computing has been shown to be an effective tool in areas such as cryptography 21 and drug discovery. 23- 25 The high quantum efficiencies and narrow emission linewidths 2,26 of LHP QDs allow for precision control of light at the quantum level and their tunable bandgap creates opportunities for qubit design. 26 Lastly, LHP QDs can achieve long quantum coherence at low 27 and even room temperatures, 8 an ideal characteristic for the practical application of quantum computing. Arranging LHP QDs into two-or threedimensional networks and placing the systems into optical cavities 28,29 allows one to design and manipulate the coupling between QDs and their light emission properties.

In this letter, we consider a systematic set of ligands that can be used to passivate LHP QD surfaces and control inter-dot coupling in QD assemblies. We show that judicious ligand design can be used to extend QD wavefunctions onto ligands, thereby enhancing inter-dot interactions. On the other hand, some ligands can introduce midgap electronic states that trap charge carriers and quench QD emission, having a strong negative effect on QD electro-optic properties. The energetics of the ligand states relative to the LHP QD frontier orbitals depend on the extent of the ligand's π-conjugated system, the presence of electron withdrawing and donating substitutents in the π-system, the binding group, and QD size. Larger π-electron conjugation lowers ligand excitation energies and brings ligand energy levels closer to the LHP QD band edges, with ligand contributions even found inside the bandgap in certain cases. The electronegative oxygen atoms in the carboxylate binding group lower ligand levels compared to the more positive ammonium binding group. Electron withdrawing substituents in a ligand's π-conjugated system also lower ligand energy levels, while electron donating groups raise the levels. Because of the quantum confinement effect, the positioning of the ligand levels relative to the QD frontier levels depends on QD size for sufficiently small QDs. For all considered ligands, only unoccupied energy levels can be brought close to the QD conduction band (CB) edge or inside the bandgap, while occupied levels of the ligands always remain deep inside the valence band (VB). Ligand binding energy calculations demonstrate that the carboxylate group binds as a bidentate ligand and tends to provide stronger binding than the ammonium group. An extra methylene group in the bridge allows for more flexibility and stronger binding. The calculated results are benchmarked against experimental data on photoluminescence (PL) quenching.

The study presented herein focuses on the all-inorganic CsPbBr3 LHP, which is one of the most common choices for synthesizing LHP QDs. Ligand binding to CsPbBr3 can occur through an ammonium cation at the A-site by replacement of a Cs cation, or through a carboxylate group by replacement of a bromide in the octahedral lead complex. Fifteen different ligands are selected for the study, fourteen of which contain π-conjugated systems, Figure 1. Two ligands (I and II) bind to CsPbBr3 through a carboxylate group. Their comparison is designed to investigate the effects of π-conjugation. The remaining ligands bind through the ammonium cation. Among them, there are large ligands with extended π-conjugation (III-V), ligands with the same π-electron system and either electron withdrawing or donating groups (VI-X), similar ligands but with longer carbon bridges to bind to the LHP (XI-XIV), and a ligand with two electronegative nitrogen atoms embedded into the π-conjugated system (XV).

Systems containing qualitatively similar ligands to the ones studied in this work have been considered previously. They include ligands with both electron withdrawing and donating groups, 17,30 ligands with π-conjugation, 5,18,19,[31][32][33] and ligands with carboxylate 18,19,34 and ammonium 4 groups at the preferred binding sites. The various electron withdrawing and donating groups will lower or raise the ligand's lowest unoccupied molecular orbital (LUMO), respectively. Longer bridges can increase flexibility in the bonding, facilitating the process. Most experimental ligands contain inert aliphatic chains which are omitted during the calculations of the system's electronic properties, since the aliphatic chains have little influence on the alignment of the frontier electronic energy levels. Selecting an accurate and efficient tool for calculating and characterizing the electronic properties of ligand-LHP systems is a challenging task. LHPs are periodic systems, while ligands are finite. LHPs exhibit significant screening of Coulomb interactions due to relatively large dielectric constants, and their exciton binding energies are relatively small. Thus, LHP electronic properties can be described reasonably well within the single-particle approximation, such as in the Kohn-Sham (KS) density functional theory (DFT). In comparison, an accurate description of electronic excitations in molecules typically require at least linear response (LR) DFT. LHPs contain heavy elements, in particular Pb, which exhibit strong spinorbit coupling (SOC) effects, while SOC effects are insignificant in organic matter. Luckily, a fortuitous cancellation of various errors enables us to use the computationally efficient Perdew-Burke-Ernzerhof (PBE) functional. 35 In particular, the PBE functional is known to underestimate energy gaps in most semiconductors due to the electron self-interaction error. However, PBE produces good bandgap values for LHPs, because the self-interaction error and neglect of SOC cancel each other. 36 The energy gaps between the highest occupied molecular orbital (HOMO) and LUMO of the ligands calculated by PBE reproduce reasonably well the lowest electronic excitation energy of the types of molecules considered here. Thus, the excitation energy of benzene, ~4.9 eV, 37 can be well represented by the HOMO-LUMO gap obtained using the PBE functional. Even if the KS HOMO-LUMO gaps can accurately represent electronic excitations in the QD and ligand systems, this does not guarantee that the relative alignment of QD and ligand levels is correct. To investigate this, we report experimental PL data for the cinnamate ligand. The data confirms the existence of a midgap ligand state predicted by the PBE calculation. This is observed through PL quenching, a key aspect of midgap states. In addition to the PBE calculations, we also report PBE+SOC results, as well as results of calculations obtained with the hybrid Heyd-Scuseria-Ernzerhof 38 (HSE) functional, and HSE+SOC data. The electronic properties obtained with HSE are reported for PBE optimized geometries. It is important to stress that the reported results should be interpreted semi-quantitatively, and that trends rather than absolute values should be considered more beneficial for analysis.

