Study of Nonadiabatic Effects in Low-Lying Electronic States of HCNH

May 15, 2013 - the dissociation of HCNH, right after its formation from electron capture by HCNH+, available in the upper atmosphere. In the present w...
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Study of Nonadiabatic Effects in Low-Lying Electronic States of HCNH with Implication in Its Dissociation to HCN and HNC Anita Das and Debasis Mukhopadhyay* Department of Chemistry, University of Calcutta, Kolkata - 700009, India ABSTRACT: Abnormal abundance of HNC in interstellar spaces has been the motivation of many experimental as well as theoretical studies of the branching ratio [HNC]/[HCN] in the dissociation of HCNH, right after its formation from electron capture by HCNH+, available in the upper atmosphere. In the present work we were interested in nonadiabatic studies involving the dissociation channel of HCNH leading to the formation of HNC. This study reports for the first time that the conical intersection (CI) between the states 12Σ+ and 22Σ+ exists only in some bent geometry (and not in the collinear geometry) where both of these states have A′ symmetry. This finding is important as this CI is crucial in the dissociation of HCNH. We further report that these two states strongly couple with 12Π, the lowest electronic state of collinear HCNH. Hence we construct a three-state Hilbert subspace (HSS), comprising of the states 12Π, 12Σ+, and 22Σ+, in a configuration space where these states interact very strongly and the adiabatic-to-diabatic transformation angles (mixing angle) yield meaningful values of topological (Berry) phase. This leads to the construction of the corresponding three-state diabatic potentials. We advocate that these diabatic potentials, considering both the linear as well as bent configurations, nicely elucidate the formation of the HNC molecule by the CH bond dissociation of HCNH molecule.

I. INTRODUCTION Highly correlated ab initio molecular orbital (MO) calculation1 has shown that hydrogen isocyanide (HNC) is much less stable than that of hydrogen cyanide (HCN). Thus one may expect a relatively negligible abundance of HNC with respect to HCN. In contrast, in many interstellar clouds the abundance ratio [HNC]/[HCN], has been found to be of the order of 1 or even more.2−4 Studies attempting an explanation for such an unusually high abundance of HNC in dense interstellar spaces have pointed out that both the species are formed from the same precursor ion HCNH+. It has been postulated that the dissociative recombination (DR) reaction

calculation. They found that the lowest dissociative states of 2 + Σ symmetry (leading to the C−H or N−H bond dissociation yield HNC or HCN respectively), cross the ground state PES of HCNH+ at its minimum, below its first vibrational level. This indicates that a direct mechanism of electronic DR process might have been effective. They also found that the ground state of HCNH+ crosses the Rydberg states (of 2Σ symmetry) of HCNH molecule at almost the same configuration between the first and second vibrational levels. Thus, the indirect mechanism may also be involved. It was noted that the bond dissociation processes involved in the DR reaction show strong nonadiabatic character and thus the necessity of a diabatic representation, over of the adiabatic one, has been felt. Twostate qausi-diabatic potentials have been formulated by Hickman et al.,10 considering the earlier results of Talbi and Ellinger9 using the block diagonalization method,17 and they used this to estimate the width of the electron capture by HCNH+ as well as the rate of dissociation process. Both of these results led to the conclusion that both HCN and HNC are produced with the same probability. In direct ab initio dynamics calculations, Tachikawa11 found that the direct mechanism for DR of HCNH+ at low temperature is favorable. The adiabatic PESs were also computed by Shiba et al.12 using the multi reference configuration interaction with single and double excitations (MRSDCI) method and derived the

HCNH+ + e− → (HCNH)* → HNC + H → HCN + H

(1)

5−8

may be the main process for the production of an almost equal amount of HCN or HNC. In the DR reaction, it has been considered that a cation, on accepting an electron enters the highest diffuse Rydberg-type excited state of the neutral molecule in a Franck−Condon manner. If this state is repulsive, the dissociation takes place through a direct process. Otherwise, an indirect process is observed where either autoionization may take place or the neutral molecule relaxes to the lower energy state and may dissociate. Several theoretical as well as experimental investigations of potential energy surfaces (PESs) of both the cationic and neutral HCNH molecule and also the dynamics of this process are available in the literature.9−16 Talbi and Ellinger9 have computed the two-dimensional adiabatic PESs for a linear HCNH molecule using a large scale configuration interaction © XXXX American Chemical Society

Special Issue: Structure and Dynamics: ESDMC, IACS-2013 Received: March 28, 2013 Revised: May 15, 2013

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form quasi-diabatic states17 where the choice of diabatization method plays an important role. A variety of diabatization methods17,18,28−32 are available in the literature, but most of them are very useful only when one considers CI(s) between the two states. It is observed that the degenerate 2Π states of linear HCNH molecule split33 into A′ (lower one) and A″ (upper one) states via the bending motion (Renner−Teller34 effect). In the bent configuration, the 2Σ+ states are designated as A′ states. In the construction of diabatic potentials, the strong coupling among these three A′ states naturally has a profound effect. In the present article, along with linear geometry we also consider the bent geometry of neutral HCNH molecule, and from the study of the potential intersections we construct the corresponding three-state diabatic potentials where these three A′ states strongly interact with each other. In section II, we briefly refer to the theoretical framework used in this article. Section III gives the numerical results to identify the CIs and demonstrates their effects on the ADT angles involving the three low-lying states 12A′, 22A′, and 32A′, as well as the corresponding three-state diabatic potentials with detailed discussion. In section IV, we summarize the findings of the investigation presented in this article.

