Article pubs.acs.org/JPCA
Subfemtosecond Quantum Nuclear Dynamics in Water Isotopomers B. Jayachander Rao and A. J. C. Varandas* Departamento de Quı ́mica and Centro de Quı ́mica, Universidade de Coimbra, 3004-535 Coimbra, Portugal ABSTRACT: Subfemtosecond quantum dynamics studies of all water isotopomers in the X̃ 2B1 and à 2A1 electronic states of the cation formed by Franck−Condon ionization of the neutral ground electronic state are reported. Using the ratio of the autocorrelation functions for the isotopomers as obtained from the solution of the time-dependent Schrödinger equation in a grid representation, high-order harmonic generation signals are calculated as a function of time. The results are found to be in agreement with the available experimental findings and with our earlier study for D2O+/H2O+. Maxima are predicted in the autocorrelation function ratio at various times. Their origin and occurrence is explained by calculating expectation values of the bond lengths and bond angle of the water isotopomers as a function of time. The values so calculated for the 2B1 and 2A1 electronic states of the cation show quasiperiodic oscillations that can be associated with the time at which the nuclear wave packet reaches the minima of the potential energy surface, there being responsible for the peaks in the HHG signals.
1. INTRODUCTION Molecules exposed to intense laser fields show processes that are of both practical and fundamental interest.1,2 High-order harmonic generation (HHG) is one such process of key relevance in strong-field (SF) physics but not yet established as a practical tool. The HHG can be understood within the framework of a semiclassical three-step model.3,4 An intense laser pulse ionizes the molecule in the first step, resulting in a correlated electron and vibrational wave packet pair. The electron gets accelerated by the laser field while the vibrational wave packet moves on the potential energy surface5 of the resulting cation. In the next step, when the field phase changes, the electron is driven back to the ion core. Finally, the third step consists of the recombination of the electron with the cation, leading to the production of high-energy radiation, which extends into the range of extreme ultraviolet6 and soft Xrays.7 The HHG technique exploits the correlation between the nuclear wave packet created in the molecular cation by the SF ionization and the electronic wave packet with a view to measure the nuclear dynamics on a subfemtosecond (subfs) time scale. Consequently, the study of nuclear dynamics and structural cationic rearrangements on the molecular time scale has become possible. As the mapping of the HHG signal is possible up to ∼1.6 fs at a wavelength of 800 nm, fs nuclear dynamics has been predicted and observed in H2/D2.8−10 Other systems with a fast nuclear dynamics have also been studied, in particular, CD4/CH4,9 where, for methane, it has been attributed to changes of the PES upon ionization.10 In fact, the important role played by the geometrical changes in the SF ionization of molecules and its consequences in nuclear dynamics have been discussed.11,12 This becomes particularly important for cations with Jahn−Teller (JT) interactions. For example, ionization of an electron from the highest occupied © 2015 American Chemical Society
molecular orbital (HOMO) at the equilibrium configuration of methane results in the formation of CH+4 in its ground state, which is known to undergo JT distortion. Accordingly, the three-fold degeneracy of the ground electronic state of the cation is split into three nondegenerate states. In general, JT effects and the resulting conical intersection (CI) of PESs will play a key role in nuclear dynamics. Indeed, subfs nuclear dynamics in CD4/CH4 has recently been investigated by Mondal and Varandas13 via ab initio quantum dynamics including the JT effect. Employing a quadratic vibronic coupling Hamiltonian in a diabatic picture and a multiconfiguration time-dependent Hartree propagation scheme, the ratio of the HHG signal for the CD4 and CH4 cations has been calculated as a function of time.13 Mondal and Varandas found the HHG signal to be largest for the heavier isotope, and this trend was enhanced up to ∼1.85 fs. From this, they predicted the structural rearrangement of CH 4+ from T d to C 2v configuration to occur in this time scale,13 with such predictions having most recently been confirmed by experiment.14 Recently, an analytical approach to calculate the short-time nuclear autocorrelation functions in the vicinity of CIs has been suggested, including nonadiabatic coupling.15 Unfortunately, no results were given for CD+4 /CH+4 that might allow a comparison with the Mondal−Varandas13 predictions. To understand and interpret the origin of the maxima observed in the HHG signals,13 we have most recently started to study the SF ionization of water.16 This has been much utilized as a prototypical model for the study of nuclear dynamics due to SF ionization to multiple electronic states, Received: March 4, 2015 Revised: April 16, 2015 Published: April 24, 2015 4856
DOI: 10.1021/acs.jpca.5b02129 J. Phys. Chem. A 2015, 119, 4856−4863
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The Journal of Physical Chemistry A
the grid for the Jacobi angle, γ, is chosen as the nodes of an Nγpoint Gauss−Legendre quadrature. The action of the exponential operator on Ψ(t=0) is then carried out by dividing the total propagation time into N steps of length Δt, with the exponential operator at each Δt being for this purpose approximated using the second-order split-operator method.27 This is used in conjunction with the fast Fourier transform method28 to evaluate the exponential containing the radial kinetic energy operator and the discrete variable representation method to evaluate the exponential containing the rotational kinetic energy operator.29 The latter is accomplished by transforming the grid wave function to the angular momentum basis (finite basis representation), multiplying it by the diagonal value of the operator (e−ij(j+1)Δtℏ/4I), and back transforming it to the grid representation. The initial wave function is then propagated for a time T. To avoid unphysical reflections or wraparounds of the high-energy components of the wave function that reach the finite-sized grid boundaries at longer times, the last 20 points of the grid along R and r are multiplied by a damping function.30 In summary, the time-dependent wave function is obtained as
both experientally and theoretically.17−24 Indeed, a similar study of the resonant Auger effect in gas-phase water, which involves vibronic coupling effects in the same electronic states of the ion (they form a Renner−Teller pair), has also appeared.25 By employing a time-dependent quantum wave packet approach, the SF ionization of water molecules has been described from the moment of ionization to dissociation. Accordingly, the nuclear autocorrelation functions were calculated for H2O+ and D2O+ in such electronic states by solving numerially the time-dependent Schrödinger equation on a grid. The HHG signals were then calculated by taking the ratio of the squared autocorrelation functions of D2O+ and H2O+ molecules as a function of time. The so-calculated HHG signals were found to be larger for D2O+ than H2O+, with the trend being enhanced up to ∼1.1 and ∼1.6 fs for 2B1 and 2A1 electronic states, respectively. On the basis of these theoretical results, the structural rearragements of H2O+ (D2O+) from C2v to near D∞h configurations in the 2A1 electronic states were shown to occur at ∼8.5 fs (∼9.3 fs) due to the strong excitation of the bending mode. In this article, we extend our recent theoretical effort16 to include the water isotopomers involving protium, deuterium, and tritium in the 2B1 and 2A1 electronic states. As will be shown, the calculated photoelectron spectra of these isotopomers compare well with the available experimental results. Moreover, the HHG signals as a function of time reveal peaks at various times in line with those from our previous study.13,16 By looking at the expectation value of the bond lengths and bond angle, which show quasiperiodic oscillations, such peaks were rationalized as due to the wave packet motion on the PESs of the cation. The paper is organized as follows. Section 2 presents a brief summary of the theoretical framework, while the results are presented and discussed in section 3. The summarizing remarks are in section 4.
