Theoretical study of molecular vibrations in electron momentum spectroscopy experiments on furan: An analytical versus a molecular dynamical approach Filippo Morini, Michael S. Deleuze, Noboru Watanabe, and Masahiko Takahashi Citation: The Journal of Chemical Physics 142, 094308 (2015); doi: 10.1063/1.4913642 View online: http://dx.doi.org/10.1063/1.4913642 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Vibrational effects on valence electron momentum distributions of CH2F2 J. Chem. Phys. 141, 244314 (2014); 10.1063/1.4904705 Binding sites and electronic states of group 3 metal-aniline complexes probed by high-resolution electron spectroscopy J. Chem. Phys. 138, 224304 (2013); 10.1063/1.4809742 Vibrational effects on valence electron momentum distributions of ethylene J. Chem. Phys. 137, 114301 (2012); 10.1063/1.4752653 Electron momentum spectroscopy study of Jahn–Teller effect in cyclopropane J. Chem. Phys. 130, 054302 (2009); 10.1063/1.3068619 An investigation of valence shell orbital momentum profiles of difluoromethane by binary (e,2e) spectroscopy J. Chem. Phys. 122, 054301 (2005); 10.1063/1.1839851

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THE JOURNAL OF CHEMICAL PHYSICS 142, 094308 (2015)

Theoretical study of molecular vibrations in electron momentum spectroscopy experiments on furan: An analytical versus a molecular dynamical approach Filippo Morini,1 Michael S. Deleuze,1,a) Noboru Watanabe,2 and Masahiko Takahashi2 1

Center of Molecular and Materials Modelling, Hasselt University, Agoralaan Gebouw D, B-3590 Diepenbeek, Belgium 2 Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan

(Received 13 January 2015; accepted 16 February 2015; published online 5 March 2015) The influence of thermally induced nuclear dynamics (molecular vibrations) in the initial electronic ground state on the valence orbital momentum profiles of furan has been theoretically investigated using two different approaches. The first of these approaches employs the principles of BornOppenheimer molecular dynamics, whereas the so-called harmonic analytical quantum mechanical approach resorts to an analytical decomposition of contributions arising from quantized harmonic vibrational eigenstates. In spite of their intrinsic differences, the two approaches enable consistent insights into the electron momentum distributions inferred from new measurements employing electron momentum spectroscopy and an electron impact energy of 1.2 keV. Both approaches point out in particular an appreciable influence of a few specific molecular vibrations of A1 symmetry on the 9a1 momentum profile, which can be unravelled from considerations on the symmetry characteristics of orbitals and their energy spacing. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4913642]

I. INTRODUCTION

About 40 years research with Electron Momentum Spectroscopy (EMS)1–6 has amply demonstrated the efficiency of this spectroscopy for experimentally investigating the electronic structure and the orbitals of atoms, molecules, and solids. By virtue of an angular analysis of binary (e,2e) electron impact ionization intensities at high enough kinetic energies, EMS enables indeed direct experimental reconstruction of electron momentum densities associated to specific ionization channels, i.e., of orbital momentum densities according to the one-electron picture of ionization. However, like with any ionization spectroscopy, EMS experiments are subject to many complications and their interpretation requires extensive theoretical work if it has to have any value at all. It is very generally considered that, for a non-coplanar symmetric kinematic setup and for electron impact energies approaching the high energy limit, the Born, electron binary encounter, weak coupling, and plane wave impulse approximations (PWIAs)1–6 can be invoked, so that the angular dependence of the measured differential cross sections can be unravelled through a mapping of all channels in the (e,2e) ionization spectrum with momentum-space electron distributions obtained as the squares of the Fourier transforms of Dyson orbitals.7–9 In the vast majority of cases (see, e.g., Refs. 10–12), Kohn-Sham orbitals represent excellent approximations to normalized Dyson orbitals and enable, therefore, quantitative insights into experimental (e,2e) momentum distributions in the high energy limit. a)Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0021-9606/2015/142(9)/094308/13/$30.00

The PWIA is most often regarded as one of the most critical and questionable assumptions in the theory underlying EMS. This approximation implies that the kinetic energies of the incoming and two outgoing electrons are so high that their interactions with the molecular target or residual ion can be neglected. Breakdowns of the latter approximation have been almost systematically invoked13–19 when facing strong discrepancies between theoretical and experimental momentum distributions, such as turn-ups of the recorded (e,2e) momentum densities at vanishing electron momenta. From EMS experiments upon atoms, such as Xe,20,21 Zn,22 and Cd,23 it is known that orbitals exhibiting a d-type topology (i.e., two perpendicular nodal planes) are most commonly subject to distorted wave effects—an observation that has been experimentally verified in many situations. Several models have been developed in order to account for distorted wave effects in EMS experiments upon atomic targets. Of relevance are the distorted-wave impulse approximation (DWIA),1,24,25 the distorted-wave Born approximation (DWBA),26,27 or the Brauner-Briggs-Klar theory (BBK),28–30 to name just a few. Due to the multi-centre nature of the scattered waves, the problem becomes particularly cumbersome when considering distorted wave effects in EMS experiments on molecular targets, and much effort has been made to account for these effects using (atomic-like) onecentered orbital depictions,31–38 or, very recently, the more sophisticated and more reliable multicenter distorted-wave (MCDW) approach.39 Accounting for distorted waves in theoretical studies of EMS experiments remains nevertheless computationally extremely expensive and still mainly intractable in the case of large molecular targets. On the experimental side, the validity of the PWIA approximation

