1 Introduction

Nanoparticles that show unique size-dependent optical and electrical properties are referred to as quantum dots (QDs) due to their quantum confinement. Reed (1986) was the first who defined the term “quantum dot” in describing ∼250 nm molecules, which were then named “spatially quantized system” within which carriers have “zero degree of freedom”. “Colloidal” quantum dots produced by wet chemical synthesis were prepared by as zero-dimensional material (Steigerwald and Brus 1989). Currently, the terms “colloidal quantum dot” and “colloidal semiconductor nanocrystal” are used interchangeably. The intricacies of QDs electronic structure depend upon the size and shape of the nanocrystal as well as the band structure of the corresponding bulk material (Klimov 2007, 2014; Nozik et al. 2010; Carey et al. 2015). By alternating the size of nanoparticles and ligand in the solution, the contributions of nanoparticle size and ligand to the energy of the highest occupied molecular orbital (EHOMO), the lowest unoccupied molecular orbital (ELUMO), and the energy gap ΔE could be systematically determined. As a result, photo-voltage, driving forces, and spectral response of highly efficient donor-linker-acceptor components could be optimized (Hagberg et al. 2007). For example, Cui et al. (2001) studied the interaction of QDs with proximate organic molecules present in solution. These ligands influence the electronic and vibrational states of the QDs, the energy of the electron donor transfer and the structural components of QDs assemblies. Moreover, the ligands participate productively in photo-physical processes that make QDs so attractive for many applications in specific photocatalytic reactions. Photo-redox catalysis of QDs is related to the combination of the electronic structure and environment at the surface because QDs catalysts both influence the energy levels of the QDs excited state and drive chemical reactions (Harris et al. 2016). For all these processes, one of the most important variables is the QDs size to tune the electronic property such as the QD EHOMO, ELUMO, and the ΔE between the EHOMO and the ELUMO. Therefore, the relationship between these electronic properties and the QDs diameter, d (nm), is quantified to predict the selectivity of photo-catalysis. ΔE is an important descriptor for the excited state property of organic molecules and usually directly affects their excited state reactions. Quantum mechanical coupling of the QDs, the EHOMO and the ELUMO controls the Raman intensity, thus the optically excited electrons in the metal (Brus 2008). The amplitude of the energy gap of the frontier orbitals, ΔE, is a way to rationally control electron transport properties between molecules (Yoshizawa 2012). Modifications to this interface can change the potential barrier in both the height and the shape of the wave functions. Therefore, they will affect the tunnelling of the electron and hole wave functions into the interfacial region (Harris et al. 2016).

The unique electrical, catalytic, magnetic, mechanical, photonic, and thermal properties of QDs results in their wide applications in industry, agriculture, business, medicine, clothing, cosmetics, and food in the twenty-first century (Oberdörster et al. 2005; Gonzalez et al. 2008; Schmalhofer et al. 2008; Yin et al. 2009), especially in environmental applications. Applications for soil, groundwater, wastewater, air remediation are some of the issues where the ability of nanotechnology to minimize pollution is in progress and could represent revolutionary changes in the environmental field (Bovea et al. 2010; Smith and Nie 2010; Crane and Scott 2012; Zhang et al. 2013). Wastewater remediation using engineered nanoparticles is one of the field among the various applications of the nanotechnology where QDs are more applied (Yadav et al. 2017; Smith and Nie 2010). Thus, the optimization in the synthesis of QDs to control morphologies and sizes must be considered fundamental. Moreover, it is also necessary to develop new methods to assess better understanding of QDs based innovation (Zhang 1997; Burda et al. 2005).

Among the quantum dots under intensive investigation, CdSe stands out as a promising candidate in different applications with its particle size as a determining factor on its optical band gap. Its unique tuneable optical, magnetic, and chemical features are widely used not only in medicine, where they improve the diagnosis, imaging, and treatment of disease but also in water treatment (Liou et al. 1996; Dahan et al. 2003; Alivisatos et al. 2005; Mahmoodi et al. 2016). To improve QDs efficiency, organic ligand phase-transfer process may modify QDs from their hydrophobic state to hydrophilic state (Yu et al. 2007).

