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Effect of Initial Vibrational-State Excitation on Sub-Femtosecond Photodynamics of Water Borker Jayachander Rao, and António J.C. Varandas J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b07593 • Publication Date (Web): 23 Oct 2015 Downloaded from http://pubs.acs.org on October 27, 2015
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Effect of initial vibrational-state excitation on sub-femtosecond photodynamics of water B. Jayachander Rao and A. J. C. Varandas∗ Departamento de Química, and Centro de Química, Universidade de Coimbra, 3004-535 Coimbra, Portugal (Dated: September 17, 2015) We discuss the effect of initial vibrational-state excitation on the sub-femtosecond photodynamics of water. Photoelectron spectra of Franck-Condon ionization to the 2 B1 state of the H2 O+ (D2 O+ ) from the ground and several vibrationally excited states of the neutral are reported. Also calculated are ratios of the high-order harmonic generation (HHG) signals as a function of time for each initial vibrational-state of the neutral molecule as predicted from the ratios of the square of the autocorrelation functions for D2 O+ and H2 O+ . They reveal maxima as a function of time for each vibrational state of the neutral molecule. In turn, the HHG signals are found to be enhanced with vibrational excitation, with the calculated expectation values of the bond lengths and bond angle revealing quasiperiodic oscillations in time for all initial vibrational-states of the neutral species. While the bond lengths show only a marginal increase, the bond angle is found to be enhanced markedly by vibrational excitation, this being therefore responsible for the observed rise in the HHG signal.
I. INTRODUCTION
The study of atoms and molecules in intense ultrashort laser fields is a subject of increasing significance in attosecond physics and chemistry [1, 2]. Due to the highly nonlinear nonperturbative nature of high-order harmonic generation (HHG) processes, the modelling of HHG in molecules is difficult [3]. Also molecules add further complexity due to the nuclear motion and coupled with other process such as charge resonance enhanced ionization [4, 5] and bond softening [6]. In particular, HHG , i.e., high-energy radiation due to the interaction of intense laser pulses with atoms and molecules, has recently received considerable attention by the scientific community, offering possible use as a source of coherent radiation. The HHG spectroscopy can be understood within the framework of a semiclassical three-step model [7, 8]. According to this model, an electron tunnels out of the atom or molecule and recombines with the parent [7, 9] or neighboring [10] ion at the end of the laser driven motion, leading to the emission of high-energy photons which extend to the ultraviolet and X-ray regions [11, 12]. Such a technique then exploits the correlation between the nuclear wave packet created in the molecular cation by the strong-field (SF) ionization and the electronic wave packet with a view to measure the nuclear dynamics on sub-femtosecond (sub-fs) and sub-angstrom scales. As a result, studies of high-resolution nuclear dynamics and structural rearrangements in molecular cations have become feasible on the sub-fs time scale. As the mapping of the HHG signal is possible up to ∼ 1.6 fs at a wavelength of 800 nm, sub-fs nuclear dynamics has been predicted and observed in D2 /H2 [13–15] and CD4 /CH4 [14] where, for the latter, it has been attributed to changes in the potential energy surface (PES) upon ionization [15]. In fact, the important role played by the geometrical changes in the SF ionization of molecules and its consequences in nuclear dynamics has been discussed earlier [16, 17]. More recently, the sub-fs nuclear dynamics in CD4 /CH4 has been investigated by Mondal and Varandas [18] via ab initio quantum dynamics using a quadratic vibronic coupling Hamiltonian in a diabatic picture and a multiconfiguration time-dependent-Hartree propagation scheme [18]. With the ratio of the HHG signals for the CD4 and CH4 cations calculated as a function of time [18], Mondal
∗
[email protected] and Varandas found the HHG signal to be largest for the heavier isotope with the ratio being enhanced up to ∼ 1.85 fs. From this, they predicted that the structural rearrangement from Td to C2v configuration in CH+ 4 to occur in this time scale [18], with such predictions having most recently been confirmed by experiment [19]. To understand and interpret the origin of the maxima observed in the HHG signals [18], we have most recently initiated a study of the SF ionization of a simpler system, water [20]. This has been much utilized as a prototype for the study of nuclear dynamics due to SF ionization, both experimentally and theoretically [21–28]. For D2 O/H2 O, it has been established that the difference between the potentials of the neutral and ground electronic state of the cation determines the speed of the nuclear dynamics [21]. Recent experiments, however, have demonstrated that the molecular SF response depends, at least for some molecules, on more than one orbital [24–27]. Superposition of several ionic states resulting from the SF ionization will then complicate simple imaging schemes, but will be a source for richer information. Similarly, conical intersections (CIs) and the ensuing nonadiabatic effects will add further complication. In fact, vibronic coupling effects in the study of the resonant Auger effect in gas-phase water which involves the 2 B1 and 2 A1 electronic states of the cation (these form a Renner-Teller pair) have been reported [29]. Indeed, a study of the influence of the initial vibrational-state and isotope dependence on the HHG signals has already been reported [28] using approximate model potentials. These authors have found that the calculated ratio of the HHG spectra for D2 O and H2 O is close to unity for the initial vibrational ground-state of the neutral, thus implying no isotopic effects in the detection window of∼ 1.6 fs for 800 nm laser pulses. However, for vibrationally excited states of the neutral, the ratio of the HHG spectra showed a clear dependence on both the initialstate and isotopic species. Moreover, an explanation for the observed rise in the HHG signals as function of initial vibrational state of the neutral has not been furnished in the above study. Later experiments [21] have, however, shown pronounced isotope effects within the ∼ 1.6 fs time window, which is attributed to the ionization of water from the 3a1 (HOMO-1) orbital.
