Proposal of pseudo-intrinsic dynamic nature of interactions: simple methods to generate the perturbed structures and to analyze for the prediction of the nature of high reliability, with the applications

Highly reliable dynamic nature of interactions is obtained with QTAIM dual functional analysis, if the perturbed structures generated by CIV, the established method, are used. Simpler and easier methods are proposed to predict the nature, with the same reliability as that with CIV. The dynamic nature using the structures generated with the partial optimization method is described as the ‘‘pseudo intrinsic dynamic nature of interactions,’’ since it satisfies the requirements. An ultimate method, called OMA


Introduction
Weak interactions determine the fine details of structures and create high functionalities to materials, while strong bonds form the molecular skeletons.][3][4][5][6][7][8][9][10][11][12][13][14][15][16] It is a most important issue to control the weak interactions, experimentally and theoretically, in chemical and biological sciences.It is the first stage of investigations to clarify the nature of the chemical bonds and interactions.In our investigation, the nature of an interaction in question will not be defined based on the interaction in formal structural features but is determined by considering the character of the interaction.Namely, the nature of an interaction is predicted to have vdW nature, if the interaction shows the typical character of the vdW interaction, for instance.In this process, it is necessary to define the scope of the typical character of each interaction, which should arise from the typical electronic structure of the interaction.
How can the scope of the nature of interactions be defined?The chemical bonds and interactions are classified using the quantum theory of atoms-in-molecules (QTAIM) approach, introduced by Bader. 17,18owever, it seems difficult to characterize the noncovalent interactions with QTAIM.0][21][22][23][24] QTAIM-DFA incorporates the QTAIM approach.1][22][23][24] However, the full treatment of QTAIM-DFA has some difficulties, especially for experimental chemists who are not familiar with such treatments.The purpose of this paper is to propose simpler and easier methods for QTAIM-DFA, of the substantially same reliability as the full treatment.
QTAIM-DFA and QTAIM approaches are explained first, followed by the suggestions to promote research in this work.

Survey of QTAIM-DFA and QTAIM approach
A chemical bond or an interaction between A and B is denoted by A-B, which corresponds to the bond path (BP) in QTAIM.0][21][22][23][24] BCP is a point along the interatomic bond path at the interatomic surface where the charge density (r) reaches a minimum, while its maximum is on the interatomic surface separating the atomic basins.The (r) values at BCPs are described by b(rc), so are other QTAIM functions at BCPs, such as the total electron energy densities Hb(rc), potential energy densities Vb(rc), and kinetic energy densities Gb(rc).Equations ( 1), (2), and (2') show the relations among the functions (cf: Virial theorem for equation ( 2)).
In QTAIM-DFA, Hb(rc) are plotted versus Hb(rc) -Vb(rc)/2 [(ћ 2 /8m) 2 b(rc)] at BCPs, instead of Hb(rc) versus  2 b(rc).This choice enables us to analyze the plots much more effectively, since the four arithmetic operations can be applied to analyze the plots by unifying the unit of both axes in the plot to energy.Figure 1 shows the QTAIM-DFA plots of Hb(rc) versus Hb(rc) -Vb(rc)/2 for various noncovalent and some weak covalent interactions.(See Table 3 for the data plotted in Figure 1, calculated under MP2/S-TZPsp, where the perturbed structures are generated with the partial optimization method of the Modredundant form (POM-M).) The plots in Figure 1 have a spiral stream as a whole and the interactions seem well separated.The interactions are expected to be well classified and characterized with QTAIM-DFA.
Recently, calculations of species containing atoms of the 5 th period have been carried out more and more.We must be careful when the dynamic nature of  3.
In our treatment, data from the perturbed structures around the fully optimized structures are employed, in addition to those that are fully optimized.1][22][23][24] Each interaction plot, containing data from both perturbed and fully optimized structures, is expressed by (p, p), where p corresponds to the tangent line of the plot and p is the curvature. and p are measured from the y-axis and the y-direction, respectively.1][22][23][24] (See Figures 1 and 2, footnotes of Tables 3 and 4, and the Appendix of Supporting Information for the definitions of the QTAIM-DFA parameters of (R, ) and (p, p)).As a result, the signs of d(Hb(rc) -Vb(rc)/2)/dr and dHb(rc)/dr, where r is the interaction distance, are employed to characterize interactions in QTAIM-DFA, while the signs of Hb(rc) -Vb(rc)/2 and Hb(rc) are used to classify them, overall.
Namely,  classifies interactions, while p characterizes them.
The reliability of the dynamic nature is controlled by the quality of the perturbed structures.We have proposed three methods to generate the perturbed structures for QTAIM-DFA, so far.They are called POM, 20,21 NIV, 22,23 and CIV. 26The perturbed structures are generated by the partial optimizations in POM with the interaction distances in question being fixed appropriately longer and shorter than the fully optimized distances (according to equation ( 3)).POM is the thermal process.POM of the Z-matrix form, called POM-Z, was used in the early stage of the investigations.The reliability of the dynamic nature will be examined carefully in this work, when the perturbed structures are generated with POM of the Modredundant form (POM-M).The normal coordinates of the (best-fitted) internal vibrations are used to generate the perturbed structures in NIV.The perturbed structures with NIV are the mathematical similarity to those appearing in the zero-point internal vibrations.NIV is the adiabatic process.
In CIV, the coordinates derived from the compliance constants Cii for the internal vibrations are employed to generate the perturbed structures, instead of the normal coordinates in NIV.The compliance constants Cij are defined as the partial second derivatives of the potential energy due to an external force, as shown in equation (R1). 26,27CIV corresponds to an improved method of NIV, therefore, it should be recognized as an adiabatic process.1][22][23][24] QTAIM-DFA with CIV has excellent potential to classify, characterize, and understand weak to strong interactions with high reliability, according to a unified form.The dynamic nature of interactions with CIV is described as the "intrinsic dynamic nature of interactions," since the coordinates in CIV are invariant to the choice of coordinate system.QTAIM-DFA with CIV were applied to the standard interactions consisting of the atoms of the 1 st -4 th periods.The rough criteria that distinguish the interaction in question from others were obtained, at the early stage of the investigations, under MP2/6-311++G(3df,3pd).QTAIM-DFA and the criteria are explained in the Appendix of the Supporting Information, using Schemes SA1-SA3, Figs.SA1 and SA2, Table SA1, and equations (SA1)-(SA7).

