1. Introduction

As the name suggests, quantum-dot molecules are the quantum-mechanical systems which are made from coupled semiconductor nanocrystals (quantum dots) and have delocalized electronic states and molecular-like wave functions [1]. Much like the naturally occurring molecules are the elementary building blocks of molecular crystals, quantum-dot molecules can serve as the building blocks of higher-order structures — quantum-dot supercrystals [2]. Owing to their unique and readjustable optical properties, which significantly differ from the properties of noninteracting quantum dots and can be individually tuned over wide ranges by changing the number, relative positions, sizes, and shapes of individual nanocrystals, quantum-dot molecules is a very promising material base for the next-generation quantum information technologies and devices. The last five years have already witnessed various experimental realizations of such molecules [1] and their multiple uses in the implementation of solar cells [3], qubits [4,5], logic gates [6], and quantum memories [7].

The optical activity of chiral molecules is naturally weak due to the smallness of molecules compared to the excitation wavelength and the relatively low density of the constituting atoms [8, 9]. While semiconductor nanocrystals are also much smaller than the optical wavelength, they can be anticipated to exhibit a much stronger optical activity due to the significantly higher density of atoms. A great deal of research efforts have therefore been recently focussed on studying the chiroptical response of individual chiral nanocrystals [10–13], achiral quantum-dot molecules [14, 15], and quantum-dot supercrystals [16, 17]. In particular, it was theoretically shown that otherwise optically inactive semiconductor nanocrystals exhibit strong optical activity in the presence of screw dislocations [18], asymmetrically positioned inside them ionic impurities [19], and chiral surface defects [20]. This activity was found to be significantly stronger than that of chiral organic molecules whereas the asymmetry in the interaction of these chiral nanocrystals with left- and right-handed light was shown to be comparable to that of chiral plasmonic complexes [21–28]. It was also shown that by assembling achiral quantum dots into helix-like structures much longer than the excitation wavelength, one can make them fully absorb light of one circular polarization and transmit the light of the other [29].

This paper presents the first theory of chiral quantum-dot molecules, which allows one to analytically treat and optimize the optical activity of basic configurations of such coupled quantum systems. We begin with a formulation of the general theoretical model of artificial molecules made from an arbitrary number of sufficiently strongly coupled semiconductor quantum dots. Each of the quantum dots is assumed to have a dipole moment associated with the fundamental interband transition between the size-quantized states of its confined charge carriers. This assumption amounts to considering quantum dots as elongated nanorods with their largest dimension along the growth direction of the crystal structure. We use the developed model to analytically calculate the wave functions, energy spectrum, rotatory strengths, dipole absorption rates, and dissymmetry factors of quantum-dot dimers, trimers, and tetramers. The obtained expressions are then thoroughly analyzed to find the optimal configurations of the quantum-dot molecules featuring the strongest chiroptical response.

2. Theoretical model

Consider a chiral molecule composed of N arbitrarily oriented semiconductor nanocrystals (quantum dots) with coordinates r₁, r₂, . . . , r_N. Let all the quantum dots be identical in terms of size, shape, and chemical composition. Then the wave function of the first excited electronic state ψ_n(r) in the nth quantum dot can be represented in the form ψ_n(r) = ψ(r − r_n). This state is assumed to be strongly localized inside the quantum dots, so that the wave functions of different nanocrystals do not overlap and are normalized according to the condition

If the interaction between the quantum dots is negligibly small, then the first excited state of the quantum-dot molecule is N-fold degenerate and has energy E of state ψ. The interdot coupling partially or fully lifts this degeneracy [30], leading to the formation of delocalized molecular states Ψ₁, Ψ₂, . . . , Ψ_N of energies E₁, E₂, . . . , E_N. These states obey the stationary Schrödinger equation ĤΨ_μ = E_μΨ_μ and can be experimentally studied by analysing the absorption and luminescence spectra of the molecule. By denoting the matrix element of the interaction between dots n and m as V_nm, we can write the Hamiltonian of the quantum-dot molecule in the form

The rotatory strength of the interband transition that creates an electron in state Ψ_μ is given by the Rosenfeld’s formula [31,32]

One can see that regardless of the nature of the interaction between the quantum dots, the rotatory strength vanishes if all the dipoles are parallel to each other. The fact that this strength also vanishes in the hypothetical situation where the displacement vectors of the quantum dots are all alike (r₁ = r₂ = . . . = r_N) implies that the rotatory strength is independent of the origin of coordinates. It should also be noted that the rotatory strength R_μ is proportional to the total optical helicity emitted by the quantum dot molecule (see Ref. [33] and, more specifically, Eq. (5) from Ref. [34]).

