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Multiconfiguration Self-Consistent Field Study on Formonitrile Imine and N‑Substituted Nitrile Imines HCN2−R: Energy Component Analysis of the Pseudo-Jahn−Teller Effect Azumao Toyota,§ Takashi Muramatsu,§ and Shiro Koseki*,†,‡ §

Environmental Education Center, Miyagi University of Education, Sendai 980-0845, Japan Department of Chemistry, Graduate School of Science, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan ‡ The Research Institute for Molecular Electronic Devices (RIMED), Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan †

S Supporting Information *

ABSTRACT: Stable geometrical structures for formonitrile imine (1) and N-substituted nitrile imines HCN2−R (R = Li, BeH, BH2, CH3, CN, CCH, C6H5, NH2, OH, and F) (2−11) were examined by using the multiconfiguration self-consistent-field (MCSCF) method followed by second-order configuration interaction (SOCI) calculations and second-order multiconfiguration quasidegenerate perturbation theory (MCQDPT2) calculations, together with the aug-cc-pVTZ basis sets. The results show that 1 suffers a pseudo-Jahn−Teller (JT) distortion from a linear C∞v structure to a C1 structure via a planar bent Cs structure. Each of the others is found to undergo pseudo-JT distortion from a symmetrical structure to a planar bent Cs structure for 2, 3, and 7 and to a C1 structure for 4, 5, 6, 8, 9, 10, and 11. At the stationary structures of 1−11, the structural characteristics were briefly discussed in terms of allenic and propargylic. To elucidate the nature of pseudo-JT distortions, energy component analyses were carried out at the MCSCF+SOCI level of theory at all of the stationary structures for the relevant molecules. In most of the molecules examined, pseudo-JT stabilizations were classified into two groups, one in which the stability arises from a lowering of the energy of the attractive term Ven and the other in which the stability results from a lowering of the energy of the repulsive terms Vnn and Vee. In addition to the above two groups, it was also found that the following three groups are responsible for the pseudo-JT stabilizations in a certain stage of the structural changes. Namely, one is a lowering of the energy of the term Vee observed in 6, another is a lowering of the energy of the terms Vee and Ven observed in 9−11, and the other is a lowering of the energy of the terms Ven and Vnn observed in 10. These energetic behaviors were accounted in terms of an elongation or a contraction of the molecular skeleton, a migration of electrons from one part of the molecule to other parts, and the combined effects arising from these two factors.

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INTRODUCTION In a previous study, we examined the most stable geometrical structures for several azide molecules R−N3 (R = H, BeH, BH2, CH3, C6H6) by using the multiconfiguration self-consistent field (MCSCF) method together with the aug-cc-pVTZ basis set.1 Except for the unknown hydridoberyllium azide, each of the azides was shown to adopt a planar bent Cs structure. Like azide molecules, nitrile imines R1−CN2−R2 are useful reagents in azaheterocyclic synthesis,2−5 because they undergo 1,3-dipolar cycloaddition reactions with multiple bonds to give various heterocyclic compounds. Much theoretical work has been devoted to the study of the bonding nature of 1,3-dipolar cycloadditions.6 On the contrary, Bertrand and co-workers showed by X-ray structural analyses that all stable nitrile imines prepared in the 1990s possess bent, helical backbones.7−10 Many theoretical studies have been undertaken on equilibrium geometries and vibrational spectra for some substituted nitrile imines including formonitrile imine to explain available experimental data.11−19 To the best of our knowledge, however, no theoretical research has been undertaken to elucidate why © XXXX American Chemical Society

the C−N−N moiety in nitrile imines examined does not take a linear structure, but a bent structure. Our concern for nitrile imines is to explicate what occurs inside the molecule as it undergoes a pseudo-Jahn−Teller (JT) distortion from a symmetrical structure to a less symmetrical structure with a bent C−N−N moiety.20−43 Our results obtained so far suggest that non-nearest neighbor interactions are important in such structural changes that the molecules bend their molecular skeletons.37−43 This is because, with skeletal bending, the nuclei and electron clouds are brought spatially close to each other and, concomitantly, the electrostatic interactions between them are greatly affected. Thus, energy component analyses of the structural changes are expected to provide significant information regarding the chemical bonding in the nitrile imines. Using the MCSCF method followed by SOCI and MCQDPT2 calculations together with aug-cc-pVTZ basis Received: February 21, 2017 Revised: March 1, 2017 Published: March 2, 2017 A

DOI: 10.1021/acs.jpca.7b01718 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

Figure 1. Geometrical distortion paths of 1. Structural symmetry, SOCI relative energy (kcal/mol) in parentheses, and imaginary frequencies and their irreducible representation in square brackets are shown for each structure, where [pd] indicates that the Hessian matrix is positive definite. The colors of arrows in Figures 1−11 are as follows, where Ven, Vee, and Vnn are the electron−nuclear attraction energy, the interelectron repulsion energy,and the internuclear repulsion energy, respectively: (red) the decrease of Vee and Vnn, and the increase of Ven; (blue) the decrease of Ven, and the increase of Vee and Vnn; (black) the decrease of Ven and Vee, and the increase of Vnn; (green) the decrease of Ven and Vnn, and the increase of Vee; (brown) the decrease of Vee, and the increase of Ven and Vnn.