The calculations in this work were performed with the Vienna Ab initio Simulation Package 39 (VASP). All structures were initially optimized using the PBE functional using 3x3x1 k-point mesh. The plane-wave basis energy cutoff was 520 eV and Grimme's DFT-D3 dispersion correction 40 was utilized. For the PBE single point calculations, a finer 5x5x1 k-point mesh was used. All other single point calculations (PBE+SOC, HSE, HSE+SOC) were carried out at the Γ-point only for computational efficiency. Both singlet and triplet calculations were performed. Because LHP QDs used in experiment are much larger than those accessible by ab initio simulation, the LHP is represented by a slab made with a 2x2x3 supercell with two exposed surfaces, once of which is used for ligand binding. The cubic phase CsPbBr3 was used to construct the supercell slab. The slab was 3 layers thick with the (100) surface exposed for binding. The LHP primitive cell was taken from the Materials Project 41 (ID: mp-600089). The ligand binding energies were calculated as:

Here, 𝐸𝐸 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐴𝐴+𝐿𝐿𝐿𝐿𝐿𝐿 is the energy of the ligand-LHP complex, 𝐸𝐸 𝐶𝐶𝐴𝐴/𝐵𝐵𝐵𝐵𝐴𝐴𝐵𝐵𝐵𝐵𝐵𝐵 is the energy of the cesium or bromine atom (which the ligand replaces upon binding), and 𝐸𝐸 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐴𝐴 and 𝐸𝐸 𝐿𝐿𝐿𝐿𝐿𝐿 are the energies of the ligand and LHP, respectively. Figures 2 and 3 illustrate the ligand binding motifs and LUMO charge densities, respectively, with four representative cases. The first case, ligand II containing a carboxylate group, binds through said group and replaces a bromine atom (representative of I and II). The second case, ligand IX, has a shorter chain connected to its ammonium group (ligands III-X), and the third case, represented by XIV, has a longer chain, allowing for more flexibility in binding to the LHP QD (ligands XI-XIV). Simultaneously, binding through these longer bridges leads to larger local distortions in the LHP QD. Lastly, the small XV ligand binds to the LHP without any bridge. Due to its size, this ligand barely extends past the LHP surface. Ligands IX, XIV and XV replace the A-site cesium cation. for systems IX and XIV to better visualize their binding motifs. For similar reasons, the polyhedral formation of the LHP QD is not visualized to better observe the charge densities located within the QD structure (IX, XIV, XV).

Figure 3 demonstrates LUMO charge densities of the four representative systems discussed in Figure 2. Three different situations are encountered. The LUMO of system II is localized on the cinnamate ligand, suggesting that it can act as an electron trap. The LUMO is supported by the π-conjugated electronic system and extends onto the carboxylate binding group. The presence of electronegative atoms of the carboxylate group lowers the LUMO energy below the perovskite conduction band minimum (CBM), Figure 4. In comparison, the LUMO of the same ligand with the ammonium binding group, ligand IV, is slightly above the perovskite CBM. The LUMO of system IX, representing the shorter-bridge category of the investigated ligands, is delocalized across the LHP, with charge density distributed primarily over lead atoms, as well as their connecting bromine atoms.

