Inverse Versus Normal Behavior of Interactions, Elucidated Based on the Dynamic Nature with QTAIM Dual-Functional Analysis

In QTAIM dual-functional analysis, Hb(rc) is plotted versus Hb(rc) − Vb(rc)/2 for the interactions, where Hb(rc) and Vb(rc) are the total electron energy densities and potential energy densities, respectively, at the bond critical points (BCPs) on the interactions in question. The plots are analyzed by the polar (R, θ) coordinate representation for the data from the fully optimized structures, while those from the perturbed structures around the fully optimized structures are analyzed by (θp, κp). θp corresponds to the tangent line of the plot, and κp is the curvature; θ and θp are measured from the y-axis and y-direction, respectively. The normal and inverse behavior of interactions is proposed for the cases of θp > θ and θp < θ, respectively. The origin and the mechanism for the behavior are elucidated. Interactions with θp < θ are typically found, although seldom for [F–I-∗-F]−, [MeS-∗-TeMe]2+, [HS-∗-TeH]2+ and CF3SO2N-∗-IMe, where the asterisks emphasize the existence of BCPs in the interactions and where [Cl–Cl-∗-Cl]− and CF3SO2N-∗-BrMe were employed as the reference of θp > θ. The inverse behavior of the interactions is demonstrated to arise when Hb(rc) − Vb(rc)/2 and when the corresponding Gb(rc), the kinetic energy densities at BCPs, does not show normal behavior.


Introduction
The quantum theory of atoms-in-molecules dual-functional analysis (QTAIM-DFA) has been proposed to analyze chemical bonds and interactions more effectively [1-3] after the QTAIM approach, introduced by Bader [4,5]. The values of QTAIM functions at the bond critical points (BCPs, * ) on the bond paths (BPs) are often employed for analyses [6]. A chemical bond or an interaction between atoms A and B is denoted by A-B, which corresponds to the BP in the QTAIM. BCP is an important concept in the QTAIM approach, in which charge density ρ(r) reaches a minimum along the interatomic (bond) path and a maximum on the interatomic surface separating the atomic basins. The ρ(r) at the BCP is described by ρ b (r c ), as well as other QTAIM functions, such as total electron energy densities H b (r c ), potential energy densities V b (r c ) and kinetic energy densities G b (r c ) at the BCPs. We use A- * -B for BP, where the asterisk emphasizes the existence of a BCP in A-B. Equations (1)- (3) show the relationships among the functions (cf.: virial theorem for Equation (2)) [4,5].
= G b (r c ) + V b (r c )/2 Interactions are usually classified by the signs of ∇ 2 ρ b (r c ) and H b (r c ), where the signs of ∇ 2 ρ b (r c ) can be replaced by those of H b (r c ) − V b (r c )/2 because (h 2 /8m)∇ 2 ρ b (r c ) = H b (r c ) Scheme 1. Species and compound numbers. Data from the fully optimized structures are analyzed using the polar coordinate (R, θ) representation [1,2], which corresponds to the static nature of the interactions. Each interaction plot, which contains data from both the perturbed and fully optimized structures, includes a specific curve that provides important information about the interaction. This plot is expressed by (θ p , κ p ), where θ p corresponds to the tangent line of the plot, and κ p is the curvature [1,2]. θ and θ p are measured from the y-axis and y-direction, respectively. The parameters are illustrated in Figure 1a, exemplified by Br- * -Br (29). R, θ, θ p and κ p are defined by Equations (4)- (7), respectively, and are given by the energy unit.
The concept of the dynamic nature of interactions has been proposed based on (θ p , κ p ). The (R, θ) and (θ p , κ p ) parameters are called the QTAIM-DFA parameters here [1][2][3]. The signs of the first derivatives of H b (r c ) − V b (r c )/2 and H b (r c ) (derived from (H b (r c ) − V b (r c )/2)/dr and H b (r c )/dr, respectively, where r is the interaction distances in question) are used in the prediction of the natures of the interactions, in addition to those of H b (r c ) − V b (r c )/2 and H b (r c ). As a result, QTAIM-DFA incorporates the classification of the interactions with the QTAIM approach. R = (x 2 + y 2 ) 1/2 (4) θ p = 90 • − tan −1 (dy/dx) (6) κ p = |d 2 y/dx 2 |/[1 + (dy/dx) 2 ] 3/2 (7) where (x, y) = (H b (r c ) − V b (r c )/2, H b (r c )). The reliability of the dynamic nature is controlled by the quality of the perturbed structures. We proposed a method to generate perturbed structures of excellent quality for QTAIM-DFA [7]. The method is called CIV, which employs the coordinates derived from the compliance constants C ii for the internal vibrations [8,9]. The dynamic nature of interactions based on the perturbed structures with CIV is described as the "intrinsic dynamic nature of interactions," as the coordinates are invariant with the choice of the coordinate system. QTAIM-DFA is applied to the standard interactions, employing the perturbed structures generated with CIV, and rough criteria that distinguish the interaction in question from others are obtained. QTAIM-DFA and the criteria are explained in the Supplementary Materials using Schemes A1-A3, Figures A1 and A2, and Table A1. The basic concept of the QTAIM approach is also explained [1 -3].
