Independent Electron Surface Hopping

• Independent Electron Surface Hopping is defined as a many-electron extension of Tully’s fewest-switches method, modeling electrons as independent via a one-electron Hamiltonian.
• It employs a stochastic hopping rule based on orbital populations and nonadiabatic couplings to capture electron–hole pair excitations in adsorbate–surface systems.
• Benchmark studies reveal that IESH accurately simulates energy loss dynamics and threshold behavior, outperforming mean-field approaches in practical applications.

Independent Electron Surface Hopping (IESH) is a mixed quantum–classical method for nonadiabatic dynamics in which nuclei move classically while the electronic subsystem is represented by noninteracting single-particle states, each electron being propagated and hopped independently under a one-electron Hamiltonian. The method was developed especially for adsorbates interacting with metal surfaces, where electron–hole pair excitations, charge transfer, inelastic scattering, vibrational relaxation, and desorption must be described in a dense manifold of electronic states; later work connected the same independent-electron logic to orbital-level formulations derived from the orbital quantum–classical Liouville equation and to electronic-thermostat variants designed to recover detailed balance in the long-time limit (Gardner et al., 2022, Ma et al., 2024, Ma et al., 21 Nov 2025).

1. Definition, scope, and the meaning of “independent”

IESH is best described as a many-electron extension of Tully’s fewest-switches surface hopping that treats electrons as independent. In this formulation, the electronic Hamiltonian is effectively one-body, electron–electron interactions beyond Pauli exclusion are neglected during dynamics, and the many-electron state remains a Slater determinant of evolving one-electron orbitals. “Independent” therefore refers to independent propagation of occupied single-particle orbitals, not to an absence of Fermi statistics or to neglect of the shared nuclear trajectory (Gardner et al., 2022, Lu et al., 18 Aug 2025).

The method is most naturally formulated for adsorbate–metal and adsorbate–semiconductor problems. In metal-surface applications it targets electron–hole pair excitations in a dense continuum of surface states; in semiconductor scattering it becomes especially valuable because explicit orbital occupations and energy-conserving hops can reproduce threshold behavior imposed by a bandgap (Gardner et al., 2022, Lu et al., 18 Aug 2025).

A recurring source of confusion is the overlap between “independent electrons” and “independent trajectories.” Conventional fewest-switches surface hopping in molecular photochemistry propagates independent nuclear trajectories, but its electronic states are often many-electron adiabatic states from correlated electronic-structure methods rather than independent electrons. For example, on-the-fly surface hopping for cyclobutanone uses XMS(3)-CASPT2(10, 8)/def2-SVPD adiabatic states and explicitly notes that the term “independent electron surface hopping” is not used there (Peng et al., 2024). Likewise, exact-factorization-based variants such as SHXF and QTSH-XF retain the independent-trajectory picture but do not introduce IESH per se (Vindel-Zandbergen et al., 2021, Dupuy et al., 2024).

Before comparing variants, it is useful to separate the main formulations that now coexist around the independent-electron idea.

Formulation	Electronic object	Stated relation to IESH
IESH	Independent occupied single-particle orbitals	Many-electron extension of FSSH treating electrons as independent (Gardner et al., 2022)
OSH	Single-particle reduced density matrix and orbital occupations	Closely connects with IESH, except that electrons hop between orbitals (Ma et al., 2024)
OSH with electronic thermostat	Orbital density matrix plus thermostat terms	Generalizes OSH to many discrete states and restores detailed balance (Ma et al., 21 Nov 2025)

2. Hamiltonians, electronic propagation, and hopping rules

The canonical IESH setting starts from a Hamiltonian of the form

$H(R,P) = H_\text{nuc}(R,P) + H_\text{el}(R),$

with classical nuclei and a one-electron electronic Hamiltonian

$$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$

For Newns–Anderson- or Anderson–Holstein-type models, $h_{ij}(R)$ describes an adsorbate impurity orbital, a discretized set of metal or semiconductor states, and coordinate-dependent couplings between them (Gardner et al., 2022, Ma et al., 2024).

At fixed nuclear geometry, one diagonalizes the one-electron Hamiltonian,

$$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$

and expands each occupied single-electron wavefunction in the adiabatic orbital basis. In the semiconductor H/Ge(111) application this is written as

$$|\phi^{(j)}(t)\rangle = \sum_n C_n^{(j)}(t)\,|\chi_n(x(t))\rangle,$$

with coefficients obeying

$$i\hbar \dot C_n^{(j)}(t) = E_n(x(t)) C_n^{(j)}(t) - i\hbar \sum_m C_m^{(j)}(t)\,\dot x(t)\, d_{nm}(x(t)),$$

where $$d_{nm}(x)=\langle \chi_n(x)|\partial_x \chi_m(x)\rangle$$ is the nonadiabatic coupling (Lu et al., 18 Aug 2025).

The single-electron hopping rule is a direct orbital analogue of Tully’s fewest-switches prescription. For electron $j$, the probability to hop from orbital $n$ to $m$ over a time step $$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$0 is

$$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$1

When a hop is accepted, nuclear velocity is rescaled to conserve total energy; if there is insufficient kinetic energy, the hop is rejected. Because electrons are treated independently, Pauli blocking must be enforced explicitly: hops into already fully occupied states are disallowed (Lu et al., 18 Aug 2025).

