Improvements to a Peak Assignment Algorithm for Two-Dimensional NMR Correlation Spectra of Zeolites Using Graph Theory

Darren H. BROUWER and E. Keith LLOYD


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1 Introduction

Nuclear magnetic resonance (NMR) spectroscopy is a very powerful technique for probing chemical structure. An NMR spectrum typically contains a number of peaks or resonances at different frequencies (chemical shifts) arising from the different electronic environments surrounding the nuclei of the atoms. The assignment of the peaks in an NMR spectrum to the atoms in a structure is a very important first step towards further structural investigation. Two-dimensional (2D) NMR correlation experiments, which exploit the nuclear spin interactions between the nuclei of atoms, provide 2D NMR spectra which reveal correlations between the peaks. This correlation data can be used in combination with chemical shift information to assign the peaks in the NMR spectrum to the atoms in the structure.
Zeolites are microcrystalline open-framework materials with well-defined pores and channels of molecular dimensions that enable them to act as "molecular sieves" towards small molecules and ions. Zeolites are very important commercial materials being used mainly for catalysis, dessicants, gas adsorption and separation, ion exchange, and detergent builders. The properties of a zeolite are mainly determined by its atomic composition and three-dimensional structure. Solid state magic-angle spinning (MAS) NMR has emerged as an important technique for investigating zeolite structures since the microcrystalline nature of most zeolites limits the application of single crystal X-ray diffraction experiments. In the case of pure silica zeolites (in which the zeolite framework is made up of silicon and oxygen atoms), 29Si MAS NMR spectra can be very highly resolved with the individual resonances arising from the crystallographically unique silicon atoms in the zeolite framework [1]. The assignment of these resonances to the inequivalent silicon atoms in the structure is a crucial first step towards investigating the three-dimensional structures of zeolite host-guest complexes involving ions or small organic molecules [2].
The assignment of peaks in the 29Si spectrum of pure silica zeolites is particularly challenging. In comparison to 1H, 13C, or 15N NMR spectra of most organic molecules and proteins, the 29Si chemical shifts provide very little direct information about the identity of the peaks, as all of the silicon atoms are found in similar tetrahedral coordination environments. The 2D "Incredible Natural Abundance DoublE QUAnTum Experiment" (INADEQUATE) NMR experiment [3] correlates pairs of peaks that correspond to pairs of atoms whose nuclei are coupled via the through-bond indirect spin-spin interaction, the so-called J-coupling. In the case of pure silica zeolites, the 2D 29Si INADEQUATE experiment probes the 29Si-29Si couplings that exist through the Si-O-Si bonds [4]. The peaks in the 29Si spectrum can be assigned by comparing the correlations observed in such a 2D spectrum to the known bonding pattern between the silicon atoms in the zeolite framework.
An algorithm has recently been described [5] that finds all possible peak assignments for which there is agreement between the experimentally observed peak correlations in a 2D NMR correlation spectrum and the known bonds between the atoms in the zeolite framework. The 2D NMR spectrum can be expressed as a graph where the vertices are the peaks and the edges are the correlations between the peaks. Likewise, the zeolite framework structure can also be expressed as a graph where the vertices consist of a set of inequivalent atoms and the edges are the bonds between the atoms. Essentially, the algorithm assigns the peaks by finding all possible labelings of the vertices of the two graphs which show that the two graphs are homomorphic. In this paper, we describe a number of improvements to this peak assignment algorithm by taking advantage of some of the properties of the spectrum and structure graphs.

