Theoretical Insight into the Optical Properties of Al _m N _n (m = 1–8, n = 1–2) Clusters under External Electric Fields

Abstract

Nonlinear optical (NLO) materials are indispensable functional components for optical communication, photonic data storage, and solid-state laser devices. Current strategies for regulating the NLO performances of aluminum-based nanoclusters mainly rely on compositional modulation, whereas modulating NLO responses without perturbing cluster stoichiometry remains a formidable experimental challenge. Herein, density functional theory (DFT) and time-dependent DFT (TD-DFT) calculations are performed to systematically investigate the NLO characteristics of Al _m N _n (m = 1–8, n = 1–2) clusters under external electric field (EEF) modulation. Structural scrutiny reveals that incremental Al stoichiometry elicits conspicuous geometric reconstruction, unveiling an explicit size-correlated structural evolution. Notably, appropriate EEF polarization enables the conversion of various non-superhalogen units into superhalogen species, which significantly improves the NLO performance of the clusters. Numerical results demonstrate that EEF regulation delivers extraordinary enhancement in the first hyperpolarizability of typical Al _m N _n clusters. The Al₈N₂ cluster presents a 669-fold improvement, the Al₂N cluster exhibits a roughly 360-fold elevation, and the Al₃N cluster obtains a 44-fold increment. Mechanistic studies demonstrate that EEF stimulation triggers intramolecular charge density redistribution, reduces the HOMO–LUMO energy gap, and consequently enhances the intrinsic NLO susceptibility. Furthermore, EEF modulation increases the dipole moment of clusters by 1–3 orders of magnitude and remarkably promotes molecular polarizability, hyperpolarizability, and electron spatial delocalization, which fundamentally accounts for the enhanced NLO activity. This work affords a robust and reliable paradigm for the rational design and precise optimization of high-efficiency AlN-based NLO functional nanomaterials.

1. Introduction

Nonlinear optical (NLO) materials have found widespread applications in optical communications, optical data storage, solid-state lasers, and related fields. − Consequently, considerable and sustained research efforts have been devoted to exploring NLO properties and predicting favorable geometries for enhanced nonlinear optical responses. − Among these efforts, aluminum clusters with symmetric structures have been shown to exhibit high NLO performance. − Particularly, as one of the ultrawide-bandgap semiconductor materials, clusters composed of aluminum and nitrogen atoms have attracted considerable attention owing to their unique electronic structures and tunable optoelectronic properties. ,,

Previous theoretical and experimental investigations have primarily focused on the geometric structures, vibrational frequencies, and thermodynamic stability of aluminum nitride clusters. − Hao et al. performed a systematic study on the structural evolution of AlN (N = 1–15) clusters, while Luo et al. investigated the electronic properties of small AlN clusters using density functional theory (DFT). However, these studies concentrated on ground-state structural and energetic properties, while the nonlinear optical responses of aluminum nitride clusters and the strategies for modulating such responses without altering their chemical composition remain largely unexplored. External electric fields (EEFs) represent a promising noninvasive strategy for tuning the electronic structure and optical response of nanoscale systems. − Despite the well-established theoretical framework for EEF effects on molecular polarizability, a comprehensive understanding of how EEFs simultaneously modulate the superhyperpolarizability and NLO responses across a series of Al _m N _n clusters with varying sizes and compositions is lacking. Specifically, the following critical questions remain unanswered: (i) how does the size-dependent structural evolution of Al _m N _n clusters influence their intrinsic NLO properties and (ii) to what extent can EEFs serve as an effective external knob to enhance and tune the NLO responses of these clusters?

