Photodissociation of HCN and HNC isomers in the 7-10 eV energy range

Abstract

The ultraviolet photoabsorption spectra of the HCN and HNC isomers have been simulated in the 7-10 eV photon energy range. For this purpose the three-dimensional adiabatic potential energy surfaces of the 7 lowest electronic states, and the corresponding transition dipole moments, have been calculated, at multi reference configuration interaction level. The spectra are calculated with a quantum wave packet method on these adiabatic potential energy surfaces. The spectra for the 3 lower excited states, the dissociative electronic states, correspond essentially to predissociation peaks, most of them through tunneling on the same adiabatic state. The 3 higher electronic states are bound, hereafter electronic bound states, and their spectra consist of delta lines, in the adiabatic approximation. The radiative lifetime towards the ground electronic states of these bound states have been calculated, being longer than 10 ns in all cases, much longer that the characteristic predissociation lifetimes. The spectra of HCN is compared with the available experimental and previous theoretical simulations while in the case of HNC there are no previous studies to our knowledge. The spectrum for HNC is considerably more intense than that of HCN, which implies a much faster destruction of HNC than HCN in astrophysical environments illuminated by ultraviolet radiation.

I. Introduction

HCN is one of the most abundant polyatomic molecules in interstellar and circumstellar media[1, 2]. This molecule has two linear isomers, HCN and HNC, the latter being ≈ 0.6 eV above in energy than the former, with a barrier between them of ≈ 2 eV. The two isomers have been observed in many diverse astrophysical environments, such as diffuse and translucent interstellar clouds[3, 4], dense interstellar clouds[5–7], star forming regions[8, 9], protoplanetary disks[10], circumstellar envelopes around evolved stars[11, 12], the circumnuclear disk of the Galactic center[13], external galaxies[14–16], as well as in comets[17] and atmospheres of planets[18], showing that HCN and HNC are ubiquitous in space. The rovibrational levels of the two isomers have been measured with diverse experimental techniques[19–25], and the isomerization process has been the subject of numerous theoretical studies[26–32].

The HCN/HNC abundance ratio shows important changes between different astrophysical environments. To understand the underlying causes of these variations it is of paramount importance to determine the rates of all processes, formation, excitation and destruction, of the two isomers. The rates of many reactive collisional processes involving HCN and HNC have been revisited recently[33], and the rates of rotational excitation through inellastic collisions with H2 and He have been also investigated[34–36]. The photodissociation, however, has been only studied for HCN (see below) but not for the HNC isomer. Photoprocesses are likely to regulate to a large extent the HCN/HNC abundance ratio in those interstellar regions exposed to an intense ultraviolet radiation field, where HCN is found to be more abundant than HNC. In particular, the destruction of both isomers in such regions is dominated by photodissociation rather than by chemical reactions with radicals or ions. The aim of this work is to present a comparative study of the photoabsorption cross section of the two isomers HCN and HNC, to provide their photo-stability in astrophysical regions exposed to intense ultraviolet radiation.

There are several experimental works on individual electronic bands of HCN isomer. Herzberg and Hines[37] studied the weak absorption bands associated to the 1¹A″ and 2¹A′ electronic states. These bands are formed by narrow peaks, decaying by either electronic predissociation to the ground state or through tunneling across potential barriers due to avoided crossing with higher electronic states. These processes have been studied in detail by either experimental techniques[37–41] or theoretical simulations[42–46]. The band associated to 3¹A′ state was measured by Mcpherson and Simons[47] and the peaks where assigned[47–49]. Also, the photodissociation cross section for HCN was studied experimentally over a wide wavelength interval, from 90 to 150 nm, by several authors[50–52]. However, in none of these publications the photodissociation of the HNC isomer was studied.

The aim of this work is to provide a comparative study of the photoabsorption cross section of the two, HCN and HNC, isomers over a wide range of photon energy. For this purpose, the potential energy surface (PES) of the 7 lowest electronic states (4 of ¹A′ and 3 of ¹A″ symmetries) are calculated. On these adiabatic PES’s, the photoabsorption cross section is calculated using a quantum wave packet approach for the two isomers, HCN and HNC. The results have been compared with previous results when possible. For the 3 electronic bound states, 4 ¹A′ and 2,3 ¹A″, the adiabatic rovibrational bound levels have been calculated and their radiative lifetimes calculated.

This work is organized as follows. In section II the ab initio methods and results obtained for the lower 7 electronic states are presented. In section III a short description is presented of the methods used to calculate the rovibrational states and photodissociation dynamics, while the results obtained are shown and discussed in section IV. Finally, some conclusions are extracted in section V.

II. Potential energy surfaces

The PESs for the ground and excited states of the HCN/HNC system were obtained with the internally contracted multireference configuration interaction method (icMRCI) which employs wave functions that explicitly depend on the electron-electron distance (icMRCI-F12)[53], as implemented in the MOLPRO suite of programs[54]. This method has been proved to be efficient for achieving near Complete Basis Set limit correlation energies[55], even with small basis sets. In this calculations the correlation consistent F12 triple zeta basis sets[56], VTZ-F12, have been used.

The reference configurations for the icMRCI-F12 are obtained from a full valence state-averaged complete active space MCSCF calculation (SA-CASSCF) with the frozen core approximation. The 1s orbitals of C and N are doubly occupied in the SA-CASSCF references and not correlated in the icMRCI calculations. The SA-CASSCF wave functions are optimized in the C_s point group, including 8 states of symmetry ¹A′ and 6 states of symmetry ¹A″. This selection allows to describe the degenerate electronic states that appear at linear geometries, HCN and HNC, corresponding to Π or Δ states in C_∞v.

Finally, icMRCI-F12 for four states of symmetry ¹A′ and three of symmetry ¹A″, including single and double excitations from the configuration state functions obtained in the SA-CASSCF calculations, have been done. With these state selections the icMRCI-F12 calculations involve a number of contracted (uncontracted) configurations of the order of 2.6 × 10⁶ (42 × 10⁶) for ¹A′ states and 2.1 × 10⁶ (40 × 10⁶) for ¹A″ states.

