Vibrational Effects on Electron Momentum Distributions of Outer

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Vibrational Effects on Electron Momentum Distributions of Outer-Valence Orbitals of Oxetane YaGuo Tang, Xu Shan, Jing Yang, Shanshan Niu, Zhe Zhang, Noboru Watanabe, Masakazu Yamazaki, Masahiko Takahashi, and Xiangjun Chen J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b06706 • Publication Date (Web): 05 Aug 2016 Downloaded from http://pubs.acs.org on August 12, 2016

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The Journal of Physical Chemistry

Vibrational Effects on Electron Momentum Distributions of Outer-Valence Orbitals of Oxetane Yaguo Tang,1 Xu Shan,1 Jing Yang,1 Shanshan Niu,1 Zhe Zhang,1 Noboru Watanabe,2 Masakazu Yamazaki,2 Masahiko Takahashi,2 and Xiangjun Chen1*

1

Hefei National Laboratory for Physical Sciences at the Microscale and Department

of Modern Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China 2

Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan

ABSTRACT Vibrational effects on electron momentum distributions (EMDs) of outer-valence orbitals of oxetane are computed with a comprehensive consideration of all vibrational modes. It is found that vibrational motions influence EMDs of all outer-valence

orbitals

noticeably.

The

agreement

between

theoretical and

experimental momentum profiles of the first five orbitals is greatly improved when including molecular vibrations in the calculation. In particular, the large turn-up at low momentum in the experimental momentum profile of 3b1 orbital is well interpreted by vibrational effects, indicating that, besides the low-frequency ring-puckering mode, C-H stretching motion also plays a significant role in affecting 1 ACS Paragon Plus Environment


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EMDs of outer-valence orbitals of oxetane. The case of oxetane exhibits the significance of checking vibrational effects when performing electron momentum spectroscopy measurements.

I. INTRODUCTION Electron momentum spectroscopy (EMS)1-5 has been demonstrated to be a robust technique for exploring the electronic structure of atoms and molecules in the last several decades due to its unique ability of imaging electron momentum distributions (EMDs) for individual orbitals. The technique is based on the high energy electron impact (e, 2e) reaction near the Bethe ridge condition. Within a series of approximations including Born-Oppenheimer, binary encounter, weak-coupling and plane wave impulse approximations, the triple differential cross section (TDCS) of the (e, 2e) reaction is proportional to the modulus square of the wavefunction of the ionized orbital in momentum space, i.e., electron momentum profile. However, careful treatments should be carried out when interpreting EMS experimental results due to the possible invalidity of these approximations and the complexity of polyatomic molecules, resulting in distorted wave effects,6-7 conformational effects,8-25 relativistic effects,26-27 Jahn-Teller effects,28-29 vibrational effects,30-40 and so on. Among those effects, vibrational effects or nuclear dynamics in the electronic ground state did not attracted researchers’ attention until recently. EMS studies on conformational isomers8-25 have already shown that EMDs are remarkably affected by 2 ACS Paragon Plus Environment

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the change of molecular geometry, suggesting that vibrational motions in molecules may also influence EMDs to some extent. Early studies on H2,41-42 H2O,43-44 and 1b3u orbital of C2H445 displayed that EMS results seem to be not sensitive to molecular vibrations. However, in recent years, several investigations30-39 proved that the vibrational motion does make a noticeable difference and plays an important role in explaining the substantial unexpected turn-up in the low momentum region of experimental momentum profiles for specific orbitals of polyatomic molecules. The attention was usually paid to the vibrational modes with very low frequencies.30-32, 35 To fully estimate vibrational effects on EMDs theoretically, Watanabe et al. proposed a harmonic analytical quantum mechanical (HAQM) approach36-37 by introducing the vibration harmonic approximation, in which all vibrational modes were taken into account and the contribution of each mode can be estimated individually. It was demonstrated that vibrational modes with extremely high frequency can also influence EMDs greatly, e.g. the effects of the v5 mode (~3000 cm-1) on 1b3g orbital for ethylene.36 Oxetane or trimethylene oxide, a four-membered ring molecule of flexible structure, is one of the simplest molecules having the ring-puckering motion.46-50 The unusual one-dimensional ring-puckering potential curve is proved to be a double-minimum function with an extremely low barrier of 15.52 ± 0.05 cm-1 and the first vibrational level is only 11.86 ± 0.05 cm-1 above the barrier in the electronic ground state.50 Though oscillating between two equivalent bent conformers, oxetane can essentially be treated as a planar ring molecule (C2v).51-54 On the other hand, many 3 ACS Paragon Plus Environment


