Normal Mode Variance in Nonadiabatic Dynamics

• Normal Mode Variance (NMV) is a technique that ranks vibrational modes by the variance of mass-weighted displacements, identifying key coordinates in nonadiabatic molecular dynamics.
• Its algorithmic workflow projects full-dimensional trajectory data onto a normal-mode basis, automating the selection of high-variance modes for reduced nuclear propagation.
• Benchmark studies comparing NMV with PCA across photoreactive molecules highlight trade-offs between reduction efficiency and the preservation of critical dynamical features.

Searching arXiv for the specified paper and any directly relevant context on NMV in reduced-dimensional nonadiabatic dynamics. Normal Mode Variance (NMV) is a dimensionality-reduction method for nonadiabatic molecular dynamics that ranks normal modes by the variance of their trajectory-projected amplitudes and retains the highest-variance subset for nuclear propagation. In the formulation used for reduced-dimensionality Trajectory Surface Hopping (TSH), NMV is defined from mass-weighted displacements relative to a reference Franck–Condon geometry, projected onto the normal modes obtained from the mass-weighted Hessian. The method was investigated as an automated coordinate-selection strategy for photochemical dynamics in trans-azomethane, butyrolactone, and furanone, where it was compared directly with Principal Component Analysis (PCA). Across all three systems, NMV was able to reproduce full-dimensional results in selected reduced spaces, but it was consistently outperformed by PCA in achievable reduction at fixed accuracy (Delmas et al., 11 Sep 2025).

1. Definition in mass-weighted normal-mode space

Let $x(t)\in\mathbb{R}^{3N}$ denote the Cartesian nuclear coordinates at time $t$ after removing overall translation and rotation, and let $x_{\mathrm{ref}}$ denote the reference geometry, identified as the Franck–Condon geometry. The mass-weighted displacement is defined as

$\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$

where $M$ is the diagonal mass matrix (Delmas et al., 11 Sep 2025).

The normal modes are obtained by diagonalizing the Hessian at $x_{\mathrm{ref}}$, yielding the mass-weighted eigenvectors $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$, with each $\ell_i$ normalized in mass-weighted space. The normal-mode coordinate of mode $i$ at time $t$ is then

$t$0

The Normal Mode Variance for mode $t$1 is

$t$2

where $t$3 denotes an average over all sampled times and over all trajectories. NMV orders the modes by descending $t$4; modes with the smallest variances are treated as least essential and may be pruned (Delmas et al., 11 Sep 2025).

This definition makes NMV a variance-ranking procedure in the basis of reference-geometry normal modes rather than in a learned collective-coordinate basis. A plausible implication is that NMV preserves the interpretability of individual vibrational directions, because each retained coordinate remains tied to a specific normal mode rather than to a linear combination constructed from trajectory covariance.

2. Algorithmic workflow

The implemented NMV workflow begins with full-dimensional mixed quantum-classical TSH in Cartesian space, using $t$5 degrees of freedom including translations and rotations. Each saved geometry $t$6 is then aligned to $t$7 by Kabsch alignment to remove overall rotation and translation. After alignment, each geometry is projected onto the normal-mode basis through

$t$8

for all modes and all sampled times (Delmas et al., 11 Sep 2025).

The variances $t$9 are computed and sorted such that

$x_{\mathrm{ref}}$0

The retained dimensionality $x_{\mathrm{ref}}$1 is then chosen by one of two criteria. Under a variance cutoff, one selects the smallest $x_{\mathrm{ref}}$2 such that

$x_{\mathrm{ref}}$3

with the paper giving $x_{\mathrm{ref}}$4 as an example. Under a fixed-dimension strategy, $x_{\mathrm{ref}}$5 is specified directly, for example by scanning a sequence $x_{\mathrm{ref}}$6. A projection matrix $x_{\mathrm{ref}}$7 of size $x_{\mathrm{ref}}$8 is then constructed to retain only modes $x_{\mathrm{ref}}$9 (Delmas et al., 11 Sep 2025).

The workflow embodies an automated pruning criterion based entirely on trajectory variance. This suggests a systematic alternative to manual coordinate selection, which the study explicitly frames as a way of avoiding human bias in reduced-dimensional photochemical simulations (Delmas et al., 11 Sep 2025).

3. Embedding in Trajectory Surface Hopping

In the reduced-dimensional TSH implementation, all electronic-structure quantities remain computed on the fly in full $\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$0 Cartesian space. Only the nuclear propagation is confined to the retained $\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$1-dimensional subspace (Delmas et al., 11 Sep 2025).

Initial Wigner conditions $\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$2 are sampled, and positions and velocities are aligned and projected into the retained normal-mode subspace:

$\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$3

$\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$4

During each time step $\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$5, the protocol computes electronic state populations and the full Cartesian gradient $\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$6, transforms the gradient to normal-mode forces according to

$\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$7

projects those forces into the retained subspace with $\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$8, and back-transforms to Cartesian forces through

$\Delta x(t) = M^{1/2}[x(t)-x_{\mathrm{ref}}],$9

Newton’s equations are then integrated in Cartesian form using $M$0 with velocity-Verlet. Because $M$1 has no components in the discarded modes, the dynamics remain within the $M$2-mode subspace. Positions and velocities are updated, and the coordinates may be realigned and reprojected to enforce confinement to the retained subspace. Surface hops are treated with the Fewest-Switches algorithm, and after a hop the velocities are rescaled along the retained-mode directions. Decoherence corrections and electronic integrator substeps are unchanged relative to full-dimensional TSH (Delmas et al., 11 Sep 2025).

This protocol separates the electronic problem from the reduced nuclear subspace. A plausible implication is that NMV reduction, as implemented here, targets the cost and tractability of nuclear dynamics without redefining the electronic-structure evaluation itself.