Although the autocorrelation function (and hence the HHG signal) can be calculated exactly for diatomics, [13, 14] it is much more challenging and prohibitively expensive for larger molecules. To circumvent the problem, reduced dimensionality models are commonly employed, [15, 18, 30] but this limits ACS Paragon Plus Environment
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the quantity and quality of the information that can be obtained Table I. Times scales for the structural transformation from the bent to from the dynamics simulations. Alternatively, such informathe quasilinear configurations in the 2 A1 electronic state for different tion can be estimated from experimental photoelectron spectra water isotopomers. Here t represents the time taken for the structural [21]. Such a procedure has been utilized for ND3 /NH3 , using a transformation between the bent (C2v or Cs ) and quasilinear configuraone-dimensional model [30, 31]. Apart from the SF correction tions (D∞h or C∞v ). and reduced-dimensionality approximation, the extracted autoIsotopomer ∼ t/fs correlation function should be exact. However, a disadvantage is that the extracted autocorrelation function, and hence the nuH2 O+ 8.5 clear dynamics, cannot be easily inverted to explore the wave D2 O+ 9.3 packet dynamics, thence the structural evolution. + HOD 9.4 In recent work, we have carried out a study of SF ionDOT+ 9.5 ization of water molecules from the moment of ionization to T 2 O+ 9.9 dissociation by using a time-dependent quantum wave packet + H18 O 8.6 2 method [20]. Accordingly, the nuclear autocorrelation functions were calculated for H2 O+ and D2 O+ by solving numerically the time-dependent Schrödinger equation on a grid [20]. suggested including nonadiabatic coupling [37]. Indeed an ab Following previous work, [18, 20, 32, 33] the HHG signals have initio quantum dynamics study to explore the sub-fs structural then been approximated by the ratio of the squared autocorrerearragement of ground state of CH+ + + 4 cation [18] has been relation functions of D2 O and H2 O molecules as a function ported by calculating the expectation values of the nuclear coof time [20]. As expected, the calculated HHG signals were ordinates as a function of time [33]. Confirming previous obfound to be larger for D2 O+ than H2 O+ , with the trend being servation, [18] the sub-fs structural evolution of CH+ 2 2 4 starts enhanced up to ∼ 1.1 fs and ∼ 1.6 fs for the B1 and A1 electhrough activation of the totally symmetric a1 and doublytronic states, respectively [20]. Substantial vibrational dynamdegenerate e modes [33]. While the former retains the origics has been found in the 2 A1 electronic state when compared inal Td symmetry of the cation, the Jahn-Teller (JT) active e to the one in 2 B1 , which is in accordance with the recent expermode conducts it to a D2d structure. The novelty [33] is that imental observations Farrell et al. [21]. Based on these results, at ∼ 1.85 fs the intermediate D2d structure is predicted to rearthe structural rearrangements of H2 O+ (D2 O+ ) from C2v to near range to the local C2v minimum geometry via JT active bending 2 D∞h configurations in the A1 electronic states were shown to vibrations of t2 symmetry, thence explaining the entire ultrafast occur by ∼ 8.5 fs (∼ 9.3 fs) due to the strong excitation of the evolution of the cation since its inset until entering the experibending mode [20]. mentally observed region of its equilibrium geometry. We have also extended such theoretical efforts [20] to inIn this work, we investigate the effect of initial vibrationalclude other water isotopomers in the 2 B1 and 2 A1 electronic state excitation and isotope dependence on the sub-fs photostates [32]. Accordingly, we have calculated the photoelectron dynamics of water. The photoelectron spectra due to Franckspectra of the water isotopomers due to SF ionization to the Condon (FC) ionization of water to the 2 B1 state of the 2 B and 2 A electronic states of the cation [32]. Such photo1 1 cation and the nuclear autocorrelation functions of the resultelectron spectra have been found to be not only in agreement ing cation are calculated numerically by solving the timewith the available experimental results [34–36] but also in line dependent Schrödinger equation. The calculated spectra are with our recent results of H2 O+ and D2 O+ [20]. In turn, the found to match the available experimental results. We have HHG signals are predicted to be larger for the heavier water isocalculated the ratio of the HHG signals from the ratio of the topomers, a trend enhanced up to ∼ 1.5 fs and ∼ 1.7 fs for the square of the autocorrelation function of D2 O+ and H2 O+ as 2 B and 2 A states of water with C symmetry (DOT+ /HOD+ ). s 1 1 a function of time for each vibrational-state of the neutral. As For T2 O+ , the corresponding times are ∼ 1.6 fs for state 2 B1 shown later, the calculated HHG signals are found to be larger 2 18 + and ∼ 1.4 fs for A1 . Similarly, the HHG signals for H2 O for the heavier isotopomer which, in turn, increase