The full dipole absorption rate upon the excitation of state Ψ_μ is given by the sum [10]

Finally, in order to complete the above model of optical activity, we approximate the coupling between the quantum dots inside molecules by the dipole–dipole interaction potential

3. Quantum-dot dimer

A quantum-dot molecule exhibits optical activity if it lacks a center or plane of symmetry. We now apply the developed general formalism to three kinds of quantum-dot molecules with the aim of optimizing their structure and achieving the strongest manifestation of optical activity in the circular dichroism (CD) spectra.

We begin by considering the simplest molecule consisting of two identical quantum dots, each of which has its own direction of the dipole moment. The dipole moment of the jth quantum dot (j = 1 or 2) is described by a pair of angles (ϑ_j, φ_j) shown in Fig. 1(a). The interaction between the dipoles lifts the double degeneracy of the molecular states, producing a pair of optically active states of energies E_1,2 = E ± δE, where δE is given by Eq. (19). The energy gap between the dimer states scales like ∝ d²/(εa³) whereas the rotatory strengths of the transitions to these states scale like ∝ σad², where a is the distance between the quantum dots. These two scaling factors are characteristic of quantum-dot molecules with the dipole–dipole coupling. The angular dependencies of the energy gap and rotatory strengths are of the forms (see Appendix A)

 

Fig. 1 Quantum-dot (a) dimer, (b) trimer, and (c) tetramer composed of equal but differently oriented semiconductor quantum dots. The molecules become optically active for certain orientations of dipole moments d_j of the fundamental interband transitons inside the quantum dots; a is the distance between the quantum dots in the dimer, trimer, and in the base of the tetramer; b is the height of the tetramer.

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Suppose that the lineshapes of the two transitions can be approximated by Lorentzians Γ(E_j, ω) = (γ/π)/[(ħω − E_j)² + γ²] of the full width at half maximum (FWHM) 2γ. Then the CD spectrum of the quantum-dot dimer has the form

Figures 2(a)–2(c) show one kind of enantiomers (with negative δφ) of the three optimal quantum-dot dimers. While dimer (a) corresponds to the absolute maximum of the CD peaks, dimers (b) and (c) are the saddle-point configurations. Therefore, there is a continuum of optimal dimer configurations, with their CD-peak intensities spanning between those of dimers (a) and (c). These configurations are schematically shown by the shaded regions in Fig. 2(d).

 

Fig. 2 Optimal configurations of quantum-dot dimers exhibiting the strongest peaks in the CD spectrum for γ ≫ δE. The dipoles of the upper quantum dots are all in the yz plane. The intensities of peaks in the CD spectrum of dimer (a) exceed the intensities of peaks in the spectra of dimers (b) and (c) by a factor of $\sqrt{3}/2$. Panel (d) shows the transformation between the optimal configurations (c) and (a), which leads to a continuum of intermediate optimal orientations of the quantum-dot dipoles.

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As we have seen, the linear growth of the rotatory strength with the distance between the quantum dots due to the increase of the angular momentum is accompanied by the much faster reduction of the energy gap between the two molecular states (like ∝ a⁻³) due to the weakening of the dipole–dipole coupling. This eventually reduces the intensity of the CD signal. The dependence of the dissymmetry factors on a is somewhat more intricate. For ϑ₁ = ϑ₂ these factors are given by the expressions (see Appendix A)

