Nano-photoluminescence of natural anyon molecules and topological quantum computation

The proposal of fault-tolerant quantum computations, which promise to dramatically improve the operation of quantum computers and to accelerate the development of the compact hardware for them, is based on topological quantum field theories, which rely on the existence in Nature of physical systems described by a Lagrangian containing a non-Abelian (NA) topological term. These are solid-state systems having two-dimensional electrons, which are coupled to magnetic-flux-quanta vortexes, forming complex particles, known as anyons. Topological quantum computing (TQC) operations thus represent a physical realization of the mathematical operations involving NA representations of a braid group Bn, generated by a set of n localized anyons, which can be braided and fused using a “tweezer” and controlled by a detector. For most of the potential TQC material systems known so far, which are 2D-electron–gas semiconductor structure at high magnetic field and a variety of hybrid superconductor/topological-material heterostructures, the realization of anyon localization versus tweezing and detecting meets serious obstacles, chief among which are the necessity of using current control, i.e., mobile particles, of the TQC operations and high density electron puddles (containing thousands of electrons) to generate a single vortex. Here we demonstrate a novel system, in which these obstacles can be overcome, and in which vortexes are generated by a single electron. This is a ~ 150 nm size many electron InP/GaInP2 self-organized quantum dot, in which molecules, consisting of a few localized anyons, are naturally formed and exist at zero external magnetic field. We used high-spatial-resolution scanning magneto-photoluminescence spectroscopy measurements of a set of the dots having five and six electrons, together with many-body quantum mechanical calculations to demonstrate spontaneous formation of the anyon magneto-electron particles (eν) having fractional charge ν = n/k, where n = 1–4 and k = 3–15 are the number of electrons and vortexes, respectively, arranged in molecular structures having a built-in (internal) magnetic field of 6–12 T. Using direct imaging of the molecular configurations we observed fusion and braiding of eν-anyons under photo-excitation and revealed the possibility of using charge sensing for their control. Our investigations show that InP/GaInP2 anyon-molecule QDs, which have intrinsic transformations of localized eν-anyons compatible with TQC operations and capable of being probed by charge sensing, are very promising for the realization of TQC.

www.nature.com/scientificreports/ indicates the NSOM run. One normal dot is denoted F1e. The dots are located within an area ~ 2 × 2 µm 2 for the same run/scan (see below) and within an area ~ 2 × 2 mm 2 for different runs. For anomalous dots the parameters measured include the AM type, the main peak energy E 0 , the s-p splitting ΔE sp , the size D, the Wigner-Seitz radius r s , the built-in magnetic field B bi , ν and the e ν configuration. The AM type, E 0 and ΔE sp , were measured from intensity distribution, position and energy splitting of the PL spectral components. The D values for most of the dots were measured/estimated directly from scanning experiments and used for calculation of r s = DN -0.5 /(2a B * ), where a B * ~ 8 nm 35 . B bi and ν was estimated from a complex analysis of the whole data set including the dependence of the PL spectra on B e , the NSOM maps and theoretical calculations/ analysis (see below). The e ν configuration was suggested from analysis of the whole data set. These parameters apply to the photo-excited state (PS), which for all dots (except D1e) have N* = N + 1 = 6 electrons, where N is number of electrons in initial (IS) state. For D1e N* = 7. Table 1 summarizes the dot parameters measured. The parameters, important for further discussion, are r s , which changes from 2.2 to 2.6 making up 20% variations and B bi , which changes from 6 to 12 T making up two times (200%) variations.
Theoretical description. The analysis of the experimental data was done using a phenomenological description within a framework of a general theory based on exact quantum mechanical calculations involving Fock-Darwin (FD), Hartree-Fock (HF) and configuration interaction (CI) approaches, developed for a few 2D electrons confined in circular potential in a perpendicular magnetic field about 20 years ago [40][41][42] .
The many-body HF and CI methods were used to calculate dependence of the energy structure together with the shell energy and electron distribution on magnetic field for specific QD studied taking into account their shape, and, thus, a circular symmetry breaking effects (see "Methods"). These were used for analysis and comparison with corresponding experimental data.
The FD spectrum was used to fit the experimental shifts of PL lines versus external magnetic field B e to estimate the electron charge e* and B bi .