To address these gaps, this study presents a systematic theoretical investigation of the NLO characteristics and electronic properties of Al _m N _n (m = 1–8, n = 1–2) clusters under different EEF modulations by means of density functional theory and time-dependent DFT (TD-DFT) calculations. The novelty of this work lies in three aspects: (i) a systematic size-dependent NLO analysis of Al _m N _n (m = 1–8, n = 1–2) clusters; (ii) elucidation of the EEF modulation mechanism on NLO responses, offering a noninvasive tuning strategy; and (iii) establishment of a structure-property relationship linking cluster size, geometry, and EEF strength to NLO performance, providing theoretical guidance for designing AlN-based NLO materials. Structural characterization shows that increasing the number of Al atoms triggers a distinct geometric evolution of Al _m N _n clusters, which transform from one-dimensional linear frameworks to three-dimensional stereoscopic configurations and present a prominent size-dependent structural evolution feature. Subsequently, the simulated IR, Raman, and absorption spectra further corroborate that EEF possesses superior tunability in regulating the spectral characteristics of Al _m N _n clusters. At the same time, frequency analysis was subsequently conducted on all optimized geometries at the same computational level to confirm that the obtained structures correspond to stable minima on the potential energy surface without imaginary frequencies. Furthermore, EEF intervention can efficiently boost the dipole moment by 1–3 orders of magnitude for most clusters, as well as polarizability, hyperpolarizability, and electronic spatial extents, with β showing far more pronounced increments that directly reflect the enhanced NLO activity of these clusters. Remarkably, a series of cluster species can be converted from non-superhalogen into superhalogens via rational EEF regulation, heightening the NLO properties of these clusters. This study provides valuable theoretical insights into the rational design of EEF-controlled optoelectronic materials based on aluminum nitride nanoclusters.

2. Theory and Calculation Methods

2.1. Quantum Chemical Calculations

A systematic search for the global energy minimum of Al _m N _n (m = 1–8, n = 1–2) clusters was carried out using the Gaussian 16 program in combination with the Becke three-parameter Lee-Yang-Parr hybrid generalized function (B3LYP) and the 6-311+G basis set. This basis set includes both diffuse functions and d-type polarization functions, providing a balanced compromise between accuracy and computational efficiency for the systematic investigation of all clusters under multiple external electric field (EEF) strengths. Frequency analysis was subsequently performed on all optimized geometries at the same computational level to confirm that the obtained structures correspond to true minima on the potential energy surface. On the basis of the optimized ground-state structures, the first hyperpolarizability (β) was evaluated using the analytical second-order derivative method, which directly computes the second derivative of the energy with respect to the applied electric field at the SCF level. In addition, the present study focuses exclusively on the second-order NLO response, i.e., the first β, which is the dominant nonlinear optical coefficient for noncentrosymmetric systems and is directly relevant to second harmonic generation and electro-optic modulation applications. The EEF was applied along the principal dipole moment axis of each cluster to maximize the field-cluster interaction, with field strengths ranging from 0 to 0.05 a.u. , At each field strength, full geometry reoptimization and subsequent frequency verification were performed to ensure that the reported properties correspond to field-adapted equilibrium structures. Previous studies have demonstrated the reliability of this theoretical level in predicting the response properties under an EEF, particularly for nonlinear optical phenomena. , Furthermore, the excited-state properties and charge transfer spectroscopic information on Al _m N _n clusters at different EEF intensities were examined by time-dependent DFT (TD-DFT) , based on the corresponding field-optimized ground-state geometries. In addition, analyses of the electronic structure, electron density, and electrostatic potential (ESP) were performed using Multiwfn 3.8. ,

2.2. Theoretical and Computational Details

The electron affinity energy (EA), vertical ionization energy (VIP), average binding energy (E _b), and chemical hardness (η) of clusters were calculated as follows ,

$$\mathit{EA}=E(X)-E(X^{-})$$ 1

$$\mathrm{V}\mathrm{I}\mathrm{P}=E(X^{+})-E(X)$$ 2

$$E_{\mathrm{b}}=[\mathit{mE}(\mathrm{Al})+\mathit{nE}(N)-E(\mathrm{A}{\mathrm{l}}_m{\mathrm{N}}_n)]/(m+n)$$ 3

$$\eta =\frac{\mathrm{V}\mathrm{I}\mathrm{P}-\mathit{EA}}{2}$$ 4

where E(X), E(X ⁺), and E(X ^–) denote the global minimum energies of the neutral, cation, and anion, respectively, and for the optimized clusters, E(Al), E(N), and E(Al _m N _n ) denote the lowest energy of the ground state of the corresponding atoms and clusters. Furthermore, E _b is defined as the ratio of the binding energy of the nucleus to the total number of nucleons contained in the nucleus. A larger average binding energy indicates a more stable nucleus. η can be regarded as a kind of resistance to charge transfer, so the relative stability of the clusters can be evaluated by E _b and η of the clusters.