⎛ iHt̂ ⎞ Ψ(t ) = exp⎜ − ⎟Ψ(t = 0) ⎝ ℏ ⎠
with the initial wave function for the ground electronic state of the neutral species assuming the Gaussian form Ψ(t = 0) ⎡ (R − R )2 (r − r0)2 ⎤ 0 ⎥ = N exp⎢ − − 2σR2 2σr2 ⎦ ⎣ ⎧ ⎡ (γ − γ )2 ⎤ ⎫ ⎡ (γ − π + γ )2 ⎤⎪ ⎪ 0 ⎥ 0 ⎥ ⎢ ⎨exp⎢ ⎬ ×⎪ + exp 2 ⎢⎣ ⎥⎦⎪ 2σγ2 ⎩ ⎢⎣ 2σγ ⎥⎦ ⎭
2. THEORETICAL AND COMPUTATIONAL DETAILS In this section, we describe briefly the theoretical formalism and computational details used in the present work. The Hamiltonian Ĥ for nuclear motion is the sum of the kinetic, TN, and potential energy, VX/A, operators and may be written as
where μR =
mO(mHa + mHb) mO + mHa + mHb
(1)
and μr =
mHa mHb 2
(3)
where R0, r0, and γ0 specify the initial location, while δR, δr, and δγ are the width parameters of the wave packet along the associated coordinates. Note that due to the poles at γ = 0, π in eq 1, such limiting values cannot be reached. This should cause no fundamental limitation and may be avoided by using, for example, hyperspherical coordinates,31,32 an issue not pursued in the present work. The complete set of numerical parameters here utilized is given in Table 1. As noted above, the SF ionization will be described as a FC transition from the ground electronic state of the neutral water isotopomer to the ground (X̃ 2B1) or first excited (Ã 2A1) electronic states of the cation, respectively. These transitions result, each per se, in two bands in the energy range of 10−18 eV in the experimental recordings.33,34 The transition to the
(J − j)2 j2 ℏ2 ∂ 2 ℏ2 ∂ 2 − + + Ĥ = − 2μR ∂R2 2μr ∂r 2 2μR R2 2μr r 2 + VX/A(R , r , γ )
(2)
; mO, mHi (i = a, b)
are the masses of the involved atoms. In turn, J and j are the total and diatomic angular momenta operators, while the subscripts in the potential energy operator refer to the ground (X ≡ 2B1) and excited (A ≡ 2A1) states of the water ion. As in previous work,16 we will consider J = 0. Within the Franck−Condon (FC) approximation, the initial nuclear wave function of the neutral’s ground electronic state is vertically pumped to the ground- or first excited-state PESs of the cation, where it is then propagated. Such a scheme has been also utilized26 for the calculation of transition-state resonances in MuH2. Accordingly, the time-dependent Schrö dinger equation (cf., eq 2) is numerically solved on a grid in the (R,r,γ) space to obtain the wave function at time t from the one at t = 0. For this, an NR × Nr spatial grid is used in the (R,r) plane defined by Rmin ≤ R ≤ Rmax and rmin ≤ r ≤ rmax. In turn,
Table 1. Grid Parameters and Initial Wave Function
a
4857
parametera
value
description
NR × Nr × Nγ Rmin, Rmax rmin, rmax ΔR, Δr R0, r0, γ0 δR, δr, δγ T Δt
128 × 128 × 48 10−4, 15.34 0.5, 15.74 0.12, 0.12 1.1, 2.86, (π/2) 0.3, 0.2, 0.1 1100 0.135
grid size extension of grid in R extension of grid in r grid spacings in R and r initial location of GWP width parameters of GWP total propagation time time step in propagation
Bond distances are in a0, angles are in radians, and time is in fs. DOI: 10.1021/acs.jpca.5b02129 J. Phys. Chem. A 2015, 119, 4856−4863
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The Journal of Physical Chemistry A lowest ionic state (2B1) occurs near 12.6 eV (B band) by the removal of an electron from the HOMO (1b1), while the transition to the lowest excited ionic A state (2A1) occurs near 14.8 eV (A band) by the removal of the electron from the second least-bound molecular orbital (HOMO−1). Following our previous work,16 the photoelectron spectrum due to SF ionization is calculated by using a suitably chosen Gaussian wave packet, which will be propagated either in the ground or first excited electronic states of water isotopomers. Finally, the photoelectron spectrum is obtained as the Fourier transform of the time autocorrelation function35,36 P(E) ≈ |λE|2 Re
∫0
∞
e i(E0 + E)t / ℏC′(t ) dt
of attosecond pulse trains for one-dimensional H2 and D2 molecules and shown that the above approximation is valid for few-cycle pulses, whereas for longer pulses, double-ionization occurs. Accordingly, the intensity of the HHG signal comes to be proportional to the squared modulus of the autocorrelation function, C(t). Thus, the ratio of the HHG signal of water heavier and lighter isotopomers can be approximately given by the ratio of the square of the autocorrelation functions8,9 autocorrelation function ratio =
|C l(t )|2
(6)
where Ch (Cl) is the time autocorrelation function of the heavier water isotopomer (lighter isotopomer). Following the above work,8,9 eq 6 assumes that the PESs of both the neutral and cationic species are, up to some constant multiplicative factor,13,16 not affected by the intense laser field. Despite being approximate, eq 6 is found to be very useful for studying subfs nuclear dynamics in molecules,10,13,16 where the exact treatment is prohibitively expensive.
(4)
where λE is the transition dipole moment between the ground electronic state of the neutral water isotopomer and its cation, which usually depends on the kinetic energy and the molecular geometry but is here assumed to be a constant. The quantity E0 is the energy of the vibrational ground level of the ground electronic state of the neutral water isotopomer, and C′(t) is a damped time autocorrelation function of the wave packet evolving on the chosen PES. This is obtained by multiplying C(t) by an exponential function C′(t) = C(t)e−t/τ, with the autocorrelation function assuming the form C(t ) = ⟨Ψ(0)|Ψ(t )⟩
|C h(t )|2
3. RESULTS AND DISCUSSION We first present the photoelectron bands of HOD, DOT, T2O, and H218O calculated with the formalism discussed in the previous section and compare the results with the available experimental data.33 Because the starting state is bound, one has C(t) = ⟨Ψ*(t/2) |Ψ(t/2)⟩, which halves the propagation time T needed to achieve the energy resolution ΔE = 2πℏ/T in the photoelectron spectrum.42,43 The initial wave packet is time-evolved for 1.1 ps, and the autocorrelation functions are calculated and damped with an exponential function, where τ = 44 fs (cf., section. 2) to account for experimental line broadening effects. It has been subsequently Fourier-transformed to calculate the vibronic spectra due to the FC transition to the 2B1 and 2A1 states of the cations. Note that the value of τ used in the damping function corresponds to a convolution of the vibronic line spectrum with a Lorentzian function of ∼15 meV (fwhm). 3.1. Photoelectron Spectra and Femtosecond Nuclear Dynamics. Because the photoelectron spectra of H2O+ and D2O+ have been given elsewhere,16 we only summarize some results in Table 2 and discuss in the following the other
(5)
where Ψ(0) is the eigenfunction of the ground-state Hamiltonian of the neutral molecule and Ψ(t) is the timedependent wave function that is evolving on a particular cationic PES. This convolutes the spectrum with a Lorentzian form of the fwhm (full width at half-maximum) Γ = 2/τ, which can be chosen to approximate the experimental broadening of the spectral features. Note that, following previous work,13,16 the scattering wave function of the outgoing electron is missing in the above theoretical formalism. This is an intricate problem as the scattering state is a plane wave at asymptotic distances, but it gets modified when interacting with the cation. Hence, a full treatment of HHG spectroscopy proves correspondingly more difficult as the cation and electron are produced under the action of an intense laser field, which is ignored in the present formalism. It should also be noted that the calculation of vibronic intensities close to the ionization threshold poses an additional problem by the fact that it should verify the Wigner threshold law.37 Although the role of the ejected photoelectron has been explicitly neglected above, this does not imply that it has a superficial role. In fact, the subfs generated electron wave packet will recollide with the cation after a certain time delay (the recollision occurs on 2/3 of a frequency period), which leads to the harmonic emission, with successively higher harmonics occurring at longer time delays.38 In fact, it is this property of the high-harmonic radiation that allows us to monitor the nuclear motion over a range of time delays. On the basis of previously reported work,10,13,16 ignoring the explicit treatment of the outgoing electron wave packet led to crucial information on the motion of the nuclei in the subfs time regime, according to the semiclassical three-step model.9 Recently, Starace and co-workers39,40 have shown that the HHG intensities can be generally factorized for linear polarization as a product of the ionization and photoemission factors within the semicalssical three-step model. Additionally, Bandrauk et al.41 have performed an exact non-Born− Oppenheimer calculation of the harmonic spectra and shapes
Table 2. Calculated Frequencies for Protonated and Deuterated Water in Both 2B1 and 2A1 Statesa molecule H2O+(2B1) D2O+(2B1) H2O+(2A1) D2O+(2A1) a
ν1/fs 10.7 14.6 16.9 18.5
(10.4) (14.0) (16.5) (n.a.b)
ν1/eV 0.386 0.283 0.245 0.223
ν2/fs 20.9 28.9 33.9 44.6
(21.2) (29.0) (33.2) (44.8)
ν2/eV 0.197 0.143 0.112 0.093
Experimental results33,44 are in parentheses. bn.a. = not available.
isotopomers. To our knowledge, no experimental results are available for most of them but HOD+44 and H218O+.33 Thus, we compare our results only with the available experimental ones. The calculated B and A photoelectron bands of the above water isotopomers are shown in Figure 1a−d, which show (from a to d) the calculated photoelectron spectra of HOD+, DOT+, H218O+, and T2O+, respectively. In this case, SF ionization of the water molecule removes an electron from the HOMO (1b1) and excites it into the ionic ground state, 2B1. Because the PES of H2O+ has a similar topography to the PES 4858
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correspond to the symmetric stretching mode (ν1) and bending mode (ν2). The average peak spacing in the absolute value of the autocorrelation function for the ν1 and ν2 progressions in the 2B1 state of HOD+ shown in panel a are on the same order, ∼12.3 and ∼25.1 fs, which correspond to energy spacings of ∼0.330 and ∼0.165 eV. In turn, the average peak spacings for the ν1 and ν2 peaks in the autocorrelation functions corresponding to the 2A1 electronic state plotted in panel a (blue lines) are ∼17.7 and ∼34.6 fs, respectively. The latter compares well with the available experimental data44 for HOD+, 35.3 fs. This implies energy spacings of ∼0.234 and ∼0.124 eV in the theoretical results and ∼0.117 eV in the experimental one for the latter. Such energy spacings can be seen from the calculated photoelectron spectra of both 2B1 and 2A1 electronic states of HOD+ in panel a of Figure 1. The corresponding results for the other isotopomers are gathered in Tables 3 and 4. Note that the calculated results for the ν2 mode of H218O+ compare well with the experimental33 data, ∼21.4 (2B1 state) and ∼34.1 fs (2A1 state).