corresponding diabatic potentials following the procedure presented by Werner and Mayer.18 In the diabatic picture, they found that valence states of HCNH molecule are wellseparated from the Rydberg states. PESs of these Rydberg states of HCNH molecule are parallel to the PES of the cation ground state and are not dissociative. They also found that the two lowest dissociative valence type 2Σ+ states of neutral HCNH molecule (leading to HCN and HNC) in diabatic representation are located beneath the ground state PES of HCNH+ and play a key role in the DR reaction. They suggested that the DR process is driven by the indirect mechanism where initial relaxation through several nonadiabatically coupled Rydberg states takes place followed by transition to respective valence states. A two-state quantum wave packet dynamics study on the same PESs, with the inclusion of Davidson’s quadruple excitation correction,19 was carried out by Ishii et al.13 The wave packets were generated from the low lying vibrational eigen states of HCNH+ and put initially on the 22Σ+ PES of a HCNH molecule. Depending on the initial vibrational quantum numbers, the branching ratio [HNC]/[HCN] varied from 0.77 to 1.32, indicating the unusual abundance of HNC in interstellar spaces. Taketsugu et al.16 considered the nonadiabatic transitions among the adiabatic states and carried out the ab initio direct trajectory simulations to show that HCN and HNC are generated to the same extent. It is important to note that most of the so far generated PESs and dynamical calculations have been restricted to the linear configuration of HCNH molecule. Since the ground state PES of HCNH+, X1Σ+, has a linear geometry at equilibrium, HCNH molecule obtained in DR process may initially take the linear geometry in Franck−Condon manner. Notably, 12Σ+ and 22Σ+ states of HCNH are stable in linear configurations.12 Nevertheless, as the equilibrium geometry of HCNH is trans planar, it is expected that there is a strong tendency of HCNH molecule to attend the bent structure soon after its formation in DR reaction. The bending of the HCNH molecule also permits the isomerization reaction between HNC and HCN with a finite possibility.16 On the other hand, the adiabatic PESs of HCNH molecule show that in the linear configuration there is symmetry-allowed Jahn−Teller (JT)20 conical intersection (CI) seams (line of CIs) between the 2Π and 2Σ+ states. The 22Σ+− 12Σ+ transition is very relevant for these cases, and in such calculations, one has to consider the bending motion. Therefore, in order to understand the reaction mechanism more clearly, knowledge of the PESs and their nonadiabatic behavior in linear as well as bent HCNH molecules is essential. The two 2Σ+ states, involved in bond breaking processes, show strong nonadiabatic effects. It is important to mention here that the various concepts of nonadiabatic effects such as coupling among the adiabatic PESs, namely, nonadiabatic coupling terms (NACTs), adiabatic to diabatic transformation (ADT) matrix,21−23 which transforms adiabatic PESs to the corresponding diabatic one, and the topological phase24 (Berry phase25) obtained from the ADT angle (mixing angle) at the end of the closed contour, all are related to each other and have the same origin, i.e., the electronic NACTs. Since the adiabatic potentials are single-valued by definition, the single-valuedness of the diabatic potentials is guaranteed only when the ADT matrix at the end of the closed contour i.e., D-matrix is diagonal26 having a value of ±1. The diagonal D-matrix is associated with the ADT angle at the end of the closed contour as a multiple of π or zero. Since for polyatomic molecules, it is not possible to form strictly diabatic states,27 one can always

II. METHODOLOGY We’d like to start this section by pointing out that in the investigations pursued in this article, the key role is being played by the electronic NACTs. We assume the existence of n-state electronic Hilbert subspace (HSS) considering the presence of CIs anywhere in the nuclear configuration space. The Schrodinger equation for an HSS may be written35,36 compactly as 1 − (∇ + τ )2 ψ + (u − E)ψ = 0 (2) 2m where τ (= τ(S)) consists of the nuclear coordinate-dependent NACTs, u (= u(S)) is the adiabatic PES, and E is the total energy of the system. For the transformation from adiabatic to diabatic basis, ψ = Aψd, where ψd is the diabatic state, and A is the ADT matrix.21−23 With ψd the Schrodinger equation becomes 1 2 d − ∇ ψ + ( v − E )ψ d = 0 (3) 2m Here the diabatic potential matrix v( = v(S)) is related to u, the adiabatic one, as v = A†uA

(4)

The ADT matrix may be shown to be an orthogonal (unitary) matrix and fulfills the first-order differential equation23 (5)

∇A + τ A = 0 23,37

Its solution may be written as an integral equation. In the case of circular contour,35,36 this solution may be written as A(φ|q , Γ) = W exp( −

∫0

φ

dφ ·τφ(φ|q , Γ))

(6)

where W is the path ordering operator, and the dot stands for a scalar dot product. Here (τφ)ki = ⟨ζk|(∂/(∂φ))ζi⟩ is the corresponding τ-matrix element with ζk and ζi as the corresponding kth and ith electronic eigenfunctions, respectively. In the coordinates used, q and φ are the radius and the B

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angle associated with the rotating atom. The angular component of NACT is given by 1/q (τφ)ki. Another matrix of interest is the topological matrix D(q, Γ), identical to the Amatrix but calculated along the closed contour and is defined as D(q , Γ) = W exp( −

∫0

γ12(φ , q) =

(7)

α12(q) =

For single-valued diabatic potential, the D-matrix elements are expected to be as follows: Dij(Γ) = ±δij ,

i = {1, n}

(τφ)12 (φ′, q) dφ′

(10)

∫0

2π

(τφ)12 (φ′, q) dφ′

(11)

The corresponding D-matrix is similar to the A-matrix given in eq 9 with γ12(φ,q) replaced by α12(q). Thus, the condition for the D-matrix to be diagonal is

(8)

Here, we are concerned with angular NACTs only, and, henceforth, for simplicity, we will drop the suffix φ from τ. II.1. Two-State HSS. For a two-state HSS, eqs 6 and 7 are simplified significantly because any (2 × 2) ADT matrix can be written as ⎛ cos(γ (φ , q)) sin(γ (φ , q)) ⎞ 12 12 ⎟ A(2)(φ , q) = ⎜⎜ ⎟ ⎝−sin(γ12(φ , q)) cos(γ12(φ , q))⎠

φ

On integrating along the circularly closed contour, this yields the topological phase24 (Berry phase25)

2π

dφ ·τφ(φ|q , Γ))

∫0

α12(q) = 2πn

(12)

where n is an integer or half integer. II.2. Three-State HSS. Since the A-matrix is an orthogonal matrix, the nine elements of the three-state HSS may be presented in terms of the three Euler angles Q12, Q13, amd Q23. Thus here the A-matrix is constructed by taking the product of these three matrices, and the product can be written in six different ways. One of such products yields:38

(9)

where γ12(φ,q) is the mixing angle or the ADT angle expressed in terms of a line integral A =Q 12Q 13Q 23

⎛ cos γ12 sin γ12 0 ⎞ ⎛ cos γ13 0 sin γ13 ⎞ ⎟ ⎜ ⎟ ⎜ =⎜−sin γ12 cos γ12 0 ⎟· ⎜ 0 1 0 ⎟· ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 1 ⎠ ⎝−sin γ13 0 cos γ13 ⎠