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excited vibrational levels in this mode are appreciably populated at room temperature since the frequency of the ring-puckering vibration is sufficiently low.55 Recently, Yang et al.35 reported their EMS study on five outer-valence molecular orbitals (MOs) of oxetane. A substantially high intensity in the low momentum region of the momentum profile of the highest occupied molecular orbital (HOMO) 3b1 was observed. The calculation at planar geometry (C2v) completely failed to interpret it, while the calculation considering the thermal abundances of planar (C2v) and bent (Cs) conformers or the thermally populated vibrational states of the ring-puckering motion only partially reproduce the experiment. On the other hand, for other outer-valence MOs, the ring-puckering motion exhibits little effect on their momentum profiles, although there were observed discrepancies between experiment and theory. As mentioned above, the vibrational modes with higher frequency may also affect EMDs noticeably. It should be noted that Yang et al.’s model with the geometry optimized potential curve actually involved the influence of other vibrational modes to some extent, which achieved better agreement with experiment than that with the pure ring-puckering potential curve. This inspires the present work to conduct a comprehensive investigation of vibrational effects on EMDs of oxetane involving all vibrational degrees of freedom to fully check the effects of other vibrational modes and the possible reason for the disagreement between experiment and theory.

II. THEORETICAL METHODS EMS is a coincidence experiment based on the kinematically complete (e, 2e) 4 ACS Paragon Plus Environment

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reaction, in which a fast incident electron is scattered by a target atom or molecule and an electron is knocked out from the target. The conservation of energy and momentum gives the binding energy εf and the momentum p of the target orbital electron as1-4

ε f = E0 − E1 − E2 , p = p0 − p1 − p2 ,

(1) (2)

where E0, E1, and E2 are energies, and p0, p1, and p2 are momenta of the incident and two outgoing electrons, respectively. Within the binary encounter, plane wave impulse and the weak-coupling approximations, the TDCS for EMS can be expressed as4

σ EMS =

d 3σ pp pp = (2π )4 1 2 f ee Σ av pf i = (2π ) 4 1 2 f ee M ( p) , (3) d Ω1d Ω 2 dE1 p0 p0

where fee is the e-e collision factor and M(p) is the relevant structure factor corresponding to the momentum profile. Σav represents an average over initial i and a sum over finial f

degenerated states. In terms of the Born-Oppenheimer

approximation, the total wavefunction of a molecular target can be described as a product of separate electronic, vibrational, and rotational wavefunctions. Ignoring rotational motions and employing the closure relation of the final vibrational eigenstates, the relevant structure factor can be simplified as35-36 M ( p ) = ∑ v pv (T ) ∫ χ iv (Q ) ρ f ( p; Q ) dQ , (4) 2

where χ iv (Q ) is the vibrational wavefunction of the eigenstate v in the ground state and pv (T ) is the related population at temperature T. ρ f ( p; Q ) is the spherically averaged EMD at a given geometry with a coordinate Q, which reads 5 ACS Paragon Plus Environment


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ρ f ( p; Q )=

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1 2 Sf (Q ) ∫ ϕf ( p; Q ) d Ω p , (5) 4π

where Sf (Q) is the pole strength or spectroscopic factor and ϕf ( p; Q ) is the normalized Dyson orbital. According to Watanabe et al.’s deduction,36 vibrational effects on momentum profiles can be written as a sum of the contribution of every vibrational mode if the rigid-rotator harmonic oscillator (RRHO) approximation is adopted and the cross-terms between vibrational modes are neglected:

χ iv (Q ) = ∏ L ξ v (QL ) , (6) L

M ( p) = ρ f ( p; Q0 ) + ∑ L ∆M L ( p) = ρ f ( p; Q0 ) + ∑ L

(∑

2

vL

pvL (T ) ∫ ξ vL ( QL ) ρf ( p; Q0 + QL qˆL )dQL − ρf ( p; Q0 )

) , (7)

where ξ vL (QL ) is the harmonic function describing molecular vibrations and ∆M L ( p ) is the contribution to vibrational effects from the L-th normal mode. QL is the displacement from the equilibrium geometry Q0, and qˆL is the unit vector along the normal coordinate. Equ. (7) is easily understood since it can be regarded as a first-order approximation of equ. (4) when the normal coordinate is introduced. Based on this, it is available to individually compute the effects of each vibrational mode on momentum profiles and to understand which kind of and how vibrational motions influence EMDs. In the case of oxetane, vibrational motions can be handled as harmonic oscillations appropriately except for the ring-puckering mode due to its double minimal shape of the potential curve when estimating vibrational effects. However, it should be noted that during the deduction of Watanabe et al. only the odd-even 6 ACS Paragon Plus Environment