The isotropic polarizability (α) and first hyperpolarizability (β) are observed via the quantum-mechanical polarizability expression ,

$$\alpha \mathit{XY}(-\omega ;\omega )=\stackrel{\wedge }{P}[X(-\omega ),Y(\omega )]\sum _{i\ne 0}\frac{{\mu }_{0i}^X{\mu }_{0i}^Y}{\mathrm{\Delta }i-\omega }$$ 5

$${\beta }_{\mathit{XYZ}}(-{\omega }_{\sigma };{\omega }_1,{\omega }_2)=\stackrel{\wedge }{P}[X(-{\omega }_{\sigma }),Y({\omega }_1),Z({\omega }_2)]\sum _{i\ne 0}\sum _{j\ne 0}\frac{{\mu }_{0i}^X\stackrel{\circledR }{{\mu }_{\mathit{ij}}^Y}{\mu }_{j0}^Z}{({\mathrm{\Delta }}_i-{\omega }_{\sigma })({\mathrm{\Delta }}_j-{\omega }_2)}$$ 6

Here, ${\mu }_{\mathit{ij}}^X=⟨i|{\stackrel{\wedge }{\mu }}^X|j⟩,\stackrel{\circledR }{{\mu }_{\mathit{ij}}^X}=⟨{\mu }_{\mathit{ij}}^X-{\mu }_{00}^X\delta \mathit{ij}⟩,\omega \sigma =\sum _i\omega i◦$

where X, Y, and Z denote the directions, and ω is the external field energy, which corresponds to the static polarizability when ω is zero. Δ denotes the excitation energy of the excited state of the system with respect to the ground state; μ _ij is the component of the x-direction of the transition dipole moment of the two states i and j.

3. Results and Discussion

3.1. Geometrical and Energetic Characteristics of Al _m N _n Clusters

Geometry optimizations were performed on the initially proposed structures of the Al _m N _n (m = 1–8, n = 1–2) clusters to obtain their energetically stable configurations, as shown in Figure , and the Cartesian coordinates of all optimized clusters are provided in the Supporting Information Table S1. As can be observed from Figure , the geometrical structures of the Al _m N _n clusters are essentially linear when the number of atoms in the clusters is less than three. Moreover, the configurations of the Al _m N _n clusters (3 < m + n < 7) are mainly planar, whereas the geometrical structures of clusters become mainly three-dimensional when the number of atoms exceeds seven. ,, It is worth noting that the Al _m N _n clusters have a high symmetry. Additionally, the structural evolution of the Al _m N _n clusters exhibits a clear size-dependent trend, progressing from one-dimensional linear to three-dimensional stereoarchitectures with increasing numbers of Al atoms.

1.

(a–p) Optimized equilibrium structures of the Al _m N _n (m = 1–8, n = 1–2) clusters.

In order to study the relative stability of the ground-state structures of Al _m N _n clusters, the electron affinity energy (EA), vertical ionization energy (VIP), average binding energy (E _b), and chemical hardness (η) of clusters were calculated. , These calculated parameters of different Al _m N _n clusters are illustrated in Figure with eqs 1–4. As displayed in Figure , the calculated EA, VIP, E _b, and η of Al _m N and Al _m N₂ (m = 1–8) clusters exhibit pronounced size-dependent evolutionary behaviors governed by incremental Al atom doping and additional N incorporation. In Figure (a), the EA curves for both cluster species feature evident oscillatory fluctuations with increasing m, where the minimum EA of Al _m N emerges at m = 3 while Al _m N₂ forms two obvious valleys at m = 2 and 6, revealing periodic modulation of the clusters′ electron-capturing capacity induced by geometric rearrangement and electronic shell filling.

2.

EA (a), VIP (b), E _b (c), and (d) η of the Al _m N _n clusters with the number of Al atoms.

