Article pubs.acs.org/JPCA
Theoretical Investigation of the Infrared Spectra of the H5+ and D5+ Cations Á lvaro Valdés and Rita Prosmiti* Instituto de Fsica Fundamental (IFF-CSIC), CSIC, Serrano 123, 28006 Madrid, Spain ABSTRACT: Reduced dimensional quantum dynamics calculations of the infrared spectrum of the H5+ and D5+ clusters are reported in both low, 300−2200 cm−1, and high, 2400− 4500 cm−1, energy regions. The proposed four-dimensional quantum model describes the motion of the proton between the two vibrating hydrogen molecules. The simulations are performed using time-dependent and time-independent approaches within the multiconfiguration time-dependent Hartree method. Propagation of the wavepackets includes an absorbing scheme to deal with vibrational dissociating states, and to assign the different spectral lines, block improved relaxation computations are performed for both bound and predissociative vibrational states of the systems. The reported computations make use of an analytical ab initio-based potential energy, and “on the fly” DFT dipole moment surfaces. The predominant features in the spectra are assigned to the excitations of the shared-proton stretch mode, and above dissociation the symmetric and antisymmetric stretching of the two H2 and the breathing mode of H3+ are also involved. The computed infrared absorption spectra for both cations are in very good agreement with the recent experimental measurements available from multiple-photon dissociation and mass-selected single-photon photodissociation spectroscopy techniques. Comparison of the present results with previous theoretical calculations on these systems is also presented. Such comparisons between different theoretical approaches and experimental measurements can serve to evaluate the approximations employed, and to guide higherorder computations.
I. INTRODUCTION The recent increased interest on research studies of the H5+ cation, and its isotopologues, is due to the postulation for their presence, although still not detected, in the interstellar medium.1,2 In particular, H5+ is the intermediate complex in the proton hop/exchange H3+ + H2 → H2 + H3+ reaction, playing a central role in molecular astrophysics.2,3 The first experimental observation of the infrared (IR) spectrum of H5+ near 4000 cm−1 has been reported by Okumura et al. in 1985 from vibrational predissociation spectroscopy,4 and later on by Bae5 for energies between 5400 and 10000 cm−1. Very recently, the IR spectra for both H5+ and D5+ have been reported by Duncan and co-workers, who scanned the ranges between 200−4500 and 550−3500 cm−1, respectively.6,7 The spectra have been measured in the gas phase by single-photon photodissociation experiments for energies above the dissociation thresholds,6 whereas in the midand far-IR regions they have been measured by resonanceenhanced multiple-photon dissociation (IR-MPD) technique using the FELIX free electron laser.7 On the basis of diffusion Monte Carlo (DMC) and vibrational configuration interaction calculations, using the reaction path version of the MULTIMODE method (MM-RPH), it has been shown that the shared-proton stretching mode plays a major role in the assignments of both H5+ and D5+ spectra.6,7 From this analysis, the spectral bands in the high-energy regime have been assigned to combinations of this mode with H2 stretching © 2013 American Chemical Society
motions, whereas for the low-energy bands the fundamental of the proton stretch vibration, and its combinations with the torsional and hydrogen bending modes, have been found in reasonable agreement with the experimental data. However, they found that the agreement between the MM-RPH data and experiment is worse for the D5+ spectra than for the H5+ ones. There is no doubt, particularly in the light of the recent experimental spectra,6,7 that the spectroscopy of these systems is a great challenge. On the theoretical side the high anharmonicity of the PES (presence of multiple minima with low barriers),8 the strong mode coupling, the predissociating dynamics involved, and the multidimensional nature of the system render the evaluation of their vibrational states in the whole spectral range difficult. For studying the energetics and structures of these complexes, different methods have been employed to deal with the nuclear part.6,9−12 Up to now, fixednode DMC calculations are limited to few selected vibrational levels,6,13 whereas the vibrational configuration interaction MM-RPH calculations6,7 are based on normal mode basis sets, which is not the most appropriate representation for describing Special Issue: Oka Festschrift: Celebrating 45 Years of Astrochemistry Received: December 11, 2012 Revised: January 15, 2013 Published: February 7, 2013 9518