with the get enhanced up to ∼ 1.6 fs in the 2 A1 state. From the calvibrational-state of the neutral. The expectation values of bond culated expectation values of the bond lengths and bond anlengths and bond angle show oscillatory behavior with time. gle in the water isotopomers, the structural rearrangement of Specifically, while the expectation values of the bond lengths HOD+ (DOT+ ) from the bent to quasi-linear configurations reveal a marginal increase, those for the bond angle substanin the 2 A1 electronic state occur in ∼ 9.4 fs (∼ 9.5 fs), while tially increase as a function of the vibrational excitation, thus + for T2 O+ and H18 2 O the corresponding times are ∼ 9.9 fs and suggesting the latter to be responsible for the observed rise in ∼ 8.6 fs. The time-scales for the bent to quasilinear configuthe HHG signals. 2 rations in the A1 electronic state of the cation are collected in The paper is organized as follows. Section II and III present a Table I, which clearly shows isotopic effects in the structural brief summary of the theoretical and computational framework dynamics of the cations. This confirms the previous speculahere utilized. The numerical results are presented and discussed tion [18, 20, 30, 32] and observation [19, 31] that the average in section IV. Some summarizing remarks are in section V. time difference between these successive maxima reflect vibrational wave packet oscillations between two extreme structural geometries, thereby time-resolving the structural motion in this II. METHOD electronic state. We have further speculated that these quasiperiodic oscillations could be observed in typical time-resolved pump-probe experiments [32]. The photoelectron spectrum is described as a FC transition to Recently, an analytical approach to calculate the short-time the 2 B1 state of H2 O+ (D2 O+ ) from the ground and several vinuclear autocorrelation functions in the vicinity ofACS CIs has been Plus brationally excited states of the neutral via a Fourier transform Paragon Environment
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of the time autocorrelation function [38, 39]: ˆ ∞ ′ P(E) ≈ |λE |2 Re ei(Ev +E0 )t/¯hC (t)dt 0
(1)
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. This is justified as the ejected electron is assumed not to correlate with electrons of the ion. In turn, Ev (E0 ) is the energy of the v-th vibrational level of the ground electronic state (ground vibrational level) of ′ the neutral water isotopomer, and C (t) is the damped time autocorrelation function of the wave packet evolving on the chosen PES. This is obtained by multiplying C(t) with an exponen′ tial function C (t) = C(t)e−t/τ , which convolutes the spectrum with a Lorentzian form of fwhm (full width at half-maximum) Γ = 2/τ chosen such as to approximate the experimental broadening of the spectral features. The undamped autocorrelation function assumes the form
C(t) = Ψv (0)|Ψv′ (t) (2) where, Ψv (0) is the eigenfunction of the ground-state Hamiltonian of the neutral molecule, and Ψv′ (t) is the time dependent wave function that is evolving on a particular cationic PES. The wave function, Ψv′ (t) on the 2 B1 state PES of H2 O+ is obtained by solving the time-dependent Schrödinger equation: Ψv′ (t) = exp(−
iHˆ X t )Ψv (t = 0). h¯
(3)
The Hamiltonian of the cation, Hˆ X (X = 2 B1 ) is here described in Jacobi coordinates as h¯ 2 ∂ 2 ∂ h¯ 2 ∂ 2 h¯ 2 1 ∂ Hˆ X =− (sin γ )+VX (R, r, γ ) − − 2µ1 ∂ R2 2µ2 ∂ r2 2I sin γ ∂ γ ∂γ (4) where, R, r and γ are the separation of the center of mass of H2 from O, the internuclear separation of H2 and the angle between the ~R and ~r. In turn, the reduced masses are defined as O mH and µ2 = m2H , where mO , mH and I denote the µ1 = m2m O +2mH masses of oxygen, hydrogen, and the moment of inertia. This is defined as 1I = µ 1R2 + µ 1r2 . We employ an ab initio PES [40] 1
2
recollide with the cation after a certain time delay (∼ 2/3 of the cycle time), which leads to the HHG, with successively higher harmonics occurring at longer time delays [45]. In fact, it is this high-harmonic radiation property that allows to monitor the nuclear motion over a range of time delays. Based on previous work [14, 15, 18, 20, 30, 32], we have ignored the explicit treatment of the outgoing electron wave packet. However, the method can still yield crucial information on the motion of the nuclei in the sub-fs time regime within the semiclassical three-step model [14]. In fact, Starace and co-workers [46, 47] have shown that, within the semiclassical three-step model, the HHG intensities can generally be split as a product of an ionization and a photoemission factors for linear polarization. Indeed, Bandrauk et al. [3] have recently performed a calculation of harmonic spectra and shapes of attosecond-pulse trains for 1-dimensional H2 and D2 molecules beyond the Born-Oppenheimer approximation. From such an analysis [3, 14, 15, 18, 20, 30, 46, 47], the intensity of the HHG signal is proportional to the squared modulus of the autocorrelation function,
η (t) = |C(t)|2 .
