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REVIEW 4 major objections 6 minor 36 references

DFT-based energy shifts screening of Na$_x$K$_{55-x}$ alloy clusters

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that in 55-atom sodium–potassium alloy clusters, atoms with fewer neighbors bind more tightly, and that this local bonding picture can be read off from computed core-level binding-energy shifts.

desk verdict Useful new DFT screening data for alkali nanoalloys, wrapped in a BOLS analysis that reaches beyond what the method supports. read the letter →

arxiv 1908.02421 v1 pith:5O3UO27L submitted 2019-08-07 cond-mat.mtrl-sci

Jing Zhao , Maolin Bo This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords sodium-potassiumalloyclustersbindingenergyshiftdensityfunctionaltheoryBOLSnotationcoordinationnumberbondsurfacesegregationnanoalloystability
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

This paper asks what controls the binding energy in 55-atom sodium–potassium alloy clusters, Na$_x$K$_{55-x}$, and answers that it is the arrangement of the atoms, not just overall composition: atoms sitting in the surface shell bind more tightly than atoms inside, because surface atoms have fewer neighbours and therefore shorter, stronger bonds. Using DFT total-energy calculations on global-minimum geometries and connecting the resulting core-level energy shifts to coordination numbers through BOLS (bond-order–length–strength) notation, the authors identify which compositions place Na in bulk or shell sites and extract per-atom bond energy ratios, bond energy densities, and local bond strains. They report that lower-coordinated atoms have higher binding energies and bond energy densities, and that their calculated Na $2p$ and K $3p$ shifts track available XPS measurements. If this screening picture is right, the same DFT-plus-BOLS recipe can rank alloy compositions by stability and provide a practical guide for preparing NaK nanoalloys with wanted surface/bulk ordering.

What carries the argument

The central object is the BOLS correlation of Eq. (2), which ties a core-level binding-energy shift $\Delta E'_v(i)$ of level $v$ at site $i$ to the site coordination number $z$ through $\Delta E'_v(i)=\Delta E_v(z)-\Delta E_v(12)$ and an inversion formula that uses bulk shift constants $\Delta E_v(12)$ taken from Na(110) and K(110) XPS measurements (2.401 eV for Na, 2.754 eV for K). This object converts the DFT-computed Kohn–Sham eigenvalue shifts into per-atom coordination numbers, and Eq. (3) then converts those numbers into the bond energy ratio $\gamma$, the bond energy density $\delta E_d$, and the local bond strain $-\varepsilon_z$. The load-bearing step is treating the DFT eigenvalue shifts as measurable core-level binding-energy shifts, and using bulk surface constants inside 55-atom alloy clusters.

What would settle it

Run explicit core-hole ($Z+1$) DFT on Na55, K55, Na7K48, and Na15K40 and compare the core-level shifts with the plain DFT eigenvalue shifts used here; if the ordering or magnitudes differ substantially, the BOLS conversion needs reparameterization. Alternatively, measure XPS of size-selected NaxK55-x clusters and check whether the predicted coordination numbers match the known structural coordination numbers from the global-minimum geometries.

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Extended reading notes

Core claim

The paper's central claim is that compositional effects in NaxK55-x nanoclusters are principally determined by the atomic arrangement within the structure, and that atoms with lower coordination numbers have higher binding energies and bond energy densities. Concretely, the DFT+BOLS analysis resolves each cluster into bulk, S2, and S1 components, and finds that the more negative (higher-magnitude) core-level binding energies always belong to the lower-coordinated surface components: in Na55 the Na $2p$ level peaks at –24.839 eV (bulk), –25.034 eV (S2), and –25.217 eV (S1), with the same ordering for K $3p$ in K55 and in the alloys. Extracted coordination numbers fall from $z=12$ in the bulk to roughly $z=3$–$7$ at the surface, while the bond energy ratio rises to 1.05–1.20 and the local bond strain to 5–17 percent. The composition scan also reveals a clear structural crossover: in K-rich clusters (x = 1–26) Na atoms occupy bulk layers; for x = 27–42 Na moves into the shell; and in Na-rich clusters K atoms sit on the shell with no inward segregation. The largest Na $2p$ binding energy occurs at Na4K51 (–25.228 eV) and the smallest at Na15K40 (–24.753 eV); K $3p$ extrema occur at Na39K16 (–15.803 eV) and Na13K42 (–15.523 eV).

