Introduction

The term “electronegativity” was first introduced by Berzelius in 18111, it is a key concept in chemistry, closely related to the physical and chemical properties of elements2. Therefore, its definition, concept, calculation, and especially the selection of different “scales” have always been a subject of intense research and debate3,4,5.

Since the beginning of the 21st century, researchers have increasingly considered network reactions as the core of chemistry6, and have started building networks from compounds to study various chemical properties from a network perspective7,8,9. In our previous study, we viewed chemical elements as a system connected by chemical bonds and constructed an undirected chemical network with 97 elements and 2198 bonds10,11. However, the undirected chemical network does not fully capture the relationships be- tween elements, as many properties related to chemical bonds and elements, such as bond polarity and electronegativity, are not represented. To address this limitation, we transformed the network into a directed one using element electronegativity12. The analysis of this directed network reveals that many chemical phenomena can be studied from the network perspective. For instance, there are significant correlations between most electronegativity scale pairs, and the electronegativity of elements is highly correlated with node degree centrality, consistent with conventional understanding. In this paper, we extend the analysis to study the relationship between electronegativity and network properties, providing another approach to study chemistry.

The remainder of the paper is organized as follows. The establishment of the directed network is discussed in Sect. “The directed network”, the relation between electronegativity of elements and degree of nodes is discussed in Sect. “Relation”, the application of relation is discussed in Sect. “Discussion and application”, before giving conclusion and future work in Sect. “Conclusions”.

The directed network

The network discussed in this paper is the same one as in our previous study. This network contains 97 elements. He, Ne, Ar and elements with atomic number greater than 100 are excluded, as binary compounds containing these elements are very rare. Nodes of the network are elements and edges are assigned based on co-occurrences of elements in binary com- pounds. The stoichiometric of elements are ignored. The reason for not considering stoichiometry is to simplify the network as it has no impact on determining the direction of edges. Furthermore, the ability of two elements to form compounds is foundational in understanding the number of possible compounds that can be formed between them, a topic of considerable research significance. Moreover, incorporating stoichiometry does not fully capture the reactivity of the elements involved in a reaction. For instance, C and H would appear more reactive than F due to the abundant presence of hydrocarbons, which contradicts established facts and theories.

In chemistry, chemists often use the electronegativity of the element to reflect the ability of its atoms to attract bonding electrons4,13. Therefore, the relative size of electronegativity can also be used to approximately judge polarities of bonds between two bonding elements2. Given element A and element B, if binary compound AiBj (i ≠ 0, j ≠ 0) exists, then A, B and an directed edge between them are added to the network. The direction of edge is defined using the following rule:

if the electronegativity of A is equal to that of B, then there are two and only two directed edges between A and B, which are Edge A→B and Edge A←B; if the electronegativity of B is bigger than that of A, then there is one and only one directed edge between A and B, which is Edge A→B .

For example, for the bond between N and H in binary compound NH3, the Pauling scale for element N is 3.04, while the Pauling scale for element H is 2.23, then the direction is H→N. Based on above rule, we get a directed network with 92 nodes and 2107 directed edges. (Note that the number of nodes in this network is smaller than that in undirected network, as electronegavitity of 5 elements cannot be found in Pauling Scale, which are element Rn, Pm, Eu, Tb and Yb). We constructed 5 directed networks using 5 electronegativity scales, which are Nagle scale χα14, Allred–Rochow scale χar15, Pauling scale χp, Allen scale χs16 and Rahm scale χr17. The top 5 elements with the biggest indegree and outdegree in each network are listed in Table 1(each network is named according to its corresponding scale).

We can see that elements with the largest indegree are those that their atoms have the largest ability to attract bonding electrons, while elements with the largest outdegree are not necessarily those whose atoms are most likely to lose bonding electrons. For example, most electronegative elements like Na, Ca and K are not included in Table 1.

Table 1 Top 5 nodes with the biggest indegree and outdegree in each network.
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If we map the indegree and outdegree of elements in the network onto a two-dimensional space, then each element has a coordinate point in the space. Figure 1 illustrates the mapping result of network χp, in which indegree of each element is the X coordinate and outdegree the Y coordinate.

Fig. 1
figure 1

Mapping of elements into the two-dimentional coordinate system.