Cremer has suggested that if bonding is investigated with the Laplacian of ρ b (r c ) via Equation (8) (both sides of Equation (3) were multiplied by 2), the situation is not clearly covered when 2G b (r c ) > |V b (r c )| > G b (r c ). Therefore, he has suggested that in such cases, it seems to be more appropriate to choose H b (r c ) as the indicator of the binding interactions [10]. Our proposed QTAIM-DFA method is a powerful tool that covers his proposal well and can also clearly separate vdW, HB, CT-MC, X 3 and CT-TBP by using perturbation structures. QTAIM-DFA has excellent potential in evaluating, classifying, characterizing, and understanding weak to strong interactions according to a unified form.
The θ p value of an interaction is usually larger than the corresponding θ value, and it is rare that the θ p value is less than the θ value [11]. As shown in Figure 1, each plot for 1-29 seems almost parallel, albeit partially, to the plot of the wide range of 16, although the plot for Me 2 Se + - * -Cl (26) seems somewhat different from others. All interactions in Figure 1 are expected to show behavior that is very similar to the behavior of 16, except for 26. The behavior was examined based on the ∆θ p (= θ p − θ) values. The ∆θ p values were positive for all interactions, except for C- * -H in H 3 C- * -H (35), for which the value was calculated to be ∆θ p = -0.2 • (< 0 • ). The prediction of the negative ∆θ p value to 35 seems curious at first glance. It is not easy to image the result based on its plot, perhaps due to the very small magnitude. One could find a very slight difference in the plot for 35 from the plots for 33 and 36. One could find the similarity to the difference of the plot for 26 from the plot for 25, for example. In the case of 26, the ∆θ p value was calculated to be a positive value of 0.6 • ; ∆θ p = 0.6 • (> 0 • ). The magnitude is also very small. The interactions of Me 2 Se + - * -Cl (26) and H 3 C- * -H (35) should be on the borderline to give the positive and negative ∆θ p values, judging from the magnitudes of ∆θ p . The ∆θ p values for 1-36 are also collected in Table S3 of  Why is θ p usually larger than θ? What is the origin of the positive and negative ∆θ p values for interaction? What are the mechanisms that cause the positive and negative ∆θ p values? Some expectations have been raised that it may be possible to elucidate the behavior of the interactions based on the ∆θ p values. Then, we began to search for interactions that have negative ∆θ p values by examining and reexamining the interactions, including those we investigated thus far.
Indeed, the standard interactions in 1-36 consist of the atoms of the first to fourth periods, but some interactions are expected to have negative ∆θ p values if the interactions contain the atoms of the fifth period. Investigations are started by examining the nature of the interactions containing the atoms of the fifth period and F (37-61), which we call new standard interactions. Some interactions are found to have negative ∆θ p values. We propose the concept of the normal behavior of interactions for ∆θ p > 0 and the inverse behavior when ∆θ p < 0. The origin of the negative ∆θ p values is clarified by analyzing the QTAIM-DFA plots for the interactions with ∆θ p < 0 over a wide range of interaction distances. The mechanisms to show the negative ∆θ p values are also elucidated by analyzing the behavior of QTAIM functions over a wide range of interaction distances. Based on the analysis of various interactions. The behavior of the interactions with the negative ∆θ p values is discussed after the application to the wider range of interactions. The inverse behavior of interactions is demonstrated to arise when H b (r c ) − V b (r c )/2, and the corresponding kinetic energy densities at BCPs, G b (r c ) do not show normal behavior. The results of the investigations are discussed.

