Vibronic Coupling Effects in Resonant Auger ... - ACS Publications

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Vibronic Coupling Effects in Resonant Auger Spectra of H2O Matthis Eroms,*,† Martin Jungen,*,‡ and Hans-Dieter Meyer*,† †

Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, INF 229, D-69120 Heidelberg, Germany Institut für Physikalische Chemie, Universität Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland

‡

ABSTRACT: We present a theoretical investigation of the resonant Auger effect in gas-phase water. As in our earlier work, the simulation of nuclear dynamics is treated in a one-step picture, because excitation and decay events cannot be disentangled. Extending this framework, we now account for the vibronic coupling in the cationic final states arising from degeneracies in their potential energy surfaces (PESs). A diabatization of the cationic states permits a correct treatment of non Born−Oppenheimer dynamics leading to a significantly better agreement with experimental results. Moreover, we arrive at a more balanced understanding of the various spectral features that can be attributed to nuclear motion in the core-excited state or to vibronic coupling effects. The nuclear equations of motion have been solved using the multiconfiguration time-dependent Hartree (MCTDH) method. The cationic PESs were recalculated using the coupled electron pair approach (CEPA) whereas previously a multireference configuration interaction method had been employed. core hole lifetime is reported to be 4.1 fs,12 which is comparable to vibrational periods or dissociative motion of the nuclei. As a result, resonant Auger spectra not only reflect the orbitals from which the electrons are emitted but also reflect the intrinsic nuclear dynamics taking place in the core-excited state and, after the electronic decay, in the cationic final states. Theoretical work on polyatomic molecules, like H2O, is still scarce.5,12 A reason lies in the computationally expensive calculation of potential energy surfaces (PESs) from which the spectrum can be obtained by either diagonalization or wave packet propagation, the latter being numerically advantageous and providing also valuable insight into the time development of the spectrum and the nuclear motion. We already reported earlier on the RAE in water13 (paper I). There, coupled equations of motion were set up to calculate the spectra in a time-dependent framework. This method was first put forward in refs 14 and 15, and it was applied to the computation of resonant Auger spectra of several diatomic molecules, e.g., O2 and N215 and HF.16 As was pointed out in paper I, the PESs for the cationic states exhibit several degeneracies. In particular, at linearity of the H2O+ cation, the X 2B1 ground state and the A 2A1 first excited state are degenerate components of a 2Πu state leading to Renner−Teller coupling. Furthermore, a conical intersection between the A 2A1 state and the B 2B2 second excited state is well-known.17,18 If the energetic separation of the PESs becomes comparable to vibrational quanta, i.e., around several hundred millielectronvolts, the Born−Oppenheimer approximation becomes invalid.

1. INTRODUCTION Over the past two decades, experimental techniques for investigating highly excited states of atoms and molecules have been greatly refined. Most notably, light sources of high spectral brightness are available at synchrotron facilities of the third generation, cf. refs 1−3. These radiation sources owe their versatility also to a wide range of tunability and narrow spectral bandwidth facilitating the excitation of short-lived resonances of, e.g., molecular samples in the gas phase. This, in turn, has opened the path for precise analysis and assignment of electronic decay spectra, in particular in the resonant Auger effect (RAE). In addition, the development of highly resolving electron energy analyzers has been essential for research activities carried out in the field of resonant core excitation and decay phenomena. Recently, results from several different experiments on the RAE in water have been published.4−10 The observations revealed that excitation in the photon energy region of 533− 537 eV leads to the emission of highly energetic electrons and a strong dependence of the spectrum on the exact choice of excitation frequency. From a theoretical point of view, first an intermediate, neutral state is resonantly created by the external field: One electron is removed from the core orbital, which corresponds mainly to the oxygen 1s orbital, and an unoccupied valence or Rydberg orbital is populated. Being in a highly excited state, the molecule quickly decays into a cationic final state by emitting an Auger electron. Electronic structure calculations suggest emission essentially from the outer valence orbitals of the H2O molecule,11 if the kinetic energy of the Auger electron exceeds 510 eV. Being able to very closely examine the electron dynamics is not the only scientific interest in this research area. The changes in molecular structure are also accomplished by its nuclear constituents as the vibrational motion evolves on a time scale similar to that of the electronic transitions. The experimental © XXXX American Chemical Society

Special Issue: Jorn Manz Festschrift Received: May 14, 2012 Revised: July 4, 2012

A

dx.doi.org/10.1021/jp304666k | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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Recollecting from the Introduction that the molecule can be in the initial (i), decaying (d), or in a cationic final state (f) at any instant of time during the resonant Auger process, the full wave function of the H2O molecule may be written as

As a result, vibrational and electronic degrees of freedom couple (vibronic coupling). Although in paper I vibronic coupling was ignored, the agreement between the computed spectra and the experimental results of refs 5, 6, and 10 was already satisfying. In a more recent work19 (paper II), we calculated the photoelectron (PE) spectrum of H2O. There, a transformation of the PESs was presented that corresponds to a diabatization of the electronic basis set. In the diabatic basis, singularities in the matrix of the kinetic energy operator vanish, and only in this basis could correct results for the PE spectrum be obtained, as a comparison of spectra stemming from the (coupled) diabatic and (uncoupled) adiabatic approaches showed. The same final states are reached both in the RAE and after direct photoionization. What can be expected for the numerical results if the RAE is revisited working in a diabatic representation? In the PE spectrum, the observed resolution is much higher than in the RAE due to the excited state dynamics. The individual line intensities of the PE spectrum are entirely determined by the Franck−Condon factors, given by the overlap of the ground and final state nuclear wave functions evaluated at the relevant geometry for the process, the Franck−Condon point. If a core-excited state is also involved, however, there is no well-defined geometry where the decay occurs, because excitation and decay are governed by the duration and shape of the excitation function and the statistical nature of the decay process. Considering the interference of the wave packet in the final state manifold with contributions arriving continuously from the intermediate state, it becomes clear that on the one hand, the spectrum is less resolved in the RAE. On the other hand, it was shown in paper I and also experimentally7 that some control over the nuclear motion in the final states can be gained through the choice of the intermediate state. Moreover, detuning of the excitation frequency has a pronounced influence on the spectral envelope.10 To summarize, the main scope of paper I was to apply the onestep model of resonant Auger decay for the first time to a polyatomic molecule starting from ab initio PESs. Detuning effects and the time development of the spectra were studied and the final vibrational distribution was analyzed. The diabatic PESs were then introduced in paper II. In the following, we will show to which extent vibronic coupling effects can be noted in resonant Auger spectra of water. The outline of this work is as follows. The main theoretical aspects, in particular the equations of motion for the nuclei and the diabatization are formulated and presented in section 2. In section 3, we address several aspects of the computational methods used for the simulation and the analysis of the results. They are presented in section 4 where some attention is devoted to the nuclear dynamics in the core-excited states before the simulations of several experimental resonant Auger spectra are shown. This is followed by an analysis and interpretation of our findings preceding our final conclusions in section 5.

