Local Diabatic Representation

• Local diabatic representation is a framework that formulates nonadiabatic molecular dynamics using locally defined electronic bases to replace singular derivative couplings.
• It utilizes discrete evaluations of adiabatic states at selected nuclear geometries, ensuring divergence-free behavior near avoided crossings and conical intersections.
• The method underpins efficient propagation schemes and accurate diatomic spectroscopy by offering robust phase handling and faster convergence in complex quantum systems.

Local diabatic representation denotes a family of constructions in which nonadiabatic molecular or quantum dynamics are formulated in locally defined electronic frames rather than in a single globally smooth diabatic basis. In the molecular setting, especially near avoided crossings and conical intersections, the representation typically uses adiabatic electronic states evaluated at discrete or locally selected nuclear geometries and replaces singular derivative couplings by off-diagonal potential couplings or bounded electronic overlaps. In this form it is free of singularities in the first and second derivative couplings, does not require a smooth gauge of electronic-wavefunction phase, and captures nonadiabatic transitions, electronic coherences, and geometric phases (Gu, 2023, Gu, 2023, Sha et al., 6 Sep 2025). In diatomic spectroscopy, the same local principle is also used to diabatize selected avoided-crossing subspaces, as in sulfur monoxide (Brady et al., 2022).

1. Adiabatic, diabatic, and local formulations

In the adiabatic representation, electronic states are eigenstates of the clamped-nuclei electronic Hamiltonian, and nuclear motion is expanded on adiabatic potential energy surfaces. For a two-state subspace, the adiabatic potential matrix is diagonal,

$${\bf V}^{\rm a}(R) = \begin{pmatrix} V_1^{\rm a}(R) & 0 \ 0 & V_2^{\rm a}(R) \end{pmatrix},$$

while nonadiabatic effects enter through derivative couplings such as

$$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$

Near avoided crossings or conical intersections, these couplings become very large and sharply peaked; in the conical-intersection case they diverge with inverse powers of the energy gap. This makes the adiabatic representation numerically ill-suited for exact wavepacket dynamics and for smooth global fitting of rovibronic models (Brady et al., 2022, Gu, 2023).

In the diabatic representation, derivative couplings are removed or minimized and nonadiabatic effects are shifted into off-diagonal potential couplings,

$${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$

For diatomics, this gives zero derivative couplings and non-diagonal diabatic couplings; for polyatomics, exact global diabatization is not generally available, and only quasi-diabatic schemes exist (Brady et al., 2023, Gu, 2023). A related statement in the geometric-integrator literature is that exact diabatization is only formally possible for systems with two electronic states and one nuclear coordinate (Roulet et al., 2019).

The local diabatic representation occupies the interval between these two limits. It does not seek a globally smooth diabatic basis over the full nuclear configuration space. Instead, it defines diabatic behavior only locally: at discrete nuclear geometries, within short propagation segments, or inside restricted avoided-crossing subspaces. In the discrete-variable form, it has been described as “a local generalization of the crude adiabatic representation with many reference geometries instead of one” (Sha et al., 6 Sep 2025).

2. Discrete local diabatic bases and the LDR ansatz

A central implementation of the local diabatic representation expands the molecular wavefunction in localized nuclear basis functions and adiabatic electronic states evaluated at the associated grid points,

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

Here $\chi_{\mathbf n}(\mathbf R)$ is a nuclear DVR basis function localized around $\mathbf R_{\mathbf n}$, and $\phi_\alpha(\mathbf r;\mathbf R_{\mathbf n})$ is the adiabatic electronic eigenstate of $\hat H_{\mathrm{BO}}(\mathbf R_{\mathbf n})$ at that geometry (Gu, 2023, Sha et al., 6 Sep 2025).

The defining structural feature is that the nuclear kinetic-energy operator never differentiates the electronic basis functions. Instead, all nonadiabatic effects are encoded in the overlap matrix

$$A_{\mathbf m\mathbf n}^{\beta\alpha} = \langle \phi_\beta(\mathbf R_{\mathbf m}) | \phi_\alpha(\mathbf R_{\mathbf n}) \rangle_{\mathbf r},$$

and the Hamiltonian matrix elements take the form

$$H_{\mathbf m\beta,\mathbf n\alpha} = A_{\mathbf m\mathbf n}^{\beta\alpha} \langle \mathbf m|\hat T_{\mathrm N}|\mathbf n\rangle_{\mathbf R} + V_\alpha(\mathbf R_{\mathbf n}) \delta_{\mathbf m\mathbf n}\delta_{\beta\alpha}.$$

Because no derivative of $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$0 with respect to nuclear coordinates appears, the construction is divergence-free at conical intersections and does not require a smooth or single-valued gauge for the adiabatic electronic states (Sha et al., 6 Sep 2025).