Methodological Details of the Calculations
Gaussian 03 and 09 programs were employed for the calculations [12,13]. The aug-cc-pVTZ and/or 6-311+G(3df,3pd) basis sets were applied to the atoms of the group's firstfourth elements in the calculations. They are called basis set system A (BSS-A) and basis set system B (BSS-B), respectively. The Sapporo-TZP basis sets with diffusion functions of the 1s1p type (abbreviated as S-TZPsp) were implemented from the Sapporo Basis Set Factory, which was called BSS-C [14,15]. The S-TZPsp basis sets were applied to the atoms of the fifth period in addition to BSS-A and BSS-B, which were called BSS-A' and BSS-B', respectively. The Møller-Plesset second-order energy correlation (MP2) level [16] was applied to the calculations. The S-TZPsp basis sets were used for I, Br and N with the 6-311+G(d,p) basis sets for F, O, S, C and H, which was called BSS-C'. The reaction processes for CF 3 SO 2 NXMe (X = Br and I) were calculated with MP2/BSS-C'. The optimized structures were confirmed by all real frequencies. The results were used to obtain the compliance constants (C ii ) and the coordinates corresponding to C ii (C i ) [7][8][9]. The optimizations were not corrected with the BSSE method.
Equation (9) explains the method used to generate the perturbed structures with CIV [7]. The kth perturbed structure in question (S kw ) is generated by the addition of the coordinates corresponding to C ii (C i ) to the standard orientation of a fully optimized structure (S o ) in the matrix representation. The coefficient g kw in Equation (9) controls the structural difference between S kw and S o [7]. The g kw is determined to satisfy Equation (10) for r, where r and r o are the interaction distances in question in the perturbed and fully optimized structures, respectively. The C i values of five digits are used to calculate S kw .
The perturbed structures are also obtained by the partial optimization method (POM), where the interaction distances in question are fixed according to Equation (10), containing a wide range of w [1,2]. The reliability of the perturbed structures with POM is substantially the same as the reliability with CIV. The IRC (intrinsic reaction coordinates) method was also applied to generate the perturbed structures, starting from the transition states, TS.
QTAIM functions were calculated with the same method as the optimizations, unless otherwise noted. The calculated values were analyzed with the AIM2000 [17] and AIMAll [18] programs. H b (r c ) is plotted versus H b (r c ) − V b (r c )/2 for the data of five points of w = 0, ±0.05 and ±0.1 in Equation (10) in QTAIM-DFA. Each plot is analyzed using a regression curve of the cubic function, shown in Equation (11), where (x, y) = (H b (r c ) − V b (r c )/2, H b (r c )) (R c 2 > 0.99999 is typical) [3].

Basic Trend in the ∆θ p Values
The H b (r c ) − V b (r c )/2 and H b (r c ) values of the QTAIM functions and the QTAIM-DFA parameters of (R, θ) and (θ p , κ p ) [1,2] for 1-36 (Scheme 1) calculated under MP2/BSS-A are collected in Table S3 of the Supplementary Materials, together with the ∆θ p and C ii values and the predicted nature. The ∆θ p values are positive for all standard interactions of 1-36, except for C- * -H of H 3 C- * -H (35), for which (θ, ∆θ p ) = (202.8 • , -0.2 • ). It is reasonable to assume that an interaction shows normal behavior if ∆θ p > 0, whereas it does the inverse behavior when ∆θ p < 0, which we propose in this work. Namely, all standard interactions in 1-36 behave normally, except for 35, which shows inverse behavior, although the magnitude of ∆θ p is very small. To elucidate the nature of the interactions based on the ∆θ p values, investigations are started by examining the basic trend in the standard interactions of 1-36, employing the ∆θ p values. Figure 2 shows the plot of ∆θ p versus θ for 1-36 (Scheme 1). The plot is analyzed using a regression curve of the eighth-order function after the addition of some fictional points, such as (θ, ∆θ p ) = (45 • , 0 • ) and (206.6 • , 0 • ), and omitting the data for 35 and Me 2 Se + - * -Cl (26). Fortunately, a smooth regression curve, described by f (∆θ p ), was obtained, and the curve passes very close to the points of (45 • , 0 • ) and (206.6 • , 0 • ). The regression curve is shown by a black solid line in Figure 2. The maximum point of the regression curve, shown by the solid line, was (θ, ∆θ p ) = (109.7 • , 48.9 • ). The regression curve was revised to a new curve by amplifying the maximum value of (109.7 • , 48.9 • ) to (109.7 • , 50.0 • ). The new curve is described by f r (∆θ p ) and shown by the dotted line. The treatment helps the discussion to be simpler, because the maximum value of 48.9 • should be tentative and change depending on the employed species. Two more curves of f r (∆θ p )/2 and −f r (∆θ p )/2 are added in Figure 2, which are shown by the blue and red dotted lines, respectively, for a better explanation of the behavior.