|Ψ(t )⟩ = |ϕi⟩|ψi(t )⟩ + |ϕd⟩|ψd(t )⟩ 3

+

∑∫ k=1

∞

0

dE |ϕf̃ (E)⟩|ψf (E ,t )⟩ k

k

(1)

In the spirit of the Born−Oppenheimer approximation, it is assumed that the dependence on electronic coordinates can be shifted to one set of functions ϕi,d,f which also depend parametrically on the nuclear configurations, whereas the time dependence of Ψ(t) is taken over by a different set of wave functions residing exclusively in the nuclear coordinate space. Any of these functions ψi,d,f can be regarded as a weight function reflecting the occupation of the respective electronic state. Under the conditions assumed here, the initial and decaying states of the molecule can be described to sufficient accuracy by only one electronic state each whereas the final state is given by the sum over three cationic states. Each of them also includes the outgoing Auger electron taking away the energy E as is indicated by the tilde symbol in ϕ̃ f(E); i.e., ϕ̃ f(E) is an antisymmetrized product of a cationic state ϕf and a wave function of the outgoing electron with kinetic energy E. Because the electron energy E is given by the difference of photon energy and binding energy of the electron and may assume an arbitrary value, a continuum is formed into which the decay from the core-excited state takes place. The full wave function Ψ(t) obeys the Schrödinger equation (atomic units are used throughout this text) iΨ̇(t ) = Ĥ (t ) Ψ(t )

(2)

with the total Hamiltonian Ĥ (t) being composed of the nuclear kinetic energy operator T̂ n and the electronic Hamiltonian Ĥ el, which together form the well-known molecular Hamiltonian Ĥ mol, and an additional term describing the interaction with the radiation field E(t) in the dipole approximation, hence mol Ĥ (t ) = Tn̂ + Ĥ el − D̂ ·E(t ) = Ĥ − D̂ ·E(t )

(3)

The term D̂ stands for the dipole operator which for a weak field only couples the states ϕi and ϕd. Direct photoionization is disregarded as its contribution is not significant in the photon energy range considered here. Because the time dependence of the system is contained in the nuclear part of the ansatz (1), equations of motion for the set of nuclear wave functions, ψ (E ,t ) = (ψi(t ), ψd(t ), ψf (E , t ), ψf (E , t ), ψf (E , t ))T 1

2

3

(4)

are derived by integrating out the electronic degrees of freedom. Thus, the dipole transition matrix element is formulated as

2. THEORETICAL BACKGROUND 2.1. Equations of Motion for the Nuclei. To take into account the possibility of interference effects in the nuclear dynamics, we aim at a simulation of the resonant Auger effect on a quantum mechanical level. This should also make transparent how the evolution of the nuclear motion in several coupled electronic states can affect the resonant Auger spectra. The derivation of the equations of motion which we will summarize in the following goes back to paper I and ref 16.

F (̂ t ) = −⟨ϕd|D̂ ·E(t )|ϕi⟩=: Fĝ (t )

(5)

The matrix elements of the molecular Hamiltonian deserve special attention. Coupling between vibrational and electronic degrees of freedom can be induced by the action of the derivative terms in T̂ n on the electronic states, eventually leading to diverging off-diagonal matrix elements for degenerate states. We assume that this vibronic coupling is negligible for the ground and decaying states in view of their energetic separation and weak B



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where ω is the excitation frequency. The sine-squared envelope is chosen for its advantage of a compact support. The pulse length is given by T, and the step functions Θ(t) and Θ(T − t) switch the pulse on and off. The spectral width of such a pulse is 8.25 eV fs T−1 fwhm. Furthermore, in the Condon approximation, the coupling element F̂ is treated as a constant, as the excitation takes place only around the equilibrium configuration of the molecule. In a similar vein, also the coupling element Ŵ fk and consequently Γ̂ as well are considered constant. From experimental studies as well as from the considerations of paper I, the value of Γ is assumed to be 150 meV corresponding to a core hole lifetime of 4.1 fs. After solving the set of eqs 10, the resonant Auger spectrum is given by the norm of the nuclear final state wave functions ψf(E,t) due to their weight character. Hence,

dependence on nuclear coordinates. The examination of the decaying states in paper I is also reminded here, which concluded that at least for the two lowest core-excited states (1A1 and 1B2), to which this study is confined, vibronic coupling is insignificant. Hence, the initial and final state nuclear wave functions evolve according to the Hamiltonians Ĥ i = Tn̂ + ⟨ϕi|Ĥ el|ϕi⟩

(6)

and Ĥd = Tn̂ + ⟨ϕd|Ĥ el|ϕd⟩

(7)

where the diagonal matrix elements of Ĥ el are the adiabatic PESs of these states. In contrast to this, vibronic coupling of the final states greatly affects the nuclear dynamics as is known from paper II. There, a diabatization of the adiabatic potential energy matrix was performed. In the diabatic basis, the kinetic energy matrix is diagonal on the expense of coupling terms in the potential energy matrix. The latter, however, are not singular and make a successful simulation of nuclear dynamics in the presence of vibronic coupling feasible. For the time being, all diagonal and off-diagonal matrix elements Ĥ mol kk′ of the molecular Hamiltonian with respect to the final states ϕfk, ϕfk′ are formally retained. Later, the diabatic Hamiltonian of the cation will enter the equations of motion explicitly. Coupling between the decaying and final states is mediated by Ŵfk(E) = ⟨ϕf̃ (E)|Ĥ el|ϕd⟩

3

σ(E ,t ) =

3

† Γf̂ k := 2πŴ fk Ŵfk

with

(9)

k=1

and is related to the lifetime of the core hole. The imaginary part of /̂ d eventually leads to a decay of the wave function ψd(t) over time. Now, the equations of motion for the nuclei assume the form

iψ̇ = Ĥ (E ,t )ψ

(10)

where the matrix Hamiltonian reads ⎛ Ĥ ⎜ i ⎜ Fĝ (t ) ⎜ ⎜ ̂ E ,t ) = ⎜ 0 H( ⎜ ⎜0 ⎜ ⎜ ⎝0

0

0

0

/̂ d 0

0

̂ mol + E H12 ̂ mol Ŵf1 H11 mol Ŵf2 Ĥ 21

mol Ĥ 22

̂ mol Ŵf3 H31

̂ mol H32

⎞ ⎟ ⎟ 0 ⎟ ⎟ mol ̂ H13 ⎟ ⎟ mol ⎟ + E Ĥ 23 ⎟ ̂ mol + E ⎟ H33 ⎠ 0

(11)

