Force-Isosurface Simulations Probe the Limits of High-Resolution AFM on Three-Dimensional Molecules

Edward J Dunn; Robert J Young; Samuel P Jarvis

REVIEW 3 major objections 6 minor 22 references

Force-Isosurface Simulations Probe the Limits of High-Resolution AFM on Three-Dimensional Molecules

T0 review · 3 major / 6 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Constant-force AFM recovers 3D molecular structure lost in conventional imaging

desk verdict Solid simulation study with a real but unevenly distributed electrostatics gap; deserves a serious referee read the letter →

arxiv 2607.07363 v1 pith:DRUKFASS submitted 2026-07-08 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.soft

Edward J Dunn , Robert J Young , Samuel P Jarvis This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.soft

keywords molecularforcethree-dimensionalacrossconstant-forceforce-isosurfaceinformationmolecules

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that extracting a force isosurface — the set of points in space where the tip–sample interaction force equals a chosen threshold — from a simulated 3D force field above a molecule preserves structural information that conventional constant-height imaging loses when the molecule is non-planar. In constant-height AFM, the tip scans a fixed geometric plane; for tilted or 3D molecules, features lying below that plane vanish from the image while features above it may destabilise the tip. The force isosurface instead lets the tip rise and fall to follow the molecular force field, maintaining contact with both high and low parts of the structure simultaneously. Using probe-particle-model simulations with Lennard–Jones interactions, the authors show that for tilted benzene and pyrrole the full molecular ring remains visible across tilt angles from 0 to 30 degrees and the tilt angle can be recovered to within 1 degree by fitting a plane to the isosurface. For larger non-planar systems — two adsorption geometries of 2H-TPP, ZnTPP, CO-FePc, and C60 — the force isosurface distinguishes conformations, reveals lower-lying structural features obscured by protruding ligands, and resolves contrast across curved molecular surfaces. The paper provides simulated target contrasts that experimentalists can compare against when performing 3D force-mapping or force-feedback measurements.

What carries the argument

The Probe Particle Model simulates a flexible CO-like tip apex interacting with molecular coordinates through Lennard–Jones potentials. A 3D force field is computed on a grid above each molecule. A custom extraction algorithm searches each vertical column from large tip–sample separation toward the molecule and records the highest z-position where the force threshold (typically 250 pN) is exceeded, producing the outermost force isosurface. Plane fitting to selected regions of this isosurface recovers molecular tilt angles.

What would settle it

If adding electrostatic interactions to the probe-particle-model simulations substantially changes the shape of the force isosurface for pyrrole, ZnTPP, or CO-FePc — for instance by shifting the contour above polar regions enough to obscure the lower-lying features the paper claims are revealed — then the predicted contrasts and tilt-angle recovery would not hold for these molecules in real experiments.

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Extended reading notes

Core claim

The central object is the force isosurface: the height contour defined not by a fixed plane but by a constant force threshold in the 3D tip–sample interaction field. The paper demonstrates that this isosurface retains submolecular and three-dimensional structural information — molecular rings, tilt angles, central metal atoms, lower-lying cores beneath protruding ligands, and adjacent faces on curved molecules — that is progressively lost in constant-height images as molecular non-planarity increases. For simple tilted fragments the isosurface recovers orientation to within 1 degree; for complex molecules it distinguishes adsorption geometries and reveals features otherwise hidden, though it

Load-bearing premise

The simulations include only Lennard–Jones (van der Waals and Pauli repulsion) interactions and omit electrostatic forces entirely. Several molecules studied — pyrrole with its polar N–H bond, ZnTPP with a central metal ion, and CO-FePc with a CO ligand carrying a significant dipole — have electrostatic contributions that could reshape the force isosurface and alter the specific contrasts and quantitative angle recovery reported.

