Discrete Variable Local Diabatic Representation

• Discrete variable LDR is a grid-based formulation for coupled electron–nuclear dynamics that uses localized nuclear basis functions to bypass divergent derivative couplings.
• It replaces traditional derivative couplings with finite electronic overlap matrices, ensuring divergence-free behavior at conical intersections and robust gauge independence.
• The method leverages discrete variable representation with Fourier grids to achieve exponential convergence and high accuracy in simulating nonadiabatic transitions.

Searching arXiv for recent and foundational papers on local diabatic representation and DVR-based constructions. The discrete variable local diabatic representation (LDR) is a grid-based formulation of coupled electron–nuclear quantum mechanics in which the total wavefunction is expanded in nuclear basis functions localized at discrete nuclear geometries and adiabatic electronic states evaluated at those same geometries. In this representation, non-Born–Oppenheimer effects are encoded by finite electronic overlap matrices rather than by first- and second-derivative couplings, so the framework is divergence-free at conical intersections, robust to local gauge choices of electronic phases, and directly compatible with adiabatic electronic states obtained from standard electronic-structure calculations (Gu, 2023, Gu, 2023, Sha et al., 6 Sep 2025).

1. Formal definition and basic equations

In LDR, the molecular wavefunction is written as

$$\Psi(\mathbf{r}, \mathbf{R}, t) = \sum_{\mathbf{n},\alpha} C_{\mathbf{n}\alpha}(t)\, \phi_\alpha(\mathbf{r};\mathbf{R}_{\mathbf{n}})\, \chi_{\mathbf{n}}(\mathbf{R}),$$

where $\chi_{\mathbf{n}}(\mathbf{R})$ is a nuclear basis function localized at grid point $\mathbf{R}_{\mathbf{n}}$, $\phi_\alpha(\mathbf{r};\mathbf{R}_{\mathbf{n}})$ is an adiabatic electronic state at that geometry, and $C_{\mathbf{n}\alpha}(t)$ are expansion coefficients. Projection of the time-dependent Schrödinger equation gives

$$i \dot{C}_{\mathbf{m}\beta}(t) = E_\beta(\mathbf{R}_{\mathbf{m}}) C_{\mathbf{m}\beta}(t) + \sum_{\mathbf{n},\alpha} T_{\mathbf{m}\mathbf{n}}\, A_{\mathbf{m}\beta,\mathbf{n}\alpha}\, C_{\mathbf{n}\alpha}(t),$$

with $T_{\mathbf{m}\mathbf{n}}$ the nuclear kinetic-energy matrix and

$$A_{\mathbf{m}\beta,\mathbf{n}\alpha} = \langle \phi_\beta(\mathbf{R}_{\mathbf{m}}) | \phi_\alpha(\mathbf{R}_{\mathbf{n}}) \rangle$$

the electronic overlap matrix (Gu, 2023, Gu, 2023).

The defining structural feature is that the nuclear kinetic operator acts only on the nuclear basis and never on the adiabatic electronic states. This removes the singular first- and second-derivative couplings that appear in the Born–Huang ansatz near conical intersections. In the continuum interpretation, the electronic overlap matrix plays the role otherwise carried by nonadiabatic derivative couplings, but without the associated divergences (Gu, 2023, Sha et al., 6 Sep 2025).

The formulation is “local” because each electronic basis is tied to a discrete nuclear geometry rather than defined as a globally smooth diabatic set over the full configuration space. This local construction is central near conical intersections, where globally smooth diabatic bases usually do not exist because of topological obstruction (Gu, 2023).

2. Discrete variable representation as the nuclear infrastructure

The “discrete variable” component of LDR refers to the use of a DVR to construct localized nuclear basis functions and analytic or semi-analytic matrix elements for coordinate and kinetic operators. In the 2023 Fourier-grid implementation, a primitive nuclear basis is used to obtain position eigenstates by diagonalizing the position operator, and a Fourier basis is chosen because it is universal and can be applied to all types of reactive coordinates (Gu, 2023).

For a one-dimensional box, the grid points are

$$x_n = a + n \Delta x, \qquad \Delta x = \frac{L}{N+1},$$

and the Colbert–Miller kinetic matrix is

$$T_{nn'} = \frac{1}{2m\Delta x^2} (-1)^{n-n'} \begin{cases} \frac{\pi^2}{3}, & n=n' \ \frac{2}{(n-n')^2}, & n\neq n' . \end{cases}$$

This makes the DVR especially attractive when the nuclear propagator is combined with split-operator methods (Gu, 2023).