Therefore, the main objectives of the current study are: 1) to develop a database of electronic properties such as EHOMO, ELUMO, and ΔE for different QDs of semiconductors and ligands; 2) to conduct regression analysis between EHOMO, ELUMO, and ΔE with the diameter of QDs and interpret the meaning of these regression equations; 3) to validate the correlation equations using other independent data set which was not used in regression analysis; and 4) to quantify the uncertainty of the predicted values by the prediction equations and the experimental data.

2 Material and Methods

2.1 Theory

The exciton wave function is the product of that for electron and hole of QDs, as follows:

$$ {\Psi}_{\mathrm{e}\mathrm{xc}}\kern0.5em =\kern0.5em {\Psi}_{\mathrm{e}}{\Psi}_{\mathrm{h}} $$
(1)

where Ψexc, Ψe and Ψh are the exciton, the single-electron and single-hole wave functions, respectively.

When the states of the QDs are close to the bulk conduction and valence band edges near k = 0, the effective mass/envelope function approximation (EMA/EFA) could be used for the electron and hole wavefunctions:

$$ {\Psi}_{\mathrm{e}}\left(\mathrm{k},\mathrm{r}\right)=\kern0.5em {\upmu}_{\mathrm{CB}}\left(\mathrm{k}\right)\ {\Psi}_{\mathrm{e}}\left(\mathrm{r}\right) $$
(2)
$$ {\Psi}_{\mathrm{h}}\left(\mathrm{k},\mathrm{r}\right)=\kern0.5em {\upmu}_{\mathrm{CB}}\left(\mathrm{k}\right)\ {\Psi}_{\mathrm{h}}\left(\mathrm{r}\right) $$
(3)

where the bloch wavefunctions of the conduction band μCB(k) and valence band μCB(k) have been separated from the electron Ψe(r) and hole Ψh(r) envelope wavefunctions. The envelope wavefunctions can be calculated by solving the Schrödinger equation (HΨ = EΨ). Considering the spherically symmetric wavefunctions for a particle-in-a-box, it is possible to simplify the equation by ignoring the radial and angular dependencies θ and ϕ. Therefore, the following treatment would be accurate for the confinement energy of a quantum dot band-edge exciton (EBEE), which represents the difference between the bulk energy gap and the lowest energy optical transition. The BEE state of 1S3/2 − 1Se is S-like but will not be accurate for non-S-like states such as the 1PS3/2 − 1Pe. If the electron of a QDs particle is considered as a particle-in-a-box, the Hamiltonian can be written as a sum of the kinetic energy, confinement potential and Coulomb potential:

$$ {V}_{conf,e}=\frac{V_{\infty, e}}{1+{e}^{-\left(r-R\right)/s}} $$
(4)
$$ {V}_{conf,h}=\frac{V_{\infty, h}}{1+{e}^{-\left(r-R\right)/s}} $$
(5)

where: V∞,e and V∞,h are the potentials outside the quantum dot for the electron or the hole, respectively; R is the quantum dot size; and s is the softness factor, whose magnitude is directly proportional to how rapidly (in terms of length) the confinement potential changes as the variable r approaches the dot radius R radiating outward from the center of the QDs. Very small values of s would most closely resemble a particle-in-a-box potential.

2.2 Database Description and Statistical Analysis

The data were collected from different peer reviewed research papers to evaluate the energy level of different kinds of QDs in the presence of different organic capping ligands. The data collected were analyzed using SPSS Regression Models™ to establish the correlation between independent and dependent factors. The size of QDs is the independent variable in all cases studied. Different dependent factors were taken into account: EHOMO, ELUMO, ΔE. Energy data were regressed using linear models to obtain the slope and the correlation coefficients. The linear equation used is as follows:

$$ y= kd+C $$
(6)

where: y is the dependent variable, such as EHOMO, ELUMO and ΔE; d is the independent variable (size in nm); k is the slope of the straight line and C is the constant.