Figure 2. Geometrical distortion paths of 2. See the caption of Figure 1.

sets,44−51 we explore here the nature of pseudo-JT distortions in nitrile imine systems by dealing with formonitrile imine (1) and N-substituted nitrile imines HCN2−R (R = Li, BeH, BH2, CH3, CN, CCH, C6H5, NH2, OH, and F) (2−11) (Figure S1). It is noted that almost all of them are still unknown compounds but that IR spectra of 1 and 8 have been obtained in an Ar matrix and in a PVC film, respectively.52−54

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METHODS OF CALCULATION MCSCF Method Followed by SOCI and MCQDPT2 Calculations. The stationary geometrical structures of 1−11 were located by using the full optimized reaction space (FORS) multiconfiguration self-consistent field (MCSCF) method with aug-cc-pVTZ basis sets.44−51 In these calculations, the MCSCF active space includes six orbitals and eight electrons corresponding to the occupied and unoccupied π and π* orbitals of the C−N−N moiety. Vibrational analyses were also performed for the purpose of determining whether each symmetrical structure is recognized as a stationary point on the singlet potential energy surface of the ground state. Then, when some imaginary frequencies appear at the structures, geometrical optimizations were performed to locate more stable structures with less symmetry. Second-order configuration interaction (SOCI) calculations58 and second-order multiconfiguration quasi-degenerate perturbation calculation (MCQDPT2)59,60 were achieved at MCSCF stationary structures to refine the energy differences among the stationary structures. Energy Partitioning Scheme. In what follows, the energy partitioning scheme61 is briefly reviewed. The total energy E of the ground state for a molecular system is partitioned as follows:

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PSEUDO-JAHN−TELLER EFFECT When the highest symmetrical structure is assumed for a closed-shell molecule, a symmetry reduction often occurs by the interaction between the ground state and low-lying electronically excited states. Such a geometrical distortion is known to be pseudo-Jahn−Teller (JT) effects.20−34 For instance, the linear structure (C∞v) of 1 should be distorted into a bent structure provided the interaction between the ground state and the low-lying Π states is strong. Because the ground states for 1−11 are totally symmetric, the pseudo-JT theory indicates that bending motions belonging to π representation derive such structural distortions. To determine whether such appropriate excited singlet states exist at a relatively low energy region, popular time-dependent density functional theory (TD DFT)55−57 calculations were performed together with aug-cc-pVTZ basis sets.44−51 The excitation energies to the lowest singlet Σ−, Δ, and Π states in 1 were calculated to be 4.89, 5.01, and 5.07 eV, respectively. Although the former two excited states are improper by symmetry, there surely exists a proper singlet state of Π symmetry at a relative low energy region. This suggests the possibility of the occurrence of pseudo-JT distortion in 1 in the range of such excitation energies.29,30 To investigate whether 1−11 suffer from the pseudo-JT effect, vibrational analyses at the stationary structures were carried out by employing the MCSCF method.

E = E el + Vnn = T + V + Vnn = T + Ven + Vee + Vnn

where Vnn is the internuclear repulsion energy and Eel is the electronic energy. The latter is given by the sum of the kinetic energy T and the potential energy V consisting of the electron− nuclear attraction energy Ven and the interelectronic repulsion energy Vee. According to the molecular virial theorem for stationary structures, the energy difference between a less symmetrical structure and a symmetrical structure is given as B


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−ΔT or ΔV/2, the quantity being negative.62−67 Accordingly, the energy difference can be described by the differences of potential energy terms only as follows: ΔE =

obtained by MCSCF geometry optimizations. An account of the appearance of each conformations is given below. In the C∞v structure for 1, 6, and 11, vibrational analyses provide two bending π modes possessing imaginary frequencies, converting the linear C∞v structure into a bent Cs structure. The resulting Cs structure has one imaginary frequency that corresponds to a nuclear deformation of a″ symmetry, and it is distorted into a C1 structure (Figures 1, 6, and 11). For 6, the energy difference between the planar Cs structure and the C1 structure is only 0.4 (SOCI) or 2.0 (MCQDPT2) kcal/mol (Table 1), so that it cannot be concluded that the PJT distortion occurs from the Cs structure to the C1 structure at the present. For 10, the structural changes take place in two stages from the C∞v structure to the C1 structure as shown in Figure 10. Three π modes possessing imaginary frequencies are found

1 (ΔVen + ΔVee + ΔVnn) 2

These energy components were calculated by using SOCI wave functions58 at the stationary structures optimized at the MCSCF level of theory. All calculations were carried out by means of the quantum chemistry code GAMESS.50,51

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RESULTS AND DISCUSSION Location of Stationary Geometrical Structures. Figures 1−11 show the paths of the structural distortions of 1−11 C


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(Figures 2, 3, and 7). The resulting Cs structure is a true energy minimum because its force-constant matrix is positive definite. As for 3, the energy difference between the C2v structure and the Cs structure is 1.7 (SOCI) or 1.5 (MCQDPT2) kcal/mol (Table 1). Because the zero-point energy correction calculated by using the MCSCF method reduces the energy difference by 0.6 kcal/mol, it is difficult to determine whether or not the PJT distortion occurs from the C2v structure to the Cs structure. It may be necessary to exclude this molecule from the present discussion until performing more sophisticated calculations.

to appear at the C∞v structure. Then, the relevant nuclear distortions lead to two different planar Cs structures, cis- and trans-bent structures with regard to an interior N−N bond (Figure 10). Each of the two different planar Cs structures has one imaginary frequency that corresponds to a nuclear deformation of a″ symmetry, and they are distorted into the same C1 structure. The C∞v structures of 2, 3, and 7 were found to have one, one, and two bending π modes possessing imaginary frequencies, respectively. The linear C∞v structure is transformed into a bent Cs structure by the relevant bending modes D