⎛1 0 0 ⎞ ⎜ ⎟ γ γ23 ⎟ 0 cos sin 23 ⎜ ⎜ 0 −sin γ cos γ ⎟ ⎝ 23 23 ⎠

⎛ cos γ12cos γ13 −cos γ12sin γ13sin γ23 + sin γ12cos γ23 cos γ12sin γ13cos γ23 + sin γ12 sin γ23 ⎞ ⎜ ⎟ =⎜−sin γ12cos γ13 sin γ12 sin γ13sin γ23 + cos γ12 cos γ23 − sin γ12 sin γ13cos γ23 + cos γ12 sin γ23 ⎟ ⎜⎜ ⎟⎟ − sin γ23cos γ13 cos γ13cos γ23 ⎝−sin γ13 ⎠

Substitution of this into eq 5 yields three coupled first order differential equations for the three corresponding ADT angles γij’s. The final set of equations as well as their solutions depends on the order of the Q-matrices. For the above product, the set of differential equations are

expressions of the two-state diabatic potential using eq 4, where u is the known 2 × 2 diagonal adiabatic potential matrix. Thus, the expressions of diabatic potentials v11(φ,q), v22(φ,q) and v12(φ,q) from the given two adiabatic PESs uj (φ,q); j = 1,2, are as follows:36,39

∇γ12 = −τ12 − (τ23 cos γ12 + τ13 sin γ12) tan γ13

v11 = u1 cos 2 γ12 + u 2 sin 2 γ12

∇γ13 = −τ13 cos γ12 + τ23 sin γ12 ∇γ23 = −(cos γ13)−1(τ13 sin γ12 + τ23 cos γ12)

(13)

v22 = u1 sin 2 γ12 + u 2 cos2 γ12

(14)

v12 =

In the chosen order of Q-matrices, we took Q12 as the leftmost matrix, and, as a consequence, the equation for the corresponding ADT angle, γ12, contains τ12 as a free term. This implies that, for the case when the other two NACTs τ13 and τ23 are zero (signifying that the coupling with the third state is negligible), the three-state HSS reduces to a two-state HSS. A similar result is expected for another possible choice of product38 such as A = Q12Q23Q13. From the ADT angles γij’s, one can get the corresponding topological phases at the end of the closed contour and construct the ADT matrix A as well as the corresponding topological D-matrix. II.3. Diabatic PES for Two-State HSS. For a two-state HSS, once the orthogonal A-matrix is obtained using the value of γ12 (φ,q) (from eqs 10 and 9), one can obtain the

1 [u1 − u 2] sin(2γ12) 2

(15)

The transformation that yields the diabatic potentials is presented explicitly to show that, in order to guarantee the single-valuedness of these potentials, the angle γ12 (φ,q) has to a value of ∼π at φ = 2π. II.4. Diabatic PES for Three-State HSS. For a three-state HSS, the orthogonal A-matrix is obtained from eq 13 using the values of ADT angles γ12 (φ,q), γ13 (φ,q), and γ23 (φ,q) obtained from eq 14. Hence, one can get the expressions of the three-state diabatic potential using the same eq 4 from the known 3 × 3 diagonal adiabatic potential matrix u. The expressions of the diabatic potentials v11(φ,q), v22(φ,q), v33(φ,q), v12(φ,q), v13(φ,q), and v23(φ,q) from the given three adiabatic PESs uj (φ,q); j = 1,2,3, are as follows:40 C

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v11 = u1 cos2 γ12 cos2 γ13 + u 2 cos 2 γ13 sin 2 γ12 + u3 sin 2 γ13

as shown in Figure 1a, both the N−H bond length (designated as RNH) and C−H bond length (designated as RCH), are used as variables. For the HCNH molecule, we also consider the bent geometry (Figure 1b) where one of the N−H or C−H bonds makes an angle θ with the axis of the molecule in its linear configuration. Here, in addition to RNH and RCH, the bending angle θ is also used as a variable. In order to avoid more complexities, we are not considering the bent geometries where both the C−H and N−H bonds make nonzero angles with the C−N bond. Ab-initio PESs for neutral HCNH molecule have been computed at the state-average CASSCF level with the lowest three electronic states 12Π, 12Σ+, and 22Σ+ in linear geometry configurations and 12A′, 22A′, 32A′, and 12A″ in bent geometry configurations with equal weights, followed by MRSDCI calculations using the MOLPRO41 package of programs. The Davidson correction19 has been used to account for the contributions of higher excitations and the size-extensive corrections to the energy. For linear configurations, the C2v point group has been adopted, whereas for the bent geometry, the planar Cs point group has been chosen. We have employed the Dunning’s correlation consistent polarized valence quadruple-ζ (cc-pVQZ) basis set. A full valence complete active space, i.e., 3σ−8σ and 1π−2π for the linear form and 3a′−10a′ and 1a″−2a″ valence orbitals for the bent geometry, has been used so that 11 valence electrons are distributed over these 10 orbitals. Natural bond orbital analysis of the HCNH+ molecule has shown that 6σ and 7σ MOs have antibonding character, whereas the 8σ MO is Rydberg-type in nature. So, in neutral HCNH molecule, the 12Σ+ and 22Σ+ states, the key states in the dissociation process, are generated with the excitation of one electron from 12Π to the 6σ orbital or 7σ orbital and have valence character. Adiabatic energies have been calculated in the following configuration space: both the bond lengths RCH and RNH have been varied from 0.7 Å to 2.0 Å keeping the C−N distance RCN fixed at 1.138 Å. For bent geometry, in addition to RCH and RNH, the angle θ has been varied from 0° to 90°. The calculation of angular NACTs have also been carried out at the same level of MRSDCI using the numerical gradient technique DDR in MOLPRO.41 In our calculations, with a choice of a “no symmetry” condition (C1 symmetry), there is the scope of considering the interactions of the A″ state with the A′ states. However, we are not considering this interaction for the following reason. With the variation of geometry considered in our calculations, the energy of the A″ state increases sharply, and thus it has now little to do with the dissociation process. Apart from this, in general, the symmetry-allowed JT-CIs involving the A′ and A″ states would always have relatively smaller effect than that of the same-symmetry JT-CIs between different A′ states. III.1. Investigation with the Plot of Energies. We start with a report on investigation of PESs of three low-lying electronic states in collinear as well as bent geometry configurations. Figure 2a presents the plot of the three electronic PESs in linear geometry configurations where both RCH and RNH are varied from 0.7 Å to 1.7 Å. All the energy values, shown in the figure, are relative to the energy of −93.75 au, i.e., henceforth the value of −93.75 au is taken as zero in the energy-axis of the figures. It is observed that all three states are strongly coupled with one another via the symmetry-allowed JT CIs. For a more clear presentation, Figure 2b is given with the two-dimensional plot of the JT CI seams for the linear geometry configurations. It is observed that the ground 12Π