Editorial extensions

If this is right

Experimental 3D force-mapping protocols can use these simulated isosurfaces as reference templates to verify that measured force contours contain genuine structural information rather than artefacts of tip relaxation or feedback instability.
Force-feedback AFM modes (PeakForce, off-resonance) that already acquire force curves during imaging could extract isosurface contrast directly, potentially bringing 3D molecular structure determination to ambient or room-temperature conditions without requiring full UHV frequency-shift tomography.
For molecules with protruding functional groups — axial ligands, metal centres, or adsorbed species — force-isosurface imaging could simultaneously resolve the protrusion and the underlying molecular framework, which is structurally important for understanding catalytic sites or on-surface reaction intermediates.
The systematic underestimation of tilt angles in the saddle geometry of 2H-TPP demonstrates that neighbouring molecular features distort the local isosurface, which means quantitative structural recovery in complex molecules will require simulation-guided interpretation rather than direct geometric reading of the contour.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main thing to know: this paper systematically shows that constant-force AFM imaging (via force isosurfaces extracted from simulated 3D force fields) retains structural information that constant-height imaging loses on non-planar molecules. The quantitative tilt-angle recovery for benzene and pyrrole (within 1°, Figure 3) is clean and convincing. The progression from simple tilted molecules to 2H-TPP, ZnTPP, CO-FePc, and C60 is well-designed and gives the AFM community useful target contrasts for force-feedback experiments. The honest reporting of the 5° systematic underestimation in the 2H-TPP saddle geometry is a good sign — they don't paper over it. The CO-FePc result, where constant-force imaging recovers the phthalocyanine core hidden behind the protruding CO ligand, is a genuinely nice demonstration. What is new here relative to Schuler et al. [10] and Hapala et al. [6] is the systematic head-to-head comparison of constant-height vs constant-force across a series of molecules with increasing non-planarity, plus the quantitative angle recovery from force isosurfaces. That is a real contribution. The soft spot is the omission of electrostatics, and the reader is right to flag it. But the severity is uneven. For benzene and pyrrole at 250 pN (repulsive regime, ~2.5–3 Å), electrostatic contributions from a CO-terminated probe are small perturbations to the Pauli/LJ-dominated force. The 1° tilt recovery is probably robust. The concern lands harder on ZnTPP and CO-FePc, where the distinguishing features involve charged metal centers and polar ligands. A rough estimate puts electrostatic forces at 50–200 pN at these distances — not negligible relative to 250 pN. The qualitative claims (Zn produces a raised central feature; constant-force reveals the FePc core) are likely directionally correct because they rest on geometry, not electrostatics. But the specific contrast patterns — the cross-like structure around Zn, the depressions in the isoindole sections — could shift if electrostatics were included. The paper provides no sensitivity analysis or bound on this. Minor issues: no code or data shipped, and the C60 results are purely qualitative with no quantitative metrics. Neither is fatal for a simulation study of this type, but shipping the extraction code would be straightforward and would help. Overall: the core findings hold within the scope of what is simulated. The paper is written for AFM experimentalists who need predicted contrasts for force-mapping experiments, and it delivers that. It deserves a serious referee. The referee should push the authors to include electrostatics for at least one molecule (ZnTPP is the obvious candidate) and to discuss how it would affect the qualitative contrasts they predict.

Referee Report

3 major / 6 minor

Summary. This manuscript uses Probe Particle Model (PPM) simulations to extract force isosurfaces from 3D force fields above molecules of increasing structural complexity, from tilted benzene and pyrrole to 2H-TPP, ZnTPP, CO-FePc, and C60. The central claim is that constant-force (force-isosurface) imaging retains submolecular and 3D structural information that is lost in constant-height imaging of non-planar molecules. For tilted benzene and pyrrole, the authors show quantitative recovery of tilt angles to within 1 degree. For larger systems, they identify characteristic contrast patterns associated with different adsorption geometries, central metal features, and curved molecular surfaces. The progression from simple to complex systems is well-designed, and the honest reporting of the 5-degree systematic underestimation for the 2H-TPP saddle geometry is commendable.

Significance. The paper provides falsifiable, molecule-specific target contrasts for force-isosurface imaging that could guide future 3D force-mapping experiments. The quantitative tilt-angle recovery for benzene and pyrrole (Figure 3, within 1 degree) is a clean validation result. The systematic comparison of constant-height versus constant-force imaging across a range of molecular complexities is a useful contribution to the field. The simulation methodology is standard and sound, and the parameter choices (250 pN threshold, 4 N/m stiffness, CO-like probe) are drawn from established PPM practice. The paper does not ship reproducible code or machine-checked proofs, but the methodology is sufficiently described for replication.

major comments (3)