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can be assessed by checking the dependence of the recorded momentum distributions upon the kinetic energy of the impinging electron.40–42 Further physical phenomena that can strongly influence the recorded momentum distributions pertain to nuclear dynamics. Following Hajgató et al.43 or Deleuze et al.,44,45 it is useful to distinguish nuclear dynamics in the initial and neutral electronic ground state, comprising thermally induced conformational rearrangements46–60 or pseudo-rotational motions,61,62 from nuclear dynamics in the final ionized state, more specifically, geometrical relaxation, vibronic coupling interactions, and ultra-fast bond cleavages43–45,63–67 induced by the ionization processes of interest. Following early studies on the H2 and H2O molecules,68–71 it has been generally considered that, under the conditions that prevail in EMS experiments, electron momentum profiles are not sensitive to molecular vibrations in the initial electronic ground state. Such views however need to be reconsidered. These motions may indeed explain numerous discrepancies between theory and experiment, precisely in cases where the PWIA was assumed to break down. For instance, two recent and pioneering studies on formamide and ethylene, by Miao et al.72 and by Watanabe et al.,73 respectively, have shown that specific vibrational motions can noticeably affect valence orbital momentum distributions, in the form of significant turn-ups of (e,2e) ionization intensities at low electron momenta. The latter study was based on a decomposition of contributions arising from each quantized vibrational eigenstates to the momentum distributions of interest within the frame of the (rigid-rotor-) harmonic oscillator approximation. This quantum-mechanical approach may fail if nonharmonic effects (anharmonicities in the vibrational potentials, couplings between vibrations and rotations, couplings between internal and external rotations due to Coriolis forces, pseudorotations, . . . ) become significant. In view of its advantages and limitations, this first methodology will be referred, from here and henceforth, to as the Harmonic Analytical Quantum Mechanical (HAQM) approach. An alternative computational approach employing the principles of Born-Oppenheimer Molecular Dynamics (BOMD)74–76 for unravelling EMS experiments has been considered recently for treating non-harmonic effects on pseudo-classical grounds and/or for studying the outcome of nuclear dynamics in the electronic ground state of structurally highly versatile compounds, such as group 6 metal hexacarbonyl compounds [W(CO)6, Cr(CO)6, Mo(CO)6],77 tetrahydrofuran,62 and 1-butene.78 The advantage of the latter approach is that it allows, in principle (depending on the duration of the MD run), a complete exploration of phase space within the frame of the Born-Oppenheimer approximation, which, by virtue of the ergodic principle, is equivalent to an ensemble average over all internal degrees of freedom of the system of interest, such as is computed in Monte Carlo simulations. The BOMD approach therefore accounts for the intrinsically chaotic nature of nuclear motions in large polyatomic molecules. The main drawback of this approach is a formally less rigorous (i.e., classical) description of molecular vibrations.

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Since neither the HAQM nor the BOMD approaches enable an exact and complete description of molecular vibrations in the electronic ground state, we wish to compare results obtained with both approaches for a molecular target that has led to noticeable and so far unexplained discrepancies between theoretical and experimental (e,2e) momentum profiles, namely, furan. The interested reader is referred to Refs. 79–81 for previous studies of the electronic structure of this compound by means of electron momentum spectroscopy and photoelectron spectroscopy, respectively. The first EMS measurements on furan79 were carried out by Takahashi et al. at a rather low electron impact energy (E0 = 0.8 keV); therefore, we also wish to present new measurements at a higher impact energy (E0 = 1.2 keV) to assess the likelihood of breakdowns of the plane wave impulse approximation. II. THE HAQM THEORY OF ELECTRON MOMENTUM SPECTROSCOPY: OUTLINE AND COMPUTATIONAL DETAILS

EMS is a high-energy binary (e,2e) electron-impact ionization experiment (M + e− → M+ + 2e−) that involves coincident detection of the two inelastically scattered and ejected electrons. The binding energy, Ebind, and the momentum of the target electron before ionization, p, can be determined with the aid of the energy and momentum conservation laws Ebind = E0 − E1 − E2,

(1)

p = p1 + p2 − p0.

(2)

Here, E j ’s and p j ’s ( j = 0, 1, 2) are the kinetic energies and momenta of the incident, inelastically scattered, and ejected electrons, respectively. In the following, we specifically discuss how vibrational motions of the target molecule affect the EMS cross section and then describe theoretical methods used to model these effects on quantum mechanical grounds. Within the PWIA, the EMS cross section is given by3 p1 p2  f ee M(p) , (3) σEMS = (2π)4 av p0 where f ee is the electron-electron collision factor and Σav represents a sum over final and an average over initial degeneracies. M(p) denotes the structure factor and Σav M(p) is the quantity that corresponds to electron momentum profile. Within the Born-Oppenheimer approximation, the structure factor for a transition from the v vibrational level of the initial neutral state i to the v’ vibrational level of the final ion state f can be written as Miv→ fv′ (p)  1 ⟨ χ ′ (Q)|  S (Q)ϕ (p; Q) | χ (Q)⟩ 2dΩ , = f f iv p fv 4π (4) where χiv(Q) and χfv′(Q) are the vibrational wave functions of the initial and final states, respectively, with Q being the nuclear coordinates. ϕf (p; Q) denotes the momentum space representation of the normalized Dyson orbital for the electronic ionization channel f ,