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We can see from Fig. 1 that elements listed closely in the Periodic Table are mainly located closely in this coordinate system. The same situation also occur in the other 4 networks, where we did not provide mapping figures due to space limitation. Therefore, the combination of indegree and outdegree of elements is related to its chemical properties.

Relation

In Table 1, F has the largest indegree and the smallest outdegree of 0, suggesting that F can attract all other elements in the network. In network χp, O has the second largest indegree, indicating that it can attract all elements except F. The outdegree of O is 1, suggesting that it can only be attracted by F. Therefore, the electronegativity of a node in the directed network is closely related to its ability to attract other nodes relative to its ability to be attracted by other nodes. Based on this observation, we define a new index, comparative attractiveness(CA), to study its relationship with electronegativity.

$$\:\text{C}\text{A}\left(\text{E}\right)=\frac{\text{i}\text{n}\text{d}\text{e}\text{g}\text{r}\text{e}\text{e}\left(\text{E}\right)}{\text{o}\text{u}\text{t}\text{d}\text{e}\text{g}\text{r}\text{e}\text{e}\left(\text{E}\right)}$$
(1)

Relation between CA and electronegativity

As the number of nodes varies across different networks—for instance, network χr has 93 nodes, while network χs has only 69 nodes—to avoid errors arising from differences in sample size, we selected the 63 nodes that are common to all networks for analysis. 6 inert gas elements, 15 lanthanide elements (15 lanthanide elements are excluded because network χar contains only one lanthanide element La, and network χs contains only one lanthanide element Lu), 14 actinide elements, as well as Po, At, Fr, Ra and Lr, are excluded. The CA and electronegativity values of 63 elements of 5 scales are provided in the supplementary files due to space limitation. The changing trend of CA and electronegativity of 63 elements vary with atomic number in 5 networks are shown Fig. 2 (the CA of F should be ∞ but is given as 100, as ∞ cannot be represented in the figure). We can see from this figure that CA and electronegativity are related. Moreover, CA and electronegativity have the same periodic variation pattern, indicating that periodicity is widely present in the properties of chemical elements.

Fig. 2
figure 2

Relation between CA and electronegativity in 5 networks.

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We use Pearson index to calculate the correlation coefficients between CA and electronegativity18, the results are listed in Table 2. Although the minimum coefficient of all elements is only 0.708, if we classify these elements by category, the minimum value reaches 0.807, indicating that the CA in different categories is highly correlated with electronegativity.

Table 2 Pearson correlation analysis of CA and electronegativity in 5 networks.
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At first glance, CA and electronegativity may seem to have different definitions, since electronegativity is a specific value, while CA is simply a ratio. However, electronegativity is also a relative concept, reflecting an element’s ability to gain or lose electrons. Therefore, the definition of electronegativity is essentially the same as that of CA.

Monotonic consistency between CA and electronegativity

We also analyzed the monotonic consistency between CA and electronegativity for each column of elements, the results showed that CA and electronegativity exhibit monotonic consistency for the 5 alkali metals and 5 alkaline earth metals across all 5 networks. The same phenomenon occurs in all columns of non-metallic elements (column with only element B is omitted), including the C, Si column, N, P, As column, O, S, Se, Te column and F, Cl, Br, I column. Table 3 lists the monotonic consistency of CA and electronegativity for the 9 columns of transition metals.

Table 3 Monotonic consistency of 9 columns of transition metals.
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In Table 3, each column contains 3 elements, group 3 is not included as it contains Lu, which is not among the 63 elements we choose. We can see out of 45 columns, 37 are monotonically consistent, accounting for 82.2%.

If we treat 15 non-metallic elements as one group and check their monotonic consistency between CA and electronegativity, there is only 8 elements that their CAs violate consistency in 5 networks, which are As, H and C in network χs, H and C in network χar and χα, P in network χs. Therefore, the proportion of consistency for network χα reaches 86.6%((15−2)/15), the proportion reaches 89.3%((75−8)/75) for 5 networks, indicating that the consistency between CA and electronegativity is commonly present in 5 networks. How C and H in network χα violate the consistency is listed in Table 4.

Table 4 Monotonic consistency of 15 non-metallic elements in network Χα.
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Correlations among CAs

In our previous paper, we found that there exist statistically significant correlations between any pair of electronegativity scales12. As CA and electronegativity are highly related, significant correlations should also exist between any pair of CAs, and this is indeed the case, as shown in Table 5. The relationship among 5 CAs also indicate that, because CA can reflect the relative size of the element’s ability to gain and lose electrons. Regardless of the electronegativity that CA uses, it is essentially consistent.