Photo-luminescence spectra. Anomalous spectral features and intensity distributions. The spectra of anomalous dots presented in Fig. 1a have a set of sharp peaks, which are a main zero-energy e 0 -peak and about five e 1 -e 5 peaks, related to anti-Stokes components (ASCs) having a splitting of ~ 0.5 meV. This is significantly different from the spectra of normal dots presented in the insert (see figure caption for their parameters), which reveal two or three ASC peaks, a few times larger splitting (3-5 meV), and order-of-magnitude larger peak broadening. While, in the normal dots, the zero-energy and ASCs peaks are related to the occupied degenerate s-p-, d-, … electron (e) shells (including a photo-excited e), in the anomalous ones they are related to single spin-polarized es and manifest B bi , as we will show below. The energy splitting between the e 0 and the e 3 peaks ΔE sp , shown by a horizontal arrow, is related to the s-p-shell splitting, which, according to HF calculations, determines a quantum confinement energy ħω 0 * = 0.7ΔE sp . Besides the ASCs, Stokes components (SCs) denoted as nw 0 (n = 1-3, ħω 0 ≈ ΔE sp ) are observed for some dots (D1d, D2b, D4e and D1e) and they are related to a centerof-mass vibration.
While the anomalous spectral features are nearly the same for all dots, the intensity distribution of ASCs shows significant variations. These variations reveal three types denoted by AM 5,1 , AM m and AM 6,0 . For the AM 5,1 type (see two lowest spectra in Fig. 1a) the ASCs are an order of magnitude weaker than e 0 . For the AM m type (four middle spectra) the intensity of the ASCs increases a few times reaching a value of up to half of e 0 . For the AM 6,0 type (see three upper spectra) the intensity of the ASC peaks increases further and becomes nearly the same as the intensity of the peak e 0 . www.nature.com/scientificreports/ Magnetic field dependence. In the magnetic fields B e = 0-10 T (see Fig. 1b) the AM 5,1 -type D1d dot reveals a very weak diamagnetic shifts of PL spectral lines (about 0.5 meV for 10 T) and a strong change in relative intensity versus B e resulting in the appearance of the type AM m at B e = 4-5 T and type AM 6,0 at 8-10 T. The emergence of these two types is accompanied by the appearance of an additional peak e 6 between peaks e 1 and e 2 . For the AM 6,0 type a SC at 2ω 0 appears, similar to the dot D1e (see Fig. 1a). For normal dots the shifts (not shown here) are an order of magnitude stronger (up to 1 meV/T) having anti-crossings and paramagnetic regions at fields < 2 T , while the intensity distributions do not show significant changes, as will be discussed elsewhere.
Spatially-resolve spectra and imaging. The dots reveal anomalous size of the emission area (EA) of the single-e PL peaks, which is seen from spatially resolved data for dots D1e and D4e (see Fig. 1c,d and Fig. 2a-e). A variation of the intensity of e 0 -e 5 peaks down to ~ 0.4% per nm is seen in spectra in Fig. 1c,d. This corresponds to an EA size d EA down to ~ 25 nm as it is seen in the PL intensity maps in Fig. 2a,b, giving d EA = 40 ± 20 nm.
The maps also show the location of the individual es, and the combined maps presented in Fig. 2c,d display molecular structures. The molecular structure is absent in the normal D0e and F1e dots, the EA of which corresponds to the dot size of ~ 90 and 110 nm, respectively. This is seen in the combined map of the area containing these four dots and presented in Fig. 2e. In the map the images of IS are presented for D1e and D4e dots. The map shows that these are neighboring dots arranged in a liner chain with separation ~ 200 nm.
The molecular structure of the QD D4e in PS in Fig. 2c is a compressed hexagon having a size ~ 120 × 70 nm and a bond length ~ 40 nm. The IS structure (see Fig. 2e) is a pentagon, in which e 0 and e 1 electrons are shifted by 10-20 nm along the (x,y) diagonal, as follows from ω 0 EAs shape.
A complicated quasi-1D molecular structure oriented along the (x,y)-diagonal and having a size ~ 150 × 70 nm 2 is observed for D1e (see Fig. 2d). In this structure e 1 , e 2 , e 3 and e 6 EAs overlap and e 0,1 and e 5,6 EAs have two locations, revealing degenerate 2e states, which gives a 7e PS of this dot. The overlapped peaks are located in the upper right corner of the map; the rest three peaks are located along the (x,y)-diagonal below at the distance ~ 70 nm and separated from each other by 40 nm.