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  ϕf (p; Q) = (2π)−3/2 Sf (Q) dr eip·rϕf (r; Q)   = (2π)−3/2eip·rΨfN −1 (r2, . . . r N ; Q) ΨiN (r, r2, . . . r N ; Q)

and Sf (Q) the associated spectroscopic strength. Here, rj is the position of the j-th electron, and ΨiN and ΨfN −1 are the electronic wave functions of the initial neutral and final ion states, respectively. (4π)−1 dΩ p represents the spherical averaging which is required for taking into account the random orientation of molecular targets in the gas phase. Since the currently achieved energy resolution of EMS does not allow one to resolve vibrational structures in binding energy spectra of polyatomic molecules, what can be measured is the total intensity of i → f ionization band. Thus, the structure factor to be considered is a quantity that can be obtained by summing up all the contributions from each vibrational level,  Pv(T)Miv→ fv′ (p) , Mi→ f (p) = (6) v,v′

with Pv(T) being the Boltzmann’s statistical population of the initial vibrational level ν at temperature T. By using the closure relation of the vibrational eigenstates for the residual  ion, v′| χfv′(Q)⟩⟨ χfv′(Q)| = 1, Mi→ f, can be written as a sum of contributions for each quantized vibrational states associated to the electronic ground state of the neutral molecule     2  ρf (p, Q) dQ, (7) (T) (Q)| | Mi→ f (p) = P χ v iv    v which enables quantum-mechanical insights into the outcome of nuclear dynamics in the initial state. Here, ρf (p, Q) represents a spherically-averaged one-electron momentum density distribution at a given molecular geometry, Q, which reads  1 Sf (Q) |ϕf (p; Q)|2dΩp. (8) ρf (p, Q) = 4π One may notice that ΣvPv(T)| χiv(Q)|2 in Eq. (7) is equivalent to the probability of finding the molecule with a geometry Q, and its integration over the nuclear coordinates yields unity. Hence, assuming that the change in ρf (p, Q) with Q is negligible, Mi→ f can be simplified into     Mi→ f (p) ≈ ρf (p, Q0)  Pv (T) | χiv (Q)|2 dQ  v    2 1 = Sf (Q0) ϕf (p; Q0) dΩ p (9) 4π where Q0 denotes the equilibrium coordinates of the neutral molecule in its initial electronic ground state. This approximation has been widely used in EMS studies for unravelling the orbital momentum distributions of molecules in their equilibrium geometry. For investigating vibrational effects on electron momentum profiles, the dependence on Q of ρf (p, Q) has to be taken into account through the interplay of Eq. (7). In the present work, this is achieved within the frame of the harmonic approximation for quantized vibrational levels (the

(5)

so-called HAQM approximation). Details of this method for vibrational averaging have been described elsewhere,73 and thus only a brief outline is given here. In the HAQM approximation, the vibrational wave function of the initial state is described as a product of harmonic oscillator functions, ξvL(QL) with QL being the normal coordinate of the L-th mode. Furthermore, ρf (p, Q) in Eq. (7) is expanded in terms of the normal vibrational coordinates at around Q = Q0. The 4th, 6th, and higherorder contributions that come from the cross-terms are dropped from the expansion under the assumption that ρf (p, Q) changes slowly with Q in the neighbourhood of the equilibrium geometry (odd terms do not contribute to the expansion, because of the harmonic approximation). As a result, the following computationally tractable expression is obtained:   2 1 Sf (Q0) ϕf (p, Q0) dΩp Mi→ f (p) = 4π  ⟨ξvL| ρf (p, Q0 + QLqˆ L) + Pv (T) v

L

− ρf (p, Q0) |ξvL⟩ .

(10)

Here, qˆ L denotes a unit vector that points along the L-th normal coordinate. The first term is equivalent to the structure factor for the molecule at the equilibrium geometry, and the second term represents vibrational effects in the electronic neutral ground state. One of the advantages of this method is that vibrational effects on momentum profiles can be analysed in detail by dividing those into contributions from each normal vibrational mode. The HAQM calculations were performed as follows. First, vibrational normal coordinates and frequencies were calculated for the electronic ground state of furan by means of Density Functional Theory (DFT) along with the Becke-3parameters-Lee-Yang-Parr (B3LYP) functional82,83 and Dunning’s augmented correlation-consistent polarization valence basis set of triple-zeta quality, aug-cc-pVTZ.84,85 The normal coordinate analysis was carried out using the Gaussian03 program.86 The calculated vibrational frequencies are presented in Table I, showing good agreement with available experimental results.87 Subsequently, ρf (p, Q)’s were calculated at several nuclear geometries distorted from equilibrium along each normal coordinate. These calculations were conducted within the target Kohn-Sham approximation.88 Briefly, for each distorted structures, the Kohn-Sham orbitals of furan were calculated with the DFT method using the B3LYP functional and the augmented correlation-consistent polarized valence-double-zeta (aug-cc-pVDZ) basis,84,85 and they were used as approximations to normalized Dyson orbitals. These orbitals were subsequently converted to one-electron spherically averaged momentum densities (SAMDs) ρf (p, Q) (Eq. (8)) with the aid of the MOMAP program developed

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TABLE I. Vibrational normal modes of furan.

Mode ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν 10 ν 11 ν 12 ν 13 ν 14 ν 15 ν 16 ν 17 ν 18 ν 19 ν 20 ν 21

Symmetry

B3LYP/aug-cc-pVTZa (cm−1)

Experimentb (cm−1)

A1 A1 A1 A1 A1 A1 A1 A1 A2 A2 A2 B2 B2 B2 B2 B2 B2 B2 B1 B1 B1

3282.5 3252.7 1508.9 1413.1 1165.0 1083.3 1012.3 886.5 898.4 740.0 617.7 3276.0 3242.2 1590.6 1292.3 1196.2 1059.7 894.1 864.9 761.8 623.3

3169.4 3139.84 1490.55 1384.51 1140.2 1067.22 994.68 870.43 864 721.50 599.6 3160.75 3130.15 1557.50 1266.70 1180.97 1042.5 873 837.59 744.65 602.85

a This b See

study. Ref. 87 and references therein.

by Brion and his coworkers,70 assuming constant and equal spectroscopic strengths [Sf (Q) = 1] for all orbitals. We thereby discard the influence of vibrations on spectroscopic strengths and consider the one-electron picture of ionization to be (essentially) valid, which is most generally the case in the outer-valence region. To illustrate practical details of the calculations mentioned above, we show an example of the Q-dependence of spherically averaged momentum densities [ρf (p, Q)] in Fig. 1, which were obtained for the 9a1 orbital at p = 0.0, 0.4, and 0.8 a.u. as a function of the ν2 normal coordinate. Also depicted in the figure is the absolute square of the associated harmonic oscillator function of the ground vibrational level, |ξv2(QL)|2. The distorted geometries used in the calculation were chosen so that |ξv2(QL)|2/|ξv2(0)|2 = 3/4, 1/2, 1/4, 1/8, 1/16, and 1/32, respectively. It is evident from Fig. 1 that the results of the calculations are satisfactorily reproduced, over the QL range to be covered, by a least-squares curve-fitting procedure using a fourth-order polynomial function, ρf (p, Q0 + Q L qˆ L ) = ρf (p, Q0) +

4 

cL, s (p) Q sL .

