Ab Initio Study of the VUV-Induced Multistate Photodynamics of

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J. Phys. Chem. A 2010, 114, 11436–11449

Ab Initio Study of the VUV-Induced Multistate Photodynamics of Formaldehyde S. Go´mez-Carrasco,† T. Mu¨ller,‡ and H. Ko¨ppel*,† Theoretical Chemistry, Institute of Physical Chemistry, UniVersity of Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany and, Ju¨lich Supercomputer Centre, Institute of AdVanced Simultation, Research Centre Ju¨lich, D-52425, Ju¨lich, Germany ReceiVed: July 21, 2010; ReVised Manuscript ReceiVed: September 9, 2010

Although formaldehyde, H2CO, has been extensively studied there are still several issues not-well understood, specially regarding its dynamics in the VUV energy range, mainly due to the amount of nonadiabatic effects governing its dynamics. Most of the theoretical work on this molecule has focused on vertical excitation energies of Rydberg and valence states. In contrast to photodissociation processes involving the lowest-lying electronic states below 4.0 eV, there is little known about the photodynamics of the high-lying electronic states of formaldehyde (7-10 eV). One question of particular interest is why the (π, π*) electronic state is invisible experimentally even though it corresponds to a strongly dipole-allowed transition. In this work we present a coupled multisurface 2D photodynamics study of formaldehyde along the CO stretching and the symmetric HCH bending motion, using a quantum time-dependent approach. Potential energy curves along all the vibrational normal modes of formaldehyde have been computed using equation-of-motion coupled cluster including single and double excitations with a quadruply augmented basis set. In the case of the CO stretching coordinate, state-averaged complete active space self-consistent field followed by multireference configuration interaction was used for large values of this coordinate. 2D (for the CO stretching coordinate and the HCH angle) and 3D (including the out-of-plane distortion) potential energy surfaces have been computed for several Rydberg and valence states. Several conical intersections (crossings between potential energy surfaces of the same multiplicity) have been characterized and analyzed and a 2D 5 × 5 diabatic model Hamiltonian has been constructed. Based on this Hamiltonian, electronic absorption spectra, adiabatic and diabatic electronic populations and vibrational densities have been obtained and analyzed. The experimental VUV absorption spectrum in the 7-10 eV energy range is well reproduced, including the vibrational structure and the high irregularity in the regime of strong interaction between the (π, π*) electronic state and neighboring Rydberg states. I. Introduction Carbonyl compounds are important in many areas of chemistry, from physical-organic to photochemical, synthetic, and biological studies. The presence of a π system and the lone pairs on the oxygen give rise to both (n, π*) and (π, π*) transitions, which are characteristic of this kind of compound1,2 and are responsible for important photochemical reactions. In particular, the 1A1 (π, π*) and the 1B1 (σ, π*) valence states play a key role in understanding the spectral differences observed in ketones and aldehydes. Formaldehyde, H2CO, is the simplest carbonyl compound, and it is also one of the polyatomic organic molecules observed in the interstellar medium.3 It is thought to be a constituent of the primitive atmosphere of Earth and other planets, and is known to be a constituent of Earth’s present stratosphere. The interest on its dynamics is due to the important role that formaldehyde plays in atmospheric chemistry: it is the most abundant carbonyl compound in the atmosphere and is one of the indicators of poor air quality because essentially every organic species in the atmosphere degrades with a reaction pathway that includes formaldehyde.4 This system does provide, therefore, a showcase for quantitative tests of models on the investigation of the * To whom correspondence should be addressed. E-mail: Horst. [email protected]. † University of Heidelberg. ‡ Research Centre Ju¨lich.

spectroscopy, photochemistry, and photophysics of polyatomic molecules, and calls for an accurate account of the character and behavior of its excited electronic states along with important nuclear motion effects governing its photodynamics. Many of the theoretical studies on formaldehyde5–16 have focused on vertical excitations. In this regard, we refer to the pioneering configuration interaction (CI) calculations performed in 1969 by Whitten and Hackmeyer5 and subsequent multireference CI (MRDCI) calculations by Buenker and Peyerimhoff.6,7 Davidson and McMurchie collected in 1982 the ab initio information available to that date.17 Later MRCI results were given by Harding and Goddard8 and Grein et al.10,13,18 Grein et al. provided also information on the perturbations of the 1A1 (π, π*) and 1B1 (σ, π*) valence states by the surrounding Rydberg states. Figure 1 displays the potential energy curves for several 1A′ electronic states calculated by Grein (taken from ref 10); they will be compared with our results in the following sections. Complete active space perturbation theory (CASPT2) was applied to formaldehyde by Merchan and Roos,11 equationof-motion coupled cluster including single and double excitations (EOM-CCSD) theory by Barlett et al.12 Recently, highlevel MRCI and multireference averaged quadratic coupled clusters (MR-AQCC) calculations based on complete active space self-consistent field (CASSCF) wave functions have been presented by Lischka et al.15,16 along with full geometry optimizations for several electronic states. They have also analyzed a conical intersection (CoIn)19 occurring between the

10.1021/jp106777z  2010 American Chemical Society Published on Web 10/08/2010

VUV-Induced Multistate Photodynamics of Formaldehyde

Figure 1. 1D potential energy curves of 1A1 symmetry of formaldehyde calculated by Grein et al., reproduced with permission from ref 10.

11B1 (σ, π*) and 21A1 (π, π*) electronic states, coupled through the out-of-plane bending motion. The electronic spectrum of formaldehyde has been also analyzed experimentally.20–22,22–30 Optical20,22,26,27,31 and electron impact techniques28–30) have been used to record the vacuum-ultraviolet (VUV) absorption region from approximately 7 to 11 eV. The spectrum shows long n f (spdf) Rydberg, π f (sp) Rydberg and σ f (p) Rydberg series, converging to the ground state of H2CO+. Moule and Walsh21 reviewed the early experimental work on the excited states of formaldehyde. Despite the large number of studies on formaldehyde, there are still not well-understood issues. In the absorption spectrum of this system, the electric dipole forbidden transition 1A2 (n, ˜ 1A1 is observed and well studied; however, the 1A1 π*) r X ˜ 1A1 transition, that is allowed and should be one of (π, π*) rX the most intense given its large oscillator strength, has not been yet experimentally detected. Likewise, the experimental assign˜ 1A1 electronic transition also remains ment of the 1B1 (σ, π*) rX still uncertain. For a long time, it was believed32 that the 1A1 (π, π*) state lies above the first ionization potential of 10.88 eV.33 Later, however, different electronic calculations predicted a vertical excitation energy between 9.47 and 9.84 eV.10–12,15 The reason for these “missing” states is associated with the interaction between the valence and Rydberg states, mainly along the rCO stretching coordinate13,14 (excitations to π* are expected to carry large changes in the CO bond length). In contrast to the missing 1A1 (π, π*) state, the 3A1 (π, π*) state of formaldehyde has been observed by electron impact studies.30 Theoretically, it has been shown13 that the 3A1 (π, π*) electronic state lies well below the Rydberg states, and, therefore, it should be free of the Rydberg mixing in contrast to its singlet counterpart. From the dynamical point of view, the photodissociation of formaldehyde has been also extensively studied both theoretically34–43 and experimentally.44–49 However, the dynamical studies have mainly focused on the photodissociation via the lowest-lying electronic states: the ground (S0), the 11A2 (S1) and the 13A2 (T1) electronic states. Excitation to the dipole-forbidden