Figure 1. Photoelectron spectra of the water isotopomers. Red and blue lines correspond to the water isotopomers with Cs and C2v symmetries, respectively.
of the neutral molecule, the FC ionization populates only a few vibronic levels in the 2B1 electronic state.16 This can be seen from the exponentially decaying FC progression between ∼11.0 and ∼14.0 eV in each panel of the photoelectron spectra. In contrast, the FC ionization from the HOMO−1 (3a1) excites the molecule into the 2A1 electronic state. Because the PES of the cation differs now very much from the ground-state PES of the neutral (the equilibrium geometry of the cation occurs in the former at linear geometries), the FC ionization leads to a strong excitation of the bending mode. As a result, the FC ionization from the HOMO−1 populates a large number of vibrational states, which can be seen from the dense photoelectron bands in panels a−d of Figure 1 between ∼14.0 and ∼18.0 eV. It can be seen from panels a−d of Figure 1 that the photoelectron bands marginally get denser, which can be attributed to the isotopic effect. The squared absolute values of the autocorrelation functions for the FC ionization to the 2B1 (2A1) electronic states of the isotopomeric water cations are shown in panels a−d of Figure 2. The red lines correspond to the 2B1 state and the blue lines to the 2A1 state. As visible from Figure 2, strong quasiperiodic recurrences occur. The absolute value of the autocorrelation function for the 2B1 electronic state plotted in panel a for HOD+ (red line) shows two kinds of oscillations. They
Table 3. Calculated Frequencies for the Water Isotopomers in the 2B1 Statea
a
molecule
ν1/fs
ν1/eV
ν2/fs
ν2/eV
HOD+ DOT+ T2O+ H218O+
12.5 16.2 17.9 10.6
0.330 0.255 0.231 0.390
25.1 32.2 37.8 21.9 (21.4)
0.165 0.130 0.110 0.190
Experimental results33,44 are in parentheses.
Table 4. Calculated Frequencies for the Water Isotopomers in the 2A1 Statea molecule +
HOD DOT+ T2O+ H218O+ a
ν1/fs
ν1/eV
ν2/fs
ν2/eV
17.7 18.0 19.9 17.2
0.234 0.230 0.210 0.241
34.6 (35.3) 35.8 40.6 33.8 (33.1)
0.120 0.115 0.102 0.122
Experimental results33,44 are in parentheses.
3.2. HHG Signals. The ratios of the squared modulus of the autocorrelation functions between DOT+ and HOD+ isotopomers for the 2B1 and 2A1 electronic states are shown in panels a−d of Figure 3 as a function of time. The results in panels a and b correspond to the calcualted HHG signals in the 2 B1 electronic state at short- and long-time intervals, respectively. Similarly, the results in panels c and d are for the 2A1 state. In agreement with our previous study,16 our calculated results predict the intensity of the HHG signal to be larger for DOT+. This is due to the much slower nuclear motion, an effect enhanced with time up to ∼1.5 and ∼1.3 fs for the 2B1 and 2A1 states, respectively. Similarly, the ratios of the squared moduli of the autocorrelation functions of T2O+ and D2O+ in the 2B1 and 2A1 electronic states are shown in panels a−d of Figure 4. Panels a and b show the results for the 2 B1 electronic state at short and long times, while c and d illustrate the same for the 2A1 state. Note that panels a and c refer to the HHG singals for the heavier isotope, T2O+, which reveal a similar behavior but enhanced up to ∼1.6 (2B1 state) and ∼1.4 fs (2A1 state). In turn, panels a−d of Figure 5 show the ratio of the squared autocorrelation functions of H218O+ and H2O+ for the 2B1 and 2A1 electronic states. As above, the results in panels a and b are for the 2B1 state at short- and long-
Figure 2. Autocorrelation functions for the 2B1 (2A1) electronic state of (a) HOD+, (b) DOT+, (c) H218O+, and (d) T2O+ as a function of time. The red lines correspond to 2B1 and the blue lines to 2A1. 4859
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until ∼1.6 fs. These results are not only in accordance with our recent explorations16 but also complement the available experimental findings of Farrell et al.,17 where the HHG signals are found to be more visible in the 2A1 state than those in 2B1. This is due to the large change in geometry of the 2A1 ionic state and hence to a stronger excitation of the vibrational bending motion. 3.3. Extracting Subfemtosecond Nuclear Dynamics. To understand the origin of the large number of peaks in the HHG signal of Figures 3−5 in the 2B1 and 2A1 electronic states (panels b and d), we calculate the expectation values of the bond lengths and bond angle for the water isotopomers here examined. Because no substantial vibrational dynamics is observed for the 2B1 state of water,16,17 we present and discuss only the results for the 2A1 state. The calculated results are shown in Figure 6 as a function of time. Also illustrated are snapshots of the atomic position
Figure 3. (a) Ratio of the squared autocorrelation functions for the 2B1 electronic state of DOT+ and HOD+. (b) Same as (a) but for longer times. (c) As in (a) but fot the 2A1 electronic state of DOT+ and HOD+. (d) Same as (c) but for longer times. Red and blue lines correspond to the 2B1 and 2A1 states, respectively.
Figure 6. Expectation values of the bond lengths and bond angle of HOD+ in the 2A1 electronic state as a function of time. Also shown are snapshots of the position coordinates of the atoms at critical points. Figure 4. Same as Figure 3 but for T2O+ and D2O+.
coordinates taken at critical points in the PES of the HOD isotopomer. As seen from Figure 6, the expectation values of the bond lengths and bond angle show quasiperiodic oscillations in time, which indicate that the HOD+ isotopomer vibrates between the bent (Cs) and a quasi-linear (quasi C∞v) structures. Such structural transformations occur within ∼9.4 fs. Figure 7 shows a similar plot as Figure 6 but for DOT+(2A1). If compared with Figure 6, the calculated attributes in Figure 7
Figure 5. (a) Same as Figure 3 but for H218O+ and H2O+.
times scales, respectively, while in panels c and d are for the 2A1 state. It is clear from panels a and b that the HHG signals are minimal for the 2B1 state, which can be attributed to the absence of any significant bending motion in this electronic state. In fact, the mass of the oxygen isotope (central atom) should have only a little effect on the HHG signal, as expected. In contrast, the results shown in panel c and d show that the HHG signals are largest for H218O+, with the trend enhanced
Figure 7. Same as in Figure 6 for DOT+ in the 2A1 electronic state. 4860
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The Journal of Physical Chemistry A indicate that the corresponding structural transformations in DOT+ occur in ∼9.5 fs. At longer times, the system oscillates between the two geometries shown in Figure 7. In turn, the expectation values of the bond lengths and bond angle for T2O+ in the 2A1 electronic state are shown in Figure 8 as a function of
Figure 8. Same as in Figure 6 for T2O+ in the 2A1 electronic state. Figure 10. Snapshots of the time-resolved vibronic structures calculated from the expectation values of the positions of the atoms in the HOD+ in the 2A1 electronic state.
time. As shown, the structural transformation from C2v to near D∞h geometries takes now ∼9.9 fs. Finally, we show in Figure 9
the (R,γ) and (r,γ) planes once averaged over r or R (left- and right-hand-side panels, respectively). Clearly, the wave packet shows substantial vibrational dynamics with time. If now compared with the expectation values plotted in Figure 6, the probability density contours in Figure 9 suggest that the wave packet oscillates between turning points in the 2A1 PES. Similar oscillations are visible from Figures 11 (for DOT+), 12 (for T2O+), and 13 (for H218O+) where the corresponding probability density snapshots are plotted. Also visible by inspecting Figures 10−13 are isotopic effects. Note that the results in Figure 12 are similar to the ones reported for H2O+,16
Figure 9. Same as in Figure 6 for H218O+ in the 2A1 electronic state.