Natures
Requirement a p-CS/vdW Guides to the new methodologies QTAIM-DFA with CIV is the excellent method to predict the dynamic and static natures of interactions.However, it is necessary to handle the complex compliance program in the process with CIV, which would be difficult for those not familiar with the theoretical treatment.Therefore, we searched for simpler and easier methodologies to generate the perturbed structures of very high quality, other than CIV.
How can new simpler and easier methods be devised?We have recognized that the perturbed structures generated with POM-Z have the quality very close to that with CIV, since CIV has been proposed. 26Using this as a clue, substantial effort was paid to inflate the image for the establishment of the simpler and easier methods to analyze the interactions effectively.Figure 3 shows the plot of Hb(rc) versus Hb(rc) -Vb(rc)/2 for the data of the fully optimized structures, exemplified by Cl-Cl--Cl -, together with those from the perturbed structures generated with CIV, POM-Z, POM-M, and NIV of the symmetric and anti-symmetric vibrations (abbreviated by NIVs and NIVas, respectively).The perturbed structures were also used to generate by changing only the interaction distance in question (called the major interaction) in Cl-Cl--Cl -.The method is called OMA. Figure 3 contains the plot with OMA.The plots are analyzed using the QTAIM-DFA parameters of (R, ) and (p, p), as explained above.The p values based on the methods to generate the perturbed structures are given in Tables 3 and 4. The reliability of the dynamic nature of the interaction can be recognized based on the p values.As shown in Figure 3 (and Table 3 and Table S3 of the Supporting Information), the p values are 178.4° for the three methods of CIV, POM-Z, and POM-M.The p values with NIVs and NIVas are 174.4°and 180.1°, respectively.The p value is 177.2° with OMA.It is demonstrated that the quality of the perturbed structures generated with POM-Z and POM-M is the same as that with CIV. 26 The p values with NIVs and NIVas are smaller and larger than that with CIV by 4.0° and 1.7°, respectively.The p value with OMA is smaller than that with CIV by 1.2°.The results show that OMA is better than NIV to generate the perturbed structures, since the magnitudes in the differences from CIV is smaller for OMA than the case of NIVs and NIVas.Indeed, POM-Z and POM-M can be recognized as the same methods as CIV, but POM-M seems superior to POM-Z, since POM-Z has some disadvantages in constructing the perturbed structures and in the optimizations, 28 especially when it is applied to large and/or complex species.POM-M would release such disadvantages in POM-Z.OMA will show good advantages, when applied to large and/or complex molecules, although some devices would be necessary.We will mainly examine POM-M and OMA, here, as the simpler and easier methods to generate the perturbed structures for QTAIM-DFA.
Four perturbed structures and a fully optimized one, with the regression curve of a cubic function, are necessary to predict the dynamic nature for an interaction, in the full treatment of QTAIM-DFA.Two perturbed structures and a fully optimized one, with a regression curve of a linear type, also give the dynamic nature for the interaction.If the predicted dynamic nature based on the latter is demonstrated to be substantially the same as that based on the former, the latter method can be recognized as the simpler and easier methodology.The reliability was also examined by comparing the two results.Indeed, the p values cannot be obtained with a linear regression curve, but it does not damage our discussion, since p is not used in the prediction of the natures.
Here, we present the results of the investigations, searching for the simpler and easier methods for QTAIM-DFA.The dynamic nature with POM (POM-Z and POM-M) we propose to term "pseudo intrinsic dynamic nature of interactions," after the establishment of the same reliability of the dynamic nature with POM as that with CIV.POM-M seems superior to POM-Z, as the simpler and easier method.The reliability of the dynamic nature obtained using two perturbed structures and a fully optimized one, with a regression curve of a linear type, is substantially the same as that in the full treatment.The reliability with OMA also seems substantially high, which should be the ultimate simpler and easier method to generate the perturbed structures.The applied results of the proposed methods are also discussed.
The purpose of this paper is to present a simpler and easier method for experimental chemists to be able to predict the nature of interactions in question with high reliability.We believe that by applying the proposed methods, experimental chemists will be released from concerns about the complex compliance program and from the frequency analysis in the prediction of the dynamic nature of interactions with QTAIM-DFA.