The data for the normal behavior of the interactions with ∆θ p > 0 appear on the upper side of the x-axis in Figure 2. Interactions show inverse behavior if the (θ, ∆θ p ) points appear downside of the x-axis. The typical data for the normal behavior of interactions should appear around the black dotted line of f r (∆θ p ), whereas data for the inverse behavior of interactions appear around the red dotted line of −f r (∆θ p )/2. An interaction is called to show weak normal behavior if the point appears between f r (∆θ p )/2 and the x-axis. The interaction in Ne- * -HF (2) seems to show borderline to weak normal behavior ( Figure 2). While the interactions in 26 are borderline between weak normal and inverse behavior, that in 35 show inverse behavior, close to weak normal behavior.
Very similar results were obtained in the plot of ∆θ p versus θ for 1-36 when calculated with BSS-B (see Table S4 and Figure S1 Figure S2 of the Supplementary Materials, together with the data for 1-36. Figure A1 is essentially the same as Figure 2 (see Figure 1 for the QTAIM-DFA plot of 16, shown by the dotted line). A detailed comparison of the two figures revealed that the plots are very close to each other when the Cl- * -Cl distances in question are longer than the optimized values, whereas the ∆θ p values for 16 are evaluated to be somewhat larger than those expected based on the data for 1-36 when the Cl- * -Cl distances in question become shorter than the optimized values of 2.300 Å. When the interaction distance becomes shorter than the optimized value, the energy curve sharply tightens. The calculation conditions for POM under the interaction distances are widely shorter than the optimized values, where the shortened distance are 0.5 Å for 20 data points. The very severe conditions would be responsible for the results.
The close similarity between the two figures can be explained as follows: Starting from interatomic distances that are long enough, chemical bonds or interactions form as the distances shorten. The processes for the interactions are similar to each other, if the interactions of the normal behavior are compared. The processes can be understood based on the H b (r c ) and H b (r c ) − V b (r c )/2 values or the plot of H b (r c ) versus H b (r c ) − V b (r c )/2. In these cases, the (θ, θ p ) values are very close to each other, if they are calculated at the points of substantially the same positions on the plots. Namely, starting from interaction distances that are far enough, stable interactions form as if they go along the similar plot of H b (r c ) versus H b (r c ) − V b (r c )/2. The drive on the plot arrives at a point of the minimum energy of the interaction, where the (θ, θ p ) values are given for the interaction. Different (θ, θ p ) values are obtained because the minimum energy is mainly determined depending on the interacting atoms. There must also be some differences in the energy curves, depending on the characteristics of the interactions, which result in the somewhat different curves of the plots. This should be the reason that the data points of various interactions appear close to the curve for the plot of (θ, ∆θ p ) of 16.
The next investigation is to search for the interactions with negative ∆θ p values after the establishment of the basic trend in the normal behavior of the standard interactions.  Table 1, where the perturbed structures are generated with CIV. Contrary to the case of Figure 1, some plots for the interactions in Figure 3 show different streams from the main (averaged) stream, which must be a reflection of the different behaviors of the interactions. The (R, θ) and (θ p , κ p ) values were similarly calculated for 37-61 under MP2/BSS-A'. Table 1 collects the values, together with the ∆θ p and C ii values and the predicted natures. The ∆θ p values were plotted versus θ for the data of 37-61 shown in Table 1. Figure 4 shows the plots. Table 1. The H b (r c ) − V b (r c )/2 and H b (r c ) values and QTAIM-DFA parameters of (R, θ) and (θ p , κ p ) for the standard interactions in 37-61, together with the C ii , ∆θ p values, and predicted natures, calculated with QTAIM-DFA under MP2/BSS-A', employing the perturbed structures generated with CIV 1 .
Species  As shown in Figure 4 (and Table 1), the data of the (θ, ∆θ p ) values drop very near the f r (∆θ p ) curve for most of 37-61, which shows the normal behavior of the interactions, although some points appear much further below the f r (∆θ p ) curve. The data of the (θ, ∆θ p ) values appear between the f r (∆θ p )/2 curve and the x-axis for Me 2 Te- * - , the data appear below the −f r (∆θ p )/2 curve. Therefore, the interactions are recognized to show inverse behavior stronger than that supposed from the behavior of the standard interactions in 1-36.
