Accurate Potential Energy Surfaces for the Three Lowest Electronic States of N(2D) + H2(X1∑g +) Scattering Reaction.

The three lowest full three-dimensional adiabatic and three diabatic global potential energy surfaces are reported for the title system. The accurate ab initio method (MCSCF/MRCI) with larger basis sets (aug-cc-pVQZ) is used to reduce the adiabatic potential energies, and the global adiabatic potential energy surfaces are deduced by a three-dimensional B-spline fitting method. The conical intersections and the mixing angles between the lowest three adiabatic potential energy surfaces are precisely studied. The most possible nonadiabatic reaction pathways are predicted, i.e., N(2D) + H2(X1∑g +) → NH2(22A') → CI (12A'-22A') → NH2(12A') → CI (12A″-12A') → NH2(12A″) → NH(X3∑-) + H(2S). The products of the first excited state (NH(a1Δ) + H(2S)) and the second excited state (NH(b1∑g +) + H(2S)) can be generated in these nonadiabatic reaction pathways too.

Introduction

Reactions between atomic nitrogen and molecule are of great interest in atmospheric chemistry and combustion processes.[1] The three lowest adiabatic electronic states (labeled 12A″/12B1, 12A′/12A1, and 22A′/12B2 in the C/C2 symmetry, respectively) of NH2 have been studied quite extensively over a number of years by various experimental[2−5] and theoretical techniques.[6−34] Jungen and co-workers[6] obtained an accurate potential energy surface for X2B1 and A2A1 states of NH2 in 1980. Carter and Handy[7] generated three-dimensional (3D) potential energy functions of the A2A1–X2B1 bent–bent Renner–Teller system of NH2. Alijah and Duxbury[8,9] used the stretch–bender approach method to study the rovibronic energies for the ground state of NH2, and the Renner–Teller and spin–orbit interactions between the A2A1 and the X2B1 states of NH2. The global potential energy surface (GPES) for 1A″ has been careful studied by Pederson and co-workers.[15] In this work, the authors found a saddle point energy of 2.3 kcal/mol for the perpendicular approach of the second-order configuration interaction surfaces, and got the collinear stationary point energy of 5.5 kcal/mol. Another GPES[24] for the ground state of NH2(X2A″) has been constructed by three-dimensional cubic spline interpolation method, and the vibrational frequencies of NH2 and its isotopomers were calculated. The NH2(12A′) GPES has been presented by Pederson et al.,[16] and the nonadiabatic quasiclassical trajectory calculations were performed. This work obtained the different products of the ground-state NH and the excited-state NH(a1Δ). The NH2(12A′) GPES and the NH2(22A′) GPES have been accurately calculated by Li and Varandas[26,28] with single-sheeted double many-body expansion method. The 22A″/22B1 state of NH2 is also well studied by Li and co-workers.[35]

Most of these works have studied the potential energy surfaces (PESs) for the ground and the first two excited states. Although the title reaction process mostly occurs in the ground state, nonadiabatic interactions with excited electronic states may also play an important role.[36] It is well-known that the diabatic potential energy surfaces can be used for modeling electronically nonadiabatic processes.[37−41]

In quantum chemistry, conical intersections (CIs) between two potential energy surfaces are very important because the potential energy surfaces are degenerated, the Born–Oppenheimer approximation breaks down, and the nonadiabatic processes can occur here. Therefore, the location and characterization of the geometry of the CIs are essential in explaining the nonadiabatic events. CIs play a very important role in nonradiative de-excitation transitions from an excited electronic state to a ground electronic state of the molecules.[42]

Up to now, no accurate diabatic PES exists for nonadiabatic study for NH2. The major goal of the present work is to report an accurate global diabatic PES including the ground and the first two excited states of NH2 based on the MCSCF/MRCI calculation method. First, the lowest three adiabatic PESs are calculated. Second, the mixing angles, which are used to transform the adiabatic PESs to diabatic PES, are fitted. Third, CIs, which are between two adjacent PESs, are accurately studied.

This paper is organized as follows: the next section outlines the theory for calculating the potential energies and mixing angles. The adiabatic potential, CIs, and diabatic potential results are presented and discussed in the third section. A simple summary is given in the last section.