§'Simulating constant-force images': The omission of electrostatic interactions is stated but its implications are not analyzed. This is a load-bearing simplification for the ZnTPP (Figure 5) and CO-FePc (Figure 6) results, where the distinguishing features involve charged or polar species (Zn2+ center, CO ligand dipole). At the ~2.5-3 Å probe-sample separations corresponding to 250 pN repulsion, electrostatic forces from a CO-terminated tip interacting with a +1e metal center are plausibly on the order of 50-200 pN, which is not negligible relative to the 250 pN threshold. The paper should either (a) provide a sensitivity analysis or order-of-magnitude bound on how electrostatics would modify the predicted contrast for ZnTPP and CO-FePc, or (b) explicitly scope the claims for these two systems as purely LJ-based predictions that may not hold quantitatively in experiment. The benzene/pyr
§'Differentiating between adsorption geometries of 2H-TPP', Figure 3c,d: The 5-degree systematic underestimation of the phenyl-ring angle for the saddle geometry is attributed to 'the influence of the surrounding molecular structure on the local force isosurface.' This attribution is plausible but not demonstrated. A simple test would be to fit the plane to a smaller or shifted region of the phenyl-ring isosurface to check whether the shoulder from the macrocycle is the actual cause, or whether the discrepancy arises from the plane-fitting procedure itself. Without this, the reader cannot assess whether the 5-degree error is a fundamental limitation of force-isosurface imaging for crowded molecules or an artifact of the fitting protocol. This matters because the paper's central quantitative claim — that molecular orientation can be recovered from force isosurfaces — is directly tested by
§'Metal-centred porphyrins and phthalocyanines', Figure 5: The claim that the Zn atom 'produces a raised central feature' with 'contrast extending towards the four surrounding nitrogen atoms' is presented without comparison to a 2H-TPP constant-force image at the same setpoint. The text references Albrecht et al. [12] showing that CuTPP produces a filled centre compared to 2H-TPP, but the reader cannot visually confirm this distinction from the figures provided, since Figure 4 (2H-TPP) and Figure 5 (ZnTPP) use different colour scales and height ranges. A direct side-by-side comparison with matched scales would substantially strengthen the claim that the central feature is attributable to the metal atom rather than to the saddle geometry itself.

minor comments (6)

Figure 2: The line profiles are labelled with force values in pN but the relationship between the profile positions and the molecular structure is not always clear. Adding molecular skeleton overlays or tick marks at atom positions would help readers interpret the profiles.
Figure 3c,d: The 2H-TPP data points are plotted alongside benzene and pyrrole but use different marker shapes (diamonds vs hexagons/pentagons). A legend entry or caption note clarifying which diamond colour corresponds to which geometry would help readers interpret the plot.
§'Differentiating between adsorption geometries of 2H-TPP': The phenyl-ring angles for the inverted geometry are reported as 3.9, 3.9, 3.8, and 3.9 degrees with error ±0.5 degrees. It is unclear whether this error is a fitting uncertainty, an RMS residual, or something else. This should be specified.
§'Metal-centred porphyrins and phthalocyanines': The C60 colour scales in Figure 7 are described as 'focused on the uppermost face' and 'adjusted to lower heights,' but the specific height ranges used for each panel are not stated. This should be specified.
The paper would benefit from a brief discussion of the experimental accessibility of the simulated force thresholds. The 250 pN setpoint is within the range of PeakForce-style methods, but it would help to note whether such thresholds are achievable under typical UHV NC-AFM conditions.
Reference [18] is cited for both the CO-FePc experimental work (Chen et al.) and the PPM methodology description. These appear to be different works and should be disambiguated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the main contributions of the paper and raises three substantive points regarding (1) the omission of electrostatic interactions, (2) the untested origin of the 5-degree underestimation in the 2H-TPP saddle geometry, and (3) the lack of a matched-scale comparison between ZnTPP and 2H-TPP. We address each point below.

read point-by-point responses

Referee: Omission of electrostatic interactions not analyzed, especially for ZnTPP and CO-FePc where charged/polar species are involved.

Authors: The referee is correct that electrostatic forces are not negligible at the probe-sample separations corresponding to 250 pN repulsion, and we should have been more explicit about the scope and limitations this introduces. We agree with option (b): the claims for ZnTPP and CO-FePc should be explicitly scoped as Lennard-Jones-based predictions. We will add a paragraph to the methodology section stating that electrostatic contributions from the Zn2+ centre and CO ligand dipole are expected to modify the predicted contrast at the ~2.5-3 Å separations involved, and that the ZnTPP and CO-FePc results should be interpreted as purely LJ-based predictions that identify which structural features are expected to dominate the short-range force contrast. We will also add an order-of-magnitude estimate of the electrostatic force from a CO-terminated tip interacting with a +1e metal centre at these separations to make clear that this is a non-negligible effect. The benzene, pyrrole, 2H-TPP, and C60 results, which do not involve charged or strongly polar species, are less affected by this omission. revision: yes
Referee: The 5-degree underestimation for 2H-TPP saddle geometry is attributed to surrounding molecular structure but not demonstrated; a test fitting to a smaller or shifted region is needed.