(5)
Note that, within the Condon approximation, the transition dipole moment matrix from the ground electronic state of the neutral to the cationic ground electronic state is assumed to be independent of nuclear coordinates. Thus, the ratio of the HHG signals of the heavier and lighter isotopomers of water can be approximately given by the ratio of the square of the autocorrelation functions: [13, 14] ratio of autocorrelation functions=
ηD2 O (t) . ηH2 O (t)
(6)
Note further that Eqs. (5) and (6) are based on the assumption that both PESs for the neutral and cationic species are, up to some constant multiplicative factor [18, 20], not affected by the intensity of the laser field. Yet, despite approximate, they can still yield very useful information when studying sub-fs nuclear dynamics in polyatomic molecules [15, 18, 20], particularly having in mind that the exact treatment is prohibitively expensive.
III.
COMPUTATIONAL DETAILS
for the 2 B1 state of the cation, which is based on multireference configuration interaction energies [41] with the augmented A. Vibrational states for the 1 A1 state of H2 O (D2 O) correlation consistent quadruple-zeta polarized basis [42], and subsequently modelled as a many-body expansion [43]. With a To calculate the photoelectron spectrum using the timeroot-mean-square deviation of ∼ 1.8 kcal mol−1 , the 2 B1 state dependent wave packet approach, we first need to obtain the PES is quite accurate [40]. vibrational eigenfunctions, Ψv (0), of the ground state of H2 O Note that the scattering wave function of the outgoing elec(D2 O) using the spectral quantization approach [48]. In this tron is missing in the above formalism [18, 20], which is a method, the pseudospectrum pertinent to the FC transition of plane wave at asymptotic distances but gets modified when ina hypothetical initial state to the final Born-Oppenheimer PES teracting with the cation. Hence a full treatment of HHG specof H2 O (D2 O) is calculated. In the time-dependent picture, the troscopy becomes correspondingly more difficult as the cation golden rule expression for the spectral intensity is Eq. (1). A and electron are produced under the action of an intense laser suitably chosen Gaussian wave packet (GWP) is treated as a field, which is ignored in the present formalism. Note also hypothetical initial state: that the calculation of vibrational intensities close to the ion (R − R0)2 (r − r0 )2 ization threshold poses an additional problem by the fact that it Ψ(t = 0) = N exp − − should verify the Wigner threshold law [44]. Although the role 2σr2 2σR2 # " #) ( " of the ejected photoelectron has been here explicitly neglected, 2 2 ( ) ) ( γ − γ γ − π + γ 0 0 it should not imply that it has an insignificant role in the dy+ exp . (7) × exp 2σγ2 2σγ2 namics. In fact, the sub-fs generated electron wave packet will ACS Paragon Plus Environment
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The Journal of Physical Chemistry This is located along the C2v geometry on the 1 A1 PES of H2 O and propagated in time to calculate the vibrational energy level spectra via Eq. (1) as a function of energy. The quantities, R0 , r0 and γ0 in the above equation specify the initial location, while δR , δr and δγ are the width parameters of the wave packet along the associated coordinates. The near-spectroscopic accuracy ab initio ground state PES of the neutral [49] has been utilized, which is partly based on complete-basis-set extrapolated MRCI energies and spectroscopic data via energy-switching (ES [50]). In turn, the numerical approach used in the propagation is well described in the literature [51, 52] and needs not being reiterated here. Thus, for calculating the vibrational states of the neutral, a 128 × 128 spatial grid has been used in the R × r plane with 0.1 a0 ≤ R ≤ 15.34 a0 and 0.5 a0 ≤ r ≤15.74 a0 . In turn, the nodes of a 47-point Gauss-Legendre quadrature (GLQ) are used for the grid along the Jacobi angle γ . The initial state has then been propagated for ∼ 1.1 ps with a time step of ∆ t = 0.135 fs. The damping function is activated at Rmask =12.94 a0 and rmask =13.34 a0, such as to absorb the components that reach the grid edges at longer times [52]. The calculated pseudospectral intensity when plotted as a function of energy yields peaks at the energy eigenvalues of the vibrational levels of the ground electronic state of the neutral. The eigenfunctions of these levels are then determined by projecting a time-evolved wave packet onto the desired veigenstate of energy Ev [48]: Ψv (Ev ) =
ˆ
Table II. Grid parameters for the photoelectron spectra calculations. Parametera) NR × Nr × Nγ Rmin , Rmax rmin , rmax ∆R, ∆r T ∆t Rmask rmask a)
value 128 × 128 × 47 0.1, 15.34 0.5, 15.74 0.12, 0.12 1100 0.135 12.94 13.34
description grid size extension of grid in R extension of grid in r grid spacings in R and r total propagation time time step in propagation starting damping point in R starting damping point in r
Bond distances are in a0 , angles in radians, time in fs.
intensity/au
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T
eiEvt/¯h Ψ(t)dt.