Methodological Details in Calculations
Calculations were performed employing the Gaussian 09 programs package. 29Structures of the selected species 1-56 were optimized, where 1-4, 6-12, 14, 15, 17, 18, 23, 24, 26, 27, 30-33, 37, 38, 40, 41, 46, 47, 49, 50, and 52-56 consist of the atoms in the 1 st -4 th periods, while 1-56 other than above contain Te, I, and Xe atoms in the 5 th period (see Table 1).The Sapporo-TZP basis set with 1s1p diffusion functions (abbreviated by S-TZPsp), as implemented from Sapporo Basis Set Factory, [30][31][32] was applied to all atoms in 1-56.Calculations were performed at the Møller-Plesset second-order energy correlation (MP2) level (MP2/S-TZPsp).The optimized structures were confirmed by frequency analysis.The tight optimization mode was applied to all species and the very tight mode, if necessary.The results of the frequency analysis were used to obtain the Cij values and the coordinates corresponding to Cii (Ci).It is necessary to use such a calculation method that reproduces well the observed structures, if the nature of the interactions are to be determined on the observed structures or very close to them. 33The MP2/S-TZPsp method will afford reliable results in the calculations of the systems, containing the atoms of the 5 th period.
In POM, the fully optimized structures are further (partially) optimized with r being fixed to satisfy equation ( 3), where r and ro are the interaction distances in question in the perturbed and fully optimized structures, respectively, and ao is the Bohr radius (0.52918 Å). 20,21 The optimized values of r' (i.e., other than r) are described by equation ( 4).We call r and r' the major and minor interactions, respectively. 34Equation ( 5) explains the method to generate the perturbed structures with CIV.An i-th perturbed structure in question (Siw) is generated by the addition of the coordinates (Ci), derived from Cii, to the standard orientation of a fully optimized structure (So) in the matrix representation. 26The coefficient giw in equation ( 5) controls the structural difference between Siw and So: giw is determined to satisfy equation ( 3) for the interaction in question.The Ci values of six digits are used to predict Siw.Data were analyzed with the AIM2000 35 and AIMAll 36 programs.r = ro + wao (w = (0), ±0.05, and ±0.1; ao = 0.52918 Å) (3) r' = ro' + w'ao (4) In the full treatment of QTAIM-DFA, Hb(rc) are plotted versus Hb(rc) -Vb(rc)/2 for the data of five points (w = 0, ±0.05, and ±0.1 in equation ( 3)).Equation ( 6) is applied when each plot is analyzed using a regression curve of the cubic function in this process (Rc 2 (square of correlation coefficient) > 0.99999 usually).Data of three points (w = 0 and ±0.05 in equation ( 3)) are used in the simpler method.The regression curve of the linear function (equation ( 7)) is similarly applied for the analysis.Only p can be obtained in this process.