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property of the harmonic functions is used.36 Due to the same odd-even property of the ring-puckering eigenfunctions resulted from the symmetry of the potential curve, equ. (7) is still valid for oxetane even if the non-harmonic ring-puckering wavefuction is included. As a result, vibrational effects can be computed according to equ. (7) where the ring-puckering mode is described by the wavefunction through solving the Schrödinger equation with the double minimal potential function and other vibrational modes described by harmonic functions from high-accuracy ab initio calculations. In this work, the effect of the ring-puckering mode on the EMDs of oxetane is computed similar to Yang et al’s work35 which is based on Chan’s model,46-47 while the contributions from other vibrational degrees of freedom are calculated employing harmonic approximation. Technically, the vibrational normal coordinates and frequencies are calculated by density functional theory (DFT) along with the Becke-3-parameters-Lee-Yang-Parr

(B3LYP)56-57

functional

and

Dunning’s

augmented correlation-consistent polarization valence basis set of triple-zeta quality (aug-cc-pVTZ)58-59 using Gaussian 03 program.60 Then

ρ f ( p; Q0 + QL qˆL ) is

calculated

target

at

B3LYP/aug-cc-pVTZ

level

where

the

Kohn-Sham

approximation61 is adopted and the pole strengths, which are generally very much the same, are assumed to be constant and identical. In fact, a symmetry adapted cluster / configuration interaction (SAC-CI)62-64 calculation performed in the Supercomputing Center of University of Science and Technology of China using Gaussian 09 program65 exhibits that one particle ionization picture is valid for outer-valence orbitals of oxetane and the pole strengths are nearly the same (0.82~0.84) for different 7 ACS Paragon Plus Environment


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orbitals

and 2

geometries.

The

displacement

QL

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is

chosen

so

that

2

ξv (QL ) / ξv (0) = 3 / 4, 1/ 2, 1/ 4, 1/ 8, and 1/16 , respectively. A fourth-order L

L

polynomial function is used to fit ρ f ( p; Q0 + QL qˆL ) as a function of QL. Finally the momentum profiles are obtained according to equ. (7) considering thermally induced nuclear motions using the Boltzmann distribution at room temperature (298 K).

III. RESULTS AND DISCUSSION There are 24 vibrational modes (8A1+4A2+6B1+6B2) for oxetane molecule, as illustrated in Figure 1. The vibrational mode characters and the corresponding vibrational frequencies calculated by B3LYP/aug-cc-pVTZ are shown in Table 1. The Herzberg’s notation66 is adopted here. It should be noted that the normal coordinate predicted by B3LYP/aug-cc-pVTZ for the ring-puckering mode is basically consistent with that of Chan’s model,47 in which C-C and C-O bond lengths and methylene angles are fixed. The calculated frequencies are compared with experimental ones,67-68 and a fairly good agreement (within 35 cm-1) will be achieved if a scaling factor 0.968 from Ref. 69 is employed. Vibrational effects on the EMDs of all eight outer valence orbitals of oxetane are computed with a comprehensive consideration of all vibrational modes. The results are shown in Figure 2-5, respectively. As a comparison, the EMS experimental results on the first five MOs of Yang et al.35 are also plotted in the figures. For HOMO (3b1) of oxetane as shown in Figure 2, the experimental momentum profile (solid circles) exhibits a huge intensity in the low momentum region which is

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expected to show a p-type character. The calculation (dashed curve) at planar geometry (C2v) completely fails to describe it, while the calculation (dotted curve) considering the thermally populated vibrational states of the ring-puckering motion improves obviously. However the large discrepancy between experiment and theory still remained.35 As depicted in Figure 2(a), the present calculation with a comprehensive consideration of all vibrational modes (solid curve) reproduces the experiment quite well, which indicates that other vibrational motions (dash-dotted curve) also play significant roles in accounting for the high intensity at the origin of momentum. In order to have insight into the influence from which kind of vibrational motions and how it works, the contribution of each vibrational mode are plotted in Figure 2(b). It can be seen that the high intensity of HOMO momentum profile at small momentum is clearly originated from the ring-puckering and α-CH2 stretching motion of B1 symmetry. From the orbital map in Figure 2(c), one can see that 3b1 orbital of C2v geometry is constituted by oxygen 2p lone pair (no) and α-CH bonds and completely anti-symmetric about the ring plane. Keep in mind that the momentum-space wavefunction ϕ f ( p; Q ) is the Dirac-Fourier transform of the corresponding position space wavefunction ϕ f (r ; Q ) and its value at p = 0 reads