A related DVR construction appears in optical-lattice Wannier theory. Using a uniform Fourier grid DVR over a supercell, the single-particle Hamiltonian becomes real symmetric, which enables strictly real eigenvectors even for asymmetric lattices. Localized functions are then obtained by diagonalizing the projected position operator,

$\chi_{\mathbf{n}}(\mathbf{R})$0

or, for double wells, by projecting onto a composite two-band subspace (Paul et al., 2016).

This suggests a close formal kinship between molecular LDR and DVR-based projected-position localization in periodic systems. In both settings, a finite subspace is selected, the position operator is diagonalized within that subspace, and the outcome is a set of localized, orthonormal states adapted to the physical geometry. The applications differ—nonadiabatic dynamics in one case, tight-binding and Bose–Hubbard reductions in the other—but the discrete-variable logic is closely aligned (Paul et al., 2016).

3. Nonadiabatic couplings, gauge structure, and topology

LDR replaces derivative couplings by the electronic overlap matrix. Because overlap elements are bounded, the method is free of the divergences that occur when adiabatic energy gaps vanish. In the adiabatic Born–Huang formulation, the first-order derivative coupling

$\chi_{\mathbf{n}}(\mathbf{R})$1

and the second-order derivative couplings, including the diagonal Born–Oppenheimer correction (DBOC), diverge at degeneracies and conical intersections. In LDR, all such effects are encoded in $\chi_{\mathbf{n}}(\mathbf{R})$2, which is always finite (Sha et al., 6 Sep 2025).

Gauge robustness follows from the fact that no nuclear derivative of the electronic wavefunction appears. Independent phase changes at each center leave the equations of motion and observables invariant, and no smooth gauge of electronic wavefunction phase is required. This is a central practical distinction from adiabatic dynamics near conical intersections, where gauge smoothing is typically a major numerical issue (Gu, 2023, Gu, 2023).

The geometric phase is represented by products of overlaps around closed loops in nuclear space. For a given electronic channel, the Wilson-loop form

$\chi_{\mathbf{n}}(\mathbf{R})$3

is gauge invariant and captures the sign change acquired by encircling a conical intersection. In wavepacket dynamics this appears as nodal lines in the nuclear density, a hallmark of Berry-phase physics (Gu, 2023, Gu, 2023).

A common misconception is that “local diabatic” means merely an approximate smoothing of adiabatic states. In the LDR literature, the point is more specific: nonadiabatic transitions, electronic coherences, and geometric phases are carried exactly by discrete overlaps between adiabatic states at different geometries, not by an attempt to construct a globally smooth diabatic basis (Gu, 2023).

4. Propagation algorithms and representative benchmarks

The first implementation introduced LDR on a two-dimensional conical-intersection model using coherent states for the nuclear basis and a 4th-order Runge–Kutta integrator. That calculation showed that the representation captures nonadiabatic transitions, electronic coherence, and geometric phase, including the nodal line in the nuclear probability density after encircling the conical intersection (Gu, 2023).

A later formulation combined LDR with uniform-grid DVR, a Fourier basis, and second-order Strang splitting for the molecular propagator,

$\chi_{\mathbf{n}}(\mathbf{R})$4

Because LDR removes derivative-coupling singularities and leaves the kinetic operator acting only on the nuclear grid, the split-operator method can be directly applied to the full molecular propagator, unlike in the adiabatic representation (Gu, 2023).

In the benchmark application, the nuclear wavefunction was expanded on a 2D Fourier DVR grid with $\chi_{\mathbf{n}}(\mathbf{R})$5 points covering $\chi_{\mathbf{n}}(\mathbf{R})$6 to $\chi_{\mathbf{n}}(\mathbf{R})$7 a.u. for each mode. The method reproduced rapid population transfer at the conical intersection and the expected geometric-phase nodal structure. It also conserved norm and energy precisely with $\chi_{\mathbf{n}}(\mathbf{R})$8 a.u., whereas previous Runge–Kutta-based implementations required $\chi_{\mathbf{n}}(\mathbf{R})$9 a.u. (Gu, 2023).

The principal methodological comparison is with the Born–Huang and crude adiabatic representations:

Representation	Nonadiabatic ingredient	Characteristic behavior
LDR	Electronic overlap matrix	Divergence-free; gauge-robust; geometric phase through overlap products
Born–Huang	First- and second-order derivative couplings, including DBOC	Diverges at conical intersections
Crude adiabatic	Electronic states fixed at one reference geometry	Avoids derivative couplings but generally converges more slowly

The comparison clarifies that LDR is not simply another diabatic parameterization. It is a discrete-basis reformulation of the coupled problem in which the problematic continuum couplings never appear explicitly (Sha et al., 6 Sep 2025).