Load-bearing premise

The load-bearing assumption is that bulk-surface XPS shift constants for Na(110) and K(110) remain valid inside a 55-atom mixed alloy cluster, and that DFT Kohn–Sham eigenvalue shifts are identical to the measured core-level binding-energy shifts; if either part fails, the derived coordination numbers and bond parameters in Table 2 are not reliable.

Editorial extensions

If this is right

  • For K-rich compositions (x = 1–26), Na atoms sit in the bulk atomic layers; for x = 27–42 they move into the shell; and for Na-rich compositions the minority K atoms sit on the shell with no inward segregation, so composition alone predicts which species will be surface-enriched.
  • Surface and shell atoms always bind more tightly than bulk atoms, meaning the alloy's core-level spectrum carries direct information about its chemical ordering pattern (bulk–shell, onion, or random).
  • The calculated Na $2p$ and K $3p$ shifts track the XPS measurements and the excess-energy trends of the reference global-minimum structures, so the screening can be used to rank candidate stoichiometries before synthesis.
  • The lower the coordinated atom, the higher its binding energy and bond energy density, giving a simple local rule that can be checked atom-by-atom in any proposed alloy geometry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to invert the procedure: given an experimental XPS spectrum of an unknown NaK nanoalloy, the BOLS formula could be used to infer the distribution of coordination numbers and hence the segregation pattern, turning the screening into a spectroscopy-based structure probe.
  • The same DFT+BOLS pipeline could be applied to other simple-metal alloys (for example Li–Na, Na–Rb, or K–Cs) to test whether the 'low coordination binds tighter' ordering and the composition-dependent surface/bulk crossover are universal.
  • The identification of DFT eigenvalue shifts with measured XPS binding-energy shifts is the main point a skeptical reader would probe; a direct test would be a core-hole ($Z+1$) calculation, which the paper does not include.
  • If the trend holds, the extreme binding energies at specific stoichiometries (Na4K51, Na15K40, Na39K16, Na13K42) could serve as composition markers in co-deposition experiments, where XPS peak positions would signal which stoichiometry has formed.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper reports DFT calculations for NaxK55-x alloy clusters (x=0–55) using VASP with PBE functionals, with starting structures taken from Aguado's Gupta-potential global minima. The authors extract Na 2p and K 3p 'binding energies' (Kohn-Sham eigenvalues) for each composition and, for four selected clusters (Na55, K55, Na7K48, Na15K40), decompose the DOS into bulk and shell components. Using BOLS theory, they convert the core-level shifts into coordination numbers, bond energy ratios, bond energy densities, and local strains (Table 2), concluding that compositional effects are determined by atomic arrangement, with lower-coordinated surface atoms having higher binding energies and bond energy densities.

Significance. The paper's strength is the systematic DFT screening of all 56 compositions and the clear shell-resolved LDOS, which qualitatively show that surface atoms have deeper core levels than bulk atoms. If the BOLS-based conversion were validated, the approach would offer a low-cost screening rule for chemical ordering in alkali alloy clusters and a direct connection to XPS. However, the central quantitative bridge—identifying Kohn-Sham eigenvalue shifts with XPS binding-energy shifts and applying bulk surface constants to finite clusters—is not justified, and Table 2 contains an internal inconsistency that undercuts the headline monotonic claim. The result is therefore a plausible but not yet established screening tool.