The longer-bridge ligands, represented by ligand XIV in Figure 3, exhibit a more significant reconstruction of the binding site. This results in an increased localization of the LUMO on the LHP surface compared to the shorter-bridge ligands. In this example, the LUMO is still localized on the LHP as in the case with the shorter bridge (IX). However, the charge is drawn to the surface, which can favor interactions with charges photogenerated in the LHP with surface species and other QDs. It is important to note that the surface state observed with system XIV, Figure 3, is part of the LHP CB, Figure 4. Therefore, it does not trap charges permanently, and charges drawn to the LHP surface by the surface state can easily escape into bulk. The final system in Figure 3, containing ligand XV, exemplifies the situation, in which the LUMO charge density is delocalized between the LHP and the ligand. Similar to the ligand XIV system, the charge migrates from within the LHP to the surface where the ligand binds, and the terminal bromine atom closest to the ligand gains charge density. The charge density also extends onto the electronegative nitrogen atoms of the ligand as well as the carbon atom separating them. The energies of the ligand states relative to the LHP VB and CB are illustrated through plotting the density of states (DOS), Figure 4. The DOS are obtained with the PBE functional, which should present the most useful results, as discussed previously and supported by the experimental data in Figure 5. DOS plots obtained with other methods are shown in Figures S1-S5, including HSE in the singlet and triplet states, HSE+SOC, and PBE in the triplet state. Different approaches give different quantitative results, and therefore, it is more valuable to consider qualitative trends rather than absolute values.

For all of the systems, except III, the ligands contribute narrow peaks to the DOS. This is indicative of negligible interaction between periodic images of the ligands in the simulation. I.e., even at a high degree of ligand coverage, ligand states maintain their molecular identity and do not form bands. Ligand III is an exception due to its large size, making the π-electron systems of periodically replicated ligand molecules interact and form narrow bands.

Various trends can be observed from the DOS of the systems, including effects caused by the extent of π-electron conjugation, inclusion of electronegative elements as well as electron donating and withdrawing groups, and ligand bridge length. As the size of the π-conjugated system increases, e.g,

systems II, IV, V, the energy of the ligand LUMO decreases. In systems II and V, the energies of the ligand orbitals decrease significantly in the CB, potentially forming shallow electron trap states not present in the other ligands studied. The presence of more electronegative elements lowers the LUMO as well, which is seen when the carboxylate group containing two oxygens in II and the aminium group containing one nitrogen in IV are compared. The effects of electron withdrawing versus donating groups is also illustrated in Figure 4. Fluorine is used as the electron withdrawing group in ligands IX, X, and XIV. The ether substituents in VIII and XIII are electron donating through resonance, while at the same time they exhibit an electron withdrawing inductive effect. The methyl groups in VI and XII, and the propyl group in VII act as electron donating groups. When electron withdrawing groups are added to the ligand, the ligand LUMO energies decrease, and in the case of ligand X which contains three electronegative fluorine atoms, drop below the LHP CBM. While the influence of the electron donating and withdrawing groups to the π-conjugated system has been studied for the ligands with the ammonium binding group, similar conclusions can be expected for carboxylate ligands. These observations provide key insights into how electron withdrawing and donating groups can be used to tune the electronic structure of the ligand-LHP QD systems.

Additional DOS plots are included in the Supporting Information (SI). The DOS was additionally calculated with SOC for both the PBE and HSE functionals, as well as for both singlet and triplet spin states without SOC. Figure S1 shows the DOS calculated with the HSE functional in the singlet state. There is a distinct increase of the band gap, by ~1 eV in some cases, e.g., IV and V. The HSE and HSE+SOC DOS are calculated only at the Γ k-point. For this reason, the DOS shown in Figures S1, S2, S4, S5 exhibit sharp peaks in the LHP component. Incorporating SOC decreases the LHP band gap, Figure S2. At the same time, the energy levels of the ligands change little upon including SOC. In particular, the energy of the cinnamate (ligand II) LUMO is significantly above the LHP CBM in the HSE+SOC calculation in Figure S2. This contradicts the experimental results discussed below and visualized in Figure 5. The triplet state PBE DOS in Figure S3 shows a slight splitting in some of the ligands' DOS, including systems II, X, and V, listed from most pronounced to least pronounced splitting. In the case of II, the shallow trap state shown in Figure 4 is realized only in one spin direction in Figure S3. On the other hand, the shallow trap state in V in Figure 4 disappears when the triplet state DOS is considered. In a few of the ligand systems studied, the HSE calculation for the triplet state exhibits midgap states that can act as charge traps, Figure S4.