The SCs maps correspond to a ~ 60 nm vertical up shift of e 4 (ω 0 map), a ~ 60 nm down shift e 3 (2ω 0 map) and a ~ 40 nm down shift of e 5 along the (x,-y)-diagonal (2ω 0 map). This results in e 0 -e 4 and e 3 -e 5 pairing and in the IS the molecular structure has nearly equilateral triangle arrangement of paired es at the vertexes, having sides ~ 80 nm and 90 nm and a bond length ~ 60 nm.
In the combined maps in Fig. 2c,d a dark-light color code of EAs corresponds to large-small separation of es from the photo-excited hole, which allows determine its location as shown in the maps. Thus, specific spectral shape observed is related to a specific distribution of such separations (see below).
Analysis of the data. Classification of energies and states in magnetic field. A general theory of circular dots having N electrons in a magnetic field distinguishes four contributions to the total energy E tot , first www.nature.com/scientificreports/ discussed for 2e in Ref. 44 , which are the kinetic energies of localization induced by the confinement potential E conf (N) and by the magnetic field, i.e., cyclotron motion, E cycl (N,B), respectively, and the Coulomb energies of electrons center-of-mass E Coul,c.m (N) and relative E Coul,rel (N,B) motion, respectively. The B-independent parts are is non-trivial and, in spite of its extremely small value ~ 0.01E Coul,c.m (see Fig. 3d below), provides ground-state transitions having discrete total angular momentum values L z (B) = N i l zi , where l zi is the angular momentum of single-es, known as magic numbers (L z MN ). These appear because of the necessity of matching the spatial structure of the e wave-functions and the spatial symmetry of the e arrangement 45 . The e state of the specific L z MN is related to a filling factor ν = L 0 /L z MN , where L 0 = N(N − 1)/2, linking it to FQHE 46 . The set of values B(L z MN ) = B ν represent a "spectrum", which for specific N value, is scaled as ~ 1/r s .

Molecular configurations and PL spectra intensity distributions.
In the QD D1d the mixing of the configurations takes place, due to non-circular shape, as is seen in electron density distributions, calculated using CI, in Fig. 3a. Nine density distributions shown for ν from 1 to 3/14 occupying B range 2.7-16 T reveal the onset of molecular structure formation near ν = 5/7 (B ~ 5 T) similar to the circular symmetry. A "pure" (6,0) and (5,1) configurations appear for ν = 5/7 and ¼ and for ν = 2/7 and 3/14, respectively. For other values of ν two mixed configurations are appeared. In the one (see ν = 1/2) a central e maximum is shifted ~ 10-20 nm down from the center, and in the other (see ν = 1/3) a pentagon with adjacent single-e maxima at the left is formed. Note, that at ν = 1/4 the calculated configuration is a compressed hexagon, exactly the same as observed experimentally in D1e, which implies B bi ~ 12 T.
Molecular configurations in Fig. 3a can be assigned to specific AM types spectra observed in Fig. 1a,b. They, together with the maps in Fig. 2c,d, show that the appearance of different AM types spectra reflects a competition between (6,0) and (5,1) isomers for a non-circular dot shape and different B bi or B. In these the former has nearly equal AM 6,0 type ASC intensity, due to the nearly equal separation of electrons from the hole, while the latter has one dominant peak AM 5,1 type owing to its location closer to the hole and the AM m -type has intermediate, mixed configurations and intensity distributions.

Magneto-electrons and spontaneous anyon molecule formation. The analysis of the calculated
B-dispersion of HF and FD shell energies, L z , E tot and their comparison with experimental data of the dot D1d shown in Fig. 3b together with the FD fit to these data in Fig. 3c and E Coul,rel and E tot , shown in Figs. 3d,e, respectively, reveal the formation of the fractionally charged e ν s.