For VIP shown in Figure b, the value of VIP globally decreases along with growing Al content with mild local oscillations, and Al _m N₂ retains higher VIP than Al _m N at small m(m ≤ 4), while it gradually converges toward the Al _m N curve for larger cluster sizes, signifying a weakened disparity in valence-electron removal energy upon cluster expansion. From the perspective of E _b in Figure c, the specific binding energies of the two cluster configurations rapidly ascend to their respective maxima near m = 2–3 and subsequently decrease monotonically with further addition of Al atoms, and Al _m N₂ persistently possesses larger E _b values across the entire size range, corroborating that the second N atom effectively strengthens interatomic bonding and improves the intrinsic structural stability of the Al–N cluster skeleton. Regarding η in Figure d, an essential descriptor for chemical inertness derived from density functional theory, the two cluster families display irregular oscillating profiles as a function of Al number. The Al _m N₂ series starts from an extraordinarily high η at m = 1 followed by a sharp attenuation, whereas the hardness peak of Al _m N is located at m = 3, and the extremal points on hardness curves correspond to cluster sizes with elevated chemical stability and suppressed reactivity. Overall, the systematic discrepancies of the four energetic parameters between Al _m N and Al_mN₂ unambiguously demonstrate that additional nitrogen doping remarkably tailors the electronic configuration, interatomic bonding nature, and energetic stability of aluminum nitride clusters, and the synchronous variation of all physical quantities convincingly manifests the dominant role of quantum size effect in regulating the fundamental physicochemical properties of Al _m N _n clusters.

The infrared (IR) and Raman spectra of the Al _m N _n clusters under different electric fields were simulated and are summarized in Figures S1–S5, with the applied field acting as an effective symmetry-breaking perturbation. , As evidenced by the IR and Raman spectral profiles in Figure S1, no imaginary frequencies are present, confirming that all the studied clusters correspond to true minima on the potential energy surface. For IR spectral profiles in Figure S1, EEF significantly increases the number of characteristic absorption peaks and enhances the molar absorption coefficients by activating abundant latent vibrational modes for the investigated Al _m N _n clusters. Under zero-field conditions, most pristine clusters exhibit only one dominant intense IR band within 700–900 cm^–1, which originates from the primary Al–N stretching vibration. After the application of EEF, field-driven intramolecular charge redistribution breaks the original geometric symmetry, activates numerous IR-silent modes, and generates abundant new absorption signals spanning 100–800 cm^–1. The enhanced IR intensity of major characteristic peaks is primarily attributed to the EEF-induced enlargement of ground-state dipole moments and strengthened dipole derivatives along the vibrational coordinates.

Consistent with the IR spectral evolution, Raman scattering features also exhibit a prominent field-dependent modulation. EEF effectively lifts the prohibition of multiple Raman-inactive phonon modes, increases the number of distinct scattering peaks, and dramatically improves the overall Raman intensity, which is closely correlated with the EEF-enhanced molecular polarizability and first hyperpolarizability. In contrast to the universal spectral enhancement of most clusters, AlN presents an exceptional variation: its IR and Raman peak positions remain nearly unchanged under EEF modulation, while the spectral intensity slightly decreases. Collectively, the IR and Raman spectroscopic results provide experimentally accessible fingerprints for identifying EEF-induced changes in the cluster symmetry, bonding, and electronic structure. In particular, the emergence of additional vibrational bands under an EEF is not a trivial size effect but is attributed to field-induced symmetry breaking, which activates otherwise silent modes via charge redistribution and enhanced polarizability.

3.2. EEF-Controlled Superhalogen of Al _m N _n Clusters

Previous work has shown that superhalogens constitute a distinctive subclass of superatomic clusters, characterized by EAs exceeding the intrinsic EA range of elemental halogen atoms (3.06–3.62 eV), ,, which endows such clusters with extraordinary electron-capturing capability and oxidability. In view of the above, we systematically explored the feasibility of employing EEFs to tailor the superhalogen characteristics of Al _m N _n clusters. In line with the intrinsic geometric configuration of each cluster, a three-dimensional Cartesian coordinate system was defined. Computational results showed that the EA of a cluster can be significantly enhanced in Table by applying EEF, verifying that AlN, Al₂N, Al₃N, Al₅N, Al₆N, Al₇N, Al₇N₂, Al₈N, and Al₈N₂ can be converted from non-superhalogen into superhalogens via rational EEF regulation, heightening the NLO properties of these clusters. For example, when an EEF of 0.020 au was applied to Al₃N in the +y direction, the EA value increased from 0.76 to 3.55 eV, giving it successful superhalogen properties. The other clusters could not be converted into superhalogens under EEF, as they failed to converge to stable geometries during optimization and underwent structural dissociation. Therefore, their electron affinities and NLO properties under EEF are not reported. In addition, to quantify structural stability variation throughout the EEF-induced superhalogen transformation, the average bond lengths of neutral and anionic Al–Al bonds and Al–N bonds of the Al _m N _n clusters with superhalogen characteristics were calculated. The comparative bond-length evolution before and after field modulation is visualized in Figure . It is clearly manifested that nearly all Al–Al and Al–N covalent bonds either remain invariant or undergo marginal elongation upon EEF loading, demonstrating that the EEF-mediated superhalogen engineering barely perturbs the skeleton geometry of target clusters and effectively circumvents the interference from unwanted structural distortion. Obviously, the EA of Al _m N _n clusters can be significantly enhanced by applying EEF, transforming it from a non-superhalogen into a superhalogen, while maintaining its geometric structure and superatomic state unchanged. This regulation is independent of the cluster’s shell filling. Furthermore, there is a quantifiable functional relationship between the EA enhancement and the electric field strength, enabling precise, continuous, and nondestructive tuning.