J. Phys. Chem. A, Vol. 114, No. 43, 2010 11437 S1 state can result in fluorescence, internal conversion or intersystem crossing, leading to H+HCO and H2+CO dissociation products, at energies below 4.0 eV. CO bond dissociation is not relevant for these low energies. Since the dissociation products CH2+O lie experimentally 7.64 eV above the ground state, rupture of the CO bond can only be achieved via absorption into electronic states lying above 7.0 eV. However, there are so far no studies on the dynamics of these high-lying electronic states. In this work we present a coupled surface photodynamics study of formaldehyde for the energy range of 7-10 eV, using a time-dependent fully quantum approach. Potential energy curves (PECs) have been computed for all the vibrational normal modes. Likewise, 2D (varying the CO stretching and the HCH bending coordinates) and 3D (including the out-of-plane angle) potential energy surfaces have been also computed and the existing CoIns have been identified and characterized. The dynamics has been investigated on five coupled 1A1 and eight 1 B2 potential energy surfaces, the latter taken to be uncoupled (see explanation below). A 5 × 5 model Hamiltonian has been constructed for the diabatization of the 2D 1A1 potential energy surfaces and the 2D nonadiabatic dynamics along the CO stretching and the HCH bending coordinates has been studied. The total 2D absorption spectrum and dynamics has been also recomputed by convolution of those for the CO stretching and the HCH angle. Along with the electronic absorption spectra, adiabatic and diabatic electronic populations have been obtained and analyzed. To the best of our knowledge this is the first full quantum dynamical treatment of more than two intersecting potential energy surfaces of the same symmetry. The paper is structured as follows: Section II describes the technical details and results of the electronic structure calculations along, and Section III is devoted to the dynamical calculations. Conclusions and suggestions for future work are finally presented in Section IV. II. Electronic Structure Details and Results A. Computational Method. The ground and excited electronic states of formaldedyde were calculated using an equationof-motion coupled cluster method with single and double excitations (EOM-CCSD) as implemented in the MOLPRO suite of programs.50 The C2 axis is aligned with the z-axis, and the molecular plane coincides with the yz-plane. The electronic configuration of the ground state can be described as:

(core)(3a1σ(CO))2(4a1σ(CH))2(1b2σ(CH))2(5a1σ(CO))2 (1b1π(CO))2(2b2n(O))2 The most relevant molecular orbitals are the HOMO (n), the HOMO - 1 (π), and the HOMO - 2 (σ). Rydberg MOs are labeled using the usual atomic-like notation (nl) as, ns, np, nd. To ensure a balanced description of the valence and Rydberg states of formaldehyde a thorough initial study with several basis sets was performed: for the valence states, the correlation consistent cc-pVDZ and cc-pVTZ bases sets51 were used for oxygen, hydrogen, and carbon. To obtain a set of converged Rydberg states in the energy range of interest, two sequences of even-tempered diffuse functions52 were added to the valence basis. The corresponding exponents, ξlk, of the even-tempered basis sets were obtained by the recursion formula l ξlk ) ξk-1 β-1 for k ) 3 (or t), 4 (or q), 5, 6

(1)

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Go´mez-Carrasco et al.

TABLE 1: Valence (V) and Rydberg (R) States of Formaldehyde Calculated with Different Basis Sets (See Text for Explanation) at EOM-CCSD Level of Theorya mc-qaVTZ+VTZ State 1 1 2 2 2 3 4 1 3 5 3 2 6 4 4 7 5 8 6 3

1

A2 (V) B2 (R) 1 B2 (R) 1 A1 (R) 1 A2 (R) 1 B2 (R) 1 B2 (R) 1 B1 (R) 1 A1 (R) 1 B2 (R) 1 A2 (R) 1 B1 (V) 1 B2 (R) 1 A1 (R) 1 A2 (R) 1 B2 (R) 1 A1 (V) 1 B2 (R) 1 A1 (R) 1 B1 (R) 1

tr.

Eexc

(n, π*) (n, 3s) (n, 3pz) (n, 3py) (n, 3px) (n, 3dx2-y2) (n, 3dz2) (n, 3dxy) (n, 3dyz) (n, 4s) (n, 3dxz) (σ, π*) (n, 4pz) (n, 4py) (n, 4px) (n, 4dz2) (π, π*) (n, 4dx2-y2) (n, 5py) (n, 4dxy)

4.04 7.21 8.03 8.16 8.38 9.05 9.24 9.24 9.29 9.35 9.38 9.33 9.55 9.58 9.65 9.88 9.80 9.99 10.18 10.03

f 0.018 0.035 0.050 0.019 0.00006 0.001 0.008 0.0005 0.0004 0.009 0.012 0.009 0.093 0.001 0.045 0.0008

mc-qaVDZ+VTZ 〈x2〉

Eexc

10.4 19.0 17.7 18.1 45.8 49.1 32.1 53.4 23.7 62.7 69.9 11.4 74.2 65.2 202.9 155.7 23.6 88.6 90.0 114.4

4.04 7.20 8.05 8.16 8.38 9.09 9.26 9.29 9.34 9.36 9.41 9.34 9.55 9.58 9.65 9.91 9.83 10.01 10.10 10.16

f 0.017 0.037 0.048 0.015 0.0004 0.0003 0.011 0.0006 0.0005 0.010 0.012 0.005 0.118 0.004 0.017 0.00002

mc-qaVDZ+VDZ 〈x2〉

Eexc

10.5 19.1 19.0 18.8 47.7 56.9 31.7 57.3 25.9 57.9 76.6 14.4 75.4 63.4 203.9 208.2 19.8 128.7 135.4 90.1

3.98 7.02 7.86 7.96 8.18 8.91 9.06 9.09 9.15 9.16 9.20 9.31 9.34 9.37 9.44 9.71 9.74 9.80 9.89 9.96

f 0.017 0.041 0.047 0.012 0.0004 0.00007 0.006 0.0007 0.0009 0.011 0.013 0.003 0.102 0.004 0.023 0.0002

exp

〈x2〉

Eexc

10.6 19.4 19.5 19.1 48.5 61.0 32.1 62.6 27.4 55.7 77.3 10.8 78.6 64.2 207.3 225.4 22.5 129.7 133.3 89.2