The E tot curve in Fig. 3b shows a nearly dispersionless E conf (N) + E Coul,c.m (N) contribution for B < 3 T, a linear increase from E cycl (N,B) and B > 4 T, and weak oscillations from E Coul,rel (B) at B ν (see also plot of E Coul,rel (B) in Fig. 3d) over the entire B-range. The L z MN states of the dot D1d are clearly visible on L z (B) curve as plateaus and weak kinks at L z = 5, 9, 15, 21, 25 30, 35 and 45. The corresponding B ν -spectrum has a ν values set from 3 to 1/3 and is very close to that of circular dot 46 . The plateaus are also seen in the region ν > 1corresponding to www.nature.com/scientificreports/ the integer (ν = 3) and SLL fractional (ν = 5/3) QHE states, demonstrating its deep connection to the localized es states in QDs. This connection is also revealed for FD states, which appears as a matching of the crossings between the (1,0) and (0,l) FD levels to L z MN transitions at ν = 3 to 5/3 and to 1, corresponding to a total spin transitions from S z = 0 to 1 to 2. The HF energies, accounting for spin and p x -p y circular symmetry distortion splittings, are shown in the figure in corresponding regions and approximately match the FD energies, neglecting the splitting. Both HF and FD energies at large fields come out to the LLL line ħω c /2, with a nearly order-of-magnitude reduction of inter level splitting, which evolves to zero in the limit B → ∞.
From Fig. 3b we can see, that the critical discrepancy between the calculated FD/HF energies and the measured ASC shifts is that the latter have negligible B-dispersion and an order of magnitude larger level splitting over the entire 0-10 T magnetic field range. This implies a reduction of ω c , as can be revealed from the FD fit, shown in Fig. 3c. The FD fit gives a general matching with experimental shifts and splittings of e 0 -e 5 peaks for B bi ~ 6 T and a three-fold reduction of ω c . The total internal field B = B bi + B e thus has the range B = 6-16 T, which includes ν = ½ and ¼.
The matching, however, does not account for the appearance of the additional peak e 6   www.nature.com/scientificreports/ Reduction of ω c can be interpreted as fractional charge and manifests a self-generation of the magneticquanta-flux vortexes by single es forming e ν s 35 . The vortex self-generation arises because of the reduction of E tot parts E Coul,c.m + E cycl caused by the decrease in charge and the dissipationless (superconducting) motion of es occupying quantum confined states for r s > 2. Such a reduction of E tot for e ν s compared to es is shown in Fig. 3e. It reveals a ~ 10 meV minimum at ν = 5/3 and a gradual decrease from 100 meV for ν~ 5/7 to 50 meV for ν ~ 1/4, which corresponds to increase of the energy drop from 10 to 80 meV.
Size dependence of molecular structure. Since the vortexes have fixed radius ~ a B * 35 , the resulting charge, i.e., ν, B bi , and AM configuration, are determined mostly by r s . The size of the e ν area d ν increases by a fraction ~ 0.1 per vortex, as can be found from simple geometrical drawing (see the corresponding cartoons for ν = ½, 1/3 and ¼ e ν areas in Fig. 3e). The d ν value of ~ 40 nm comes from the calculations for ν = 1/4 is in agreement with the measured d EA in the dot D4e. Thus, this dot can be considered as 6e 1/4 and 5e 1/4 AM in PS and IS, respectively. For the dot D1e a pairing of EAs in IS indicates 3e 2/7 AM, which corresponds to B bi ~ 11 T. For B bi ~ 6 T of the dot D1d the PS state is 2e 3/5 AM.
In Fig. 3f we present expected schematic arrangements of e ν s in these three AMs obtained by placing vortexes onto the maxima of their e distribution. The e 3/5 and e 2/7 are realizations of the complex e ν s, in which ν = n/p and n > 1, consisting of n-es fused by a "coupling" vortex (see below). The figure accounts for different e distribution size, i.e., r s , and demonstrates matching of ν and r s . This justifies our analysis, which, being extended to the rest of the dots, results in the values ν and B bi and the e ν -AM configurations in the PS given in Table 1. The analysis shows factor-of-two increase of B bi for a 20% increase of r s in agreement with the experiment (see Table 1).

Magneto-electrons and topological quantum computations. While the AM configurations
observed and analyzed in 6e states show symmetric, homogenous molecular structures compatible with distorted five-and sixfold circular symmetry and with the calculated electron density distributions, for 7e state they are not. The observed anomalous 1D composite configuration for the 7e state in the dot D1e (see Fig. 2d) can be assigned to a decomposition of a symmetrical 3e 2/7 AM of a 6e state after an extra e is added. In this case, the additional e 2/7 needed for symmetrical arrangement, cannot be generated, since it requires pair of es. Thus, we can suppose that the additional e creates e 1/3 , which results to a transition from 21 to 24 vortexes state. The corresponding configuration could then consists of a 3e 1/3 AM and a single e 4/15 . The corresponding arrangements of the e ν s are presented in Figs. 4a,b. In the figures the coupling vortexes are shown by dashed circle.