2 Definitions

A graph G(V, E) consists of a set of vertices V and a set of edges E which connect these vertices. A vertex vi is adjacent to a vertex vj if there exists an edge between them. The valency of a vertex is the number of incident edges. A circuit of length n is a path along a sequence of n consecutive edges which ends at the vertex from which it began. An adjacency matrix of a graph A(G) is a matrix representation of a graph in which each i, j element is the number of edges between vertices vi and vj.
The graph G(V2, E2) is a homomorphic image of the graph G(V1, E1) if there is a map j : V1 ® V2 which has the property that vertices j(vi) and j(vj) are joined by an edge in V2 if and only if vi and vj are joined by an edge in V1. The map j is called a homomorphism.
A homomorphism j is sometimes called an epimorphism if each vertex in V2 is the image of at least one vertex in V1. In that case |V1| >= |V2|, where |V| denotes the number of vertices in V. If, in addition, |V1| = |V2| then distinct vertices of V1 map to distinct vertices of V2 and the epimorphism is called an isomorphism. Isomorphic graphs have exactly the same structure, but the names given to the vertices may well be different. When V1 = V2, an isomorphism is called an automorphism.
It is possible to express the 2D NMR correlation spectrum and the zeolite framework structure as graphs. The structure graph of the zeolite framework G(A, B) consists of a set A of crystallographically inequivalent atoms as the vertices and a set B of bonds between these atoms as the edges. The graph of the 2D NMR spectrum is called the spectrum graph G(P, Q); it consists of a set P of peaks as the vertices and a set Q of peak correlations as the edges.
The task of finding all possible assignments of the set A of atoms in the zeolite structure to the set P of peaks in the NMR spectrum can therefore be expressed as finding all homomorphisms j : A ® P from the structure graph to the spectrum graph.
Some care is needed in extending the above concepts to graphs with multiple edges, but in the case of isomorphisms, this could be done by requiring that distinct edges map to distinct edges and changing the main condition to: "j (vi) and j (vj) are joined by the same number of edges in V2 as vi and vj are joined in V1".

3 Results and Discussion

As a first example, we present the peak assignment of the 29Si MAS NMR spectrum of the host-guest complex of the zeolite ZSM-5 with p-xylene molecules adsorbed in the cavities of the zeolite framework [4]. The structure of the ZSM-5 framework is illustrated in Figure 1(a). There are 12 crystallographically unique silicon atoms in the structure. The structure graph G(A, B) of the zeolite framework is represented in Figure 1(b) where A is the set of 12 silicon atoms and B is the set of Si-O-Si bonds which exist between the atoms. The graph has been drawn in a way that emphasizes that it has a symmetry or automorphism which interchanges pairs of vertices. The 1D 29Si MAS NMR spectrum and the 2D 29Si INADEQUATE spectrum of this host-guest complex are presented in Figure 1(c) and Figure 1(d) respectively. An analysis of the relative peak areas in the 1D 29Si spectrum reveal that there are 12 resonances corresponding to the 12 crystallographically inequivalent silicon atoms in the structure. Ten of the 12 peaks are resolved from each other and a pair of resonances are indistinguishable from each other. The spectrum graph G(P, Q) of the 2D 29Si INADEQUATE spectrum is presented in Figure 1(e) where P is the set of 11 resolved peaks and Q is the set of observed correlations between the peaks. Since |A| = 12 > |P| = 11, the task of peak assignment in this case will be to find epimorphisms from the structure graph to the spectrum graph.


Figure 1. (a) Illustration of the framework structure of the zeolite ZSM-5 in the orthorhombic phase with 12 crystallographically unique Si atoms (adapted from Ref. [4]) (b) The corresponding structure graph emphasizing the symmetry or automorphism of the graph. (c) 1D 29Si MAS NMR spectrum and (d) 2D 29Si INADEQUATE spectrum of the zeolite ZSM-5 with adsorbed p-xylene molecules (adapted from Ref. [4]). (e) The corresponding spectrum graph of the 2D correlation spectrum.

