Nonadiabatic Force Matching

• Nonadiabatic force matching is a methodology that constructs effective nuclear forces to accurately reproduce electron–nuclear interactions beyond the traditional adiabatic approximation.
• It employs frameworks such as exact factorization, Floquet QCLE, and trajectory-based methods to derive mean forces, friction tensors, and dissipation functionals from high-level theories.
• These techniques enable the calibration and validation of simplified force models against detailed many-body simulations and benchmark studies using methods like FN-DMC and variational wavefunction interpolation.

Searching arXiv for recent and foundational papers directly relevant to nonadiabatic force matching, exact-factorization forces, Floquet/QCLE friction, NaF, and variational interpolation of correlated wavefunctions. Nonadiabatic force matching, in the literature considered here, is used in closely related senses, suggesting a common program: constructing, calibrating, or validating effective nuclear forces so that they reproduce forces, force correlations, or force-derived observables implied by coupled electron–nuclear dynamics beyond the Born–Oppenheimer approximation. In this setting, the reference object may be an exact-factorization time-dependent potential energy surface, a Floquet quantum–classical Liouville description with friction and diffusion, an explicitly correlated electron–ion wavefunction, or a variationally interpolated many-body state from which analytic forces and nonadiabatic couplings are obtained. The underlying issue is always the same: when electronic and nuclear motion cannot be cleanly separated, the force on nuclei is no longer exhausted by the gradient of a single adiabatic potential energy surface (Agostini et al., 2014, Mosallanejad et al., 2023, Atalar et al., 2024, Rosa-Raíces et al., 19 Aug 2025).

1. Scope and central objects

Within this broad program, the matched quantity depends on the formalism. Some approaches match the instantaneous force itself, others the force law in a mixed quantum–classical reduction, others the friction/noise tensors that emerge after electronic elimination, and others the dissipation functional associated with nonequilibrium transformations. What unifies them is the use of a higher-level nonadiabatic description as the reference from which a lower-complexity force model is derived or against which it is tested.

Framework	Matched object	Representative source
Exact factorization	Scalar and vector-potential-driven nuclear force	(Agostini et al., 2014, Agostini et al., 2014, Cohen et al., 11 Feb 2025)
Floquet QCLE / electronic friction	Mean force, friction tensor, diffusion, Lorentz-like term	(Mosallanejad et al., 2023, Chen et al., 2023, Wang et al., 2023)
Trajectory-based MQC	Conditioned force assignment, decoherence-consistent force, random-force correction	(Feng et al., 2013, Wu et al., 2024, Wu et al., 11 Apr 2025, Chen et al., 2024)
Correlated wavefunction interpolation	Analytic energies, forces, NAC vectors from many-body states	(Atalar et al., 2024, Tubman et al., 2016)
Nonequilibrium free-energy estimation	Dissipation functional of a nonadiabatic potential	(Rosa-Raíces et al., 19 Aug 2025)

At the most general level, the coupled problem replaces the adiabatic picture of nuclear motion on a clamped-ion surface $E_{\mathrm{BO}}(R)$ by a wavefunction $\Psi(R,r)$ or density operator in which nuclear coordinates $R$ and electronic coordinates $r$ remain entangled. In the explicitly electron–ion Hamiltonian emphasized in fixed-node diffusion Monte Carlo work, one treats

$$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$

so that nonadiabatic effects are encoded directly in the many-body state $\Psi(R,r)$ rather than appended perturbatively (Tubman et al., 2016).

2. Exact-force formulations beyond Born–Oppenheimer

The most direct formulation of nonadiabatic force matching comes from exact factorization. The full electron–nuclear wavefunction is written as

$$\Psi(\mathbf r,\mathbf R,t)=\Phi_{\mathbf R}(\mathbf r,t)\,\chi(\mathbf R,t),$$

with the partial normalization condition on $\Phi_{\mathbf R}$. The nuclear equation then contains a scalar potential $\epsilon(\mathbf R,t)$, the time-dependent potential energy surface (TDPES), and a vector potential $\mathbf A_\nu(\mathbf R,t)$, yielding a formally exact nuclear Hamiltonian

$\Psi(R,r)$0

In the mixed quantum–classical limit derived from this structure, the nuclear force law contains not only the scalar contribution $\Psi(R,r)$1 but also $\Psi(R,r)$2, Lorentz-like terms involving $\Psi(R,r)$3, and inter-nuclear terms mediated by the electronic vector potential (Agostini et al., 2014).

For the Shin–Metiu nonadiabatic charge-transfer problem, propagation of an ensemble of independent classical trajectories on the exact TDPES yields dynamics that are essentially indistinguishable from the exact quantum dynamics. The critical features are the step and bump structures of the exact potential after the avoided crossing. The gauge-invariant part of the TDPES aligns piecewise with different Born–Oppenheimer surfaces, while the gauge-dependent part develops an opposite-sign step; their sum leaves a smaller residual step and a localized bump. This structure is necessary for correct nuclear wavepacket splitting, because the step ties different spatial branches to different adiabatic surfaces and the bump regulates trajectory transfer between them (Agostini et al., 2014).