To obtain 3D PESs for the HCN/HNC system, ab initio calculations have been performed over a large grid in Jacobi coordinates (r, R), being r the CN vector and R the vector joining the CN center-of-mass (we have used the masses of the ¹²C and ¹⁴N) and the H. The Jacobi angle γ is defined by the scalar product as r R cos(γ) = r · R. With this selection, γ = 0 corresponds to linear C-N-H (HNC isomer) and γ = π to linear H-C-N (HCN isomer). The (r, R, γ) grid of 15504 points used is defined as

$$r=0.5,0.9,0.95,1.0,1.05,1.10,1.153,1.2,1.3,1.4,1.5,1.6,1.8,2.1,2.5and3.0a_0$$ (16 values),

$$R=0.00-5.00 a_0$$ (51 values),

$$\gamma =0-\pi$$ (19 values).

Using this grid, the ab initio icMRCI-F12 energies for the seven singlet electronic states have been fitted using 3D cubic splines, using the DB3INK/DB3VAL subroutines based on the method of de Boor[57] and distributed by GAMS[58]. Sathyamurphy and Raff[59] investigated the use of 1D, 2D and 3D splines fit of ab initio data, presenting the first full 3D cubic splines fit to a triatomic surface. They study the adequacy of a 3D cubic spline PES in quasiclassical trajectory studies, and found that average magnitudes, like total reaction cross sections or energy partitioning distributions, were in good agreement with that obtained with the full analytic PES used as benchmark.

To analyze the accuracy of the cubic splines we have performed several tests with less ab initio geometries, using the non-fitted points to estimate the root-mean-square (rms) error. Using about 2800-3100 non-fitted geometries with an energy lower than 12 eV (taking the zero of energy in the HCN minima of the ground electronic state $X^1{\sum }_g^{+}$), we have obtained an estimated rms error of 0.022, 0.029, 0.040, 0.027 eV for the X²A′, 2²A′, 3²A′ and 1²A″ electronic states, respectively. The higher excited electronic states have many avoided crossings and, as a consequence, the estimated rms error grows up to 0.058-0.074 eV.

For describing the transition dipole moments between the ground and excited electronic states, we have also used a 3D cubic splines. The sign of the transition dipole moments depends on the relative phase of the two electronic wavefunction involved. To obtain a continuous transition dipole moments, we have corrected the relative sign of the eigenstates among successive points along the γ coordinate, calculating the overlap of the wave functions in two grid points and imposing that the diagonal matrix elements of overlap matrix to be positive. The transition dipole moments vary a lot with the internal coordinates, and present sudden changes in the regions of avoided crossings and conical intersections, as shown in Fig. 2. This introduces an extra difficulty in the interpolation. We have estimated the rms error using 900 extra points, not included in the interpolation, finding an error of ≈ 0.09 ea₀. In any case, it should be noted that the photodissociation cross section is only affected by by the value of the transition moments in the regions regions covered by the bound eqigenfunctions on the X electronic state. Several interpolation and fitting schemes have been checked, all finding rather similar absorption spectra.

Fig. 2.

Angular dependence of the potential of the lower electronic states of HCN for r = 1.153 Å and R = 1.7 Å(bottom panel) and of the transition electric dipole moments between the ground and few excited electronic states (top panel). The symmetry at the two linear geometries are also shown.

A. Topology of the PES

Schwenzer et al. study low lying electronic states for HCN[60] and HNC[61], using Configuration Interaction that include all single excitations with respect to a single configuration selected to describe each electronic state. With this approach they obtained the geometry and electronic excitation energies T_e for both isomers. Peric et al.[62] obtained the PES’s of the valence-type singlet electronic states of HCN, using Multireference Single and Double excitation Configuration Interaction (MRDCI) approach. Later they used this PES’s to study the vibrational structure of the 1¹Σ⁻ ← X¹Σ⁺ and 1¹Δ ← X¹Σ⁺[49] and 1¹Π ← X¹Σ⁺[42] transitions. More recently, Xu et al. obtain the 1¹A″[43, 44] and 2¹A′[45] PES, and use them to study the predissociation dynamics of the HCN/DCN. These PES’s do not include the region of the hydrogen isocyanide HNC.

The main electronic configurations of the four ¹A′ and three ¹A″ electronic states are shown in Table I. Dissociative electronic states are 2,3¹A′ and 1¹A″ states, whose main electronic configuration correspond to the excited electron on a 7a′ orbital. Electronic bound states are 4¹A′ and 2,3¹A″, corresponding to excitations to the 8a′, 2a″ or higher molecular orbitals. These results are in agreement with those reported by Schwenzer et al.[60, 61] and Nayak et al.[63]. Also the electronic excitation energies for HCN, T₀, are compared with the experimental results[64], showing a good agreement. Because we are interested in study the correlation with the linear conformations, we have included the C_∞v symmetry for both isomers, HCN and HNC. The minimum of the ground electronic state, X¹Σ⁺, corresponds to the HCN linear conformation (γ = π), and has an electronic configuration … 5a′²6a′²1a″² ≡ … 5σ²1π⁴. In the HNC isomer (γ = 0), the electronic configuration is … 5a′²6a′²1a″² ≡ … 1π⁴5σ².

Table 1.

Electronic excitation energies and correlation of the PES with the linear HCN and HNC isomers.