Table 5 Pearson correlation analysis of 5 CAs.
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Discussion and application

The relationship between CA and electronegativity suggests that CA can be regarded as an internal chemical property of an element. Since this analysis is conducted purely from a network perspective, the results can be used to address traditional chemical problems in a simpler way. For example, the prediction of potential chemical compounds, which is typically based on experimental fieldwork, is often costly and time-consuming. By predicting based on established rules and focusing on compounds most likely to exist, experimental costs can be significantly reduced if the predictions are accurate. We use the CA of C to illustrate this prediction. Out of the 5 networks, 3 display abnormal CA values for C relative to the consistency requirement, indicating that the CA of C needs to be updated. For example, in network χα, the indegree of C is 61, the outdegree of C is 5, and CA of C is 12.2 (61/5). To maintain monotonic consistency, CA of C should be greater than 13.17 (CA of S) and less than or equal to CA of Br, which is 21.0 (84/4), as shown in Table 4. The outdegree reaches its maximum because the electronegativity of C is smaller than that of five elements: F, O, N, Cl, and Br. Therefore, the outdegree of C cannot be increased, and its indegree should be larger. Following this guidance, we can search the literature to identify potential edges. For elements without edges to C, we can investigate the possibility of having edges to C. Several edges, such as C—Au and C—Cu, which are not yet included in the current network, have been found19, thereby validating the reliability of this prediction method.

It is worth noting that while the value of electronegativity determines the direction of the edges, which in turn influences the indegree, outdegree, and CA of elements, statistical uncertainties in electronegativity values do not necessarily alter the relative magnitude of CA, which is the most critical factor. For instance, in the case of F, a statistical uncertainty in its electronegativity value does not affect its ranking among all elements, and thus, its indegree, outdegree, and CA remain unchanged. However, for elements with very close electronegativity values, such as C (2.55) and Br (2.56) in Table 4, statistical uncertainties may lead to changes in the direction of the edge between them. For example, if the electronegativity of C increases by 0.2 to 2.57 and the electronegativity of Br decreases by 0.2 to 2.54, the direction of the C-Br edge would reverse. Consequently, the indegree of C would increase by 1, while its outdegree would decrease by 1, and its CA would change from 12.2 (61/5) to 10.5 (62/4). Similarly, the CA of Br would change from 21 (84/4) to 16.6 (83/5). Despite these changes, the CA of Br remains greater than that of C, demonstrating that the relative ranking of CA is preserved.

In contrast to electronegativity, which is a single value, CA provides a more intuitive representation by incorporating the relative ability of an element to gain electrons compared to its ability to lose them. This distinction is clearly illustrated in the modified Table 4.

This method can also be used to calibrate electronegativity scales. For example, in the Rahm scale, the electronegativity of H and S is both 13.6, which is not reasonable as it fails to explain the polarity of the S—H bond in in H2S. From the perspective of CA, the CA of S is 6.91, while the CA of H is 6.4. Therefore, the electronegativity of S should be higher than that of H, as suggested by CA values.

Conclusions

The main contribution of this paper is the definition of the new index CA, which represents the comparative attractiveness of elements using their ingree and outdegree in the directed network. Since the indegree and outdegree are related to the chemical properties of elements, the correlation analysis between CA and electronegativity, along with the monotonic consistency between CA and electronegativity, demonstrates that CA can be considered an internal chemical property of elements.

Because CA is derived from a directed network and is influenced by all elements in the network, it differs from traditional electronegativity scales. It can be used for tasks that traditional electronegativity cannot address, such as predicting potential compounds and calibrating electronegativity scales. It should be noted, however, that although the compound data we collected are limited, the results are still reasonable.

Future work will focus on ternary networks, quaternary networks, and higher-order networks to obtain CA values for binary functional groups, ternary functional groups, and higher-order functional groups.

It is worth emphasizing that the network constructed in this study establishes a system based on binary chemical relationships, offering a novel platform for analyzing the properties of elements and compounds using network- based tools. Analyzing electronegativity represents just one application of this approach. We aspire to leverage this platform to address a broader range of challenges that traditional methods cannot resolve.