1. Variations of EA (eV) and the Corresponding EEF (a.u.) Parameters for Al _m N _n (m = 1–8, n = 1–2) Clusters Acquiring Superhalogen Behaviors .

	AlN	Al2N	Al3N	Al5N	Al6N	Al7N	Al7N2	Al8N	Al8N2
EA non‑EEF	3.01	2.40	0.76	1.91	2.33	1.84	2.53	2.49	1.92
EA superhalogen	3.28	3.80	3.55	3.52	3.45	3.52	3.67	3.53	3.59
OEEFdirection	F+x	F+x	F+y	F+y	F+y	F+y	F+y	F+x	F+x
OEEFintensity	0.005	0.025	0.020	0.018	0.020	0.020	0.020	0.015	0.015

¹F_+x and F_+y denote EEF applied along the positive x-axis and positive y-axis, respectively.

3.

Comparative plots of the averages of Al–Al and Al–N bonds for the neutral (a) and anionic structures (b) of the Al _m N _n clusters before and after EEF-driven superhalogen modulation.

To further elucidate the EEF-driven transition of Al _m N _n clusters from non-superhalogen to superhalogen behavior, which is governed by the increase in EA accompanying the downward shift of the LUMO. Hereby, field-induced intramolecular charge redistribution within Al _m N _n frameworks was investigated. The Al₃N cluster was chosen as a representative prototype, owing to its high symmetry and homogeneous atomic arrangement, which permit unambiguous interpretation of the field-induced charge redistribution. The qualitative mechanism, EEF-driven Al-to-N electron transfer accompanying LUMO downshift and EA enhancement, is general across the Al _m N _n frameworks, although quantitative details may vary with cluster size and stoichiometry. Its field-dependent charge evolution was quantified through natural bond orbital (NBO) population analysis [Figure a]; the corresponding natural population charges are summarized in Table . Geometrically, the Al₃N skeleton can be partitioned into two layers, a top layer carrying charge Q _t (comprising Al atom No. 4) and a bottom layer carrying charge Q _l (comprising N₁, Al₂, and Al₃). As the EEF strength along the +y direction increases, Q _t decreases monotonically while Q _l becomes progressively more negative, indicating a directional interlayer electron migration. The upper layer continuously accumulates surplus electrons, whereas the bottom layer suffers electron depletion. Although the interlayer charge transfer per unit field is small (∼0.003e per 0.005 a.u.), it is physically meaningful for a finite-size cluster; the limited valence-electron count and the spatially localized LUMO render the orbital energy highly sensitive to even minute charge redistribution. The transfer accumulates quasi-linearly to ∼0.012e at 0.020 a.u., concomitant with a pronounced dipole moment enhancement (Table ) and the emergence of a new UV absorption band (Section ). Such sub-millielectron charge displacements sufficient to drive substantial electronic and optical changes are consistent with prior findings in small molecular systems, , confirming that the observed magnitude is adequate to account for the EEF-controlled superhalogen transition. Quantitatively, applying an external field of 0.005 a.u. yields an electron gain of 0.003 e for the top layer accompanied by an equivalent 0.003 e loss from the bottom layer. In essence, EEF serves as an effective thermodynamic driving force to rearrange intramolecular charge density and facilitate directional electron migration from the bottom atomic layer toward the top layer of Al₃N.

4.

(a) Quantitative evolution of layer-resolved NBO charges for layers of Al₃N under different OEEF intensities along the +y direction. (b) Electron-density difference contour map of Al₃N before and after the application of 0.020 au of OEEF along the +y direction.