State			Main configurations*		Te(eV)	T0(eV)		Te(eV)
Cs	HCN	HNC			HCN	Calc.	Exp.a	HNC(Calc.)
11A′	X1Σ+	X1Σ+	… 5a′26a′21a″2		0	(0.430)b	—	(0.323)c
21A′	11∆	21Σ+	…5a′26a′ 1a″27a′		7.04	6.88	6.77	6.47
			…5a′6a′2 1a″27a′
			…5a′26a′2 1a″	2a″
31A′	11Π	11Π	…5a′26a′1a″27a′		8.29	8.16	8.14	7.94
			…5a′6a′21a″27a′
			…5a′26a′21a″2	2a″
41A′	21Π	21Π	…5a′26a′ 1a″2	8a′	9.36
11A″	11Σ–	11Π	…5a′6a′21a″7a′		6.56	6.41	6.48	6.19
21A″	11∆	21Π	…5a′26a′1a″2	2a″	7.81	7.70		7.48
			…5a′26a′21a″	7a′
31A″	11Π	31Π	…5a′6a′21a″2	2a″	9.05	9.01	8.88	8.25

^*…refers to doubly occupied orbitals 1a^′22a^′23a^′24a^′2.

^aExperimental values taken from Ref.[64].

^bZero point energy from Ref.[31].

^cZero point energy from Ref.[31], taken from the minimum of the HNC isomer, 0.657 eV over the HCN minimum.

Therefore, the symmetry of the 6a′ orbital changes from π in HCN to σ in HNC (while the 5a′ experience the opposite behavior), as illustrated in Fig. 1, and their energies also cross. Such change, explains the change of the electronic configuration from HCN and HNC, and, therefore, the transition dipole moments involving the ground electronic state, as explained below.

Fig. 1.

Molecular orbitals of HCN at r = 1.153 Å and R = 1.7Å for different angles, γ.

On the contrary, the 7a′ and 2a″ keep the same character for HCN and HNC, as can be seen in Fig. 1. In these two orbitals, of π character at linear geometries, the 1s orbital of H can not contribute. This changes at bent geometries, where symmetry restrictions disappears for the a′ orbitals, and the 1s orbital of H has a notorious contribution. As a consequence, the states with a dominant configuration including the 7a′ orbital, have negative charge on the H atom, while for the others this charge is positive, calculated with Mulliken population analysis. This difference in the electronic population on H produces a change of sign in the permanent electric dipole of the different electronic states.

The changes of the molecular orbitals produce a complex structure of crossings of electronic states, as shown in Fig. 2 as a function of the angle. It can be seen that the projection of the electronic orbital angular momentum Λ changes from HCN to HNC. For example, at HCN the third state is of Δ symmetry, while this is not the case for HNC. This is due to a sequence of avoided crossings clearly seen in the 0 < γ < 20° interval. This has important implications because Σ⁺ → Σ⁻, Δ transitions are forbidden for electric dipole transitions. Thus for the HCN isomer the 1¹A″ electronic state, correlating to ¹Σ⁻ and the 2¹A′ and 2¹A″ electronic states, correlating to ¹Δ, show a very small electric dipole transition moments from the ground electronic state, while for the HNC this is not the case. This change of symmetry from the HCN isomer to the HNC isomer is a consequence of the orbital crossing between the 1π and 5σ orbitals. For example, the first 1¹A″ excited state (… ¹a″7a′ electronic configuration) correlates with 1¹Σ⁻ HCN electronic state (… 5σ²1π³2π) and 1¹Π HNC electronic state (… 1π⁴5σ2π).

The HCN/HNC PES for several electronic states of ¹A′ and ¹A″ symmetry are shown in Fig. 3 as a function of R for r=1.153 Å and HCN (left panel) and HNC (right panel). The middle panel shows the electronic states of CN as a function of CN internuclear distance, r, which in the other two panels was fixed at the equilibrium distance of CN in the ground electronic state of HCN(X¹Σ⁺). The ground 1¹A′ state correlates to CN(X²Σ⁺) + H(²S) and is deeply bound, by ≈ 5.5 eV, presenting the deepest well for the HCN isomer and γ=π.

Fig. 3.

Potential energy cuts of isolated CN as a function of internuclear distance, r,(central panel), and linear HCN (left panel) and HNC (right panel) as a function of the distance, R, between H and the center-of-mass of CN (kept frozen at r = 1.153 Å).

The first two excited electronic states, 2¹A′ and 1¹A″, correlates with the first excited state CN(A²Π) + H(²S). These two states are essentially dissociative down to distances of R ≈ 1.5 Å, where they cross with other excited states correlating to excited states of CN, D²Π or higher. This avoided crossing originates a barrier with a local well, whose minimum is higher than the dissociation energy. The crossing depends on the angle γ leading to different well depths for HCN and HNC isomers.

The 3¹A′ state correlates to CN(B²Σ⁺) + H(²S) and is repulsive for HCN and attractive for HNC, leading to avoided crossings with the higher electronic states at rather different distances. These avoided crossings give rise to a rather deep well in the HNC isomer, and not for HCN isomers.

The higher electronic states correlate with excited electronic states of CN, leading to dissociation threshold above 12 eV over the minimum of the HCN(X¹A′) state. These states present deep wells for HCN and HNC isomers.

The wells of the different electronic states are clearly seen in Fig. 4, where the contour plots of the lower 7 potential energy surfaces calculated here are shown as a function of X = R cos γ and Y = R sin γ. The C and N atoms are placed in the X axis (Y =0) at X = −0.621 Å and X = 0.532 Å, respectively.

Fig. 4.

Contour plots of the potential energy surfaces calculated in this work as a function of Rcos γ and Rsin γ, for fixed CN internuclear distance, r = 1.153Å. Energies (in eV) are referred to the minimum of the HCN well of the ground state.

There are many avoided crossings among the adiabatic states as shown in Fig. 3. At these crossings non adiabatic couplings are expected to play an important role in the photodissociation dynamics, which will be neglected in the present study. These effects can be classified as bound-bound interaction, producing a shift and broadening of peaks, and bound-free transitions, producing the dissociation of adiabatically bound levels, and hence the broadening. These two aspects are mentioned when discussing the dynamical results below.

