.\]

Equivalently, the real-space Hamiltonian block can be written as

\[H_{ij}^{\mathbf{R}} = \sum_v C_{vij}^{\mathbf{R}} V_v(\Delta r) V_v^0.\]

The implementation stores the geometry-dependent part

\[h_{vij}^{\mathbf{R}} = C_{vij}^{\mathbf{R}} V_v(\Delta r),\]

so that Hamiltonian construction reduces to the linear expression

\[H^{\mathbf{R}} = \sum_v V_v^0 h_v^{\mathbf{R}}.\]

Parameter Sharing

For multiple structures, each structure may contain a local parameter ordering. The model therefore stores a global parameter vector and a mapping from each structure to the relevant global parameter indices.

For structure $n$,

\[V^{(n)} = V[\mathrm{params\_ per\_ strc}[n]].\]

This allows different structures and bases to share the same optimized parameters consistently.

Spin Basis

If spin-orbit coupling or an explicit spin basis is requested, the Hamiltonian blocks are expanded into spin space. In that case, each orbital degree of freedom is duplicated for spin-up and spin-down components before additional spin-dependent terms are added.

Optimization

The TB parameters are optimized by differentiating the loss with respect to the Hamiltonian.

Given

\[\frac{\partial L}{\partial H_R},\]

the gradient with respect to a TB parameter $V_v$ is

\[\frac{\partial L}{\partial V_v} = \sum_R \sum_{ij} h_{vR,ij} \frac{\partial L}{\partial H_{R,ij}}.\]

This is a direct consequence of the linear relation

\[H_R = \sum_v V_v h_{vR}.\]

Regularization terms can be added to the parameter gradient,

\[\frac{\partial L_{\mathrm{tot}}}{\partial V_v} = \frac{\partial L}{\partial V_v} + \frac{\partial L_{\mathrm{reg}}}{\partial V_v}.\]

Only parameters marked as updateable are modified during optimization.

Parameter Initialization

TB parameters can be initialized from:

• ones
• random values
• zeros
• an external parameter file

When parameters are read from a file, they are matched by their parameter labels before being inserted into the model.