ϕ f ( p; Q ) p =0 = ( 2π )

−3/2

∫ϕ

f

(r ; Q )dr . Therefore, it is reasonable to have zero intensity

at zero momentum for 3b1 orbital with anti-symmetry character about the ring plane. Yang et al.’s natural bond orbital analysis35 indicated that the different charge transfer from no to the nearby C-H bonds due to hyperconjugative interactions breaks the symmetry (Figure 2(d)) when the molecule is puckered, and consequently contributes 9 ACS Paragon Plus Environment


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dominantly to the high intensity in the low momentum region. On the other hand, the α-CH2 stretching motion of B1 symmetry shortens or elongates the C-H bonds above and below the ring plane oppositely (Figure 2(e)), which also understandably results in a significant intensity at zero momentum. For the next two orbitals, 8a1 and 4b2, the effects of vibrational motions are complicated (Figure 3). There is no distinct influence for the ring-puckering mode, while other vibrational modes have considerable effects on momentum profiles. Calculations considering vibrational motions predict slightly higher intensities at zero momentum for 8a1 and 4b2 orbitals. For 8a1 orbital, vibrational motions increase the intensity at p = 0.5~1.3 a.u., improving the agreement between theoretical and experimental momentum profiles, while for 4b2 orbital the intensity in the region of p < 1.0 a.u. is increased to some extent. Obvious discrepancies between theory and experiment still exist in Figure 3 (a1) and (b1). To explain this, a comparison between the experimental and calculated momentum profiles for the sum of 8a1 and 4b2 is made, as shown in Figure 3 (c). As we can see, the sum of the computed momentum profiles of 8a1 and 4b2 orbitals agrees the experimental result well except in the low momentum region. It suggests that the uncertainty from the deconvolution process should be responsible for the remained discrepancy in the momentum region of p > 0.25 a.u. in Figure 3 (a1) and (b1) noting that the ionization bands of 8a1 and 4b2 orbitals are embedded in one envelope in the binding energy spectrum.35 As for 2b1 and 1a2 orbitals, vibrational effects from other vibrational modes other than ring puckering contribute to a visible turn-up in momentum profiles, which are in 10 ACS Paragon Plus Environment

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line with the experimental results (Figure 4). It is easily understood with the help of orbital maps in Figure 4 and vibrational displacement vectors in Figure 1. The orbital map in position space in Figure 4 shows that 2b1 orbital exhibits anti-symmetry about the ring plane. The v13 mode of β-CH2 stretch and v14 mode of α-CH2 stretch of B1 symmetry are mainly responsible for the high intensity at zero momentum for 2b1 orbital. The 1a2 orbital consisted of α-CH bonds displays a d-orbital character. The v9 mode of α-CH2 stretch of A2 symmetry increases the intensity of the 1a2 momentum profile at p ~ 0, as the C-H bonds oscillate in a way that breaks the symmetry about both the ring and the β-CH2-O planes. Though an improved agreement is achieved with experiment, a distinct underestimation of the turn-up still exists. Similar to the case of 1b3g orbital of ethylene6 and the π* orbital of oxygen,7 the d-type 1a2 orbital of oxetane also likely suffers distorted wave effect, resulting in a high turn-up and accounting for the remained discrepancy between the experiment and the theoretical calculation considering vibrational effects. Since the ionization bands of 2b1 and 1a2 orbitals are also embedded in one ionization envelope,35 deconvolution processes may also lead to large uncertainty. The sum of the measured 2b1 and 1a2 momentum profiles displays a much smoother and reliable momentum distribution. The underestimate of the intensity at low momentum then certainly be attributed to the distorted wave effect in ionization from 1a2 orbital. The distorted wave effect may also be the reason for the underestimation of the turn-up in the summed momentum profile of 8a1 and 4b2 orbitals. 11 ACS Paragon Plus Environment