4. Benchmark behavior in three photoreactive molecules

The method was evaluated on trans-azomethane (tAZM), butyrolactone (Bulac), and furanone (Fur). For each molecule, the study defined two electronic and two geometric descriptors, expressed as timescales and yields, and measured the absolute relative error against full-dimensional TSH as

$M$3

The reported performance establishes the extent to which NMV can reproduce full-dimensional dynamics in reduced spaces (Delmas et al., 11 Sep 2025).

Molecule	Full dimensionality	NMV result
trans-azomethane (tAZM)	$M$4 modes	ARE $M$5 only down to $M$6
butyrolactone (Bulac)	$M$7 modes	geometric accuracy lost at $M$8; fails rapidly for $M$9
furanone (Fur)	$x_{\mathrm{ref}}$0 modes	no consistent ARE $x_{\mathrm{ref}}$1 for any $x_{\mathrm{ref}}$2

For tAZM, the monitored descriptors were the $x_{\mathrm{ref}}$3 crossing time, final $x_{\mathrm{ref}}$4 population, half-cis time, and final cis yield. NMV maintained ARE below $x_{\mathrm{ref}}$5 only down to $x_{\mathrm{ref}}$6 modes, while at $x_{\mathrm{ref}}$7 the electronic errors rose above $x_{\mathrm{ref}}$8. The recommended NMV reduction was therefore 18 modes (Delmas et al., 11 Sep 2025).

For Bulac, the descriptors were the $x_{\mathrm{ref}}$9 inflection time, final $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$0 population, half-bond-break time, and final C–O break yield. NMV already lost geometric accuracy at $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$1, where modest bond re-formation appeared, and it failed rapidly for $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$2. The recommended NMV reduction was 24 modes out of a full space of 30 (Delmas et al., 11 Sep 2025).

For Fur, the descriptors were the half-$L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$3 decay time, final $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$4 population, half-bond-break time, and final C–O break yield. NMV did not achieve consistent ARE below $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$5 for any $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$6, and the competition with ring-puckering was reported as washed out in reduced NMV spaces. No reduced-dimensional NMV recommendation was made for Fur (Delmas et al., 11 Sep 2025).

These results indicate that NMV can be adequate for moderate reductions in some systems, but its reliability is system-dependent and especially sensitive where low-variance motions remain mechanistically important.

5. Relation to PCA

The study compared NMV directly with PCA across all three benchmark systems and found PCA superior in each case. PCA retained all four descriptors for tAZM with ARE below $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$7 down to $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$8 modes, kept Bulac below the same threshold down to $L=[\ell_1,\ell_2,\dots,\ell_{3N-6}]$9 modes, and maintained geometric ARE below $\ell_i$0 for Fur down to $\ell_i$1, with electronic ARE remaining small for all $\ell_i$2 (Delmas et al., 11 Sep 2025).

Several contrasts were identified. First, PCA captured a given fraction of total variance with fewer dimensions than NMV; for tAZM at $\ell_i$3, PCA retained approximately $\ell_i$4 of the variance, whereas NMV retained approximately $\ell_i$5. Second, PCA errors remained below the $\ell_i$6 threshold at lower retained variances than NMV in all three systems. Third, NMV sometimes discarded low-variance but mechanistically critical modes, with torsion-bending in tAZM given as an example. By contrast, PCA constructs linear combinations of all modes and thereby retains subtler couplings with fewer dimensions. Finally, NMV was characterized as conceptually simpler, because it only sorts and prunes by $\ell_i$7, whereas PCA requires diagonalizing the covariance of all trajectory displacements (Delmas et al., 11 Sep 2025).

The comparison highlights a central limitation of variance ranking in a fixed normal-mode basis: small amplitude does not necessarily imply dynamical irrelevance. This suggests that NMV is most appropriate when the reaction coordinate is strongly aligned with a subset of high-variance normal modes, and less appropriate when essential motion is distributed across coupled or individually low-variance directions.

6. Interpretation, scope, and limitations

NMV functions as a one-shot variance criterion over reference-geometry normal modes. Its principal strength is procedural simplicity: the method requires normal-mode analysis at the reference geometry, projection of trajectory data, variance estimation, and pruning by descending $\ell_i$8. This yields a transparent hierarchy of retained and discarded motions, and the study identifies this simplicity explicitly as an ease-of-use advantage relative to PCA (Delmas et al., 11 Sep 2025).

Its principal limitation is that the criterion neglects mode coupling. The paper states that NMV can remove motion essential to the reaction coordinate, even when that motion has low variance. The observed failures in tAZM, Bulac, and Fur provide concrete manifestations of this issue: electronic errors can rise sharply once specific modes are removed, geometric mechanisms may be altered through bond re-formation, and competing structural pathways such as ring-puckering may be suppressed inappropriately (Delmas et al., 11 Sep 2025).

A common misconception would be to treat variance retention alone as a sufficient indicator of dynamical fidelity. The reported comparison rejects that equivalence. Although NMV is built from physically interpretable normal modes, mechanistic relevance is not guaranteed by amplitude statistics alone. In this context, low-variance coordinates can still control nonadiabatic transitions, product branching, or barrier-crossing topology.

Within the scope of the reported work, NMV remains a viable automated filter for “quiet” modes, but not a generally optimal one. The broader significance of the study is the demonstration that automated dimensionality reduction can reproduce full-dimensional TSH in lower-dimensional spaces while avoiding human bias; however, the evidence in the same benchmarks indicates that PCA provides a more robust balance between dimensionality reduction and accuracy than NMV (Delmas et al., 11 Sep 2025).