General model of nonradiative excitation energy migration on a spherical nanoparticle with attached chromophores

Theory of multistep excitation energy migration within the set of chemically identical chromophores distributed on the surface of a spherical nanoparticle is presented. The Green function solution to the master equation is expanded as a diagrammatic series. Topological reduction of the series leads to the expression for emission anisotropy decay. The solution obtained behaves very well over the whole time range and it remains accurate even for a high number of the attached chromophores. Emission anisotropy decay depends strongly not only on the number of fluorophores linked to the spherical nanoparticle but also on the ratio of critical radius to spherical nanoparticle radius, which may be crucial for optimal design of antenna-like fluorescent nanostructures. The results for mean squared excitation displacement are provided as well. Excellent quantitative agreement between the theoretical model and Monte–Carlo simulation results was found. The current model shows clear advantage over previously elaborated approach based on the Padé approximant.

By further calculating, we observe that the -th sum in the equation for  ̂ () (not explicitly shown there) involves an -fold product of  1 1   1  2 . . .   1 .Using the definition of the  matrix, we can see that each element   consists of a single term (when  = ) or a sum of  terms (when   ).
Thus, each -fold product of   terms, with  of them being diagonal ( = ), generates ( − 1)  products of -element transfer rates       .Summing all these products (as  → ∞) to obtain the corresponding Green function requires a special computational technique.The proper method for this purpose is the diagrammatic method.
Let us introduce a diagrammatic representation, (a graph), to depict the transfer rates       .In this representation, the vertices of the graph are marked with the donor number, and directed arrows represent the transfer rates       .We adopt the following convention: continuous arrows       represent the transfer rate from donor molecule  to donor molecule , while two arrows corresponding to (−      ) are drawn with a solid line followed by a dashed line (Fig. S1).The sign (+ or −) assigned to the arrow       indicates an increase or decrease in the probability of the respective transition.In the graph, a path is defined as a sequence of vertices connected by edges, where each vertex is connected to the next vertex in the sequence.Essentially, a path represents a traversal of the graph, moving from one vertex to another along the edges, without revisiting any vertex.Refer to Figure S2 for an illustration.Let us assign the value  −1 to the vertices corresponding to the donors.To each multigraph, a numerical value is assigned calculated by performing an ensemble average of the product formed by a factor of  −1 for each donor vertex, a factor of       for each solid arrow, and a factor of (−1) for each dashed arrow.The number that we refer to as the value of the multigraph is given by Two multigraphs are considered topologically inequivalent if, after labeling the vertices with the same set of labels, there is no permutation that leads to the equivalence of the multigraphs.
To calculate the values of individual multigraphs, we choose an arbitrary representative from each equivalence class and assign fixed labels (any labels) to its vertices.For example, the initially excited donor molecule at  = 0 is labeled as 1.
Let us consider modifications of diagram IV by assigning the vertex 3 loops with an increasing number of donors, as illustrated in Figure S4.∫  12 ( 12 ) ∫  13 ∫  14  −1   3  1   −1 (−  4  3  ) −1   2  3   −1 (10) Finally, the diagrammatic series for the Green function  ̂ ()is as follows:  ̂ () =  −1 + sum of all multigraphs consisting of loops starting in the vertex number 1, as shown in Figure S5.In a similar manner, we can define a diagrammatic representation of Green function  ̂ (, ) = sum of all different (topologically) multigraphs consisting of paths starting in a vertex number 1 and ending in a vertex number 2, as shown in Figure S6.We can renormalize the multigraphs containing loops by eliminating the maximum subgraph that forms a loop within the given multigraph.During this process, we merge the start and end vertices of the loop into a single vertex, effectively collapsing the loop.This transformation yields a representative of the class encompassing all multigraphs that share the same donors interconnected by arrows and loops.To compute the value of this transformed multigraph, which no longer contains loops, we make appropriate adjustments to the expressions assigned to its vertices.These modifications accounts for the elimination of the loop and ensures the accurate numerical value of the renormalized multigraph.
If we compare the value of multigraph IV from Fig. S3 (without loops), with the value of multigraph I from the same figure, we can observe that multigraph I can be generated from multigraph IV by replacing the factor  −1 corresponding to vertex 3 with an expression representing the value of a certain multigraph from the diagrammatic series of  ̂ ().This expression is one of the components of the  ̂ () function.The above analysis can be generalized into a theorem stating that the value of any multigraph without loops from the diagrammatic expansion of the Green's function  ̂ (, ) is computed by assigning to each donor vertex the factor  ̂ () instead of  −1 .The next topological property of the multigraphs in the series representing the Green's function  ̂ (, ) is the presence of nodes.A node is defined as a vertex in the multigraph that separates it into two disjoint subgraphs.In the renormalized multigraphs (without loops) currently under consideration, each circle (donor) can have at most one node.However, it is possible for a multigraph to contain multiple nodes, with the number of nodes limited by the number of vertices in the multigraph.An example of a multigraph containing nodes is shown in Figure S7.If we analyze the topological structure of the multigraph (VI) in Figure S8, we notice that it is the disjoint union of two subgraphs, denoted as (VII) and (VIII), connected at the node labeled as .The value of the multigraph (VI) is equal to The values of the multigraphs (VII) and (VIII) are respectively Let us note that the value of the multigraph (VI) consisting of three vertices is equal to the product of the values of its subgraphs (VII) and (VIII) with two vertices.However, to the node , which was formed by identifying the final vertex of the first subgraph (VII) with the initial vertex of the second subgraph (VIII), we now assign the factor 1/ ̂ ().In summary, the value of any multigraph (without loops) containing nodes can be approximated by multiplying the values of its individual subgraphs, to which it is divided.Notably, we assign the factor 1/ ̂ () to the donor nodes.
Continuing the analysis of the diagrammatic representation of the Green's function  ̂ (, ), which can contain any number of nodes (but without loops), let us express the function  ̂ (, ) as follows Here, the function  ̂  (, ,  ̂ ()) represents the sum of all loop-free multigraphs from the series in  ̂ (, ) that contain  nodes.
The method of calculating the values of multigraphs suggests the introduction of the following fundamental series: Σ ̂ (, ,  ̂ ()), which represents the sum of all multigraphs without loops and without nodes starting from specific donor (e.g., donor 1) and ending at a different specified donor (e.g., donor 2).Let's denote the numerical value of this series as Σ  .
Furthermore, we can denote When calculating the individual terms of the series in  ̂ (, ), we can express them as follows:  ̂0  =   ,  ̂1  = Σ *  Σ  ,  ̂2  = Σ *  Σ *  Σ  , and so on, where each term can be seen as forming an infinite geometric series.By summing this series, we obtain: Following the previous analysis, we have determined that knowing the values of the diagrammatic series Σ ̂ (, ,  ̂ ()) is sufficient for determining the time decays and stationary values of the observables in the system.However, this series cannot be exactly summed, and we must rely on approximate methods.In the two-body approximation, we calculate the function Σ ̂ (, ,  ̂ ())only for two-body multigraphs.Similarly, in the three-body approximation, we consider both two-body and three-body multigraphs.To obtain results for the fundamental observables, we apply the self-consistent method.This involves finding an expression for Σ ̂ (, ,  ̂ ()) in the appropriate approximation and treating the function  ̂ () as the variable we are searching for.We determine this variable by substituting the Green's function  ̂ (, ), which depends on Σ ̂ (, ,  ̂ ()), into the self-consistent equation  ̂ () +  ̂ ( = , ) =  −1 .
This self-consistent procedure greatly improves the accuracy of calculations and is simpler compared to other methods that involve approximating both the functions  ̂ () and  ̂ (, ) using appropriate expansions into diagrammatic series.