The absolute values of both dissymmetry factors are seen to grow like ∝ a for a ≪ 1/σ when the electric-dipole absorption is stronger than the magnetic-dipole absorption, and to decrease like ∝ 1/a when a becomes large and the magnetic-dipole absorption starts dominating the electric-dipole one. The maximal values g₁ = − χ₁ cot(δφ/2) and g₂ = (2/χ₂) sin δφ are attained for ${\chi }_1=\sqrt{2(1-\text{cos}\delta \phi )}$ and ${\chi }_2=\sqrt{4{\text{csc}}^2{\vartheta }_1-2(1-\text{cos}\delta \phi )}$. For example, $g_1=\mp 1/\sqrt{2}\approx \mp 0.71$ and g₂ = ±1 are the maximal dissymmetry factors for the dimers whose parameters are given in Eq. (13). The analogous values for the dimers shown in Figs. 2(b) and 2(c) are $g_1=-1/g_2=-1/\sqrt{2-\surd 2}\approx -1.31$ and $g_1=-1/g_2=-1/\sqrt{2+\surd 2}\approx -0.54$ respectively. These values are about 10³ − 10⁴ times larger than the typical g-factors of chiral organic molecules and can exceed the dissymmetry factors of intrinsically chiral nanocrystals by more than an order of magnitude [11,36,37].

It is instructive to estimate the typical values of the scaling factors σ, η, and ρ. By assuming the following material parameters: ε = 5, E_g = 2 eV, d = 200 D, and a = 7 nm, we get σ⁻¹ ≈ 75 nm, η ≈ 14.6 meV, and ρ ≈ 1.88 × 10⁻³³ erg×cm³. Hence, the rotatory strengths of quantum-dot molecules can be up to 10⁴ times larger than the rotatory strengths of chiral semiconductor nanocrystals doped with impurity ions [19], and can exceed the rotatory strengths of chiral molecules by five to six orders of magnitude [38–40]. The obtained estimate also justifies the approximation of broad spectral lines (γ ≫ δE), which was employed to study the optimal dimer configurations. Indeed, the energy parameter η is much smaller than the typical linewidths (∼ 100 meV) of interband transitions in semiconductor quantum dots at room temperature [41,42].

4. Quantum-dot trimer

Consider next three quantum dots located in the vertices of an equilateral triangle shown in Fig. 1(b). Let the direction of the dipole moment of the first quantum dot be set by the polar angle ξ and the azimuthal angle φ whereas the dipole moments of the second and third quantum dots have the same polar angle ϑ and azimuths φ + 2π/3 and φ − 2π/3 respectively. This molecule has three generally nondegenerate and optically active states (see Appendix B) of energies E₁ = E − B and E_2,3 = E + δE_∓, where $\delta E_{\pm }=B/2\pm \sqrt{2A^2+{\left(B/2\right)}^2}$ and coefficients A and B given in Eq. (33) are the functions of ξ, ϑ, and φ. The rotatory strengths of all the states vanish in essentially achiral molecules with φ = π/2 or 3π/2.

Trimers with ξ = −ϑ are of particular interest, because the sign of their rotatory strength R₁ is different to the signs of the other two rotatory strengths regardless of the specific directions of the quantum-dot dipoles. Unless φ is specifically chosen such that δE₋ = −B, there can be up to three peaks in the CD spectrum of such trimers, the strongest of which is often produced by the first molecular state. If this peak is much narrower than the energy spacing between states Ψ₁ and Ψ₂, which occurs for γ ≪ B + δE₋, then its intensity scales in direct proportion to the rotatory strength R₁ ∝ sin 2ϑ cos φ. In this case, the optimal trimers are described by parameters ϑ = π/4 or 3π/4 and φ = 0 or π. One of the optimal trimer configurations is shown in Fig. 3(a).

 

Fig. 3 Quantum-dot trimers with the strongest CD peaks for (a) γ ≪ B + δE₋ and [(c)–(f)] γ ≫ A. Trimers in panels (c) through (e) correspond to three critical points of surface R₁ A shown in panel (b). Transformation between the two optimal configurations, shown by the green curve in panel (b), is illustrated by the shaded regions in panel (f). The vertical scale in (b) is from −0.6 to 0.6 and the six contour lines are −0.5, −0.3, . . . , 0.5.

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Another kind of trimers that deserves special consideration is the one with ξ = ϑ. Such trimers have a doubly degenerate state of energy E_1,2 = E − A and a nondegenerate state of energy E₃ = E + A. As we have seen earlier from Eq. (12), the intensity of the CD peaks produced by the population of these states is determined for γ ≫ A by the product of the rotatory strength and the energy gap between the states, i.e. by R₁ A. This product is plotted as a function of angles ϑ and φ in Fig. 3(b). Three kinds of optimal trimer configurations correspond to the following three types of critical points of this function [see Eqs. (41) and (44)]:

The analysis of trimers with ξ ≠ ±ϑ is more complicated and can be performed using the general expressions given by Eqs. (33) through (38) in Appendix B.