v22 = u3 cos 2 γ13 sin 2 γ23 + u1(cos γ23 sin γ12 − cos γ12 sin γ13 sin γ23)2 + u 2(cos γ12 cos γ23 + sin γ12 sin γ13 sin γ23)2 v33 = u3 cos2 γ13 cos2 γ23 + u 2( −cos γ23 sin γ12 sin γ13 + cos γ12 sin γ23)2 + u1(cos γ12 cos γ23 sin γ13 + sin γ12 sin γ23)2 v12 = v21 = u3 cos γ13 sin γ13 sin γ23 + u1 cos γ12 cos γ13(cos γ23 sin γ12 − cos γ12sin γ13sin γ23) − u 2 cos γ13 sin γ12(cos γ12cos γ23 + sin γ12 sin γ13 sin γ23) v13 = v31 = −u3 cos γ13 cos γ23 sin γ13 − u 2 cos γ13 sin γ12( −cos γ23 sin γ12 sin γ13 + cos γ12 sin γ23) + u1 cos γ12 cos γ13(cos γ12 cos γ23 sin γ13 + sin γ12 sin γ23) v23 = v32 = −u3 cos2 γ13 cos γ23 sin γ23 + u1(cos γ12 cos γ23 sin γ13 + sin γ12 sin γ23) (cos γ23 sin γ12 − cos γ12 sin γ13 sin γ23) + u 2( −cos γ23 sin γ12 sin γ13 + cos γ12 sin γ23) (cos γ12 cos γ23 + sin γ12 sin13 sin γ23)

(16)

Similarly, with the order of product taken as A = Q12Q23Q13, one can form the other set of diabatic potentials. It is important to note the two set of equations give the same value of diabatic potentials.

III. RESULTS AND DISCUSSION We assume the four atoms of the tetra-atomic HCNH molecule to be located along the z-axis with the origin at some point on the axis (Figure 1a). For all the calculations performed here, the C−N bond length is fixed at 1.138 Å, the equilibrium C−N bond length of the HCNH+ molecule. In linear configuration,

Figure 1. The schematic picture of the different arrangements of four atoms in HCNH with N−H bond length (designated as RNH) and C− H bond length (designated as RCH) used as variables; the C−N bond length fixed at 1.138 Å, the equilibrium C−N bond length of the HCNH+ molecule. (a) Collinear configuration of HCNH; (b) bent configuration of HCNH where the C−H bond makes an angle θ with respect to its linear configuration. D

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Figure 3. PECs of a HCNH molecule in different arrangements to show the position of intersections among the different potentials as a function of RCH, where both RNH and RCN are fixed at 0.8 Å and 1.138 Å, respectively. (a) Symmetry-allowed (1,2) JT CI between the two potentials 12Π, 12Σ+ in the collinear arrangement at RCH = 1.35 Å. (b) Same-symmetry (2,3) CI between the potentials 22A′ and 32A′ at RCH = 1.15 Å with θ = 30°.

Figure 2. Plots to show the locations of symmetry-allowed JT CI among the three low-lying electronic states 12Π, 12Σ+, and 22Σ+ in the linear HCNH molecule. (a) Three-dimensional PESs of HCNH molecule where both RCH and RNH vary from 0.7 Å to 1.7 Å. For proper resolution in energy-axis, all the energy values, shown in Figure, are relative to the energy of −93.75 au, i.e., the energy −93.75 is taken as zero. (b) Two-dimensional plot of the positions of JT CI seams between 12Π, 12Σ+ states and 12Π, 22Σ+ states, respectively.

ground 12Π state forms (1,2) CI with the first excited 12Σ+ state at RCH ∼ 1.35 Å, and two 2Σ+ states are close to each other around RCH ∼ 1.05 Å. Interestingly, strong avoided crossing between 12Σ+ and 22Σ+ states is observed once the bending angle θ is ∼30° and RCH ∼ 1.15 Å. In bent geometry configuration, both of these states have A′ symmetry. Figure 3b shows the four low-lying PECs in the bent geometry configuration at θ = 30° where the degenerate 12Π states split into A′ and A″ states. It is also observed that the 12A″ state rises steeply with increase in RCH and also with the increase in the bending angle θ. III.2. Investigation with the Calculations of Angular NACTs. The study of PECs as described in section III.1 has given a preliminary idea about the range of configuration space where the JT CIs among the three adjacent electronic states may be observed. In order to ascertain the position of these CIs, we relied on the D-matrix analysis. For this purpose, we calculated the ADT angles as well as the diagonal elements of the corresponding ADT matrix along a circularly closed contour as a function of the rotation angle φ. III.2.A. JT CIs Involving Two-State HSSs. I. The (1,2) Collinear CI. Motivated by the investigation of adiabatic potentials, as presented in Figure 3a, we have chosen the configuration space with RNH fixed at 0.8 Å in its linear geometry configuration and a point on the collinear axis at RCH = 1.35 Å as the center of the circular contour of radius q = 0.05 Å. The angular NACTs between all three states are calculated along this contour. Figure 4a shows that the NACT τ12(φ), between the two states 12A′ and 22A′, is large in comparison to the other two NACTs τ13(φ) and τ23(φ), obtained from considering the coupling between the states 12A′, 32A′ and 22A′, 32A′, respectively. This implies that in this configuration space, two-state HSS is sufficient to explain the nonadiabatic