(8)
0
B. Photoelectron spectra
-10.0
-9.5
-9.0
-8.5 -8.0 energy/eV
-7.5
-7.0
-6.5
Once the vibrational states of the neutral water are obtained, Figure 1. Vibrational spectrum of H2 O computed with GWP No. 2 within the FC approximation, the initial nuclear wave func(c.f. Table II). Intensity is in arbitrary units as a function of energy of the vibrational level, with the zero of energy corresponding to the tion of the ground-state of the neutral is vertically shifted by asymptotically separated O + H2 . a FC transition to the ground-state PES of the cation, where it is then propagated in time. Accordingly, the time-dependent Schrödinger equation (c.f. Eq. (3)) is numerically solved on a IV. RESULTS AND DISCUSSION grid in the (R, r, γ ) space to obtain the wavefunction at time t from the one at t = 0. For this, a NR ×Nr spatial grid is used in the (R, r) plane defined by Rmin ≤ R ≤ Rmax and rmin ≤ r ≤ rmax . A. Vibrational states of H2 O (D2 O) In turn, the grid for the Jacobi angle γ , is chosen as the nodes of a Nγ -point GLQ. The action of the exponential operator on We present next the vibrational energy level spectrum of H2 O Ψv (t = 0) is then carried out by dividing the total propagation and D2 O as calculated via spectral quantization [48]; see the time into N steps of length ∆t, with the exponential operator at previous section. The initial parameters and average energies each ∆t being for this purpose approximated using the secondof the GWPs here utilized are given in Table III. As expected, order split-operator method [53]. This is used in conjunction the deep potential energy well (∼ −10.06 eV) of the H2 O (1 A1 ) with the fast Fourier transform method [54] to evaluate the exground electronic state at a geometry with C2v symmetry supponential containing the radial kinetic energy operator, and the ports a large number of bound states corresponding to the vidiscrete variable representation method to evaluate the expobrational energy levels of H2 O. nential containing the rotational kinetic energy operator [55– Fig. 1 shows a typical vibrational spectrum of H2 O calcu57]. The latter is accomplished by transforming the grid wave lated using the GWP No. 2 of Table III. The intensity in arfunction to the angular momentum basis (finite basis represenbitrary units is plotted as a function of the energy of the vibratation), multiplying it by the diagonal value of the operator tional level; energy is measured from the bottom of the well of (e−i j( j+1)∆t h¯ /4I ), and back transforming it to the grid representhe 1 A1 PES and the zero of the energy corresponds to asymptation. The initial wave function is then propagated for a time totically separated H2 + O reactants. The peaks correspond to T . To avoid unphysical reflections or wraparounds of the highthe vibrational levels of H2 O, and the energy at the peak maxenergy components of the wave function that reach the finiteimum to the vibrational energy eigenvalue. As already noted, sized grid boundaries at longer times, the last 20 points of the the initial WP has been time evolved for ∼ 1.1 ps which leads grid along R and r are multiplied by a damping function [52]. to a Environment spectral resolution of 4 meV. This is obtained by effectively The grid parameters used are summarized in Table II. ACS Paragon Plus
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(0,2) (c4)
(3,0,0)
>
(5,0,0) 0.5
1.0 1.5 R/a0
γ /rad
3.0
2.0
90
90 (5,0,0)
0 0.0
0.5
(0,5) 1.0 1.5 R/a0
2.0
1.6
1.8 2.0 rOH/a0
0 2.2
Figure 2. Probability densities for the eigenfunctions of the spectrum of H2 O in Fig. 1. The contours are plotted in the (R, r) (first column), (R, γ ) (second column) and (rOH , θ ) (right most column) planes.
-10.0
-9.5
-9.0
-8.5 -8.0 energy/eV
-7.5
-7.0
-6.5
Figure 3. Same as Fig. 1 but for D2 O.
doubling the propagation time via calculation of C(2t) from the time evolving wave function. It can be seen from Fig. 1 that well resolved peaks are obtained at low energies near the bottom of the PES. The number of such peaks increases with energy, with the 0-0 line shown in Fig. 1 occurring close to the minimum of the PES (−10.06 eV)) at ∼ −9.5 eV. Thus, the zero-point energy of H2 O is estimated to be ∼ 0.56 eV.