5. Quantum-dot tetramer

As the last example, we study the optical activity of four coupled quantum dots located in the vertices of a tetrahedron shown in Fig. 1(c). The base of the tetrahedron is an equilateral triangle formed by the quantum dots whose dipole moments are assumed to be in the plane of this base. The azimuths of dipoles d₁, d₂, and d₃ are α + 2π/3, α − 2π/3, and α respectively, and the direction of the fourth dipole is set by the polar angle ϑ and azimuth φ. The energies and wave functions of this tetramer states have the simplest forms for ϑ = 0 and ϑ = π/2. These two special cases are considered separately in Appendix C.

Tetramers with ϑ = 0 have two nondegenerate optically active states of energies E_1,2 = E − δE_±, and two doubly degenerate optically inactive states of energies E_3,4 = E + A, where $\delta E_{\pm }=A\pm \sqrt{A^2+3B^2}$ and coefficients A and B are given in Eq. (50). The rotatory strengths of the nondegenerate states are proportional to sin 2α, vanishing for achiral tetramers with α = ±π/2, as well for chiral ones with α = 0 and ±π — in which the fourth dipole decoupled from the rest three. Figure 4(a) shows how E_j − E and R_j vary with α. The rotatory strengths heavily depend on height b of the tetrahedron, turning zero when either b = 0 or b → ∞ due to the absence of coupling between the dipoles in the basic triangle and in the vertex. It is easy to show using Eqs. (50) and (54) that the strength of this interaction peaks for $b=a/\left(2\sqrt{3}\right)$, where a is the distance between the quantum dots in the base, and that so do the rotatory strengths and the product R₁(δE₊ − δE₋), which determines the intensities of relatively broad lines in the CD spectrum. Since R₁(δE₊ − δE₋) ∝ sin 2α, the CD signal is also the strongest for tetramers with α = ±π/4 or ±3π/4.

 

Fig. 4 Energies (upper panels) of four tetramer states and rotatory strengths (lower panels) of the respective interband transitions as functions of azimuth α for a = b and different polar angles ϑ, which determine orientations of the quantum dots constituting the tetramer [see Fig. 1(c)]. The azimuth φ of the fourth dipole affects neither energies nor rotatory strengths.

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When the direction of the fourth dipole deflects from the vertical, the degeneracy of states Ψ₃ and Ψ₄ is lifted and the fourth state, too, becomes optically active. Figures 4(b) through 4(g) show how the α-spectra of energies and rotatory strengths transform with ϑ. Note that the shapes of these spectra are independent of the azimuth φ of the fourth dipole, which affects only the wave functions of the tetramer states. Four anticrossings in the energy spectra of states Ψ₁, Ψ₂, and Ψ₄ are seen to result in the renormalization of the rotatory strengths’ spectra. As ϑ increases from zero to π/4, four anticrossings merge into two and then split back into four when ϑ is increased still further. At the same time, the rotatory strengths change drastically, and spectra R₂(α) and R₄(α) swap their places. It should also be noted that the variation of ϑ neither affects the energy of state Ψ₃ nor makes this state optically active.

Figure 4(h) shows the energies and rotatory strengths of four nondegenerate states of the tetramer with ϑ = π/2. The states’ energies are E₁ = E − 2A, E₂ = E + δE₋, E₃ = E + A, and E₄ = E + δE₊, where $\delta E_{\pm }=A/2\pm \sqrt{{\left(A/2\right)}^2+2\left(B^2+3C^2\right)}$ and coefficients A, B, and C are given in Eqs. (50), (56), and (57). The optically active states are Ψ₂ and Ψ₄. Much like in the case of ϑ = 0, the rotatory strengths vanish for α = 0, ±π/2, and ±π. There is also an optimal tetrahedron height, because R₂ ∝ b for b → 0 and R₂ ∝ 1/b² for b → ∞, but its depends on α and does not have a simple analytical expression. From Eqs. (56), (57), (59), and (64) it follows that R₂(δE₊ − δE₋) ∝ sin 2α, so that the strongest CD signal is achieved for tetramers with α = ±π/4 or ±3π/4. The same equations also show that the CD signal is the strongest for $b=a/\left(2\sqrt{3}\right)$.

