The new competitive mechanism of hydrogen bonding interactions and transition process for the hydroxyphenyl imidazo [1, 2-a] pyridine in mixed liquid solution

The new competitive mechanism of intermolecular and intramolecular hydrogen bond can be proposed with an improved mixed model. Upon the photoinduced process, the twisting intramolecular charge transfer (TICT) structure of the hydroxyphenyl imidazo [1, 2-a] pyridine (HPIP) can be obtained. TICT character prompts the fluorescent inactivation via non-radiative decay process. For exploring the photochemical and photophysical properties, the electronic spectra and the infrared (IR) vibrational spectra of titled compounds have been detailedly investigated. In addition, the frontier molecular orbitals (MOs) analysis visually reveals that the unbalanced electron population can give rise to the torsion of molecular structure. To further give an attractive insight into the non-radiative decay process, the potential energy curves have been depicted on the ground state (S0), the first excited state (S1) and the triple excited state (T1). Minimum energy crossing point (MECP) has been found in the S1 and T1 state. On the MECP, the intersystem crossing (ISC) might be dominant channel. The density functional theory (DFT) and the time-dependent density functional theory (TDDFT) methods have been throughout employed in the S0 state, T1 state and S1 state, respectively. The theoretical results are consistent with experiment in mixed and PCM model.

depends on its bond length, the bond angle, the local dielectric constant, the electronegativity of the donor and acceptor groups, temperature, and pressure, etc. 27,28 . The hydrogen bonding interaction and corresponding dynamical behaviour have been interpreted via a variety of photophysical and photochemical phenomena 16,[29][30][31][32][33] . For example, the ESIPT reaction, the intramolecular charge transfer (ICT) and the TICT, etc. 19,28 .
The tautomer structures can be obtained by the ESIPT reaction 28,[34][35][36] . The ultrafast hydrogen bond strengthening can offer the driving forces for the ESIPT process. Upon the photo-induced process, the electron population has an obvious change for the molecule, which converts from the π character to the π* character. The electron density of proton donor group and acceptor group will reduce and increase, respectively 29 . Therefore, the proton transfer process will be facilitated in the S 1 state 11,37 . However, the ESIPT process can be understood as a four-level circular loop model and it is a great important process in the photochemistry, biochemistry and so on refs 38, 39. As discussed by Toshiki Mutai et al. isomer of HPIP originated from ESIPT process had shown an extremely weak fluorescence in the polar solvent tetrahydrofuran (THF). They have drawn a conclusion that the fluorescence yield can be widely enhanced by changing the surroundings from the polar liquid state to the solid state 40 . Actually, the non-radiative decay process plays a prominent role in the biological systems 41,42 . The long lifetime deactivation process of HPIP will occur via the TICT character in the polar solvent, and then the ISC between the S 1 state and the T 1 state might be dominant channel for the non-radiative decay. On the MECP, the molecular structures have almost identical energy in the S 1 and T 1 state 41 . Subsequently, the radiationless decay will jump from the T 1 state to the S 0 state. However, the torsion of HPIP isomer can be prohibited in the solid state surroundings, the nearly co-planar form will emit a drastic fluorescence 40 . In this study, we will carefully investigate the fluorescence quenching mechanism of deactivation process. As Zhao and Liu et al. discussed 33,43 .
In the present work, we utilize a mixed solvent model 44 to investigate the reaction where a THF molecule bounds to imino group of HPIP, the other solvent can be substituted by polarizable continuum model (PCM). The complex can be established by hydrogen bond interaction. As Fileti et al. have discussed that for alcohol-water complexs the stability of complex constitution depends on not only different donor and acceptor molecules, but binding energy of hydrogen bond also is influenced by spatial configurations of complex 45 . Especially, Fileti et al. 46 and Malaspina et al. 47 have accurately provided with the most stable complex of pyridine and water molecule via combining Monte Carlo computer simulation and first-principles quantum mechanical calculations method in an aqueous environment 48 . Therefore, in this study HPIP complexs have been certified as most stable structures when considering different spatial configurations of complex. The TICT character can be clearly revealed when we consider the interaction between THF molecule and imine group of HPIP molecule. As Wang et al. discussed multiple proton transfer via an intermolecular hydrogen-bonded water wire can clearly exhibited the effect of hydrogen bonding dynamics for 3-hydroxypyridine 29 . Otherwise it cannot be primely explained in the conventional PCM model.