What are the differences in the nature of the interactions among the three groups? The (θ, ∆θ p ) values are examined in more detail for the interactions shown in Table 1 Te- * -X 2 > Me 2 (X)Te- * -X > Me 2 Te + - * -X, as a whole, if those of the same X are compared. The strength of Te-X is in the order of Me 2 Te- * -X 2 < Me 2 (X)Te- * -X < Me 2 Te + - * -X; therefore, the strength of Te-X in the species operates to decrease the ∆θ p values for the species. The ∆θ p values for the interactions in 41-44 become more positive, as X goes from F to Cl, then Br, and then to I, where the character (or atomic number) of X becomes closer to Te in 41-44. The ∆θ p values in 49-52 and 55-58 show the trends, similar to that in 41-44, for X = Cl, Br and I. The strength of Te-X operates to decrease the ∆θ p values also for the species. However, the ∆θ p values in 49 and 55 become more negative, when X goes F to Cl, contrary to the case in 41. The differences in the strength of Te-F in 41, 49 and 55 would result in smaller differences in the plot. In the case of H 3 C- * -X, the ∆θ p values become more negative in the order of X = Cl ≥ Br > I >> F. The behavior of the interactions should be controlled by the differences in the characters (or atomic numbers) of X. The character of C in H 3 C- * -X should be closer to those of Cl and Br, relative to the case of I and F. The calculated results are well explained based on Equations (5) and (6).
What are the behavior of the E- * -E' bonds in the neutral, monoanionic, monocationic, and dicationic forms of [MeE- * -E'Me]* (E, E' = S, Se, and Te; * = null, −, +, and 2+)? The species are optimized, first. The optimized structures of the neutral form have C 2 symmetry for E = E' and close to C 2 symmetry for E = E'. Here, the structures are called the C 2 type. The structures of the C 2 and trans types are optimized for the monoanionic form. The structures of the C 2 type are only discussed here because the ∆θ p values are larger than 20 • for both forms ( Table 2). The structures of the trans type are optimized for the monoand dicationic forms. Table 2 collects the (θ, θ p , ∆θ p ) values of the species calculated under MP2/BSS-C. Numbers for the species are shown in Table 1. The calculated results, other than those in Table 2, are collected in Table S8 Equation (16) shows the common order for ∆θ p of E- * -E' in the neutral, mono-and dicationic forms of [MeE- * -E'Me]* (E, E' = S, Se, and Te), where the equation is arranged in the ascending order of ∆θ p . The order can also be understood under the guidance of the conclusions derived from Equations (5) and (6). That is, the ∆θ p values are more negative as the difference in the atomic numbers of A and B in A- * -B becomes larger. However, it is not so clear that the ∆θ p values for the same A- * -B become more negative if the interaction in question becomes stronger.   Table 2 Table S8 of the Supplementary Materials, and the values are plotted in Figure 6. Some (data) points of (∆θ p , θ) appear below the -f r (∆θ p )/2 curve for S- * -Te in 3E Me 0 , 3E H 0 , 3E Me + and 3E Me 2+ , and 3E H 2+ with Se- * -Te in 5E Me 0 and 5E H 0 . The results show that some S- * -Te and Se- * -Te interactions show stronger inverse behavior than that supposed from the behavior for the standard interactions in 1-36.  Table S8 of the Supplementary Materials. Figure 7 shows the plot of ∆θ p versus θ, for the data in Table S8. A lot of data are well visualized. After recognizing the interactions with ∆θ p < 0, the next extension is to clarify the origin and mechanisms that cause the negative ∆θ p values.

Requirements for the Positive to Negative Values of ∆θ p for the Interactions in Question
The ∆θ p value for an interaction is ∆θ p > 0, ∆θ p = 0 or ∆θ p < 0, depending on the dynamic nature of the interaction. The dynamic nature was considered based on the characteristics of the plot around the BCP. Figure 8 illustrates the requirements for the QTAIM-DFA plot of H b (r c ) versus H b (r c ) − V b (r c )/2, where the arrows on the plot lines indicate the direction in which the interaction distance becomes shorter. As shown by the black plot in Figure 8, the ∆θ p value for an interaction is positive when its plot line crosses the line for θ in the direction of clockwise rotation, viewed from the origin. The ∆θ p value is negative when its plot line crosses the line for θ in the direction of counterclockwise rotation. The ∆θ p value is zero when it is parallel to the line for θ, as shown by the blue plot.