Computational Methods

Adiabatic Potential Energies Calculation

All ab initio calculations have been carried out at the MCSCF/MRCI level with the MOLPRO 2012 package[43] using the large basis sets (aug-cc-pVQZ) of Dunning.[44] This involves 9 electrons in 1 closed orbital (1A′ + 0A″), 6 active orbitals (5A′ + 1A″), and 165 external orbitals (104A′ + 61A″). Similar to our earlier works,[45−48] we calculated the three lowest states in atom–diatom (N–HH) Jacobi coordinate (r(H–H), R(N–HH), and θ) (see Figure ). For θ = 0.0–40.0°, the angle grid is 5.0°. In this range, r is calculated from 0.4 to 12.0 Å with 72 points and R is calculated from 0.0 to 18.0 Å with 107 points in different step sizes. Therefore, 61 633 geometries are generated for each state. For θ = 40.0–90.0°, the angle grid is 10°. In this range, r is calculated from 0.4 to 5.0 Å using 37 points and R is calculated from 0.0 to 18.0 Å using 77 points. Therefore, 17 094 geometries are generated in this range. A total of 78 727 geometries are calculated for each state. The PESs, which are fitted using this range and large number points, are accurate for further study. Inside the whole scan field, the procedures interpolate the surfaces are made using the three dimension B-spline method.[49,50]

Figure 1

Jacobi coordinate.

Mixing Angles (α) Calculation

It is generally known that the nonadiabatic coupling terms and the avoided crossing point between two adiabatic PESs can be well-calculated in one-dimensional (1D) coordinate;[51−53] the mixing angles are predicted in the same method in the present work. The details are shown as follows. The mixing angles (α) between states (12A″, 12A′, and 22A′) are calculated using the corresponding adiabatic potential energies. The details for calculating the α12 between E1 and E2 is as follows. First, r and θ values are fixed and then the potential energies E1 and E2 are calculated with the function R and the derivatives of dE1/dR and dE2/dR. Second, a point where dE1/dR = dE2/dR is found and the mixing angle α = 45.0° is assigned, which is the crossing point of the two diabatic potential energy surfaces. Third, tangents are made from the crossing point to the adiabatic curves and then α is set at 90.0 and 0.0°. Finally, the other values of α are fitted between these two points. Using the same method, the α23 between the states of 12A′ and 22A′ can be achieved. After calculating all α12 and α23 at the needed range of r and θ, the global three-dimensional (3D) mixing angles can be derived by 3D-B-spline method.

Adiabatic-to-Diabatic Transformation

Considering three coupling states of NH2, the diabatic energies Hiid can be obtained in terms of our fitted adiabatic energies Eia as follows:

when α12 is larger than 45.0°, the diabatic energies H11d can be calculated using eq .when α12 is less than 45.0° and larger than 0°, the diabatic energies H11d can be calculated using eqs and 3.The second and the third diabatic potential energies can be calculated using eqs and 5.The coupling potential energies between two diabatic states are shown as follows

Results and Discussion

The adiabatic potential energies of the lowest three states of the NH2 system are calculated and transformed to the lowest three diabatic potential energies. For the ease of discussion of these PESs, the lowest state energy of N(2D) + H2(X1∑g+) is shifted to 0.000 eV.

Two-Dimensional Adiabatic Potential Energy Surfaces

To clearly show the change of potential energy as a function of R and r, the two-dimensional ground-state adiabatic potential energy surfaces (2D-PESs) of NH2 are plotted in Figures –5 for θ = 0.0, 30.0, 60.0, and 90.0°, respectively. The upper part of each figure is 2D-PES, and the lower part is the contour plot of the corresponding 2D-PES. The difference of energy between two adjacent curves is 0.20 eV in the contour plot parts, and short distance between two adjacent curves corresponds to big energy changes with distance (R or r) and vice verse.

Figure 2

Adiabatic potential energy surface (in eV) and contour plots of the potential energy surface for the ground state of NH2 as a function of distances r and R (in Å) in C∞ symmetry.