Results and Discussion
In this study, we primarily investigate the photophysics and photochemistry properties of HPIP. The HPIP in the THF solvent phase has a dual emitting in the PCM solvation, but the fluorescence of keto form isomer will be totally quenched via the TICT character in the mixed liquid model. However, we are interested in the fluorescence quenching process, and we have speculated that the TICT structure might go through an ISC process from S 1 state to T 1 state on the MECP 41, 49-51 . Then, the nonradiative decay will be dominant in the T 1 → S 0 state as shown in Fig. 1. Chu et al. has introduced the competition mechanism between the intermolecular hydrogen bonding interaction and the ESIPT process 52 . However, in this study a new competitive mechanism of intramolecular and intermolecular hydrogen bond will be investigated in the torsional process.
Geometric structures and spectra analysis. Four stable structures have been found in the S 0 state, the HPIP (a), the keto form of HPIP (k-HPIP) (b), the trans-keto form HPIP (trans-k-HPIP) (c) and the open-ended intramolecular hydrogen bond HPIP (o-HPIP) (d) have been shown in the Fig. 2. It should be noted that the k-HPIP has not occurred torsion. However, Four stable structures in the S 1 state, the cis-HPIP (a), in PCM the k-HPIP (ik-HPIP) (b), the nearly vertical isomer form of HPIP (v-HPIP) (c) and the trans-k-HPIP (d) have been shown in the Fig. 3. The k-HPIP form cannot exist in the S 1 state. After finishing the ESIPT process of HPIP molecule the isomer form has been directly optimized into v-HPIP form. The ik-HPIP and the v-HPIP forms present the rotations to ~33° and ~80° between benzothiazole and phenyl group, respectively. For illustrating the reliability of our computation, the absorption and emission spectra have been calculated in the mixed liquid model. The absorption peak values of the cis-HPIP are located in 331 nm. The emission peak values of the cis-HPIP are located in 377 nm. In addition, the fluorescence peak values of the ik-HPIP are located in 602 nm. The above those peak values have coincided with the computational results of Toshiki Mutai et al., which have been shown in Fig. 4. Herein, we have proved the availability of the computing method. The fluorescence of the tautomer form in the S 1 state exists a Stokes' shift beyond 200 nm, which indicates that the ik-HPIP form has a drastic change compared with cis-HPIP form. However, the stable structures have not been investigated exclusively in the mixed liquid model. In the Fig. 5, we have calculated the absorption and fluorescence spectra of the cis-HPIP (a) and trans-k-HPIP (b). As shown in Fig. 5(a), the absorption peak values of the cis-HPIP are located in 328 nm. The emission peak values of the cis-HPIP are located in 374 nm. The emission peak value of v-HPIP form is nearly nonluminous located in the about 847 nm. As shown in Fig. 5(b), the absorption peak is about 497 nm and the fluorescence peak is about 595 nm for the trans-k-HPIP. Although the trans-k-HPIP and k-HPIP cannot be directly gained in the S 0 state, the two structures can be received in the radiationless decay way from the S 1 state to the S 0 state. The reaction mechanism has been exhibited in the Fig. 1. Subsequently, for above structures we have dissected carefully the changes of crucial bond parameters in the S 0 and S 1 state. As shown in the Tables 1-4, the intriguing bond parameters have been revealed. In Table 1   bond angle δ(O 1 -H 1 -N 2 ) increases from 145.4° to 149.4°. There is a definite conclusion that intramolecular hydrogen bond O 1 -H 1 ···N 2 of the cis-HPIP can be strengthened in the S 1 state. The dihedral angle δ(N 2 -C 3 C 8 -C 9 ) is 4.6° in the S 0 state, but it reduces to 1.0° in the S 1 state. The structure of cis-HPIP is near planar in the S 1 state.  The bond parameters of ik-HPIP form has been shown in the Table 2, the bond lengths O 1 ···H 1 , H 1 -N 2 and δ(O 1 -H 1 -N 2 ) of ik-HPIP change from 2.150 Å, 1.013 Å and 118.5° to 1.644 Å, 1.055 Å, and 137.9° in the S 1 → S 0 state, which indicates that the intramolecular hydrogen bonds O 1 ···H 1 -N 2 of ik-HPIP form is weaker in the S 1 state than that in the S 0 state. In addition, we can find that the dihedral angle δ(N 2 -C 3 C 8 -C 9 ) of ik-HPIP form decreases from 33° to 0.01° in the S 1 → S 0 state from this Table. In the Fig. 2(d), the o-HPIP molecular structure has been optimized with mixed liquid model. We find that the o-HPIP form has the torsion of about 52.4° in the The bond parameter (bond length (Å), length angle (°) and dihedral angle (°)) of crucial moiety for the cis-HPIP in the S 0 and S 1 state.