The same framework underlies the exact conical-intersection dynamics method that combines the local diabatic representation, Strang splitting, and a Fourier basis. In that formulation, the localized nuclear basis is built from discrete variable representation with uniform grids, the electronic states are adiabatic states at local geometries, and nonadiabatic effects are mediated through overlap matrices between grid points. The method is explicitly described as capturing all nonadiabatic effects, including nonadiabatic transitions, electronic coherences, and geometric phases, while remaining free of singularities in the first and second derivative couplings (Gu, 2023).

The convergence properties of this discrete-variable LDR have been analyzed for coupled oscillator models. For weak vibronic couplings, LDR shows similar convergence rate to the exact Born–Huang representation including first-order derivative couplings, diagonal Born–Oppenheimer corrections, and second-order derivative couplings. For strong vibronic couplings, LDR shows a significant faster convergence rate with respect to the number of grid points than the exact Born–Huang representation, whereas the crude adiabatic representation generally shows a much slower convergence rate (Sha et al., 6 Sep 2025).

3. Gauge structure, topology, and geometric phase

In the adiabatic representation, conical intersections are accompanied by nontrivial geometric phases. For a two-state system, encircling a conical intersection changes the sign of a real adiabatic electronic state, and the associated derivative couplings carry the full geometric-phase topology (Izmaylov et al., 2016). This is one reason a globally smooth adiabatic gauge is not available around a conical intersection.

The local diabatic representation replaces the continuous gauge problem by a discrete one. At each grid point $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$1, the electronic states may be assigned arbitrary phases, and the equations of motion remain invariant because those phase changes are compensated by corresponding changes in the overlap matrix and in the coefficients $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$2 (Gu, 2023). The local construction is therefore robust to arbitrary, even discontinuous, phase conventions of the adiabatic electronic wavefunctions. A smooth gauge is not required (Gu, 2023).

Geometric phase is not eliminated by this procedure; it is re-expressed. In the discrete formulation, it appears through holonomy of the overlap matrix. For a closed loop of grid points around a conical intersection, the Wilson-loop product

$$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$3

is gauge invariant and encodes the geometric phase of adiabatic state $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$4 (Gu, 2023). This is consistent with numerical demonstrations in which LDR reproduces the nodal line and destructive interference associated with Berry phase in two-dimensional conical-intersection dynamics (Gu, 2023).

A related point emerges from diabatic model studies of geometric phase. A modified diabatic representation based on the absolute value of diabatic couplings can remove the geometric phase while preserving adiabatic potential energy surfaces and the conical intersection itself, allowing dynamical effects arising from the geometric phase presence to be isolated (Izmaylov et al., 2016). This makes clear that “diabatic” does not mean “geometric-phase free”; the topological content depends on how the representation is defined.

4. Local diabatization in diatomic spectroscopy and avoided crossings

In diatomic spectroscopy, local diabatic representation is often applied not to all electronic states globally but to selected near-degenerate pairs. An ab initio rovibronic study of $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$5 computed 13 electronic states up to $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$6, together with 13 potential energy curves, 23 dipole and transition dipole moment curves, 23 spin-orbit curves, and 14 electronic angular momentum curves. A diabatic representation was then built by removing the avoided crossings between the spatially degenerate pairs $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$7 and $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$8 through a property-based diabatisation method (Brady et al., 2022).

That study explicitly describes the construction as diabatic, more precisely locally diabatic, because it is restricted to selected avoided-crossing subsystems rather than imposed globally over all 13 states. The aim is to obtain a numerically stable, phase-consistent spectroscopic model and to include nonadiabatic effects without explicit derivative couplings. The work also reports nonadiabatic couplings and diabatic couplings for the avoided-crossing systems, defines all phases for the coupling curves consistently, and presents the first fully reproducible spectroscopic model of SO covering the wavelength range longer than $$\phi_{12}(R) := \left\langle \psi_1(R) \left| \frac{d}{dR} \right| \psi_2(R) \right\rangle .$$9 (Brady et al., 2022).

A complementary diatomic result concerns the relation between adiabatic and diabatic calculations. For two-state diatomic problems, a strict diabatic representation was constructed and compared directly to the adiabatic Born–Huang treatment for YO and CH. Rovibronic energies and wavefunctions were computed using a new diabatic module implemented in DUO, and the calculations demonstrated numerical equivalence between the adiabatic and diabatic representations. The same study showed that it is important to include both the diagonal Born–Oppenheimer correction and non-diagonal derivative couplings, and that convergence can strongly depend on the chosen nuclear-motion representation; no one representation is best in all cases (Brady et al., 2023).

These diatomic studies establish a practical version of local diabatization: one isolates the troublesome subspace, removes avoided-crossing singular behavior from that subspace, enforces phase consistency, and then embeds the result in a larger spectroscopic Hamiltonian.

5. Propagation schemes based on local or quasi-diabatic frames

Local diabatic ideas have also been incorporated directly into propagation algorithms. In conical-intersection wavepacket dynamics, the combination of LDR, Strang splitting, and a Fourier basis permits direct split-operator propagation of the full molecular propagator, something that is not available in the adiabatic representation because the Hamiltonian there is not separable in the same way (Gu, 2023).