In solids, exact factorization is recast in phonon coordinates $\Psi(R,r)$4, where the effective nuclear potential is written as

$\Psi(R,r)$5

The corresponding force is

$\Psi(R,r)$6

so the nonadiabatic correction is carried by the geometric term. Within the exact-factorization-based DFT/DFPT framework, first- and second-order corrections to electronic states and energies are expressed in terms of standard DFPT components such as $\Psi(R,r)$7 and $\Psi(R,r)$8, making nonadiabatic force corrections available in terms of electron–phonon couplings and phonon frequencies (Cohen et al., 11 Feb 2025).

3. Open-system, Floquet, and friction-based force matching

In periodically driven open systems, nonadiabatic force matching is formulated at the level of Floquet quantum master equations, Floquet QCLE, and the resulting Fokker–Planck or Langevin reductions. After partial Wigner transformation over nuclear degrees of freedom, the Floquet QCLE produces a nuclear Fokker–Planck equation with a mean force $\Psi(R,r)$9, an electronic friction tensor $R$0, and a momentum-diffusion tensor $R$1: $R$2 The friction tensor is the first-order nonadiabatic correction to Born–Oppenheimer dynamics, and in one-body Floquet form it is written in terms of Floquet Green’s functions, $R$3, $R$4, and $R$5. In this hierarchy, force matching means choosing effective classical forces and transition rates so that trajectory-based dynamics reproduces the Floquet QME/QCLE populations, relaxation, and steady states (Mosallanejad et al., 2023).

For driven quantum transport near metal surfaces, the Floquet electronic friction framework makes the matched force structure explicit: $R$6 Here $R$7 is the mean force, $R$8 the Floquet electronic friction tensor, and $R$9 the random force. The tensor is not necessarily symmetric; its antisymmetric part

$r$0

acts as a Lorentz-like force and can generate circular nuclear motion. The paper shows that periodic driving alone can produce this antisymmetric component even at equilibrium, while finite bias can enhance it further (Chen et al., 2023).

In the Anderson–Holstein setting under Floquet driving, fast electronic relaxation and fast driving reduce the Floquet classical master equation to a Floquet Fokker–Planck equation with a potential of mean force $r$1, a Floquet electronic friction coefficient

$r$2

and a diffusion coefficient $r$3. Because $r$4 under strong Floquet driving, the second fluctuation–dissipation theorem is violated, which produces heating effects. This gives a concrete coarse-grained force-matching route: derive drift, friction, and noise from the underlying FCME/FQME rather than fitting them ad hoc (Wang et al., 2023).

4. Trajectory-based force assignment and conditioned dynamics

Several trajectory formalisms reinterpret nonadiabatic force matching as the problem of assigning the correct force to a classical trajectory conditioned on an evolving electronic state. In the quantum-trajectory mean-field picture, classical nuclei continuously measure the electronic subsystem through the distinct forces associated with different Born–Oppenheimer surfaces. The electronic density matrix $r$5 evolves by a stochastic quantum trajectory equation with coherent, decoherence, and measurement-backaction terms, while the nuclear force is taken as

$r$6

In the weak-measurement regime this reduces to a mean-field force, whereas in the strong-measurement regime $r$7 collapses onto a single adiabatic state and the force becomes single-surface, thereby bridging Ehrenfest and surface hopping without an external hopping algorithm (Feng et al., 2013).

The Nonadiabatic Field (NaF) program makes the nonadiabatic force term explicit. In the adiabatic representation, the Ehrenfest-like mapping dynamics yields a nuclear force decomposition

$r$8

where $r$9 are population-like weights and $$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$0 encode electronic coherences. NaF replaces the mean-field adiabatic term by the gradient of a single adiabatic surface while retaining the nonadiabatic term. The resulting force is

$$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$1

with

$$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$2

This construction is designed to recover single-surface asymptotics while preserving explicit nonadiabatic back-reaction in the coupling region (Wu et al., 2024). The later NaF development extends this picture to adiabatic and diabatic representations, efficient integrators, and a broad benchmark suite, and emphasizes that the nuclear force in NaF involves the nonadiabatic force arising from nonadiabatic coupling in addition to the adiabatic force contributed by a single adiabatic electronic state (Wu et al., 11 Apr 2025).

A complementary stochastic correction is the Ehrenfest-plus-random-force scheme. Standard Ehrenfest dynamics uses the mean-field force

$$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$3

and omits the fluctuating component

$$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$4

E$$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$5 replaces this missing term by a classical random process with matched correlation function, in Markovian or non-Markovian form, so that detailed balance is restored while retaining Ehrenfest-level efficiency. In this setting, force matching is literal correlation matching: the stochastic force is constructed so that its statistics reproduce those of the omitted quantum force operator (Chen et al., 2024).

