Tight-Binding Propagation Method Overview

• TBPM is a computational method that propagates quantum states using a time-dependent tight-binding Hamiltonian for large-scale material simulations.
• It leverages localized atomic orbitals and Chebyshev expansion to efficiently compute spectral properties, bonding, and charge transfer.
• Its parameterization from ab initio data and energy decomposition enables scalable, multiscale modeling that bridges DFT insights with interatomic potentials.

The Tight-Binding Propagation Method (TBPM) is a computational technique for simulating large-scale quantum systems using the tight-binding (TB) approximation, with key applications in electronic structure, quantum dynamics, and material modeling. TBPM leverages the locality and transferability of tight-binding Hamiltonians—where interactions are parameterized in terms of physically motivated on-site energies, hopping integrals, and, where needed, additional correction terms—to obtain electronic and structural properties with linear scaling in system size. Its foundational principles, recent algorithmic variants, and major application areas are grounded in the theoretical and methodical analysis of quantum bonding, charge transfer, and the physical decomposition of total energy in atomistic systems.

1. Fundamental Principles of the Tight-Binding Propagation Method

TBPM is constructed on the real-space, time-dependent propagation of quantum states governed by an approximate, localized Hamiltonian. In TB, the electronic wave function is written as a linear combination of atomic-like orbitals: $$\psi_{AB}(\mathbf{r}) = c_A\phi_A(\mathbf{r}) + c_B\phi_B(\mathbf{r} - \mathbf{R})$$ where coefficients $c_A$ and %%%%1%%%% are determined through the secular equation: $$\left| \begin{array}{cc} H_{AA} - E & H_{AB} - ES \ H_{BA} - ES & H_{BB} - E \end{array} \right| = 0$$ with $H_{AB}$ representing inter-site (bond or hopping) matrix elements and $S$ the orbital overlap.

TBPM does not require the explicit diagonalization of $H$ but rather computes time-evolution operators (e.g., via Chebyshev polynomial expansions) to obtain observables such as densities of states (DOS), local densities (LDOS), and spectral functions. The propagation of an initial state $|\psi(0)\rangle$ is given by

$|\psi(t)\rangle = e^{-iHt/\hbar}|\psi(0)\rangle$

with spectral properties calculated from correlation functions (e.g., $\langle\psi(0)|\psi(t)\rangle$).

2. Bonding, Charge Transfer, and Energy Decomposition

In TBPM’s foundational framework (Pettifor, 2011), chemical bonding and charge transfer are encoded in the structure of the Hamiltonian:

• The bond integral $\beta = H_{AB} - \bar{E}S$, with $\bar{E} = \frac{1}{2}(H_{AA} + H_{BB})$, sets the magnitude of bonding–antibonding splitting.
• The on-site energy mismatch $\Delta = H_{BB} - H_{AA}$ drives charge transfer and defines ionicity.

For the $AB$ $s$-valent dimer, the eigenstates have populations: $$N_A = 1 + \frac{\hat{\Delta} - S}{\sqrt{1 + \hat{\Delta}^2}}, \quad N_B = 1 - \frac{\hat{\Delta} + S}{\sqrt{1 + \hat{\Delta}^2}}$$ with normalized mismatch $\hat{\Delta} = \Delta / (2|\beta|)$, and the transferred "Mulliken" charge

$q = \frac{\hat{\Delta}}{\sqrt{1+\hat{\Delta}^2}}$

giving $q \to 0$ in the covalent limit ($\Delta\to0$), $q\to1$ in the ionic limit.

The binding energy decomposes as: $U = U_\mathrm{band} - U_\mathrm{dc} + \frac{1}{R}$ with $U_\mathrm{band}$ capturing electronic contributions and $U_\mathrm{dc}$ correcting for double counting. The total binding energy—after subtracting atomic reference energies—is dissected into

$$U_{be}(R, q) = U_\mathrm{rep}(R) + U_\mathrm{cov}(R, q) + U_\mathrm{ionic}(R,q)$$

where

• $U_\mathrm{rep}(R)$ includes overlap repulsion and electrostatic terms,
• $U_\mathrm{cov}(R,q) = -2|\beta_0(R)|\sqrt{1-q^2}$,
• $U_\mathrm{ionic}(R,q) = -\Delta_0 q + J(R)q^2$ (with $J(R)$ from Coulomb integrals).

This decomposition provides a physically motivated template for constructing effective interatomic potentials to be used in TBPM.