state forms a (1,2) CI seam with the first excited 12Σ+ state in the region where RCH is varied from 0.7 Å to 1.37 Å and RNH is varied from 0.7 Å to 1.23 Å. So, within this configuration space, the three lowest electronic states are 12Π, 12Σ+, and 22Σ+, and beyond this line, the 12Π state becomes the first excited state. Now, this 12Π state forms another JT type (2,3) CI seam with the 22Σ+ state in the region with 1.35 Å ≤ RCH ≤ 1.7 Å and 1.22 Å ≤ RNH ≤ 1.7 Å. So, beyond this line, the ordering of the three electronic states becomes 12Σ+ < 22Σ+ < 12Π. With the RNH varying from 0.7 Å to 1.7 Å, keeping RCH fixed at 1.35 Å, two JT CIs are observed, one at RNH ∼ 1.004 Å and another at RNH ∼ 1.565 Å. A similar situation is also observed when RCH is varied from 0.7 Å to 1.7 Å keeping RNH fixed at 1.225 Å. We’d like to emphasize here that the two adiabatic states 12Σ+ and 22Σ+ never show any avoided crossing in the linear geometry configurations. As we confirmed the absence of any avoided crossing between the states 12Σ+ and 22Σ+ of the HCNH molecule in its linear configuration, we considered the bent geometry of the molecule and undertook a detailed investigation of the situation where RNH is fixed at 0.8 Å, and RCH as well as the angle θ (which was explained earlier) are used as variables. Figure 3a shows the three adiabatic potential energy curves (PECs) in the linear geometry configuration (θ = 0°), where only the RCH is used as a variable. It is observed that in the adiabatic picture 12Π and 22Σ+ states are bound, whereas the 12Σ+ state is dissociative in nature. Therefore, once the HCNH molecule is in its 12Σ+ state, then only the C−H bond will dissociate and HNC will form. This implies that the C−H bond dissociation is possible only if two 2Σ+ states intersect with each other. The E

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Figure 5. Summary of results for the confirmation of (2,3) CI in the bent configuration with center at RCH = 1.15 Å and θ = 30°. Panels a− c are same as in Figure 4 for a two-state HSS comprising the states 22A′ and 32A′, except q = 0.1 Å. In panel a, the magnified NACTs τ12(φ) and τ13(φ) are plotted in the inset to show the two-state picture.

Figure 4. Summary of results for the confirmation of (1,2) CI. Calculations are carried out with a two-state HSS comprising the states 12A′ and 22A′, along a circular contour of radius q = 0.05 Å, in the collinear arrangement with center at RCH = 1.35 Å with both the RNH and RCN fixed at 0.8 Å and 1.138 Å, respectively. Panel a presents the three angular NACTs τ12(φ), τ13(φ), and τ23(φ) between the three states 12A′, 22A′ ,and 32A′ along with magnified τ13(φ) and τ23(φ) in the inset; panels b and c respectively present the ADT angle γ12(φ) and the diagonal element of the 2 × 2 ADT matrix, A11(φ), as a function of φ.

large and is elliptical in nature. The nature of these NACTs indicates that the 22A′ and 32A′ states form a two-state HSS. The two-state ADT angle γ23(φ) and the corresponding diagonal element of the ADT matrix are presented in Figure 5b,c, respectively. The topological phase α23 ∼ π and the (2) diagonal element of the corresponding D-matrix D(2) 22 = D33 = −1 clearly indicate the presence of (2,3) CI and that the twostate HSS is sufficient to form the single-valued diabatic potential. The strong asymmetric nature of the corresponding NACT τ23(φ) with respect to the chosen circular contour indicates that this (2,3) CI is situated only in some nonsymmetric bent geometry of the molecule. III.2.B. Three-State HSS. After knowing the positions of these two CIs (12A′, 22A′) and (22A′, 32A′) and confirming these by the corresponding NACTs calculations, our next motivation was to investigate whether in a bending situation these two CIs are coupled with each other. If they are strongly coupled, it will lead to the formation of a three-state HSS. In such a situation, the model for study of the dissociation of C−H or N−H bond should consider all of these three states. For this purpose, we considered the same configuration space as in the case of (2,3) CI, i.e., we fixed the center of the circular contour at RCH = 1.15 Å with θ = 30°, but chose the radius of

effect. The NACT τ12(φ) is highly symmetric in nature. By integrating this NACT τ12(φ), the ADT angle (mixing angle) γ12(φ) is obtained and hence the topological Berry phase α12, as given in eq 11. Figure 4b,c presents the results of the ADT angle γ12(φ) and the diagonal element of the ADT matrix A11(φ), respectively. The well-defined topological Berry phase α12 ∼ π and the sign change of the diagonal elements of the D(2) matrix D(2) 11 = D22 = −1 are clear indications of the presence of (1,2) CI. II. The (2,3) Bent CI. The adiabatic potentials as given in Figure 3b, indicate that, like (1,2) CI, the 22A′ and 32A′ states are also associated with a CI. This CI is situated in the configuration space with RNH fixed at 0.8 Å in its linear geometry configuration and the center of the circular contour chosen at RCH = 1.15 Å, where the C−H bond makes an angle θ = 30° with respect to its linear geometry configuration. Here the radius of the circular contour is chosen at 0.1 Å. The three angular NACTs τ12(φ), τ13(φ), and τ23(φ) are presented in Figure 5a. Among these three NACTs, only τ23(φ) is quite F

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the circular contour, q = 0.7 Å, large enough so that the contour may contain both (1,2) CI and the (2,3) CI. In the next step, we performed a three-state HSS calculation. Figure 6a shows the three low-lying electronic states 12A′, 22A′, and 32A′ along this circular contour. As in planar Cs symmetry configurations, all the states have A′ symmetry, and the crossing between all