a similar plot but for H218O+, where the structural resolution is predicted to be ∼8.6 fs. We emphasize that the quasiperiodic vibrations seen in Figures 6−9 can be attributed to the oscillations of the water isotopomer from the bent geometries to the quasi-linear ones (C2v to near D∞h symmetries for T2O or Cs to near C∞v for HOD+ and DOT+), a trend similar to the one reported16 for H2O+ and D2O+. The peaks seen in the HHG signals for longer times can therefore be rationalized from the wave packet motion on the PES of the ion. We are then led to speculate that the observed bent to quasi-linear oscillations may in the future be observed from time-resolved ultrashort pump−probe experiments. 3.4. Time-Dependent Dynamics in the 2A1 Electronic State. To get a picture of the nuclear dynamics associated with HHG, in panels b and d of Figures 3−5, we show snapshots of the wave packet evolving on the 2A1 state of the H2O+ isotopomers at different times. Because the vibrational dynamics is minimal for the 2B1 state, the corresponding snapshots are omitted for brevity. Panels a−l of Figure 10 show the HOD+(2A1) snapshots as probability density contours in
Figure 11. Same as in Figure 10 for DOT+. 4861
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larger for the heavier water isotopomers, a trend enhanced up to ∼1.5 and ∼1.7 fs for the 2B1 and 2A1 states of water with Cs symmetry (DOT+/HOD+). For T2O+, the corresponding times are ∼1.6 fs in state 2B1 and ∼1.4 fs in state 2A1. Similarly, the HHG signals for H218O+ get enhanced up to ∼1.6 fs in the 2A1 state. From the calculated expectation values of the bond lengths and bond angle in the water isotopomers, the structural rearrangement of HOD+ (DOT+) from the bent to quasi-linear configurations in the 2A1 electronic state occur in ∼9.4 fs (∼9.5 fs), while for T2O+ and H218O+, the corresponding times are 9.9 and 8.6 fs. This confirms our previous speculation13 and observation14,16 that the average time difference between these successive maxima reflects wave packet oscillations between two extreme structural geometries. It is hoped that the signature of these quasiperiodic oscillations between such turning point geometries may be observed in future timeresolved experiments.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
Figure 12. Same as in Figure 10 for T2O+.
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ACKNOWLEDGMENTS This work is supported by Fundaçaõ para a Ciência e a Tecnologia, Portugal, under Contracts PTDC/CEQ-COM/ 3249/2012 and PTDC/AAG-MAA/4657/2012. The support to the Coimbra Chemistry Centre through the Project PEstOE/QUI/UI0313/2014 is also acknowledged.
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REFERENCES
(1) Posthumus, J. H. The Dynamics of Small Molecules in Intense Laser Fields. Rep. Prog. Phys. 2004, 67, 623−665. (2) Bandrauk, A. D.; Ivanov, M. Quantum Dynamic Imaging; Springer: New York, 2011. (3) Corkum, P. B. Plasma perspective on strong field multiphoton ionization. Phys. Rev. Lett. 1993, 71, 1994−1997. (4) Lewenstein, M.; Balcou, P.; Ivanov, M. Y.; L’Huillier, A.; Corkum, P. B. Theory of High-Harmonic Generation by Low-Frequency Laser Fields. Phys. Rev. A 1994, 49, 2117−2132. (5) Paniagua, M.; Martinez, R.; Gamallo, P.; González, M. Potential Energy Surfaces and Quasiclassical Trajectory Study of the O + H+2 → OH+ + H, OH + H+ Proton and Hydrogen Atom Transfer Reactions and Isotopic Variants (D+2 , HD+). Phys. Chem. Chem. Phys. 2014, 16, 23594−23603. (6) Ferray, M.; L’Huillier, A.; Li, X.; Lompre, A. L.; Mainfray, G.; Manus, C. Multiple-Harmonic Conversion of 1064 nm Radiation in Rare Gases. J. Phys. B 1998, 21, L31−L35. (7) Popmintchev, T.; Chen, M.-C.; Popmintchev, D.; Arpin, P.; Brown, S.; Alis̃auskas, S.; Andriukaitis, G.; Balc̃iunas, T.; Múcke, O. D.; Pugzlys, A.; et al. Bright Coherent Ultrahigh Harmonics in the keV Xray Regime from Mid-Infrared Femtosecond Lasers. Science 2012, 336, 1287−1291. (8) Baker, S.; Robinson, J. S.; Haworth, C. A.; Teng, H.; Smith, R. A.; Chirila, C. C.; Lein, M.; Tisch, J. W. G.; Marangos, J. P. Probing Proton Dynamics in Molecules on an Attosecond Time Scale. Science 2006, 312, 424−427. (9) Lein, M. Attosecond Probing of Vibrational Dynamics with HighHarmonic Generation. Phys. Rev. Lett. 2005, 94, 053004−053007. (10) Patchkovskii, S. Nuclear Dynamics in Polyatomic Molecules and High-Order Harmonic Generation. Phys. Rev. Lett. 2009, 102, 253602. (11) Saenz, A. On the Influence of Vibrational Motion on StrongField Ionization Rates in Molecules. J. Phys. B: At. Mol. Opt. Phys. 2000, 33, 4365.
Figure 13. Same as in Figure 10 for H218O+.
which is not surprising because the larger isotopic mass of the central oxygen atom has a minimal role on the dynamics. As could be anticipated, due to being the heaviest isotope (T), the wave packet in T2O+ takes a bit longer to reach the right-handside turning point, followed by DOT+ and HOD+.
4. SUMMARY AND CONCLUSIONS We have calculated the photoelectron spectra of water isotopomers due to SF ionization to the 2B1 and 2A1 electronic states of the cation. The calculated photoelectron spectra are found to be not only in agreement with the available experimental results33,44 but also in line with our recent results of H2O+ and D2O+.16 The HHG signals are predicted to be 4862