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than the C2v structure, the distortion path via this nonplanar structure had better be postponed until performing more sophisticated calculations.23,66,67 The C3v structure of 5 has two bending (e) modes possessing imaginary frequencies, and the pseudo-JT distortions bring about two different Cs structures, staggered and eclipsed conformations with regard to spatial arrangement of the methyl group (Figure 5). The staggered Cs structure has one imaginary frequency of a″ symmetry, and the relevant deformation leads to a C1 structure. On the contrary, the eclipsed Cs structure has two imaginary frequencies of a″ symmetry. The nuclear deformations along these two vibrational modes provide a C1

In the C2v structure of 4, vibrational analyses give four imaginary frequencies consisting of two bending b2 and b1 modes that convert the C2v structure into a planar Cs structure and a nonplanar Cs structure, respectively (Figure 4). The planar Cs structure has one imaginary frequency of a″ symmetry, and it is distorted into a C1 structure. On the contrary, the nonplanar Cs structure has two imaginary frequencies of a″ symmetry that correspond to a C−H bending mode and a BH2 rotational mode, respectively, and this structure is only 1.0 kcal/mol lower than the C2v structure at the SOCI level of calculation (Table 1). Because the MCQDPT2 result suggests that the Cs structure is higher E


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given below in terms of allenic and propargylic.10,54,69 On thre basis of the geometrical parameters, it is readily assessed that 1, 2, 5, 8, and 9−11 take allenic structures with longer C−N bonds and shorter N−N bonds, whereas 4, 6, and 7 take propargylic structures with short C−N triple bonds and N−N single bonds. Of these nitrile imines, the stable geometrical structures for 1 and 8−11 agree sufficiently with those reported previously.54,69 In this context, it is remarked for 1 that the planar Cs structure with a propargylic form corresponds to the transition state for inversion of the C1 structure with an allenic form. For 2 and 7, because each of them assumes a planar Cs structure, the linear C∞v structure corresponds to the transition state for bending of the Cs structure with a propargylic form. As to 4, the planar Cs structure with a propargylic form corresponds to the transition state for inversion of the C1 structure with a propargylic form. In this connection, it is added that, according to the calculation results by Muchall et al., the C1 structure for 4 is not a stationary point and that the planar Cs structure is an energy minimum.6 Except for 2, 4, and 7, it is concluded that the Cs structure corresponds to the transition state for inversion (rotation) of the C1 structure with an allenic form, in good agreement with the available literature information.54,69 At the end of this section, it is noted that an enantiomer exists for the C1 structures of 1, 4, 5, and 8−11. Energy Component Analyses of the Total Energy in Ground States. Table 1 shows the total energies and their components obtained by using the SOCI wave functions at the stationary structures optimized by the MCSCF method. The differences of the total energies and the internuclear repulsion energies between the symmetric and distorted structures are also shown in this table. Mulliken atomic populations70−73 were obtained at the SOCI level of calculations (Table S3). The changes of larger than 0.01 Å in bonded and nonbonded atomic distances at a less symmetric structure relative to those at a symmetric structure were given in Table S4. Formonitrile Imine (1). The structural changes from the C∞v structure to the C1 structure take place in two steps via the planar Cs structure (Figure 1). In the first step (C∞v → Cs), because the changes in bond distances correspond to an expansion of the molecular skeleton, the attractive interactions (Ven) are weakened and the repulsive energies Vnn and Vee are decreased. In the second step (Cs → C1), the molecular skeleton is slightly expanded in the C1 structure relative to the Cs structure and, concomitantly, is folded in a helical fashion by twisting of C−H and N−H bonds relative to the C2−N3−N5 plane. Because the skeletal folding

structure and one of these is expected to reach the staggered structure via a C1 structure. In the C2v structure of 8, two imaginary frequencies correspond to each of the nuclear deformations of b1 and b2 symmetry, respectively. The relevant deformations provide planar and nonplanar Cs structures (Figure 8), which is similar to those of 4. In this case, the both Cs structures are explicitly lower in energy than the C2v structure, and the planar structure has one imaginary frequency of a″ symmetry and the nonplanar structure has two imaginary frequencies of a″ symmetry. The deformations arising from the planar and nonplanar Cs structures derives the same C1 structure shown in Figure 8. Note that, just like the case of 5 in the previous paragraph, one of the deformations started at the nonplanar structure is expected to reach the planar structure via a C1 structure. In the C2v structure of 9, two and three imaginary frequencies appear that correspond to the nuclear deformations of b2 and b1 symmetry, respectively. Relevant geometry optimizations derive a planar Cs structure for the former and two different nonplanar Cs structures for the latter, cis- and trans-bent structures with regard to spatial arrangement of the amino group (Figure 9). Furthermore, the planar Cs structure has three a″ modes with imaginary frequencies, while two different nonplanar Cs structures have one a″ mode with an imaginary frequency. The nuclear motions possessing the largest two imaginary frequencies at the planar Cs structure derive two different C1 structures, C1 (lower) and C1 (higher), depicted in Figure 9, respectively.68 The nuclear motions possessing the third imaginary frequency is the rotation of the amino group and provide both C1 structures after the bifurcation of the distortion path. The nonplanar Cs structures, trans- and cis-bent structures, are distorted into the C1 (higher) and C1 (lower) structures, respectively, along the nuclear motions possessing imaginary frequencies. Thus, in the present investigation, the two different C1 structures, C1 (higher) and C1 (lower), were located at the MCSCF level of theory,54 but our consecutive SOCI calculations suggest that no transition state exists between the MCSCF C1 structures and only one C1 structure should be obtained (see ref 69), even though SOCI geometry optimization is not feasible at the present. Structural Characteristics. The geometrical parameters for the various structures of 1−11 were optimized by the MCSCF method (Table S2). Because geometrical structures of nitrile imines were reported to be reflected in experimental observables such as infrared spectra, a brief discussion of the structural characteristics of the stable geometrical structures is F