Processes to Arise the Negative ∆θ p Values around the Optimized Structures
The processes to give the negative ∆θ p values are examined for some interactions from the sufficiently distant states to around the optimized structures. H b (r c ) values are plotted versus H b (r c ) − V b (r c )/2 for the reaction processes over wide ranges of the interaction distances in question. It is more suitable if the singlet state is retained throughout the process, which relieves us from the trouble of considering the effect of the change on the multiplicity in the process. The structural change is also expected to be limited to the minimum extent if the singlet state is retained throughout the processes. The analysis is much simpler under these conditions, together with the discussion. As a result, the mechanisms to give the negative ∆θ p values are more clearly understood as the reactions proceed if the singlet state is retained throughout the reaction processes. Such processes seem rather rare and satisfy the above conditions.
The negative ∆θ p value of −4.2 • is calculated for [F-I- * -F] − (46), as shown in Table 1, where the reaction process is shown by Equation (17). The reaction shown by Equation (18) has been reported to proceed in the singlet state under metal-free conditions [19,20]. A negative value of −22.8 • was also calculated for Tf-N- * -IMe (Tf = SO 2 CF 3 ) ( Table 3). The reaction processes for [F-I- * -F] -(46) in Equation (17) and Tf-N- * -IMe in Equation (18) could be good candidates to achieve this purpose. Therefore, the reaction processes to give the negative ∆θ p values are examined, exemplified by the reactions shown in Equations (17) and (18).    Figure 9 contains the plot for [Cl-Cl- * -Cl] − , similarly calculated, for comparison (see also Table 1). The plot for [Cl-Cl- * -Cl] − shows a smooth and monotonic curve, starting from a point close to the origin. The plot for [Cl-Cl- * -Cl] − forms a spiral curve overall. As a result, the curve of the plot for [Cl-Cl- * -Cl] − satisfies the requirements for θ p > θ (∆θ p > 0) throughout the reaction process, as illustrated in Figure 9. This is the reason that [Cl-Cl- * -Cl] − typically shows the normal behavior of interactions.  Figure 9. The curve gradually changes to show the downwardly convex character to satisfy the requirements for θ p = θ (∆θ p = 0). The optimized structure, shown by w = 0.0 in Figure 9, appears in the range satisfying the requirements for θ p < θ (∆θ p < 0), illustrated in Figure 8. Indeed, the plot in Figure 9 shows the outline to give the negative ∆θ p value (-5.5 • ) for [F-I- * -F] − (46), but it is better if the requirement for ∆θ p < 0 is illustrated more clearly in the plot for a process. Next, this process was examined and exemplified by the ligand exchange reaction at N via TS [MeBr- * -N(Tf)- * -IMe]. Table 3  The reaction processes shown in Equation (19) were examined before investigations on those shown in Equation (18) (Figures 1 and 9).  Figure 10 shows the QTAIM-DFA plots for the process shown by Equation (18). Figure 10 is briefly explained to avoid misunderstanding. It is not a plot for the energy profile, but a QTAIM-DFA plot for both Br- * -N and N- * -I, so two plots appear corresponding to the two bonds, while a single transition state contributes to the reaction. (This is also the case for the plots in Figure S3 of the Supplementary Materials.) The forward and reverse processes for the reaction shown by Equation (18) are clearly specified in Figure 10. In spite of the forward and reverse processes for the reaction, the plot for Br- * -N is drawn by the black dots in Figure 10, which corresponds to the reaction process of Equation (18). The process forms MeI + Tf-N- * -BrMe, starting from Tf-N- * -IMe + MeBr via TS [MeBr- * -N(Tf)- * -IMe]. The ∆θ p value of Br- * -N changes from 7.9 • for Tf-N- * -BrMe to 42.6 • for TS [MeBr- * -N(Tf)- * -IMe], then to 0.0 • , with no interaction state, if the reaction proceeds in the direction shown in Equation (18).
The plot for Br- * -N in Figure 10 also shows a smooth, monotonic and spiral curve, which is very similar to the curve for [Cl-Cl- * -Cl] − (Figures 1 and 9) and the curves for Br- * -N illustrated in Figure S3 Figure 10. The difference between the plots clarifies the mechanism to give the negative ∆θ p value for Tf-N- * -IMe.