5. Convergence, accuracy, and computational scaling

The 2025 convergence study examined LDR eigenvalue problems for coupled oscillator models and compared them with the exact Born–Huang representation and the crude adiabatic representation. The main result was exponential convergence with respect to both the number of nuclear grid points and the number of electronic states. For weak vibronic couplings, LDR showed a convergence rate similar to the exact Born–Huang representation when the latter included not only first-order derivative couplings but also the DBOC and second-order derivative couplings. For strong vibronic couplings, LDR showed a significantly faster convergence rate with respect to the number of grid points than the exact Born–Huang representation, while the crude adiabatic representation generally showed a much slower convergence rate for all cases (Sha et al., 6 Sep 2025).

In the detailed benchmarks, approximately $\mathbf{R}_{\mathbf{n}}$0 nuclear grid points and a few electronic states ($\mathbf{R}_{\mathbf{n}}$1–$\mathbf{R}_{\mathbf{n}}$2) sufficed for errors as low as $\mathbf{R}_{\mathbf{n}}$3. The study also emphasized that DBOC and second-order derivative couplings are important in the Born–Huang framework; neglecting them leads to sizeable errors. In LDR those effects are encoded through the overlap matrix rather than added as separate singular terms (Sha et al., 6 Sep 2025).

Direct evaluation of all overlaps becomes expensive in high dimensions. The reported scaling for all overlaps is $\mathbf{R}_{\mathbf{n}}$4, and the linked product approximation (LPA) was introduced to approximate long-range overlaps as products of nearest-neighbor overlaps along a path. The reported result was that LDR with LPA remained almost as accurate as exact LDR, even for stringent benchmarks (Sha et al., 6 Sep 2025).

Accuracy considerations also arise in the DVR-based lattice analogue. In optical lattices, tunneling matrix elements computed from DVR-based Wannier functions agreed to machine precision, $\mathbf{R}_{\mathbf{n}}$5, in symmetric lattices, and within $\mathbf{R}_{\mathbf{n}}$6 in asymmetric lattices. The paper described the resulting localized functions as accurate to better than ten significant digits when using double-precision arithmetic (Paul et al., 2016). This suggests that projected-position localization in a DVR can deliver numerically robust local bases not only for conical-intersection dynamics but also for tight-binding reductions.

6. Relation to other diabatic constructions and scope

LDR belongs to a broader family of diabatic or quasidiabatic strategies developed to remove singular adiabatic couplings, but it differs from model-based adiabatic-to-diabatic transformations. In Li$\mathbf{R}_{\mathbf{n}}$7, the two lowest electronic states were transformed to a diabatic representation by fitting adiabatic potential-energy surfaces to a cubic $\mathbf{R}_{\mathbf{n}}$8 Jahn–Teller model and then constructing a coordinate-dependent transformation matrix

$\mathbf{R}_{\mathbf{n}}$9

with gauge parameter chosen as $\phi_\alpha(\mathbf{r};\mathbf{R}_{\mathbf{n}})$0 for correct Berry-phase properties. That representation is free from singular adiabatic couplings near the conical intersection, but it is anchored to a fitted local analytic model rather than to a discrete nuclear basis and overlap matrix (Ghassemi et al., 2013).

In SO spectroscopy, a property-based diabatisation removed avoided crossings through a unitary mixing-angle transformation

$\phi_\alpha(\mathbf{r};\mathbf{R}_{\mathbf{n}})$1

with

$\phi_\alpha(\mathbf{r};\mathbf{R}_{\mathbf{n}})$2

and parameters chosen so that diabatic potential curves were as smooth as possible. That framework yields smooth diabatic couplings and phase-consistent coupling curves, but again it is a local two-state diabatization scheme rather than the discrete-variable overlap-based LDR used for exact conical-intersection dynamics (Brady et al., 2022).

The scope of LDR is therefore specific. It is especially suited to exact or systematically convergent nonadiabatic calculations on low-dimensional grids, particularly around conical intersections. Its principal limitation, explicitly noted in the Fourier-grid implementation, is exponential scaling with system size due to the direct-product grid. Proposed future directions include sparse grids or hierarchical bases (Gu, 2023).

Within that scope, the discrete variable LDR occupies a distinct position. It uses adiabatic electronic states directly, avoids singular derivative couplings, does not require global phase smoothing, and incorporates nonadiabatic transitions, electronic coherences, geometric phases, DBOC effects, and second-order coupling effects through finite overlap data on a nuclear grid (Gu, 2023, Gu, 2023, Sha et al., 6 Sep 2025).