High similarities between the perturbed structures generated with POM-M, POM-Z and CIV
The structures of the selected species (1-56) were optimized with MP2/S-TZPsp.The interaction distances in question for 1-56 calculated with MP2/S-TZPsp are collected in Table S1 of the Supporting Information.The perturbed structures at w = ±0.05 and ±0.1 in equation (3) were generated with POM-M, POM-Z and CIV.The w' values for the most important minor interactions in the perturbed structures are calculated at w = 0.1 of the major interactions, according to equation (4).The w'/w ratios should be the reflection from the effects of the major interactions on the minor ones and closely related to the perturbed structures, as pointed out before. 28The ratios are calculated in the perturbed structures generated with POM-M ((w'/w)POM-M), POM-Z ((w'/w)POM-Z), and CIV ((w'/w)CIV).The (w'/w)POM-M, (w'/w)POM-Z, and (w'/w)CIV values are collected in Table S2 of the Supporting Information.The (w'/w)POM-M, (w'/w)POM-Z, and (w'/w)CIV values are less than approximately 0.4, which shows that the magnitudes of the displacements of the minor interactions are less than about 0.   7)  S3 of the Supporting Information.Figure 1 shows the plots of Hb(rc) versus Hb(rc) -Vb(rc)/2 for CS of vdW, t-HBnc, t-HBwc, CT-MC, X3 -, and CT-TBP and SS of Cov-w, where the perturbed structures are generated with POM-M.The plots for Cov-w and Cov-s of SS are similarly drawn in Figure S2 of the Supporting Information.

High similarities between QTAIM-DFA parameters for 1-56, calculated with POM-M, POM-Z and CIV under MP2/S-TZPsp
Figure 4 shows the plots of p:POM-M versus p:CIV and p:POM-Z versus p:CIV, evaluated under MP2/S-TZPsp.The correlations are excellent, which are collected in Table 2 (entries 4 and 5, respectively).The p:POM-M values are also plotted versus p:POM-Z, although not shown in a figure.The plot is also excellent, which is given in Table 2 (entry 6).The plots can be substantially recognized as the direct proportion described by y = x (Rc 2 = 1.00).The results show that the behavior of p:POM-M, p:POM-Z, and p:CIV are the same with each other for 1-56 under MP2/S-TZPsp, although some magnitudes of p amount to 0.6-0.7°.It must be the reflections of the excellent similarities in the perturbed structures generated with POM-M, POM-Z, and CIV (see entries 1-3 in Table 2).The results lead the excellent correlations shown in Table 2 (entries 4-6).The high reliabilities of p:POM-M, p:POM-Z, and p:CIV are established equally based on the excellent correlations among them.The high reliabilities of p:POM-M, p:POM-Z, and p:CIV are similarly established judging from the very good correlations.However, the reliabilities in p seem higher than those in p.This would be curious, at first glance, since the same perturbed structures with the common regression curve are used to evaluate p and p (see equation (SA6) in the Appendix of the Supporting Information).How can the differences be explained?The differences may be mainly attributable to the much more complex route to evaluate p (= [d 2 y/dx 2 ]/[1 + (dy/dx) 2 ] 3/2 ), relative to the case of p (= 90° -tan -1 (dy/dx)), as pointed out before. 37The (very) small differences in the QTAIM functions for the perturbed structures generated with POM-M, POM-Z, and CIV would be magnified in the second derivatives to evaluate p.The process would magnify the differences more in p:POM-M, p:POM-Z, and p:CIV, relative to the case in p:POM-M, p:POM-Z, and p:CIV.
The (R, , p) parameters are employed to classify and characterize the interactions in question, whereas p are not used.Therefore, the reliabilities of the predicted natures using the perturbed structures generated with POM-M, POM-Z, and CIV are clearly demonstrated to be the same with each other.The common (R, ) parameters are employed in the processes.

Prediction of the natures for the interactions in 1-56 under MP2/S-TZPsp
The natures of the interactions in question of 1-56 are predicted based on the (R, , p) values calculated with MP2/S-TZPsp (Table 3), under the guidance of the requirements (criteria) shown in Table 1 and the explanation in the section of "Survey of QTAIM-DFA and QTAIM approach", established for the standard interactions consisted of the atoms of the 1 st -4 th periods calculated with MP2/6-311++G(3df,3pd).The basic values of (, p) described in bold are superior to the ones given tentatively in plain, in the classification and characterization of interactions.The p values calculated based on the perturbed structures generated with POM-M are employed for the following discussion.