4.07 7.09 7.98 8.13 8.37 8.88

f 0.032,0.038 0.019 0.036,0.042 0.018,0.013

9.26 9.22 9.59 9.63 9.61 9.85 9.85 10.13 9.85

0.038,0.032

0.023,0.013

a The configurational assignments are given in parentheses. Vertical excitation energies in eV, oscillator strengths (f) and the spatial extent of the wave functions (〈x2〉) in au. Experimental data taken from refs 27 and 20.

where l indicates the angular momentum (s, p, d, ... functions). The first two exponents, ξ1l and ξ2l , are taken from the d-augcc-pVXZ (X ) D or T) basis sets. An average value of β ) 3.5 was chosen, estimated from the ratios (ξ1l /ξ2l ). The diffuse functions were placed in the center of mass of the molecule (mc). Altogether, the basis sets have been denoted as mckaVXZ+VXZ, where k ) t, q, 5, 6 and X ) D, T. For each of the basis sets indicated above, the geometry of the electronic ground state was first optimized and, subsequently, the excitation energies (Eexc), oscillator strengths (f) and the spatial extent of the wave functions (〈x2〉) were calculated for several excited states of formaldehyde. Analyzing the results obtained with all the basis sets, it was concluded that at least quadruply augmented (mc-qaVXZ) diffuse functions are required to get converged and well-balanced results for the Rydberg states, either for VDZ or for VTZ valence functions. A comparison of the results for three of the basis sets is shown in Table 1, for several valence and Rydberg states (calculated up to an energy threshold slightly above 10 eV). The states are characterized by symmetry, valence (V) or Rydberg (R) character, nature of the transition (tr.), vertical excitation energy (Eexc), oscillator strength (f) and spatial extent perpendicular to the molecular plane (〈x2〉) along with experimental data as far as available. The presence of a large number of Rydberg states interspersed among the valence states requires a fairly large number of electronic states to be optimized simultaneously. Increasing the valence basis (mc-qaVDZ+VDZ versus mcqaVDZ+VTZ) yields somewhat higher excitation energies for the Rydberg states (by ≈0.2 eV), and the excitation energies agree in most of the cases better with the experimental ones. Small differences are found, however, in the calculated values of f and 〈x2〉 for both basis sets. Increasing the Rydberg basis at a fixed valence basis (3rd and 4th column of Table 1) shows that the basis sets are practically saturated regarding diffuse functions. Finally, a triple-ζ basis set (VTZ) for the valence states plus quadruple-augmented double-ζ (mc-qaVDZ) diffused functions for the Rydberg states was selected for our calculations.

Altogether, the basis set contains a total of 124 contracted functions. With this basis, the calculated ground-state optimized equilibrium geometry of formaldehyde is planar with rCO ) 2.2766 au, rCH ) 2.0800 au and ∠HCH ) 116.5°, in good agreement with the experimental values, that is, rCO ) 2.2733 au, rCH ) 2.0806 au and ∠HCH ) 116.2°.53 Table 2 shows the calculated valence and Rydberg excited electronic states along with the data of previous studies10,11,15 and, again, experimental values20,27 for comparison. The agreement of the vertical excitation energies with the experimental data and previous calculations is quite good although our results for the B2 states appear in general shifted toward higher energies (≈0.10 eV for most of the states but up to 0.21 eV for the 31B2 state), compared with the experimental values and the calculated ones by Mu¨ller and Lischka.15 Following the symmetry rules, the transition to the 1A2 electronic states is dipole-forbidden. From the analysis of the oscillator strengths, it follows that the transitions to the B1 states are only very weakly allowed (see Tables 1 and 2 below) and henceforth are neglected in the dynamical calculations. Since we are mainly interested on the VUV absorption spectrum, our dynamical calculations will focus on the A1 and B2 symmetries, although also 1A2 and 1B1 electronic states have been computed and will be shown in some of the potential energy curves. B. Potential Energy Surfaces and Diabatic Hamiltonian. Further electronic structure calculations have been carried out to compute PECs for all the vibrational modes of formaldehyde: the CO stretching (a1), the symmetric (a1) HCH bending, the antisymmetric (b2) HCO bending (or wagging motion), the symmetric (a1) and antisymmetric (b2) CH stretching and the out-of-plane (b1) bending (or rocking motion). The symmetry of the normal modes is indicated in parentheses. For the electronic structure calculations, the C2V point group of symmetry is used for the rCO stretching, the symmetric HCH bending and for the symmetric CH stretching coordinates. The Cs point group of symmetry is used for the out-of-plane motion, and the C1 point group for the asymmetric CH stretching and the wagging motions.


J. Phys. Chem. A, Vol. 114, No. 43, 2010 11439

TABLE 2: Valence (V) and Rydberg (R) States of Formaldehydea Greinb state 1 1 2 2 2 3 4 1 3 5 3 2 6 4 4 7 5 8 6 3

1

A2 (V) B2 (R) 1 B2 (R) 1 A1 (R) 1 A2 (R) 1 B2 (R) 1 B2 (R) 1 B1 (R) 1 A1 (R) 1 B2 (R) 1 A2 (R) 1 B1 (V) 1 B2 (R) 1 A1 (R) 1 A2 (R) 1 B2 (R) 1 A1 (V) 1 B2 (R) 1 A1 (R) 1 B1 (R) 1

tr.

Eexc

(n, π*) (n, 3s) (n, 3pz) (n, 3py) (n, 3px) (n, 3dx2-y2) (n, 3dz2) (n, 3dxy) (n, 3dyz) (n, 4s) (n, 3dxz) (σ, π*) (n, 4pz) (n, 4py) (n, 4px) (n, 4dz2) (π, π*) (n, 4dx2-y2) (n, 5py) (n, 4dxy)

4.05 7.15 8.05 8.10 8.32 9.05 9.25 9.32 9.25 9.26 9.34 9.35 9.47 9.41 9.67 9.89 9.60 9.80 9.87 9.94

Merchan-Roosc f

0.005 0.021 0.039

0.005

〈x 〉

Eexc

0.006 0.001 0.0003 0.003

10.5 23.7 21.1 19.9 46.5 46.6 47.0 58.3 25.3

3.98 7.12 7.94 8.16 8.38 8.99 9.14 9.29 9.29

0.012

60.8 10.0

9.41 9.32

Eexc 3.91 7.30 8.09 8.12 8.32 9.13 9.31 9.23 9.24 9.31 9.09

Mu¨ller-Lischkad

f 0.006 0.023 0.041

2

Eexc

0.010 0.000 0.000 0.018

10.5 21.8 21.3 19.1 45.0 44.9 44.7 46.1 24.4

0.000

61.5 23.1

4.04 7.20 8.05 8.16 8.38 9.09 9.26 9.29 9.34 9.36 9.41 9.34 9.55 9.58 9.65 9.91 9.83 10.01 10.10 10.16

0.018 0.040 0.043

0.001 0.000 9.77

0.277

13.5

9.83

0.100

16.7

expe,f

This Work

〈x 〉

f

2

f 0.017 0.037 0.048 0.015 0.0004 0.0003 0.011 0.0006 0.0005 0.010 0.012 0.005 0.118 0.004 0.017 0.0000

〈x 〉

Eexc

10.5 19.1 19.0 18.8 47.7 56.9 31.7 57.3 25.9 57.9 76.6 14.4 75.4 63.4 203.9 208.2 19.8 128.7 135.4 90.1

4.07 7.09 7.98 8.13 8.37 8.88

2

f 0.032,0.038 0.019 0.036,0.042 0.018,0.013

9.26 9.22 9.59 9.63 9.61 9.85 9.85 10.13 9.85

0.038,0.032

0.023,0.013

a

Comparison of the present results (This Work) with previous ones. The configurational assignments are given in parentheses. Vertical excitation energies in eV, oscillator strengths (f) and 〈x2〉 in au. b MRDCI calculations from ref 10. c CASPT2 calculations from ref 11. d MR-AQCC/LRT calculations from ref 15. e Reference 27. f Reference 20.