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As for the final three outer-valence orbitals, 3b2, 7a1, and 1b1, where no experimental results are available, theoretical calculations (Figure 5) also shows that the vibrational motions significantly affect momentum profiles. 3b2 orbital of C2v geometry is anti-symmetric about the β-CH2-O plane, thus yielding zero-intensity at p ~ 0. The vibrational modes like v21, v23 will break this symmetry and make a positive contribution to the momentum intensity at p ~ 0. Theoretical calculation considering vibrational effects predicts an evident higher intensity in the low momentum region. It is interesting that a vibronic coupling occurs for 7a1 and 1b1 orbitals. According to Morini et al.,38-39 the strong vibronic couplings can happen between orbitals with relevant symmetries and small energy separation. The SAC-CI calculation with the C2v geometry gives the binding energies of the two orbitals at 16.5 eV and 17.1 eV, that is to say, only a small energy difference of 0.6 eV exists. The direct products of the form A1 (7a1)⊗B1 (v18, v13) and B1 (1b1)⊗ B1 (v18, v13) create B1 and A1 symmetries, respectively. In a consequence, a coupling between 7a1 and 1b1 orbitals can easily occur through vibrational modes of B1 symmetry. As Figure 5 (b) and (c) show, the related vibronic coupling lowers the intensity of the momentum profile of 7a1 orbital while notably increases that of 1b1 orbital in the low momentum region.

IV. CONCLUSION Vibrational effects on EMDs of the outer-valence orbitals of oxetane involving all vibrational modes are investigated with a combination of Chan’s ring-puckering model35, 47 and Watanabe et al.’s HAQM approach.36-39 Turn-up phenomena in the 12 ACS Paragon Plus Environment

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low momentum region of momentum profiles caused by vibrational motions appear in the cases of 3b1, 2b1, 1a2, and 3b2 orbitals. The analysis through comparison of calculations considering different vibrational modes give a rational explanation about the discrepancy between experimental and theoretical 3b1 momentum profiles. Furthermore, computations taking all vibrational modes into account give an improved agreement between theory and experiment for the next four orbitals. Theoretical calculation also indicates a coupling of 7a1 and 1b1 orbitals taking place through certain vibrational variations of B1 symmetry. In a word, molecular vibrations in the electronic ground state have a considerable impact on EMDs of all outer-valence orbitals of oxetane. The case of oxetane demonstrates again the significant influence of molecular vibration on EMS cross sections for structurally versatile molecules, emphasizing the necessity of a careful check on vibrational effects when interpreting the results of EMS experiments.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. *Telephone: +86-551-63601170 Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENT 13 ACS Paragon Plus Environment


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This work was supported by the National Natural Science Foundation of China (Grant Nos. 11534011, 11327404) The authors also gratefully acknowledge Professor C. E. Brion from the University of British Columbia (UBC) in Canada for giving us the HEMS programs. The SAC-CI calculation with Gaussian 09 program was performed on the supercomputing system in the Supercomputing Center of University of Science and Technology of China.

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Phys. 2008, 21, 515-520. (21) Luo, Z. H.; Ning, C. G.; Liu, K.; Huang, Y. R.; Deng, J. K. Electron Momentum Spectroscopy Study of A Conformationally Versatile Molecule: n-Propanol. J. Phys. B 2009, 42, 165205. (22) Yan, M.; Shan, X.; Wu, F.; Xia, X.; Wang, K. D.; Xu, K. Z.; Chen, X. J. Electron Momentum Spectroscopy Study on Valence Electronic Structures of Ethylamine. J. Phys. Chem. A 2009, 113, 507-512. (23) Shojaei, S. H. R.; Filippo, M.; Bálazs, H.; Michael, S. D. Photoelectron and Electron Momentum Spectroscopy of 1-Butene at Benchmark Theoretical Levels. J. Phys. B 2011, 44, 235101. (24) Shi, Y. F.; Shan, X.; Wang, E. L.; Yang, H. J.; Zhang, W.; Chen, X. J. Experimental and Theoretical Investigation on the Outer Valence Electronic Structure of Cyclopropylamine by (e, 2e) Electron Momentum Spectroscopy. J. Phys. Chem. A

2014, 118, 4484-4493. (25) Shi, Y. F.; Shan, X.; Wang, E. L.; Yang, H. J.; Zhang, W.; Chen, X. J. Electron Momentum Spectroscopy of Outer Valence Orbitals of 2-Fluoroethanol. Chin. J. Chem. Phys. 2015, 28, 35-42. (26) Li, Z. J.; Chen, X. J.; Shan, X.; Xue, X. X.; Liu, T.; Xu, K. Z. Experimental Observation of Relativistic Effects on Electronic Wavefunction for Iodine Lone-Pair Orbital of CF3I. Chem. Phys. Lett. 2008, 457, 45-48. (27) Liu, K.; Ning, C. G.; Deng, J. K. Combining Relativistic Quantum-Chemistry Theories and Electron-Momentum Spectroscopy to Study Valence-Electron 17 ACS Paragon Plus Environment