Parameters of bonds
The bond parameter (bond length (Å), length angle (°) and dihedral angle (°)) of crucial moiety for the ik-HPIP in the S 0 and S 1 state.

Parameters of bonds Electronic state trans-k-HPIP
able 3. The bond parameter (bond length (Å), length angle (°) and dihedral angle (°)) of crucial moiety for the trans-k-HPIP in the S 0 and S 1 state. S 0 state, since the intramolecular hydrogen bond has been destroyed via reversing the bond O 1 -H 1 . Therefore, we can conclude that the intramolecular hydrogen bond O 1 -H 1 ···N 2 can preclude the torsion of HPIP structure. In the Table 3, the bond parameters of trans-k-HPIP form have been shown. The dihedral angle δ (N 2 -C 3 C 8 -C 9 ) of trans-k-HPIP compared with that of cis-HPIP has twisted about 180°. In addition, a strong intermolecular hydrogen bond N 2 -H 1 ···O 2 has formed. In the S 0 state, the bond lengths O 1 -H 1 , H 1 -N 2 and O 2 ···H 1 are 4.954 Å, 1.030 Å and 1.811 Å, respectively. The corresponding bond lengths are 4.945 Å, 1.025 Å and 1.853 Å in the S 1 state. Finally, the bond parameters of ik-HPIP and v-HPIP form have been shown in the Table 4, it should be noted that the bond length O 1 -H 1 increases from 2.150 Å to 3.389 Å with torsion of the molecule system, which indicates the torsional behavior has given rise to weakening of the intramolecular hydrogen bond N 2 -H 1 ···O 1 . In addition, the intermolecular hydrogen bond N 2 -H 1 ···O 2 has been found in the v-HPIP form, and the bond length O 2 -H 1 is 1.831 Å. The intermolecular hydrogen bonding interaction can compete with the intramolecular hydrogen bonding interaction of the ik-HPIP form in the mixed liquid model. Yan et al. have concluded that the slightly weaker hydrogen bond allows the competition with other type of interaction 53 . Non-coplanar ik-HPIP form results in the intramolecular hydrogen bond is extremely weak. So when the THF molecule approaches to the imino group (=N-H) of ik-HPIP, the intermolecular hydrogen bond will be strengthen and the intramolecular hydrogen bond will be further weaken, the torsion of HPIP molecule can be facilitated as shown in the Fig. 6. We can propose a viewpoint that the intermolecular hydrogen N 2 -H 1 ···O 2 can give rise to the weakening of intramolecular hydrogen bond N 2 -H 1 ···O 1 . Furthermore, we have found that a nonatomic ring structure has been generated in the v-HPIP form linked by two intermolecular hydrogen bonds, which are the strong hydrogen bond N 2 -H 1 ···O 2 and the weak hydrogen bond C 14 -H 12 ···O 1 . The reduced density gradient (RDG) isosurfaces have been shown in the Fig. 7. Herein, the intensity of hydrogen bond N 2 -H 1 ···O 2 can be visibly compared with that of hydrogen bond C 14 -H 12 ···O 1 .