III. Dynamical methods

The dynamical calculations have been performed in Jacobi coordinates, as described above. A body-fixed frame, defined by three Euler angles (φ, θ, χ), is used in which the three atoms are in the x-z plane, with the z-axis being parallel to R, and r_x = sin γ being possitive. Thus the wave functions are expressed as

$${\text{Ψ}}_k^{J\lambda M\epsilon }\left(\mathrm{r,}R\right)={\displaystyle \sum _{\Omega }\frac{{\text{Φ}}_k^{J\epsilon \lambda \text{Ω}}\left(r,R,\hspace{0.17em}\text{γ}\right)}{rR}}{W}_{M\text{Ω}}^{J\epsilon }\left(\text{φ},\theta ,\chi \right)|\lambda \rangle ,$$ (1)

where J is the quantum number associated to the angular momentum operator J, with projections M and Ω on the space-fixed and body-fixed z-axis, respectively, and $|\lambda \rangle$ is the adiabatic electronic wave function. ε = ±1 is the parity under spatial inversion of coordinates. The number k denotes all the required quantum numbers to characterize a state. Finally

$$\begin{array}{l}{W}_{M\text{Ω}}^{J\epsilon }\left(\phi ,\theta ,\chi \right)=\sqrt{\frac{2J+1}{16{\pi }^2\left(1+\delta \text{Ω,0}\right)}}\\ \times \left[D_{M,\text{Ω}}^{J_{*}}\left(\phi ,\theta ,\chi \right)+\epsilon {\left(-1\right)}^{J+\text{Ω}}{D}_{M,-\text{Ω}}^{J_{*}}\left(\phi ,\theta ,\chi \right)\right]\end{array}$$ (2)

are parity adapted functions, with $D_{M,\Omega }^J$ being Wigner rotation matrices[65] corresponding to a total angular momentum J. The ${\Phi }_k^{J\epsilon \lambda \Omega }\left(\mathrm{r,\; R,}\gamma \right)$ coefficients are represented in grids: for the radial variables they consist of equi-spaced points, while for γ a Gauss-Legendre quadrature is used. These coefficients are calculated numerically with the program MADWAVE3[66, 67] as implemented for photo-initiated processes, for infrared[68], electronic[69] and photodetachment[70] processes in one or several electronic states. Here only some details of the calculations are presented, separately for bound states and time propagation calculations.

A. Bound states

The bound states are calculated in the mixed grid/basis representation using a non-orthogonal iterative Lanczos procedure[71] in two steps. The eigenvalues $E_k^{\lambda }$ are first obtained with a non-orthogonal Lanczos procedure following the method of Cullum and Willoughby[72]. The eigenstates are then obtained iteratively using the conjugate gradient method[73, 74]. The grid is formed by 64 and 256 equally spaced points in r and R, respectively, in the intervals 0.5 ≤ r ≤ 3 Å and 0.001 ≤ r ≤ 6 Å. A Gauss-Legendre quadrature is chosen for the angle γ formed by 200 points. Most of the calculations presented below are restricted to J=0 and 1. Also, the calculations are performed on a single adiabatic electronic state $|\lambda \rangle$.

The use of a grid facilitates the calculation of very excited states. The action of the Hamiltonian on this grid representation is performed using Sine Fourier transforms for the radial derivatives and the angular kinetic operator is evaluated as described previously[75], as implemented in the MADWAVE3 code described elsewhere[66, 67].

B. Wave packet calculations

The total absorption cross section for electric dipole transitions from initial state k, in the electronic state $|{\lambda }_i\rangle$ and angular momentum J_i, to an excited electronic state $|\lambda \rangle$ and total angular momentum J, in a first order perturbative treatment is given by

$$\sigma \left(hv\right)=Ahv{\displaystyle \sum _{Mi}\frac{1}{2J_i+1}}{\displaystyle \sum _{\alpha }|\langle {\Psi }_{E\alpha }^{J\lambda M\epsilon }|\mathbf{\text{d}}\cdot \mathbf{\text{e}}{|{\Psi }_k^{J_i{\lambda }_i{M}_i{\epsilon }_i}\rangle |}^2,}$$ (3)

where $E=E_k^{{\lambda }_i}+hv,A=1/{\hslash }^2{\epsilon }_{0}c,$ d is the electric dipole of the molecule, e is an unitary vector parallel to the electric field of the radiation, and ${\Psi }_{E\alpha }^{J\lambda M\epsilon }$ is a dissociative wave function of energy E leading to H+CN fragments in the rovibrational level α ≡ v, j of the CN fragments in the electronic state correlating to state λis the electric transition dipole moment of HCN. Within a time-dependent framework this expression transforms to

$$\sigma (hv)=\frac{Ahv}{2\pi \hslash }\frac{2J+1}{3}{\displaystyle \int dt{e}^{iEt/\hslash }\langle {\Psi }_{t=0}^{J\lambda \epsilon }|{\Psi }_t^{J\lambda \epsilon }\rangle },$$ (4)

where linear polarization of the radiation has been assumed. The components of the initial wave packet in Eq. (1) are given by[68, 69]

$$\begin{array}{l}{\Phi }_{t=0}^{J\epsilon \lambda \Omega }\left(r,R,\text{γ}\right)={\displaystyle \sum _{q\Omega i}{\Phi }_k^{J_i{\epsilon }_i{\lambda }_i{\Omega }_i}(r,R,\text{γ})d_q^{\lambda {\lambda }_i}(r,R,\text{γ})}\\ \times \frac{{\left(-1\right)}^{q-{\Omega }_i}(1-\epsilon {\epsilon }_i)}{2\sqrt{\left(1+\delta {\Omega }_0\right)\left(1+\delta {\Omega }_i\right)}}\\ \times \left\{\left(\begin{array}{ccc}J& 1& J_i\\ \Omega & -q& -{\Omega }_i\end{array}\right)+{\epsilon }_i{\left(-1\right)}^{j_i}\left(\begin{array}{ccc}J& 1& J_i\\ \Omega & -q& {\Omega }_i\end{array}\right)\right\},\end{array}$$ (5)

where $d_q^{\lambda {\lambda }_i}\left(r,R,\text{γ}\right)$ is the electric transition dipole moment matrix elements between the electronic states $|{\lambda }_i\rangle$ and $|\lambda \rangle$

The wave packet is propagated with a modified Chebyshev integrator [66, 76–81]. The grid used is that used for the bound state calculations described above, except in the number of points in the grid of R, which is duplicated using the same radial intervals. The wave packet is absorbed in the interval 5 < R < 12Å, with the function exp[−0.0001(R − 5)⁴], to avoid reflexions when the wave packet reaches the edges of the grid. For dissociative electronic states the propagation is performed for 50000 Chebyshev iterations, while for bound electronic states the propagation is continued until 100000 iterations. The autocorrelation function in Eq.(4) is multiplied by an exponential function e^−2Γt, leading to a lorentzian broadening of the narrow lines of the spectra, of width Γ ≈ 0.5 meV in the present case.