Figure 5

Adiabatic potential energy surface (in eV) and its contour plots for the ground state of NH2 as a function of distances r and R (in Å) in the C2 symmetry.

From Figure , one can see that with the decrease in the distance R, the potential energies increase. There is no minimum for these linear geometries, but there exists a transition state. The energy of this transition state is nearly 0.663 eV (r = 1.08 Å, R = 1.65 Å). After crossing this transition state, the system can reach its product part. Figure shows the adiabatic potential energy surface for the ground state of NH2 with the angle θ = 30.0°. The character of this figure is similar to the linear symmetry: first, with the decrease in the distance between the atom N and the molecule H2, the potential energy is increased; second, after the transition (the energy of this transition state is 0.533 eV), the system will reach its product part. The different character of this figure is that after the transition state, whose energy is 0.225 eV, there is a shallow minimum in this figure. The minimum energy is 0.134 eV. Similar to our earlier work,[47] this is not a real minimum; if this geometry is optimized, the C2 symmetry geometry will be obtained, which has been frequently pointed out by Poveda and Varandas.[54]

Figure 3

Adiabatic potential energy surface (in eV) and its contour plots for the ground state of NH2 as a function of distances r and R (in Å) at an angle θ = 30.0° in the Jacobi coordinate.

The 2D-PES and its contour plot for θ = 60.0° is shown in Figure . From this figure, one can know that when the N atom attacks the H2 molecule along the angle θ = 60.0°, the energies are increased with decrease in the distance R; after crossing this transition state (the energy of this transition state is 0.145 eV), the system will reach its first minimum. The energy of this minimum is −0.415 eV. After this minimum, the lowest potential energy reaction pathway will cross another transition state (the energy of this transition state is 0.265 eV) and then will reach its product part.

Figure 4

Adiabatic potential energy surface (in eV) and its contour plots for the ground state of NH2 as a function of distances r and R (in Å) at angle θ = 60.0° in the Jacobi coordinate.

The 2D-PES of θ = 90.0° is shown in Figure . Similar to the HN2 system,[54] there are two minima and two transition states in this figure. When using the N atom to attack H2 molecule along the θ = 90.0°, the system should overcome 0.117 eV energy to reach the first minimum, whose potential energy is −0.962 eV (R = 1.06 Å, r = 0.96 Å); then, the ground-state system crossing the second transition state reaches the global minimum. The energy of the global minimum is −4.022 eV, which is 5.154 eV lower than that of the reactants of N(2D) + H2, which is in good agreement with the results of Qu et al.[17] (5.47 eV). The geometry of this minimum is R = 0.32 Å and r = 1.88 Å.

Comparing the energy of the minima of θ = 0.0, 30.0, 60.0, and 90.0°, we can reach the conclusion that the minimum with the C2 symmetry is the global minimum. The detailed information is given in Figure .

Figure 6

2D and contour plots for the ground-state PES (in eV) as a function of R (in Å) and θ (in degree) at fixed r = 0.96 Å.

The 2D-PESs for the first excited state and the second excited state are plotted in Figure S1–S6 (see the Supporting Information).

Three-Dimensional Diabatic Potential Energy Surfaces

Conical Intersections (CIs)

The CIs between two potential energy surfaces, where the nonadiabatic reaction processes take place, are very important. To clearly understand the nonadiabatic event of the NH2 system, the geometries of CI1 for 12A″–12A′, and CI2 12A′–22A′ states are optimized using the CASSCF method with 6-311G(d,p) basis set. The optimized structures are shown in Figure . The left panel of Figure shows the CI1 geometry. The figure shows that the geometry of this conical intersection is C2 symmetry, the bond length of the two H atoms is stretched to 1.304 Å, and R(N–HH) is 0.989 Å. For ease of understanding the change of these two potential energy surfaces near the CI1 part, the 2D-PESs of the ground state (12A″) and the first excited state (12A′) are plotted in Figure . In this figure, the geometry of the avoid crossing point for these two adiabatic PESs is R = 0.906 Å and r = 1.192 Å, the corresponding adiabatic energies are E1 = −0.681 eV and E2 = −0.583 eV, the difference of energy for these two states is 0.098 eV. The geometry of CI2 are plotted in the right panel of Figure and the 2D-PESs near CI2 are plotted in Figure . The geometry of CI2 is different from CI1. CI2 is near the product part, the data of this geometry are r = 1.693 Å, R = 1.153 Å, and θ is nearly 60.0°. The avoid crossing part for these two adiabatic states is in good agreement with the optimized geometry; our results are R = 1.202 Å and r = 1.808 Å, and the potential energies are E2 = 1.434 eV and E3 = 1.439 eV, with the difference between these two energies being 0.005 eV. Because the energies of these two conical intersections are not too high, these two conical intersections may play an important role in the title reaction.