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The Journal of Physical Chemistry A (12) Goll, E.; Wunner, G.; Saenz, A. Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses. Phys. Rev. Lett. 2006, 97, 103003. (13) Mondal, T.; Varandas, A. J. C. On Extracting Sub-femtosecond Data from Femtosecond Quantum Dynamics Calculations: Methane Cation. J. Chem. Theory Comput. 2014, 10, 3606−3616. (14) Wörner, H. J. Private communication to A. J. C. Varandas. September 2, 2004. (15) Patchkovskii, S. Short-Time Dynamics at a Conical Intersection in High-Harmonic Spectroscopy. J. Phys. Chem. A 2014, 118, 12069− 12079. (16) Jayachander Rao, B.; Varandas, A. J. C. Sub-femtosecond Quantum Dynamics of the Strong-Field Ionization of Water to the X̃ 2B1 and à 2A1 States of the Cation. Phys. Chem. Chem. Phys. 2015, 17, 6545−6553. (17) Farrell, J. P.; Petretti, S.; Förster, J.; McFarland, B. K.; Spector, L. S.; Vanne, Y. V.; Decleva, P.; Bucksbaum, P. H.; Saenz, A.; Gühr, M. Strong Field Ionization to Multiple Electronic States in Water. Phys. Rev. Lett. 2011, 107, 083001. (18) Wong, M. C. H.; Brichta, J. P.; Bhardwaj, V. R. High-Harmonic Generation in H2O. Opt. Lett. 2010, 35, 1947−1949. (19) Zhao, S. F.; Jin, C.; Lucchese, R. R.; Le, A. T.; Lin, C. D. HighOrder-Harmonic Generation Using Gas-Phase H2O Molecules. Phys. Rev. A 2011, 83, 033409. (20) McFarland, B. K.; Farrell, J. P.; Bucksbaum, P. H.; Gühr, M. High Harmonic Generation from Multiple Orbitals in N2. Science 2008, 322, 1232−1235. (21) Akagi, H.; Otobe, T.; Staudte, A.; Shiner, A.; Turner, F.; Dorner, R.; Villeneuve, D. M.; Corkum, P. B. Laser Tunnel Ionization from Multiple Orbitals in HCl. Science 2009, 325, 1364−1367. (22) Smirnova, O.; Mairesse, Y.; Patchkovskii, S.; Dudovich, N.; Villeneuve, D.; Corkum, P. B.; Ivanov, M. Y. High Harmonic Interferometry of Multi-electron Dynamics in Molecules. Nature 2009, 460, 972−977. (23) Petretti, S.; Saenz, A.; Castro, A.; Decleva, P. Water Molecules in Ultrashort Intense Laser Fields. Chem. Phys. 2013, 414, 45−52. (24) Falge, M.; Engel, V.; Lein, M. Vibrational-State and Isotope Dependence of High-Order Harmonic Generation in Water Molecules. Phys. Rev. A 2010, 81, 023412. (25) Eroms, M.; Jungen, M.; Meyer, H.-D. Vibronic Coupling Effects in Resonant Auger Spectra of H2O. J. Phys. Chem. A 2012, 116, 11140−11150. (26) Yu, H. G.; Varandas, A. J. C. Three-Dimensional TimeDependent Wavepacket Calculation of the Transition-State Resonances for MuH2 and MuD2: Resonance Energies and Widths. J. Phys. Chem. 1996, 100, 14598−14601. (27) Feit, M. D.; Fleck, J. A.; Steiger, A. Solution of the Schrödinger Equation by a Spectral Method. J. Comput. Phys. 1982, 47, 412−433. (28) Kosloff, R. A. Fourier Method Solution for the Time dependent Schrödinger Equation as a Tool in Molecular Dynamics. J. Comput. Phys. 1983, 52, 35−53. (29) Light, J. C.; Hamilton, I. P.; Lill, J. V. Generalized Discrete Variable Approximation in Quantum Mechanics. J. Chem. Phys. 1985, 82, 1400−1409. (30) Krishnan, G. M.; Ghosal, S.; Mahapatra, S. A Theoretical Study of the Electronic Nonadiabatic Transitions in the Photoelectron Spectroscopy of F2O. J. Phys. Chem. A 2006, 110, 1022−1030. (31) Johnson, B. R. The Quantum Dynamics of Three Particles in Hyperspherical Coordinates. J. Chem. Phys. 1983, 79, 1916−1925. (32) Varandas, A. J. C.; Yu, H. G.; Xu, Z. R. Vibrational Spectrum of Li3 and Statistical Analysis of the Energy Levels. Mol. Phys. 1999, 96, 1193−1206. (33) Karlsson, L.; Mattsson, L.; Jadrny, R.; Albridge, R. G.; Pinchas, S.; Bergmark, T.; Siegbahn, K. Isotopic and Vibronic Coupling Effects in the Valence Electron Spectra of H216O, H218O, and D216O. J. Chem. Phys. 1975, 62, 4745−4752. (34) Brundle, C. R.; Turner, D. W. High Resolution Molecular Photoelectron Spectroscopy II: Water and Deuterium Oxide. Proc. R. Soc. London, Ser. A 1968, 307, 27−36.
(35) Köppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular Dynamics Beyond the Born−Oppenheimer Approximation. Adv. Chem. Phys. 1984, 57, 1−169. (36) Heller, E. J. The Semiclassical Way to Molecular Spectroscopy. Acc. Chem. Res. 1981, 14, 368−375. (37) Wigner, E. P. On the Behavior of Cross Sections near Thresholds. Phys. Rev. 1948, 73, 1002−1009. (38) Mairesse, Y.; de Bohan, A.; Fransinski, L. J.; Merdji, H.; Dinu, L. C.; Monchicourt, P.; Breger, P.; Kovacev, M.; Taïrb, R.; Carre, B.; et al. Attosecond Synchronization of High-Harmonic Soft X-rays. Science 2003, 302, 1540−1543. (39) Frolov, M. V.; Manakov, N. L.; Sarantseva, T. S.; Emelin, M. Y.; Ryabikin, M. Y.; Starace, A. F. Analytic Description of the High-Energy Plateau in Harmonic Generation by Atoms: Can the Harmonic Power Increase with Increasing Laser Wavelengths? Phys. Rev. Lett. 2009, 102, 243901. (40) Frolov, M. V.; Manakov, N. L.; Sarantseva, T. S.; Starace, A. F. High-order-harmonic-generation spectroscopy with an elliptically polarized laser field. Phys. Rev. A 2012, 86, 063406. (41) Bandrauk, A. D.; Chelkowski, S.; Kawai, S.; Lu, H. Effect of Nuclear Motion on Molecular High-Order Harmonics and on Generation of Attosecond Pulses in Intense Laser Pulses. Phys. Rev. Lett. 2008, 101, 153901. (42) Manthe, U.; Meyer, H.-D.; Cederbaum, L. S. Wave-Packet Dynamics within the Multiconfiguration Hartree Framework: General Aspects and Application to NOCl. J. Chem. Phys. 1992, 97, 3199− 3213. (43) Engel, V. The Calculation of Autocorrelation Functions for Spectroscopy. Chem. Phys. Lett. 1992, 189, 76−78. (44) Dixon, R. N.; Duxbury, G.; Rabalais, J. W.; Åsbrink, L. Rovibronic Structure in the Photoelectron Spectra of H2O, D2O and HDO. Mol. Phys. 1976, 31, 423−435.
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Subfemtosecond Quantum Nuclear Dynamics in Water Isotopomers B. Jayachander Rao and A. J. C. Varandas* Departamento de Quı ́mica and Centro de Quı ́mica, Universidade de Coimbra, 3004-535 Coimbra, Portugal ABSTRACT: Subfemtosecond quantum dynamics studies of all water isotopomers in the X̃ 2B1 and à 2A1 electronic states of the cation formed by Franck−Condon ionization of the neutral ground electronic state are reported. Using the ratio of the autocorrelation functions for the isotopomers as obtained from the solution of the time-dependent Schrödinger equation in a grid representation, high-order harmonic generation signals are calculated as a function of time. The results are found to be in agreement with the available experimental findings and with our earlier study for D2O+/H2O+. Maxima are predicted in the autocorrelation function ratio at various times. Their origin and occurrence is explained by calculating expectation values of the bond lengths and bond angle of the water isotopomers as a function of time. The values so calculated for the 2B1 and 2A1 electronic states of the cation show quasiperiodic oscillations that can be associated with the time at which the nuclear wave packet reaches the minima of the potential energy surface, there being responsible for the peaks in the HHG signals.