2. Layer-Resolved Charges Q_t (|e|) and Q _l (|e|) of the Al₃N Cluster at Different EEF Intensities (a.u.).

	Al3N
OEEF	0.000	0.005	0.010	0.015	0.020
Qt	0.81347	0.81065	0.80784	0.80498	0.80351
Q l	–0.81347	–0.81065	–0.80784	–0.80498	–0.80351

3. Variation of H-Lgap (eV) for Al _m N _n Clusters before and after OEEF-Induced Superhalogen Functionalization.

	AlN	Al2N	Al3N	Al5N	Al6N	Al7N	Al7N2	Al8N	Al8N2
H-Lgap non‑EEF	1.49	2.10	3.75	1.35	1.60	2.75	2.45	1.68	1.73
H-Lgap superhalogen	1.49	2.09	3.42	1.16	1.51	2.44	2.34	1.50	1.52

To visualize the field-modulated electron-density rearrangement, we computed the electron density difference map of the Al₃N cluster with and without an EEF of 0.020 a.u. applied along the +y axis [Figure b]. Solid contour lines around the apex Al atom indicate electron accumulation upon field application, while dashed contours surrounding the central N atom and the two basal Al atoms signify pronounced electron depletion consistent with the NBO charge trends discussed above. Furthermore, the ESP isosurface distributions of Al _m N _n in Figure corroborate the field-driven electrostatic transformation. The red and blue regions on the ESP isosurfaces denote electron accumulation (negative electrostatic potential) and electron depletion (positive electrostatic potential), respectively. Upon application of the EEF, the initial concentrated red positive-potential domain gradually evolves into a localized blue negative-potential region. In the absence of an external field, the ESP distributions generally exhibit symmetry consistent with the geometric structures of the respective clusters. Upon application of an EEF along specific axes, significant polarization effects are observed across all cluster sizes. The external field induces a distinct redistribution of electron density, leading to an asymmetric accumulation of positive potential (red regions) on one side of the cluster and negative potential (blue regions) on the opposite side, aligned with the direction of the applied field. This visual evidence demonstrates that the EEF effectively modulates the electronic structure and surface charge distribution of the Al _m N _n clusters, breaking the intrinsic symmetry of the cluster and enhancing their dipole moment. This charge redistribution is central to understanding the tunable superhalogen characteristics of Al _m N _n clusters under external perturbations. This difference can be attributed to the structural characteristics of the clusters. The nine superhalogen clusters possess compact cage-like structures with an appropriate Al/N ratio, enabling stable field-induced charge separation between Al-rich and N-containing regions. In contrast, the other clusters adopt open or low-dimensional configurations with insufficient bonding rigidity, making them prone to structural collapse under an EEF.

5.

ESP isosurface distributions of Al _m N _n clusters without (left) and with (right) the EEF. Red and blue regions denote electron accumulation (negative ESP) and electron depletion (positive ESP), respectively.

3.3. EEF-Modulated Absorption Spectra of the Al _m N _n Clusters

Analysis of electronic absorption spectra reveals variations in the molecular electronic structures. The absorption spectra of Al _m N _n clusters are compiled in Figure . As illustrated by the overall spectral profiles, it predominantly spans the near-ultraviolet to visible spectral region, demonstrating their promising prospect as efficient UV–visible light-harvesting functional materials. Notably, EEF triggers distinct spectral reconfiguration. Distinctly, Al₃N exhibits the most characteristic variation: a brand-new absorption peak emerges within the ultraviolet range after applying a directional external field. Further orbital-resolved transition decomposition clarifies that this emerging absorption originates from the HOMO to LUMO–6 transition, with a contribution of 86.3%. This effect can be attributed to the increase in the transition dipole moment caused by the EEF. The oscillator strength increased from 0.000 to 0.164, which sufficiently activates this formerly optically forbidden transition and yields the newly observable absorption band. From the point of view of charge transfer, the absorption peak is mainly due to the local excitation of the Al atom and the charge transfer excitation of the N→Al atom.

6.

(a–r) UV–vis absorption spectra of the Al _m N _n clusters under different OEEF, in which the green curve denotes the conventional UV–vis curve, the blue and pink curves are the contribution curves of electron redistribution within fragment 1 (Al atoms) and fragment 2 (N atoms), respectively, and the red and brown curves are the contribution of electron transfer from fragment 1 to fragment 2, and fragment 2 to fragment 1, respectively, to the absorption.

