8 GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s52>GLYPH<s48>GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s78>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48> GLYPH<s100>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s49>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s32> GLYPH<s32> GLYPH<s69> GLYPH<s105> GLYPH<s103> GLYPH<s101> GLYPH<s110> GLYPH<s118> GLYPH<s97> GLYPH<s108> GLYPH<s117> GLYPH<s101> GLYPH<s115> GLYPH<s32>GLYPH<s99> GLYPH<s111> GLYPH<s117> GLYPH<s110> GLYPH<s116>GLYPH<s115> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s78>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48> GLYPH<s100>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s32> GLYPH<s32> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s51>GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s78>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48> GLYPH<s100>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s32> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s40>GLYPH<s99> GLYPH<s41> GLYPH<s45>GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s78>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48> GLYPH<s100>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s49>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s32> GLYPH<s32> GLYPH<s69> GLYPH<s105> GLYPH<s103> GLYPH<s101> GLYPH<s110> GLYPH<s118> GLYPH<s97> GLYPH<s108> GLYPH<s117> GLYPH<s101> GLYPH<s115> GLYPH<s32>GLYPH<s99> GLYPH<s111> GLYPH<s117> GLYPH<s110> GLYPH<s116>GLYPH<s115> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s45>GLYPH<s53>GLYPH<s48> GLYPH<s48> GLYPH<s53>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s78>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48> GLYPH<s100>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s32> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s45>GLYPH<s50>GLYPH<s53> GLYPH<s48> GLYPH<s50>GLYPH<s53> GLYPH<s53>GLYPH<s48> GLYPH<s55>GLYPH<s53> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s78>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s50>GLYPH<s48> GLYPH<s100>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s51>GLYPH<s48>GLYPH<s32> GLYPH<s110>GLYPH<s109> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s32> GLYPH<s40>GLYPH<s99> GLYPH<s41> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s48> GLYPH<s54> GLYPH<s48> GLYPH<s48> GLYPH<s56> GLYPH<s48> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s82> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s50> GLYPH<s48> GLYPH<s32>GLYPH<s110> GLYPH<s109> GLYPH<s78>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s50> GLYPH<s48> GLYPH<s100> GLYPH<s32>GLYPH<s40> GLYPH<s110> GLYPH<s109>GLYPH<s41> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s40> GLYPH<s97> GLYPH<s41> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s45> GLYPH<s52> GLYPH<s48> GLYPH<s48> GLYPH<s45> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s40> GLYPH<s98> GLYPH<s41> FIG. 6: (Color online) (a) Decay rates and (b) energy shifts vs. distance for 20 dipoles in C20 configuration around Ag NP. Each line corresponds to a system eigenstate and is sim- ilarly marked in both graphs. The dash-dotted line corre- sponds to three degenerate superradiant states, the dashed line is the darkest subradiant states dominated by dipole- dipole interactions, and solid lines correspond to the rest of subradiant states. FIG. 5: (Color online) Distribution of decay rates for 20 dipoles in C20 configuration around Ag NP at several average (with 10% fluctuations) distances to its surface. FIG. 4: (Color online) Distribution of energy shifts for 20 dipoles in C20 configuration around Ag NP at several average (with 10% fluctuations) distances to its surface. inary parts of complex eigenvalues of Σ$_{jk}$ for N = 20 molecules at the sites of C20 fullerene at three different molecule-surface distances. The system spectrum repre- sents several sets of degenerate eigenvalues indicating a high degree of lattice symmetry. For all distances, the histograms show a single eigenvalue with a large posi- tive energy shift (Fig. 2), which corresponds to the di- rect dipole-dipole interaction between nearest-neighbor molecules. On the other hand, there are three degenerate eigenvalues with the largest decay rate, corresponding to predominantly superradiant states while the smaller de- cay rates are those of predominantly subradiant states (Fig. 3). With increasing distance, the mixing between superradiant and subradiant states decreases and, for d = 30 nm, decay rates of all but three eigenstates nearly vanish; note that in our approximation, pure subradiant states should have zero decay rate. Such behavior is due to diminishing contribution of higher- l plasmons at larger distances [see Eqs. ((23) and 24)]. Importantly, direct in- teractions between close molecules result only in energy shift of subradiant states, without affecting superradiant states. We therefore conclude that dipole-dipole interac- tions do not destroy cooperative emission in plasmonic systems. This main conclusion remains unchanged when fluctu- ations (up to 10%) of molecules positions in radial direc- tion are included into simulations (Figs. 4 and 5). The spatial disorder lifts lattice symmetry, so that superradi- ant states now have different, however close, decay rates. Note that without interactions, i.e., when molecules an- gular positions are completely random, the spread of su- perradiant decay rates is considerably higher. 10 To elucidate the structure of collective states, we cal- culate the distance dependences of complex eigenvalues, Δ$_{J}$ − i $_{2}$Γ$_{J}$ , for C20, C60, and C80 configurations of dyes, as shown in Figs. 6, 7, and 8, respectively. For C20 con- figuration (Fig. 6), there are five sets of eigenvalues with 3, 4, 4, 7, and 1-fold degeneracies, in descending order of Γ$_{J}$ magnitudes. Down to the distance of d = 5 nm, the largest decay rates, corresponding to three predom- inantly superradiant states, are well separated from the rest. The steep rise of Γ$_{J}$ at small distances is due to increasing contribution of high- l plasmon modes close to NP surface [see Eqs. (23) and (24)]. The interplay be- tween various coupling mechanisms is especially revealing