As in paper I, the excitation function g(t) is given by ⎛ πt ⎞ g (t ) = exp( −iωt ) sin 2⎜ ⎟ Θ(t )Θ(T −t ) ⎝T ⎠

k

(13)

gives the intensity of electrons with energy E recorded after a time t has passed since the onset of the process. For large values of t, σ(E,t) corresponds to the spectral intensity accessible in the experiment. 2.2. Discussion of the Matrix Elements of the Electronic Hamiltonian. To perform the integration of (10), the Hamiltonians Ĥ i and Ĥ d as well as the matrix elements Ĥ mol kk′ of the diabatic states have to be known. For the ground and decaying states, the results from paper I have been adopted. The ground state PES was taken from the literature20 as a spectroscopically determined surface. The calculation of the core-excited states was already detailed in paper I. We decided to recalculate the cationic states using the coupled electron pair approach (CEPA). More precisely, the MCCEPA program by Fink and Staemmler21 was used because it leads to more accurate results than the multireference configuration interaction method employed for the results of paper I. In total, now eight cationic states were documented in either the A′ or A″ representations of Cs symmetry group. The calculations were carried out on a product grid, using 19 values for the O−H distances, r1 and r2, between 1.1 a0 and 10.8 a0 and 19 values for the H−O−H angle θ between 0° and 180°. As we are only interested in the participator Auger spectrum, only the three lowest lying, principal states of H2O+ are considered, namely the X 2A″, A2A′, and B 2A′ states. For future reference, we introduce the labels VX, VA, and VB for their adiabatic PESs and ϕX, ϕA, and ϕB for the corresponding adiabatic electronic wave functions. To discuss the improvements attained with the new PESs, we compare photoelectron (PE) spectra obtained with old and new surfaces in Figure 1. The computation of a PE spectrum is comparatively simple. Adopting the Condon approximation, we create the initial wave function ψ(t = 0) by placing the vibrational and electronic ground state of H2O vertically onto the coupled cationic PESs. For the propagation hence only the lower 3 × 3 block of the matrix in eq 11, or eq 25 below, is used with E = 0. To obtain the PE spectrum, the Fourier transform of the autocorrelation function c(t) = ⟨ψ(0)|ψ(t)⟩ is evaluated. To arrive at an autocorrelation length of twice the propagation time, we use the so-called t/2-trick and evaluate the autocorrelation as c(t) = ⟨ψ*(t/2)|ψ(t/2)⟩. The calculation using the new PESs reproduces correctly especially the vibrational peak pertaining to bending motion in the X state, as is marked in the figure, whereas the spectral

Formally, the decaying state depends on its own history through the coupling Ŵ †fk(E). For a large energetic separation of the decaying and final states, which is fulfilled here, an effective Hamiltonian /̂ d = Ĥ d−iΓ̂/2 can be introduced in a local approximation.16 The operator Γ̂ is given by Γ̂ :=∑ Γf̂ k

k

k=1

(8)

k

∑ ⟨ψf (E , t )|ψf (E , t )⟩

(12) C



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Figure 2. Cut through the PESs of the three lowest lying adiabatic states of the water cation. Both O−H distances are kept fixed at 2.0 a0 and the bond angle is varied. The X and A states are Renner−Teller coupled at 180°, and the A and B states exhibit a conical intersection. These degeneracies of the PESs induce coupling of the vibrational and electronic degrees of freedom (vibronic coupling). Figure 1. Calculated and experimental18 photoelectron spectra of H2O. The spectral bands X, A, and B can be related to electron emission from the three principal states of the water cation. The accuracy of the numerical results is improved using the new set of PESs instead of the previous ones, especially at the peak from the bending vibrational mode in the X band marked as (0, 1, 0) and regarding the exact position of the A band. The alternating line structure in the A band is reproduced utilizing even and odd values for the vibronic quantum number K, which is introduced due to the Renner−Teller effect.

Furthermore, the VX and VA curves become degenerate at 180°, inducing Renner−Teller coupling of the X and A cationic states which is discussed in more detail below. 2.3. Diabatization of the Cationic States. The transformation to diabatic states for the cation was put forward in paper II and evolves in two consecutive steps which are formally performed by acting with two unitary transformations S1 and S2 onto the adiabatic electronic states or onto the adiabatic matrix V which assumes the form

intensity calculated from the old PESs is too low. Another improvement is the position of the A band. Note that in the spectrum from the old PESs, this band is situated at slightly higher binding energies than in the experiment and as compared to the newly obtained spectrum. In an adiabatic calculation, a shift of the VA PES could have been introduced. This procedure is unfavorable, however, if vibronic coupling is included as such a shift would lift the degeneracy of the X and A states at linear geometry. Using the new PES for the A state, the A band is simulated at the correct spectral position. The Renner−Teller coupling of the X and A states makes it necessary to introduce a vibronic quantum number K. As is pointed out below in section 2.3, the alternating line structure of the bending states in the A band which can be observed in Figure 1, can only be simulated if K takes on both even and odd values. To illustrate the origin of the vibronic coupling in the final states, Figure 2 displays a cut through their adiabatic PESs at r1 = r2 = 2.0 a0. The H−O−H angle is varied between 20° and 180° (the equilibrium angle of the neutral ground state is 105°). Two regions are of interest. Around 75°, there is a conical intersection of the VA and VB curves. According to the theory of conical intersections, summarized, e.g., in ref 22, this point is part of an intersection seam which in the case of the water molecule is a one-dimensional line in the three-dimensional nuclear coordinate space. The seam lies in the plane of the two so-called tuning modes, i.e., symmetric stretch of the two O−H distances and bending, whereas the third mode (asymmetric stretch) accomplishes the coupling of the two electronic states. Thus, the conical intersection occurs only in C2v symmetry (see also Appendix A of paper II).

⎛ VX 0 0 ⎞ ⎜ ⎟ V = ⎜ 0 VA 0 ⎟ ⎜ ⎟ ⎝ 0 0 VB ⎠

(14)

First, the conical intersection was considered and the PESs VA and VB were transformed, leaving the adiabatic PES VX unaltered, hence

D1 =

S1VS1†

⎛ VX 0 0 ⎞ ⎜ ⎟ = ⎜ 0 D11 D12 ⎟ ⎜ ⎟ ⎝ 0 D12 D22 ⎠

(15)

yielding diabatic PESs D11 and D22, as well as a coupling matrix element, D12. Details can be found in the Appendix A of paper II. Note that there, the letter W was used for the diabatic matrix elements. Second, we considered the kinetic energy operator (KEO) of the nuclei in more detail. Setting the Renner−Teller effect aside for the time being, we modified the exact KEO, which is given in eq 3 of paper II, following the lines of ref 23. Invoking the coupled states or centrifugal sudden approximation, the Coriolis coupling of nuclear states possessing different values of k can be disregarded. Working in a body-fixed (bf) frame of reference, k is the eigenvalue of jẑ , the projection of the nuclear angular momentum operator onto the bf z-axis. The decoupling turns k into a good quantum number and a more simple expression of the KEO which depends parametrically on k is obtained (compare with paper II): D


The Journal of Physical Chemistry A Tn̂ (k) =

p1̂ 2 2μ1 −

+

p1̂ pθ̂ mOr2

p2̂ 2

2

+

2μ2 −

2

ĵ

2μ1r12

p2̂ pθ̂ mOr1

Article

+

ĵ

2μ2 r2 2 2

−

+

nearly linear configurations23 and postulate the validity of eq 21 for all relevant geometries. Thus, the states ϕ± are subject to different values of k, as can be seen from eq 19:

p1̂ p2̂ cos θ mO

2

cos θj ̂ + j ̂ cos θ 2mOr1r2

k = Jẑ − lẑ = K ∓ 1.