In addition to peak generation, EEF manipulation induces pronounced wavelength shifts of intrinsic main absorption bands, featuring either blue-shift or red-shift depending on specific cluster composition. Taking Al₆N as a representative blue-shift case, its dominant zero-field absorption peak is located at 777.42 nm. , When EEF was applied, the absorption peak shifted to 703.91 nm; the absorption peak was blue-shifted. In contrast, typical clusters including Al₅N, Al₇N, and Al₈N₂ undergo an evident red-shift of characteristic absorption maxima under equivalent field modulation. Such divergent shift behaviors stem from an EEF-tunable HOMO-LUMO band gap and varied transition dipole evolution for different cluster configurations. As revealed by the multicomponent decomposed spectra in Figure , the overall absorption envelope (green curve) can be quantitatively decoupled into four physical contributions: intra-fragment electron redistribution within the Al moiety (blue curve), intra-fragment electron redistribution within the N moiety (magenta curve), inter-fragment electron transfer from the Al fragment toward the N fragment (red curve), and reverse N→Al charge transfer (brown curve). The relative weight of each component varies drastically after EEF polarization, directly reflecting field-driven intramolecular charge rearrangement and polarization effects. Collectively, EEF serves as a facile extrinsic tuning knob to flexibly tailor the excited-state transition nature, oscillator strength, and peak position of Al _m N _n clusters, enabling customizable modulation of their absorption properties toward target wavelengths. These findings validate EEF as an efficient strategy to rationally design high-performance opto-absorbing and nonlinear optical cluster materials.

The HOMO-LUMO gap (H-L_gap) is recognized as a core descriptor to evaluate the optoelectronic performance of cluster-based materials. − Table summarizes the H-L_gap of the Al _m N _n clusters in pristine and EEF-modified configurations. Uniformly, EEF regulation induces conspicuous bandgap narrowing for all target clusters. Taking Al₇N as a typical instance, the H-L_gap of Al₇N decreases from 2.75 eV (zero field) to 2.44 eV after EEF modulation, corresponding to a total reduction of 0.31 eV. The narrowed energy gap effectively reduces the energy barrier for electronic transitions from the HOMO to LUMO, which simultaneously enhances the nonlinear optical response of the material. Consequently, the EEF-induced bandgap narrowing of Al _m N _n clusters provides a reliable theoretical basis for improving their electroluminescent performance. This finding demonstrates that EEF can act as an efficient extrinsic modulation strategy to tailor and optimize the optoelectronic properties of emerging cluster-type electroluminescence materials.

3.4. EEF Modulation on NLO Responses of Al _m N _n Clusters

On the basis of the foregoing discussions, we further explore the manipulation effect of EEF on the NLO characteristics of Al _m N _n clusters. In the study of NLO materials, the dipole moments, isotropic polarizability (α), first hyperpolarizability (β), and electronic spatial extents (R²) serve as core physical parameters for describing microscopic molecular structures and predicting their macroscopic NLO properties. The numerically calculated dipole moments, α, β, and R² for Al _m N _n clusters under different EEF are summarized in Table with eqs 5 and 6. In the absence of an external electric field, the high intrinsic geometric symmetry of most bare Al _m N _n configurations suppresses molecular charge separation, leading to negligibly small or near-zero equilibrium dipole moments. Upon EEF application, significant symmetry breaking occurs within the cluster frameworks, leading to a drastic increase in dipole moments by 1–3 orders of magnitude for most clusters relative to their zero-field counterparts, along with a pronounced enhancement of the intrinsic molecular polarity. Notably, the AlN cluster exhibits an opposite trend, with its dipole moment decreasing from 4.95 D at zero field to 2.50 D under an EEF, which may be attributed to the unique geometric and electronic structure of the mononuclear AlN cluster. This field-induced charge polarization fundamentally modulates the molecular dipole moment, laying the foundation for the subsequent quantitative analysis of the field-dependent evolution of polarizability and hyperpolarizability.