Figure 7

Conical intersection structures, which are optimized with 6-311(d,p) basis set using MOLPRO program, for the 12A″–12A′ and 12A′–22A′ states of NH2.

Figure 8

Adiabatic PESs for the states of 12A″ and 12A′ of NH2 near the conical intersection area with the function of R and r (in Å), the angle is fixed at θ = 90.0°.

Figure 9

Adiabatic PESs for the states of 12A′ and 22A′ of NH2 near the conical intersection area with the function of R and r (in Å), the angle is fixed at θ = 60.0°.

Taking into account all of the adiabatic PESs and CIs, the nonadiabatic reaction pathways can be speculated. Figure shows the most possible nonadiabatic reaction pathways for the title system. For N(2D) + H2(X1∑g+) reaction, the system can form the NH2(22A′) isomers, a part of these isomers will separate and generate the second excited state products (NH(b1∑g+) + H(2S)). The energy of the second excited state products is slightly higher than that of the reactants. The other part of the second excited state isomers can pass CI1 to transfer to the first excited state isomers. The energy of CI1 is nearly equal to that of the isomer (NH2(22A′)), so the transformation is quite easy. Some part of these first excited state isomers can separate to the first excited state products (NH(a1Δ) + H(2S)); the others may cross CI2 to transfer the isomers of the ground state; at last, the isomers of the ground state can lead to the ground-state products (NH(X3∑–) + H(2S)). The reverse reaction can also occur simultaneously in these reaction pathways. It should be pointed out that the main aim of Figure is to show the nonadiabatic process, so the complexes and the transition states for the title system are not included in this figure.

Figure 10

Nonadiabatic reaction pathway for N(2D) + H2(X1∑g+).

Diabatic Potential

Using formulas –11, the global diabatic potentials can be derived. To clearly show the most important part of the diabatic potentials, the 1D adiabatic and diabatic potential near the CI points are plotted in Figures and 12, and the enlarged avoid crossing parts are plotted in the same figure. To ensure that the diabatic PESs are accurate, atomic units are chosen to show these potential energy surfaces. Figure shows the three lowest adiabatic energies (E1, E2, and E3), diabatic energies (H11, H22, and H33), and nondiagonal elements (H12 and H23) in the function of R values with the angle fixed at θ = 90.0° and r = 1.30 Å. For the adiabatic potential energy surfaces, there are three avoid crossing points in this figure. The first one lies between the ground-state PES and the first excited-state PES. The geometry of the CI point is similar to the minimum of the first excited state. The energy of this CI is nearly −0.6 eV, meaning that the electrons easily hop between the ground state and the first excited state. There are two adiabatic avoid crossing points between the first excited state and the second excited state. The crossing geometry of the diabatic PES, R, is nearly equal to 1.27 Å, and the potential energy is nearly −55.53 au for the first adiabatic avoid crossing point. The R value is nearly 0.5 Å for the second diabatic crossing point; the energy (near −55.39 au) of this point is higher than that of the previous one (−55.53 au). The energies of these two avoid crossing points are higher than that of the conical intersection between these two excited states, so they are not the main surface hopping part. Figure shows the lowest three adiabatic potential energies and the corresponding diabatic potential energies near the conical intersection between the first excited state and the second excited state. This figure shows that near the CI point, the difference in the energy between the first excited state and the second excited state is very small, and the geometry of the CI point is close to the minimum geometry of the second excited state. On this condition, the electrons hop easily between these two states. In other words, from this CI point, the NH2 system is easily transformed between 12A′ and 22A′ states. There is a second crossing part between the first excited state and the second excited state; the R value for this diabatic energy crossing part is nearly 0.5 Å, but the energy of this crossing point is much higher than that of the first crossing point, which is not important for this system. In this figure, there is crossing point between diabatic energies H11 and H22; however, since the energy difference between these two states is large, this is not the main surface hopping part. From Figures and 12, we could find that the diabatic PESs are very smooth even near the crossing part. And near the crossing part, the nondiagonal elements H12 and H23 are large and other parts are zero.