In the initial description of this algorithm [5], the possible assignments of each of the peaks were set to all atoms at the start of the algorithm and a branching "tree-search" (breadth-first search) was employed to find all homomorphisms between the structure and spectrum graphs, testing the subgraphs for homomorphisms at each step along the way. In this paper, we present a number of ways to analyze the spectrum and structure graphs prior to the start of the search so that the initial assignments of the peaks are narrowed down from all atoms. By doing this initial analysis, fewer subgraphs need to be analyzed for homomorphisms at each step in the search, potentially leading to a large gain in the efficiency of the peak assignment algorithm.
1. Valencies of vertices. An inspection of the structure graph in Figure 1(b) reveals that 7, 9, 10, and 12 each have valencies of three compared to the remainder of the silicon atoms which have valencies of four. This difference in valency arises from the fact that each of these silicon atoms has a "self-connectivity" (i.e. 7-7, 9-9, 10-10, 12-12, see Figure 1(a)) which cannot be observed in the two-dimensional INADEQUATE experiment since, in most circumstances, it is not possible to observe J-couplings between identical nuclei. An inspection of the spectrum graph in Figure 1(e) reveals that peaks A, B, E, and G have valencies of three, while the other peaks have valencies of four. This simple analysis narrows down the possible assignments of several of the peaks, revealing that peaks A, B, E, and G must each be assigned to one of {7, 9, 10, 12}, while the remainder of the peaks must each be assigned to one of {1, 2, 3, 4, 5, 6, 8, 11} (see column 1 of Table 1).

Table 1. Possible assignments of the peaks in the 29Si MAS NMR spectrum of the ZSM5/p-xylene complex at each stage of the analysis of the structure and spectrum graphs.
1. Possible peak2. Possible peak3. Possible peak4. Peak5. Symmetry
Peakassignments after examining valenciesassignments after examining valencies of adjacent verticesassignments after symmetry reductionassignments after searchrelated assignment
A7, 9, 10, 129, 109910
B7, 9, 10, 129, 1010109
CD1, 2, 3, 4, 5, 6, 8, 111, 2, 3, 4, 5, 61, 2, 3, 4, 5, 61, 25, 6
E7, 9, 10, 127, 127, 12127
F1, 2, 3, 4, 5, 6, 8, 111, 3, 4, 61, 3, 4, 661
G7, 9, 10, 127, 127, 12712
H1, 2, 3, 4, 5, 6, 8, 118, 118, 11811
I1, 2, 3, 4, 5, 6, 8, 111, 3, 4, 61, 3, 4, 643
J1, 2, 3, 4, 5, 6, 8, 111, 3, 4, 61, 3, 4, 634
K1, 2, 3, 4, 5, 6, 8, 111, 2, 3, 4, 5, 61, 2, 3, 4, 5, 652
L1, 2, 3, 4, 5, 6, 8, 118, 118, 11118