TWO-AND THREE-BODY APPROXIMATIONS: GENERAL FORMULAS.
As derived from the previous chapter, to obtain the time decays and stationary values of observables characterizing the studied system, knowledge of the values of the diagrammatic series Σ ̂ (, ,  ̂ ()) is sufficient.Unfortunately, this series cannot be precisely summed, necessitating the use of approximate methods.The two-body approximation involves restricting calculations of the function Σ ̂ (, ,  ̂ ()) exclusively to two-body multigraphs.Similarly, in the three-body approximation, we only consider two-body and three-body multigraphs (refer to Figure S9).The two-body approximation relies on a rigorous solution to the two-body problem.To achieve this, we expand the Green function  ̂ (, ) in a series with respect to the concentration of molecules for a system consisting of  donors positioned on a sphere with a radius  and surface   ̂ (, , ) = ( − 1)   2  (, ) + ( − 1)( − 2)  2  3  (, ) + Based on the equation provided, we can conclude that The function  2  (, ), representing two-body multigraphs, corresponds to multigraphs without nodes, as a system of two molecules cannot have nodes.These multigraphs may contain loops.
The explicit form of the  −  matrix for the considered system of molecules with  = 2 is as follows: Next, when calculating the elements of the inverse matrix, we obtain: where we used the transfer rates symmetry,   2  1  =   1  2  .
By averaging over the configuration of molecules, and removing loops,  −1 →  ̂ (), we obtain: The function Σ ̂3  (, ,  ̂ ()), which represents the three-body approximation for Σ ̂ (, ,  ̂ ()), is defined as the sum of the values of all multigraphs without nodes and without loops, formed by three donors.The most straightforward approach to calculate it is to first determine the sum of all three-body multigraphs A ̂3  (, ), and then subtract from it the multigraphs with loops L ̂3  (, ) and nodes N ̂3  (, ).After performing the renormalization,  −1 →  ̂ (), we obtain the expression for the function Σ ̂3  (, ,  ̂ ()).