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Structures of Molecules. Phys. Rev. A 2009, 80, 022716. (28) Li, Z. J.; Chen, X. J.; Shan, X.; Liu, T.; Xu, K. Z. Electron Momentum Spectroscopy Study of Jahn-Teller Effect in Cyclopropane. J. Chem. Phys. 2009, 130, 054302. (29) Zhu, J. S.; Miao, Y. R.; Deng, J. K.; Ning, C. G. The Jahn-Teller Effect in the Electron Momentum Spectroscopy of Ammonia. J. Chem. Phys. 2012, 137, 174305. (30) Liu, K.; Ning, C. G.; Luo, Z. H.; Shi, L. L.; Deng, J. K. An Experimental and Theoretical Study of the HOMO of W(CO)6: Vibrational Effects on the Electron Momentum Density Distribution. Chem. Phys. Lett. 2010, 497, 229-233. (31) Hajgató, B.; Morini, F.; Deleuze, M. S. Electron Momentum Spectroscopy of Metal Carbonyls: A Reinvestigation of the Role of Nuclear Dynamics. Theor. Chem. Acc. 2012, 131, 1-15. (32) Miao, Y. R.; Deng, J. K.; Ning, C. G. Vibrational Effects on the Electron Momentum Distributions of Valence Orbitals of Formamide. J. Chem. Phys. 2012, 136, 124302. (33) Shojaei, S. H. R.; Morini, F.; Deleuze, M. S. Photoelectron and Electron Momentum Spectroscopy of Tetrahydrofuran from a Molecular Dynamical Perspective. J. Phys. Chem. A 2013, 117, 1918-1929. (34) Shojaei, S. H. R.; Vandenbussche, J.; Deleuze, M. S.; Bultinck, P. Electron Momentum Spectroscopy of 1-Butene: A Theoretical Analysis Using Molecular Dynamics and Molecular Quantum Similarity. J. Phys. Chem. A 2013, 117, 8388-8398. 18 ACS Paragon Plus Environment

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(35) Yang, J.; Shan, X.; Zhang, Z.; Tang, Y. G.; Zhao, M. F.; Chen, X. J. Ring-Puckering Effects on Electron Momentum Distributions of Valence Orbitals of Oxetane. J. Phys. Chem. A 2014, 118, 11780-11786. (36) Watanabe, N.; Yamazaki, M.; Takahashi, M. Vibrational Effects on Valence Electron Momentum Distributions of Ethylene. J. Chem. Phys. 2012, 137, 114301. (37) Watanabe, N.; Yamazaki, M.; Takahashi, M. Vibrational Effects on Valence Electron Momentum Distributions of CH2F2. J. Chem. Phys. 2014, 141, 244314. (38) Morini, F.; Deleuze, M. S.; Watanabe, N.; Takahashi, M. Theoretical Study of Molecular Vibrations in Electron Momentum Spectroscopy Experiments on Furan: An Analytical Versus A Molecular Dynamical Approach. J. Chem. Phys. 2015, 142, 094308. (39) Morini, F.; Watanabe, N.; Kojima, M.; Deleuze, M. S.; Takahashi, M. Electron Momentum Spectroscopy of Dimethyl Ether Taking Account of Nuclear Dynamics in the Electronic Ground State. J. Chem. Phys. 2015, 143, 134309. (40) Farasat, M.; Shojaei, S. H. R.; Morini, F.; Golzan, M. M.; Deleuze, M. S. Electron Momentum Spectroscopy of Aniline Taking Account of Nuclear Dynamics in the Initial Electronic Ground State. J. Phys. B 2016, 49, 075102. (41) Dey, S.; McCarthy, I. E.; Teubner, P. J. O.; Weigold, E. (e, 2e) Probe for Hydrogen-Molecule Wave Functions. Phys. Rev. Lett. 1975, 34, 782-785. (42) Lermer, N.; Todd, B. R.; Cann, N. M.; Zheng, Y.; Brion, C. E.; Yang, Z.; Davidson, E. R. Electron Momentum Spectroscopy of H2 and D2: Ground and Excited Final States. Phys. Rev. A 1997, 56, 1393-1402. 19 ACS Paragon Plus Environment

Ionization to


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(43) Bawagan, A. O.; Brion, C. E.; Davidson, E. R.; Feller, D. Electron Momentum Spectroscopy of the Valence Orbitals of H2O and D2O: Quantitative Comparisons Using Hartree-Fock Limit and Correlated Wavefunctions. Chem. Phys. 1987, 113, 19-42. (44) Leung, K. T.; Sheehy, J. A.; Langhoff, P. W. Vibrational Averaging Effects on the