Pseudo intrinsic dynamic nature of interactions based on the perturbed structures generated with POM
The dynamic nature predicted with CIV is described as the "intrinsic dynamic nature," as aforementioned.The reliability of dynamic nature predicted with POM-M and POM-Z is demonstrated to be the same as that with CIV.The perturbed structures with POM-M and POM-Z are formulated based on the thermal process, whereas those with CIV would correspond to the adiabatic process.Therefore, the dynamic nature of interactions predicted with POM-M and POM-Z seems unsuitable to describe as the "intrinsic dynamic nature," irrespective of the same reliability of the natures with the three methods.
We propose that the dynamic natures of interactions predicted with POM-M and POM-Z are to be describe as the "pseudo-intrinsic dynamic nature of interactions."The proposal using POM will release experimental chemists from worrying about the Compliance program, when they study their theme, concerning the chemical bonds and interactions, in more detail.It will also release from the frequency analysis of the optimized structures, where the difficulty of the frequency analysis increases as the species become larger and/or more complex.
Our aim is completely achieved to generate the perturbed structures of the same quality as that with CIV for QTAIM-DFA, by applying with POM-M and POM-Z.The slight differences between p:POM and p:CIV will not damage our discussion, since p are not used to characterize the interactions, as mentioned above.Some difficulties are often encountered with POM-Z, especially when it is applied to large and/or complex species, in our experience.However, POM-M would solve the problem.POM-M seems more easily formulated and converged more effectively, than the case of POM-Z.The frequency analysis is necessary in CIV but not in POM, which is very important, especially when the method is applied to large and/or complex species at the MP2 level.
The simpler and easier method to predict the pseudo intrinsic dynamic nature of interactions, other than above, is proposed, next.