(16)

with the reduced masses μ1 = μ2 = mOmH/(mO + mH). The momentum operators are defined as ̂ = −i p1,2

∂ ∂r1,2

pθ̂ = −i

∂ sin θ ∂θ

The eigenvalue K is varied over a small range (up to K = 5) and the simulations consist of separate calculations for each value of K. In general, K can also be negative. However, it suffices to take into account only non-negative values of K and weight each calculation where K > 0 twice, because the KEO depends on K solely through the k2-term in j2̂ . As is outlined in paper II, K should at least take on the values 0 and 1 for a simulation of the alternating bending vibrational lines in the A band where even values of the bending quantum number are related to odd values of K and vice versa. According to eq 20, the second unitary transformation, S2, can be given explicitly:

(17)

∂ 1 ∂ k2 + sin θ (18) ∂θ sin θ ∂θ sin 2 θ The Renner−Teller vibronic coupling, however, couples nuclear (jẑ ) and electronic (lẑ ) angular momenta and only the projection 2

ĵ = −

Jẑ = jẑ + lẑ

(19)

of the total angular momentum is conserved. The eigenvalue of jẑ is called K and, as usual, one may use the labels Σ, Π, Δ, Φ equivalently for K = 0−3. The Renner−Teller effect shifts the vibrational energies through the term k2/sin2 θ in j2̂ , which only contributes significantly close to linear nuclear arrangements where the states ϕX and ϕA become degenerate. At such geometries, the relation lẑ ϕA/X = ±iϕX/A holds, which implies the following choice of the diabatic electronic states for the Renner−Teller subsystem: 1 (ϕ ± iϕX ) ϕ± = (20) 2 A

1 S2 = 2

D = S2D1S2†

(21)

0 /̂ d Ŵf1 Ŵf2 Ŵf3

(23)

⎛ D11 + VX D12 ⎜ 2 2 ⎜ ⎜D D22 = ⎜ 12 ⎜ 2 ⎜D − V D X 12 ⎜ 11 ⎝ 2 2

D11 − VX ⎞ ⎟ 2 ⎟ ⎟ D12 ⎟ 2 ⎟ D11 + VX ⎟ ⎟ ⎠ 2

(24)

Substituting the diabatic matrix into the equations of motion (11) yields the concrete form of the matrix Hamiltonian in the diabatic representation

strictly only holds for linear arrangements of the nuclei, but one can assume that Renner−Teller coupling is only considerable for ⎛ Ĥ ⎜ i ⎜ Fĝ (t ) ⎜ ⎜ ⎜0 ̂ E ,t ) = ⎜ H( ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎝

⎛i 1 0 ⎞ ⎟ ⎜ ⎜0 0 2 ⎟ ⎟ ⎜ ⎝− i 1 0 ⎠

and the final diabatic matrix is found to be

The eigenvalue equation

lẑ ϕ± = ±ϕ±

(22)

⎞ ⎟ ⎟ 0 0 0 ⎟ ⎟ D V D D V + − X 12 11 X Tn̂ (K − 1) + 11 +E ⎟ 2 2 2 ⎟ ⎟ D12 D12 ⎟ Tn̂ (K ) + D22 + E 2 2 ⎟ ⎟ D11 − VX D12 D + VX Tn̂ (K + 1) + 11 + E⎟ ⎠ 2 2 2 0

0

Note that the state-dependent KEO T̂ n(K) enters the matrix Hamiltonian with a value of k = K−1 for the state ϕ+ and k = K + 1 for the state ϕ−. For the third cationic state (corresponding to D22), a value of k = K is always assumed as this state does not involve a change of nuclear angular momentum. The initial nuclear wave function ψi was chosen as the total (vibrational and electronic) ground state of the H2O molecule. Assuming a temperature of the gas sample of up to 300 K, vibrational excitation of the neutral molecule is negligible. The initial wave functions differ only by their nuclear angular momentum k as the diabatization requires separate calculations

0

(25)

starting from states with k = K ± 1 to simulate decay to the Renner−Teller coupled states or k = K for decay to the D22 state. The spectra obtained from these calculations are weighted by thermal weight factors due to the Boltzmann distribution of the k-states besides the overall factor of 2 to account for negative values of K. The Auger electron energy E is the main parameter in the calculations. For a good resolution of the spectra, E was increased in steps of 50 meV over the whole range. Employing the time-dependent Schrödinger equation for the investigation of the nuclear dynamics in the resonant Auger E



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process is necessary due to the inclusion of the excitation function g(t). It is also advantageous to work in the timedependent framework as it is numerically more efficient than time-independent methods. It also allows for direct insight into the physics of the process as the time evolution of the nuclear wave packets can be observed. Nuclear motion evolves in a more complex manner already in the decaying state and proceeds in the diabatic coupled final states. Therefore, as will be shown in section 4, the resonant Auger spectrum displays a much more involved appearance than the photoelectron spectrum, bearing signatures of both the excited and final state nuclear dynamics.

3. COMPUTATIONAL METHODS Several issues shall be addressed in the following regarding the numerical computation of the quantities which are relevant in this work. 3.1. Wave Packet and Surface Representation. As in the previous studies, reported on in papers I and II, the multiconfiguration time-dependent Hartree (MCTDH) method was used to solve the highly complex coupled differential equations. In MCTDH, the nuclear wave function is expanded in Hartree products formed by time-dependent one- or low-dimensional single-particle functions (SPFs). These, in turn, are represented by linear combinations of time-independent primitive basis functions, e.g., using a discrete variable representation (DVR). The efficiency of MCTDH is essentially due to the time dependence of the SPFs, which adapt themselves flexibly to the time evolution of the wave function. Thorough reviews of this method can be found, e.g., in refs 24−30. A summary is also given in section III of paper II. The Heidelberg MCTDH program package can be obtained from the authors upon request.31 The MCTDH method requires that the PESs, too, are expanded in products of one-dimensional functions, so-called natural potentials, to speed up the integration of the MCTDH working equations. For this purpose, the POTFIT program which is part of the MCTDH package provides the necessary transformation of the quantum chemically obtained energy values to the product representation. The underlying working equations of POTFIT are laid out in more detail in, e.g., refs 32 and 33. Still, after the transformation of the PESs to the product form suitable for MCTDH, they are only known on the coarse product grid of the ab initio calculations. In a wave packet propagation, however, this grid would not allow for a satisfying representation of higher nuclear momenta. Therefore, the chnpot interpolation utility that is also implemented in the MCTDH program package was used to provide the PESs on a fine grid by interpolating the one-dimensional natural potentials utilizing splines or polynomial fits. To represent the coordinates r1 and r2, 90 grid points were used and the angle θ required the usage of 70 grid points. 3.2. Dissociation Analysis. Anticipating the discussion of nuclear dynamics in the core-excited states in section 4.1, Figure 3 illustrates the different shapes of the two decaying state PESs. The ground state is added for comparison. One expects that the wave packet which evolves in the 1A1 core-excited state will be accelerated toward the end of the grid due to the repulsive force exerted in this state. This rises numerical complications as the wave packet incurs artificial reflection upon reaching the last grid point. Hence, interference with components traveling with different momenta will lead to an incorrect time evolution of the wave packet. This, in turn, also affects the final state wave functions and thus the spectrum. It is therefore beneficial to remove the wave packet close to the end of