The plot for N- * -I in red shows a distorted Z shape. This is of great interest because the plot for N- * -I seems to largely overlap the plot for N- * -Br in the range of In the plot for N- * -I in Figure 10, H b (r c ) decreases monotonically, whereas H b (r c ) -V b (r c )/2 shows abnormal behavior when H b (r c ) < 0. Therefore, it is strongly suggested that the strange behavior of H b (r c ) -V b (r c )/2 should be responsible for the negative ∆θ p value.
The process for N- * -I to give the negative ∆θ p value is well clarified by the plot for N- * -I in Figure 10. However, it would be difficult to visually recognize the value of ∆θ p , positive or negative, from the plot. We searched for a typical example that can be recognized as a clear negative ∆θ p value visually from the plot. The (θ, θ p , ∆θ p ) values for [HS- * -TeH] 2+ are given in Table S8 of the Supplementary Materials, of which ∆θ p is a negative value of −9.7 • . The process for [HS- * -TeH] 2+ was calculated with POM by elongating the S- * -Te distance, starting from the structure around the optimized one. The calculations were performed in the range where the rational structures were optimized at the singlet state. Figure 11 draws the reaction process for [HS- * -TeH] 2+ , although the interaction distance is limited to the distance around the optimized structure, where the perturbed structures are rationally optimized at the singlet state. The plots for N- * -I and N- * -Br in the reaction processes shown in Equation (18) are added to Figure 11 for reference. The partial plot for [HS- * -TeH] 2+ , shown in Figure 11, seems to correspond to the plot for N- * -I in the range of H b (r c ) < −0.005 au. The lines of ∆θ p = 0 • for the plots of N- * -I and [HS- * -TeH] 2+ are drawn by the red and blue dotted lines, respectively. The tangent directions from the origin to the curves correspond to the lines of ∆θ p = 0 • (see also Figure 8). As discussed above, the plot for N- * -Br satisfies the requirement for ∆θ p > 0 • in the whole range of the reaction process, while the plot for N- * -I should satisfy the requirement for ∆θ p < 0 • around the optimized structure. However, it seems visually unclear from the plot, as mentioned above. In the case of the plot for [HS- * -TeH] 2+ , there exists an area that clearly satisfies the requirements for ∆θ p < 0 • , after the blue dotted line for ∆θ p = 0 • . The area with ∆θ p < 0 • is the area around the optimized structure in this case. The crossing point on the plot with the tangent line corresponds to ∆θ p = 0 • ; therefore, the perturbed structure of which the S- * -Te distance is shorter than the value at ∆θ p = 0 • should show positive ∆θ p values, namely the area of ∆θ p ≥ 0 • .
After clarification of the origin for the positive and negative ∆θ p values through the plots, the next extension is to elucidate the mechanisms based on the behavior of G b (r c ) and V b (r c ).

Mechanisms for the Origin of the Negative ∆θ p Values
The reaction processes can be directly expressed using the interaction distances. However, the ρ b (r c ) values can also be used for the analyses, where the values decrease exponentially as the interaction distances increase. The distances are also expressed by w in Equation (10). The relation between ρ b (r c ) and w is confirmed by the plot of ρ b (r c ) versus w for an interaction. Such a plot is shown in Figure S6    In the case of [F-I- * -F] − , the plots of G b (r c ) versus ρ b (r c ) and V b (r c ) versus ρ b (r c ) go upside and downside, respectively, smoothly and monotonically, starting from a point close to the origin. However, the plots show convex upward and downward shapes, respectively, and the magnitude of the former seems 0.8 times larger than the magnitude of the latter. As a result, the plot of H b (r c ) − V b (r c )/2 versus ρ b (r c ) is mainly controlled by the plot of G b (r c ) versus ρ b (r c ), whereas the plot of H b (r c ) versus ρ b (r c ) is almost controlled by the plot of V b (r c ) versus ρ b (r c ) (cf; Equations (1) and (2)   The results support the following statements: The inverse behavior of the N- * -I interaction must originate based on the convex then concave curve in the plot of H b (r c ) − V b (r c )/2 versus ρ b (r c ). The loose Z-shaped plot of H b (r c ) versus H b (r c ) − V b (r c )/2 for N- * -I is observed, starting from a point close to the origin. The N- * -Br and N- * -I interactions are demonstrated through the above discussion, which show typical normal and inverse behavior, respectively.
Applications of the proposed methodology to wider range of bonds and interactions, such as a bird's-eye view of the periodic table, are in progress. The results will be discussed elsewhere, together with the results, related to ours, reported so far by others.