Reliability of dynamic nature with three data points versus that with five data points
A trial was made for further simplification to predict the dynamic nature of interactions.The dynamic nature for an interaction is tried for calculations by analyzing the plot for the data of the two perturbed structures and a fully optimized structure using the regression curve of a linear function.For the full analysis of the interactions with QTAIM-DFA, four perturbed structures and a fully optimized structure, with the regression curve of a cubic function, are necessary to predict the nature of an interaction.The p values derived from the regression curves of cubic and linear functions are denoted by p-Cb and p-Ln, respectively, here.The p-Cb and p-Ln values are denoted as p-POM-M-Cb and p-POM-M-Ln, respectively, if they are calculated with POM-M.Table 4 collects the p-POM-M-Cb and p-POM-M-Ln values for the standard interactions in 1-56, calculated under MP2/S-TZPsp.
To confirm the reliability of p:POM-M-Ln further, the p-POM-M-Ln values are plotted versus p:POM-M-Cb.Figure 6 shows the plot, which gives an excellent correlation.The correlation is shown in Table 2 (entry 10).The results confirm again the excellent reliability of p:POM-M-Ln.The p:POM-M-Ln values, based on the three data points, are demonstrated to be very close to the p:POM-M-Cb values, derived from the five data points.As a result, p:POM-M-Ln can be employed for the discussion of high reliability, in place of p:POM-M-Cb.The results show that the reliability of p:POM-M-Ln is the same as that of p:POM-M-Cb.The results also show that the dynamic nature derived from the three data points calculated with POM-M retains the very high reliability very close to that derived from the five data points calculated with POM-M.Namely, the dynamic nature based on the three data points with POM-M can be used with the very high reliability.The reliability of p:POM-M-Ln can be recognized as substantially the same as that of p:CIV-Cb through p:POM-M-Cb.cubic and linear functions, respectively, as the regression curves (cf: equations ( 6) and ( 7)).d Data from w = 0, ±0.025, and ±0.5 were employed, since the (3, -3) attractor appeared at the center of the perturbed structure for w = -0.1.
Reliability of dynamic nature with three data points using perturbed structurers generated by OMA The simplification was further examined.The perturbed structures are generated with OMA, the ultimate method, where the major interactions are only fixed appropriately longer and shorter than those in the optimized structures.(No further optimizations are made for the perturbed structures in OMA.)In this work, the p value for an interaction was calculated by analyzing the regression curve of a linear function for the plot of the data from the two perturbed structures and the fully optimized structures.The p value is named p:OMA-Ln.Table 4 collects the p:OMA-Ln values calculated under MP2/S-TZPsp.The p:OMA-Ln values are very close to the p:POM-M-Cb values, respectively, for the same species.The magnitudes in the differences between p:OMA-Ln and p:POM-M-Cb (p:POM-Ln-Cb = p:OMA-Ln -p:POM-M-Cb) are less than or equal to 1.5° for all interactions, except for Me2O--HOH (10), Me2S--Cl2 ( 14), and Me2Se--Cl2 (17), of which p:POM-Ln-Cb values are -3.2°,-3.3°, and -2.9°, respectively.Among the interactions with p:POM-Ln-Cb less than or equal to 1.5°, The p:POM-Ln-Cb values are less than or equal to 1.0° for 41 interactions and less than or equal to 1.5° for 12 interactions.The magnitudes of p:POM-Ln-Cb seem large when the minor interactions are placed just the backside of the major ones.Namely the three atoms in the major and minor interactions are aligned linearly.In such case, the minor interactions will be affected much from the major ones, in the calculation process.3.As a result, p:OMA-Ln can also be employed for the discussion with high reliability, in place of p:POM-M-Cb.The p:OMA-Ln values will be effectively applied to very lsrge and/or complex species, containing the biological species.
Figure 8 shows the observed structures of 59a-59c. 39The structures of 57a-59c are optimized under MP2/S-TZPsp, which are shown in Figure S3 of the Supporting Information.The observed structures are well reproduced by the optimizations, as shown in Table S4 of the Supporting Information.Figure 9 illustrates the molecular graphs of the species, exemplified by 59a-59c.The molecular graphs, other than 59a-59c, are shown in Figure S4 of the Supporting Information.All BPs with BCP are clearly detected, containing those for the noncovalent 1 E--2 E interactions.S5 of the Supporting Information.The interactions in Table 5 are all predicted to have the r-CS/t-HBwc nature based on all methods.
The magnitudes in the deviations of p:POM-M-Ln and p:CIV-Ln (and p:POM-M-Cb) from p:CIV-Cb are less than 0.1° for all interactions, except for p:POM-M-Ln and p:POM-M-Cb for O=S--S=O (57c) and p:POM-M-Cb for O=S--Se=O (58c), of which magnitudes are 0.2°.It is demonstrated that p:POM-M-Ln and p:CIV-Ln (and p:POM-M-Cb) can be widely applied to predict the natures of interactions in the usual molecules, such as 57a-59c, with the excellently high reliability, in addition to p:CIV-Cb, again.
In the case of p:OMA-Ln, the magnitudes in the differences between p:OMA-Ln and p:CIV-Cb are less than 0.7° for all interactions, except for S--S (57a (CC: C2)) and Se--Se (59a (CC: C2)) of which magnitudes are 1.5° and 3.5°, respectively.The magnitudes are reduced to 0.6° and 0.8°, respectively, if the AB conformers are calculated for 57a (C1) and 59a (C1) (57a&59a (AB: C1)).Why the differences between p:OMA-Ln and p:CIV-Cb in CC (C2) are larger than the case in AB (C1)?The differences in perturbed structures generated with OMA and CIV must be responsible for energy surface around 1 E--2 E of 57a and 59a.The CC (C2) conformer should have unique energy surface, which must affect much on the perturbed structures.The vibrational motions corresponding to compliance force constants for 1 E--2 E of 57a (CC: C2), 57a (AB: C1), 59a (CC: C2), and 59a (AB: C1), are illustrated in Figure S5 of the Supporting Information.As shown in the figure, whereas the 1 E--2 E stretch in the 1 E-2 E-C direction mainly contributes to of (AB: C1), lots of stretches, other than the 1 E--2 E stretch in the 1 E-2 E-C direction seem to contribute in (CC: C2).As a result, the energy surface around 1 E--2 E would be very different between those of AB and CC.
The reason for the calculated results in 57a&59a (AB: C1) versus 57a&59a (CC: C2) would be clarified by considering the style of the 1 E--2 E interactions.The style was examined based on the NBO analysis. 40The results of the NBO analysis for 57a-59c are summarized in Table S6 of the Supporting Information.It is noteworthy that the np( 1 E)→*( 2 E-CMe) interaction mainly contributes to (AB: C1), as expected, whereas the ns( 1 E)→*( 2 E-CMe) interaction contributes substantially to (CC: C2).The perturbed structures should be close to the optimized structure in (AB: C1), since the np( 1 E)→*( 2 E-CMe) interaction will maintain the perturbed structures close to the optimized structure in (AB: C1).On the other hand, the ns( 1 E)→*( 2 E-CMe) interaction in (CC: C2) will have less tendency to keep the perturbed structures, close to the optimization structure, since the atomic s orbital has no direction.As a result, the perturbed structures in (AB: C1) will be substantially the same as the optimized structure, irrespective of the calculation methods, however, those in (CC: C2) will be somewhat different with each other, depending on the calculation methods.Only the interaction distance of 1 E--2 E changes in the perturbed structures, if they are generated with OMA.Namely, the perturbed structures generated with OMA will be very similar to the optimized structures in (AB: C1) and (CC: C2).However, the differences in the perturbed structures generated with (NIV, CIV, and POM) and OMA will be (much) larger in (CC: C2), relative to the case in (AB: C1), since all bond distances, containing 1 E--2 E, will change in the perturbed structures, if generated with NIV, CIV, and POM.This is the reason for the much larger differences of the magnitudes between p:OMA-Ln and p:CIV-Cb in (CC: C2), relative to the case in (AB: C1), shown in Table 5.
The results imply that the applicability and the reliability of p:OMA-Ln will also be very high, if the specific cases, such as the CC (C2), are carefully considered, as discussed above.The p:OMA-Ln value is expected to be applied to large molecules, such as biomolecules, although some devices would be necessary.