Table III. Parameters for the different choices of initial GWP in the calculation of the vibrational spectrum of H2 O. GWP R0 /a0
r0 /a0 γ0 /rad δR0 /a0 δr0 /a0 δγ0 /rad Ev /eV
1 2 3 4
2.8623 2.3623 1.8623 1.3623
1.08111 1.58111 2.08111 2.58111
π /2 π /2 π /2 π /2
0.30 0.30 0.30 0.30
1.0 0.0
(0,3) (c5)
(4,0,0)
(0,4)
180 (a6)
3.0 2.0
(c4)
(b5)
(4,0,0) 4.0
180 (c6)
(0,2) (b4)
(3,0,0) (a5)
(0,4) (b6)
r/a0
>
(4,0,0) 180 (a6)
(c3)
(2,0,0)
(3,0,0)
(c5)
(0,1) (b3)
(a4)
(0,3) (b5)
(4,0,0) 4.0
(a3) (2,0,0)
(b4)
(3,0,0)
(1,0,0)
(5,0,0) 0.5
1.0 1.5 R/a0
2.0
>
(c3)
(2,0,0) (a4)
r/a0
(0,1) (b3)
(2,0,0)
(a5)
(1,0,0)
180 (c6)
(b6) 90
90 (0,5)
(5,0,0) 0 0.0
0.5
θ /rad
(a3)
(c2)
(b2)
>
(1,0,0)
(0,0)
(0,0,0) (a2)
>
(1,0,0)
1.0 0.0
(c2)
(b2)
(c1)
(b1)
(0,0,0)
(0,0)
(0,0,0) (a2)
(a1)
θ /rad
(0,0,0)
2.0
(c1)
(b1)
>
(a1)
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γ /rad
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1.0 1.5 R/a0
2.0
1.6
1.8 2.0 rOH/a0
0 2.2
Figure 4. Same as Fig. 2 but for D2 O.
The eigenfunctions of these vibrational levels are calculated via Eq. (8) and their probability density (|Ψv (E)|2 ) contours are plotted in Fig. 2 in (R, r), (R, γ ) and (rOH , θ ) planes. The results in the first and second columns are averaged over γ and r, respectively. The results in the right most column are plotted for a fixed value of γ = π2 . As shown, these reveal nodal progressions along the v = ν2 mode of H2 O. The nodal progressions are indicated in each panel of Fig. 2 using the corresponding coordinates. Clearly, the eigenfunction plotted in panels a1, b1 and c1 refer to water in its ground vibrational state. Indeed, it does not contain any node along any of the coordinates. As expected, this level occurs at ∼ −9.5 eV and about ∼ 0.56 eV above the minimum of the well on the 1 A1 PES, which corresponds to the zero-point energy of H2 O. The WP results plotted in other panels are the vibrationally excited states of the H2 O along the ν2 mode as shown in Fig. 2 (right most column). For example, the eigenfunction plotted in panels a2, b2 and c2 contain a node along the R (in the first two columns) and θ (in the right most column) of Fig. 2. Such a level occurs at ∼−9.302 eV, which is approximately ∼ 0.1976 eV above the ground vibrational state. Such a result is in excellent agreement with the reported value of ∼ 0.198 eV [58]. The energies of the calculated vibrational states and the most accurate reported values from the Ref. [58] are shown in Table IV. Clearly, the agreement is excellent. Fig. 3 shows a typical vibrational spectrum of D2 O. The intensity in arbitrary units is plotted as a function of the energy of the vibrational levels of ground-state water. It is clear that well resolved peaks occur here too near the minimum, with the number of the peaks increasing as one moves up in energy. Also seen is the fact that the density of peaks in the vibrational spectrum of D2 O is larger when compared to H2 O in Fig. 1, which can be attributed to the isotopic effect. The 0-0 line of the spectrum in Fig. 3 occurs closest to ∼ −9.65 eV, namely ∼ 0.41 eV above the minimum, thus corresponding to the zero-point energy of D2 O.
The probability density contours of the eigenfunctions of these vibrational levels calculated via Eq. (8) are plotted in Fig. 4 in the (R, r), (R, γ ) and (rOH , θ ) planes. The results plotted in the first two columns are averaged over γ and r, while the last column ones are for a fixed value of γ = π2 . They reveal nodal progressions along the ν = ν2 mode of D2 O, which ACS Paragon Plus Environment
0.25 0.25 0.25 0.25
0.10 0.10 0.10 0.10
-8.410 -7.740 -5.530 -2.816
6
The Journal of Physical Chemistry Table IV. Vibrational levels for ground state H2 O as compared with the available values from the literature [58]. (0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (0, 5)
-9.500 -9.302 -9.110 -8.922 -8.740 -8.565
– 0.1976 0.3906 0.5784 0.7603 0.9349
– 0.198 0.391 0.578 0.760 0.935
>
0.8 0.6 0.4
H2O
0.2
D2O
v=0
2
(rOH , θ ) Evpresent /(eV) ∆E = (Ev − E0 )/eV ∆E/eV [58]
1.0
|C(t)|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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0.0
0
25
+ +
(a) 50
75
100 0
v=1 25
(b) 50
75
100 0
v=2 25
(c) 50
75
100
1.0 (d)
v=3
0.8
(e)
v=4
(f)
v=5
0.6 0.4
Table V. Vibronic levels of the D2 O as compared with the available values from the literature [59, 60]. present
(rOH , θ ) Ev (0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (0, 5)
/(eV) ∆E = (Ev − E0 )/eV ∆E/eV [59, 60]
-9.650 -9.510 -9.360 -9.220 -9.080 -8.950
– 0.140 0.290 0.430 0.569 0.700
– 0.146 0.289 – – –
0.2 0.0
0
25
50
75
100 0
25
50 time/fs
75
100 0
25
50
75
100
>
Figure 6. Autocorrelation functions for the 2 B1 electronic state of H2 O+ (red solid lines) and D2 O+ (red dashed line) for the initial vibrational-state of ν2 = (a) 0, (b) 1, (c) 2, (d) 3, (e) 4 and ( f ) 5.