Comparison of the arrangements and particle displacements in the scheme of Fig. 4a,b reveals transformations and interchange of e ν s, which correspond to elementary topological quantum computing operations (TQCOs). These involve first, two unfusions of (e 0 -e 4 ) and (e 3 -e 5 ) e 2/7 anyon pairs resulting in four e 1/3 single anyons; second, two braids of (e 0 ,e 4 ) and (e 1 ,e 3 ) pairs; and, third, two fusions, which are a fusion of a e 1/3 (e 3 ,e 6) pair forming a e 2/7 anyon and a fusion of a e 2/7 (e 1 -e 2 ) and a (e 3 -e 6 ) pair forming e 4/15 anyon.
The above configuration transformation involves spatially separated e ν -anyons localized on the specific levels of QD confinement potential and a full description of such a transformation should specify a localization energy, www.nature.com/scientificreports/ i.e., include the energy coordinate. Along these lines we add an energy coordinate to x-y plots and present the energy/space diagrams in Fig. 4c,d overlaid on the confinement potential. The diagrams outline a few meV difference in the confinement energy of coupled e ν s. They show that the coupling vortex should be presented by a pair of half-flux vortexes synchronously generated in gapped e ν -states. These vortexes should have twice size (not shown in the figure) and thus embrace one of the neighboring vortexes. While such a representation of complex e ν s seems unusual in the framework of conventional theories of FQHE states, assuming degenerate states and zeros of many-electron wave-functions for the vortex description 34 , it is supported by our data and by the experimental observations of 2/5 anyons by Aharonov-Bhom interferometry in quantum Hall bars 48 , giving independent evidence of their existence.
TQCOs resulting from the PL process in the dot D1e, presented in the Fig. 4a,b, are described by a B 7 braid group diagram, i.e., time-position world lines, shown in Fig. 4e. The braid of e 4 -e 0 and e 4 -e 0 e 1/3 -anyons should result in a /3 phase change 23 , which according to the diagram, is encoded in the unfusion of the corresponding e 2/7 anyon and accompanied by a local charge redistribution. This allows one to use local charge sensing to control TQCOs, which can be provided with high sensitivity by a single-e transistor (SET) 49 . Since control of phase is a key step in TQCOs and a technology and methodology of SET fabrication and measurements are well developed 50 , using SETs opens up a new possibility for physical realization of TQGs.
The design of the TQG based on InP/GaInP2 e ν -AMs will include few SETs and nano-electrode gates having sizes ~ 50 nm adjacent to individual QDs formed by semiconductor manufacturing technology using e-beam lithography 51 . The arrays of QDs (see Fig. 2e)] can be used to realize circuits. Since we demonstrated that e ν -AM structure is sensitive to small variation of r s , using nano-gates changing local potentials near QD gives possibility of the fine tuning of the initial state.
In Fig. 4a,b we draw three SETs and a gate near D1e QD, which can be used to realize a prototype of the device for electrical testing of TQCOs shown in Fig. 4e. In the device RF SETs acting simultaneously and utilizing a single RF line by means of carrier multiplexing 52 will be used to detect "charge triangulation". Each of the devices is biased at the slope of SET transfer characteristic and employs charge cancellation technique 53 to minimize effects of direct capacitive coupling to the pulsing gate. In this way signals obtained from SETs (Us i ) under variation of r s or injection/removing of the electrons by the gate voltage (U G ), i.e., Us i (U G )-functions, will be used for designing of topological quantum computing processing.
Finally, we should point out, that the QDs considered do not directly involve NA anyons and MZMs, as suggested in initial TQC proposals. However, our magneto-PL measurements and the preliminary analysis of N ~ 8 InP/GaInP 2 dots having r s ~ 1.5 (similar to D04 in the insert in Fig. 1a), which will be published elsewhere, reveal B bi ~ 2 T close to ν ~ 5/2. This can indicate e 1/4 anyon supporting MZMs, similar to that discussed for corresponding SLL FQHE state. The possibility of forming of the corresponding state in an appropriate QD naturally follows from e ν -E tot of in Fig. 3e for ν> 1. Moreover, for such dot types we also observed a quasi 1D shape of EA 36 , similar to that in Fig. 2b, which can be a signature of TQCOs. We already demonstrated Coulomb blockade control of N in these dots in the range 8-18 having ΔU G ~ 0.3 meV per electron and the possible formation of a spin-polarized state for N > 15 54.