The following grids have been used:

CO str. ) [1.52-3.40] au (220 points) ∠(HCH) ) [80-160]° (21 points) ∠(HCO) ) [90-150]° (7 points) CHs str. ) [1.70-2.65] au (11 points) CHa str. ) [2.08-2.65] au (17 points) ∠(out-of-plane) ) [120-180]° (8 points) For the 1D calculations, the rest of the coordinates are kept fixed at the equilibrium geometry of the ground electronic state. In all the CCSD calculations the T1 diagnostic values were checked. The T1 diagnostic gives a qualitative assessment of the nondynamical correlation and ensures the reliability of the method (the T1 diagnostic is the norm of the t1 amplitudes in the CC wave function expansion divided by the square root of the number of correlated electrons. Large t1 amplitudes (single excitations) indicate that the Hartree-Fock orbitals are not wellsuited for describing electron correlation). For closed-shell systems, the T1 values should be below 0.02.54,55 The T1 diagnostic indicated substantial multireference character of the electronic reference state only for the CO bond distances beyond 2.4 bohr. To describe properly the potential energy curves for long CO distances in the 1D calculations, additional stateaveraged complete active space self-consistent field (SACAS)SCF calculations followed by internally contracted multireference configuration interaction (icMRCI) calculation were performed up to rCO ) 25 bohr. The (SA-CAS)SCF calculation included all molecular orbitals arising from the valence atomic orbitals (12 electrons in 10 orbitals). The 1s electrons have been kept frozen. The number of states included in the (SA-CAS)SCF

calculations has been selected from the singlet states correlating with the asymptotic limits, CH2 (3B1, 1A1) + O(3P, 1D), in C2V symmetry. Figure 2 displays the 1D adiabatic PECs along the six degrees of freedom (dofs) of formaldehyde. The CCSD ab initio energies are plotted for all the dofs except for the CO stretching coordinate for which the SA-CASSCF/MRCI energies are shown for CO distances beyond 2.4 bohr. The symmetry of the vibrational modes in the molecular point group of symmetry, C2V, is also indicated in brackets. The vertical dashed lines indicate the equilibrium geometry of the ground electronic state, that is, the Franck-Condon (FC) point line. For the C2V symmetry-conserving modes, the A1 or B2 electronic states are plotted on the right and left panels, respectively. The nature of the electronic transitions is only given for the potential energy curves along the CO stretching coordinate. On the other hand, the out-of-plane bending mode lowers the symmetry of the molecule (from C2V to Cs), coupling the A1/B1 states and the A2/B2 states. Those pairs of electronic states correlate with the A′ or A′′ irreducible representations in the Cs point group, respectively. Several A′ (A1 + B1) and A′′ (A2 + B2) electronic states are plotted on the right and left panels, respectively, along the out-of-plane bending coordinate. Likewise, the asymmetric CH stretching and the wagging motions lower the symmetry of the system to the C1 point group. Therefore, all the symmetries in C2V appear in the lowest panels for the asymmetric CH stretching (left panel) and the wagging motion (right panel) of the figure. The 1A1 states show a very complex shape along the CO stretching coordinate due to various avoided crossings among the electronic states, either Rydberg or valence states.10,13,14,18 In particular, note that the 1A1 (π, π*) valence state undergoes several avoided crossings with all the Rydberg states. In contrast to the 1A1 states, there is only one avoided crossing within the B2 manifold, located at a larger distance and higher energy with respect to the corresponding minima. Therefore, the B2 electronic states have been treated uncoupled. Not much excitation is expected along the symmetric and antisymmetric CH stretching

11440



Figure 2. 1D adiabatic potential energy curves of formaldehyde for all the vibrational degrees of freedom at the EOM-CCSD/mc-qaVDZ+VTZ and icMRCI/mc-qaVDZ+VTZ levels of theory (for details see text). The symmetry assignment of the normal modes is given in brackets. For the totally symmetric vibrational modes, the electronic states are shown in the C2V point group of symmetry. In particular, only A1 and B2 symmetries are plotted. Cs and C1 point groups are used for the electronic states along the nontotally symmetric vibrational modes. Note that there is no correlation in the color coding between the states in C2V and those ones in Cs or C1.

vibrational modes and neither along the asymmetric HCO bending, because these are similar in shape and have their energetic minima at or near the center of the FC zone (indicated by the vertical dashed lines in the various panels). However, some excitation is predicted for the symmetric HCH bending mode either for the 1A1 and for the 1B2 electronic states. Concerning the importance of the out-of-plane mode in the B1 states see discussion of Figure 5. To account for nonadiabatic effects on the dynamics of a system in the presence of avoided crossings or CoIns, it is often beneficial to shift from the adiabatic to a diabatic representation.56–59 The diabatic approach amounts to choosing a different

representation of the electronic states which minimizes, in contrast to the adiabatic one, the divergent kinetic couplings and replaces them with slowly varying potential couplings, which are much easier to handle, computationally speaking, than the former. For the five adiabatic 1A1 electronic states, shown on an enlarged scale in the top panel in Figure 3, the diabatic Hamiltonian reads

H ) TN + W

(2)


J. Phys. Chem. A, Vol. 114, No. 43, 2010 11441 TABLE 3: Energy and Geometry Locations of the Avoided Crossings (1D PES) and the Corresponding CoIns among the 1A1 Electronic Statesa (a) 1D PES 1

2 A1/3 A1

31A1/1A4,5 1

41A1/51A1

51A1/61A1

2.513 116.54 8.68 0.0317

2.340 116.54 9.47 0.1693

2.328 116.54 9.64 0.0000

2.230 116.54 10.10 0.0257

CoIn 1 (21A1/31A1)

“CoIn 2” (31A1/1A4,5 1 )

CoIn 3 (41A1/51A1)

CoIn 4 (51A1/61A1)

rCO/au ∠HCH/° energy/eV ∆V/eV

2.485 103.24 8.91 0.0253

(b) 2D PES 2.343 126.00 9.46 0.1700

2.328 116.54 9.63 0.0000

2.306 152.00 10.56 0.0000

rCO/au ∠HCH/° energy/eV ∆V/eV kij (eV/°)