major comments (4)
  1. [Sec. 3.3, Table 2] The row for Na15K40(K) S1 lists z=3.23, γ=1.202, and δEd=108.92%. Using Eq. (3) with the bond contraction relation of Eq. (1) (m=1) gives δEd = γ C^{-3} ≈ 1.202 × (1/0.832)^3 ≈ 209%, not 108.92%. The printed value also violates the paper's stated monotonic trend, since the S2 row with higher coordination (z=4.63) has δEd=151.82%, larger than the S1 value. Because the central conclusion rests on this table, the error must be corrected and the monotonic claim re-examined.
  2. [Secs. 2.1 and 3.1] The manuscript does not state how the Na 2p and K 3p binding energies are extracted from VASP: which eigenvalues are used, how the DOS peak positions are defined, and what energy reference (e.g., vacuum level in the supercell) is applied. No convergence tests are reported for the 400 eV cutoff, Gamma-only sampling, or the 15 Å vacuum spacer. These omissions make the numerical shifts in Table 1 and Fig. 2 irreproducible; at minimum, a brief description of the extraction procedure and a convergence check for one cluster are needed.
  3. [Sec. 3.3, Eq. (2)] The conversion of DFT eigenvalue shifts into coordination numbers via Eq. (2) relies on two unverified assumptions: (i) Kohn-Sham core-level eigenvalue differences are quantitatively equal to XPS binding-energy shifts, including final-state effects; and (ii) the bulk shift constants ΔEv(12)=2.401 eV (Na) and 2.754 eV (K) from Na(110) and K(110) XPS measurements transfer unchanged to finite mixed clusters. The derived z, γ, δEd, and εz in Table 2 are therefore entirely model-dependent. The authors should validate at least one case by comparing derived coordination numbers with the known shell structure of the DFT-optimized clusters.
  4. [Secs. 3.2 and 4] The statement that the DFT results 'are consistent with the XPS data reported by Tchaplyguine et al.' is not backed by a quantitative comparison or a discussion of reference alignment (Fermi vs vacuum, initial vs final state). The absolute eigenvalue positions (e.g., Na 2p at −25.034 eV for Na55) are not directly comparable to measured binding energies without such corrections. The concluding claim that compositional effects are 'principally determined by the arrangement of the different atoms within its structure' is also stronger than the evidence, because the detailed shell-resolved analysis is limited to four clusters.
minor comments (6)
  1. [Table 1] The Na 2p binding energy for Na36K19 is listed as '24.877' without the minus sign used for all other entries.
  2. [Table 2] The cluster label 'Na15K4' should be 'Na15K40'.
  3. [Fig. 4] Panel (c) uses the label 'Core' for the bulk-like component, whereas other panels use 'B'; the notation should be made uniform.
  4. [Eq. (2)] Equation (2) is very hard to read as typeset; please ensure all subscripts, exponents, and the 'ln' argument are clearly rendered.
  5. [Fig. 5] The axis label 'Relative change(%)' is ambiguous; please specify which quantity is plotted on the x-axis and what the relative change refers to.
  6. [Table 1] The heading 'Energy (eV)' should state that this is the DFT total energy without entropy, not a binding energy.

Circularity Check

1 steps flagged · score 4.0 of 10

BOLS coordination numbers are one-to-one transforms of the DFT shifts, making the low-coordination/high-binding-energy conclusion definitional; the underlying DFT energies remain non-circular.

  1. self definitional [Sec. 3.3, Eq. (2), Eq. (3), and Table 2]
    "Eq. (2) can be used to calculate the coordination number, z, of an atom from its binding energy. The results of such calculations are shown in Table 2 for selected NaxK55–x alloy nanoclusters. Calculations carried out using Eq. (3) reveal that atoms with lower coordination numbers have higher binding energies and bond energy densities."

    The coordination number z is not taken from the DFT geometry; it is computed by inverting Eq. (2), which monotonically maps each DFT binding-energy shift ΔE'_v(z) to z using the bulk constants ΔEv(12) from ref. [30]. Hence 'lower z' and 'larger ΔE'_v' are the same variable expressed two ways, so the Sec. 3.3 finding that lower-coordinated atoms have higher binding energies is a restatement of the inversion formula, not an independent result. Table 2 confirms this: the Na S1 sites in Na7K48 and Na15K40 have identical ΔE'_v=0.122 eV and identical z=6.76, as a one-to-one transform requires. The γ, δEd, and εz values follow algebraically from z via Eq. (3) and inherit the same definitional status.