4. Calculated Dipole Moments, Polarizabilities, and Electronic Spatial Extents (R²) of the Al _m N _n Clusters under Different OEEF Conditions.

cluster	OEEF/au	dipole moment/debye	α/au	β/au	R2/au
AlN	F+x = 0.000	4.95	78.71	5313.14	88.26
	F+x = 0.005	2.50	77.96	5097.64	93.49
Al2N	F+x = 0.000	0.000	127.64	0.00001	348.50
	F+x = 0.025	15.45	127.68	359.72	366.45
Al3N	F+y = 0.000	0.002	128.25	0.84	589.63
	F+y = 0.020	9.16	129.88	37.21	609.91
Al5N	F+y = 0.000	0.71	249.52	403.12	1444.72
	F+y = 0.018	32.52	270.34	665.58	1701.58
Al6N	F+y = 0.000	0.26	284.52	439.46	1523.36
	F+y = 0.020	14.58	285.31	595.94	1689.32
Al7N	F+y = 0.000	0.13	273.24	197.49	1720.19
	F+y = 0.020	19.74	289.30	240.05	1807.62
Al7N2	F+y = 0.000	1.12	334.91	6696.26	2309.78
	F+y = 0.020	16.71	358.06	8719.84	2606.77
Al8N	F+x = 0.000	1.11	352.81	1322.63	2378.76
	F+x = 0.015	17.47	358.65	1365.72	2490.54
Al8N2	F+x = 0.000	0.006	381.70	4.44	3183.08
	F+x = 0.015	24.71	396.21	2971.52	3245.67

As listed in Table and illustrated in Figure , both α and β exhibit an overall monotonic increasing trend with the application of EEF for most Al _m N _n clusters, with β showing far more pronounced increments that directly reflect the enhanced NLO activity of the clusters. For instance, the β value of the Al₂N cluster increases from 0.00 at zero field to 359.72 au under an EEF of 0.025 au, while that of the Al₈N₂ cluster rises from 4.44 to 2971.52 au under an EEF of 0.015 au, corresponding to an increase of more than three orders of magnitude. The electronic spatial extent R² quantitatively describes the degree of delocalization of the valence electron cloud around the nuclear centroid, and a robust positive correlation between R² and molecular polarizability has been widely validated in various NLO materials. Our calculation results in Table demonstrate that EEF application consistently increases the R² values of all the investigated Al _m N _n clusters. The field-induced electron delocalization expands the spatial distribution range of valence electrons, enabling more efficient charge displacement in response to external electric field perturbations, which in turn further enhances the α and β of the Al _m N _n clusters and consequently the NLO response.

7.

Calculated α (a), R² (b), and β (c) of the Al _m N _n clusters under different EEF conditions.

4. Conclusions

In summary, the evolution of the geometrical structures and relative stabilities of the Al _m N _n (m = 1–8, n = 1–2) clusters was systematically investigated by means of DFT calculations. Meanwhile, TD-DFT was utilized to elucidate the modulation mechanism of the EEF on the optoelectronic properties of Al _m N _n clusters. The structural analysis reveals that the geometric configuration of Al_mN_n clusters undergoes a distinct transition from one-dimensional linear structures to three-dimensional spatial structures, while the structural evolution of the Al _m N _n clusters exhibits a clear size-dependent trend with increasing cluster size. The simulated IR, Raman, and absorption spectra results further verify that EEF possesses excellent tunability over the spectral characteristics of Al _m N _n clusters. Subsequently, the introduction of EEF serves as an effective modulation strategy for regulating the intrinsic properties of the Al _m N _n clusters. Significantly, the AlN, Al₂N, Al₃N, Al₅N, Al₆N, Al₇N, Al₇N₂, Al₈N, and Al₈N₂ can be converted from non-superhalogen into superhalogens via rational EEF regulation, heightening the NLO properties of these clusters. Mechanistically, EEF stimulation rearranges the internal charge distribution of the clusters, which significantly reduces the H-L gap and further boosts the nonlinear optical response of the material. Furthermore, EEF can effectively enhance the dipole moment and molecular polarizability of clusters, thereby strengthening the molecular polarity and ultimately improving the NLO response performance of the AlN systems. This evidence confirms that EEF modulation is a reliable and effective external means to precisely tune the electronic structure and macroscopic nonlinear optical performance of cluster materials, providing a feasible route for the design of high-efficiency NLO functional nanosystems.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c08367.

• Infrared (red) and Raman spectra (blue) of AlN, Al2N, Al3N, Al5N, Al6N, Al7N, Al7N2, Al8N, and Al8N2 clusters under different EEF conditions and the calculated parameters for aluminum nitride systems (PDF)

Q.Z. and Q.W. performed the theoretical calculation, analyzed the data, and commented on this paper.

The authors declare no competing financial interest.