3. TBPM Workflow and Modeling Strategies

In practical TBPM simulations, the Hamiltonian is constructed by parameterizing:

• On-site energies (potentially including environment- and strain-dependence),
• Hopping integrals between atomic sites (distance and angular dependence),
• Overlap parameters, if needed (may be incorporated to improve unoccupied band description).

These parameters may be set empirically, via ab initio mapping, or refined by fitting key experimental observables (as in (Phan et al., 30 Jan 2024, Zargar et al., 29 Aug 2024)). The process generally follows:

1. Construct $H$ in the desired atomistic geometry.
2. Prepare an appropriate initial state $|\psi(0)\rangle$ (randomized, spatially localized, etc.).
3. Evolve $|\psi(t)\rangle$ via a high-order Chebyshev expansion or similar.
4. Post-process correlation functions (e.g., via Fourier transform) to extract target observables.

For finite-temperature or transport calculations, suitable Fermi–Dirac averaging or Green’s function techniques can be integrated into the TBPM framework.

4. Physical Interpretation: Relation to DFT and Interatomic Potentials

TBPM builds upon a coarse-grained version of the DFT one-electron equations, assuming that the relevant physics is captured by a minimal basis set and appropriately parameterized TB matrix elements. The electronic structure—bonding vs. antibonding states, charge asymmetry, covalency/ionicity—is thereby directly connected to physically intuitive parameters ($\beta$, $\Delta$, $q$) that reflect the underlying DFT potential landscape.

These parameters permit the explicit construction of transferable, chemically motivated interatomic potentials:

• Covalent potentials depend on both the magnitude of the bond integral and the bond order (encoded via $\sqrt{1-q^2}$),
• Ionic potentials arise from the energy-level mismatch and Coulomb self-energies.

In TBPM, physically accurate propagation thus hinges on the quality of these potentials, which may be extracted directly from first-principles data or calibrated against DFT.

5. Implications for Large-Scale and Multiscale Simulation

The explicit separation of the total energy and site population given by the underlying TB formalism is crucial for TBPM’s efficiency and transferability:

• The locality of the TB matrix and the exponential decay of its elements (and energy derivatives) enables parallelization and simulation of very large systems.
• The construction of the effective Hamiltonian from molecular (or atomic cluster) parameters enables simulation of heterogeneous materials, interfaces, and disordered systems without a significant loss in accuracy.
• The TBPM framework is extensible to multi-scale approaches: coarse-graining and local truncation are both justified by the exponential decay properties that permit hybrid or region-specific methods (e.g., combining quantum and classical force fields).

Furthermore, the bond–ionicity decomposition provides a physical handle for connecting quantum and classical domains in multi-scale embedding schemes.

6. Practical Considerations and Limitations

TBPM’s practical efficacy hinges on rigorous and physically motivated parameterization. Important considerations include:

• Parameter accuracy: The bond integral and on-site energies must be tuned—ideally against DFT or experiment—to match the material’s electronic structure and energy landscape. The explicit dependence on $\beta$, $\Delta$, and their spatial variations is essential for quantitative accuracy.
• Range of validity: Minimal basis sets (e.g., $s$-orbitals for $AB$ dimers) are sufficient for many systems, but for accurate modeling of more complex materials, basis set extension (e.g., inclusion of $p$, $d$ orbitals) may be necessary.
• Transferability: Potentials constructed in the TBPM framework are physically justified and may be transferable across chemically similar systems, but users should always validate parameters for the target chemical, structural, or physical regime.

The TBPM can, in principle, accommodate higher-order corrections (overlap terms, environment/strain-induced shifts, many-body effects), but this may complicate both parameter determination and computational workflow.

7. Connections to Advanced Theoretical and Simulation Frameworks

The analysis in (Pettifor, 2011) establishes TBPM as a robust link between DFT-based quantum chemistry and effective, physically interpretable interatomic potentials. This methodology underpins a variety of modern computational materials science approaches, including:

• Parameterized TB for large-scale atomistic simulation,
• First-principles-based TBPM for accurate electronic/transport property computation,
• Multi-scale modeling efforts that combine DFT, TB, and classical force fields.

The explicit covalent–ionic separation provided by TB theory is central to many emerging quantum and atomistic simulation packages employing TBPM as a core engine.

---

In summary, the Tight-Binding Propagation Method is an efficient, physically grounded approach for quantum simulations in materials modeling, with the explicit parameterization of bonding and charge transfer at its core. By providing a direct and interpretable connection between chemical bonding motifs and electronic structure—mirrored in DFT and encapsulated in covalent and ionic interatomic potentials—TBPM delivers a scalable and transferable methodology for the simulation of complex materials and their emergent properties.