Figure 3. Cut through the PESs of the 1A1 and 1B2 core-excited states and of the ground state (GS). One O−H distance is kept fixed at r1 = 1.8 a0 and the angle at θ = 105°. The energy scale and the GS curve are shifted for convenience (the actual separation of the GS and 1A1 curves is around 531 eV). The repulsive shape of the 1A1 curve leads to dissociation of the molecule. In contrast, the 1B2 state supports vibrationally bound states.

the grid in the intermediate and the final states. This can conveniently be achieved with the aid of a complex absorbing potential (CAP), which is added to the diagonal elements of the matrix in eq 25. The form of the CAP used here is Cα = −iηΘ(rα − r0)(rα − r0) p

α = 1, 2

(26)

The CAP is switched on by the step function at r1,2 = r0. The strength η, the CAP length given by the starting point r0, and the order p have to be carefully evaluated before the start of the actual calculation to reduce unwanted reflection of the wave packet from the CAP and to ensure complete absorption by it. Here, the values η = 0.021 15 au, r0 = 8.8 a0, and p = 3 were used. A detailed investigation of CAP properties and a discussion of its various parameters are provided in refs 34 and 35. Although unphysical reflection can be prevented by the CAP, the decrease of the final state population due to absorption of the wave function reduces the spectral intensity, as eq 13 shows. The final state population changes due to the CAP according to 2

d |ψ (t )||2 = −2 ∑ ⟨ψf (t )|iCα|ψf (t )⟩ dt f α=1

(27)

Evaluation of the matrix element on the right-hand side of eq 27 and integration over time yields the amount of population that is destroyed by the CAP, Δ∥ψf∥. This is conveniently done with the flux utility of the MCTDH package. The population and hence the spectrum are restored by adding Δ∥ψf∥ to the remaining population. 3.3. Vibrational Analysis. To determine the vibrational distribution in the decaying and final states of the molecule, the nuclear wave functions ψd and ψf can be examined more closely by projecting them onto vibrational eigenstates of the respective state. In H2O, the notation (ν1, ν2, ν3) is commonly used to indicate the number of quanta of the symmetric stretching (ν1), bending (ν2) and asymmetric stretching (ν3) modes in respective order. To generate the eigenstates ψ(ν1,ν2,ν3), the improved relaxation algorithm that is implemented in the MCTDH package is employed. It combines diagonalization and propagation in imaginary time and is described in detail in refs 27, 36, and 37. After the relaxation, the vibrational eigenfunctions have been studied with the help of reduced density plots in the (r1,θ)- or F



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Figures 5 and 6 illustrate these differences in the excited state dynamics. The reduced density of the wave function with respect

(r1,r2)-plane to assign the proper mode label to them. Calculating the overlap |⟨ψ(ν1,ν2,ν3)|ψ(t)⟩|2 then reveals to what degree the mode (ν1,ν2,ν3) contributes to the nuclear wave function ψ(t).

4. COMPUTATIONAL RESULTS In this section, we present several theoretically obtained resonant Auger (RA) spectra of water and interpret them in comparison to the experiment and our previous calculations that disregarded vibronic coupling. We recall eq 13, which shows that the spectral outcome is directly related to the final state nuclear dynamics. The latter not only depends on the cationic PESs but also depends on the time evolution in the excited state. To arrive at a balanced viewpoint regarding the vibronic coupling effects in the RA spectrum, the excited state dynamics, being a fundamental aspect of the resonant Auger effect, will be studied in more detail in the following. 4.1. Nuclear Motion in the Decaying State. According to eq 11, the nuclear wave packet in the core-excited state is generated by a time-dependent excitation from the ground state. For a very short duration of the excitation (broad-band), the population of the core-excited state increases fast and then decays exponentially. Going to longer pulses (narrow-band), the development of the population starts to resemble the sine-squared shaped pulseform as can be seen from Figure 4 for the 1B2 core-excited state.

Figure 5. Time evolution of the reduced wave packet density ∥ψd(r2)∥2 in the 1A1 core-excited state is shown in the broad-band case for time steps between 4 and 10 fs. The dissociation of the molecule due to the repulsivity of the core-excited state sets in at about 6 fs.

Figure 6. Time evolution of the reduced wave packet density ∥ψd(r2)∥2 in the 1B2 core-excited state is shown for time steps between 20 and 70 fs. The pulse length is T = 80 fs. In contrast to nuclear motion on the 1A1 core-excited state, the wave packet remains close to the Franck− Condon region.

to the O−H distance r2 is plotted for several time steps in both figures. In the 1A1 state, note how the wave packet first spreads out and then splits up into a part that remains around the equilibrium position of the ground state (where it was first generated) and a second part moving to higher O−H separations where it is eventually absorbed by a CAP. Note that the density on the core-excited state is attenuated with increasing time due to the Auger decay. For this plot, a broad-band pulse was used. In the 1B2 state, excited state dynamics depend more strongly on the tuning of the frequency and the pulse lengths than in the lower state. Using a narrow bandwidth and tuning to the absorption maximum of this state at around 536 eV, the wave packet remains closer to the Franck−Condon zone, and no dissociation can be observed. Taking into account the discussion of paper I, we note that for higher excitation energies or in the case of broad-band excitation, also predissociation may occur. The effects of excited state dynamics in the resonant Auger process can be seen in Figure 7, where the experimental RA spectrum (decay from the 1A1 state) is shown together with the PE spectrum. The most prominent effect is a broadening of spectral features. Individual vibrational lines that are best resolved in the X band appear broader in the RA spectrum. This broadening is determined by the lifetime of the core-excited state and the bandwidth of the exciting pulse and can therefore considerably be reduced using a pulse of very narrow bandwidth. Still, a spectral resolution as in PE spectra cannot be achieved yet.

Figure 4. Population of the core-excited 1B2 state (∥ψd(t)∥2, blue solid lines) for excitation with different pulse durations. Using a very short pulse (broad-band excitation), the population decreases nearly exponentially, as the lifetime of the core-excited state is about 4 fs. Going to longer pulses (narrow-band excitation), the population curves begin to resemble the pulse envelope function (green dashed lines).