MULTIGRAPHS A ̂3 𝐷𝐷𝐷 (𝒌, 𝜖)
To sum the values of all three-body multigraphs A ̂3  (, ), we once again utilize the expansion in Eq. (S20) which represents the expansion of the Green's function  ̂ (, ) in a power series with respect to the concentration of donors on a sphere with a radius  and surface .By interpreting the individual components of this series, we can conclude that the function  3  (, ) corresponds to the three-body multigraphs A ̂3  (, ).By substituting the appropriate value for , we obtain Using the definition of the Green's function  ̂ (, ), we obtain To compute  3  (, ), we consider the  −  matrix that appears in the definition of the Green's function  ̂ (, ,  = 3)).In this case, the  −  matrix is a 33 matrix, and it takes the following form Using Eq. (S26), we obtain Based on Eq. ( S26) and (S28), we can derive the final result as follows After applying the renormalization procedure,  −1 →  ̂ (), to the above formula, we obtain the equation that describes the desired function A ̂3  (, ) When deriving the above formulas, we used the transfer rates symmetry,       =       .

MULTIGRAPHS L ̂3 𝐷𝐷𝐷 (𝒌, 𝜖)
The sum of the values of all three-body multigraphs with loops L ̂3  (, ) is determined using the principle of mathematical induction.This involves adding together their respective values in a standard manner.The multigraphs under consideration, which consist of three donors, are illustrated in Figure S10.These multigraphs can consist of a loop starting in either donor 1 or donor 3, and they have the same value.Upon analyzing the topological structure of the multigraphs in the series L ̂3  (, ), we observe that they have the same form as the generating multigraphs Σ ̂2  (, ).However, the number of multigraphs differs, and the donor nodes are assigned the function  2  () instead of  −1 .Here,  2  () represents a linear component in the diagrammatic expansion of the function  ̂ () which is expanded in powers of the donor concentration (i.e., the numerical value of the loop being considered).

MULTIGRAPHS N ̂3 𝐷𝐷𝐷
As it is easy to see for a system of three donors, there can only be one node (no loops).This situation is depicted in Fig. S11.

Figure S3 :
Figure S3: Examples of multigraphs representing  ̂ (, ) containing loops.Multigraphs (I) and (II) consist of a single loop starting at vertex  and ending at ; multigraph (III) consists of two consecutive loops sharing a common vertex .Multigraph (IV) is obtained by removing the loops from (I)-(III).
) S where ∏       and ∏(−1) denote the respective products of factors       and the number−1.The factor (−1) appears as many times as there are dashed arrows in the given multigraph.If a given multigraph consists of m circles , the value of such a multigraph does not depend on the numbering of the molecules.As a result, there are ( − 1)( − 2). . .( − ) numerically equivalent multigraphs.Therefore, we can simplify the diagrammatic series by considering multigraphs with unlabeled vertices and keeping only the topologically inequivalent ones.The value of a topologically inequivalent multigraph is equal to ( − 1)( − 2). . .( − ) times the value of the corresponding original multigraph.

Figure S4 :
Figure S4: Examples of multigraphs representing  ̂ (, ) containing loops The values of the individual multigraphs are as follows

Figure S7 :
Figure S7: Example of multigraph containing nodes, labeled with Greek letters.After eliminating loops from the multigraphs, we analyze the nodes.In the renormalized multigraphs, each donor (circle) can have at most one node.

Figure S8 :
Figure S8: Multigraphs from which the multigraph (VI) can be constructed.The multigraph (VI) is the topological sum of two multigraphs (VII) and (VIII), which are its subgraphs connected at the node .The value of the multigraph (VI) is equal to the product of the values of its subgraphs (VII) and (VIII), with the node  assigned the factor 1/ ̂ ().

Figure S9 :
Figure S9: The simplest multigraphs from the diagrammatic series Σ ̂ (, ,  ̂ ()).They do not contain loops and nodes.The vertices corresponding to the donors (circles) are marked with the symbol (*) to indicate that they are assigned to the function  ̂ () after eliminating the loops.Multigraphs containing two donors are referred to as two-body multigraphs, those containing three donors are called three-body multigraphs, and so on.

Fig
Fig. S10.The multigraphs representing L ̂3  (, ) correspond to a system of three molecules, allowing for one loop.The loop can start in either donor number 1 (a) or donor number 3 (b).