full rationale

The genuinely first-principles part of the paper is the DFT calculation of Na 2p and K 3p eigenvalue shifts and shell-resolved LDOS for NaxK55-x clusters (Sec. 3.1-3.2). Those shifts are computed from optimized Gupta-potential structures and are compared with external XPS data from Tchaplyguine et al. and with Aguado et al. trends; that part is self-contained and not circular. The circularity is confined to the BOLS interpretation: Eq. (2) assigns each DFT shift a coordination number z using bulk Na(110)/K(110) XPS constants, and Eq. (3) then converts z into bond-energy ratios, densities, and strains. Since z, γ, δEd, and εz in Table 2 are monotonic algebraic transforms of the same input binding-energy shifts, the paper's 'reveal' that lower coordination correlates with higher binding energy is built into the formula used to generate z. I do not treat the self-citation of refs. [26], [30], [31] as independent circularity because the bulk XPS constants are external measurements, not results derived from the target clusters. The physical identification of Kohn-Sham eigenvalue shifts with XPS core-level binding energies, and the transfer of bulk surface constants to 55-atom mixed clusters, are real correctness risks but are not circularity. Overall the central DFT observation stands, but the BOLS-derived coordination-number conclusion partially reduces by construction, giving a moderate score of 4.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are introduced or fitted in this paper. The numerical inputs are DFT settings, published Gupta-potential geometries, and published BOLS bulk shift constants. The central quantitative interpretation rests on transferring the BOLS bulk-surface relation to mixed 55-atom alloy clusters, which is an unvalidated domain assumption.

assumptions (4)
  • domain assumption BOLS bond contraction and strengthening relations in Eq. (1) apply to these clusters with the metal bond-nature index m = 1.
    The paper uses the BOLS model from ref [26] to relate bond length, bond energy, and coordination number; this is an established but phenomenological model.
  • domain assumption Core-level binding energy shifts are related to coordination number through Eq. (2), using bulk shift constants for Na (2.401 eV) and K (2.754 eV) measured on Na(110) and K(110) surfaces.
    This empirical relation from ref [30] is transferred to mixed 55-atom alloy clusters without independent validation of transferability.
  • domain assumption PBE/PAW DFT at the Gamma point with a 400 eV cutoff and 15-A vacuum gives sufficiently accurate core-level binding energies for comparison with experimental XPS trends.
    Standard DFT settings are used, but no convergence checks for binding energies are reported.
  • domain assumption The Gupta-potential global-minimum structures from Aguado et al. are the relevant stable configurations for these clusters.
    The paper uses these published geometries as inputs and does not independently search for alternative minima.

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Pith. "Pith review of DFT-based energy shifts screening of Na$_x$K$_{55-x}$ alloy clusters." pith.science (2026). https://pith.science/paper/5O3UO27L

@misc{pith2026190802421,
  author       = {Pith},
  title        = {Pith review of: DFT-based energy shifts screening of Na$_x$K$_55-x$ alloy clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5O3UO27L}},
  note         = {Machine review of arXiv:1908.02421}
}
read the original abstract

Compositional effects in NaK alloy clusters have been studied using bond order length strength notation and density functional theory calculations. The results reveal binding energy shifts of the NaK alloy clusters under different elemental compositions. Atomic arrangements that can be used to predict the structures of stable experimental NaK alloys were also obtained. Our study of these alloy nanoclusters has uncovered a trend correlating atomic position and composition with binding energy. We believe this data will help in the experimental preparation of alloy nanoclusters.

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    Introduction Alloy nanoclusters are widely studied not only because of size effects but also because their physical and chemical properties are subject to compositional effects[1]. Size effects influence the proportion of surface atoms with low coordination numbers, which in turn affects magnetic[2, 3], thermal[4, 5], catalytic[6, 7], optical[8] and elect...

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    Principles and methods of calculation 2.1 DFT Calculations In this study, we used DFT to calculate the structures of NaxK55–x alloy nanoclusters in which x, the number of Na atoms in the cluster, takes values from 0 to

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