In a time span limited by the core hole lifetime, the nuclear wave packet can evolve on the PES of the intermediate state. The curves of the 1A1 and 1B2 core-excited states displayed in Figure 3 suggest distinctly different evolution of the nuclear wave packet in these two states. The lower state, which in paper I was found to be repulsive, leads to dissociation of the H2O molecule whereas vibrationally bound states do exist in the higher state. G



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Figure 7. Experimental resonant Auger5 and photoelectron spectra18 of H2O, normalized at the first peak at 12.6 eV binding energy. The RA spectrum is distinguished from the PE spectrum by the effects of nuclear dynamics in the core-excited state, e.g., by a broadening of vibrational lines and the overlap of spectral bands.

The extension of the three spectral bands depends on the vibrational excitation in the cationic states. Because the equilibrium geometries of the X state and the neutral ground state are nearly identical to each other, only very few vibrational lines appear in the PE spectrum. Decay from the repulsive 1A1 state, however, occurs with high probability also into highly excited vibrational states of the X state leading to an overlap of the X and A bands in the RA spectrum. The widths of the A and B bands are comparable in both spectra, as the equilibrium geometries of the A and B states and of the ground state differ more strongly from each other. Furthermore, the spectral intensity in the binding energy region between 19 and 24 eV is enhanced by resonant Auger decay in the fragmented molecule: Because the decay process is of a statistical nature, the transition to the final states can also take place at a time much larger than the core hole lifetime. In the case of the 1A1 intermediate state, Auger decay can occur after fragmentation, leading to the appearance of fragment bands, which were discussed in detail in paper I. They can be distinguished from the molecular bands, e.g., by detuning the excitation frequency as they do not disperse with photon energy. 4.2. Resonant Auger Spectrum. The theoretical results for narrow-band resonant Auger spectra after decay from the 1A1 or 1 B2 core-excited states are presented in this section. In each case, we examine the results more closely to demonstrate the relevance of the diabatic approach as well as the impact of the decaying state dynamics on the RA spectra. 4.2.1. Excitation of 1A1 State. Figure 8 displays the resonant Auger spectrum of H2O calculated from the diabatic and adiabatic PESs. For comparison, experimental results5 are shown in the center part of the figure. In the experiment, the photon bandwidth was 100 meV (fwhm). In the simulation, a pulse length of 40 fs was used to achieve the same level of spectral resolution (see paper I for a discussion of the bandwidth). Assuming that the experiment was performed at room temperature,38 the highest value of K contributing significantly due to its thermal weight factor is K = 5. To save numerical effort, however, only K = 0 and K = 1 have been adopted in the diabatic calculations to correctly reproduce the even and odd progressions of the bending mode in the A band. We convinced ourselves that the inclusion of higher angular momentum states in the calculation does not increase the accuracy of the RA spectrum. The overall agreement of the two calculated spectra with the experiment is satisfying. In the X band, the main difference lies in

Figure 8. Resonant Auger spectrum of H2O after excitation to the 1A1 state. The upper curve was calculated using the coupled diabatic PESs, the lower one from the uncoupled adiabatic PESs. Experimental data is taken from ref 5. The numerical results strongly disagree in the B band, where an artificial vibrational structure appears if vibronic coupling is neglected. The agreement with the experiment is much improved if the diabatic PESs are used.

the second peak, at about 13 eV, which shows up considerably stronger in the simulation than in the experiment. Because the PE spectrum was more accurately reproduced here (cf. Figure 1), this discrepancy cannot be attributed to errors in the cationic PESs. Instead, we suggest that the experiment was performed at an excitation energy slightly below the absorption maximum of the 1A1 state at 534.1 eV. Detuning reveals the role of the coreexcited state in the resonant Auger process which was examined more closely in paper I. The probability to populate higher excited vibrational states in the cation is reduced if the excitation frequency is tuned below resonance. In our simulation we were able to excite exactly at resonance frequency, whereas a small shift or drift of the experimentally used frequency on the order of 20 meV is possible.5 In the A band, no significant improvement can be achieved at this level of spectral resolution if the diabatic coupled PESs are employed, but we refer the reader to section 4.2.2 below, where vibronic coupling effects are definitely observable owing to a narrower excitation bandwidth. The calculated spectra differ most notably in the B band. Pronounced vibrational structure is visible in the spectrum from the adiabatic uncoupled PESs whereas the diabatic coupled PESs lead to a rather broad, unstructured feature that agrees well with the experiment. To eliminate the artificial structure in the B band, we found that the decisive step is to use the diabatic PESs instead of the adiabatic ones even when the coupling element D12 is neglected. The shapes of the adiabatic and diabatic PESs differ significantly from each other around the equilibrium geometry of the B state leading to a wide discrepancy in the possible vibrational excitations which we extrapolated from a simulation of the PE spectrum with and without the diabatic coupling. The additional effect of the coupling element is a population transfer between the D22 and D11 states as is visualized in Figure H



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resonances in the B state and further reduce the vibrational resolution in the B band. Turning to the fragment bands addressed in section 4.1, we found that adiabatic uncoupled and diabatic coupled spectra agree well with each other. This can be understood from the low probability of vibronic coupling in diatomic fragments (the resonant Auger decay after fragmentation takes place within an OH-molecule), where the occurrence of degeneracies or neardegeneracies of electronic states is considerably less likely due to the presence of only one nuclear coordinate. 4.2.2. Excitation of 1B2 State. Figure 11 displays calculated and experimental Auger spectra of H2O after excitation to the 1B2

9, where the populations of the two diabatic states are shown at an Auger electron energy of E = 515.10 eV. This corresponds to a

Figure 9. State populations of the D11 and D22 diabatic states at an Auger electron energy of 515.10 eV. The sum of the two contributions saturates at 40 fs, and an internal population exchange can be detected until 60 fs and beyond. This behavior can be related to the shape and position of the PESs involved, cf. Figure 10.

peak in the B band at a binding energy of 18.7 eV, cf. Figure 8. At first, only the D22 state is significantly populated. Population of the D11 state sets in about 8 fs later. The maximum is reached in the higher state after around 30 fs, then the state populations start to oscillate. The sum of these two contributions, however, remains constant after 40 fs as the decay has ceased until then. It is also instructive to look at the surfaces of the D11, D22, and core-excited states. In Figure 10, they are shown for r1 = 1.8 a0 and θ = 105°.