these A′ states should be avoided in nature. It is observed that the 32A′ state is very close to the 22A′ state around θ ∼ 65°, whereas the 22A′ state shows a strong nonadiabatic effect with the 12A′ state around θ ∼ 92°. Thus the three states are strongly interacting with each other. Figure 6b gives the φdependent NACTs τ12(φ), τ13(φ), and τ23(φ), and all these curves are highly asymmetric in nature. For the calculation of three-state ADT angles, we considered τ12(φ) as the free term and solved eq 14. The solution of the ADT angles γ12(φ), γ 13(φ), and γ23 (φ) is presented in panel c, and the corresponding diagonal elements of the ADT matrix are shown in panel d. Here the topological Berry phases α(3) 12 = 3.38 (3) rad, α(3) 23 = 3.32 rad, and α13 = 0.098 rad, indicate that in order to form the diabatic potentials, we have to consider at least three states. The values and the signs of the diagonal elements (3) of the D-matrix are given by D(3) 11 = −0.97, D22 = +0.96, and (3) D33 = −0.98; these results confirmed that the chosen configuration space contains (1,2) CI as well as (2,3) CI. From the PESs as well as NACTs calculations, one can conclude that in bent geometry configuration, (1,2) CI strongly influences the (2,3) CI. III.3. Effect of (1,2) CI on (2,3) CI. After confirming the situation that in bent geometry, the three low lying electronic states 12A′, 22A′, and 32A′ of neutral HCNH molecule are strongly interacting with one another and a three-state HSS may be formed, our next intention was to present the results of a detailed investigation of the effect of collinear (1,2) CI on the bent (2,3) CI. For this purpose, we took the position of the (2,3) CI as the center of the circular contour and kept on increasing q, the radius of the circle and performed the twostate as well as three-state HSSs calculations. It is noted that this systematic study helps to produce well-defined diabatic potentials. Figure 7a presents the plot of angular NACT τ23(φ) as a function of the rotation angle φ for several fixed q values. Since the center of all circular contours are located at the position of (2,3) CI, it is obvious that, irrespective of the radius of the circle, the NACT τ23(φ) should be quite large. From the figure, it is observed that as q increases, the coupling between the two states 22A′ and 32A′ increases, and it constantly shifts toward the lower φ value. Figure 7b shows the plot of the NACT τ12(φ) as a function of φ for the same q values. Since in the configuration space chosen, the distance between the (1,2) CI and (2,3) CI is ∼0.675 Å, as long as the radius q is small enough, the 22A′ and 32A′ states form a two-state HSS and the coupling between the 22A′ and 12A′ states is relatively small. However, as q increases, the magnitude of τ12(φ) increases and thus while q = 0.7 Å, τ12(φ) plays a significant role. Here also with increasing q, the NACT τ12(φ) shifts toward the lower φ value. It is to be noted that the plots of NACTs give only a qualitative idea about the interstate couplings in the given configuration space. Therefore, in order to understand its effect in a quantitative way, one has to consider the ADT angles and the corresponding D-matrix analysis. Figure 7c,d presents the three-state ADT angles γ23(φ) and γ12(φ), respectively, as a function of φ, as obtained from eq 14. From eq 15 for the twostate model, it is evident that once the mixing angle reaches the value of nπ/2 (n = integer), the two diabatic curves cross each other. In fact, this is true in general, and thus we may set the attainment of the value of the mixing angle close to nπ/2 as a criterion for the construction of well-behaved diabatic surface. With this insight, it is important to observe from Figure 7c, that irrespective of the radius of the circular contours, at φ = π,

Figure 6. Summary of results for the combination of (1,2) CI and (2,3) CI. Calculations are with a three-state HSS comprising the states 12A′, 22A′, and 32A′, along a circular contour of radius q = 0.7 Å, with center at RCH = 1.15 Å and θ = 30°, where both RNH and RCN are fixed at 0.8 Å and 1.138 Å, respectively. Panel a presents the three abovementioned PECs along the circular contour as a function of φ, the rotation angle. Panel b presents the angular NACTs τ12(φ), τ13(φ), and τ23(φ) along with magnified τ13(φ) in the inset; panels c and d present the ADT angles γ12(φ), γ13(φ), and γ23(φ), and the diagonal elements of the 3 × 3 ADT-matrix as a function of φ. G

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Table 1. Topological Berry Phases and Diagonal Elements of D-Matrix for Two- and Three-State HSS as a Function of q q/Å

α(2) 23 / rad

α(3) 23 / rad

α(3) 12 / rad

D(3) 11

D(3) 22

D(3) 33

0.1 0.3 0.5 0.7 0.9

3.1516 3.0481 2.9362 0.0379 0.4287

3.1548 3.0682 2.9818 3.3183 2.8878

−0.0028 0.0187 0.0828 3.3770 3.0249

1.0000 0.9981 0.9857 −0.9677 −0.9454

−0.9999 −0.9972 −0.9858 +0.9613 +0.9703

−0.9999 −0.9955 −0.9765 −0.9797 −0.9214

sufficient to form the single-valued diabatic potential as indicated by the value of the topological phase α(2) 23 ∼ π. However, as q increases, the Berry phase with the two-state HSS deviates from its quantized value of π. By including the effect of the ground 12A′ state and performing the three-state calculation, the three-state model stands out to be a better choice. Again, with q = 0.7 Å or more, the two-state model completely fails, thus indicating the presence of strong coupling among the three adjacent states. From Table 1, it is observed that the presence of two adjacent CIs may be concluded only from the three-state HSS calculation. Here all the topological phases and the corresponding diagonal elements of the Dmatrix have meaningful values. It is important to mention that as q increases, the Berry (topological) phase obtained with the three-state HSS model also slightly deviates from its quantized value. This deviation may be due to the influence of higher states. However, the inclusion of the effect of higher states increases the computational cost to a larger extent. So, considering all the aspects, one can easily neglect this slight deviation. III.4. Construction of Three-State Diabatic Potentials. One can employ eq 16 to construct the three-state diabatic potentials from the knowledge of the corresponding adiabatic PESs along with that of the three-state ADT angles γ12(φ), γ13(φ), and γ23(φ), obtained from eq 14 for several fixed q’s. Panels a−d of Figure 8 present the adiabatic PESs, uj (R|θ); j = 1,2,3 and the corresponding three-state diabatic PESs, vj (R|θ); j = 1,2,3 as a function of the C−H bond distance for four different θ values, θ = {0°,10°,20°,30°}, respectively. Since strictly diabatic states do not exist, 27 one can form quasidiabatic17 ones, where the choice of diabatization method plays a key role. A procedure of diabatization may be preferred where the diabatic potentials remain as close as possible to the corresponding adiabatic one, and irrespective of the restriction of symmetry, at the point of CI, the diabatic states clearly cross one another. It is important to note that in the adiabatic picture, HCNH molecule contains two adjacent CIs, located in different geometry configurations, namely, (1,2) CI in the collinear geometry and (2,3) CI in bent (θ = 30°) configuration. So, the corresponding diabatic potentials should cross each other at the respective CI points. It is observed that in panel a of Figure 8, the two diabatic potentials v3 and v1 cross one another exactly through the collinear (1,2) CI point. A similar feature is also observed in panel d of the same figure, where the two diabatic potentials v3 and v2 cross exactly through the bent (2,3) CI. Therefore, one can conclude that the three-state HSS picture nicely produces the corresponding three-state diabatic potentials where two adjacent CIs are strongly interacting with one another. In all four panels (a−d), it is observed that in the diabatic picture, the potential v3 is dissociative, whereas the potentials v1 and v2 are bound in nature. Thus, as depicted in Figure 8a, the diabatic picture shows that once HCNH molecule (in its linear geometry

Figure 7. Panels a−d represent the plots of angular NACTs τ23(φ), τ12(φ) and the three-state ADT angles γ23(φ) and γ12(φ), respectively, as a function of the rotation angle φ for five circular contours with center at the position of (2,3) CI.