The narrow resonances can be assigned from the analysis of the wave function at the energy of the peak using a pseudo-spectral method[68, 82, 83], which in the case of using a modified Chebyshev propagator is implemented in Refs.[66, 84].

C. Radiative lifetimes

The excited electronic bound states present deep wells, with bound or quasi-bound states. Without considering non-adiabatic transitions, these states can only disappear by non stimulated emission towards lower electronic states. The radiative lifetime of these states are given by the inverse of the Einstein coefficients, which are calculated in a first-order perturbative treatment as

$$\begin{array}{l}{A}^{J\lambda \epsilon k}={\displaystyle \sum _{M{M}_f}{\displaystyle \sum _{J_f{\lambda }_f{\epsilon }_f{k}_f}\frac{1}{3\pi {\epsilon }_0{\hslash }^4}{\left(\frac{hv}{c}\right)}^3}}\\ \times {|\langle {\Psi }_{k_f}^{J_f{\lambda }_f{M}_f{\epsilon }_f}|\text{d}\cdot \text{e}|{\Psi }_k^{J\lambda M\epsilon }\rangle |}^2.\end{array}$$ (6)

Here we shall only consider transitions from excited electronic states to the ground one, between the J = 0 and J_f = 1. In addition, we will classify the final levels on the ground electronic state corresponding to the HCN and CNH isomers. Thus, it will be possible to study the photoisomerization, i.e. the isomerization after electronic excitation and subsequent emission.

Iv. Dynamical results and discussion

A. Vibrational levels on the ground state

There are several previous PES’s for the ground electronic state of HCN[29, 31, 85, 86]. This state has a double minimum at collinear geometries corresponding to HCN and HNC isomers and its spectroscopy has been widely studied theoretically[29–31]. The shape of this potential has been also studied with a valence-bond curve crossing model, and the angular dependence explained as the result of avoided-crossings among the three lower diabatic curves[87].

The angular minimum energy path (MEP) from HCN to HNC isomers for the ground PES obtained in this work is compared, in Fig.5, to those obtained using the two most recent PES’s of Mourik et al.[29] (hereafter VQZANO+ PES) and Varandas and Rodrigues[31] (hereafter VR PES). All the three MEP’s show a rather similar behavior. The present PES has a barrier height slight lower than the VR PES, and both of them lower than the VQZANO+ PES. The HNC minimum energy of the VQZANO+ and VR are very close, and both slightly lower than the present PES.

Fig. 5.

Minimum energy path for the HCN/HNC isomerization in the HCN(X) PES of the present work, compared with those obtained from the PES of Refs.[29] and [31]. The points correspond to the 270 energy levels calculated with the VR PES [31] for J = 0, and the angle corresponds to the average of γ of each vibrational state.

The bound states on the three PES’s have been calculated. The energies and assignment of the bound states obtained with the VQZANO+ and VR PES’s are essentially equal to those reported by those authors[29, 31], with differences of tenths of wavenumbers or less for the first levels. The results obtained with the present PES are slightly different. The lower HCN bound state are 0.4306, 0.4316, 0.4239 eV for the AR, VQZANO+ and the present PES, repectively. For HNC the energies of the first level are 1.0749, 1.0746, 1.0845 eV for the AR, VQZANO+ and the present PES, repectively, with respect to the zero of energy corresponding to the HCN equilibrium geometry.

The present results are slightly different from the previous ones. The differences arise from the ab initio calculation. In the present work we have calculated several electronic states, optimizing the molecular orbitals with a CASSCF procedure for an average of all them together. This generates a very large active space and it is necessary to reduce it by using a frozen core approximation. For this reason hereafter we shall consider the bound states on the VR PES, which is considered to be the most accurate one.

In Fig.5 we also show the energies of the first 270 vibrational levels as a function of the average value (γ), calculated with the VR PES. For energies below ≈ 2.3 eV, the average value of angle remains in either the HCN or the HNC wells, and varies with the bending excitation of the state. The bound states can be assigned to each isomer even above the isomerization barrier, of ≈ 2.1 eV, because of the zero-point energy. Above 2.4 eV the bound states start to be located on both wells but there are still some of them being associated to one of the two isomers. Note, that the (γ) can be used to assign the bound state to each of the isomers.

B. Photoabsorption of the electronic dissociative states

The photodissociation spectra from the HCN (k=1, at E=0.43 eV) and HNC (k=11, at E= 1.07 eV) vibrational states on the ground HCN(X¹A′) state towards each of the 6 excited electronic states are shown in Fig. 6, for the J_i = 0 → J = 1 rotational transition.

Fig. 6.

Absorption spectrum for the J_i = 0 → J = 1 transition from the ground electronic state to each of the 6 excited electronic states of HCN. Transitions from the first vibrational level of HCN and HNC isomers (vibrational levels k=1 and 11, respectivelly) are shown in red and blue lines, respectively. The arrow indicates the dissociation energy leading to CN(v=0,j=0) fragments. The zero of energy is at the HCN equilibrium configuration of the ground electronic state.