2. Valencies of adjacent vertices. It is also useful to examine the valencies of the vertices that are adjacent to a vertex. Such an inspection of the structure graph reveals that the vertices can be further divided into several groups. For example the vertices of valency 3 can be differentiated further as the valencies of the vertices adjacent to each of 9 and 10 are (3, 4, 4), while the valencies of the vertices adjacent to each of 7 and 12 are (4, 4, 4). Similarly, the vertices of valency 4 can be differentiated further into {8, 11}, {1, 3, 4, 6}, and {2, 5} based on the valencies of adjacent vertices being (3, 3, 3, 4), (3, 4, 4, 4), and (4, 4, 4, 4) respectively. Examination of the valencies of adjacent vertices in the spectrum graph and comparison with the analysis of the structure graph narrows down the possible assignments of the peaks even further: peaks A and B must each be assigned to one of {9, 10}, peaks E and G to one of {7, 12}, peaks H and L each to one of {8, 11}, peaks F, I, and J each to one of {1, 3, 4, 6}, and peak K to one of {1, 2, 3, 4, 5, 6} and CD to two of {1, 2, 3, 4, 5, 6} (see column 2 of Table 1).
3. Automorphism of the structure graph. As mentioned above, the structure graph presented in Figure 1(b) has a symmetry or "automorphism". As a consequence of the automorphism of the structure graph for this particular example, there will be a minimum number of two possible peak assignments which will be related by a symmetry which interchanges pairs of vertices: (1 6) (2 5) (3 4) (7 12) (8 11) (9 10). This property of automorphism can be exploited to increase the efficiency of the peak assignment algorithm, since only half of the possible peak assignments need to be searched; the other half can be generated by the symmetry relationship at the end. This is implemented by assigning one of the peaks to half of its possible assignments. For example, peak A can be assigned to silicon atom 9 since 10 will be generated in the end by the symmetry relationship. Consequently peak B is then assigned to atom 10 with atom 9 being generated in the end by symmetry (see column 3 of Table 1).
Table 1 presents the possible assignments of each of the peaks after each stage of this initial analysis of the structure and spectrum graphs. After the initial analysis (columns 1-3), these possible assignments were used as input to the search algorithm described in Ref. [5] to find all possible homomorphisms from the structure graph to the spectrum graph (column 4). At the end of the search, the symmetry related solutions are generated according to the automorphism of the structure graph (column 5). Of these two possible assignments, the second (column 5) can be shown to be the correct assignment as it gives a good correlation between the 29Si chemical shifts and the mean Si-Si distances [6] obtained from the X-ray diffraction structure [7]a.
There is a large gain in efficiency by incorporating this initial analysis of the structure and spectrum graphs. Without any initial analysis, the search algorithm examined 7662 subgraphs for homomorphisms in 4.8 s to find the same two possible peak assignments. Incorporating the symmetry properties of the structure graph led to an increase in efficiency of a factor of two with the search examining 3831 subgraphs for homomorphisms in 2.4 s to find one of the peak assignments, the second being generated by symmetry. When the initial analysis of the valencies in the structure and spectrum graphs was incorporated, the search for peak assignments required only 0.04 s during which only 32 subgraphs were examined for homomorphisms during the various steps of the search algorithm. For this particular example, this is a gain in efficiency of about two orders of magnitude.
As an additional example, we present the peak assignment for the 29Si MAS NMR spectrum of the room-temperature monoclinic phase of the zeolite ZSM-5 [4]. This a very difficult spectrum to assign since there are 24 inequivalent silicon atoms in the structure, many of the peaks are overlapping and there is little direct information outside of the 2D 29Si INADEQUATE spectrum to identify the peaks. This example again illustrates the large gains in efficiency of the peak assignment algorithm that can be achieved by examining the structure and spectrum graphs prior to starting the search for homomorphisms.
The monoclinic ZSM-5 framework is illustrated in Figure 2(a) along with a subgraph of its corresponding structure graph in Figure 2(b) which has been presented to emphasize the short circuits and the automorphism that exist in the graph. The 1D 29Si MAS NMR and 2D 29Si INADEQUATE spectra are presented in Figure 2(c) and Figure 2(d) respectively. The relative intensities of the peaks in the 1D 29Si MAS NMR spectrum indicate that there are 24 unique silicon atoms which is in agreement with the structure. However, only 15 of the 24 resonances are individually resolved from each other and there are two groups of closely overlapping resonances. The spectrum graph (a subgraph of which is shown in Figure 2(e)) therefore consists of 17 vertices. A dashed edge represents uncertainty in the existence of a correlation between two peaks. Since |A| = 24 > |P| = 17, the task of peak assignment in this case will be to find epimorphisms from the structure graph to the spectrum graph.


Figure 2. (a) Illustration of the framework structure of the zeolite ZSM-5 in the monoclinic phase with 24 crystallographically unique Si atoms (adapted from Ref. [4]) (b) A subgraph of the corresponding structure graph emphasizing the automorphism and circuits of lengths 3 and 4. Note that not all of the edges have been drawn. (c) 1D 29Si MAS NMR spectrum and (d) 2D 29Si INADEQUATE spectrum of the zeolite ZSM-5 in the monoclinic phase (adapted from Ref. [4]). (e) A subgraph of the corresponding spectrum graph of the 2D correlation spectrum emphasizing the circuits of length 4. Only the edges which are unambiguously present have been drawn along with a few ambiguous edges (denoted by the dashed edges).