Figure 11

Adiabatic (E1, E2, and E3), diabatic (H11, H22, and H33) PESs and the nondiagonal elements (panel B: H12 and panel A: H23) (in au) as the function of R (in Å) with fixed θ = 90.0° and r = 1.30 Å. The crossing part of every two diabatic PESs are enlarged plotted in this figure.

Figure 12

Adiabatic (E1, E2, and E3), diabatic (H11, H22, and H33) PESs and the nondiagonal elements (panel C: H12 and panels A and B: H23) (in au) as the function of R (in Å) with fixed θ = 60.0° and r = 1.70 Å. The most important crossing part of every two diabatic PES is enlarged plotted in this figure.

What should we point out here is that these diabatic potential energies only considered three adiabatic potential energies; if more adiabatic potential energy surfaces were considered, the diabatic energy of H11 would have been much higher with the increase of the R value; there should be no transition state point in Figure .

Because the angles of CI1 and CI2 are 60.0 and 90.0°, the 2D diabatic PESs are plotted at angles 60.0 and 90.0°, respectively. The details are shown in Figures –7. Figures and 14 show the diabatic PESs of H11, H22, and H33 with fixed θ = 90.0° near the CI point of the ground state and the first excited state. Figure shows the different diabatic potential energy between H11−H22 and H11−H33. Figures and 17 show the diabatic PESs of H11, H22, and H33 with fixed θ = 60.0° near the CI point of the first excited state and the second excited state. These four figures show that the diabatic PESs of each state are very smooth; the diabatic crossing points are nearly in the same line for each two diabatic PESs (see Figure ).

Figure 13

2D plots of diabatic PESs of the H11 (blue surface) and H22 (magenta surface) for the NH2 system with function R and r (in Å) for fixed θ = 90.0°.

Figure 14

2D plots of diabatic PESs of the H11 (blue surface) and H33 (cyan surface) for the NH2 system with function R and r (in Å) for fixed θ = 90.0°.

Figure 15

Contour plots of different diabatic potential energies in eV (panel a shows H11–H22, panel b shows H11–H33) with function R and r (in Å) for fixed θ = 90.0°.

Figure 16

2D plots of diabatic PESs of the H11 (blue surface) and H22 (magenta surface) for the NH2 system with fixed θ = 60.0°.

Figure 17

2D plots of diabatic PESs of the H11 (blue surface) and H33 (cyan surface) for the NH2 system with fixed θ = 60.0°.

Conclusions

The three lowest electronic adiabatic state PESs have been reported for NH2 on the B-spline fit method for the ab initio MCSCF/MRCI energies calculated using AVQZ basis sets. The conical intersections between the ground state and the first excited state and between the first excited state and the second excited state have been accurately studied using MOLPRO program. The mixing angles, which can be used to calculate the diabatic PESs with adiabatic PESs, have been precisely studied in the present work. The three diabatic PESs of H11, H22, H33 and the coupling potential energies between every two diabatic states have been calculated and discussed carefully. After carefully studying the adiabatic PESs, CIs, and diabatic PESs, the authors make a conclusion that if the title reaction starts with the second excited state, some part of the reactant can reach the second excited state products, and the other part of the system should cross CI1, the electrons hopping from the second excited state to the first excited state, then form the isomer in the first excited state. The same with the second excited state, some of the first excited state isomer can separate to the first excited state products, and the others may get the enough energy then it can pass CI2, the electrons can hop to the ground state, the isomer on the ground state can be formed, at last, the ground state NH(X3∑–) + H(2S) products can be obtained. A full dynamic study of these global diabatic potential energy surfaces can be used to prove this speculation.