Conclusion
We have introduced a novel system, which can be used for the realization of TQG. These are e ν -anyon molecules (e ν -AM) naturally formed in self-organized many electron InP/GaInP 2 QD structures. We demonstrated their promising TQC prospects using high-spatial-resolution magneto-PL measurements of five-to-seven electron QDs having r s ~ 2. Our data, together with the quantum mechanical calculations of their electronic structure in a magnetic field and analysis demonstrate a self-formation of e ν AMs, having ν ~ 3/5-1/4, corresponding to a built-in magnetic field of 6-12 T. Our measurements of PL spectra and the mapping of the intensity of individual PL lines gave a direct imaging of molecular configurations and allow the observation of a transformation of a e ν arrangement in a e 2/7 -AM under photo-excitation involving fusion and braiding of the anyons. The observed transformation reveals a significant redistribution of the fractional charge within the dot, which suggests the use of single electron transistor charge sensing to control TQC operations. Our investigations show that InP/GaInP 2 AM QDs having intrinsic anyon localization at zero external magnetic field combined with charge sensing control of anyon's states open up novel directions for the realization of TQC.

Methods
Magneto-PL measurements and data processing. Spatially-resolved magneto-PL spectra were measured using NSOM operating at 10 K and magnetic fields of up to 10 T and using optical fiber probes having an aperture size of 50-300 nm in a collection-illumination mode. The spectra were excited by the 514.5 nm Ar-laser line and measured using a CCD (multi-channel) detector together with a 280 mm focal length monochromator. The excitation power measured before the fiber coupler was ~ 5 µW, which provided a power density of ~ 0.5 W/ cm 2 . The spectral resolution of the system is 0.2-0.4 meV.
For the Lorenz contour deconvolution of PL spectra, we used a multi-peak fitting procedure from Origin 8.0 graphic software.
The spatially-resolved PL intensity at the selected wavelengths (image) was generated using the spectra taken in a square grid having a mesh of 50 nm. We plotted the experimental data using a contour plot option of Origin 8.0 and the division of the intensity data into 20 levels. The size of the emission area was estimated as a size of a NSOM image at the emission intensity level of 0.9, i.e. two upper contour plot levels. NSOM image of ASC are related to PS Stokes components images correspond to a difference of electron positions in the IS and PS 36 . www.nature.com/scientificreports/ Configuration interaction and Hartree-Fock many-body calculations. The states of N confined electrons (N ~ 6 and r s ~ 2.5) in a strong perpendicular magnetic field B z = B are modeled theoretically by configuration interaction (CI), using a method described in more detail in Ref. 36 . We assume a quasi-2D effectivemass approximation with Hamiltonian (in atomic units, 4πǫ 0 = m e = ℏ = |e| = 1) where the magnetic vector potential is given by A x, y = (B z /2) −y, x, 0 , and the effective mass m * = 0.077 and dielectric constant κ = 12.61 are taken to be the values for bulk InP. The effective 2D confining potential V ext (r) in the x-y plane is assumed to be harmonic near the center, with frequency parameter ℏω 0 , and to develop a "hard wall" near the physical boundary of the dot on the substrate. The confining potential also has a slight angular deformation, indicated by the potential contours in Fig. 3a, which was chosen to be typical of the dots synthesized experimentally. The spin Zeeman term g * µ B B z S zi is usually not included explicitly. The first step in the CI calculation is to calculate a single-particle basis set |i� using spin-polarized Hartree-Fock (HF), The lowest N HF orbitals |a� (where a = 1, . . . , N ) are occupied, with one electron per orbital assuming complete spin polarization. The occupied orbitals contribute to the self-consistent HF potential V HF , which is defined as where |i� and |j� are general states (occupied or unoccupied). The eigenvalues ǫ a of the occupied orbitals, at least for zero to moderate B z , may be expected to give approximations to the removal energies of electrons from the system, according to Koopman's theorem from atomic and molecular physics 54 . In the calculated electron density distributions the dot size (D c ) was estimated as the size of the area containing 96% of electron density.