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Table 1. Total and Partitioned Energies (hartree) and Energy Differences (kcal/mol) between the Symmetrical and Distorted Structures of 1−11 Obtained by Using the MCSCF+SOCI/6-31G(d,p) Methoda 1 (R = H) E ΔEb T Ven Vee Vnn ΔVnnd

C∞v −148.039942 0 147.954022 −472.107741 113.219513 62.894264 (0)

Cs −148.092007 −32.7 (−30.3)c 148.026123 −470.471298 112.353748 61.999421 (−561.5)

E ΔE T Ven Vee Vnn ΔVnn

C∞v −154.986194 0 154.884392 −501.597466 123.011679 68.715200 (0)

Cs −154.996567 −6.5 (−5.4) 154.903561 −500.937769 122.630332 68.407309 (−193.2)


C∞v −162.772470 0 162.679359 −540.229248 134.827255 79.950165 (0)

Cs −162.775210 −1.7 (−1.5) 162.683984 −539.773807 134.569632 79.744980 (−128.8)


C2v −173.366236 0 173.272748 −591.132970 151.151935 93.342051 (0)

Cs (planar) −173.390738 −15.4 (−13.8) 173.309388 −592.461428 151.862528 93.898774 (349.3)


C3v −187.078736 0 186.979817 −651.680671 170.258587 107.363532 (0)

Cs (eclipsed) −187.128164 −31.0 (−30.5) 187.044203 −654.315158 171.554588 108.588203 (768.5)


C∞v −239.794968 0 239.702677 −823.866012 212.741329 131.627037 (0)

Cs −239.843373 −30.4 (−24.9) 239.753314 −825.875481 213.848107 132.430687 (504.3)

E ΔE T Ven Vee

C∞v −223.729710 0 223.637618 −784.411591 205.953257

Cs −223.741411 −7.3 (−22.6) 223.646332 −786.564669 207.207673

C1 −148.099363 −37.3 (−35.2) 148.030696 −470.126685 112.116207 61.880418 (−636.2) 2 (R = Li)

3 (R = BeH)

4 (R = BH2) Cs (nonplanar) −173.367863 −1.0 (+0.5) 173.274777 −591.226256 151.262774 93.320843 (−13.3)

C1 −173.390892 −15.5 (−14.0) 173.310160 −592.406212 151.825835 93.879325 (337.1)

Cs (staggered) −187.129959 −32.1 (−31.7) 187.046149 −654.547900 171.664604 108.707187 (843.2)

C1 −187.140122 −38.5 (−38.2) 187.054910 −654.024542 171.295613 108.533897 (734.4)

5 (R = CH2)

6 (R = CN) C1 −239.844121 −30.8 (−26.9) 239.757516 −825.822871 213.770561 132.450673 (516.8) 7 (R = CCH)

G


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The Journal of Physical Chemistry A Table 1. continued 7 (R = CCH) Vnn ΔVnn

131.091005 (0)

131.969252 (551.1) 8e (R = ph)

E ΔE T Ven Vee Vnn Vnn ΔVnn

E ΔE T Ven Vee Vnn ΔVnn E ΔE T Ven Vee Vnn ΔVnn

C2v −377.587592 0 377.257600 −1636.063954 503.293083 377.925678 377.925678 (0) C2v −203.005999 0 202.903130 −692.381190 177.980811 108.491250 (0) C1 (higher) −203.114081 −67.8 (−65.7) 203.028273 −693.624406 178.427607 109.054445 (353.4)

Cs (planar) −377.637143 −31.1 (−30.7) 377.334268 −1648.658197 509.671058 384.015728 384.015728 (3821.6) Cs (planar) −203.084637 −49.3 (−47.1) 203.000239 −694.857361 179.189722 109.582764 (684.9) C1 (lower) −203.114676 −68.2 (−66.1) 203.027483 −693.760880 178.486028 109.132693 (402.5)

Cs (nonplanar) −377.632181 −28.0 (−26.2) 377.318740 −1646.723828 508.742677 383.030230 383.030230 (3203.2)

C1 −377.643441 −35.0 (−48.8) 377.333839 −1647.988523 509.150441 383.860802 383.860802 (3724.3)

9 (R = NH2) Cs (nonplanar) (trans bent) −203.105259 −62.3 (−58.1) 203.029371 −693.377102 178.507840 108.734632 (152.7)

Cs (nonplanar) (cis bent) −203.099572 −58.7 (−54.7) 203.022758 −693.588538 178.541346 108.924863 (272.1)

10 (R = OH) E ΔE T Ven Vee Vnn ΔVnn

C∞v −222.707099 0 222.600319 −739.608531 186.320421 107.980693 (0)


C∞v −246.759379 0 246.651952 −791.668711 193.784933 104.472447 (0)