(11)

s=1

Hence, we have used the corresponding least-squares fitted function to calculate Mi→ f. While paying an attention to properties of the harmonic oscillator that ⟨ξvL|QL2|ξvL⟩ = (1/2)(~/ωL)(2nvL + 1) and ⟨ξvL|QL4|ξvL⟩ = (3/4)(~/ωL)2 (2nvL2 + 2nvL + 1), substitution of Eq. (11) into Eq. (10) gives the following relation:

FIG. 1. Computed spherical averaged momentum density ρ f (p, Q0 + Q L qˆ L ) for the 9a1 orbital as a function of the ν 2 normal coordinate. The upper panel presents the results calculated at p = 0.0, 0.4, and 0.8 a.u. Broken lines are fourth-order polynomial functions fitted to the data. The lower panel shows the absolute square of the associated harmonic oscillator function of the ground vibrational level.

  2 1 Mi→ f (p) = Sf (Q0) ϕf (p, Q0) dΩp 4π    ~ (2nvL + 1) cL,2 (p) + Pv (T)   2ω L v  L  ( ~ )2 
 2 2nvL + + 2nvL + 1 cL,4 (p)  3 . 2ω L L  (12) Here, ωL and nvL denote the angular frequency and vibrational quantum number of the L-th normal coordinate, respectively, and ~ = h/(2π) with h being the Plank constant. It has been found that cLs(p)’s can practically be obtained with high accuracy by fitting a fourth-order polynomial function to ρf (p, Q)’s calculated at distorted geometries chosen so that |ξvL(QL)|2/|ξvL(0)|2 = 3/4, 1/2, 1/4, and 1/8, respectively, and thus computations were performed at such nuclear coordinates for all the normal modes. Finally, the momentum profile was obtained according to Eq. (12) with Pv(T) being the Boltzmann distribution at standard room temperature (298 K).

III. THE BOMD APPROACH OF ELECTRON MOMENTUM SPECTROSCOPY: COMPUTATIONAL DETAILS

BOMD74–76 is a method of choice for unravelling the effects of thermally induced nuclear motions on molecular properties, such as static dipole-dipole electric polarizabilities89,90 or electron momentum distributions,62,77,78 to cite just a few. Molecular properties are thermally averaged by

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classical propagation of the nuclear degrees of freedom based on quantum mechanically computed forces (ab initio BOMD91,92). BOMD is a highly powerful “brute force” approach of nuclear dynamics; the theoretical foundations of which are well-established and do not need to be reviewed in detail again. We simply wish to note that, among others, BOMD has recently enabled amazingly accurate first-principle calculations of electron impact mass spectra of a large variety of organic compounds, including pharmaceutically active molecules.93 The BOMD approach accounts for all basic physical effects induced by the temperature, except the quantum dynamics of the nuclei and non-Born–Oppenheimer (BO) effects, namely, the coupling of nuclear and electronic motions in, for instance, the vicinity of conical intersections. The BOMD calculations described in this work were performed using DFT in conjunction with the ωB97XD94 functional and the aug-cc-pVDZ84,85 basis set. The BulirschStoer method was used for the integration scheme,95,96 along with an integration step size of 0.2 fs, and using a fifthorder polynomial fit in the integration-correction scheme. The trajectory step size was set to 0.250 a.u. (amu1/2 bohr), and atomic coordinates were dumped at time intervals of approximately 1 fs. Thermalization of the BOMD trajectories to standard room temperature (298 K) was enforced by setting the initial rotational energy from a thermal distribution assuming a symmetric top. The BOMD simulations were performed for a microcanonical (NVE) ensemble, and the equilibration time was set to 0.1 ps. Thermalization was checked by monitoring the time-dependence of the kinetic energies and potential energies obtained at each point of the computed trajectories. The total runtime was 1.451 ps, resulting in the generation of 1500 thermally distorted structures. All molecular structures produced by the BOMD//ωB97XD/aug-cc-pVDZ simulations were subsequently used as input geometries in single point DFT calculations employing the B3LYP functional82,83 and Dunning’s aug-cc-pVTZ basis set.84,85 Homemade C-shell scripts have been used to automatically convert the molecular coordinates output of the BOMD//ωB97XD/aug-cc-pVDZ runs into input for further B3LYP/aug-cc-pVTZ calculations of the electronic structure, using the GAUSSIAN09 package of programs,97 and to ultimately combine results for thermally averaging (e,2e) electron momentum profiles obtained by means of the MOMAP program by Brion and coworkers.89 In order to enable physically meaningful comparisons with experiment, the theoretical spherically averaged momentum distributions have been convoluted with the experimental momentum resolution, using the approach by Migdall et al.98