2.487 102.80 8.91 0.0253 -0.0049

(c) 2D Model 2.343 2.334 126.00 116.54 9.46 9.63 0.1700 0.0000 0.0000 -0.0119

2.268 153.26 10.56 0.0000 0.0014

rCO / au ∠HCH / ° energy / eV Wij0 / eV

1

a

In the 1D case, Wij0 is one-half of the smallest energy difference at the avoided crossings. ∆V is the same quantity for the 2D case. The quantities kij are the inter-state couplings. Note that the indices i and j refer to the diabatic surfaces. Figure 3. 1D adiabatic (top) and diabatic (bottom) potential energy curves of formaldehyde of A1 symmetry along the CO stretching coordinate, rCO, at the EOM-CCSD/mc-qaVDZ+VTZ level of theory. A icMRCI/mc-qaVDZ+VTZ treatment is used, however, for CO distances beyond 2.4 bohr (for details see text). The vibrational levels are also displayed in the top panel.

where TN is the (harmonic) kinetic energy operator for all relevant modes and W is the 5 × 5 diabatic potential energy matrix given by

W ≡ W5×5

(

W11 W12 W13 W14 W15 W22 0 0 0 W ) 0 0 33 W44 0 h.c. W55

)

(3)

The diagonal elements of W5×5 correspond to the diabatic surfaces. In a 1D model (along the CO stretching coordinate), the diabatic states were first estimated graphically connecting the electronic energy points of configurations with the same character. The diabatic surfaces are displayed in the bottom panel of Figure 3, so that W11 ) (π, π*), W22 ) (n, 3py), W33 ) (n, 3dyz), W44 ) (n, 4py) and W55 ) (n, 5py). In contrast to the adiabatic potential energy curves, the diabatic ones are smooth functions of the CO coordinate. Note that only the W11 )(π, π*) is coupled directly to the Rydberg states but, however, there is no direct interaction among the Rydberg states, yielding the zero off-diagonal elements in eq 3. In the simplest model, the coupling terms are taken to be constants with a value of onehalf of the smallest energy difference at the avoided crossing region, Wij ) W0ij (i * j). Table 3 shows the location of the avoided crossings along with the values of the coupling constants (W0ij). The W0ij, in the order as they appear in the last row of 0 0 , · ,W15 ) Table 3a, correspond to the off-diagonal elements (W12 of eq 3, in the same order. Note that the W44 ) (n, 4py) electronic

state is uncoupled from the rest of electronic states, W14 ) 0. The (π, π*) valence electronic state has a minimum located at ≈2.95 au, in contrast to the 1A1 Rydberg states whose minima are close to that of the ground state. The dissociation of the (π, π*) state yields the CH2 (1A1) + O(1D) products with a dissociation energy, relative to the minimum of the ground state, of 10.20 and 10.36 eV, with and without geometry relaxation of the CH2 fragment, respectively. To account for the effect of other dofs, 2D potential energy surfaces for the five 1A1 and eight 1B2 electronic states have been calculated along the CO stretching and the symmetric HCH bending using the grids mentioned above. When doing this, the avoided crossings on the 1A1 PES, shown on the top panel of Figure 3, turn into CoIns.19 The energies and geometric locations of the CoIns are also given in Table 3b (2D PES). Note in Table 3 that when going from the 1D (a) to the 2D (b) picture, the values of the CO distances are almost unchanged at the avoided crossings and at the corresponding CoIns. The change in the HCH angle is, however, larger. This allows us to set up a 2D model Hamiltonian for the diabatization of the 2D potential energy surfaces, in the spirit of the well-established linear vibronic coupling (LVC) model.19,60 In this model, the diabatic potentials are built as a sum of two 1D terms, the CO potential (Wrii) plus the ∠HCH potential (∆Wθii); moreover the coupling terms are only HCH angle-dependent. The 2D diabatic Hamiltonian is given then by

(

r,θ W5×5 ) r θ 0 0 0 0 W11 + ∆W11 W12 + k12∆θ W13 + k13∆θ W14 + k14∆θ W15 + k15∆θ r θ W22 + ∆W22

h.c.

0

0

0

r θ + ∆W33 W33

0

0

r θ + ∆W44 W44

0 r θ + ∆W55 W55

)

(4)

where the superscript r stands for the CO stretching, θ for the symmetric HCH angle bending, and ∆θ ) θ - θFC is the angular

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Figure 4. 2D adiabatic (left panels) and diabatic (right panels) potential energy surfaces of A1 symmetry of formaldehyde, using the 2D model Hamiltonian of eq 4 (contour line spacing of 0.2 eV ranging from 8.0 to 11.0 eV). Crosses indicate the location of the CoIns. The ground-state initial wave packet is also shown in dashed lines.

displacement with respect to its value at the FC zone (θFC ) 116.54°). The W0ij (i * j) terms (see Table 3a) are again onehalf of the energy difference at the avoided crossings, as just explained for the 1D treatment along the CO stretching. On the other hand, since the coupling terms must vanish at the CoIns (when θ ) θCoIn), the interstate coupling constants kij are given by kij ) -(W0ij)/(∆θCoIn). The values of the interstate coupling constants are also given in Table 3c. To describe properly the energy gap between the adiabatic surface 31A1 and the surface resulting from the diabatization of surfaces 41A1 and 51A1 (which

has been denoted as 1A4,5 1 ), the corresponding coupling constant has been set to zero, that is, k13 ) 0. In contrast to the 1D treatment, the 41A1 electronic state is now coupled to the rest of states. Table 3c shows the energetic and geometrical locations of the CoIns resulting from the model (2D MODEL). The model reproduces the positions of the CoIns in the 2D ab initio surfaces. The adiabatic and diabatic 2D potential energy surfaces (for the 1A1 electronic states only) are plotted in Figure 4 in the right and the left panels, respectively. The points of intersection are indicated by crosses. The ground state initial wave packet


J. Phys. Chem. A, Vol. 114, No. 43, 2010 11443 functions in the atomic basis sets in the electronic structure calculations. The inclusion of the out-of-angle in the dynamical treatment is postponed to subsequent work. III. Dynamical Calculations A. Computational Method. Dynamical calculations for the rCO stretching coordinate and for the HCH valence angle have been performed on five coupled 1A1 and eight uncoupled 1B2 potential energy surfaces using a wave packet method implemented in the Heidelberg MCTDH package of programs.61–64 Photoexcitation out of the ground state wave function is considered and the latter is generated by an energy relaxation method of a suitable guess wave function. The ground state wave function is then excited vertically to the various potential energy surfaces (see Figure 4) and the photodynamics is studied by following its evolution by quantum wave packet propagation. Regarding the underlying basis functions, a fast Fourier transform representation was used for describing both coordinates and, after a study of convergence, 1024 and 100 equally spaced grid points within the range [1.89-9.44] au and [100-140]° were included for rCO stretching and the HCH angle, respectively. In the MCTDH method, the total wave function, Ψ(t), is described by an expansion in Hartree products (or configurations). These Hartree products are built as products of a set of time-dependent functions known as single-particle functions (SPFs). For more details, see ref 63 and references therein. In the case of vibronically coupled states, the wave function is expressed as a linear combination of several wave functions, each corresponding to a specific electronic state,