Conclusions
The QTAIM-DFA parameters of (R, θ) and (θ p , κ p ) are obtained by analyzing the QTAIM-DFA plots of H b (r c ) versus H b (r c ) − V b (r c )/2. The θ p value is usually larger than θ, ∆θ p (= θ p − θ) > 0, for an interaction, as confirmed by the standard interactions, but it is sometimes negative. The prediction of the nature of interactions is confused when ∆θ p < 0 because the criteria to predict the nature are formulated assuming positive ∆θ p values. The negative ∆θ p value for an interaction must be a sign of its special nature.  We proposed QTAIM-DFA by plotting H b (r c ) versus H b (r c ) -V b (r c )/2 (= (h 2 /8m)∇ 2 ρ b (r c )) [1], after the proposal of H b (r c ) versus ∇ 2 ρ b (r c ). Both axes in the plot of the former are given in energy unit, therefore, distances on the (x, y) (= (H b (r c ) -V b (r c )/2, H b (r c )) plane can be expressed in the energy unit, which provides an analytical development. QTAIM-DFA incorporates the classification of interactions by the signs of ∇ 2 ρ b (r c ) and H b (r c ). Scheme A2 summarizes the QTAIM-DFA treatment. Interactions of pure CS appear in the first quadrant, those of regular CS in the fourth quadrant and SS interactions do in the third quadrant. No interactions appear in the second one. In our treatment, data for perturbed structures around fully optimized structures are also employed for the plots, together with the fully optimized ones (see Figure A1) [1,2, [22][23][24]. We proposed the concept of the "dynamic nature of interaction" originated from the perturbed structures. The behavior of interactions at the fully optimized structures corresponds to "the static nature of interactions", whereas that containing perturbed structures exhibit the "dynamic nature of interaction" as explained below. The method to generate the perturbed structures is discussed later. Plots of H b (r c ) versus H b (r c ) -V b (r c )/2 are analyzed employing the polar coordinate (R, θ) representation with (θ p , κ p ) parameters [1,2, [22][23][24]. Figure A1 explains the treatment. R in (R, θ) is defined by Equation (A4) and given in the energy unit. Indeed, R does not correspond to the usual interaction energy, but it does to the local energy at BCP, expressed by [(H b (r c )) 2 + (H b (r c ) -V b (r c )/2) 2 ] 1/2 in the plot (cf: Equation (A4)), where R = 0 for the enough large interaction distance. The plots show a spiral stream, as a whole. θ in (R, θ) defined by Equation (A5), measured from the y-axis, controls the spiral stream of the plot. Each plot for an interaction shows a specific curve, which provides important information of the interaction (see Figure A1). The curve is expressed by θ p and κ p . While θ p , defined by Equation (A6) and measured from the y-direction, corresponds to the tangent line of a plot, where θ p is calculated employing data of the perturbed structures with a fully-optimized structure and κ p is the curvature of the plot (Equation (A7)). While (R, θ) correspond to the static nature, (θ p , κ p ) represent the dynamic nature of interactions. We call (R, θ) and (θ p , κ p ) QTAIM-DFA parameters, whereas ρ b (r c ), ∇ 2 ρ b (r c ), G b (r c ), V b (r c ), H b (r c ) and H b (r c ) -V b (r c )/2 belong to QTAIM functions. k b (r c ), defined by Equation (A8), is an QTAIM function but it will be treated as if it were an QTAIM-DFA parameter, if suitable.  Table A1. The plot of H b (r c ) -V b (r c )/2 versus w in Figure A2 is essentially the same as that of ∇ 2 ρ b (r c ) versus d(H-F) in X-H-F-Y, presented by Espinosa and co-workers [25]. . Typical hydrogen bonds without covalency and typical hydrogen bonds with covalency are abbreviated as t-HB without cov. and t-HB with cov., respectively, whereas Cov-w and Cov-s stand for weak covalent bonds and strong covalent bonds, respectively. Table A1. Proposed definitions for the classification and characterization of interactions by the signs H b (r c ) and H b (r c ) -V b (r c )/2 and their first derivatives, together with the tentatively proposed definitions by the characteristic points on the plots of H b (r c ) versus H b (r c ) -V b (r c )/2. The tentatively proposed definitions are shown by italic. The requirements for the interactions are also shown.

ChP/Interaction
Requirements by H b (r c ) and V b (r c ) Requirements by G b (r c ) and V b (r c )