Cs (trans bent) −222.920877 −134.1 (−127.3) 222.842803 −740.116713 186.493305 107.859729 (−75.9)

Cs (cis bent) −222.911831 −128.5 (−122.0) 222.832659 −739.973302 186.325259 107.903554 (−48.4)

C1 −222.935184 −143.1 (−137.4) 222.850817 −740.123242 186.310099 108.027142 (29.1)

11 (R = F) Cs −246.910961 −95.1 (−87.8) 246.847843 −794.029309 195.103669 105.166836 (435.7)

C1 −246.921325 −101.6 (−96.2) 246.856308 −794.206256 194.990305 105.438318 (606.1)

a Energies are in hartree. bDifferences of the total energy in kcal/mol relative to that of the fully symmetrical structure. cDifference of the total energy obtained by the MCQDPT2 method. dDifferences of the internuclear repulsion energy in kcal/mol relative to that of the fully symmetrical structure. e The external space of the SOCI calculations includes only 200 orbitals that have the lowest eigenvalues of the standard MCSCF Fock operator, due to our computer resource limitations.

attractive term Ven is increased, it is presumed that the contribution from the skeletal expansion is greater than that from the skeletal folding. Owing to the competition between the two effects, the change in energy of each of the three terms is small in the second step compared with that in the first step. A direct explanation is made for the variation of internuclear repulsion energy Vnn, because it is given by a sum of the term vij = ZiZje2/Rij over all pairs of atoms in a molecule. In the initial step (C∞v → Cs), because the nonbonded distances, H1−H5,

brings about an enhancement of the electrostatic interactions among the nuclei and the electron clouds in bonds, the energy of the repulsive terms Vnn and Vee should be increased and that of the attractive term Ven should be decreased. Hence, the overall energy variation of the attractive and repulsive terms is affected by the opposing two factors mentioned above. Such being the case, it is difficult to predict the energy variation of each of the three terms. However, because the energy of the repulsive terms Vnn and Vee is decreased and that of the H


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the heavy atoms, C2−B5 and N3−B5, are considerably shortened. Analyses of the repulsive term Vnn reveal that the increase in repulsion energy Vnn is chiefly caused by the large change of the nuclear repulsions v35 and v25. Concurrently, a marked proximity of the non-nearest-neighbor heavy atoms enhances the electrostatic interactions in the Cs structure, in conjunction with a high contraction of the electron cloud around N4. This is because an electron migration is observed from C2, N3, and B5 to N4, with the result that N4 acquires 0.849 electrons. It is thus understood that the energy of the attractive term Ven is decreased and that of the repulsive term Vee is increased. In short, the Cs structure owes its stability to a lowering of the energy of the attractive term Ven. It might be useless to discuss the next step (Cs → C1), because the stabilization energy is an extremely small value of 0.1 (SOCI) or 0.2(MCQDPT2) kcal/mol and it cannot be concluded whether or not this distortion occurs at the present state of our investigation. N-Methylnitrile Imine (5). The structural changes from the C3v structure to the C1 structure take place through two paths (Figure 5), one via the eclipsed Cs structure and the other via the staggered Cs structure. Because the individual energy terms are similarly changed in both the paths, an explanation will be given only for the energy variations of the former path. In the first step (C3v → Cs), the origin of energy variations will be ascribed to the following two factors; one is a change in bond angle at N4 from 180 to 115.7°. Concomitantly, a bend of the N3−N4−C5 moiety leads to an appreciable shortening of the nonbonded distances C2−N5 and N3−N5, although the bond distances N3−N4 and N4−C5 are slightly lengthened. Owing to a marked proximity between the heavy atoms, the associated electrostatic interactions should be highly enhanced in the eclipsed Cs structure. The other factor is an electron migration from H1, C2, N3, and C5 to N4. Namely, N4 acquires 0.672 electrons, with a contraction of the electron cloud toward it. Thus, the combined effects arising from the proximity of nonbonded atoms and the electron migration greatly influence on the energy changes of the three terms; the energy of the repulsive terms Vnn and Vee is increased and that of the attractive term Ven is decreased. In short, the stability of the staggered Cs structure can be attributed to a lowering of the energy of the attractive term Ven. In the next step (Cs → C1) (Figure 5), the energy variations of the three terms are opposite to those observed with the first step. However, the energy lowering of the repulsive term Vnn is rather small. This is understood from analyses of the repulsive term: the increment of repulsion energy is 817.3 kcal/mol, whereas the decrement of repulsion energy is 851.4 kcal/mol. Because the balance of the two quantities comes to a rather small value of −34.1 kcal/mol, the associated structural change is regarded as a slight lengthening of atomic distances. The energy variations of the terms Vee and Ven are partly due to an electron migration from N4 to the terminal H1. This is because in concert with a slight skeletal expansion, the electron cloud is spread in the outer side of the molecular skeleton. Hence, the C1 structure is stabilized by a lowering of the energy of the repulsive terms Vnn and Vee. N-Cyanonitrile Imine (6). The structural change from the C∞v structure to the C1 structure takes place in two steps via the planar Cs structure (Figure 6). For the first step (C∞v → Cs), almost all of the bonded atomic distances are lengthened, but the nonbonded distances N3−C5, C2−C5, and N3−N6 are markedly shortened. Also, electrons migrate from H1, C2, C5,