Figure 5. The upper panel shows a 2D cut (for ∠HCH ) 122.89°) of the 21A′ adiabatic potential energy surface along the CO stretching and the out-of-plane bending coordinates calculated at the EOM-CCSD level of theory. The CoIn between the 1 1B1 and 2 1A1 electronic states is indicated by a cross. Contour energies are from 7.95 to 9.10 eV, in steps of 0.05 eV. The bottom panel displays a cut of the 1 1B1 and 2 1 A1 electronic states along the CO stretching coordinate for ∠HCH ) 122.89° in the planar configuration.

is also plotted. Note that it lies practically at the potential minimum of the Rydberg states but on the repulsive slope of the (π, π*) state. We have computed also 3D potential energy surfaces, including the out-of-plane angle for all the electronic states shown in Table 2. As mentioned before, this mode couples the 1 A1 and 1B1 electronic states and the 1A2 and 1B2 ones, respectively. Dallos and et al.16 found a CoIn between the 11B1 (σ, π*) and the 21A1 (π, π*), which was carefully analyzed in their work. Likewise, we have also first analyzed the corresponding intersection space (or seam) between the (σ, π*) and (π, π*) electronic states, in the restricted space of higher symmetry, C2V, varying the CO stretching and the symmetric HCH bending coordinates. The energetic minimum of the seam is located at rCO ) 2.81 bohr, ∠HCH ) 122.89°, and rCH ) 2.0800 bohr, the value at the minimum of the ground state. The calculations performed by Dallos et al. yielded rCO ) 2.89 bohr, ∠HCH ) 133.00°, and rCH ) 2.04 bohr. The top panel of Figure 5 shows a plot of the 2A′ surface as a function of the CO distance and the out-of-plane angle. The results obtained by Dallos et al. are qualitatively well reproduced for CO distances larger than ≈2.5 bohr. For shorter distances, however, the results are different due to the presence of the (n, 3py) Rydberg electronic state (see bottom panel of Figure 5). This state was not computed by Dallos et al., since they did not include diffuse

n

Ψ(t) )

∑ ΨR(t)|R〉

(5)

R

where n is the number of electronic states. The spectral intensity distribution P(E) for the optical transition from the ground state to five 1A1 interacting electronic states and to eight adiabatic 1B2 states has been obtained as the Fourier transform of the time autocorrelation function C(t),

P(E) ∝

∫ exp(iEt)C(t) dt

(6)

where the latter is defined as

C(t) ) 〈Ψ(0)|Ψ(t)〉

(7)

To reduce artifacts from the Gibbs phenomenon arising from a finite propagation time, C(t) is first multiplied by a damping function cos 2(πt/2T),65,66 where T denotes the final time of the autocorrelation function. Moreover, another damping function, exp [-(t/τd)2], is used to simulate the experimental line broadening, where τd is the damping parameter. Likewise, the evolution in time of the electronic populations PR(t) and the reduced densities FR(qi,t) for the R -electronic state can be also obtained:

PR(t) ) 〈ΨR(t)|ΨR(t)〉

(8)

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Figure 6. Absorption spectrum of formaldehyde: (a) Experimental spectrum, (b) calculated 2D spectrum, (c) contribution of the 1A1 electronic states to the total spectrum, and (d) contribution of the 1B2 electronic states to the total spectrum. The experimental spectrum reproduced with permission from Figure 1 of ref 27.

FR(qi, t) )

∫ Ψ*(t)Ψ R R(t) ∏ dql

(9)

l*i

From the definition of the Hamiltonians of eqs 3 and 4, the electronic populations refer to the diabatic electronic states. Adiabatic electronic populations have been also obtained from the diabatic ones by means of a rotation (at each grid point), where the transformation matrix is given by the eigenvector matrix of the diabatic potential W of eq 4.67,68 B. VUV Electronic Absorption Spectrum. The 2D electronic absorption spectrum of formaldehyde obtained for the rCO stretching and HCH angular coordinates is presented in Figure 6. The two upper panels show the total calculated absorption spectrum and the experimental one from ref 27 for comparison. The independent contributions to the total spectrum

from the 1A1 and 1B1 electronic states are also shown in the lowest panels of this figure. Although in the electronic structure calculations the transition dipole moments have been also calculated as a function of all the degrees of freedom, in this work its coordinate dependence has not been taken into account to compute the absorption spectra. Rather, the transition dipole moments have been considered constants and equal to their values at the Franck-Condon zone (given in Table 1 and 2). To identify every electronic transition, the spectra of the 1A1 diabatic states and the 1B2 adiabatic states follow the same colorcode used to plot the 1A1 diabatic potential energy curves in the bottom panel of Figure 3, and the B2 potential energy curves in the top right panel of Figure 2. For the calculated total spectrum, a damping parameter τd ) 60 fs has been used to simulate the experimental line broadening. However, a larger



Figure 7. Calculated spectra of formaldehyde for the 1A1 electronic states: (a) 2D convoluted spectrum, (b) 1D coupled diabatic spectrum for the CO stretching coordinate, (c) 1D adiabatic spectrum and (d) 1D uncoupled diabatic spectrum. Note that for panels a, b, and d, the intensities of the part of the spectrum corresponding to the interaction region (between 120 and 145 nm) have been multiplied by a factor of 3.

value of τd ) 250 fs has been used to resolve the individual spectra. Both calculated spectra are rather similar and, therefore, we will analyze next just the one from Figure 6b. The calculated total spectrum can be divided clearly into three regions: at low energies, we find the transition to the 11B2 (n, 3s) Rydberg state, whose calculated excitation energy is slightly larger that the experimental one (7.20 vs 7.09 eV, as shown in Table 2). The structure of this band is found to be caused by a vibrational excitation in the HCH angle, although the experimental vibrational spacing is smaller, almost half of the calculated one. It was proposed by Grein et al.69 that the diffuseness and the complex vibrational structure of this Rydberg state 11B2 (n, 3s) is due to the perturbation of a nonplanar 2 1A′ state, resulting from 1B1 (σ, π*)/1A1 (π, π*) mixing. Because such a 1B2/1A′ perturbation can only take place by further lowering the symmetry to C1, participation of nontotally

symmetric vibrational modes is expected along with all the totally symmetric ones. Further calculations should be performed to clarify this situation. Note finally that the calculated intensity of the peak is smaller than the experimental one since the calculation underestimates the oscillator strength for the corresponding transition. The next region, located between 145 and 160 nm corresponds also to transitions to Rydberg states: the peak at longer wavelengths (≈154 nm) corresponds to a transition to the 21B2 (n, 3pz) Rydberg state and as shown in the bottom panel. At ≈152 nm we find the vibrational progression on the HCH angle belonging to the transition to the 21A1 (n, 3py) electronic state. The relative energetic separation between both states, 21B2 and 21A1, is smaller than the experimental one since, as commented above, most of the 1B2 state appear shifted to higher energies (see Table 2). The third region is the most striking part of the