Authors: The referee makes a fair point: our attribution of the 5-degree underestimation to the shoulder from the macrocycle is plausible but untested. We will perform the suggested test by fitting the plane to smaller and shifted regions of the phenyl-ring isosurface, specifically excluding the region where the macrocycle shoulder is expected to contribute. This will allow us to determine whether the discrepancy is reduced when the neighbouring structure is excluded, or whether it arises from the plane-fitting procedure itself. We will report the results of this test in the revised manuscript, either confirming or revising our attribution. If the discrepancy persists with reduced fitting regions, we will reframe the discussion to acknowledge that the origin is not fully established. revision: yes
Referee: ZnTPP central feature claim lacks a side-by-side comparison with 2H-TPP at the same setpoint and matched colour scales.

Authors: This is a reasonable request and we will implement it. We will produce a side-by-side comparison of the 2H-TPP and ZnTPP constant-force images at the same 250 pN setpoint with matched colour scales and height ranges, so that the reader can directly assess whether the central raised feature in ZnTPP is attributable to the metal atom rather than to the saddle geometry itself. This comparison will be added as a new figure or panel in the revised manuscript. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found; the paper is a forward-looking simulation study with transparent validation methodology.

full rationale

The paper simulates force isosurfaces using the Probe Particle Model (PPM, cited to external authors Hapala et al. [6]) with parameters (250 pN, 4 N/m) selected from a parameter scan (Figure S1), not fitted to target results. The tilt-angle recovery for benzene and pyrrole (Figure 3) is an explicit validation test: known tilt angles are inputs to the simulation, and the paper checks whether the isosurface preserves them. This is standard methodology, not circular reasoning. The 2H-TPP coordinates come from self-citation [13] (Jarvis et al., 2015), but these are structural inputs — the force isosurfaces are computed from them via PPM and are not equivalent to the coordinates by construction. The Sader-Jarvis formalism [22] is mentioned only as an experimental force-conversion method and is not load-bearing for any simulation result. The CO-FePc coordinates come from external authors [18]. No step in the derivation chain reduces to its own inputs by definition, no uniqueness theorem is invoked, and no ansatz is smuggled through self-citation. The one minor self-citation [13] provides independent structural data and does not raise circularity concerns.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

No new entities are postulated. The paper uses established molecular structures and an established simulation model.

free parameters (3)

Force threshold = 250 pN
Selected from a parameter scan (Figure S1) as lying within a stable range while retaining high-resolution contrast. Not fitted to reproduce target results, but chosen by the authors.
Lateral probe-particle stiffness = 4 N/m
Standard PPM parameter, selected from Figure S1 parameter scan.
Probe particle type = CO-like
Standard choice for high-resolution AFM simulations, representing a CO-functionalized tip.

assumptions (3)

domain assumption Lennard-Jones potentials alone adequately represent the short-range tip-sample force for the purpose of comparing constant-height and constant-force imaging contrast.
Stated in 'Simulating constant-force images': 'Electrostatic interactions were omitted, so that the comparison was based on the short-range force contrast arising from the molecular structures.' This assumption underlies all results but is not tested.
domain assumption The Probe Particle Model with a CO-like probe and 4 N/m stiffness accurately reproduces the force fields above these molecules.
The PPM is an established model (Hapala et al. 2014), but its accuracy for strongly non-planar molecules like C60 or molecules with metal centers like ZnTPP is not independently validated in this paper.
domain assumption The force isosurface extraction algorithm (searching from largest separation toward the molecule for the highest z where the force threshold is exceeded) produces a unique and physically meaningful surface.
Described in 'Simulating constant-force images'. For molecules with complex force fields, multiple crossings of the threshold could exist, but this is not discussed.