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consists of an electron gun, an electrostatic lens system, a spherical analyzer, and a pair of position-sensitive detectors. Since a spherical analyzer maintains the azimuthal angles of the electrons, both the energies and angles can be determined from their arrival positions at the detectors. It is therefore possible to sample the (e,2e) cross sections over a wide range of electron binding energies and momenta in parallel. The experimental results were obtained by accumulating data at an ambient sample gas pressure of 3.0 × 10−4 Pa for two weeks runtime. During the measurements, the electron gun produced an electron beam of typically 20 µA in the interaction region. In order to achieve higher energy resolution, two outgoing electrons having energies in the order of 0.6 keV were decelerated to 2.7:1 using an electrostatic lens system prior to energy analysis. The resulting instrumental energyand momentum-resolution employed was 1.7 eV full width at half maximum (FWHM) and about 0.16 a.u. at p = 1 a.u., respectively. A single-point-normalization was performed for rescaling experimental (e,2e) ionization intensities onto theoretical ones. Namely, the experimental momentum profile for the second ionization band (B) at ∼13.3 eV due to the 9a1−1, 8a1−1, 6b2−1, 5b2−1, and 1b1−1 one-electron ionization lines was height-normalized to the associated HAQM calculation at p = 0.66 a.u. The correspondingly obtained scaling factor was subsequently applied to all other experimental momentum profiles. V. RESULTS AND DISCUSSION

The newly acquired momentum-space integrated (e,2e) ionization spectrum is reported in Fig. 2. This spectrum has been obtained by integrating contributions obtained at azimuthal angle differences ranging from −40◦ to +40◦. It is clear that the current resolution in energy (FWHM = 1.7 eV) does not allow us to disentangle the contributions of the 11 outermost valence molecular orbitals of furan, at electron binding energies ranging from ∼9 to ∼22 eV. As usual, the (e,2e) ionization spectrum has therefore been deconvolved

IV. EXPERIMENTAL SETUP AND DETAILS

The EMS experiment on furan was carried out at E0 = 1.2 keV in the symmetric noncoplanar geometry, with the two outgoing electrons having equal energies (E1 = E2) and making equal polar angles of 45◦ with respect to the incident electron beam axis. As is well-known,3 the magnitude of the target electron momentum can be determined from the out-of-plane azimuthal angle difference between the two outgoing electrons (∆φ = φ2 − φ1 − π). Details of the spectrometer used have been given elsewhere.99 Briefly, it

FIG. 2. Momentum-space integrated (e,2e) ionization spectrum of furan. The Gaussian curves are estimated from previous PES studies while the red curve represents their summation.

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(Fig. 2) by means of 11 Gaussian curves, the location of which has been determined based on previous estimations of orbital ionization energies from photoelectron studies.100,101 As indicated in Fig. 2, each dashed curve under the experimental fitting represents an orbital. At least four major structures (A-D) can be identified at ∼9.6 eV (A), ∼13.3 eV (B), ∼16.6 eV (C), and ∼23.6 eV (D), respectively. In line with benchmark calculations by Ehara et al.80 using the SAC-CI general R scheme102–105 and a polarized basis set of triple zeta quality, the A, B, and C spectral bands can be ascribed to the following sets of orbitals: {1a2, 2b1}, {9a1, 8a1, 6b2, 5b2, 1b1}, and {7a1, 6a1, 4b2}, respectively. We note that, according to the latter calculations, the one-electron picture of ionization is verified for most of the orbitals contributing to bands A and B, except for the 1b1 orbital, which appears merely in the SAC-CI spectrum in the form of three ionization lines at electron binding energies equal to 14.03, 15.84, and 17.97 eV, with spectroscopic strengths equal to 0.26, 0.45, and 0.10, respectively. In the SAC-CI spectrum, only a single oneelectron ionization line is essentially found for the 7a1 orbital at 17.77 eV (Sf = 0.75). In this spectrum, the 6a1 orbital merely appears in the form of two pairs of rather close-lying shakeup ionization lines with comparable intensities at 18.40 eV [Sf = 0.31] and 18.69 eV [Sf = 0.33], respectively—a splitting of the ionization intensity which nicely corroborates the larger width of the associated Gaussian curve in the deconvolved (e,2e spectrum). A very similar observation can be made for the 4b2 orbital, for which the SAC-CI calculations merely predict two very close-lying shake-up lines at 19.37 and 19.46 eV, with Sf = 0.24 and Sf = 0.34, respectively. All in all, the SAC-CI calculations by Ehara et al. therefore fully justify the proposed Gaussian partitioning of e,2e ionization intensities, as is displayed in Fig. 2 (see in particular Table I in Ref. 80). At electron binding energies above 20 eV (band D), these SAC-CI calculations indicate a complete breakdown of the orbital picture of ionization in the form of a dispersion of the ionization intensity over countless and totally undistinguishable sets of shake-up lines of low intensity (Sf < 0.15) arising from the 3b2, 5a1, and 4a1 orbitals. Any attempt to investigate this binding energy region in terms of individual orbital contributions is, most obviously therefore, condemned to failure. The new (e,2e) ionization spectrum is in line with what was already obtained by Takahashi et al.79 on the same molecule. However, two major factors have to be considered when the newly experimentally inferred momentum profiles are going to be analysed. The first is that, despite the energy resolution remains relatively low, the better statistical accuracy of the newly developed experimental setup allows us to perform a truly quantitative comparison of experimental and theoretical momentum profiles. The second factor is the substantial increase in the energy of the impinging electron. The previous experiment79 was performed at E0 = 800 eV, while the present one is conducted at E0 = 1200 eV. By evaluating the dependence of the recorded momentum profiles upon this parameter, we are in the position to evaluate the extent of distorted wave effects. The reader is referred to Fig. 3 for a comparison between the theoretical and experimental momentum profiles