For many-electron dynamics, the full electronic state is reconstructed statistically from the set of occupied orbitals. A standard description is that the many-electron state remains a Slater determinant of the evolving occupied single-particle orbitals, while observables such as impurity occupation, current, scattering probability, or energy loss are obtained by trajectory averaging over many independent electronic realizations with initial occupations drawn from Fermi–Dirac statistics (Gardner et al., 2022).

3. From heuristic independent electrons to orbital quantum–classical Liouville theory

A major development after the original IESH constructions was the derivation of orbital surface hopping (OSH) from the orbital quantum–classical Liouville equation (o-QCLE). In this framework, for a noninteracting many-electron Hamiltonian quadratic in fermionic operators, the full many-electron dynamics are exactly encoded in the single-particle density matrix. After a partial Wigner transform of the nuclear degrees of freedom and truncation of the Wigner–Moyal expansion at first order, one obtains an o-QCLE for the adiabatic orbital density matrix $$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$2, from which OSH is derived as a stochastic mixed quantum–classical algorithm (Ma et al., 2024).

The OSH electronic equation in the adiabatic orbital basis is

$$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$3

and its hopping probability from an occupied orbital $$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$4 to an unoccupied orbital $$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$5 is

$$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$6

with the probability set to zero if $$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$7 is already occupied. This puts the independent-electron logic on a more explicit density-matrix foundation and avoids explicit many-electron determinant manipulations (Ma et al., 2024).

The same work shows that the relation between OSH and IESH is not merely qualitative. For noninteracting electrons, derivative couplings between many-electron Slater determinants vanish unless the determinants differ by a single orbital, so the many-electron hopping structure collapses onto single-electron orbital hops. Benchmarks against a surface-hopping method with a full Configuration Interaction wavefunction show strong agreement with both IESH and OSH for molecular orbital populations and kinetic-energy relaxation, while OSH is more efficient because it does not require determinant overlaps for every possible hop (Ma et al., 2024).

This orbital-QCLE perspective also clarifies how IESH fits into the broader QCLE genealogy of surface hopping. From the conventional QCLE, fewest-switches dynamics emerge only after dropping decoherence terms and restricting momentum changes to population-state transitions. That result does not derive IESH directly, but it makes clear that any independent-electron hopping method inherits the same structural issues: decoherence, mean-surface segments, and momentum jumps are not optional technicalities but part of the underlying quantum–classical Liouville dynamics (Kapral, 2016).

4. Representative model systems and benchmark behavior

In metal-surface applications, IESH has been implemented for model Hamiltonians, for full-dimensional atomistic systems, and for biased nonequilibrium setups. The efficient implementation in NQCDynamics.jl was presented as a transparent, reliable, and efficient implementation of IESH, with tests ranging from model Hamiltonians to full-dimensional atomistic systems and extensions to an external bias potential. Those studies emphasized scattering and desorption probabilities, vibrational relaxation, and energy loss to electron–hole pairs as the central observables (Gardner et al., 2022).

The strongest semiconductor benchmark in the supplied literature is hyperthermal H scattering from $$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$8 Ge(111), modeled with a first-principles parameterized Haldane–Anderson Hamiltonian. The model contains a discretized valence band, a conduction band, and a bandgap of $$H_\text{el}(R) = \sum_{ij} h_{ij}(R)\, c_i^\dagger c_j.$$9 around the Fermi level. In this system, IESH predicts negligible nonadiabatic energy loss for $h_{ij}(R)$0, because hops that would require energy $h_{ij}(R)$1 are energetically forbidden, while for $h_{ij}(R)$2 nonadiabatic energy loss rises sharply as valence-to-conduction excitation channels open (Lu et al., 18 Aug 2025).

That semiconductor example is notable because the contrast with mean-field methods is explicit. Ehrenfest dynamics yields weak nonadiabatic energy loss that is independent of the initial kinetic energy in the same model, whereas IESH qualitatively agrees with the experimental observation that strong inelastic energy loss only appears once the projectile kinetic energy exceeds the bandgap. Convergence tests at $h_{ij}(R)$3 used 150 bath states and 75 electrons, and the reported mean inelastic kinetic energy loss in IESH is about $h_{ij}(R)$4–$h_{ij}(R)$5, with narrow confidence intervals after $h_{ij}(R)$6 trajectories (Lu et al., 18 Aug 2025).

For smaller benchmark systems where exact many-electron surface hopping remains feasible, OSH and IESH were compared against FCI-based surface hopping for equilibrium electron transfer in a Newns–Anderson model and for vibrational relaxation of NO on Au(111). In those studies, OSH and IESH showed strong agreement with FCI-SH for impurity populations and short-time kinetic-energy relaxation, while OSH also displayed higher efficiency, including a reported speed-up of up to $h_{ij}(R)$7 at $h_{ij}(R)$8 in one benchmark and about $h_{ij}(R)$9 at $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$0 in another (Ma et al., 2024).