For guaranteeing these structures are the true most stable, the corresponding IR vibrational frequency has been calculated. Meanwhile, the anharmonic effects have been considered in stretching frequencies and ΔZPE correction by means of multiplying correction factors 0.991 and 0.977 that Truhlar et al. have fitted 54 . The IR vibrational spectra of hydrogen bond have been shown in the Fig. 8. In the Fig. 8(a), the vibrational frequency of O 1 -H 1 is about 3200 cm −1 (the anharmonic frequency is 3171.2 cm −1 ) in the S 0 state and is about 2729 cm −1 (2704.4 cm −1 ) in the S 1 state for cis-HPIP form. The 471 cm −1 (466.8 cm −1 ) red shift indicates that the hydrogen bond O 1 -H 1 ···N 2 is stronger in the S 1 state. In the Fig. 8(b), the vibrational frequency of the N 2 -H 1 is tremendously blue-shift 680 cm −1 (673.8 cm −1 ) from 2926 cm −1 (2899.7 cm −1 ) to 3606 cm −1 (3573.5 cm −1 ) in the S 0 → S 1 state for the ik-HPIP form, which has indicated that the hydrogen bond N 2 -H 1 ···O 1 is stronger in the S 0 state. The IR vibrational spectrum of the trans-k-HPIP has been revealed in the Fig. 8(c). The blue-shift 68 cm −1 (67.4 cm −1 ) from 3302 cm −1 (3272.3 cm −1 ) in the S 0 state to 3370 cm −1 (3339.7 cm −1 ) in the S 1 state has indicated that the intermolecular hydrogen bond N 2 -H 1 ···O 2 is stronger in the S 0 state. Upon anharmonic effects the energy of ΔZPE correction for cis-HPIP between the S 0 and S 1 state is 0.16 eV within the error range.   The ESP and the frontier molecular orbitals (MOs) analysis. We guess the rotation is connected with relative displacement of the electron clouds and the nucleus in molecule under the photo-induced. The centre of gravity of positive and negative charges is tipped, which results in the drastic change of dipole moment. In order to prove our conjectures, the ESP values of HPIP have been calculated by the Multiwfn program 55 and the ESP surface has been shown in the Fig. 9. The corresponding maxima and minima have been exhibited on the figure, we can clearly found that the negative and positive electrostatic potential exist a drastic polarization distribution on the surface. Moreover, in the Fig. 10, the distribution ratio of the different ESP regions has been  quantificationally calculated. The ESP values ranged from −20 Kcal/mol to 20 Kcal/mol widely distribute on the surface area. Thereinto, the negative ESP values mainly origin from the π-electron cloud of aromatic nucleus, the aromatic nucleus of C-H hydrogens mainly contribute to the positive areas. In the Fig. 11, the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) have been revealed. Upon the photo-induced process, the electron distribution mainly changes from the π character on the HOMO to the π* character on the LUMO. Herein, we only analyze the HOMO and LUMO, because transient excitation mainly stems from the contribution of HOMO → LUMO transition. The transition contribution of ππ* character and corresponding oscillator strength have been listed in the Table 5. In this table, Oscillator strength (f) of ik-HPIP and v-HPIP is 0.1334 and 0.0026, respectively. Their fluorescence compared with the fluorescence of cis-HPIP has been quenched. In the Fig. 11(a), electron redistribution in the PCM is nearly identical to that in the mixed solvation. For further studying the relationship between torsion of molecule structure and electron redistribution, we have obtained non-coplanar HPIP (np-HPIP) form by factitiously twisting the dihedral angle δ(N 2 -C 3 C 8 -C 9 ) in the theoretical calculation, in which the np-HPIP molecule cannot be obtained actually in the S 0 state. The MOs graphs of np-HPIP form have been shown in the Fig. 11(b). Its electron redistribution of the HOMO → LUMO compared with that of cis-HPIP still is not significant change. Therefore, we draw a conclusion that factitious torsion of molecule structure cannot result in the drastic electron redistribution on the np-HPIP. However, the HOMO and LUMO of the ik-HPIP, v-HPIP and trans-k-HPIP have been depicted in the Fig. 11(c,d) and (e). The electron redistribution of these isomeric forms is great different from that of cis-HPIP. It should be noted that electron population of the isomeric forms is unbalanced for HOMO → LUMO. These isomeric forms are zwitterions that the imino group carries positive electric charges and the ketonic oxygen atom carries negative electric charges. ICT character of these zwitterions can lead to the increasing of the dipole moments in the S 1 state, so the corresponding electron population will be unbalanced in the molecule. Moreover, the unbalanced electron population can further result in the torsion of isomeric forms. It is worth noting that TICT character of  the v-HPIP is shown in the Fig. 11(d), the electron population almost entirely transfers from the phenyl group to the benzothiazole group in the HOMO → LUMO.