Conclusions
In QTAIM-DFA, the signs of d(Hb(rc) -Vb(rc)/2)/dr and dHb(rc)/dr (r: the interaction distance) are employed to predict the dynamic and static natures of interactions, in addition to those of Hb(rc) -Vb(rc)/2 and Hb(rc).The prediction is achieved by analyzing the QTAIM-DFA plots of Hb(rc) versus Hb(rc) -Vb(rc)/2 with (R, ) and (p, p).
The treatment enables us to classify and characterize the CS interactions of vdW, t-HBnc, t-HBwc, CT-MC, X3 -, and CT-TBP, and the SS interactions of Cov-w and Cov-s.The reliability of the dynamic nature is controlled by the quality of the perturbed structures.The perturbed structures generated with CIV has been demonstrated to have the excellent quality for QTAIM-DFA.However, it seems difficult to handle freely the compliance program, necessary to operate CIV, for non-specialists in this field.We searched for the simpler and easier methods to generate the perturbed structures, of which quality is substantially the same as that with CIV.The perturbed structures generated with POM (POM-M and POM-Z) are shown to satisfy the requirements.The dynamic nature of interactions with POM is described as the "pseudo-intrinsic dynamic nature of interactions," after the "intrinsic dynamic nature of interactions," with CIV.As an ultimate method, OMA is proposed to generate the perturbed structures, of which quality seems very good.In the full treatment of QTAIM-DFA, data from the fully optimized structure with four perturbed structures around it are analyzed, employing the regression curve of a cubic function, which is sometimes troublesome.Data from a fully optimized and two perturbed structures with the regression curve of a linear type are demonstrated to give the same reliability of the dynamic nature for an interaction as that derived from the five data points.The applicability of the proposed methods is examined, employing the noncovalent interactions at the naphthalene 1,8-positions.
The "pseudo-intrinsic dynamic nature of interactions" is obtained with POM, which helps us to predict the highly reliable dynamic nature for the interactions.In this case, the frequency analysis is not necessary.It will release the experimental chemists from worrying about the complex compliance program in CIV, when they study their theme, concerning the chemical bonds and interactions, in more detail.Analyzing the data from the three points will also release chemists from excessive effort.OMA is expected to open the door to apply QTAIM-DFA to large molecules, such as the biomolecules, although some devices would be necessary.

Figure 1 .
Figure 1.QTAIM-DFA plots of Hb(rc) versus Hb(rc) -Vb(rc)/2 for 1-48 of vdW, HB, CT-MC, X3 -, CT-TBP of CS, and some Cov-w of SS, where the perturbed structures are generated employing POM-M, under MP2/S-TZPsp.The numbers and colors in the figure are the same as those in Table3.