Figure 11. Resonant Auger spectrum of H2O after excitation to the 1B2 state. The upper curve was calculated using the coupled diabatized PESs; the lower one stems from the uncoupled adiabatic PESs. Experimental data are taken from ref 3. The narrower photon bandwidth (compared to Figure 8) enhances the vibrational resolution. The vibronic coupling effects are more pronounced, as can be seen both from the B band and from the correct reproduction of the line structure in the A band in the diabatic approach.

state. The theoretical results were obtained from the diabatic and adiabatic PESs, the experimental data were taken from ref 3. Here, the experimental photon bandwidth was about 50 meV and we used a pulse length of T = 80 fs to obtain a similar spectral resolution in our simulation. The frequency was tuned to the absorption maximum at 535.95 eV (see also Figure 12 below where the absorption spectrum is shown). At this level of resolution, we can expect to unambigously observe several vibronic coupling effects. The most obvious aspect can be perceived again in the B band. The artifical vibrational structure is even more pronounced in the adiabatic spectrum due to the narrower bandwidth. The calculation relying on the diabatic coupled PESs yields the correct band envelope compared to the experiment. The form and position of the B band was experimentally studied for different excitation frequencies in ref 3. We simulated this by varying ω as shown in Figure 12. On the left are shown the experimental and calculated absorption spectra of H2O in the photon energy range of core-excitation to the 1B2 state. The

1

Figure 10. Cut through the diabatic and core-excited A1 PESs at an O− H distance of r1 = 1.8 a0 and an angle θ = 105°. The curves of the diabatic D11 and D22 states are shifted by the Auger electron energy of 515.10 eV mentioned in the text. The energy scale on the right refers to the diabatic coupling element D12. Note how the curve of the core-excited state first touches the shifted curve of the D22 state at 2.0 a0 and at about 3.0 a0 it crosses the curve of the D11 state. The arrangement of the PESs is related to the population transfer shown in Figure 9.

The cationic states are shifted by the Auger electron energy of 515.10 eV. Obviously, the decay takes place more favorably into the D22 state, as the core-excited state and the D22 state touch at around 2.0 a0. Only after the wave packet in the core-excited state has reached the region around 3.0 a0 can it also decay more strongly into the D11 state. However, the D11 and D22 states are also directly coupled by the diabatic coupling element D12, which is also shown in Figure 10, leading to the oscillations visualized in Figure 9. They prevent the build-up of stable vibrational I



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Figure 12. Absorption spectra (left) and resonant Auger spectra (right) of H2O at three different excitation frequencies around the 1B2 resonance. Calculated results are shown in blue solid lines, experimental spectra3 in green dashed lines. The absorption spectra are shown together with overlaps (brown vertical lines) of the wave functions in the core-excited state with its vibrational eigenstates for different values of ω. Different eigenstates can predominantly but not exclusively be populated in the 1B2 state at different frequencies due to the finite core hole lifetime.

Figure 13. Resonant Auger spectrum of H2O after excitation to the 1B2 state, focus on the A band. In the upper graph, experimental3 (green dashed lines) and calculated spectra (blue solid lines) are displayed together. In the four lower graphs, the separate calculations with values of the vibronic quantum number K = 0−3 are shown to underline that bending states with even and odd quantum numbers (cf. the mode labels printed at the top) and the Renner−Teller splitting (Σ−Δ and Π−Φ) of these states can be observed. The arrows connect the mode labels with the positions in the separate four spectra where the contribution of the respective mode to nuclear motion in the cation was most pronounced, as is explained in the text.

vibrational structure in the absorption spectrum is further confirmed by calculating the overlap of the nuclear wave function in the core-excited state with a number of its vibrational eigenfunctions. Due to the short lifetime, none of them can be exclusively selected by the radiation even at very narrow excitation bandwidth. The overlaps, however, reveal that at different excitation frequencies a tendency for lower or higher eigenfunctions to show up can be observed. This is reflected in the B band of the RA spectra shown on the right, which are plotted here against the Auger electron energy E. For increasing values of ω, the spectral envelope is shifted to lower Auger electron energies corresponding to higher binding energies of the emitted Auger electron as also the population of higher lying vibrational states in the cation is increased. The vibronic structure in the A band deserves special attention. As was pointed out in section 2.3, even values of K generate bending vibrational states with odd mode labels and vice versa. The bending states are also Renner−Teller split: In paper II, we evaluated the Renner−Teller splitting to be 14 meV for the Σ−Δ (K = 0, 2) states and 17 meV for the Π−Φ (K = 1, 3) states. We were able to perform simulations for all of these Kvalues over a small binding energy range using a lower electron energy increment (10 meV instead of 50 meV as in the other calculations). The results, compared to the experiment, are shown in Figure 13. In contrast to the less resolved experiment shown before (decay from the 1A1 state, Figure 8), it is possible here to discern the even and odd vibrational peaks in the A band. As the A band partly overlaps with the X band, the peaks in the spectra do not necessarily match the (0, ν2, 0) bending modes but might be formed also by vibrational states of the X band or progressions of the (1, ν2, 0) modes. The mode assignment was

therefore done with the help of overlaps with the vibrational eigenfunctions as detailed in section 3.3. For example, to detect the correct position for the (0, 10, 0) mode label in the spectrum calculated with K = 1, we first determined the Auger electron energy E where the overlap of the final state wave function with the corresponding eigenfunction is maximal. This is possible, because E is a parameter in the simulations. This procedure yielded two different energies as the spectrum is the sum of two calculations where initially k = 0 and k = 2. Finally, these two energies were averaged which yielded the location of the (0,10,0) Π mode label. Comparing the (0, ν2, 0) mode positions in Figure 13 to the corresponding ones from our PE spectra, we found that they do not fully coincide with each other. We suggest that the energy differences of these mode positions in the RA spectrum are due to dissipation of a small amount of excitation energy in the nuclear dynamics in the core-excited state. Although the radiation is tuned to the absorption maximum of the 1B2 state and is rather narrow, excitation into dissociating states is also possible, which takes away part of the excitation energy. Finally, the K-substructure in the X band arising from the Renner−Teller effect which can be observed in PE spectra still cannot be resolved due to the more complicated decay dynamics and the finite excitation bandwidth. J