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large amplitude motions, and predissociating states. Just recently, full-dimensional anharmonic treatments, using the multiconfiguration time-dependent Hartree (MCTDH) approach,14,15 have been reported, although they are only extended up to an energy of 1000 cm−1 for H5+ and D5+.12,16 This energy corresponds almost to halfway up to the dissociation threshold of these systems, and for reaching the higher energy regime, further improvements in the reported MCTDH computations,16 involving an efficient and accurate representation of the potential form, should be considered. Therefore, for overcoming the current limitations, approximated models have been developed, and reduced dimensional quantum calculations have also been carried out to simulate the spectra to provide theoretical assignment of them.13,17−21 In these studies two- and three-dimensional quantum simulations of the IR photodissociation spectra of H5+ and D5+ have been presented19−21 using the two most recent potential energy surfaces (PESs).22,23 By focusing on a lower-dimensional subspace, it is possible to describe selected vibrational bands, and a comparative study with the experimental recorded IR spectra6,7 requires at least a four-dimensional quantum treatment, which should judiciously include the proton and the H2 stretching coordinates, to cover the whole spectral region. The aim of this study is to simulate the anharmonic IR spectra of the H5+ and D5+ cations in the 200−4500 and 550− 3500 cm−1 energy range, respectively, recorded by the recent experiments.6,7 The calculation and assignment of the observed vibrational bands is done in terms of quantum 4D timedependent and time-independent computations within the framework of the MCTDH methodology. As in most of the previous studies, we employed here the surface by Xie et al.,24 whereas the dipole moment surface (DMS) is obtained from DFT calculations.20 The present results at both low- and highenergy regimes, together with their comparison with previous spectroscopic investigations, allow us to extract meaningful conclusions on the cluster dynamics and could serve as a useful guide for future spectral studies. The paper is organized as follows. In section II we present the 4D model Hamiltonian, and we describe some computational and methodological details of the time-dependent and time-independent MCTDH computations. The results of the spectral simulations and the assignment of the vibrational bands are discussed in section III, in comparison with the experimental data, as well as with previous theoretical calculations available in the literature. The last section contains some concluding remarks.
Figure 1. Internal coordinates of the 4D model Hamiltonian used to describe the H5+ system.
ℏ2 2 ℏ2 2 ℏ2 ℏ2 ∂R − ∂z − ∂ R12 − ∂R 2 Ĥ = − 2μR 2m 2μ1 2μ2 2 + V (R ,z ,R1 ,R 2)
(1)
where μR = mH/4, m = mH/5, and μ1 = μ2 = mH/2, with mH being the mass of the H atoms. For the calculation of the vibrational states we employed the so-called block improved relaxation (BIR) procedure26 implemented within the Heidelberg MCTDH package.25 Such a time-independent approach could provide accurate calculation of eigenstates up to a given energy (depending on the problem to handle). However, at high energies when the density of states increases, it is not practical from the computational point of view to calculate the whole spectrum.27 Further, here we are treating with predissociating states, in the R coordinate; thus the IR spectrum is obtained in the timedependent representation by employing absorbing boundary conditions to prevent the wave function to reach the grid boundaries. In this approach, the IR spectrum is computed from the Fourier transform of the autocorrelation function of an initial wave function including the dipole moment operator, given by I (E ) =
E 9 3cε0ℏ2
∫0
∞
e i(E + E0)t / ℏC(t ) dt (2)
where C(t) = ⟨Ψ(0)|Ψ(t)⟩ = ⟨Ψμ(0)|e−iHt/ℏ|Ψμ(0)⟩, E0 the energy of the initial state |Ψ0⟩, μ̂ the dipole-moment operator, and |Ψμ(0)⟩ = μ̂ |Ψ0⟩. The Hamiltonian operator and the wave function are represented on a grid in the internal coordinates. The range of the grid and the type of the primitive basis are listed in Table 1. For the calculations of the vibrational states we employed a
II. HAMILTONIAN OPERATOR AND COMPUTATIONAL DETAILS
Table 1. Number, Type, and Grid Range for the Primitive Basis Set for Each Coordinate Used in the MCTDH Calculationsa
All reported calculations were performed using the Heidelberg MCTHD package of codes.25 In Figure 1 we depict the four internal coordinates involved in the present study, namely R, z, R1, and R2, to describe the H5+ system, with R being the distance between the centers of mass of both H2 molecules, z the position of proton with respect the center of mass of the H2−H2 system, and R1 and R2 the bond lengths of each H2 molecule. The reduced 4D Hamiltonian of H5+ in these coordinates can be written as
coord
no./type of DVR basis
grid range
R z R1 R2
50/sin 41/HO 15/HO 15/HO
[3.25, 8.25] [−2.6, 2.6] [0.9, 2.4] [0.9, 2.4]
a
sin refers to the sine discrete variable representation (DVR) basis, and HO stands for the harmonic oscillator DVR basis.