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FIG. 3. Theoretical and experimental momentum profiles for the reported sets of orbitals. (a) 1a2, (b) 2b1, and (c) 9a1 + 8a1 + 6b2 + 5b2 + 1b1 (band B). The current experimental results at E 0 = 1200 eV are shown along with those previously taken at E 0 = 800 eV.79

associated to the 7 outermost valence orbitals of furan (bands A and B). Rather remarkably, this figure shows that the HAQM and BOMD approaches provide virtually identical results. The two Gaussian curves contributing to band A are sufficiently well-resolved for allowing us to examine separately the momentum distributions of the two outermost orbitals, 1a2 and 2b1, which are located at 8.83 eV [Γ = 0.92] and 10.10 eV [Γ = 0.87], respectively, in the SAC-CI spectrum by Ehara et al.80 In line with the symmetry characteristics of these orbitals, the corresponding momentum profiles (Figs. 3(a) and 3(b)) are of p-type character. Band B yields also a ptype momentum profile (Fig. 3(c)), reflecting essentially the dominance of contributions due to the 8a1, 6b2, 5b2, and 1b1 orbitals. A comparison between the former experimental results by Takahashi et al.79 at E0 = 0.8 keV and the present ones at E0 = 1.2 keV shows a dramatic improvement in the statistics

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FIG. 4. Contour plots of the seven outermost orbitals of furan: (a) 1a2, (b) 2b1, (c) 9a1, (d) 8a1, (e) 6b2, (f) 5b2, and (g) 1b1 (B3LYP/aug-cc-pVDZ results).

of the data which have been acquired for the 1a2 and 2b1 momentum profiles (Figs. 3(a) and 3(b)). Both measurements show significant differences in intensity at low momenta when compared with theoretical results. The momentum profiles that were correspondingly obtained with the HAQM and BOMD approaches suggest that nuclear dynamics in the initial state cannot provide a viable explanation to these turn-ups. Indeed, these profiles are almost equal to the profile obtained using the fixed equilibrium molecular geometry. Only a negligible increase of intensity is observed in the low momentum region when turning on thermally induced nuclear motions in the initial state. Note that distorted wave effects are likely to occur for the 1a2 and 2b1 orbitals (Fig. 4), both of which exhibit a d-type topology (two perpendicular nodal planes). Ultra-fast nuclear dynamics in the final ionized state and non adiabatic effects may also represent viable explanations for the experimental turn-ups of the 1a2 and 2b1 (e,2e) ionization intensities near p ∼ 0 a.u., compared with the HAQM and BOMD theoretical predictions. Indeed, the 2A2 and 2B1 states formed by ionization of the outermost π orbitals in furan are known to interact vibronically through totally antisymmetric b2 vibrational modes, and the adiabatic potential energy surface associated to these states exhibit a conical intersection slightly above the minimum of the upper (2B1) state.106 An ultra-fast symmetry breaking in the final ionized state is therefore likely to explain the rather large and significant deviation from zero intensity at p ∼ 0.0 a.u. of the experimental (e,2e) momentum profiles characterizing orbitals 1a2 and 2b1. Further simulations of these momentum profiles accounting for multi-state nuclear dynamics107 will be required for checking this latter hypothesis in details. For band B (orbitals 9a1, 8a1, 6b2, 5b2, and 1b1), a comparison between the former experimental results by Takahashi et al.79 at E0 = 0.8 keV and the present ones at E0 = 1.2 keV also shows (Fig. 3(c)) a clear improvement in the statistics of the newly acquired data. Comparing the HAQM and BOMD, theoretical data with estimates of the

associated momentum profile obtained assuming no departure from the equilibrium geometry of furan in its electronic neutral ground state shows that thermally induced nuclear dynamics has some minor influence on this profile at electron momenta lower than 0.8 a.u. (an influence which is more specifically ascribable to the 9a1 orbital, see further). Some discrepancies remain nevertheless in this momentum range with experiment, which may in this case most likely be the result of rather strong overlap effects with the neighbouring spectral bands (band C, in particular). The momentum profile which has been experimentally inferred for band C is compared in Fig. 5 with theoretical simulations. In these simulations, specific rescaling factors (0.91, 0.80, and 0.82) have been applied onto the contributions from the 7a1, 6a1, and 4b2 orbitals, in order to take into account the higher dispersion of ionization intensity into shake-up and correlation bands, based on the SAC-CI results for spectroscopic strengths by Ehara et al.80 Again, there is no influence of nuclear dynamics in the initial electronic neutral ground state, and the agreement between theory and experiment is undeniably excellent.

FIG. 5. Theoretical and experimental momentum profiles for the reported sets of orbitals (band C).

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FIG. 6. Individual theoretical momentum profiles for the 1a2, 2b1, 9a1, and 8a1 orbitals (experimental momentum resolution is not accounted for).

In Figs. 6 and 7, we compare theoretical momentum profiles obtained for individual orbitals by means of the HAQM and BOMD approaches. Thermally induced nuclear dynamics in the initial state has no influence at all on the theoretical momentum profiles associated to the 1a2 and 2b1 orbitals (Figs. 6(a) and 6(b)). Whereas, the BOMD approach indicates almost no influence of nuclear dynamics on the 8a1 momentum profile (Fig. 6(d)), the HAQM approach indicates that thermally induced molecular vibrations may have a minor influence on this profile, in the form of an increase of the intensity of the peak at p ∼ 0.45 a.u., relative to the intensity of the peak at p ∼ 1.2 a.u. All in all, the two approaches nevertheless provide qualitatively very similar results for this profile. Both the HAQM and BOMD approaches demonstrate that nuclear dynamics in the initial state has a significant influence on the spherically averaged momentum profile characterizing the 9a1 orbital (Fig. 6(c)). Whereas, the momentum profile obtained assuming that the molecule remains in its equilibrium geometry exhibits only one maximum at p ∼ 1.25 a.u., both approaches consistently point out to the emergence of a second peak of densities at p ∼ 0.45 a.u. Proceeding further (Fig. 7), the HAQM and BOMD approaches are found again to produce very consistent results. More specifically, inspection of Fig. 7 reveals an increase of the 6b2 and 5b2 momentum densities (Figs. 7(a) and 7(b)) versus a decrease of the 1b1 and 6a1 momentum densities (Figs. 7(c) and 7(e)) in the intermediate momentum region (i.e., around p ∼ 0.5 a.u.) region, compared with the results obtained for equilibrium geometry, whereas the 7a1 momentum density is found to slightly increase at vanishing electron momenta (Fig. 7(d)). Nuclear dynamics in the initial electronic ground-state also results in a rather significant