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(8) Hjelte, I.; Piancastelli, M. N.; Fink, R. F.; Björneholm, O.; Bässler, M.; Feifel, R.; Giertz, A.; Wang, H.; Wiesner, K.; Ausmees, A.; et al. Chem. Phys. Lett. 2001, 334, 151−158. (9) Naves de Brito, A.; Feifel, R.; Mocellin, A.; Machado, A. B.; Sundin, S.; Hjelte, I.; Sorensen, S. L.; Björneholm, O. Chem. Phys. Lett. 1999, 309, 377−385. (10) Piancastelli, M. N.; Kempgens, B.; Hergenhahn, U.; Kivimäki, A.; Maier, K.; Rüdel, A.; Bradshaw, A. M. Phys. Rev. A 1999, 59, 1336−1340. (11) Levine, I. N. Quantum Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1991. (12) Takahashi, O.; Odelius, M.; Nordlund, D.; Nilsson, A.; Bluhm, H.; Pettersson, L. G. M. J. Chem. Phys. 2006, 124, 064307. (13) Eroms, M.; Vendrell, O.; Jungen, M.; Meyer, H.-D.; Cederbaum, L. S. J. Chem. Phys. 2009, 130, 154307. (14) Cederbaum, L. S.; Tarantelli, F. J. Chem. Phys. 1993, 98, 9691− 9706. (15) Pahl, E.; Meyer, H.-D.; Cederbaum, L. S. Z. Phys. D 1996, 38, 215−232. (16) Pahl, E.; Cederbaum, L. S.; Meyer, H.-D.; Tarantelli, F. Phys. Rev. Lett. 1998, 80, 1865−68. (17) Dehareng, D.; Chapuisat, X.; Lorquet, J.-C.; Galloy, C.; Raseev, G. J. Chem. Phys. 1983, 78, 1246−1264. (18) Reutt, J. E.; Wang, L. S.; Lee, Y. T.; Shirley, D. A. J. Chem. Phys. 1986, 85, 6928−6939. (19) Eroms, M.; Jungen, M.; Meyer, H.-D. J. Phys. Chem. A 2010, 114, 9893−9901. (20) Polyansky, O. L.; Jensen, P.; Tennyson, J. J. Chem. Phys. 1994, 101, 7651−7657. (21) Fink, R.; Staemmler, V. Theor. Chim. Acta 1993, 87, 129−145. (22) Worth, G. A.; Cederbaum, L. S. Annu. Rev. Phys. Chem. 2004, 55, 127−158. (23) Haxton, D. J.; Rescigno, T. N.; McCurdy, C. W. Phys. Rev. A 2007, 75, 012711. (24) Meyer, H.-D.; Manthe, U.; Cederbaum, L. S. Chem. Phys. Lett. 1990, 165, 73−78. (25) Manthe, U.; Meyer, H.-D.; Cederbaum, L. S. J. Chem. Phys. 1992, 97, 3199−3213. (26) Meyer, H.-D. Multiconfiguration time-dependent Hartree method. The Encyclopedia of Computational Chemistry; Wiley: Chichester, U.K., 1998; pp 3011−3018. (27) Meyer, H.-D.; Worth, G. A. Theor. Chem. Acc. 2003, 109, 251− 267. (28) Beck, M. H.; Jäckle, A.; Worth, G. A.; Meyer, H.-D. Phys. Rep 2000, 324, 1−105. (29) Multidimensional Quantum Dynamics: MCTDH Theory and Applications; Meyer, H.-D., Gatti, F., Worth, G. A., Eds.; Wiley-VCH: Weinheim, 2009. (30) Meyer, H.-D. Wiley Interdisciplinary Reviews: Computational Molecular Science 2012, 2, 351. (31) Worth, G. A.; Beck, M. H.; Jäckle, A.; Meyer, H.-D. The MCTDH Package, Version 8.2, (2000). H.-D. Meyer, Version 8.3 (2002), Version 8.4 (2007). See http://mctdh.uni-hd.de/. (32) Jäckle, A.; Meyer, H.-D. J. Chem. Phys. 1996, 104, 7974−7984. (33) Jäckle, A.; Meyer, H.-D. J. Chem. Phys. 1998, 109, 3772−3779. (34) Riss, U. V.; Meyer, H.-D. J. Phys. B 1993, 26, 4503−4535. (35) Riss, U. V.; Meyer, H.-D. J. Chem. Phys. 1996, 105, 1409−1419. (36) Meyer, H.-D.; Le Quéré, F.; Léonard, C.; Gatti, F. Chem. Phys. 2006, 329, 179−192. (37) Joubert Doriol, L.; Gatti, F.; Iung, C.; Meyer, H.-D. J. Chem. Phys. 2008, 129, 224109. (38) Shigemasa, E.; Kaneyasu, T.; Matsushita, T.; Tamenori, Y.; Hikosaka, Y. New J. Phys. 2010, 12, 063030.

5. CONCLUSION Resonant Auger spectra have been computed using a diabatization scheme introduced before for a simulation of the photoelectron spectrum of H2O. A set of cationic PESs has been recalculated to increase the accuracy of the nuclear dynamics simulations. The nuclear motion evolves both in a core-excited state of the neutral molecule and in the three principal cationic final states, which are known to exhibit vibronic coupling. The A and B cationic states are coupled via a conical intersection and the X and A states of the cation are coupled by the Renner−Teller effect. The effect of these different types of nuclear motion are reflected in various spectral features in the RA spectrum. It was the aim of the present study to reveal how much the spectrum is influenced by the vibronic coupling in the final states. Taking it into account considerably increases the computational expense, e.g., due to the Renner−Teller coupling, which involves varying the nuclear angular momentum. The additional expenditure is justified, however, because the agreement with experimental results has considerably improved in the B band where artificial vibrational lines are now quenched using the diabatic PESs around the conical intersection. In the A band, the simulations relying on different nuclear angular momenta reproduce the alternating bending lines, most clearly shown in the photoelectron spectrum, Figure 1. At high enough experimental resolution, mode labels that also reveal the Renner−Teller splitting can be assigned to the A band of the resonant Auger spectrum. The results for the fragment band are similar using either the adiabatic uncoupled or the diabatic coupled PESs as vibronic coupling is less probable if Auger decay occurs in a diatomic fragment.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: M.E., [email protected]; M.J., [email protected]; H.-D.M., hans-dieter.meyer@pci. uni-heidelberg.de. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the DFG is gratefully acknowledged. M.J. thanks the Ernst Miescher Foundation (Basel, Switzerland) for partial support.

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REFERENCES

(1) Feifel, R.; Piancastelli, M. J. Electron. Spectrosc. Relat. Phenom. 2011, 183, 10−28. (2) Piancastelli, M. N.; Simon, M.; Ueda, K. J. Electron. Spectrosc. Relat. Phenom. 2010, 181, 98−110. (3) Ueda, K. J. Phys. Soc. Jpn. 2006, 75, 032001. (4) García, E. M.; Kivimäki, A.; Pettersson, L. G. M.; Ruiz, J. A.; Coreno, M.; de Simone, M.; Richter, R.; Prince, K. C. Phys. Rev. Lett. 2006, 96, 063003. (5) Hjelte, I.; Karlsson, L.; Svensson, S.; De Fanis, A.; Carravetta, V.; Saito, N.; Kitajima, M.; Tanaka, H.; Yoshida, H.; Hiraya, A.; et al. J. Chem. Phys. 2005, 122, 084306. (6) De Fanis, A.; Tamenori, Y.; Kitajima, M.; Tanaka, H.; Ueda, K. J. Electron. Spectrosc. Relat. Phenom. 2004, 137, 271−276. (7) De Fanis, A.; Nobusada, K.; Hjelte, I.; Saito, N.; Kitajima, M.; Okamoto, M.; Tanaka, H.; Yoshida, H.; Hiraya, A.; Koyano, I.; et al. J. Phys. B 2002, 35, L23−L29.

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NOTE ADDED AFTER ASAP PUBLICATION This article was published ASAP on July 19, 2012, with an error in eq 27 and ref 13. A correction to eq 27 was published ASAP on July 20, 2012, and the final version with a change to ref 13 was published ASAP on July 24, 2012. K


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