1. INTRODUCTION Molecules exposed to intense laser fields show processes that are of both practical and fundamental interest.1,2 High-order harmonic generation (HHG) is one such process of key relevance in strong-field (SF) physics but not yet established as a practical tool. The HHG can be understood within the framework of a semiclassical three-step model.3,4 An intense laser pulse ionizes the molecule in the first step, resulting in a correlated electron and vibrational wave packet pair. The electron gets accelerated by the laser field while the vibrational wave packet moves on the potential energy surface5 of the resulting cation. In the next step, when the field phase changes, the electron is driven back to the ion core. Finally, the third step consists of the recombination of the electron with the cation, leading to the production of high-energy radiation, which extends into the range of extreme ultraviolet6 and soft Xrays.7 The HHG technique exploits the correlation between the nuclear wave packet created in the molecular cation by the SF ionization and the electronic wave packet with a view to measure the nuclear dynamics on a subfemtosecond (subfs) time scale. Consequently, the study of nuclear dynamics and structural cationic rearrangements on the molecular time scale has become possible. As the mapping of the HHG signal is possible up to ∼1.6 fs at a wavelength of 800 nm, fs nuclear dynamics has been predicted and observed in H2/D2.8−10 Other systems with a fast nuclear dynamics have also been studied, in particular, CD4/CH4,9 where, for methane, it has been attributed to changes of the PES upon ionization.10 In fact, the important role played by the geometrical changes in the SF ionization of molecules and its consequences in nuclear dynamics have been discussed.11,12 This becomes particularly important for cations with Jahn−Teller (JT) interactions. For example, ionization of an electron from the highest occupied © 2015 American Chemical Society
molecular orbital (HOMO) at the equilibrium configuration of methane results in the formation of CH+4 in its ground state, which is known to undergo JT distortion. Accordingly, the three-fold degeneracy of the ground electronic state of the cation is split into three nondegenerate states. In general, JT effects and the resulting conical intersection (CI) of PESs will play a key role in nuclear dynamics. Indeed, subfs nuclear dynamics in CD4/CH4 has recently been investigated by Mondal and Varandas13 via ab initio quantum dynamics including the JT effect. Employing a quadratic vibronic coupling Hamiltonian in a diabatic picture and a multiconfiguration time-dependent Hartree propagation scheme, the ratio of the HHG signal for the CD4 and CH4 cations has been calculated as a function of time.13 Mondal and Varandas found the HHG signal to be largest for the heavier isotope, and this trend was enhanced up to ∼1.85 fs. From this, they predicted the structural rearrangement of CH 4+ from T d to C 2v configuration to occur in this time scale,13 with such predictions having most recently been confirmed by experiment.14 Recently, an analytical approach to calculate the short-time nuclear autocorrelation functions in the vicinity of CIs has been suggested, including nonadiabatic coupling.15 Unfortunately, no results were given for CD+4 /CH+4 that might allow a comparison with the Mondal−Varandas13 predictions. To understand and interpret the origin of the maxima observed in the HHG signals,13 we have most recently started to study the SF ionization of water.16 This has been much utilized as a prototypical model for the study of nuclear dynamics due to SF ionization to multiple electronic states, Received: March 4, 2015 Revised: April 16, 2015 Published: April 24, 2015 4856
DOI: 10.1021/acs.jpca.5b02129 J. Phys. Chem. A 2015, 119, 4856−4863
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The Journal of Physical Chemistry A
the grid for the Jacobi angle, γ, is chosen as the nodes of an Nγpoint Gauss−Legendre quadrature. The action of the exponential operator on Ψ(t=0) is then carried out by dividing the total propagation time into N steps of length Δt, with the exponential operator at each Δt being for this purpose approximated using the second-order split-operator method.27 This is used in conjunction with the fast Fourier transform method28 to evaluate the exponential containing the radial kinetic energy operator and the discrete variable representation method to evaluate the exponential containing the rotational kinetic energy operator.29 The latter is accomplished by transforming the grid wave function to the angular momentum basis (finite basis representation), multiplying it by the diagonal value of the operator (e−ij(j+1)Δtℏ/4I), and back transforming it to the grid representation. The initial wave function is then propagated for a time T. To avoid unphysical reflections or wraparounds of the high-energy components of the wave function that reach the finite-sized grid boundaries at longer times, the last 20 points of the grid along R and r are multiplied by a damping function.30 In summary, the time-dependent wave function is obtained as
both experientally and theoretically.17−24 Indeed, a similar study of the resonant Auger effect in gas-phase water, which involves vibronic coupling effects in the same electronic states of the ion (they form a Renner−Teller pair), has also appeared.25 By employing a time-dependent quantum wave packet approach, the SF ionization of water molecules has been described from the moment of ionization to dissociation. Accordingly, the nuclear autocorrelation functions were calculated for H2O+ and D2O+ in such electronic states by solving numerially the time-dependent Schrödinger equation on a grid. The HHG signals were then calculated by taking the ratio of the squared autocorrelation functions of D2O+ and H2O+ molecules as a function of time. The so-calculated HHG signals were found to be larger for D2O+ than H2O+, with the trend being enhanced up to ∼1.1 and ∼1.6 fs for 2B1 and 2A1 electronic states, respectively. On the basis of these theoretical results, the structural rearragements of H2O+ (D2O+) from C2v to near D∞h configurations in the 2A1 electronic states were shown to occur at ∼8.5 fs (∼9.3 fs) due to the strong excitation of the bending mode. In this article, we extend our recent theoretical effort16 to include the water isotopomers involving protium, deuterium, and tritium in the 2B1 and 2A1 electronic states. As will be shown, the calculated photoelectron spectra of these isotopomers compare well with the available experimental results. Moreover, the HHG signals as a function of time reveal peaks at various times in line with those from our previous study.13,16 By looking at the expectation value of the bond lengths and bond angle, which show quasiperiodic oscillations, such peaks were rationalized as due to the wave packet motion on the PESs of the cation. The paper is organized as follows. Section 2 presents a brief summary of the theoretical framework, while the results are presented and discussed in section 3. The summarizing remarks are in section 4.
⎛ iHt̂ ⎞ Ψ(t ) = exp⎜ − ⎟Ψ(t = 0) ⎝ ℏ ⎠
with the initial wave function for the ground electronic state of the neutral species assuming the Gaussian form Ψ(t = 0) ⎡ (R − R )2 (r − r0)2 ⎤ 0 ⎥ = N exp⎢ − − 2σR2 2σr2 ⎦ ⎣ ⎧ ⎡ (γ − γ )2 ⎤ ⎫ ⎡ (γ − π + γ )2 ⎤⎪ ⎪ 0 ⎥ 0 ⎥ ⎢ ⎨exp⎢ ⎬ ×⎪ + exp 2 ⎢⎣ ⎥⎦⎪ 2σγ2 ⎩ ⎢⎣ 2σγ ⎥⎦ ⎭
2. THEORETICAL AND COMPUTATIONAL DETAILS In this section, we describe briefly the theoretical formalism and computational details used in the present work. The Hamiltonian Ĥ for nuclear motion is the sum of the kinetic, TN, and potential energy, VX/A, operators and may be written as
where μR =
mO(mHa + mHb) mO + mHa + mHb
(1)
and μr =
mHa mHb 2
(3)
where R0, r0, and γ0 specify the initial location, while δR, δr, and δγ are the width parameters of the wave packet along the associated coordinates. Note that due to the poles at γ = 0, π in eq 1, such limiting values cannot be reached. This should cause no fundamental limitation and may be avoided by using, for example, hyperspherical coordinates,31,32 an issue not pursued in the present work. The complete set of numerical parameters here utilized is given in Table 1. As noted above, the SF ionization will be described as a FC transition from the ground electronic state of the neutral water isotopomer to the ground (X̃ 2B1) or first excited (Ã 2A1) electronic states of the cation, respectively. These transitions result, each per se, in two bands in the energy range of 10−18 eV in the experimental recordings.33,34 The transition to the
(J − j)2 j2 ℏ2 ∂ 2 ℏ2 ∂ 2 − + + Ĥ = − 2μR ∂R2 2μr ∂r 2 2μR R2 2μr r 2 + VX/A(R , r , γ )
(2)
; mO, mHi (i = a, b)
are the masses of the involved atoms. In turn, J and j are the total and diatomic angular momenta operators, while the subscripts in the potential energy operator refer to the ground (X ≡ 2B1) and excited (A ≡ 2A1) states of the water ion. As in previous work,16 we will consider J = 0. Within the Franck−Condon (FC) approximation, the initial nuclear wave function of the neutral’s ground electronic state is vertically pumped to the ground- or first excited-state PESs of the cation, where it is then propagated. Such a scheme has been also utilized26 for the calculation of transition-state resonances in MuH2. Accordingly, the time-dependent Schrö dinger equation (cf., eq 2) is numerically solved on a grid in the (R,r,γ) space to obtain the wave function at time t from the one at t = 0. For this, an NR × Nr spatial grid is used in the (R,r) plane defined by Rmin ≤ R ≤ Rmax and rmin ≤ r ≤ rmax. In turn,
Table 1. Grid Parameters and Initial Wave Function
a
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parametera
value
description
NR × Nr × Nγ Rmin, Rmax rmin, rmax ΔR, Δr R0, r0, γ0 δR, δr, δγ T Δt
128 × 128 × 48 10−4, 15.34 0.5, 15.74 0.12, 0.12 1.1, 2.86, (π/2) 0.3, 0.2, 0.1 1100 0.135
grid size extension of grid in R extension of grid in r grid spacings in R and r initial location of GWP width parameters of GWP total propagation time time step in propagation
Bond distances are in a0, angles are in radians, and time is in fs. DOI: 10.1021/acs.jpca.5b02129 J. Phys. Chem. A 2015, 119, 4856−4863
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The Journal of Physical Chemistry A lowest ionic state (2B1) occurs near 12.6 eV (B band) by the removal of an electron from the HOMO (1b1), while the transition to the lowest excited ionic A state (2A1) occurs near 14.8 eV (A band) by the removal of the electron from the second least-bound molecular orbital (HOMO−1). Following our previous work,16 the photoelectron spectrum due to SF ionization is calculated by using a suitably chosen Gaussian wave packet, which will be propagated either in the ground or first excited electronic states of water isotopomers. Finally, the photoelectron spectrum is obtained as the Fourier transform of the time autocorrelation function35,36 P(E) ≈ |λE|2 Re
∫0
∞
e i(E0 + E)t / ℏC′(t ) dt
of attosecond pulse trains for one-dimensional H2 and D2 molecules and shown that the above approximation is valid for few-cycle pulses, whereas for longer pulses, double-ionization occurs. Accordingly, the intensity of the HHG signal comes to be proportional to the squared modulus of the autocorrelation function, C(t). Thus, the ratio of the HHG signal of water heavier and lighter isotopomers can be approximately given by the ratio of the square of the autocorrelation functions8,9 autocorrelation function ratio =
|C l(t )|2
(6)
where Ch (Cl) is the time autocorrelation function of the heavier water isotopomer (lighter isotopomer). Following the above work,8,9 eq 6 assumes that the PESs of both the neutral and cationic species are, up to some constant multiplicative factor,13,16 not affected by the intense laser field. Despite being approximate, eq 6 is found to be very useful for studying subfs nuclear dynamics in molecules,10,13,16 where the exact treatment is prohibitively expensive.