Valence-Shell

Electron

Momentum

Distributions

in

H2O

Employing

Hartree-Fock-Limit Wavefunctions. Chem. Phys. Lett. 1989, 157, 135-141. (45) Hollebone, B. P.; Neville, J. J.; Zheng, Y.; Brion, C. E.; Wang, Y.; Davidson, E. R. Valence Electron Momentum Distributions of Ethylene: Comparison of EMS Measurements with Near Hartree-Fock limit, Configuration Interaction and Density Functional Theory Calculations. Chem. Phys. 1995, 196, 13-35. (46) Chan, S. I.; Zinn, J.; Fernandez, J.; Gwinn, W. D. Trimethylene Oxide. I. Microwave Spectrum, Dipole Moment, and Double Minimum Vibration. J. Chem. Phys. 1960, 33, 1643-1655. (47) Chan, S. I.; Zinn, J.; Gwinn, W. D. Trimethylene Oxide. II. Structure, Vibration-Rotation Interaction, and Origin of Potential Function for Ring-Puckering Motion. J. Chem. Phys. 1961, 34, 1319-1329. (48) Chan, S. I.; Borgers, T. R.; Russell, J. W.; Strauss, H. L.; Gwinn, W. D. Trimethylene Oxide. III. Far-Infrared Spectrum and Double-Minimum Vibration. J. Chem. Phys. 1966, 44, 1103-1111. (49) Kydd, R. A.; Wieser, H.; Danyluk, M. Ring Puckering Potential Functions for Normal and Deuterated Trimethylene Oxides. J. Mol. Spectrosc. 1972, 44, 14-17. 20 ACS Paragon Plus Environment

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(50) Jokisaari, J.; Kauppinen, J. Vapor‐Phase Far‐Infrared Spectrum and Double Minimum Potential Function of Trimethylene Oxide. J. Chem. Phys. 1973, 59, 2260-2263. (51) Creswell, R. A. Molecular Structure of Oxetane. Mol. Phys. 1975, 30, 217-222. (52) Mollere, P. D.; Houk, K. N. Photoelectron Spectroscopy of Heterocyclobutanes: Electronic Structure of Small Ring Compounds and Ramifications for Reactivity. J. Am. Chem. Soc. 1977, 99, 3226-3233. (53) Roszak, S.; Kaufman, J. J.; Koski, W. S.; Barreto, R. D.; Fehlner, T. P.; Balasubramanian, K. Experimental and Theoretical Studies of Photoelectron Spectra of Oxetane and Some of Its Halogenated Methyl Derivatives. J. Phys. Chem. 1992, 96, 7226-7230. (54) Walker, I. C.; Holland, D. M. P.; Shaw, D. A.; McEwen, I. J.; Guest, M. F. The Valence Shell Electronic States of Trimethylene Oxide Studied by Photoabsorption and ab Initio Multireference Configuration Interaction Calculations. Mol. Phys. 2009, 107, 1473-1483. (55) Lesarri, A.; Blanco, S.; López, J. C. The Millimetre-Wave Spectrum of Oxetane. J. Mol. Struct. 1995, 354, 237-243. (56) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula Into A Functional of The Electron Density. Phys. Rev. B

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(58) Dunning, T. H., Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007-1023. (59) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron Affinities of the First-Row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96, 6796-6806. (60) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Rob, M. A.; Cheeseman, J. R.; Jr., J. A. M.; Vreven, T.; Kudin, K. N.; Burant, J. C., et al. Gaussian 03; Gaussian, Inc.: Wallingford, CT, 2003. (61) Duffy, P.; Chong, D. P.; Casida, M. E.; Salahub, D. R. Assessment of Kohn-Sham Density-Functional Orbitals as Approximate Dyson Orbitals for the Calculation of Electron-Momentum-Spectroscopy Scattering Cross Sections. Phys. Rev. A 1994, 50, 4707-4728. (62)

Nakatsiji,

H.;

Hirao,

K.

Cluster

Expansion

of

the

Wavefunction.