The analysis of potential energy curves and MECP. For further studying the non-radiative decay process, comprehending the optical properties and pathways of electronic transition, the potential energy curves of the HPIP in the S 0 and S 1 state have been carefully investigated. The energy of structure is a function of bond length O 1 -H 1 and dihedral angle δ(N 2 -C 3 C 8 -C 9 ), respectively. The potential energy curves will be drawn with the bond length and dihedral angle increased by the fixed step sizes, respectively. The reactive potential barriers and stable structures have been obtained in the potential curves. Firstly, we study the potential energy curves in the S 0 state. As shown in the Fig. 12, the two potential energy curves are the reactive pathways of (a) intramolecular proton transfer and (b) twisting dihedral angle, respectively. In the Figure the k-HPIP form is a stable structure and it has been used as the original structure in dihedral angle scan. Proton transfer cannot spontaneously occur in the S 0 state, because the process needs to cross a potential barrier 7.81 Kcal/mol in the Fig. 12(a). Similarly, the dihedral angle torsion cannot also spontaneously occur, and the process needs to cross an energy barrier 9.92 Kcal/ mol in the Fig. 12(b). However, the energy instantly reduces about 3.17 Kcal/mol in the torsion process from the Fig. 12(b), because the intermolecular hydrogen bond N 2 -H 1 ···O 2 takes shape in torsion process. Moreover, when the dihedral angle turns to about 180°, the trans-k-HPIP will be a stable structure. Therefore, cis-HPIP, k-HPIP    Table 5. The corresponding oscillator strength (f), contribution index (CI) and orbital transition for the different molecular structures in the S 0 → S 1 state. and trans-k-HPIP forms can exist in the S 0 state, but the energy barriers of the proton transfer and structural torsion process are so high that the two processes cannot spontaneously occur in the S 0 state. Secondly, we study the potential energy curves in the S 1 state. In the mixed liquid model the keto form will directly be optimized into the v-HPIP form, so the v-HPIP form has been used as the initial structure in dihedral angle scan. In the Fig. 13 the potential energy curve of twisting dihedral angle has indicated that the v-HPIP form located in about 80° is stable structure, and the stable tran-k-HPIP form located in about 180°. Moreover, the torsion from the v-HPIP to the trans-k-HPIP must get over an energy barrier 3.71 Kcal/mol. The reversed torsion process needs to get over a small energy barrier 0.93 Kcal/mol, which indicates that the trans-k-HPIP can spontaneously revert to v-HPIP form in the S 1 state. The corresponding potential barrier is the critical point of cis-and trans-isomer.
To sum up, we have fully explained the reactive pathways for the HPIP complexes via analyzing the corresponding potential energy curves in the S 0 and S 1 state. However, the non-radiative transition process is still ambiguous, so we have further calculated the potential energy curves of the T 1 state. Because the oscillator strength of v-HPIP we have calculated is 0.0026, fluorescence of the v-HPIP has been totally quenched. We speculate that the v-HPIP structure might undergo a non-radiative transition from the S 1 state to the S 0 state. Therefore, herein searching the MECP has become mainly work for us. As shown in the Fig. 14, the potential energy curves of the torsional dihedral angle in the T 1 state and S 1 state have been exhibited. It should be noted that the MECP locates in about 90° in the figure. On this point, the ISC process might be dominant channel in the S 1 → T 1 process. In the potential energy curves of the T 1 state, the MECP structure is extremely unstable, so this structure will fast slide to the k-HPIP form located in about 0° or will get over a negligible energy barrier 0.33 Kcal/mol and then fast slide to the trans-k-HPIP form located in about 180° along with the potential energy curve of T 1 state. The k-HPIP and trans-k-HPIP existed in the S 0 state could be obtained from the radiationless decay process. As Harvey et al. reported that the reliability of hybrid method has been testified about searching the MECP 56 . Therefore, for prove the reliability of MECP, the sobMECP suite 57 has been used in this study. It should be noted  that the point sobMECP has sought out is about 0.34 Kcal/mol higher than MECP we have confirmed, and their geometries are nearly consistent.
In addition, the functions of the potential energy curve versus the dihedral angle δ(N 2 -C 3 C 8 -C 9 ) of o-HPIP and np-HPIP forms have been depicted on the Fig. 15. The negligible energy barrier of o-HPIP is 0.52 Kcal/mol and the energy barrier of reversed twisting is 4.43 Kcal/mol in the Fig. 15(a), it is clearly indicated that twisting of o-HPIP can spontaneously occur in the S 0 state, the energy of o-HPIP form is gradually reduced with the torsional process and the stable structure located in about 180°. On the contrary, in the Fig. 15(b) the energy barrier of np-HPIP is 8.91 Kcal/mol and the energy barrier of reversed twisting is 1.65 Kcal/mol. The stable np-HPIP form locates in about 143.4° and the energy of np-HPIP form is gradually increased with the torsional process. Therefore, the intramolecular hydrogen bond can preclude the torsion of HPIP structure.