(4)
where λE is the transition dipole moment between the ground electronic state of the neutral water isotopomer and its cation, which usually depends on the kinetic energy and the molecular geometry but is here assumed to be a constant. The quantity E0 is the energy of the vibrational ground level of the ground electronic state of the neutral water isotopomer, and C′(t) is a damped time autocorrelation function of the wave packet evolving on the chosen PES. This is obtained by multiplying C(t) by an exponential function C′(t) = C(t)e−t/τ, with the autocorrelation function assuming the form C(t ) = ⟨Ψ(0)|Ψ(t )⟩
|C h(t )|2
3. RESULTS AND DISCUSSION We first present the photoelectron bands of HOD, DOT, T2O, and H218O calculated with the formalism discussed in the previous section and compare the results with the available experimental data.33 Because the starting state is bound, one has C(t) = ⟨Ψ*(t/2) |Ψ(t/2)⟩, which halves the propagation time T needed to achieve the energy resolution ΔE = 2πℏ/T in the photoelectron spectrum.42,43 The initial wave packet is time-evolved for 1.1 ps, and the autocorrelation functions are calculated and damped with an exponential function, where τ = 44 fs (cf., section. 2) to account for experimental line broadening effects. It has been subsequently Fourier-transformed to calculate the vibronic spectra due to the FC transition to the 2B1 and 2A1 states of the cations. Note that the value of τ used in the damping function corresponds to a convolution of the vibronic line spectrum with a Lorentzian function of ∼15 meV (fwhm). 3.1. Photoelectron Spectra and Femtosecond Nuclear Dynamics. Because the photoelectron spectra of H2O+ and D2O+ have been given elsewhere,16 we only summarize some results in Table 2 and discuss in the following the other
(5)
where Ψ(0) is the eigenfunction of the ground-state Hamiltonian of the neutral molecule and Ψ(t) is the timedependent wave function that is evolving on a particular cationic PES. This convolutes the spectrum with a Lorentzian form of the fwhm (full width at half-maximum) Γ = 2/τ, which can be chosen to approximate the experimental broadening of the spectral features. Note that, following previous work,13,16 the scattering wave function of the outgoing electron is missing in the above theoretical formalism. This is an intricate problem as the scattering state is a plane wave at asymptotic distances, but it gets modified when interacting with the cation. Hence, a full treatment of HHG spectroscopy proves correspondingly more difficult as the cation and electron are produced under the action of an intense laser field, which is ignored in the present formalism. It should also be noted that the calculation of vibronic intensities close to the ionization threshold poses an additional problem by the fact that it should verify the Wigner threshold law.37 Although the role of the ejected photoelectron has been explicitly neglected above, this does not imply that it has a superficial role. In fact, the subfs generated electron wave packet will recollide with the cation after a certain time delay (the recollision occurs on 2/3 of a frequency period), which leads to the harmonic emission, with successively higher harmonics occurring at longer time delays.38 In fact, it is this property of the high-harmonic radiation that allows us to monitor the nuclear motion over a range of time delays. On the basis of previously reported work,10,13,16 ignoring the explicit treatment of the outgoing electron wave packet led to crucial information on the motion of the nuclei in the subfs time regime, according to the semiclassical three-step model.9 Recently, Starace and co-workers39,40 have shown that the HHG intensities can be generally factorized for linear polarization as a product of the ionization and photoemission factors within the semicalssical three-step model. Additionally, Bandrauk et al.41 have performed an exact non-Born− Oppenheimer calculation of the harmonic spectra and shapes
Table 2. Calculated Frequencies for Protonated and Deuterated Water in Both 2B1 and 2A1 Statesa molecule H2O+(2B1) D2O+(2B1) H2O+(2A1) D2O+(2A1) a
ν1/fs 10.7 14.6 16.9 18.5
(10.4) (14.0) (16.5) (n.a.b)
ν1/eV 0.386 0.283 0.245 0.223
ν2/fs 20.9 28.9 33.9 44.6
(21.2) (29.0) (33.2) (44.8)
ν2/eV 0.197 0.143 0.112 0.093
Experimental results33,44 are in parentheses. bn.a. = not available.
isotopomers. To our knowledge, no experimental results are available for most of them but HOD+44 and H218O+.33 Thus, we compare our results only with the available experimental ones. The calculated B and A photoelectron bands of the above water isotopomers are shown in Figure 1a−d, which show (from a to d) the calculated photoelectron spectra of HOD+, DOT+, H218O+, and T2O+, respectively. In this case, SF ionization of the water molecule removes an electron from the HOMO (1b1) and excites it into the ionic ground state, 2B1. Because the PES of H2O+ has a similar topography to the PES 4858
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correspond to the symmetric stretching mode (ν1) and bending mode (ν2). The average peak spacing in the absolute value of the autocorrelation function for the ν1 and ν2 progressions in the 2B1 state of HOD+ shown in panel a are on the same order, ∼12.3 and ∼25.1 fs, which correspond to energy spacings of ∼0.330 and ∼0.165 eV. In turn, the average peak spacings for the ν1 and ν2 peaks in the autocorrelation functions corresponding to the 2A1 electronic state plotted in panel a (blue lines) are ∼17.7 and ∼34.6 fs, respectively. The latter compares well with the available experimental data44 for HOD+, 35.3 fs. This implies energy spacings of ∼0.234 and ∼0.124 eV in the theoretical results and ∼0.117 eV in the experimental one for the latter. Such energy spacings can be seen from the calculated photoelectron spectra of both 2B1 and 2A1 electronic states of HOD+ in panel a of Figure 1. The corresponding results for the other isotopomers are gathered in Tables 3 and 4. Note that the calculated results for the ν2 mode of H218O+ compare well with the experimental33 data, ∼21.4 (2B1 state) and ∼34.1 fs (2A1 state).
Figure 1. Photoelectron spectra of the water isotopomers. Red and blue lines correspond to the water isotopomers with Cs and C2v symmetries, respectively.
of the neutral molecule, the FC ionization populates only a few vibronic levels in the 2B1 electronic state.16 This can be seen from the exponentially decaying FC progression between ∼11.0 and ∼14.0 eV in each panel of the photoelectron spectra. In contrast, the FC ionization from the HOMO−1 (3a1) excites the molecule into the 2A1 electronic state. Because the PES of the cation differs now very much from the ground-state PES of the neutral (the equilibrium geometry of the cation occurs in the former at linear geometries), the FC ionization leads to a strong excitation of the bending mode. As a result, the FC ionization from the HOMO−1 populates a large number of vibrational states, which can be seen from the dense photoelectron bands in panels a−d of Figure 1 between ∼14.0 and ∼18.0 eV. It can be seen from panels a−d of Figure 1 that the photoelectron bands marginally get denser, which can be attributed to the isotopic effect. The squared absolute values of the autocorrelation functions for the FC ionization to the 2B1 (2A1) electronic states of the isotopomeric water cations are shown in panels a−d of Figure 2. The red lines correspond to the 2B1 state and the blue lines to the 2A1 state. As visible from Figure 2, strong quasiperiodic recurrences occur. The absolute value of the autocorrelation function for the 2B1 electronic state plotted in panel a for HOD+ (red line) shows two kinds of oscillations. They
Table 3. Calculated Frequencies for the Water Isotopomers in the 2B1 Statea
a
molecule
ν1/fs
ν1/eV
ν2/fs
ν2/eV
HOD+ DOT+ T2O+ H218O+
12.5 16.2 17.9 10.6
0.330 0.255 0.231 0.390
25.1 32.2 37.8 21.9 (21.4)
0.165 0.130 0.110 0.190
Experimental results33,44 are in parentheses.
Table 4. Calculated Frequencies for the Water Isotopomers in the 2A1 Statea molecule +
HOD DOT+ T2O+ H218O+ a
ν1/fs
ν1/eV
ν2/fs
ν2/eV
17.7 18.0 19.9 17.2
0.234 0.230 0.210 0.241
34.6 (35.3) 35.8 40.6 33.8 (33.1)
0.120 0.115 0.102 0.122
Experimental results33,44 are in parentheses.