B. Photoelectron spectra and fs nuclear dynamics
In the following, we present the photoelectron bands of H2 O and D2 O, and compare the theoretical results with the available experimental ones [61]. Since we start from an initial boundstate wave function, we calculate C(t) = hΨ∗ (t/2)|Ψ(t/2)i, which halves the propagation time T required to achieve the energy resolution of ∆E = 2Tπ h¯ in the photoelectron spectrum [62, 63]. The initial WP is time evolved on the 2 B1 PES of water isotopomer for the duration of ∼ 1.1 ps, and the calculated C(t) finally damped after multiplying by an exponential function (τ =220 fs) to account for the experimental line broadening effects. The damped autocorrelation functions are finally Fourier transformed to get the photoelectron spectra. The value of τ employed corresponds to a convolution of the vibrational line spectrum with a Lorentzian of ∼ 3 meV fwhm. The calculated and experimental [61] photoelectron spectra of H2 O and D2 O are shown in Fig. 5(a-d). Panels a and c of Fig. 5 depict the experimental [61] and theoretical results of H2 O, while panels b and d depict corresponding results for D2 O. As shown, the calculated spectra in panels c and d compare well not only with the experimental counterparts [61] in panels a and b, but also with our recent theoretical results [20]. SF ionization of the H2 O molecule removes an electron from the HOMO (1 b1 ) orbital and excites the molecule into the 2 B1 ionic ground state, leaving H2 O+ with a PES quite similar to that of the neutral ground electronic state. As a result, FC ionization populates only a few vibronic levels in the 2 B1 electronic state, as implied from the exponentially decaying FC progression [61] in Fig. 5. In turn, the FC ionization from the HOMO-1 orbital (3 a1 ) pumps the molecule into the 2 A1 state, with the resulting PES of the cation differing significantly from the neutral ground-state PES. The equilibrium geometry of the cation occurs now at linear geometries. This leads to a strong excitation of the bending mode in the 2 A1 state. Ionization from the HOMO-1 populates therefore a large number of vibrational states. Note that FC ionization to 2 B1 electronic Figure 5. Photoelectron spectra of H2 O and D2 O calculated by emstate of the cation is only taken into consideration in the present ploying the method of Section II: panels (a) and (c) refer to experiinvestigation. However, as demonstrated experimentally [21], mental and theoretical results of H2 O, while (b) and (d) are for D2 O. the SF response of water molecules depends on more than one ACS Paragon Plus Environment
shows that such states represent the vibrationally excited states along the bending mode. Conversely, the eigenfunction plotted in the first row (panels a1, b1 and c1) does not show any node, and hence represents the ground vibrational state. It occurs at ∼ −9.650 eV, thence ∼ 0.41 eV above the minimum. The WP results plotted in other panels are vibrationally excited states along the bending mode. For example, the eigenfunction plotted in panels a2, b2 and c2 contains nodes along the R (in the first two panels) and θ (in the right most column) of Fig. 4. This level occurs at ∼ −9.510 eV, which is ∼ 0.140 eV above the ground vibrational level, thence in excellent agreement with the reported value of ∼ 0.145 eV [59, 60]. The energies of the calculated vibrational states and available reported ones [59, 60] are gathered in Table V.