It is also of interest to investigate the adsorption energies of the ligand-LHP systems. The adsorption energy describes the strength, preference, and likelihood of the ligand binding to the LHP QD. The more negative the adsorption energy, the more likely the ligand is to bind. Adsorption energies were computed for all ligand-LHP systems using Eq. ( 1). The results are then summarized in Table 1. Ligands I and II, which bind to the LHP through the carboxylate group, show large negative adsorption energies, indicating a strong binding to the LHP. Ligand XV, the smallest ligand, exhibits one of the weakest adsorptions. This is consistent with the shape of the ligand and the binding motif in Figure 2. The ligand does not contain a tail as in the other examples, thereby promoting steric hindrance as the ligand cannot bind to the LHP in an effective way. The less favorable adsorption energy is exacerbated by the nitrogen atoms lying within the π-conjugated ring and therefore at suboptimal positions for binding compared to the systems that have ammonium ions at terminal or branched points (III-XIV). The LHP QD system containing ligand X also has a characteristically high adsorption energy. The presence of three strongly electron withdrawing fluorine atoms in X is likely the cause of its less favorable adsorption. Ligand III exhibits the strongest absorption, likely because of its large size, allowing for ligand-ligand interactions at high ligand coverage, stabilizing the ligand-LHP system. In order to provide an experimental verification of the reported calculations, and differentiate between the various DOS calculations that give qualitatively different results, experimental studies were performed with the cinnamate (II) ligand, as summarized in Figure 5. The experimental methods are described in the SI. Figure 5a reports both UV-Vis and photoluminescence (PL) spectra for the cinnamate-LHP QD system with varying ligand concentrations. As the concentration increases, the PL intensity decreases, indicating that cinnamate creates a trap state that quenches PL. This experimentally observed trap state is consistent with the predicted DOS using the PBE functional as shown in Figure 4. In comparison, both the HSE and HSE+SOC calculations, Figures S1 and S2, respectively, do not show ligand states inside the LHP band gap. The experimental data confirms that the PBE level of theory provides more reliable results. The UV-Vis and PL spectra of the ligand-LHP QD system qualitatively similar to ligand I can be found in Figure S6. Oleate is an aliphatic hydrocarbon containing no πconjugation. Its HOMO and LUMO orbitals are widely separated in energy (as illustrated in the DOS calculations), and none of the calculations predict midgap states for oleate. The experimental data show that the PL intensity increases with increased oleate concentration, likely due to better passivation of intrinsic defects on LHP surface. The dependence of the PL quantum yield on ligand concentation for cinnamate and oleate ligands is compared in Figure 5b, confirming the above conclusions. Lastly, Figure S6 contains additional analysis showing an average nanoparticle size of 5.4 nm. Particles of this size contain many thousands of atoms, extending beyong capabilities of current ab initio electronic structure methods, justifying the slab model used in the present studies.

In conclusion, we have screened fifteen different ligands, elucidating their binding motifs and influence on the electronic properties of the bound ligand-LHP QD systems. Various ligand characteristics have been studied, including extent of the π-electron conjugated system, electron withdrawing and donating groups, ligand size, bridge length, and binding type. Several levels of electronic structure calculations have been considered and benchmarked with experimental measurements. Comparison with experiment has indicated that the simple PBE level of theory provides the best results due to fortuitous cancelation of several systematic errors. The more advanced HSE functional and inclusion of SOC effects make the calculations less reliable overall at an increased computational expense, further supporting the preference of the PBE functional for accurate calculations. The calculations reveal several important trends that can guide design of LHP ligands. The carboxylate group tends to provide stronger binding than ammonium because it acts as a bidentate ligand. The electronegative oxygen atoms present in the carboxylate group lower orbital energy levels of the ligand compared to the ammonium binding group. Ligand binding can induce notable local distortions in the LHP around the binding site (most clearly observed in ligands containing large bridges), and the distortion tends to increase the local LHP band gap. It is possible to design ligands with electronic state energies located inside the LHP bands, close to the CB edge, and inside the fundamamental band gap. None of the considered ligands exhibited states near the VB edge. The energy of the ligand LUMO can be lowered by increasing the size of the ligand's πelectron conjugated system and through introducing electronegative, electron withdrawing atoms and substituents. One can easily obtain systems with frontier orbitals localized on either the LHP or ligand. More careful design is needed to have frontier orbitals extend between the LHP and ligands. Extension of LHP electronic states onto ligands promotes interactions between LHP QDs and makes charges more readily available for transport and chemical reactivity. One can also design ligands that localize frontier orbitals onto the LHP surface without extending said orbitals onto the ligands. This occurs through local surface distortions at the binding site. Such surface states do not act as charge traps. Charges can easily escape into the LHP bulk, while at the same time they are available on the surface to perform chemical reactions or participate in charge or exciton transport. At high ligand coverage and with large ligands, ligand-ligand interactions can become important, influencing ligand binding and electronic properties. A general strategy for design of ligand-LHP systems can be suggested by first considering a combination of the π-electron conjugated system and binding group, and then fine-tuning the ligands' electronic energy levels by electron withdrawing or donating substituents to the π-system. The trends found with ligands to the CsPbBr3 perovskite should be applicable generally to other metal halide perovskites as well. The reported results provide important guidelines and insights into ligand design for LHP applications in photovoltaics, optoelectronics, quantum information systems, and beyond.