6. Conclusion

We have developed an elegant theoretical model which allows one to calculate the rotatory strengths and dissymmetry factors of optical transitions upon the excitation of the lowest-order states of quantum-dot molecules. This model was used to analytically study the optical activity of molecules composed of two, three, and four semiconductor nanocrystals. We found the optimal geometries of such dimers, trimers, and tetramers resulting in the strongest CD signals when the widths of the CD peaks significantly exceed the energy gaps between them. It was also shown that the rotatory strengths of quantum-dot molecules can exceed the typical rotatory strengths of naturally occurring chiral molecules by five to six orders of magnitude. The derived analytical expressions and the discovered optimal geometries are useful for engineering optically active quantum-dot materials and devices for advanced nanophotonics applications.

Appendix A: Quantum-dot dimer

The positions and dipole moments of the two quantum dots in Fig. 1(a) are given by r₁ = (0, 0, 0), r₂ = (0, 0, a), and d_j = d (sin ϑ_j cos φ_j, sin ϑ_j sin φ_j, cos ϑ_j). The energies of the two molecular states and the coefficients in Eq. (3) are found to be give by

(18)$$E_1=E-\delta E,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_2=E+\delta E,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_1=\frac{1}{\sqrt{2}}(-1,1),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_2=\frac{1}{\sqrt{2}}(1,1),$$

(19)$$\delta E=\left(\text{sin}{\vartheta }_1\text{sin}{\vartheta }_2\text{cos}\delta \phi -2\text{cos}{\vartheta }_1\text{cos}{\vartheta }_2\right)\eta ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta \phi ={\phi }_1-{\phi }_2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\eta =d^2/\left(\epsilon a^3\right).$$

The rotatory strengths and dissymmetry factors are given by

(20)$$R_1=-R_2=-\rho \text{sin}{\vartheta }_1\text{sin}{\vartheta }_2\text{sin}\delta \phi ,$$

(21)$$g_{1,2}=\frac{4\chi \text{sin}{\vartheta }_1\text{sin}{\vartheta }_2\text{sin}\delta \phi }{2\left(\text{sin}{\vartheta }_1\text{sin}{\vartheta }_2\text{cos}\delta \phi +\text{cos}{\vartheta }_1\text{cos}{\vartheta }_2\mp 1\right)\mp {\chi }^2{\text{sin}}^2{\vartheta }_2},$$

where ρ = (χ/2) d², χ = aσ, and the minus or plus sign corresponds to state 1 or 2.

Optimal ϑ₁, ϑ₂, and δφ for broad CD peaks

In the case of broad spectral lines, the conditions of the extrema of the CD signal — ${\left(R_1\delta E\right)}_{{\vartheta }_1}^{\prime }=0$, ${\left(R_1\delta E\right)}_{{\vartheta }_2}^{\prime }=0$, and ${\left(R_1\delta E\right)}_{\delta \phi }^{\prime }=0$ — lead to the following system of equations:

(22)$$2\text{cos}2{\vartheta }_1\text{cot}{\vartheta }_2=\text{cos}\delta \phi \text{sin}2{\vartheta }_1,$$

(23)$$2\text{cos}2{\vartheta }_2\text{cot}{\vartheta }_1=\text{cos}\delta \phi \text{sin}2{\vartheta }_2,$$

(24)$$2\text{cos}\delta \phi \text{cot}{\vartheta }_1\text{cot}{\vartheta }_2=\text{cos}2\delta \phi ,$$

in deriving which we have taken into account that R₁ ≠ 0.