7
1.0
10 (a)
(b) v=0 v=1 v=2 v=3 v=4 v=5
8
6
(t) /η
H2O
(t)
0.8
2
D2O
0.6 +
η
orbital [21]. Hence it is essential to consider FC ionization to both 2 B1 and 2 A1 electronic states of the cation, which will be examined elsewhere. The squared absolute value of the autocorrelation functions for the FC ionization of water (heavy water) to the 2 B1 state of H2 O+ (D2 O+ ) for different values of the initial vibrationalstate of the neutral isotopomer are shown in Fig. 6(a- f ). The results in panel (a) of this Figure depict the autocorrelation functions of H2 O+ and D2 O+ as a function of time for the initial vibrational-state ν2 = 0 of the neutral. Similarly, the results in panels (b) to ( f ) of Fig. 6 illustrate the calculated autocorrelation functions of H2 O+ and D2 O+ as a function of time for ν2 = 1 to 5. Note that the autocorrelation functions plotted as a function of time in Fig. 6 compare well with Fig. 3 of Ref. [28] for the vibrational quantum numbers 0 and 1. But the fall in the autocorrelation functions in Ref. [28] for vibrational quantum number 2 is quite large as compared to the quantum number 3, which could be due to the use of simple model potentials in their theoretical calculations. With the increase in the number of nodes in the initial vibrational state of the neutral, the overlap between the intial vibrational-state of the neutral and the time evolving wave packet on the cationic PES is expected to decrease with increase in the vibrational quantum number. With the accurate ab initio potentials employed in the present investigation, the autocorrelation functions plotted in Fig. 6 show the correct decay behavior as a function of the initial vibrationalstate of the neutral. This is then expected to influence also the HHG signal behavior as discussed below. As visible from Figs. 6(a-f), strong quasiperiodic recurrences occur with time. The results from panel (a) show two types of periodic oscillations. They correspond to the symmetric stretching (ν1 ) and bending (ν2 ) modes as seen in panel (a). The average peak spacings for the ν1 and ν2 modes in the 2 B1 state of the water cation shown in panel (a) are ∼ 10.54 fs and ∼ 21.36 fs. The corresponding results for D2 O+ in panel (a) are ∼ 10.58 fs and ∼ 21.39 fs. The quasiperiodic oscillations for the ν1 and ν2 modes for H2 O+ and D2 O+ for different initial vibrational-states as shown in panels (b) to (c) are collected in Table VI along with the available experimental values [20]. As shown, the calculated values for the ν1 and ν2 frequencies compare well with the existing data [34, 35]. Fig. 6 and Table VI also suggest that the calculated ν1 frequency for both H2 O+ and D2 O+ show only a marginal increase as a function of the initial vibrational-state of the neutral while those of ν2 reveal a substantial increase.
|C(t)|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
The Journal of Physical Chemistry
v = 0 (H2O )
4
+
v = 1 (H2O ) +
v = 2 (H2O )
0.4
+
v = 3 (H2O )
2
+
v = 4 (H2O ) +
v = 5 (H2O )
0.2
1
0
2
4 3 time/fs
5
6
0
0
2
4 6 time/fs
8
10
Figure 7. (a) Square of the absolute value of the autocorrelation functions for H2 O+ (D2 O+ ) for the various initial vibrational states. (b) As in panel (a) but for the ratio of the squared autocorrelation functions of D2 O+ and H2 O+ .
2.10 v=0 v=1 v=2 v=3 v=4 v=5
2.05
2.00
/a0
Page 7 of 10
1.95
1.90
1.85
1.80
0
5
15
10
20
25
time/fs
Figure 8. Expectation values of the included bond length of H2 O+ as a function of time for various initial vibrational-state of the neutral.
= 4 and ∼ 3.90 fs for ν2 = 5, as shown in Figs. 7(a-b). These results shown in Fig. 7 imply that the HHG signals increase as a function of the initial vibrational-state of the neutral, which is Panel (a) of Fig. 7 shows the HHG signals for water and in accordance with available theoretical results [28]. heavy water as a function of time for different vibrational states as indicated in the panel. The results for various ν2 values of To understand the increase in the HHG signals as a function of ν2 , Figs. 8 and 9 show the expectation values of the bond H2 O+ are plotted with different colours. The corresponding relength and bond angle for H2 O+ as a function of time for each sults for D2 O+ are shown in dashed (but not indicated in the initial vibrational-state of the neutral (corresponding results for key for brevity). Fig. 7(b) depicts the ratio of the HHG signals D2 O+ are not shown for brevity.) The expectation values of the of the two isotopomers for various vibrational states. As shown, bond length shown in Fig. 8 show quasiperiodic oscillations as the calculated results in both panels predict the HHG signals to a function of time for each initial vibrational-state of the neube enhanced for the D2 O isotopomer irrespective of vibrational tral. These results show a quasiperiod of ∼ 10.63 fs for ν2 = 0. state in agreement with our previous results [20, 32]. This is Similarly, the results in Fig. 9 show a quasiperiod of ∼ 24.20 fs due to the much slower nuclear motion in D2 O+ , an effect enfor ν2 = 0. The results for the oscillations depicted in Fig. 8 hanced with time up to ∼ 1.51 fs for v2 = 0, ∼ 1.68 fs for ν2 = ν 1, ∼ 1.78 fs for ν2 = 2, ∼ 2.01 fs for ν2 = ACS 3, ∼ 4.51 fs for and 9 for other vibrational states are collected in Table VII. It 2 Paragon Plus Environment C. HHG signals and sub-fs nuclear dynamics
8
The Journal of Physical Chemistry V.
SUMMARY AND CONCLUSIONS
2.50
We have calculated the photoelectron spectrum of water (heavy water) due to SF ionization to the 2 B1 electronic state of the cation for various initial vibrational states of the neutral.
v=0 v=1 v=2 v=3 v=4 v=5
2.25
0.8 v= 0 v= 1 v= 2 v= 3 v= 4 v= 5
ionization
energy/eV
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
1.0
The Journal of Physical Chemistry
10
0.6
1
H2O ( X A1 ) 2 1 v=0
0 60
0.4
90
H2O
D2O
ACS Paragon Plus Environment
120 150 bond angle/deg
+
0
1
2
+
3 time/fs
4
5
6
0.2
square of the autocorrelation function/unitless
20