Conclusions
For illustrating the reliability of our computation, we have compared the electronic spectra with that of Toshiki Mutai et al. Both of the computed results are tremendous coincidental. The hydrogen bonding strengthening mechanism has been proved via analyzing the bond parameters of hydrogen bond in the S 0 and S 1 state. In addition, the analysis of IR vibrational spectra can further illustrate the above strengthening mechanism. In this study, competitive mechanism that intermolecular hydrogen bond N 2 -H 1 ···O 2 can weaken interaction of intramolecular hydrogen bond N 2 -H 1 ···O 1 has been primely proved. MOs analysis has indicated that the unbalanced electron population of the keto forms can give rise to the torsion of structures. For further studying the model of non-radiative decay process, comprehending the optical properties and pathways of electronic transition, the potential energy curves have been studied. Herein, we draw a conclusion that the k-HPIP and trans-k-HPIP cannot spontaneously occur in the S 0 state resulted from calculating the reactive energy barriers. These two structures may be obtained by the non-radiative decay process from S 1 → T 1 → S 0 state. The MECP has further been sought out via sobMECP program 57 . Moreover, we have demonstrated that the intramolecular hydrogen bond N 2 -H 1 ···O 1 Figure 14. Constructed the potential energy curves of the HPIP: red line: the energies of different structures versus the dihedral angles δ(N 2 -C 3 C 8 -C 9 ) in the S 1 state; blue line: the energies of different structures versus the dihedral angles δ(N 2 -C 3 C 8 -C 9 ) in the T 1 state. The numerical value in the graph stands for the energy barrier of the reaction. The MECP and the corresponding structure have been shown. can preclude the torsion of HPIP form via analyzing the o-HPIP form and np-HPIP form and corresponding potential energy curves.
Computational details. For providing with available information of geometrical configurations, such as the minimum energy, potential energy surface, electron spectra and infrared vibrational spectra, etc. 44 . The density functional theory (DFT) and the time-dependent density functional theory (TDDFT) have been throughout employed in the S 0 state and the S 1 state, respectively [58][59][60] . Both of them have been performed at Gaussian 09 program 61 . In this study, we only select to use Becke's three-parameter hybrid exchange function with the Lee-Yang-Parr gradient-corrected correlation functional (B3LYP) [62][63][64] , the Pople's 6-31 G (d) and 6-31 + G (d) triple-ξ quality basis sets are used in this level computation 65 . Herein, the functional B3LYP had been extensively applied in the past few decades, which indicates that B3LYP method is greatly reliable for calculating theoretically in the S 1 state 15,16,30,31,[66][67][68][69][70][71][72][73][74][75][76][77][78][79][80] . We have investigated the solvent effect based on the integral equation formalism polarizable continuum model (IEF-PCM) [81][82][83] . The molecular electrostatic potential (ESP) has been calculated for predicting the nucleophilic and electrophilic sites of target molecule. The ESP surface has been visually portrayed via the Visual Molecular Dynamics (VMD) software 84 . and we have also applied the RDG function to investigate the weak interaction types in the Multiwfn program 55 . The visual software Chemcraft has been applied to visualize these figures, such as the molecular structures, the MOs and the RDG 85 . The MECPs were sought out via sobMECP, which is a modified version of Harvey's MECP program 56 by Tian Lu. The sobMECP is a wrapper of Harvey's MECP program to simplify the operation of the MECP program 55,86 .
For performing fully optimized in our molecular system, the hybrid functional method has been carried out in computed point energies and geometries and corresponding gradients. The effective gradients f and g have been defined as: where terms E 1 and E 2 are energies on the two potential energy curves, respectively. The ∂ ∂ E q / n are corresponding partial derivatives of relative nuclear coordinates q.
The energy difference E 1 − E 2 can reduce gradually in the vector f direction and E 1 can reduce in vector g direction, which the two vectors f and g are orthogonal. The MECP can be sought out by optimizing with effective force f + g of molecular system. E 1 − E 2 of two states presented differ spin status can be reduced with descending total energy.
As hybrid method will be used to find MECP, the spin-orbit coupling will not be taken consideration. In two 2*2 Hessian matrices H 1 and H 2 electronic coupling matrix elements H 12 = H 21 become zero. Therefore, when the diagonal matrix elements of two matrices are equivalent, the MECP will be a real minimum. However, MECP is not a stable point in 3N-6 dimensions when the hybrid method is taken a consideration. The energy of MECP will be corrected via second-order Taylor expansion for two states: where ∆q is the displacement along touching hyperline, which is perpendicular to different gradient x 1 , the H eff is diagonal matrix elements of effective Hessian. For nonlinear molecular system, the MECP can be testified as indeed minimum point in 3N-7 dimensions. So this program can approximatively research dynamic measures in nonadiabatic surfaces 56 .