increase of the 4b2 momentum density at p ∼ 0.0 a.u. (Fig. 7(f)). The influence of nuclear dynamics in the initial electronic ground state on electron momentum profiles can be analyzed in detail within the frame of the HAQM approach, by computing the contributions from each vibrational normal mode ν to the expansion of Mi→ f(p) given in Eq. (10). To elucidate the origin of the significant influence of molecular vibrations on the 9a1 momentum profile, we have performed such an analysis. Figure 8 shows contributions from each normal mode to the correspondingly obtained 9a1 momentum profile. The corresponding B3LYP/aug-cc-PVTZ vibrational frequencies were supplied in Table I. It is clear from Fig. 8(a) that many of the A1 modes have a strong influence on this profile. In particular, taking into account the v3 vibration, related to C-H bending (Fig. 9) and whose frequency is 1509 cm−1, leads to a significant enhancement in the intensity at p ∼ 0.45 a.u. Besides, contributions from the v2 and v4 C-H stretching and bending modes, also of A1 symmetry, are also appreciable in this momentum region. Figure 8 further indicates that contributions from the B2 modes are relatively small, though non-negligible, compared to those from the A1 modes, and that the A2 and B2 modes play only minor, almost negligible roles. Significant contributions to the structure factor of Eq. (9) from the high frequency C-H stretching and bending v2, v3, and v4 vibrational normal modes may seem surprising at first glance. Indeed, according to the computed populations at standard room temperature (298 K), furan essentially resides in the ground vibrational state (nL = 0) of these modes. The results of the above analysis thus suggest that even the zero point vibrations of these vibrational modes can

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FIG. 7. Theoretical momentum profiles for the 6b2, 5b2, 1b1, 7a1, 6a1, and 4b2 orbitals (experimental momentum resolution is not accounted for).

considerably affect the 9a1 momentum profile. To confirm this suggestion, theoretical momentum profiles of the 9a1 orbital have been calculated, as shown in Fig. 9 at geometries distorted along the ν3 normal coordinates (solid lines). The distorted geometries were chosen so that |ξvL(QL)|2 = |ξvL(0)|2/4 with ξvL(QL) being the associated harmonic oscillator function of the ground vibrational level. For a comparison, results for the equilibrium geometry calculation are also depicted in this figure (dashed line). Also presented are theoretical electron density distributions calculated at the distorted molecular geometries. It is evident from Fig. 9 that deformation of the molecular structure, even within the extent of the zero-point vibrational wave function, may lead to significant change of the 9a1 momentum profile. The same analysis has been performed for the v2 and v4 modes (See Figs. 10 and 11). It can be seen from Figs. 9-11 that deformations of the molecular geometry along the associated normal coordinates result in considerable variations of the electron density around the two CH-bonds adjacent to the oxygen atom and around the opposite CC-bond. Structural

distortions associated to the ν4 C-H bending mode (Fig. 11) also strongly influences the 9a1 orbital density in the oxygen lone pair region, as a result of anomeric interactions between the O(2pz) orbital and the neighbouring σ*(C-H) orbitals. In addition, C-C stretching along the ν4 mode results also into significant variation of the 9a1 orbital density in the σ(C−C) bond region lying opposite to the oxygen atom. All in all, such changes in the molecular orbital pattern lead to a substantial enhancement in the intensity of the momentum profile at p ∼ 0.45 a.u. A similar observation can be made for the v6 and v8 vibrations, both of which are A1 modes. Interestingly, in spite of differences in their character, all these A1 vibrations appear to affect the 9a1 orbital in similar ways. The perturbative approach proposed by Herzberg and Teller108 provides a rational explanation to these findings. According to this famous approach, change in the Coulomb potential (V ) due to a molecular distortion is treated perturbatively, and the electronic wave function of, for instance, a given ion state at a distorted geometry Q0 + Q′ can, to first-order, be expressed as follows:

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ΨfN−1 (r1, . . . r N −1; Q0 + Q′) = ΨfN−1 (r1, . . . r N −1; Q0)   N−1 (∂V/∂Q )  ΨN−1 L Q→ Q0 Ψf j (r1, . . . r N −1; Q0) , + Q L ΨN−1 j E − E f j L, j, f

(13)

where ΨN−1 is the wave function of the j-th ion state and QL denotes the displacement from Q0 along normal coordinate of the j L-th vibrational normal mode. ∂V/∂QL describes the change in the Coulomb potential due to the molecular distortion along the L-th normal coordinate. We remind that Q0 represents the equilibrium geometry of the initial electronic neutral ground state. Similarly, application of standard orbital perturbation theory to Kohn-Sham orbitals yields straightforwardly ϕ Kf S (r; Q0 + Q′) = ϕ Kf S (r; Q0)    ϕ Kj S |(∂[vext + vCoul + vXC]/∂Q L )Q→ Q0|ϕ Kf S + Q L ϕ Kj S (r; Q0) , ε − ε f j L, j, f

(14)

with vXC the relevant state-independent non-local exchange-correlation functional. Equations (13) and (14) demonstrate that molecular distortions lead to mixings of various ion-state electronic wave functions or the Kohn-Sham orbitals with the zeroth order terms, ΨfN−1(r1, . . . , rN−1; Q0) or ϕ Kf S (r; Q0). It should be noted that such couplings will effectively take place between states or orbitals with the relevant symmetries and with small energy separations because of the energy denominators that are involved in the first-order perturbation corrections. From considerations based on symmetry point group theory, it is clear that only a1 orbitals can couple to the 9a1 orbital through A1 vibrations. Hence, deformations of the molecular structure along any A1 normal coordinates should result essentially in admixture into the 9a1 orbital of the adjacent 8a1 orbital (note indeed that the energy separations of other a1 orbitals from the 9a1 orbital are considerable). As a consequence, the intensity of the 9a1 momentum profile increases at p ∼ 0.45 a.u., where the 8a1 momentum profile has its highest intensity (Fig. 6(d)).