C2−H5, and N3−H5, are considerably shortened, the structural change should bring about an enhancement of the repulsive interactions. In contrast to expectation, however, this is not true because the nuclear repulsion is proportional to the product of two nuclear charges as well. Analyses of the repulsion energy Vnn indicate that the increment of repulsion energy arising from such terms as v23, v25, and v35 amounts to 368.1 kcal/mol, whereas the decrement of repulsion energy arising from such terms as v24, v34, and v45 is 929.6 kcal/mol. Thus, the balance of the two quantities comes to −561.5 kcal/ mol, where the dominant terms in the decrease of repulsion energy Vnn are the repulsions v24 and v34 between the heavy atoms. For the second step (Cs → C1) (Figure 1), similar analyses reveal that the increment of repulsion energy arising from such terms as v13, v14, and v34 amounts to 308.3 kcal/mol, whereas the decrement of repulsion energy arising from such terms as v12, v23, and v23 is 383.0 kcal/mol. After all, the internuclear repulsive term Vnn contributes to the pseudo-JT stabilization because the balance of the two quantities comes to −74.7 kcal/ mol. Note that this quantity is small in comparison with that observed in the first step. Also note that in the distortion from the C∞v structure to the Cs structure, an electron migration takes place from C2 and N4 to N3 and H5 with a relaxation of charge alternation. The energy changes of the terms Vee and Ven are thus partly attributable to the electron migration. In the distortion from the Cs structure to the C1 structure, an electron migration from C2, N4, and H5 to H1 and N3 also takes place, indicating that the electron cloud is spread over the entire molecule to give a rather uniform charge distribution. A combination of the electron-cloud expansion and the heavyatom skeletal expansion leads to a decrease in energy of the repulsive term Vee and, inversely, an increase in the energy of the attractive term Ven. Thus, the energetic stability of the C1 structure arises from a decrease in energy of the repulsive terms Vnn and Vee. N-Lithionitrile Imine (2). In the distortion from the C∞v structure to the Cs structure, because almost all of the bonds are lengthened (Figure 2), the energy of the repulsive terms Vnn and Vee is decreased. In addition, the decrease in energy of the latter is partly responsible for an electron migration from N4 to N3 and the terminal H1. Thus, it can be said that the two repulsive terms Vnn and Vee are responsible for the energetic stability of the Cs structure. N-Borylnitrile Imine (4). In the structural change from the C2v structure to the C1 structure, the planar and nonplanar Cs structures appear as the stationary structures within the MCSCF calculations (Figure 4). However, as mentioned in the previous section, because the distortion path via the nonplanar C s structure is excluded from the present investigation because of a very small energy difference, only the planar Cs structure is considered as a stationary structure in this paragraph. In the initial step (C2v → Cs), because the structural change brings about an expansion of the molecular skeleton, the stability of the Cs structure is expected to arise from a lowering of the energy of the repulsive terms Vnn and Vee. However, the attractive term Ven is found to be responsible for the pseudo-JT stabilization, namely, the variations of individual energy terms for 4 were exactly the opposite of those observed for 1−3. Because this distinction needs to be solved, an explanation is given below on how the energy of the repulsive term Vnn is increased. Owing to a change in bond angle at N4 from 180 to 119.4°, the atomic distances between I


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effects give rise to an increase in energy of the repulsive terms Vnn and Vee and a decrease in energy of the attractive term Ven. Therefore, the C1 structure owes its energetic stability to a lowering of the energy of the attractive term Ven. N-Aminonitrile Imine (9). The structural changes from the C2v structure to the C1 structure take place through three paths, one via the planar Cs structure, another via the nonplanar (trans bent) Cs structure, and the other via the nonplanar (cis bent) Cs structure (Figure 9). (i) For the first path via the planar Cs structure, the structural change of the initial step (C2v → Cs) corresponds to a bend of the N3−N4−N5 moiety with a slight elongation of the N−N bonds. The change in bond angle at N4 from 180 to 112.8° leads to a marked proximity of the atomic distances between N5 and N3 (C2), and N4 acquires 1.353 electrons by migration from C2 and N5. Because the above two factors enhance the electrostatic interactions in the planar Cs structure, the energy of the attractive term Ven should be decreased and, inversely, that of the repulsive terms Vnn and Vee should be increased. Hence, the stability of the planar Cs structure is attributed to a lowering of the energy of the attractive term Ven. For the next step (Cs → C1), the relevant nuclear motions with imaginary frequencies at the planar Cs structure derive the C1 (lower) and C1 (higher) structures, though the two C1 structures might be collapsed into one C1 structure at higher levels of calculation as described in the previous section. For these distortion paths, the structural change is taken as a slight expansion of the skeleton accompanied by a slight migration of electrons from N4 to the terminal H1. Hence, the energy variations of the three terms go in the opposite direction to those observed for the first step. Namely, the energy of the repulsive terms Vnn and Vee should be decreased and that of the attractive term Ven should be increased. (ii) For the second path via the nonplanar (trans bent) Cs structure (Figure 9), the structural changes and the associated energy variations of the three terms in the initial step (C2v → Cs) are similar to those observed for the first path. Hence, the stability of the trans-bent Cs structure results from a lowering the energy of the attractive term Ven. For the next step (Cs → C1), the relevant nuclear motion with an imaginary frequency at the trans-bent Cs structure provides the C1 (higher) structure at the MCSCF level of calculation. In this structural change, the bonded and nonbonded N−N distances of the N3−N4−N5 moiety are slightly shortened. Through the electrostatic interactions among the heavy N atoms, the energy of the attractive term Ven should be decreased, whereas that of the repulsive terms Vnn and Vee should be increased. However, the energy of the repulsive term Vee as well as the energy of the attractive term Ven is found to be decreased. This anomaly should be responsible for expansion of the electron cloud, because a relatively large electron migration takes place from N4 to the terminal H1. Noticeable is the finding that the combination of the energy terms Vee and Ven is responsible for the energetic stability of the C1 (higher) structure. (iii) For the third path via the nonplanar (cis bent) Cs structure (Figure 9), the energy variations of the three energy terms are similar to those observed for the second path. For this reason, we give below the energy terms responsible for the energetic stability of the Cs and C1 structures. In the initial step, the stability of the nonplanar (cis bent) Cs structure arises from a lowering of the energy of the attractive term Ven. For the next step (Cs → C1), the relevant nuclear motion with an imaginary frequency at the cis-bent Cs structure provides the C1 (lower)