To solve this system under the condition R₁δE ≠ 0, we notice that the last equation requires that cos δφ ≠ 0. For cos 2δφ = 0 this equation reduces to cos ϑ₁ cos ϑ₂ = 0 and, together with the first pair of equations, gives cos ϑ₁ = cos ϑ₂ = 0. Taking into account that δφ is between −π and π, we thus get the first set of critical points

(25)$$\delta \phi =\pm \frac{\pi }{4},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\vartheta }_1={\vartheta }_2=\frac{\pi }{2}.$$

Because cos δφ ≠ 0, Eqs. (22) and (23) require that cos 2ϑ₁ ≠ 0 and cos 2ϑ₂ ≠ 0. Using the interchangeability of ϑ₁ and ϑ₂ in our system and assuming that all the multipliers in it are nonzero, we can eliminate cot ϑ₂ from Eqs. (22) and (24) to get cos 2ϑ₁ = cos 2ϑ₂ = 1 − sin⁻² δφ. This result reduces Eq. (24) to $\left(2\text{cos}\delta \phi +1\right)\left(\sqrt{2}\text{cos}\delta \phi +1\right)=0$, leading to another five sets of critical points

(26)$$\delta \phi =\pm \frac{2\pi }{3},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\vartheta }_{1,2}=\left\{\frac{1}{2}\text{arccos}\left(-\frac{1}{3}\right),\pi -\frac{1}{2}\text{arccos}\left(-\frac{1}{3}\right)\right\};$$

(27)$$\delta \phi =\pm \frac{3\pi }{4},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\vartheta }_1={\vartheta }_2=\frac{\pi }{2}.$$

The obtained six sets of critical points are given in Table 1. One can see that two absolute maxima and two absolute minima $R_1\delta E=\pm w/\sqrt{3}$, where w = ρη, are achieved for δφ = ±2π/3 and ϑ₁ = ϑ₂ = (1/2) arccos(−1/3). There is also another pair of optimal dimer configurations, with δφ = {±π/4, ±3π/4} and ϑ₁ = ϑ₂ = π/2, which have close values of peak CD signals corresponding to R₁δE = ±w/2. Note that the dimers with different signs of δφ are mirror images of each other, and that ϑ₁ = ϑ₂ = ϑ₀ and ϑ₁ = ϑ₂ = π − ϑ₀ describe the same dimer.

Table 1. Critical points of function R₁δE; ϑ₀ = (1/2) arccos(−1/3).

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Appendix B: Quantum-dot trimer

The trimer shown in Fig. 1(b) is described by the parameters

(28)$${\mathbf{r}}_1=\left(0,\frac{a\sqrt{3}}{2},0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{r}}_2=\left(-\frac{a}{2},0,0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{r}}_3=\left(\frac{a}{2},0,0\right),$$

(29)$${\mathbf{d}}_1=d\left(\text{sin}\xi \text{cos}\phi ,\text{sin}\xi \text{sin}\phi ,\text{cos}\xi \right),$$

(30)$${\mathbf{d}}_2=d\left(\text{sin}\vartheta \text{cos}(\phi +2\pi /3),\text{sin}\vartheta \text{sin}(\phi +2\pi /3),\text{cos}\vartheta \right),$$

(31)$${\mathbf{d}}_3=d\left(\text{sin}\vartheta \text{cos}(\phi -2\pi /3),\text{sin}\vartheta \text{sin}(\phi -2\pi /3),\text{cos}\vartheta \right).$$

The Hamiltonian of this molecule is of the form

(32)$$\left(\begin{array}{ccc}E& A& A\\ A& E& B\\ A& B& E\end{array}\right),$$

where

(33)$$A=\left(\text{cos}\vartheta \text{cos}\xi -\frac{1}{4}\left(6\text{cos}2\phi -1\right)\text{sin}\vartheta \text{sin}\xi \right)\eta ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}B=\left(1-\frac{3}{4}\left(1+2\text{cos}2\phi \right){\text{sin}}^2\vartheta \right)\eta .$$

The solution to the eigensystem problem is given by

(34)$$E_1=E-B,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_2=E+\delta E_{-},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_3=E+\delta E_{+},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta E_{\pm }=\frac{1}{2}\left(B\pm \sqrt{8A^2+B^2}\right),$$

(35)$$c_1=\frac{1}{\sqrt{2}}(0,-1,1),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_2=\frac{(-\delta E_{+},A,A)}{{\left(\delta E_{+}^2+2A^2\right)}^{1/2}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_3=\frac{(-\delta E_{-},A,A)}{{\left(\delta E_{-}^2+2A^2\right)}^{1/2}},$$

and the rotatory strengths are

(36)$$R_1=\rho \frac{\sqrt{3}}{2}\text{sin}2\vartheta \text{cos}\phi ,$$

(37)$$R_2=\rho \sqrt{3}A\frac{2\delta E_{+}\text{sin}(\vartheta +\xi )-A\text{sin}2\vartheta }{4A^2+B\delta E_{+}}\text{cos}\phi ,$$