A closely related construction is the quasi-diabatic propagation scheme for polariton chemistry. There, the adiabatic-Fock states at a reference nuclear geometry are treated as the locally well-defined diabatic basis for a short time segment,

$${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$0

and the basis is updated at every nuclear time step. Because the basis is fixed over each short segment, derivative couplings vanish by construction during that segment, allowing diabatic quantum-dynamics methods to be interfaced directly with adiabatic polariton information (Hu et al., 2022).

Surface-hopping work in the diabatic basis provides a further downstream use. A two-level Schrödinger equation of the form

$${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$1

assumes a diabatic or quasi-diabatic input Hamiltonian with smooth multiplicative couplings and no derivative coupling terms. On that basis, semiclassical frozen-Gaussian and stochastic-hopping algorithms have been developed and shown good performance in weak-coupling and avoided-crossing regimes (Fang et al., 2017). The asymptotic analysis of a diabatic surface-hopping algorithm later justified the correct scaling of the transition rate in the Marcus regime for the spin-boson model and derived strong-coupling behavior matching a type of mean-field description (Cai et al., 2022).

The diabatic representation is also structurally advantageous for geometric integrators. With

$${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$2

the Hamiltonian is separable into a kinetic term depending only on nuclear momenta and a matrix-valued potential depending only on nuclear coordinates. This makes split-operator propagation explicit and efficient, including higher-order symmetric compositions that preserve unitarity, norm, inner products, symplecticity, symmetry, and time-reversibility (Roulet et al., 2019).

6. Broader usages and terminological variants

The phrase “local diabatic representation” is also used outside molecular vibronic wavepacket dynamics. In many-body quantum control, it can denote an approximate diabatic frame generated by local counter-diabatic terms. For a time-dependent Hamiltonian $${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$3, counter-diabatic driving adds $${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$4, where $${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$5 is the adiabatic gauge potential. Because the exact $${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$6 is generally nonlocal, variational constructions restrict it to local operators, producing what the literature describes as local counter-diabatic protocols or a local diabatic representation (Sels et al., 2016).

In Ising spin models, this local truncation is realized by ansätze such as

$${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$7

and by nearest-neighbor or all-to-all two-spin extensions. These approximate adiabatic gauge potentials are optimized variationally and used to construct local or few-body counter-diabatic Hamiltonians that improve ground-state preparation fidelity relative to bare adiabatic evolution (Hartmann et al., 2021). In the infinite-range Ising model, a mean-field treatment yields the explicitly local counter-diabatic term

$${\bf V}^{\rm d}(R) = \begin{pmatrix} V_1^{\rm d}(R) & V_{12}^{\rm d}(R) \ V_{12}^{\rm d}(R) & V_2^{\rm d}(R) \end{pmatrix}.$$8

which remains well-defined in the limit to the critical point (Hatomura, 2017).

A distinct electronic-structure usage appears in fragment-based dynamics. The local group diabatic Fock representation partitions Löwdin orthogonalized atomic orbitals into molecular groups, diagonalizes each group block of the Fock matrix, and uses the resulting group-localized orbitals as the working basis. The diagonal blocks then contain local eigenenergies and the off-diagonal blocks encode inter-group couplings, enabling real-time analysis of site-local excitations and charge migration in molecular aggregates (Yonehara et al., 2017).

Finally, quasi-diabaticity can also arise from least-transformed block-diagonalization in electronic-structure algorithms. State-average orbital-optimized variational quantum eigensolver has been analyzed as producing an ab initio quasi-diabatic representation “for free” without an explicit adiabatic-to-diabatic transformation, with formaldimine near a conical intersection used as the case study (Illésová et al., 25 Feb 2025).

Domain	Local object	Representative source
Conical-intersection wavepacket dynamics	Adiabatic states at discrete nuclear geometries coupled by overlaps	(Gu, 2023)
Discrete-variable exact dynamics	LDR with DVR grid, Strang splitting, Fourier basis	(Gu, 2023)
Diatomic spectroscopy	Property-based diabatization of selected avoided crossings	(Brady et al., 2022)
Polariton dynamics	Adiabatic-Fock states at a reference geometry used as locally well-defined diabatic states	(Hu et al., 2022)
Many-body counter-diabatic driving	Local adiabatic gauge-potential ansätze	(Sels et al., 2016)
Fragment electronic dynamics	Group-localized diabatic Fock basis	(Yonehara et al., 2017)

Across these usages, the shared principle is local replacement of problematic global structure—singular derivative couplings, nonlocal gauge potentials, or delocalized canonical states—by locally defined basis functions or locally acting couplings. The precise mathematical realization, however, depends on whether the target problem is rovibronic spectroscopy, conical-intersection dynamics, polariton propagation, many-body shortcuts to adiabaticity, or fragment-localized electron dynamics.