5. Decoherence, detailed balance, and conservation laws

The strengths of IESH are also the source of its main formal difficulties. Because the metallic continuum is discretized, the simulated electron system is closed unless an additional bath is introduced. The 2025 OSH thermostat paper states this directly: discretization of the metallic electronic continuum typically results in a closed-system representation that fails to capture the open-system nature of the true physical process, producing artifacts in the dynamical evolution and a violation of detailed balance. In response, that work adds an electronic thermostat at the orbital-density-matrix level,

$$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$1

so that orbital occupations relax toward Fermi–Dirac equilibrium and long-time dynamics recover detailed balance (Ma et al., 21 Nov 2025).

The thermostat construction also yields explicit bath-induced orbital transition probabilities. For orbital $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$2, if it is occupied, decay into the bath occurs with rate $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$3; if it is unoccupied, filling from the bath occurs with rate $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$4. In the model calculations reported there, standard OSH without a thermostat quickly reaches a quasi-equilibrium that differs significantly from benchmark HEOM equilibrium for both impurity population and nuclear kinetic energy, whereas the thermostat-corrected OSH reproduces accurate dynamics and detailed balance in long time (Ma et al., 21 Nov 2025).

A second structural issue concerns momentum conservation. Standard fewest-switches surface hopping ignores electronic momentum and indirectly equates nuclear momentum with total momentum, but the ETF/ERF analysis shows that this does not conserve either nuclear linear or nuclear angular momentum. Electron translation factors and electron rotation factors modify derivative couplings so that rigid translations and rigid rotations do not spuriously drive hops or momentum rescaling, and the resulting dressed couplings are constructed to remain size-consistent (Athavale et al., 2023).

An even stronger statement appears in the phase-space momentum-conservation analysis. For systems with spin–orbit coupling and an odd number of electrons, the standard FSSH algorithm does not conserve total linear or angular momentum, not so much because of the hopping direction alone but because it propagates adiabatic dynamics along surfaces that are not time reversible. The proposed remedy is to propagate on eigenvalues of a phase-space electronic Hamiltonian $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$5 with an electronic nuclear coupling $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$6, subject to explicit translational and rotational constraints on $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$7. This provides a formal route toward momentum-conserving hopping algorithms in which nuclear, electronic orbital, and electronic spin degrees of freedom are all coupled together (Wu et al., 2023).

6. Relation to neighboring methods and current directions

IESH occupies a distinct place among mixed quantum–classical methods because it combines discrete orbital bookkeeping with classical nuclei. Against Ehrenfest dynamics, its main advantage is the discrete treatment of electronic excitation events. In the Ge(111) semiconductor benchmark, Ehrenfest produces gradual, nonzero energy loss even below the bandgap, whereas IESH retains the sharp opening of excitation channels only once the projectile kinetic energy exceeds the bandgap (Lu et al., 18 Aug 2025). Against electronic-friction approaches, IESH resolves explicit orbital occupations and therefore does not rely on the metallic-continuum, linear-response assumptions that smear threshold behavior in gapped materials (Lu et al., 18 Aug 2025).

It is equally important to distinguish IESH from conventional molecular surface hopping built on many-electron states. The cyclobutanone photodissociation study is explicit that its electronic states are many-electron XMS-CASPT2 wavefunctions and that there is no independent-electron approximation; the only “independent” approximations there are independent nuclear trajectories and the independent atom model for diffraction (Peng et al., 2024). This distinction matters because it marks the natural domain of IESH: systems where a one-electron Hamiltonian and a Slater-determinant picture remain adequate.

Beyond adsorbate–surface models, two methodological extensions are especially relevant. First, reciprocal-space surface hopping reformulates fewest-switches dynamics entirely in $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$8-space and is directly compatible with band-structure calculations, making it natural for independent carriers in solids and for Brillouin-zone truncation when dynamics remain localized in reciprocal space (Krotz et al., 2021). Second, recent phase-space electronic Hamiltonians that depend on both $$H_\text{el}(R)\,|\phi_\alpha(R)\rangle = E_\alpha(R)\,|\phi_\alpha(R)\rangle,$$9 and $$|\phi^{(j)}(t)\rangle = \sum_n C_n^{(j)}(t)\,|\chi_n(x(t))\rangle,$$0 aim to absorb electronic inertial effects into the surfaces themselves, eliminating at least to very high order nonadiabatic transitions during pure nuclear translational and rotational motion (Bian et al., 2024).

The main limitations remain those repeatedly stated across the literature: the independent-electron approximation neglects electron–electron correlations beyond mean field; nuclei are classical; detailed balance is not automatic in a discretized continuum; and the correct treatment of decoherence, Berry forces, spin, and angular momentum requires more structure than bare fewest-switches hopping usually provides (Gardner et al., 2022, Ma et al., 21 Nov 2025, Wu et al., 2023). A plausible implication is that future independent-electron surface-hopping schemes will borrow simultaneously from orbital-QCLE derivations, electronic thermostats, and phase-space or exact-factorization ideas, while preserving the defining feature that made IESH useful in the first place: many-electron nonadiabatic dynamics reduced to a tractable ensemble of one-electron hopping problems.