The regression equations are reported and discussed in terms of their correlation coefficients and slopes. The regression equations were evaluated and compared. The k value, as slope of the equation line, determines through the sign the concordance or discordance of the variation of the variables; moreover, the higher is the value of k, the higher is the influence of the size on the dependent variable. The constant C, according to other parameters, can be an indicator of the dependent variable level at the lower values of size; thus, it can judge the similarity between the different kinds of particles used or between the ligands. The model was further confirmed by using of MATLAB (matrix laboratory) as multi-paradigm numerical computing environment. The residues, as differences between the observed value and the estimated value were evaluated. The equations are presented in graphs to compare the different parameters obtained by the different QDs and ligands used in the experiments.

3 Results and Discussion

Figure 1 describes statistics on QDs size found in Table 1. Figure 2 analyzes EHOMO level frequencies found in Table 2. Frequency analysis was performed to detect if the datasets could be compared. Figure 1a depics the histogram plot of distribution of different QDs size at different diameters (nm) related to Table 1 dataset. The QDs size spreads from 2 to 8 nm. The majority of available data belong to particles with 3.9 nm size as indicated by the mean value with a standard deviation of 1.65 nm. The second dataset (Table 2) is analysed in Fig. 1b, which shows a range of available sizes between about 2 and 10 nm. The mean value QDs size is 4.1 nm with a standard deviation of 1.49 nm. The comparison between the two histograms suggests that the two datasets can be taken as sample cases because they show similar distribution in the available sizes of particles used to perform the experiments.

Fig. 1
figure 1

Histograms of QDs size distribution of two different sets of databases: a Table 1; b) Table 2

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Table 1 EHOMO and QD size
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Fig. 2
figure 2

EHOMO (eV) frequencies distribution of two different sets of databases: a Table 1; b) Table 2

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Table 2 EHOMO, ELUMO, and QD size
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Figure 2a and b show the calculated frequencies of EHOMO level evaluated in its absolute value for the data of Table 1 and Table 2, respectively. The descriptive statistics of the EHOMO values are evaluated on two independent datasets and appear to be comparable since the two histograms show similar mean values equal to 5.52 eV and 5.26 eV, respectively, with standard deviations 0.35 nm and 0.40 nm, respectively.

The hypothesis is that there are correlations between the EHOMO, ELUMO, or ΔE with the QDs size. First, a linear model was applied to the dataset reported in Table 1. The results show that the EHOMO is simultaneously influenced by both the type and size of QDs and the associated ligand. The R2 values shown in Table 3 suggest that expecially for PbS QDs a linear correlations model can confidently be applied. The range of dimension of the QDs goes from 2 nm to 9 nm, the absolute value of EHOMO ranges from 4.80 eV to 5.80 eV. All the samples show higher EHOMO values for smaller sizes, as expected. The slope of the model equation, k, indicates how the size impacts on EHOMO energy level. The negative values indicate that with a decrease in size an increase of the energy level is recoded, as expected from the hypothesis. All the modelled equations have slopes k ranging from −0.025 to −0.066, which suggest that the change rate of EHOMO with the QDs size d is negative. All constant C values are strikingly similar regardless of the type of QDs or different ligands. Therefore, when the QD size approaches a minimal size, the EHOMO will become constant regardless of what type of ligand is present (Table 3).

Table 3 Correlation between HOMO energy level of first dataset (Table 1) and size of QDs
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In Table 4, it is evaluated if there is any absolute correlation between size and HOMO energy level independently of the kind of particle considered (PbS and PbSe QDs). The linear model equation results in a R2 value of 0.72, which confirms the hypothesis of good correlation between the size of particles and the energy level, despite, in this case, of the differences between QDs considered. The analysis was performed also for all the other particles, but the result obtained was not good. Only for the reported ones an R2 > 0.5 was found.

Table 4 Correlation between Energy of HOMO and QD size
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The same regression model was applied to the second independent dataset of Table 2: the same kind of QDs but different ligands were used in the second dataset (Table 5). Good correlation was achieved for all data collected; calculated R2 in all cases showed values higher than 0.91, which implies high correlation between variables. It is important not to be confused by the slope obtained in this analysis: they are negative as expected, the higher value of the rate for the CdSe with 1-decanethiol ligand is due to the narrower range considered in the dimension of QDs.