For this particular example, an analysis of the valencies in the structure and spectrum graphs does not provide any additional information about the identity of any of the peaks since all of the valencies are equal as there are no "self-connectivities" in the structure which would give rise to unobserved correlations in the 2D 29Si INADEQUATE spectrum.
However, it is useful to analyze the graphs for circuits of lengths 3 or 4. An examination of the structure graph reveals that there are two circuits of length 3 containing vertices (4, 5, 13) and (16, 17, 1) and five circuits of length 4: (9, 10, 22, 21), (4, 5, 11, 19), (16, 17, 23, 7), (5, 11, 10, 13), and (17, 23, 22, 1). Note that 5, 10, 11, 17, 22, 23 each lie in two circuits of length 4. An examination of the spectrum graph (considering only definite edges between vertices which represent resolved peaks) reveals that there are two circuits of length 4 involving peaks (V, D, W, T) and (V, U, X, S). Therefore, each of peaks V, D, W, T, U, X, S must be assigned to one of {4, 5, 7, 9, 10, 11, 16, 17, 19, 21, 22, 23}. Furthermore, peak V is present in both of these circuits indicating that peak V must be assigned to one of {5, 10, 11, 17, 22, 23}.
Again, it is also useful to take advantage of the symmetry or automorphism of the structure graph (see Figure 2(b)). The possible assignments of peak V can be reduced to one of {5, 10, 11} prior to the start of the search algorithm, with any symmetry related assignments being generated at the end of the search.
In this manner, we have a starting point for the search which consists of only three possibilities compared to 24 if this initial analysis of the structure and spectrum graphs was not performed. Again, this leads to a large gain in efficiency of the algorithm. The previous version of the algorithm [5] required approximately 300 s to search for the possible peak assignments, whereas with the information gleaned from the initial analysis of the structure and spectrum graphs, the search algorithm required only 30 s to search for the possible peak assignments. Of the two possible assignments, only one (shown in Figure 2(c)) gives a strong correlation between 29Si chemical shifts and mean Si-Si distances [6] obtained from a single crystal X-ray diffraction structure [7]b.

4 Conclusion

We have shown that an initial analysis of some of the features of the spectrum and structure graphs can narrow down the possible identities of some of the peaks and this can lead to very large gains in the efficiency of the peak assignment algorithm. For zeolites, peak assignment is an important first step towards further structural investigation of zeolite host-guest complexes and establishing relationships between the NMR data (e.g. chemical shifts) and structural data. For unknown zeolite structures, the algorithm could be useful for testing a series of proposed structures against the experimental 2D NMR correlation spectrum. This peak algorithm is general enough to be applied to many types of systems, although it is particularly well-suited for zeolites and similar three-dimensional network structures.

5 Experimental

The algorithm was implemented using Mathematica 5.0 running on a Linux machine equipped with an AMD 2.0 GHz processor and 512 MB of RAM. The graphs were represented as adjacency matrices. Routines were developed to test the adjacency matrices for valencies of the vertices, valencies of adjacent vertices, circuits of a specified length, and automorphisms. Two graphs were tested for homomorphism by comparing perturbed (rearranged rows and columns) adjacency matrices as described in Ref. [5]. The search algorithm for finding all homomorphisms from the structure graph to the spectrum graph was as described in Ref. [5]. The Mathematica notebook and sample input files can be obtained by contacting either of the authors at E.K.Lloyd@soton.ac.uk or D.Brouwer@soton.ac.uk.

DHB acknowledges the Natural Science and Engineering Research Council of Canada for financial support in the form of a Post-Doctoral Research Fellowship.

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