and C6 to N3 and N4, with N4 acquiring 0.871 electrons. These aspects suggest that the energy of the repulsive terms Vnn and Vee is increased and that of the attractive term Ven is decreased. In summary, the attractive term Ven is responsible for the energetic stability of the planar Cs structure. As mentioned above, the stabilization energy of the second step (Cs → C1) is too small to analyze the energy components, so that the detailed discussion is not performed here. N-Ethynylnitrile Imine (7). In the distortion from the C∞v structure to the Cs structure (Figure 7), almost all of the bonded atomic distances are lengthened, but the nonbonded distances C2−C5, C2−C6, N3−N6, etc. are shortened appreciably. Additionally, electrons are highly concentrated at N4 by migration from the remaining atoms, namely, it acquires 1.846 electrons. It then follows that the energy of the repulsive terms Vnn and Vee is increased and that of the attractive term Ven is decreased. Thus, the stability of the Cs structure arises from a lowering of the energy of the attractive term Ven. N-Phenylnitrile Imine (8). The structural changes from the C2v structure to the C1 structure take place through two paths (Figure 8), one via the planar Cs structure and the other via the staggered Cs structure. (i) In the former path, the structural change in the first step (C2v → Cs) is accompanied by such energy variations that the energy of the attractive term Ven is decreased and that of the repulsive terms Vnn and Vee is increased. These energetic behaviors are explained by the following two factors; one is a bend of the N3−N4−C5 moiety, leading to a marked shortening of the atomic distances between the nitrile imine and phenyl groups. In more detail, because the bond angle at N4 is deviated from 180 to 117.1°, the distances such as those of N3−C6, C2−C6, and N3−C8 are shortened appreciably, the magnitudes being in the range 0.638−1.038 Å. Owing to the proximity between the heavy atoms, the associated electrostatic interactions are greatly enhanced in the planar Cs structure. The other factor is an electron migration from H1 and N3 to C2 and N4; namely, electrons are highly concentrated at C2. Accordingly, the combined effects arising from a contraction of the molecular skeleton and an electron migration influence greatly on the energy changes of the terms Vee and Ven; the energy of the former is increased and that of the latter is decreased. Then, the stability of the planar Cs structure is due to a lowering of the energy of the attractive term Ven. In the next step (Cs → C1), the energetic behavior is opposite to that observed in the first step; namely, the energy of the two repulsive terms is decreased and that of the attractive term is increased. Note, however, that the energy lowering of the term Vnn is rather small. In brief, the stability of the C1 structure arises from a lowering of the energy of the repulsive terms Vnn and Vee. (ii) For the path via the staggered Cs structure (Figure 8), the energy variations in the first step (C2v → Cs) are similar to those observed in the former path. In this case, however, the energy variations are mainly responsible for the shortening of nonbonded atomic distances between the heavy atoms. This is because the bond angle at N4 is deviated from 180 to 115.4°. Hence, the stability of the Cs structure arises from a lowering of the energy of the attractive term Ven. In a similar manner, the energy variations of the three terms in the next step (Cs → C1) go in the same direction as those obtained for the first step. This energetic behavior is ascribed to the combined effects arising from a contraction of the molecular skeleton and an electron migration from H1 and N3 to C2, N4, and C10. These J


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term Ven decreases. Thus, the stability of the Cs structure arises from a lowering of the energy of the attractive term Ven. In the second step (Cs → C1) (Figure 11), the energy variations take place such that the energy of the repulsive term Vnn is increased and that of the attractive term Ven as well as the repulsive term Vee is decreased. Because the variation of the repulsive term Vee is unusual, emphasis is placed on this energy variation. Despite the nonbonded atomic distance N3−F5 being slightly shortened, overall changes in bond distances seem to be effectively nil in the C1 structure. The finding that the molecular structure is folded in a helical fashion by twisting of C−H and N−F bonds relative to the C2−N3−N4 plane is notable. The skeletal folding leads to an enhancement of electrostatic interactions by virtue of the proximity among the nuclei and the electron clouds in bonds. Such structural changes bring about an increase in energy of the repulsive terms Vee and Vnn and a decrease in energy of the attractive term Ven. Moreover, a slight electron migration takes place from N4 to the terminal H1 accompanied by an expansion of the electron cloud. Whether the energy of the repulsive term Vee is decreased depends on the competition between the two opposing contributions arising from the above two effects. Because the energy of the repulsive term Vee is indeed decreased, it can be presumed that the contribution of the electron migration is slightly greater than that of the skeletal folding. Such being the case, the energy change of the repulsive term Vee results in rather small. It is thus concluded that the combination of the energy terms Vee and Ven is responsible for the energetic stability of the C1 structure.