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Figure 8. Diabatic (top panel) and adiabatic (bottom panel) populations as a function of the propagation time for the five 1A1 potential energy curves. A comparison between the 1D and the 2D model is also given. The initial wave packet is excited vertically to the (π, π*) state. The inset displays a detail at short propagation times. For the line codings see Figure 3.

spectrum, since it corresponds to the region of strong interaction. The most noticeable feature is the intensity redistribution of the (π, π*) state due to the interaction with the “surrounding” Rydberg states, showing a vibronic structure that becomes denser at higher energies. The two peaks at 139.55 and 137.20 nm in the experimental spectrum were assigned experimentally to 3d transitions27 and theoretically are also associated to the HCH vibrational excitation of the transition to the 3 1B2 (n, 3dx2-y2) state although the intensities are not well described (see panel d of Figure 6). At higher energies we find a rather intense peak that theoretically corresponds to the contribution of two states, the 41A1 (n, 4py) and 61B2 (n, 4pz) transitions. The combined intensity is only slightly lower than the experimental one at the same energy and the experimental assignment also involves excitation to a 4p Rydberg orbital. Likewise, the peak theoretically assigned as a transition to the 61A1 (n, 5py) electronic state is also in good agreement with the experimental result. Given the complex shape of the absorption spectrum, and the high excitation energies involved, we consider the overall agreement theory-experiment very satisfactory. For the sake of further analysis, the 2D total spectrum was also calculated by convoluting70 the 1D spectra resulting from independent calculations for the CO stretching and the HCH bending coordinates. In contrast to the 1D spectrum for the CO stretching mode, the spectrum for the HCH angle has been calculated within an adiabatic picture. The 2D convoluted spectrum for the 1A1 electronic states is shown in panel (a) of Figure 7. The picture follows the color codings from the top panel of Figure 3. For a better visualization of the interaction region of the spectrum, all the intensities in that region have been amplified by a factor of 3. It turns out that the 2D convoluted spectrum agrees quite well with the global 2D spectrum from panel (c) of Figure 6. Moreover, the 1D spectrum

for the CO stretching coordinate is also displayed in panel (b) of Figure 7. The main effect of the inclusion of the CH2 bending mode is the vibrational excitation in the 2 A1 electronic state. Regarding the 1D spectrum for the CO stretching coordinate, we mention that Grein et al.14 performed a 3 × 3 diabatization involving the 21A1, 31A1, and the 41A1 adiabatic states. These adiabatic states are shown in Figure 1, which has been taken from ref 13. They focused on 3 states, since the basis set did not include enough diffuse functions to properly describe states above the 3d Rydberg states. Note, for example, the differences between the (n, 4py) electronic state computed by Grein (shown in Figure 1) and the one obtained in this work (shown in the top panel of Figure 3). Their diabatization was carried out using the electric transition moments as a molecular property to follow the character of the electronic states. They computed an absorption spectrum, which is shown in Figure 7 of ref 14. It corresponds just to a part of the experimentally observed one (between 130 and 150 nm). Comparing with our 1D calculated spectra, we find a more irregular structure in this wavelength range, in better agreement with the experimental one. It is also instructive to analyze the effect of the couplings on the absorption spectrum. For this purpose, two different tests were performed. First, panel (c) of Figure 7 shows the adiabatic spectrum computed for the A1 electronic states (see colorcodings from the bottom panel of Figure 3). In contrast to the adiabatic 2 1A1 electronic state, which practically remains unaffected by including the coupling (since it lies energetically quite below from the rest of the states in the FC region), the interaction region of the spectrum changes dramatically. This change is specially noticeable for the (π, π*) electronic state, showing a simple and regular vibrational progression in the adiabatic picture but a much more irregular and broader band in the full one, due to the coupling with the other states (Figure 7a) and the repulsive slope that the wave packets feels after



Figure 9. Diabatic reduced densities (dotted lines) on the different diabatic 1A1 electronic states (solid lines) as a function of the propagation time (in fs) after excitation from the ground electronic state to the (π, π*) state. The zero of the density is located at the minimum of the potential of each state. The PEC and line codings as in the bottom panel of Figure 3.

vertical excitation. A second calculation has been also performed on the diabatic PECs but turning off the coupling, and the corresponding spectrum is shown in the panel (d) of Figure 3. This displays the same broad band but shows a very regular progression of lines, and again cannot explain the experimental findings. Summarizing, we have not only been able to give a good overall account of the complex high-energy absorption spectrum of formaldehyde, but also established the impact of the interstate couplings on its shape. C. Electronic Populations and Vibrational Densities. A deeper insight into the nonadiabatic effects on the dynamics of the system can be obtained by analyzing time-dependent quantities such as the electronic populations, shown in Figure 8 up to a propagation time of 300 fs. Figure 8 displays the

probability of the wave packet to be located in any of the five A1 electronic states after excitation to the (π, π*) state. Results for the 2D convolution as for the full 2D model are plotted on the left and right panels, respectively. For each case, the upper panel displays the adiabatic populations while the bottom panel shows the diabatic ones. In every panel, the inset shows a detail for short propagation times. Note that the dynamics is quite similar in both cases, specially for short times as can be seen in the insets. The dynamics of the system can be also followed in the snapshots of Figure 9, where the vibrational densities on the diabatic states are depicted for different propagation times along the CO stretching coordinate. Analyzing the diabatic populations and the vibrational densities, the dynamics of the system can be described as follows: the system, after initial photoexcitation, 1