The arrow in each panel indicates the dissociation threshold on each electronic state and allows to classify the spectra in three groups, according to the Franck-Condon region where the initial wave packet lies (see Fig. 4). For the 2¹A′ and 1¹A″ states, the absorption spectra are above the CN(A²Π,v=0,j=0)+H(²S) dissociation threshold (see Fig.3). The spectra for 3¹A′ is formed by a broad envelope of narrow peaks associated to resonances (above the threshold) and bound states (below the threshold) within the Born-Oppenheimer approximation. Finally, the spectra for the 4¹A′, 2¹A″ and 3¹A″ electronic bound states are well below their corresponding dissociation threshold and consist of delta functions associated to bound states, because non-adiabatic transitions are neglected in this work. These peaks are broadened by ≈ 0.5 meV obtained by multiplying the autocorrelation function by a exponential function, as described above.

The main difference between HCN and HNC spectra is in the intensity for the 2¹A′, 1¹A″ and 2¹A″ states. In these three cases the absorption cross section for HCN isomer is nearly 2 orders of magnitude lower than for the HNC isomer (note that in the figure the corresponding spectra for HCN are multiplied by 50). This lower intensity is due to the symmetry of the electronic states at linear HCN and HNC geometries, leading to very different transition dipole moments. The 1¹A″ state correlates to a ¹Σ⁻ in the HCN linear configuration while it correlates to ¹Π state in the HNC one (see Fig. 2). At collinear geometries, Σ⁺ → Σ⁻ transitions are forbidden while Σ⁺ → Π are allowed, explaining why the spectrum of HCN(1¹A″) is less intense than that of HNC(1¹A″). Similar arguments hold for 2¹A′ correlating to ¹Δ and ¹Σ⁺ for HCN and HNC isomers, and for 2¹A″ correlating to ¹Δ and ¹Π symmetries, as shown in Fig. 2.

The weak absorption bands between 200 and 160 nm from HCN isomer was first assigned by Herzberg and Hines[37] to bending progressions in the 1¹A″ state, due to a linear to bent transition. These bands corresponds to predissociative peaks[37–41]. Below the CN(A²Π,v=0)+ H(^S) threshold these resonances can only decay by non-adiabatic electronic predissociation and they are extremely narrow. Above this threshold, the predissociation is through a barrier and their widths are larger, depending on the nature of the resonance. These bands have been studied extensively as reviewed in a series of work by Xu et al.[43–46].

The absorption spectrum to the 1¹A″ state for the HCN isomer is shown in the top panel of Fig. 7, and can be compared with that in Fig. 4 in Ref.[44]. The differences are attributed to differences in the PES’s and the transition electric dipole used. For example, Xu et al.[44] neglected the overall rotation and the 2¹A′ ← 1¹A′ transition was approximated by a simple sum of the x and y contributions. Also, here we use as zero of energy the CN(A²Π, v = 0, j = 0) + H threshold, and the autocorrelation function is multiplied by a decaying exponential function which introduces a broadening in the lower peaks, below 1 eV of kinetic energy, of approx. 0.5 meV. This broadening reduces the height of the narrowest peaks, and this effect depends on the real width of the levels, but the total integrated intensity along the whole absorption profile for each transition remains the same. The peaks above 1 eV are broader, indicating a fast predissociation rate. As a consequence, as energy increases the wave function for the resonances have a larger continuum contribution, as can be observed in the right panels of Fig. 7 for large R values.

Fig. 7.

Absorption spectrum for the Ji = 0 → J = 1 transition from the HCN isomer in the ground electronic state to the 1¹ A″ from the HCN (top panel) and HNC (bottom) isomers. Energy is referred to the CN(A²Π, v = 0, j = 0) threshold, at E=6.82 eV with respect to HCN equilibrium configuration on the ground electronic state. The contour plots of the wave functions at the energies of some of the resonances and r = 1.2Å are shown in the middle panels. The distribution on r, R for each of the resonances are shown in the right panels, integrating over the angle γ.

As suggested by other authors before[37–41, 43–46], the peaks of the HCN(1¹A″) are a bending progression, on the HCN side. The peaks labeled with b=1, 5 and 8 show the corresponding nodal structure in the middle-right panels of Fig. 7. Also, the distribution in r and R show little excitation. For kinetic energies below 0.7 eV, the lower peaks also show a bending progression but with some excitation in r or R. As energy increases, all the motions start mixing, and that’s why for b =5 and 8, in the right panels there is a small contribution from a vibrational quantum in R.

A similar analysis has been done for the HNC isomer on the same 1¹A″ state in the bottom panels of Fig. 7. As commented above, this spectrum is much more intense and could be also assigned to a bending progression. However, the analysis of the wave function at the peaks demonstrate that they correspond to highly excited bending levels on the HNC side but they also show a wide distribution on r and R. In this case the motions on the three internal coordinates are more mixed, making more difficult their assignment. Note, that the angular distribution in the middle panels are on the HNC side, and do not coincide with the levels reached for the HCN isomer. The potential of the 1¹A″ does not show a well for r=1.153Å in Fig.4, but for longer r values it does.

The spectra for HCN and HNC isomers on the 3¹A′ state are both of similar intensity and formed of relatively narrow peaks, distributed below and above the CN(B²Σ⁺, v = 0) + H(²S) dissociation threshold. The spectrum for HCN isomer has been measured by Mcpherson and Simons[47] who assigned the peaks to bending and H-CN stretch. The spectrum simulated in this work for HCN isomer is shown in the left top panel of Fig. 8, together with the plot of the probability associated to some selected resonances, in the right panels. In all the cases, the nodal structure of the wave functions correspond to the ground level of the CN vibration, and different excitation on γ and R coordinates, in good agreement with previous assignments[47, 48]. Also, the possition and spacing of the peaks approximately coincides with the experimental ones.

Fig. 8.

Absorption spectrum for the J_i = 0 → J = 1 transition from the HCN isomer in the ground electronic state to the 3¹A′ from the HCN (top panel) and HNC (bottom) isomers. Energy is referred to the CN(B²Σ⁻, v = 0, j = 0) threshold, at 8.93 eV with respect to HCN equilibirum configuration on the ground electronic state. The contour plots of the wave functions at the energies of some of the resonances and r = 1.2Å are shown in the middle panels. The distribution on r, R for each of the resonances are shown in the right panels, integrating over the angle γ.