Table 5 Correlation between the EHOMO and the QD size of second dataset (Table 2)
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The energy bandgap is a function of both EHOMO and ELUMO values; regression analysis was performed on ELUMO energy level against the size of QDs. The results are summarized in Table 6. Good correlations are achieved for all cases with R2 values greater than 0.78. Moreover, as expected from the hypothesis, in most cases, the ELUMO level increased with the size of the QDs. Indeed the k values are always positive, except for CuInS2. The values are in all cases lower than unit, which means that for a unit decrease in particle size (1 nm) the energy level decreases less than one unit value (eV). A comparison was made on the energy band gap as difference between the EHOMO and ELUMO level against the size the QDs, by considering the different capping ligands used.

Table 6 Correlation between the ELUMO and the QDs size
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The energy band gap ΔE (eV) is equal to the difference between EHOMO and ELUMO, as follows:

$$ \varDelta E={E}_{HOMO}-{E}_{LUMO} $$
(7)

Statistical analysis on the level of energy band gap ΔE is reported in Table 7. The data show good correlation against size confirming the hypothesis previously made. The R2 value was in all cases higher than 0.5, indicating good correlation especially for CdSe QDs. The slope, k, index of rate of correlation, shows in all cases, except for CuInS2, negative values, which means that with the decrease of QDs size the energy bandgap increases. The slope, k, varies from −0.13 to −1.67. Therefore, the higher value of the slope shown by CdSe QDs in 1-decanethiol is due to the narrowed range of size considered, thus data available belongs to a lower array of values. Apparently, the constant C values are significantly different from each other, which implies that the ligands have different impact on the regression between the ΔE and QDs size.

Table 7 Correlation between ΔE in specific ligands with the QDs size
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Matlab was used to present the residuals of the regression model used in the analysis for the band gap energy values ΔΕ calculated using Eq. (7). Figure 3 shows the more interesting cases evaluated in terms of R2 values and k values about ΔE regression analysis evaluated as size dependent variable. The graphs were divided according to the ligand used in the synthesis of the QDs.

Fig. 3
figure 3figure 3

Model equations and residues for regression analysis of QDs with the related ligand. From top graph to bottom: oleate; 2-mercaptoethanol; TOPO/dithiol/Au; TOPO

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The size of CdSe QDs can be considered as a fundamental parameter for the tunnelling of energy band gap evaluated as difference between EHOMO and ELUMO values. The energy band gap is also clearly influenced by organic ligands used as capping materials. Figure 4 shows that for CdSe QDs, with 1-decanethiol as capping ligand, the angular coefficient is −1.7: with low decrease in size we can notice higher decrease in band gap value with respect to the bisector 1:1. For the other ligands capping values belong to the range of −0.1 to −0.3, which means that not only achieved similar results, but confirm the hypothesis that a unit decrease in size is associated with an increase of energy band gap lower with respect to the bisector. The later observations demonstrate that size of CdSe strongly influence the energy band gap. CdS in TOPO and in TOPO/dithiol/Au has stronger capping ligands, as they show comparable size used in available data and comparable behaviour in previous regression tests. Results of simultaneous regression analysis are shown in Table 8 and Fig. 4.

Fig. 4
figure 4

Model equations and residues for regression analysis of chosen CdSe in multiple ligands

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Table 8 Correlation between ΔE for CdSe and the QDs size in multiple lignads
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Table 8 and Fig. 4 show that good correlation was achieved by the data available, with an R2 value of 0.63. The slope of the linear regression equation, k, shows that a unit value decrease in size corresponds to 17% increase in energy level in the condition treated, in particular with size range between 2 and 7 nm and with selected ligands and capping.

The electronic structure of a quantum dot, strictly connected to photo-physical and chemical processes, differs from bulk sand purely molecular systems. The major reasons are the size of QDs which makes electrons quantum-confined, and the properties of the surface which defines directly the interface to the environment that surrounds the QDs. Quantum dots exhibit opto-electronic properties that are intermediate between bulk semiconductors and discrete molecules, thus they turn to be function of both size and shape (Smith and Nie 2010). QDs have sizes comparable to the de Broglie wavelength, thus the excitons result to be squeezed defining quantum confinement quantified as strong, weak or intermediate, according to the difference in size (Ramirez et al. 2010). The discrete topology of the energy spectrum, mirror of opto-electronic properties of materials, is highly influenced by the confining dimension. Therefore, the band gap becomes size-dependent. The characteristic delocalization length, quantified through the “Bohr radius”, is influenced by the size (Rossetti and Brus 1982; Efros and Rosen 2000). Differences in opto-electronic response arise mainly from the number of free charges (electrons and holes) and their free path (Wertheim et al. 1986). Moreover, demonstrations in correlations between polarizability and size was performed by (Ghanty and Ghosh 1993). Ionization potentials and EHOMO level showed linear correlation according to (Zhang and Musgrave 2007).