3.2. HHG Signals. The ratios of the squared modulus of the autocorrelation functions between DOT+ and HOD+ isotopomers for the 2B1 and 2A1 electronic states are shown in panels a−d of Figure 3 as a function of time. The results in panels a and b correspond to the calcualted HHG signals in the 2 B1 electronic state at short- and long-time intervals, respectively. Similarly, the results in panels c and d are for the 2A1 state. In agreement with our previous study,16 our calculated results predict the intensity of the HHG signal to be larger for DOT+. This is due to the much slower nuclear motion, an effect enhanced with time up to ∼1.5 and ∼1.3 fs for the 2B1 and 2A1 states, respectively. Similarly, the ratios of the squared moduli of the autocorrelation functions of T2O+ and D2O+ in the 2B1 and 2A1 electronic states are shown in panels a−d of Figure 4. Panels a and b show the results for the 2 B1 electronic state at short and long times, while c and d illustrate the same for the 2A1 state. Note that panels a and c refer to the HHG singals for the heavier isotope, T2O+, which reveal a similar behavior but enhanced up to ∼1.6 (2B1 state) and ∼1.4 fs (2A1 state). In turn, panels a−d of Figure 5 show the ratio of the squared autocorrelation functions of H218O+ and H2O+ for the 2B1 and 2A1 electronic states. As above, the results in panels a and b are for the 2B1 state at short- and long-
Figure 2. Autocorrelation functions for the 2B1 (2A1) electronic state of (a) HOD+, (b) DOT+, (c) H218O+, and (d) T2O+ as a function of time. The red lines correspond to 2B1 and the blue lines to 2A1. 4859
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until ∼1.6 fs. These results are not only in accordance with our recent explorations16 but also complement the available experimental findings of Farrell et al.,17 where the HHG signals are found to be more visible in the 2A1 state than those in 2B1. This is due to the large change in geometry of the 2A1 ionic state and hence to a stronger excitation of the vibrational bending motion. 3.3. Extracting Subfemtosecond Nuclear Dynamics. To understand the origin of the large number of peaks in the HHG signal of Figures 3−5 in the 2B1 and 2A1 electronic states (panels b and d), we calculate the expectation values of the bond lengths and bond angle for the water isotopomers here examined. Because no substantial vibrational dynamics is observed for the 2B1 state of water,16,17 we present and discuss only the results for the 2A1 state. The calculated results are shown in Figure 6 as a function of time. Also illustrated are snapshots of the atomic position
Figure 3. (a) Ratio of the squared autocorrelation functions for the 2B1 electronic state of DOT+ and HOD+. (b) Same as (a) but for longer times. (c) As in (a) but fot the 2A1 electronic state of DOT+ and HOD+. (d) Same as (c) but for longer times. Red and blue lines correspond to the 2B1 and 2A1 states, respectively.
Figure 6. Expectation values of the bond lengths and bond angle of HOD+ in the 2A1 electronic state as a function of time. Also shown are snapshots of the position coordinates of the atoms at critical points. Figure 4. Same as Figure 3 but for T2O+ and D2O+.
coordinates taken at critical points in the PES of the HOD isotopomer. As seen from Figure 6, the expectation values of the bond lengths and bond angle show quasiperiodic oscillations in time, which indicate that the HOD+ isotopomer vibrates between the bent (Cs) and a quasi-linear (quasi C∞v) structures. Such structural transformations occur within ∼9.4 fs. Figure 7 shows a similar plot as Figure 6 but for DOT+(2A1). If compared with Figure 6, the calculated attributes in Figure 7
Figure 5. (a) Same as Figure 3 but for H218O+ and H2O+.
times scales, respectively, while in panels c and d are for the 2A1 state. It is clear from panels a and b that the HHG signals are minimal for the 2B1 state, which can be attributed to the absence of any significant bending motion in this electronic state. In fact, the mass of the oxygen isotope (central atom) should have only a little effect on the HHG signal, as expected. In contrast, the results shown in panel c and d show that the HHG signals are largest for H218O+, with the trend enhanced
Figure 7. Same as in Figure 6 for DOT+ in the 2A1 electronic state. 4860
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The Journal of Physical Chemistry A indicate that the corresponding structural transformations in DOT+ occur in ∼9.5 fs. At longer times, the system oscillates between the two geometries shown in Figure 7. In turn, the expectation values of the bond lengths and bond angle for T2O+ in the 2A1 electronic state are shown in Figure 8 as a function of
Figure 8. Same as in Figure 6 for T2O+ in the 2A1 electronic state. Figure 10. Snapshots of the time-resolved vibronic structures calculated from the expectation values of the positions of the atoms in the HOD+ in the 2A1 electronic state.
time. As shown, the structural transformation from C2v to near D∞h geometries takes now ∼9.9 fs. Finally, we show in Figure 9
the (R,γ) and (r,γ) planes once averaged over r or R (left- and right-hand-side panels, respectively). Clearly, the wave packet shows substantial vibrational dynamics with time. If now compared with the expectation values plotted in Figure 6, the probability density contours in Figure 9 suggest that the wave packet oscillates between turning points in the 2A1 PES. Similar oscillations are visible from Figures 11 (for DOT+), 12 (for T2O+), and 13 (for H218O+) where the corresponding probability density snapshots are plotted. Also visible by inspecting Figures 10−13 are isotopic effects. Note that the results in Figure 12 are similar to the ones reported for H2O+,16
Figure 9. Same as in Figure 6 for H218O+ in the 2A1 electronic state.
a similar plot but for H218O+, where the structural resolution is predicted to be ∼8.6 fs. We emphasize that the quasiperiodic vibrations seen in Figures 6−9 can be attributed to the oscillations of the water isotopomer from the bent geometries to the quasi-linear ones (C2v to near D∞h symmetries for T2O or Cs to near C∞v for HOD+ and DOT+), a trend similar to the one reported16 for H2O+ and D2O+. The peaks seen in the HHG signals for longer times can therefore be rationalized from the wave packet motion on the PES of the ion. We are then led to speculate that the observed bent to quasi-linear oscillations may in the future be observed from time-resolved ultrashort pump−probe experiments. 3.4. Time-Dependent Dynamics in the 2A1 Electronic State. To get a picture of the nuclear dynamics associated with HHG, in panels b and d of Figures 3−5, we show snapshots of the wave packet evolving on the 2A1 state of the H2O+ isotopomers at different times. Because the vibrational dynamics is minimal for the 2B1 state, the corresponding snapshots are omitted for brevity. Panels a−l of Figure 10 show the HOD+(2A1) snapshots as probability density contours in
Figure 11. Same as in Figure 10 for DOT+. 4861
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larger for the heavier water isotopomers, a trend enhanced up to ∼1.5 and ∼1.7 fs for the 2B1 and 2A1 states of water with Cs symmetry (DOT+/HOD+). For T2O+, the corresponding times are ∼1.6 fs in state 2B1 and ∼1.4 fs in state 2A1. Similarly, the HHG signals for H218O+ get enhanced up to ∼1.6 fs in the 2A1 state. From the calculated expectation values of the bond lengths and bond angle in the water isotopomers, the structural rearrangement of HOD+ (DOT+) from the bent to quasi-linear configurations in the 2A1 electronic state occur in ∼9.4 fs (∼9.5 fs), while for T2O+ and H218O+, the corresponding times are 9.9 and 8.6 fs. This confirms our previous speculation13 and observation14,16 that the average time difference between these successive maxima reflects wave packet oscillations between two extreme structural geometries. It is hoped that the signature of these quasiperiodic oscillations between such turning point geometries may be observed in future timeresolved experiments.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
Figure 12. Same as in Figure 10 for T2O+.
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ACKNOWLEDGMENTS This work is supported by Fundaçaõ para a Ciência e a Tecnologia, Portugal, under Contracts PTDC/CEQ-COM/ 3249/2012 and PTDC/AAG-MAA/4657/2012. The support to the Coimbra Chemistry Centre through the Project PEstOE/QUI/UI0313/2014 is also acknowledged.
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REFERENCES
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Figure 13. Same as in Figure 10 for H218O+.
which is not surprising because the larger isotopic mass of the central oxygen atom has a minimal role on the dynamics. As could be anticipated, due to being the heaviest isotope (T), the wave packet in T2O+ takes a bit longer to reach the right-handside turning point, followed by DOT+ and HOD+.
4. SUMMARY AND CONCLUSIONS We have calculated the photoelectron spectra of water isotopomers due to SF ionization to the 2B1 and 2A1 electronic states of the cation. The calculated photoelectron spectra are found to be not only in agreement with the available experimental results33,44 but also in line with our recent results of H2O+ and D2O+.16 The HHG signals are predicted to be 4862
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