(38)$$R_3=\rho \sqrt{3}A\frac{2\delta E_{-}\text{sin}(\vartheta +\xi )-A\text{sin}2\vartheta }{4A^2+B\delta E_{-}}\text{cos}\phi .$$

It is easy to verify that R₁ + R₂ + R₃ = 0 and that for ϑ = ξ we have R₁ = R₂ = −R₃/2.

Optimal ϑ and φ for broad CD peaks

For ϑ = ξ we have A = B, and the strongest CD signal is achieved when product R₁ A is maximal. In this case the critical trimer parameters ϑ and φ obey the system of equations

(39)$$\left(3\text{cos}3\phi -2\text{cos}\phi \right)\text{cos}2\vartheta =3\left(\text{cos}3\phi +2\text{cos}\phi \right)\text{cos}4\vartheta ,$$

(40)$$3\left(3\text{sin}3\phi +2\text{sin}\phi \right){\text{sin}}^2\vartheta =4\text{sin}\phi .$$

First of all, we notice that a simple solution to this system — corresponding to sin φ = 0 and 9 cos 4ϑ = cos 2ϑ — is given by

(41)$$\phi =\{0,\pi \},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vartheta =\left\{{\vartheta }_{\pm },\pi -{\vartheta }_{\pm }\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\vartheta }_{\pm }=\frac{1}{2}\text{arccos}\left(\frac{1\pm \sqrt{649}}{36}\right).$$

Next, for sin φ ≠ 0 this system reduces to

(42)$$20{\text{cos}}^2\phi =7,$$

(43)$$12{\text{sin}}^2\vartheta =5,$$

giving the following critical angles:

(44)$$\phi =\left\{{\phi }_{\pm },2\pi -{\phi }_{\pm }\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vartheta =\left\{{\vartheta }_0,\pi -{\vartheta }_0\right\},$$

(45)$${\phi }_{\pm }=\text{arccos}\left(\pm \frac{1}{2}\sqrt{\frac{7}{5}}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\vartheta }_0=\text{arcsin}\left(\frac{1}{2}\sqrt{\frac{5}{3}}\right).$$

Equations (41), (44), and (45) give 16 critical points of function R₁ A in the domain 0 ≤ φ < 2π and 0 ≤ ϑ ≤ π [see Fig. 3(b)]. Table 2 shows six different types of these critical points and the associated values of rotatory strength, energy parameter, and their product. The rest of critical points belong to one of these types, representing either dimers whose parameters are given in the table or their mirror-image enantiomers.

Table 2. Critical points of function R₁ A; angles ϑ_±, φ_±, and ϑ₀ are defined in Eqs. (41) and (45).

View Table | View all tables in this article

Appendix C: Quantum-dot tetramer

The positions and dipole moments of the quantum dots comprising the tetramer shown in Fig. 1(c) are given by

(46)$${\mathbf{r}}_1=\left(-\frac{a}{2},-\frac{a}{2\sqrt{3}},0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{r}}_2=\left(\frac{a}{2},-\frac{a}{2\sqrt{3}},0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{r}}_3=\left(0,-\frac{a}{\sqrt{3}},0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{r}}_4=(0,0,b),$$

(47)$${\mathbf{d}}_1=d\left(\text{cos}(\alpha +2\pi /3),\text{sin}(\alpha +2\pi /3),0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{d}}_2=d\left(\text{cos}(\alpha -2\pi /3),\text{sin}(\alpha -2\pi /3),0\right),$$

(48)$${\mathbf{d}}_3=d\left(\text{cos}\alpha ,\text{sin}\alpha ,0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{d}}_4=d\left(\text{sin}\vartheta \text{cos}\phi ,\text{sin}\vartheta \text{sin}\phi ,\text{cos}\vartheta \right).$$

There are two special cases in which the rotatory strengths of optical transitions are given by simple formulas.