Organic ligands are useful as growth surfactants and solubilizes in various media, but especially they participate in photo-physical processes that make QDs very attractive for applications such as optical sensing, fluorescent tagging, and photocatalysis. Indeed in addition to size, the modifications of surface interface by changing the molecules coordinated by the QDs lead to changes in height and shape of the potential barrier, and therefore, the degree of tunnelling of the electron and hole wave functions into the interfacial region (Harris et al. 2016; Kroupa et al. 2017). The electronic and vibrational interactions of QDs with proximate organic molecules, such as ligands, could modify the surface states, the energy and the electrons, the donors and acceptors of the QDs core and structural components. The surface ligands can stabilize the QDs geometry and prevent the formation of dangling bonds on the surface, which could otherwise act as trap states for the photo-generated carriers. The anchoring geometry of the ligand molecules plays an important role on the QDs optical properties (Azpiroz and De Angelis 2015). Passivation of the surface with organic ligands stabilizes the structure of QDs, prevents this surface relaxation, and leads to further change in absorption spectra relative to the bare QDs (Inerbaev et al. 2009).

Changes in opto-electronic properties are linked to confinement, thus QDs size and ligand-type also affect these properties. Despite the complexity, through the manipulation of the QDs, it is possible to control the optical response and phonon-driven dynamics (Kilina et al. 2016). Therefore, many optical and electrical properties of semiconductor become function of size, for example the band gap ΔE, the absorption and fluorescence wavelengths, the ionisation potentials and electron affinities, the availability of electrons for forming bonds or getting involved in redox reactions, and thus, the catalytic activity and selectivity (Roduner 2006).

An increasing concern focusing on contaminants in the environment was registered in the past century. The methods mainly applied to eliminate pollution are physical adsorption, chemical degradation or advanced oxidation technologies. QDs are usually synthesized by elements from II–VI (e.g., CdSe, CdTe, CdS) and III–V or IV–VI (e.g., PbSe) groups. These materials, with their rare nanometer-scale crystals, have fast become a hotspot for various applications, because of their superior optical and electro-chemical properties.

In particular, CdSe QDs were reported to be a conventional semiconductors, consisting an excellent material to apply in advanced water treatment technologies. However, this material has also some disadvantages, such as photo-corrosion or photo-dissolution, that may restrict its stability and applicability. The defects of CdSe QDs can be improved by further modification using some capping-ligands (Ma et al. 2015; Mahmoodi et al. 2016). The unique properties that make QDs such an interesting material for wastewater treatment are strictly connected with their properties and structures. In our research, we try to correlate the morphological properties, in particular the size, to the electronic energy level for CdSe and other kind of QDs in presence of different ligands. The values considered were withdrawn from a dataset present in the literature and real case studies.

4 Conclusions

We have analyzed correlations between EHOMO, ELUMO, or ΔE and the QDs size in the presence of different ligands. Linear regression was found to be suitable for many of them in terms of correlation coefficient R2, which was considered as an indicator of the goodness of model fitting. It was proved that both size and nature of organic ligand capping influence the energy levels of QDs. Optimal result was found for CdSe QDs. This class of QDs have stronger correlation between frontier molecular orbital energy levels and QDs size. The R2 value is greater than 0.63 for a regression equation with coefficient value (slope) of −0.17 and constant 2.9. This result implies that a unit decrease of size, in a range lower than 7 nm, involves an increase in the band gap of 17%. CdSe QDs are well known as suitable materials for advanced water treatment; this is due to their tunnable optical and electronic propeties which we demonstrated in correlation to the size and surface morfology of the particles.