structure at the MCSCF level of calculation. Therefore, the stability of the C1 (lower) structure arises from a lowering of the energy of the attractive term Ven and the repulsive term Vee. N-Hydroxynitrile Imine (10). The structural changes from the C∞v structure to the C1 structure take place through two paths (Figure 10), one via the planar (trans bent) Cs structure and the other via the planar (cis bent) Cs structure. The variations of individual energy terms resemble each other for both the paths, so that an explanation is given only for the former path. In the first step (C∞v → Cs), the structural change corresponds to an elongation of the bond distances, so that the energy of the repulsive terms Vnn and Vee should be decreased and that of the attractive term Ven should be increased. The magnitude being not so large, the energy of the repulsive term Vnn is indeed decreased. This is ascribed to the proximity between the heavy atoms, because the bond angle at N4 deviates from 180 to 107.3°. The variations of the energy terms Vee and Ven go in the reverse direction to our expectations. This anomaly is accounted for through the following two factors; one of the factors is the concentration of electrons at N4; namely, it acquires 1.196 electrons by migration from C2, N3, O5, and H1. This indicates a situation with the electron clouds being markedly contracted around N4. The other factor is a shortening of the nonbonded atomic distances C2−O5 and N3−O5. Therefore, the energy of the repulsive term Vee is increased, whereas that of the attractive term Ven is decreased. Note that the contribution due to a contraction of the electron cloud is larger than that due to an expansion of the molecular skeleton. Worth noting is the finding that the combination of the energy terms Ven and Vnn is responsible for the energetic stability of the planar (trans bent) Cs structure. In the second step (Cs → C1) (Figure 10), the energy of the repulsive term Vnn is increased, whereas that of the terms Vee and Ven is decreased. The increase in energy of the repulsive term Vnn is due to a shortening of the bond distances and, concomitantly, the energy of the attractive term Ven is decreased. From this viewpoint, the energetic behavior of the repulsive term Vee seems to be unusual. However, the origin is ascribed to an electron migration from N4 to the terminal H1, because the electron migration leads to a weakening of the electrostatic interactions. It should also be noted that a small lowering of the energy of the attractive term Ven is partly responsible for the electron migration. Then, the combination of the energy terms Vee and Ven is responsible for the energetic stability of the C1 structure. N-Fluoronitrile Imine (11). Distortions from the C∞v structure to the C1 structure take place in two steps via the planar Cs structure (Figure 11). In the first step (C∞v → Cs), the structural change is accompanied by an elongation of the bond distances, so that the energy of the repulsive terms Vnn and Vee should be decreased and that of the attractive term Ven should be increased. However, contrary to our expectations, the actual energy variations are the reverse for the three terms. Presumably, this discrepancy is due to the effects arising from the following two factors; a shortening of the nonbonded atomic distances C2−F5 and N3−F5 and a contraction of the electron cloud through an electron migration from C2 to N4 and F5. Accordingly, the electrostatic interactions are highly enhanced in the planar Cs structure; namely, the energies of the repulsive terms Vnn and Vee increase and that of the attractive

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CONCLUSION An exploration of the most stable geometrical structures for 1− 11 was performed using the MCSCF method, followed by the single-point calculations at the second-order configuration interaction (SOCI) and second-order multiconfiguration quasidegenerate perturbation theory (MCQDPT2) level of theory, where the aug-cc-pVTZ basis set44−51 was employed. It was shown that 2 and 7 assume a planar bent Cs structure, whereas the remaining molecules adopt a C1 structure in the ground state. For most of the present molecules, the pseudo-Jahn− Teller (JT) stabilizations are found to be classified into two groups, one arising from a lowering of the electron−nuclear attraction energy Ven and the other arising from a lowering of the internuclear and interelectronic repulsion energies Vnn and Vee. In such a case, the structural change is regarded as a contraction or an expansion of the molecular skeleton, because the associated energy changes of the attractive and repulsive terms are out of phase with each other. On the basis of the present series of our investigations,1,37−43 as the energy terms responsible for the pseudo-JT stabilization combine, no instances other than the above two groups have been, in fact, observed. Accordingly, we have been interested in the question of whether the energy variations of the two repulsive terms Vnn and Vee should always be in the same direction. As a combination of the energy terms of interest, this study showed that there are three other groups responsible for the pseudo-JT stabilization in a certain stage of the structural changes for 6 and 9−11. One is an energy lowering of the term Vee observed in 6, a second is an energy lowering of the terms Vee and Ven observed in 9−11, and a third is an energy lowering of the terms Ven and Vnn observed in 10. In such novel cases, it was observed that the energy variations of the three terms are K


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greatly influenced by the combined effects arising from a structural change and an electron migration.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b01718. Tables showing Cartesian coordinates of stationary structures, geometrical parameters, atomic populations, and largest changes in atomic distances, and figure showing numbering of atoms (PDF)

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AUTHOR INFORMATION

Corresponding Author

*S. Koseki. E-mail: [email protected]. Tel.: +81-72-2549702. ORCID

Shiro Koseki: 0000-0003-1208-4574 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.T. thanks Toshiaki Toyoda, Graduate School of Engineering, Tohoku University, Sendai, Japan, for helpful suggestions and encouragement during the course of this work.

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REFERENCES

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