The peaks below the zero of energy, must dissociate by non-adiabatic transitions towards CN(X²Σ⁻ or A²Π) + H(²S) states. Above the zero of energy, the width of the peaks increases considerably indicating a fast dissociation on the same adiabatic 3¹A′ leading to CN(B²Σ⁻) fragments. The width of the peaks vary depending on the nature of the bending of H-CN stretching as found experimentally[47]. For example, the peak labeled b = 4 has a halfwidth-halfmaximum of Γ=0.6 meV, close to the artificial broadening introduced by multiplying the autocorrelation function by a decaying exponential function. This indicates that the predissociation rate is rather low, lower than this value. The peak labeled b = 5, is broader with Γ=1.1 meV, and the peak labeled with b = *, at 0.273 eV, has a width of Γ=1.3 meV. All these states present a rather long lifetime because as bending excitation increases the predissociation slows down, as found previously[47, 48]. The lower peaks of the spectrum in Fig. 8, correspond to H-CN excitation, or combination of all the modes, and present broader widths.

The resonances reached from the excitation of HCN(X) to the 3¹A′ state are supported by a small well that is show in Fig. 4 for R cos γ < 0, the HCN side. There is a deeper well on the HNC side (R cos γ > 0) but is placed at longer R distances, R >2Å. For this reason the spectrum of the HNC isomer to the 3¹A′ state is shifted to higher energies, being nearly all the peaks above the CN(B²Σ⁻, v = 0, j = 0) + H(²S) threshold. Nevertheless, the peaks below 0.5 eV can be nicely associated to a bending progression. Above 0.5 eV, all the modes are more mixed and the contribution from the continuum is more difficult to separate, making difficult their assignment.

These quasi-bound states can predissociate, and their widths increase with energy. Level b = 1, at 9.127 meV above the threshold, has a width of Γ = 2.6 meV, while for b = 3 is Γ = 3.5 meV. The peak labelled with * is mixed with some other lower peaks and it is difficult to estimate its width. Nevertheless it is broader, indicating that as available energy increases the width also increases.

The absorption spectrum from HCN and HNC isomers to the 3¹A′ state in Fig. 8 show a comparable intensity, and the bending progression have similar spacings, being ≈ 0.095 eV and 0.109 eV for HCN and HNC isomers. However, the resonances reached in the excitation from the two isomer are different, due to Franck-Condon arguments.

C. Photoabsorption to the electronic bound states

The absorption towards the 4¹A′ and 2,3 ¹A″ electronic bound states is always below their adiabatic dissociation threshold, i.e., towards the bound states on each adiabatic surface. In all the cases there are separate wells corresponding to HCN and HNC isomers, but the MEPs, in Fig. 9, are in general more complex. For the 4¹A′ there are many minima and maxima as a consequence of the high number of curve crossing, and the two isomers with approximately the same well depth. The 2¹A″ state presents a double well similar to that of the ground state, with HCN being the deepest one. However the energy difference between HCN/HNC isomers is smaller and the barrier is approximately a half. Finally, the 3¹A″ state presents a third well for γ ≈ π/2, less deep than the HCN and HNC wells, which have nearly the same energy.

Fig. 9.

Minimum energy path for the HCN/HNC isomerization for 4¹A′ and 2,3 ¹A″ electronic bound states and first 100 bound states as a function of the angle (γ)

The energy of the first 120 bound states are also shown in Fig. 9. As for the ground electronic state, these bound states are clearly localized either for the HCN or the HNC isomers, except for the 3¹A″ where few states are located in the well at γ ≈ π/2. In all the cases, the energies of the bound states are not as regular as in the ground electronic states. The reason is that the vibrations are not so separable, presenting a more complex distribution of nodes. These make more difficult the assignment of approximate quantum numbers.

These localized character of the bound states makes that the absorption from one of the two HCN or HNC isomers on the ground state reaches bound states on the excited state corresponding to the same isomer. These states should decay by through non-adiabatic couplings towards the lower dissociative states to produce CN in X(²Σ⁺), A(²Π) or/and B(²Σ⁺), currently under study. Alternatively, they could decay through non-stimulated emission towards lower states as described below.

D. Total Absorption Spectra

There are several experimental works on the HCN absorption spectrum in a broad interval of wavelengths[50–52], from 90 to 150 nm. Unfortunately there is no experimental spectrum for the HNC isomer. These spectra may differ from each other by the experimental conditions, such as temperature and monochromator bandwith as a function of the wavelength[50], which may reduce the height of narrow peaks, specially those appearing at longer wavelengths. The simulated total spectrum for HCN is compared with the experimental one[52] in the bottom panel of Fig. 10 as a function of the photon energy.

Fig. 10.

Simulated spectrum versus experimental one, from Ref.[52], for HCN (bottom panel). Top panel shows the simulated spectrum for HNC isomer.

For energies below 8.4 eV, the narrow peaks of the simulated spectra are absent in the experimental one probably because they are much narrower than the experimental bandwidth.

In the 8.5-9 eV interval, the experimental and simulated peaks show similar progressions at the same energies, but the intensities are different. Several fittings of the transition dipole moment for the 1¹A′ → 3¹A′ transition have been tried, but the intensities obtained did not varied significantly. Since the position of the peaks are very similar, it is expected that the adiabatic potential for the 3¹A′ is essentially correct. The differences are attributed to non-adiabatic or Coriolis couplings to other states, which allow to share the oscillator strength and change the widths of the resonances, thus reducing the maximum height of individual peaks.