γ23(φ) is close to π/2. This is expected as the contours with any q encompass the (2,3) CI. Again, as q increases, γ12(φ) also increases. Even for q = 0.5 Å, where the contour does not contain the (1,2) CI, at φ = π, γ12(φ) is relatively lower than π/ 2. However, once the contour contains (1,2) CI, γ12(φ) crosses the value π/2. For larger q value, at φ = π, it is close to π/2 and therefore for the construction of well-behaved diabatic surfaces, even corresponding to the adiabatic ones 22A′ and 32A′, one has to consider the state 12A′. Thus we get the proper diabatization only with the three-state HSS comprising the states 12A′, 22A′, and 32A′. For a more clear presentation, Table 1 depicts the relative adequacy of the two- and three-state HSSs with increasing q. It summarizes the results of topological Berry phases between the 22A′ and 32A′ states for two-state HSS (α(2) 23 ) as well as for the three-state HSS (α(3) ), that between the 12A′ and 22A′ states 23 (3) for three-state HSS (α12 ) ,and the corresponding diagonal elements of the three-state D-matrix. It is important to note that for small q values, up to q = 0.1 Å, two-state HSS is H

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irrespective of the ADT angles, two different sets of equations give the same values.

IV. CONCLUSION The investigations pursued in the present work are based on the systematic study of the adiabatic PESs of the three low-lying electronic states (12Π, 12Σ+, and 22Σ+) and the nonadiabatic effects among them for the planar HCNH molecule in linear as well as bent geometry configurations. Our main concern has been to reveal the exact location of the JT type CI between the two 2Σ+ states, through one of which the C−H bond of HCNH dissociates, leading to HNC, and through another the N−H bond dissociates to produce HCN. We, for the first time, report four new results: (a) Two Σ+ states 12Σ+ and 22Σ+, in some bent geometry of the HCNH molecule, evolve as 22A′ and 32A′ and only then couple strongly via (2,3) CI. (b) These two states are strongly coupled with the lower state 12A′, evolved from the linear state 12Π via (1,2) CI in a given region of configuration space and necessitates the consideration of three-state HSS. So, in order to form single-valued diabatic potentials, we have to consider at least three states. (c) The well-defined three-state diabatic potentials may be constructed by accounting for the effect of two adjacent CIs considering both the linear as well as bent geometry configurations of the neutral HCNH molecule. (d) For the DR reaction (HCNH+ + e−), consideration of only the linear geometry of the HCNH molecule is not sufficient; one has to consider the effect of bending motion in order to explain the bond dissociation process in diabatic picture properly. Several works have so far been performed based on the study of adiabatic and diabatic PESs of neutral HCNH molecule. The dynamical calculations of the bond dissociation processes on these diabatic potentials are also available in the literature. Most of the results obtained, to date, are only for linear HCNH molecules. The main reason for considering only the linear geometry has been that the equilibrium geometry of HCNH+ is linear, and the expectation that HCNH+ after absorbing an electron will produce the linear HCNH following Franck− Condon principle. In linear geometry (considering C2v point group) of HCNH molecule, the Π−Σ type of crossing is always symmetry-allowed in nature. So, in the diabatic picture for Σ−Σ type of crossing, the two-state diabatic model has been sufficient and has been explored to explain the abnormally almost equal abundance of HNC in comparison to HCN. However, in our present work, from the calculations of the adiabatic PESs for linear HCNH molecule, it has been concluded that the linear HCNH molecule contains only different symmetry (Π−Σ type) CIs and for observing the same symmetry (Σ−Σ type) CI, which play a key role in the dissociation process, and one must consider the bending motion. Since the equilibrium geometry of HCNH molecule is trans planar, the HCNH molecule, after its formation from HCNH+ by capturing a single electron, must have a strong tendency to attain the bent configuration. For bent geometry, the present study is limited for the configuration space where the C−N−H fragment is kept in its linear geometry configuration with fixed C−N distance RCN = 1.138 Å and fixed N−H distance RNH = 0.8 Å; the C−H bond length, RCH, as well as the bending angle, θ (the angle of the C−H bond

Figure 8. Panels a−d represent the adiabatic (uj(R|θ); j = 1,2,3) and diabatic (vj(R|θ); j = 1,2,3) PECs as a function of the C−H bond distance for four different θ values: θ = {0°,10°,20°,30°}, respectively, with the configuration space, where both RNH and RCN are fixed at 0.8 Å and 1.138 Å, respectively.

configuration) is in second excited v3 state, it will easily cross the first excited v2 state through a nonadiabatic transition around RCH ∼ 1.05 Å in virtue of the presence of bent (2,3) CI. Then again due to the presence of collinear (1,2) CI at RCH ∼ 1.35 Å, it crosses the ground v1 state. Finally, due to the dissociative nature of the v3 state, the C−H bond of the HCNH molecule dissociates, leading to the formation of HNC. Such dissociation is also possible in other bending configurations, as represented by Figures 8b−d. The adiabatic picture cannot explain this type of dissociation process. Since (2,3) CI is present in the bending geometry of the molecule, the probability of the formation of the HNC molecule through the C−H bond dissociation in such bent configurations is appreciable and acceptable from the consideration of energetics. It is important to mention here that the other set of equations for the construction of diabatic potentials for different combination of the ADT matrices, A = Q12Q23Q13, gives different values of three-state quantized ADT angles γ12(φ), γ13(φ), and γ23(φ). For the diabatic potentials, I