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two-mode combination scheme with 180 and 30 single-particle functions (SPFs) for the [R, z] and [R1, R2] modes in the BIR procedure, whereas for the propagation of the wavepacket a smaller set of 20 SPFs in each mode is used for obtaining the spectrum in the energy range of 0−5000 cm−1. The POTFIT approach is used to write the potential energy operator expansion as the sum of products of single-particle operators required by the MCTDH method.28,29 The dipole moment operated ground state |Ψμ(0)⟩ was propagated for 1000 fs, providing autocorrelation function C(t) for 2000 fs, and the spectrum resolution is of about 14 cm−1.15 As we mentioned above, an absorption scheme is employed, and a complex absorbing potential (CAP) is introduced to absorb the wavepacket as it is approaching at the end of the grid. Its form is given by −iW(R) = −iη(R − Rc)bΘ(R − Rc), with the parameters being the starting point Rc = 6.0 bohr, the strength η = 0.005 au, and the order b = 3, and Θ denotes the Heaviside step function. For comparison reasons, the simulations reported here make use of the ab initio PES by Xie et al.,24 which has been previously used in most studies, whereas the electric DMS has been computed by performing “on the fly” DFT/B3(H) calculations20 on the whole grid of points.
Figure 3. Simulated 4D-MCTDH spectrum (with assignments) for H5+ (bottom, black line) in the range 2400−3500 cm−1, and its comparison with the corresponding experimental action spectrum (top, blue line).6
motion. As has been previously discussed,7 due to the MPD mechanism the intensities of the IR-MPD experimental signal are significantly weaker in lower-frequency region, below 600 cm−1, whereas above this energy the three- and two-photon processes yield more intense bands. In Table 2 we list the energies of the H5+ vibrational states obtained from the 4D BIR calculations with respect the zeropoint energy (ZPE), whereas in Figures 4 and 5 we display contour plots with their probability density distributions. By analyzing their node structure, we present a tentative assignment for each of them in the last column of Table 2. The vibrational energies calculated from the full-dimensional MCTDH and MM-RPH approaches are also given in Table 2. As can be seen, the vz = 1 state is found at 505.0 cm−1 in the 4D study, whereas the coupling with the rest of the degrees of freedom lower its energy at 365.4 cm−1 in the 9D calculations. Also the ZPE and the first four vibrational levels are found in good accord with the previously reported data using a reduced 5D model Hamiltonian, including the four stretching modes and the torsional motion, although different PES17 for H5+. Such agreement indicates once again the weak coupling between the stretching and torsional motions, and justifies further the use of the 4D model. Thus, for comparing with the experimental data, we shifted the computed 4D-MCTDH spectra of H5+ (Figures 2 and 3) by the energy difference of this state, n = 1, in the 4D and 9D results. This shift is reasonable for the proton transfer fundamental, although somehow problematic for other states. One can see in Table 2 for the n = 2 state a difference of about 10 cm−1 with the 9D results is found. As we mentioned above, the four bands in the 4DMCTDH spectra (Figure 2) are assigned to the n = 1, 3, 5, and 8 states, which correspond to the fundamental and excitations of the proton stretching (Figure 4) with vz = 1, and vR = 1, 2, and 3, respectively, and the experimental band at 1180 cm−1 should be attributed to the bending modes of H5+, which is not included in the 4D model. We should also point out that, due to the reduced dimensional treatment, the calculated bands are still shifted to higher energies than the experimental ones. As we can see, the n = 3 state has been found 116 cm−1 lower in energy in the recent 9D MCTDH calculations.16
III. RESULTS AND DISCUSSION In the following sections the different parts of the spectrum, and tentative assignment of the peaks are discussed for both H5+ and D5+ cations. H5+ Spectra. The 4D-MCTDH spectrum in the range of 200−2200 and 2400−4500 cm−1 for H5+ is depicted in Figures 2 and 3, respectively, and compared to the recent experimental data.6,7 In Figure 2 the computed low-energy region of the spectrum, below the predicted dissociation threshold, shows the stronger absorption at the energy of the proton stretch fundamental, because this motion induces a large change in the dipole moment of the cation. The next three peaks of decreasing intensity are also associated with the shared proton
Figure 2. Simulated 4D-MCTDH spectrum for H5+ (bottom, black line) in the range 200−2200 cm−1, and its comparison with the experimental IR-MPD measurements (top, red line).7 The weak bands (red asterisks) have been attributed to an artifact of higher harmonics of FELIX.7 The calculated spectrum is shifted to match the position of the most intense peak of the MCTDH spectrum with the energy of the vz = 1 state from the 9D MCTDH calculations.16 The assignment of the bands is also shown (Table 2). 9520
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Table 2. Calculated Vibrational Energies (cm−1 with Respect to the ZPE Value) for the Indicated n States of H5+, and Their Comparison with Previous MCTDH, MM-RPH, and DMC Calculationsa energy n
4D-MCTDH (shifted values)
0 1 2 3 4 5 6 7 8 D0 14 22
4115.6 505.0 829.3 1189.1 1420.1 1746.5 1803.3 2048.4 2207.9
23
3553.8 (3414.2)
28
9D-MCTDH12,16
7205 369 673 983
± ± ± ±
5 5 5 5
MM-RPH6,7
expt6,7
assignment
7244 381
379
952, 989c
940−945
1331c
1399−1402
1848c 2749 3560
1714−1723 2450,e