Figure 2 .
Figure 2. Areas for the weak to strong interactions which appear in the plot of Hb(rc) versus Hb(rc) -Vb(rc)/2, calculated under MP2/6-311++G(3df,3pd).Data for Cl-Cl--Cl -over a wide range of w in equation (3) being employed for the plot.(a) Whole picture, (b) magnified one for the p-CS region, and (c) magnified one for the r-CS region.The white asterisk in (c) corresponds to the optimized structure.The 1 (= 45°) and 2 (= 206.6°) values correspond to the limited values.The definitions of (R, ) and (p, p) are illustrated (b).First and second bending points of the plot (BD-1 and BD-2, respectively) are also shown.
4 times larger than those of the fixed major ones.The (w'/w)POM-M and (w'/w)POM-Z values are plotted versus (w'/w)CIV.The plots are shown in Figure S1 of the Supporting Information.The plots are analyzed assuming the linear correlations of y = ax + b (a: correlation constant, b: y-intercept, and Rc 2 : the square of the correlation coefficient).The plots gave excellent correlations.The (w'/w)POM-M values are also plotted versus (w'/w)POM-Z, although not shown in a figure.The plot also gives an excellent correlation.Table 2 collects the correlations (entries 1-3, respectively).The perturbed structures generated with POM-M, POM-Z, and CIV under MP2/S-TZPsp are shown to be very close with each other for 1-56.The results are the firm basis for the starting point of this investigation.

a
Omitting the data from diatomic molecules for 46-48 and 56.b Omitting the data from 10, 14, and 17.QTAIM functions and QTAIM-DFA parameters calculated with POM-M, POM-Z, and CIV QTAIM functions are calculated under MP2/S-TZPsp for the interactions in question of 1-56 at the BCPs with POM-M, POM-Z and CIV under MP2/S-TZPsp.The values of b(rc), (ħ 2 /8m) 2 b(rc) (= Hb(rc) -Vb(rc)/2), Hb(rc), kb(rc) (= Vb(rc)/Gb(rc)), and Cii are collected in Table of the Supporting Information, according to equations (SA3)-(SA7) in the Appendix of the Supporting Information.The (p, p) values calculated with POM-M, POM-Z and CIV are denoted by (p:POM-M, p:POM-M), (p:POM-Z, p:POM-Z), and (p:CIV, p:CIV), respectively.Table3collects the values, together with the (R, ) values.The (p, p) values may change depending on the methods to generate the perturbed structures, however, the differences in the parameters calculated with the three methods are negligibly small.The next step is to establish the high reliability of the dynamic nature for 1-56 with POM-M and POM-Z under MP2/S-TZPsp, after calculating the QTAIM-DFA parameters with POM-M, POM-Z, and CIV under MP2/S-TZPsp.

Figure 4 .
Figure 4. Plots of p:POM-M versus p:CIV and p:POM-Z versus p:CIV for 1-56, calculated under MP2/S-TZPsp.The correlations are shown in the figure.

Figure 5 .
Figure 5. Plots of p:POM-M versus p:CIV and p:POM-Z versus p:CIV for 1-56, calculated under MP2/S-TZPsp.The correlations are shown in the figure.

Figure 6 .
Figure 6.Plots of p:POM-M-Ln versus p:POM-M-Cb, calculated with MP2/S-TZPsp.The correlation is shown in the figure.Colors for the interactions are the same as those in Table3.

Figure 7
Figure 7 shows the plot of p:OMA-Ln versus p:POM-M-Cb, which gives an excellent correlation.The correlation is shown in Table 2 (entry 11).The correlation is also calculated by omitting the data from Me2O--HOH (10), Me2S--Cl2 (14), and Me2Se--Cl2 (17).The correlation is also shown in Table 2 (entry 12).The p:OMA-Ln values, based on the three data points, are demonstrated to be very close to the p:POM-M-Cb values, which are derived from the five data points.The results confirm again the very high reliability of p:POM-M-Ln.As a result, p:OMA-Ln can also be employed for the discussion with high reliability, in place of p:POM-M-Cb.The p:OMA-Ln values will be effectively applied to very lsrge and/or complex species, containing the biological

Table 2 .
Correlations between the parameters, evaluated using the perturbed structures generated with POM-M, POM-Z, and CIV under MP2/S-TZPsp

Table 3 .
QTAIM-DFA parameters of (R, ) and (p, p) for the standard interactions in 1-56, calculated with QTAIM-DFA under MP2/S-TZPsp, employing the perturbed structures generated with POM-M, POM-Z, and CIV a  c p:POM-M d p:POM-M e p:POM-Z d p:POM-Z e p:CIV d p:CIV e

Table 3 .
Continued Species (X--Y) R b  c p:POM-M d p:POM-M e p:POM-Z d p:POM-Z e p:CIV d p:CIV e