The narrow and intense series of peaks starting in 9.1 eV correspond to the 3¹A″ state. In this case, the position of the peaks are shifted from the experimental ones. In this electronic state, the bound vibrational levels reached can not decay in the adiabatic approximation. Their apparent widths in the spectrum is only the 0.5 meV introduced artificially as described above. This explains why the peaks are so narrow and intense. Another factor affecting the width of the observed peaks is the average over several rotational transitions, due to the presence of several initial rotational states under thermal conditions. In this work we only consider a single rotational transition, J_i = 0 → J = 1, and the simulated spectra are then narrower. Several rotational transition will be considered in a future work.

The bound levels of the 3¹A″ state can only decay by emitting photons or through the couplings to other dissociative electronic states. These couplings have been attributed to non-adiabatic, when belonging to the same symmetry, Coriolis couplings, between ¹A′ and ¹A″, or spin-orbit couplings, for different multiplicities. In any case, when considering these couplings, the levels would become broader and could share their oscillator strength with other close lying bound states.

For energies above 10 eV there are higher Rydberg states which also contribute and must be included to describe that region of the spectrum. Also the couplings to other electronic states should be accounted for, to properly describe the spectra. Work in these two directions are now being conducted.

The absorption spectrum of the HNC isomer is shown in the top panel of Fig. 10. It is much broader and intense that the spectrum of HCN. The main reason is that the electronic transitions are allowed in HNC while for many of the lower electronic states they are forbidden from the HCN isomer, as manifested by the angular dependence of the transition dipole moment in Fig. 2. This has important implication in astrochemistry, since this finding predicts a faster destruction of HNC than HCN in illuminated regions of space.

E. Radiative lifetimes

In order to determine their radiative lifetimes, the Einstein coefficients of the adiabatic bound levels for the 4 ¹A′, 2 ¹A″ and 3 ¹A″ electronic states have been calculated according to Eq. (6). The first 120 bound levels on each of these 3 electronic states have been calculated and can be clearly assigned to either HCN or HNC isomers, as shown in Fig. 9. According to Franck-Condon arguments, these states would decay towards levels on the same isomer. In fact here we consider J = 0 → J′ = 1 transitions from the excited to the ground electronic states. The radiative lifetimes, τ = 1/A, are shown in Fig. 11. For the 4 ¹A′ and 2 ¹A″ the lifetimes for the HNC isomer are clearly shorter than those of the HCN isomer. This propensity rule is due to the value ot the transition dipole moment, which are larger for γ ≈ 0. In the 3 ¹A″ state, the situation is reversed, and the radiative lifetimes of HNC become longer. The cases out of scale in the 3¹A″ correspond to those levels located in the middle well of Fig. 9 at γ=90° with very low Franck-Condon factors with the calculated bound rovibrational states on the X electronic state.

Fig. 11.

Radiative lifetimes of approximately the first 100 bound states of 4¹A′ (bottom)and 2 ¹A″ (middle) and 3 ¹A″ (top) electronic bound states. HCN/HNC isomers have been distinguished as HCN for cos γ < 0 or HNC for cos γ > 0, according to Fig. 9.

All the radiatime lifetimes are longer than 10 ns, more than 4 orders of magnitude longer than the lifetime associated to the peaks of the 1¹A″ and 3¹A′ states, shorter than 1 ps. Assuming that the levels of the electronic bound state have similar (or shorter) dissociative lifetimes, the non stimulated emission of photons seems to be rather inefficient.

This is supported by the measurements of the branching ratio of the excited CN(A²Π) and CN(B²Σ⁺), by detecting their emission to the ground CN(X²Σ⁺) state[50]. The quantum yield for CN(B²Σ⁺) emission account for about 30%, at short wavelengths, to ≈ 10% at longer ones. For the CN(A²Π) emission the proportion is in the range of 40-10 % from high vibrational levels. Adding all of them, and considering the possible direct dissociation towards the ground CN(X²Σ⁺), we can conclude that dissociation is the major channel, and that emission is negligible.

V. Conclusions

The photoabsorption spectra of HCN and HNC isomers have been studied for photon energies in the 7-10 eV range. For this, the three-dimensional adiabatic PES’s of the lower 4 and 3 states of ¹A′ and ¹A″ symmetry, respectively, have been calculated. The transition electric dipole moments for the transition from the ground electronic state shows drastic changes with the angle, with zero value at collinear HCN geometry for some excited electronic states due to symmetry. As a consequence, the X¹A′ → 2¹A′, X¹A′ → 1¹A″ and X¹A′ → 2¹A″ transitions are very weak from the HCN isomer, while they are intense from the HNC isomer.

The absorption toward the 2¹A′, 3¹A′ and 1¹A″ valence states are through resonances supported by wells arising from avoided crossing with the excited electronic bound states. Most of these resonances dissociate through tunneling, whose lifetimes depend on the energy and nodal structure, some of them estimated to be of 1 ps or less. However, the resonances around the adiabatic dissociation threshold are extremely narrow, and they probably dissociate via electronic transitions. The resonances reached from HCN and HNC are different as has been identified, corresponding to different potential wells.

The electronic 4¹A′, 2¹A″ and 3¹A″ states have deep wells for HCN and HNC isomers, and the absorption adiabatic spectra is composed by discrete lines associated to bound-bound transitions. Their radiative lifetimes have been estimated to be between 0.01-100 ms, much longer than the dissociation lifetimes. Also, the radiative transitions are among states in either the HCN or the HNC isomers, because of Franck-Condon factors.

The total absorption cross section for HCN in the 7-10 eV, is in qualitative agreement with the experimental ones, with one of the main difference been the too narrow resonances found in the simulations because of the neglect of electronic transitions made in the present adiabatic approach.

The absorption spectrum for HNC is much more intense and broad than that of HCN, what may introduce important differences between the photo-stability of the two isomers. A detailed calculation of the photodissociation rates for different astrophysical environments using our results is under progress. From preliminary calculations it appears that HNC is photodissociated at a rate factor 3 larger than HCN for clouds submitted to the normal galactic field. In order to completely asses its important it is necessary to cover the whole spectral region of astrophysical interest, from 6-14 eV. Work in this direction is being conducted together with the non-adiabatic transitions.