FIG. 8. Contributions from each vibrational normal mode to the 9a1 momentum profile.

FIG. 9. Theoretical momentum profiles of the 9a1 orbital calculated at the equilibrium geometry (dashed line) as well as for geometries distorted along the ν3 normal coordinates (solid lines). The distorted geometries were chosen so that |ξ vL(Q L)|2 = |ξ vL(0)|2/4 with ξ vL(Q L) being the associated harmonic oscillator function of the ground vibrational level.

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FIG. 10. Theoretical momentum profiles of the 9a1 orbital calculated at the equilibrium geometry (dashed line) as well as for geometries distorted along the ν2 normal coordinates (solid lines). The distorted geometries were chosen so that |ξ vL(Q L)|2 = |ξ vL(0)|2/4 with ξ vL(Q L) being the associated harmonic oscillator function of the ground vibrational level.

FIG. 11. Theoretical momentum profiles of the 9a1 orbital calculated at the equilibrium geometry (dashed line) as well as for geometries distorted along the ν4 normal coordinates (solid lines). The distorted geometries were chosen so that |ξ vL(Q L)|2 = |ξ vL(0)|2/4 with ξ vL(Q L) being the associated harmonic oscillator function of the ground vibrational level.

Similarly, the B2 vibrations induce mixing of the 6b2 and 9a1 orbitals. However, the relatively large energy difference between these two orbitals leads to smaller contributions of the B2 modes (Fig. 8(b)) compared to those of the A1 modes (Fig. 8(a)). The large energy separations of nb1 and na2 orbitals from the 9a1 orbital explain the very small contributions of the B1 and A2 modes to its momentum profile (Fig. 8(c)). Further analysis has been performed to get specific experimental information on the momentum profile related to the one-electron 9a1−1 ionization state, upon focusing specifically on the overlap region between bands A and B (Fig. 2), at electron binding energies ranging from 11.0 to 12.1 eV, where the contributions from this state dominate. In the analysis, the Gaussian bands accounting for the 1a2−1 and 2b1−1 contributions have been subtracted from the measured (e,2e) ionization spectra. The experimental momentum profile for the 9a1 orbital has been produced by plotting as a function of the electron momentum p the total intensities recorded in the subtracted spectra within the selected binding energy range. The correspondingly obtained experimental momentum profile is shown in Fig. 12 and compared with various theoretical simulations. In these simulations, one accounts for the fact that according to the Gaussian deconvolution about 19% of the 9a1−1 and 3.7% of the 8a1−1 band intensities are

estimated to be involved in the [11.0–12.2 eV] binding energy range. Despite larger experimental intensities due to band overlap effects, it is evident from this figure that while the equilibrium geometry calculation substantially underestimates the experimental momentum profile at small electron momenta, taking into account vibrational effects significantly reduce the

FIG. 12. Theoretical momentum distribution for the 9a1 orbital, compared with a momentum profile inferred from a specific angular analysis of (e,2e) ionization intensities in the [11.0–12.2 eV] binding energy range (see text for details).

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extent of deviations from the experiment, an observation which validates both the HAQM and BOMD predictions for the 9a1 orbital.

VI. CONCLUSIONS

In this study, we have theoretically investigated the influence of thermally induced nuclear dynamics (molecular vibrations) in the initial electronic ground state on the electron momentum profiles associated to the valence orbitals of furan, using two different approaches. The first of these approaches, referred to as the HAQM approach, is based on an analytical decomposition of contributions arising from each quantized vibrational eigenstates within the frame of the harmonic oscillator approximation, whereas the second one employs the principles of BOMD for averaging out the effect of temperature on momentum profiles. The BOMD approach allows us to cope with non-harmonic effects and chaos, whereas the HAQM approach accounts for the quantum nature of nuclei, within the frame of the (Born-Oppenheimer) adiabatic approximation. In spite of their differences, the two approaches enable consistent insights into the electron momentum distributions inferred from new measurements employing electron momentum spectroscopy. Both approaches point out in particular an appreciable influence of nuclear dynamics in the initial ground state, on the 9a1 momentum profile. The prominent role of a few CH-stretching and bending vibrations of A1 symmetry has been enlightened and unravelled from considerations on the symmetry characteristics of orbitals and their energy spacing, along the lines of the Herberg-Teller principle. Such considerations offer a concrete example to guide our reasoning about how molecular vibrations and further nuclear motions in the initial electronic ground state may affect electron momentum profiles and to discriminate these effects from further physical phenomena such as distorted wave effects or nuclear dynamics in the final state (comprising non adiabatic effects).

ACKNOWLEDGMENTS

This work has been supported by the FWO_Vlaanderen, the Flemish branch of the Belgian National Science Foundation, the “Bijzonder Onderzoeks Fonds” of Hasselt University, and the Japanese Ministry of Education, Culture, Sports, Science, and Technology, Grant-in-Aid for Challenging Exploratory Research (No. 25620006). F.M. is post-doctoral fellow from the FWO at Hasselt University (Grant No. 1202413N). F.M. and M.S.D. acknowledge access to the Flemish Supercomputer, which is funded by the Hercules foundation. 1I.

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