The special case of ϑ = 0

For ϑ = 0 the Hamiltonian of the tetramer is of the form

(49)$$\left(\begin{array}{cccc}E& -A& -A& B\\ -A& E& -A& B\\ -A& -A& E& B\\ B& B& B& E\end{array}\right),$$

where

(50)$$A=\frac{1}{4}\left(6\text{cos}2\alpha -1\right)\eta ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}B=\frac{27a^{4}b\text{sin}\alpha }{{\left(a^2+3b^2\right)}^{5/2}}\eta .$$

The solution to the eigensystem problem is given by

(51)$$E_1=E-\delta E_{+},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_2=E-\delta E_{-},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_3=E_4=E+A,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta E_{\pm }=A\pm \sqrt{A^2+3B^2},$$

(52)$$c_1=\frac{\left(\delta E_{+},\delta E_{+},\delta E_{+},-3B\right)}{{\left(3\delta E_{+}^2+9B^2\right)}^{1/2}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_2=\frac{\left(\delta E_{-},\delta E_{-},\delta E_{-},-3B\right)}{{\left(3\delta E_{-}^2+9B^2\right)}^{1/2}},$$

(53)$$c_3=\frac{1}{\sqrt{2}}(-1,0,1,0),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_4=\frac{1}{\sqrt{2}}(-1,1,0,0),$$

and the rotatory strengths are

(54)$$R_1=-R_2=\rho \frac{\sqrt{3}B}{\sqrt{A^2+3B^2}}\text{cos}\alpha ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}_3=R_4=0.$$

The special case of ϑ = π/2

For ϑ = π/2 the Hamiltonian of the tetramer is given by the matrix

(55)$$\left(\begin{array}{cccc}E& -A& -A& B+C\\ -A& E& -A& -B+C\\ -A& -A& E& -2C\\ B+C& -B+C& -2C& E\end{array}\right),$$

where A is given in Eq. (50),

(56)$$B=9a^3\frac{\left(a^2-6b^2\right)\text{sin}(\alpha -\phi )+3a^2\text{sin}(\alpha +\phi )}{4{\left(a^2+3b^2\right)}^{5/2}}\eta ,$$

(57)$$C=3\sqrt{3}{a}^3\frac{\left(a^2-6b^2\right)\text{cos}(\alpha -\phi )-3a^2\text{cos}(\alpha +\phi )}{4{\left(a^2+3b^2\right)}^{5/2}}\eta .$$

The solution to the eigensystem problem is given by

(58)$$E_1=E-2A,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_2=E+\delta E_{-},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_3=E+A,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_4=E+\delta E_{+},$$

(59)$$\delta E_{\pm }=\frac{1}{2}\left(A\pm \sqrt{A^2+8\left(B^2+3C^2\right)}\right),$$

(60)$$c_1=\frac{1}{\sqrt{3}}(1,1,1,0),$$

(61)$$c_2=\frac{1}{\sqrt{N_{+}}}\left(B+C,-B+C,-2C,-\delta E_{+}\right),$$

(62)$$c_3=\frac{1}{\sqrt{6\left(B^2+3C^2\right)}}\left(B-3C,B+3C,-2B,0\right),$$

(63)$$c_4=\frac{1}{\sqrt{N_{-}}}\left(B+C,-B+C,-2C,-\delta E_{-}\right),$$

where $N_{\pm }=2B^3+6C^2+\delta E_{\pm }^2$. The rotatory strengths are

(64)$$R_1=R_3=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}_2=-R_4=-\sigma \frac{\sqrt{3}B\text{cos}(\alpha -\phi )-3C\text{sin}(\alpha -\phi )}{\sqrt{A^2+8\left(B^2+3C^2\right)}}b{d}^2.$$

It can be shown that B² + 3C² and $\sqrt{3}B\text{cos}(\alpha -\phi )-3C\text{sin}(\alpha -\phi )$ are independent of φ, so that the energy spectrum and rotatory strengths can be calculated by setting φ = 0.

Funding

Ministry of Education and Science of the Russian Federation (14.B25.31.0002).

Acknowledgments

A.S.B. and I.D.R. gratefully acknowledge the financial support from the Ministry of Education and Science of the Russian Federation through its Scholarship of the President of the Russian Federation for young scientists and its Grant of the President of the Russian Federation for young scientists.

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