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undergoes a (π, π*) f (n, 3dyz) transition of the order of 5 fs (see inset of Figure 8). Although that propagation time is not shown in Figure 9, note that at 20 fs the (n, 3dyz) state is already populated. The transfer is very fast owing to the proximity of the crossing to the FC point. Approximately 40% of the wave packet is transferred initially between these two states. The part of the wave packet transferred to the (n, 3dyz) state evolves on this state and subsequently reaches again the interaction region, transferring some population back to the (π, π*) electronic state. At ≈25 fs the system reaches the second interaction region between the (π, π*) and (n, 3py) states (see also the corresponding panel of Figure 9 at 20 fs), populating slightly the latter owing to the smaller W12 coupling constant compared to W13. After that, the system remaining on the (π, π*) state keeps evolving, bouncing, and then traveling back reaching again the intersection (π, π*)/(n, 3dyz). In 1D, the dynamics can be mainly seen as the population transfer among the (π, π*) and (n, 3dyz) states, with the corresponding oscillation periods of ≈100 fs and ≈5-10 fs on the (π, π*) and on the (n, 3dyz) states, respectively. In 2D the (n, 3py) state becomes more strongly populated along the propagation. On the other hand, when exciting to the (π, π*) state, the intersection between the (π, π*) and the highest state is practically not accessible and, hence, not populated. When looking at the adiabatic populations, instead, the picture is more difficult to analyze since all the adiabatics states are involved in the dynamics. IV. Conclusions In this work, we have performed an ab initio study of the multistate photodynamics of formaldehyde for excitation energies of 7-10 eV. Optimized geometries, excitation energies, and oscillator strengths have been computed with several basis sets, including diffuse functions in order to get a balanced description of both valence and Rydberg states. It has been concluded that, at least, quadruply augmented diffuse functions are needed for that purpose. The results have been also compared with previous theoretical work and available experimental data, leading in general to good agreement. Further electronic structure calculations have been carried out to obtain the potential energy curves for all degrees of freedom of formaldehyde. Those calculations are based in the EOMCCSD method, combined with a SA-CASSCF/MRCI treatment for long CO distances, where a single-reference method is not applicable. The most relevant vibrational mode is the CO stretching coordinate, owing to several avoided crossings that are present in the potential energy curves along this mode. Subsequently, 2D (including the CO stretching and the HCH bending coordinates) potential energy surfaces have been calculated. When including these two degrees of freedom, the previous avoided crossings turn into CoIns. These CoIns have been characterized and analyzed. The presence of the abovementioned avoided crossings and CoIns calls, in fact, for a multistate dynamical treatment adopted in this work. To this end, a diabatic 2D model Hamiltonian, inspired by the wellknown linear vibronic coupling model,19,60 has been constructed and the corresponding PESs been found to well reproduce the ab initio data. The resulting photodynamics of formaldehyde was studied by means of a quantum wave packet method as implemented in the MCTDH Heidelberg package of programs. The total electronic absorption spectrum has been computed and the contribution from the different involved electronic states has been also analyzed. Moreover, 1D absorption spectra have been also obtained for the CO stretching and the HCH bending

Go´mez-Carrasco et al. coordinates. In order to establish the importance of the couplings, the uncoupled-surface spectrum was also computed for comparison, and found to look very different from the coupled-surface spectrum. In addition, time-dependent electronic populations and reduced densities were shown, which also reveal the nonadiabatic nature of the nuclear motion. For the first time, the complex shape of the VUV photoabsorption spectrum of formaldehyde could be qualitatively well reproduced by theory: strong vibronic coupling of the bright (π, π*) state with several neighbouring dark Rydberg states in the FC region causes the bright state to transfer intensity to the latter and thus yields a very irregular vibrational progression. To the best of our knowledge this is the first quantum dynamical treatment of more than two intersecting potential energy surfaces of the same symmetry. Further improvement should be possible by extending the present modeling of the vibronic coupling Hamiltonian, for example, by utilizing the concept of regularized diabatic states.71,72 Work along these lines is currently underway in our group. Acknowledgment. S. Gomez-Carrasco thanks the Alexander von Humboldt Foundation for financial support. The authors are indebted to H. Lischka for useful discussions and to COST ACTION D37/Working Group Photodyn for financial support. References and Notes (1) Calvert, J. G.; Pitts. J. N. In Photochemistry; Wiley: New York, 1966. (2) Robin, M. B. In Higher Excited States of Polyatomic Molecules; Academic Press: New York, 1985; Vol. III. (3) Snyder, L. E.; Buhl, D.; Zuckerman, B.; Palmer, P. Phys. ReV. Lett. 1969, 22, 679. (4) Finlayson-Pitts, B. J.; Pitts, J. N. In Atmospheric Chemistry; Wiley: New York, 1986. (5) Whitten, J. L.; Hackmeyer, M. J. J. Chem. Phys. 1969, 51, 5584. (6) Buenker, R. J.; Peyerimhoff, S. D. J. Chem. Phys. 1970, 53, 1368. (7) Peyerimhoff, S. D.; Buenker, R. J.; Kammer, W. K.; Hsu, H. Chem. Phys. Lett. 1971, 8, 129. (8) Harding, L. B.; Goddard, W. A. J. Am. Chem. Soc. 1977, 99, 677. (9) Fitzgerald, G.; Schaefer, H. F. J. Chem. Phys. 1985, 83, 1162. (10) Hachey, M. R. J.; Bruna, P. J.; Grein, F. J. Phys. Chem. 1995, 99, 8050. (11) Merchan, M.; Roos, B. O. Theor. Chem. Acc. 1995, 92, 227. (12) Gwaltney, S. R.; Barlett, R. J. Chem. Phys. Lett. 1995, 241, 26. (13) Grein, F.; Hachey, M. R. J. Int. J. Quant. Chem: Quant. Chem. Symp. 1996, 30, 1661. (14) Peric´, M.; Grein, F.; Hachey, M. R. J. J. Chem. Phys. 2000, 113, 9011. (15) Müller, T.; Lischka, H. Theor. Chem. Acc. 2001, 106, 369. (16) Dallos, M.; Müller, T.; Lischka, H.; Shepard, R. J. Chem. Phys. 2001, 114, 746. (17) Davidson, E. R.; McMurchie. L. E. In, Excited States; Academic Press: New York, 1982; Vol. 5. (18) Hachey, M. R. J.; Bruna, P. J.; Grein, F. J. Mol. Spectrosc. 1996, 176, 375. (19) Domcke, W.; Yarkony, D. R.; Ko¨ppel, H., Eds.; Conical Intersections: Electronic Structure, Dynamics and Spectroscopy; Word Scientific: Singapore, 2004. (20) Metall, J. E.; Gentiu, E. P.; Krauss, M.; Neumann, D. J. Chem. Phys. 1971, 55, 5471. (21) Moule, D. C.; Walsh, A. D. Chem. ReV. 1975, 75, 67. (22) Lessard, C. R.; Moule, D. C. J. Chem. Phys. 1977, 66, 3908. (23) Moore, C. B.; Weisshaar, J. C. Annu. ReV. Phys. Chem. 1983, 34, 525. (24) Clouthier, D. J.; Ramsay, D. A. Annu. ReV. Phys. Chem. 1983, 34, 31. (25) Clouthier, D. J.; Moule; D. C. In, Topics in Current Chemistry; Springer-Verlag: Berlin, 1989; Vol. 150, p 167. (26) Brint, P.; Connerade, J. P.; Mayhew, C.; Sommer, K. J. Chem. Soc. Faraday Trans. 2 1985, 81, 1643. (27) Suto, M.; Wang, X.; Lee, L. C. J. Chem. Phys. 1986, 85, 4228. (28) Weiss, M. J.; Kuyat, C. E.; Mielczraek, S. J. J. Chem. Phys. 1971, 54, 4147. (29) Chutjian, A. J. J. Chem. Phys. 1974, 61, 4279. (30) Taylor, S.; Wilden, D. G.; Comer, J. J. Chem. Phys. 1982, 70, 291.

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Ab Initio Study of the VUV-Induced Multistate Photodynamics of

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