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Cite this review

Pith. "Pith review of Force-Isosurface Simulations Probe the Limits of High-Resolution AFM on Three-Dimensional Molecules." pith.science (2026). https://pith.science/paper/DRUKFASS

@misc{pith2026260707363,
  author       = {Pith},
  title        = {Pith review of: Force-Isosurface Simulations Probe the Limits of High-Resolution AFM on Three-Dimensional Molecules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRUKFASS}},
  note         = {Machine review of arXiv:2607.07363}
}

read the original abstract

High-resolution atomic force microscopy has transformed molecular imaging by revealing intramolecular structure directly in real space. A major remaining challenge is to extend this capability from largely planar molecules to non-planar molecular systems, where the most important structural information may be distributed across different heights above the surface. Here we use probe-particle-model simulations to predict the constant-force contours expected above molecules with increasing structural complexity. By extracting force isosurfaces from simulated three-dimensional force fields, we compare the molecular information retained in constant-height and constant-force images. For tilted benzene and pyrrole, constant-force images preserve the molecular framework across a range of adsorption angles and allow the molecular orientation to be recovered quantitatively. For larger non-planar and three-dimensional systems, simulations identify characteristic force-isosurface contrast associated with adsorption geometry, lower-lying molecular structure and curved molecular surfaces. These results provide target contrasts for force isosurfaces that could be extracted from three-dimensional force-mapping experiments, evaluating the molecular information retained by ideal force-isosurface imaging across progressively non-planar systems.

Figures

Figures reproduced from arXiv: 2607.07363 by the authors.

**Figure 1.** A schematic comparison of constant-height and constant-force imaging in an arbitrary force field. The upper panels show the force experienced by a tip following constant-height (top left) and constant-force (top centre) trajectories, with the corresponding paths shown on force contour maps below. The right panel illustrates the extraction of a constant-force image from a three-dimensional force field. At each x-y po… view at source ↗

**Figure 2.** Simulated constant-height and constant-force images of benzene tilted by 0◦ , 10◦ , 20◦ and 30◦ , shown in rows 1–4, respectively. Line profiles show the dependence of the apparent features on the selected height or force threshold. Constant-height profiles were calculated at height increments of 0.1 ˚A, while constant-force profiles were calculated at 10, 20, 50, 75, 100 and 250 pN. Pale regions of the profiles ind… view at source ↗

**Figure 3.** Plane fitting of simulated force isosurfaces for benzene, pyrrole and 2H-TPP. A plane was fitted to a selected region of each force isosurface to determine the molecular tilt angle. Panels (a) and (b) show the fit for benzene tilted by 30◦ , with the grey points representing the simulated force isosurface and the red points representing the fitted plane. The measured tilt angles for benzene (black hexagons) and py… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Constant-force images of 2H-TPP in the ‘inverted’ (a) and ‘saddle’ (c) adsorption geometries. In the inverted geometry, the pyrrole-like units at the centre of the molecule protrude strongly from an otherwise near-planar structure. Plane fitting to the phenyl-ring regi…

**Figure 5.** Figure 5: Constant-force image of ZnTPP in the ‘saddle’ adsorption geometry. The zinc atom occupies the centre of the porphyrin macrocycle and produces a raised cross-like feature extending towards the surrounding nitrogen atoms. nine (CO-FePc). FePc is a relatively planar mol…

**Figure 6.** Figure 6: Simulated constant-height and constantforce images of CO-FePc. In the constant-height image simulated at 300 pm (b), the contrast is dominated by the protruding CO ligand and the FePc core is only weakly resolved. The corresponding force profile (a) shows the larger…

**Figure 7.** Figure 7: Simulated constant-force images of C60 in hexagon-up and pentagon-up orientations. With the colour scale focused on the uppermost hexagonal face (a), the top face is clearly resolved. Adjusting the colour scale to lower heights reveals the six neighbouring faces, wit…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.

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Adsorption ge- ometry determination of single molecules by atomic force microscopy

Schuler B, Liu W, Tkatchenko A, Moll N, Meyer G, Mistry A, et al. Adsorption ge- ometry determination of single molecules by atomic force microscopy. Phys Rev Lett. 2013;111(10):106103

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Hapala P, Kichin G, Wagner C, Tautz FS, Temirov R, Jel´ ınek P. Mechanism of high-resolution STM/AFM imaging with functionalized tips. Phys Rev B. 2014 Aug;90:085421

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Direct Identification and De- termination of Conformational Response in Adsorbed Individual Nonplanar Molecular Species Using Noncontact Atomic Force Mi- croscopy

Albrecht F, Bischoff F, Auw¨ arter W, Barth JV, Repp J. Direct Identification and De- termination of Conformational Response in Adsorbed Individual Nonplanar Molecular Species Using Noncontact Atomic Force Mi- croscopy. Nano Lett. 2016;16:7703-9

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High resolution SPM imag- ing of organic molecules with function- alized tips

Jel´ ınek P. High resolution SPM imag- ing of organic molecules with function- alized tips. J Phys: Condens Matter. 2017;29(34):343002

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Atomic Force Mi- croscopy for Molecular Structure Elucida- tion

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