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(5) Watson, W. D. Ion−Molecule Reactions, Molecule Formation, and Hydrogen-Isotope Exchange in Dense Interstellar Clouds. Astrophys. J. 1974, 188, 35−42. (6) Ziurys, L. M.; Turner, B. E. HCNH+: A New Interstellar Molecular Ion. Astrophys. J. 1986, 302, L31−L36. (7) Herbst, E. What are the Products of Polyatomic Ion-Electron Dissociative Recombination Reactions? Astrophys. J. 1978, 222, 508− 516. (8) Brown, R. D.; Burden, F. R.; Cuno, A. Production of Interstellar HCN & HNC. Astrophys. J. 1989, 347, 855−858. (9) Talbi, D.; Ellinger, Y. Potential Energy Surfaces for the Electronic Dissociative Recombination of HCNH+: Astrophysical Implications on the HCN/HNC Abundance Ratio. Chem. Phys. Lett. 1998, 288, 155− 164. (10) Hickman, P.; Miles, R. D.; Hayden, C.; Talbi, D. Dissociative Recombination of e + HCNH+: Diabatic Potential Curves and Dynamics Calculations. Astron. Astrophys. 2005, 438, 31−37. (11) Tachikawa, H. Reaction Mechanism of the Astrochemical Electron Capture Reaction HCNH+ + e− → HNC+ H: A Direct AbInitio Dynamics Study. Phys. Chem. Chem. Phys. 1999, 1, 4925−4930. (12) Shiba, Y.; Hirano, T.; Nagashima, U.; Ishii, K. Potential Energy Surfaces and Branching Ratio of the Dissociative Recombination Reaction HCNH+ + e−: An Ab initio Molecular Orbital Study. J. Chem. Phys. 1998, 108, 698−705. (13) Ishii, K.; Tajima, A.; Taketsugu, T.; Yamashita, K. Theoretical Elucidation of the Unusually High [HNC]/[HCN] Abundance Ratio in Interstellar Space: Two-Dimensional and Two-State Quantum Wave Packet Dynamics Study on the Branching Ratio of the Dissociative Recombination Reaction HCNH+ + e− → HNC/HCN + H. Astrophys. J. 2006, 636, 927−931. (14) Semaniak, J.; Minaev, B. F.; Derkatch, A. M.; Hellberg, F.; Neau, A.; Rosen, S.; Thomas, R.; Larsson, M.; Danared, H.; Paal, A.; Ugglas, M. Dissociative Recombination of HCNH: Absolute Cross-Sections and Branching Ratios. Astrophys. J. Suppl. Ser. 2001, 135, 275−283. (15) Tachikawa, H.; Iyama, T.; Fukuzumi, T. Decomposition Dynamics of Interstellar HCNH: Ab-Initio MO and RRKM Studies. Astron. Astrophys. 2003, 397, 1−6. (16) Taketsugu, T.; Tajima, A.; Ishii, K.; Hirano, T. Ab Initio Direct Trajectory Simulation with Nonadiabatic Transitions of the Dissociative Recombination Reaction HCNH+ + e− → HNC/HCN + H. Astrophys. J. 2004, 608, 323−329. (17) Pacher, T.; Cederbaum, L. S.; Köppel, H. Approximately Diabatic States from Block Diagonallzation of the Electronic Hamiltonian. J. Chem. Phys. 1988, 89, 7367−7381. (18) Werner, H. -J.; Meyer, W. MCSCF Study of the Avoided Curve Crossing of the Two Lowest 1Σ+ States of LiF. J. Chem. Phys. 1981, 74, 5802−5807. (19) Langhoff, S. R.; Davidson, E. R. Configuration Interaction Calculations on the Nitrogen Molecule. Int. J. Quantum Chem. 1974, 8, 61−72. (20) Jahn, H. A.; Teller, E. Stability of Polyatomic Molecules in Degenerate Electronic States I - Orbital Degeneracy. Proc. R. Soc. London, Ser. A 1937, 161, 220−235. (21) Hobey, W. D.; Mclachlan, A. D. Dynamical Jahn−Teller Effect in Hydrocarbon Radicals. J. Chem. Phys. 1960, 33, 1695−1703. (22) Smith, F. T. Diabatic and Adiabatic Representations for Atomic Collision Problems. Phys. Rev. 1969, 179, 111−123. (23) Baer, M. Adiabatic and Diabatic Representations for AtomMolecule Collissions: Treatment of the Collinear Arrangement. Chem. Phys. Lett. 1975, 35, 112−118. (24) Longuet-Higgins, H. C. The Intersection of Potential Energy Surfaces in Polyatomic Molecules. Proc. R. Soc. London, Ser. A 1975, 344, 147−156. (25) Berry, M. V. Quantal Phase Factors Accompanying Adiabatic Changes. Proc. R. Soc. London, Ser. A 1984, 392, 45−57. (26) Baer, M.; Alijah, A. Quantized Non-adiabatic Coupling Terms to Ensure Diabatic Potentials. Chem. Phys. Lett. 2000, 319, 489−493.

with respect to its linear configuration), are used as variables. This study is applicable for several fixed N−H distances where the position of (2,3) CI shifts toward the higher bending angle as well as to the higher C−H distance, and the position of (1,2) CI shifts according to Figure 2a. It is observed that in the HCNH molecule where both the RNH (= 0.8 Å) and RCN (= 1.138 Å) are fixed in its linear configurations, the ground state 12Π forms (1,2) CI with the first excited 12Σ+ state at RCH = 1.35 Å with θ = 0°, whereas two 2 + Σ states, evolved as 22A′ and 32A′, form (2,3) CI at RCH = 1.15 Å with θ = 30°. The consideration of bending motion slightly complicates the overall situation. This is because, in planar bent configuration, i.e., in the Cs point group, the degenerate 12Π states split into A′ and A″ states, whereas 2Σ+ states have A′ character. So, it is expected that, in bent geometry, all the same symmetry A′ states would strongly interact with each other, i.e., diabatization should be considered with a choice of the threestate model space. This complicacy is nicely tackled if one considers the nonadiabatic effects using the D-matrix analysis, where one can easily incorporate the effect of more than two states, when it is important to incorporate. From the present study, it is observed that the three-state HSS picture nicely explains the formation of three-state well-defined diabatic potentials up to a quite large region q ≤ 1.0 Å. The three-state diabatic potentials considering both the linear as well as bent configurations elucidate the formation of HNC molecule by the C−H bond dissociation of the corresponding HCNH molecule. So, finally one can conclude that in the construction of a diabatic framework for dissociation of the HCNH molecule, in addition to the linear configuration, the bending configurations also play a major role in the formation of the HNC molecule.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.D. acknowledges CSIR, India, for a Senior Research Fellowship, and D.M. acknowledges BRNS, India, through project No. 2009/37/42/BRNS/2265.

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REFERENCES

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dx.doi.org/10.1021/jp403068v | J. Phys. Chem. A XXXX, XXX, XXX−XXX