5. Correlated many-body benchmarks and analytic force generation

A distinct line of work treats nonadiabatic force matching as a benchmark-generation problem: obtain highly accurate nonadiabatic energies or forces from correlated electron–ion wavefunctions and use them to validate or parametrize simpler models. In fixed-node diffusion Monte Carlo for diatomics, the central object is an explicitly electron–ion trial wavefunction

$$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$6

The interpolated ansatz approximates the $$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$7-dependence of the determinant coefficients by linear interpolation between reoptimized values at nearby bond lengths. For CH, this improves both clamped-ion and dynamic-nuclei energies relative to the dragged-node approximation, lowers the dynamic FN-DMC energy, and reduces the apparent nonadiabatic contribution from about $$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$8 mHa in earlier dragged-node work to about $$\hat H=\hat T_e+\hat T_N+\hat V_{ee}+\hat V_{eN}+\hat V_{NN},$$9 mHa with interpolation. The paper does not compute forces explicitly, but it provides a more accurate nonadiabatic potential energy curve and an analytic form for $\Psi(R,r)$0, which is directly relevant to any force-matching program based on FN-DMC energetics (Tubman et al., 2016).

A more direct route to forces and nonadiabatic couplings is variational interpolation of correlated many-body wavefunctions. In the eigenvector-continuation framework, a set of training states $\Psi(R,r)$1 defines a geometry-independent subspace, and the interpolated electronic states at geometry $\Psi(R,r)$2 are obtained from the generalized eigenvalue problem

$\Psi(R,r)$3

Because the overlap metric is geometry-independent, Hellmann–Feynman differentiation yields analytic forces, and transition reduced density matrices yield analytic nonadiabatic coupling vectors. The active-learning protocol based on the Hamiltonian-distance criterion culminates in application to photoexcited H$\Psi(R,r)$4, where 22 DMRG calculations of training states are used to infer multi-state energies, forces, and nonadiabatic coupling vectors at 12,000 geometries along an ensemble of trajectories (Atalar et al., 2024).

This suggests a stringent form of nonadiabatic force matching: rather than fitting a classical potential directly, interpolate the many-body wavefunction and derive the matched force and NAC field analytically from the same variational model. The matched quantities are then internally consistent by construction.

6. Variational dissipation matching, limitations, and outlook

The phrase “nonadiabatic force matching” is also used for nonequilibrium alchemical free-energy estimation. In that formulation, the reference object is a nonadiabatic potential

$\Psi(R,r)$5

which measures the lag between the actual nonequilibrium density $\Psi(R,r)$6 and the instantaneous equilibrium density $\Psi(R,r)$7. The free-energy difference is written as the average work minus a dissipation functional of $\Psi(R,r)$8, and a parameterized $\Psi(R,r)$9 is trained by minimizing

$$\Psi(\mathbf r,\mathbf R,t)=\Phi_{\mathbf R}(\mathbf r,t)\,\chi(\mathbf R,t),$$0

Here force matching no longer targets a mechanical force on nuclei, but the gradient field $$\Psi(\mathbf r,\mathbf R,t)=\Phi_{\mathbf R}(\mathbf r,t)\,\chi(\mathbf R,t),$$1 that matches dissipation along nonequilibrium alchemical trajectories and defines a diffusion–denoising reverse process (Rosa-Raíces et al., 19 Aug 2025).

Across the field, several limitations recur. Fixed-node diffusion Monte Carlo remains subject to fixed-node bias, and the linear interpolation of CI coefficients is justified only near the sampled bond-length region (Tubman et al., 2016). Floquet QCLE, Floquet QME, and electronic-friction reductions rely on weak system–bath coupling, Markovian baths, slow nuclei, and finite Floquet truncation; the mapping to friction and diffusion is asymptotically controlled only within that regime (Mosallanejad et al., 2023). In the Floquet Fokker–Planck setting, the friction/noise description is adequate when electronic motion and driving are fast compared with nuclear motion, but full FCME/FSH or FQME becomes necessary for slow driving or weak electronic coupling (Wang et al., 2023). E$$\Psi(\mathbf r,\mathbf R,t)=\Phi_{\mathbf R}(\mathbf r,t)\,\chi(\mathbf R,t),$$2 improves detailed balance relative to Ehrenfest dynamics, but the quality of the correction depends on the fidelity of the precomputed force-correlation kernel and the validity of Gaussian noise models (Chen et al., 2024). Variational interpolation of correlated wavefunctions is limited by training-set coverage, the scaling of transition reduced density matrices, and the need for reliable high-level training solvers (Atalar et al., 2024).

A plausible implication is that nonadiabatic force matching is evolving toward a layered methodology rather than a single algorithm. Exact factorization supplies formally exact reference forces and their decomposition; Floquet QCLE and electronic-friction theories provide reduced force laws with mean, dissipative, and geometric components; trajectory formalisms such as QTMF, NaF, and E$$\Psi(\mathbf r,\mathbf R,t)=\Phi_{\mathbf R}(\mathbf r,t)\,\chi(\mathbf R,t),$$3 specify how those forces are assigned along individual paths; and correlated-wavefunction interpolation supplies high-level benchmark data and analytic force/NAC fields at scales inaccessible to brute-force many-body dynamics. In that combined sense, nonadiabatic force matching is best understood as the problem of making reduced nuclear dynamics faithfully reproduce the force content of a coupled electron–nuclear theory beyond Born–Oppenheimer.