Symmetry-Adapted-Cluster Expansion, Its Variational Determination, and Extension of Open-Shell Orbital Theory. J. Chem. Phys. 1978, 68, 2053-2065. (63) Nakatsuji, H. Electronic Structures of Ground, Excited, Ionized and Anion States Studied by the SAC/SAC-CI Theory. Acta Chim. Hung. 1992, 129, 719-776. (64) Nakatsuji, H. In Computational Chemistry: Reviews of Current Trends; Leszczynski, J., Eds.; World Scientific: Singapore, 1997; Vol. 2, pp 62-124. (65) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A., et al. 22 ACS Paragon Plus Environment

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Gaussian 09; Gaussian, Inc.: Wallingford, CT, 2009. (66) Herzberg, G. Molecular Spectra and Molecular Structure: II. Infared and Raman Spectra of Polyatomic Molecules; D. Van Nostrand Company, Inc.: Princeton, 1945. (67) Kydd, R. A.; Wieser, H.; Kiefer, W. Vibrational Assignments for Trimethylene Oxide and Several Deuterated Analogues. Spectrochim. Acta A 1983, 39, 173-180. (68) Bánhegyi, G.; Pulay, P.; Fogarasi, G. Ab Initio Study of the Vibrational Spectrum and Geometry of Oxetane—I. Interpretation of the Vibrational Spectra. Spectrochim. Acta A 1983, 39, 761-769. (69) Fábri, C.; Szidarovszky, T.; Magyarfalvi, G.; Tarczay, G. Gas-Phase and Ar-Matrix SQM Scaling Factors for Various DFT Functionals with Basis Sets Including Polarization and Diffuse Functions. J. Phys. Chem. A 2011, 115, 4640-4649.



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

Table 1. Vibrational frequencies of oxetane. modea

mode characterb

1

β-CH2 stretch (A1)

calculated

experimentc

(cm-1)

(cm-1)

3074

2979.0

2

α-CH2 stretch (A1)

3023

2893.9

3

α-CH2 scissor (A1)

1541

1505.0

4

β-CH2 scissor (A1)

1495

1452.0

5

α-CH2 wag (A1)

1370

1343.1

6

ring breathing (A1)

1039

1032.7

7

ring deformation (A1)

918

908.7

8

ring deformation (A1)

810

784.5

9

α-CH2 stretch (A2)

3055

10

β-CH2 twist (A2)

1234

1230d

11

α-CH2 twist (A2)

1159

1096d

12

α-CH2 rock (A2)

837

842

13

β-CH2 stretch (B1)

3123

3006

14

α-CH2 stretch (B1)

3053

2938.7

15

α-CH2 twist (B1)

1197

1183d

16

α-CH2 rock (B1)

1147

1137d

17

β-CH2 rock (B1)

768

703

18

ring puckering (B1)

48

19

α-CH2 stretch (B2)

3014

2887.1

20

α-CH2 scissor (B2)

1514

1480

21

α-CH2 wag (B2)

1311

1289.0

22

β-CH2 wag (B2)

1261

1230

23

C-O stretch (B2)

1022

1008.3

24

C-C stretch (B2)

939

937.3

66

a. The Herzberg’s notation is adopted here. b. The notation of α, β-CH2 is in line with Chan et al.47 c. Ref. 67 d. Ref. 68


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Figure Captions: Figure 1. Main nuclear displacement vectors for vibrational normal modes of oxetane calculated by B3LYP/aug-cc-pVTZ method. The red sphere represents oxygen, yellow one represents carbon, and grey one is the hydrogen atom.

Figure 2. Electron momentum profiles of 3b1 orbital of oxetane. (a) Experimental momentum profile35 with theoretical ones considering vibrational effects. (b) Contributions from each vibrational mode on 3b1 momentum profile. (c) Orbital map of 3b1 orbital with C2v geometry. (d) Orbital map of 3b1 orbital distorted along the ring-puckering mode. (e) Orbital map of 3b1 2

2

orbital distorted along the v14 α-CH2 stretch mode ( ξ vL (QL ) / ξvL (0) = 1/ 4 ). The two

different colors in the orbital maps mean opposite phases of the position-space wavefunctions.

Figure 3. Experimental35 and theoretical electron momentum profiles for (a1) 8a1, (b1) 4b2, and (c) 8a1+4b2 orbitals and contributions from each vibrational mode on (a2) 8a1 and (b2) 4b2 orbitals. The position space orbital maps are calculated with C2v geometry. The two

different colors in the orbital maps mean opposite phases of the position-space wavefunctions.

Figure 4. Experimental35 and theoretical electron momentum profiles for (a1) 2b1, (b1) 1a2, and (c) 2b1+1a2 orbitals and contributions from each vibrational mode on (a2) 2b1 and (b2) 1a2 orbitals. The position space orbital maps calculated with C2v geometry are shown at 25 ACS Paragon Plus Environment


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

bottom-right and the position space orbital maps for the distorted structures along specific vibrational modes are shown as insets in (a2) and (b2) with the coordinate QL meeting 2

2

ξ v (QL ) / ξv (0) = 1/ 4 . The two different colors in the orbital maps mean opposite L

L

phases of the position-space wavefunctions.

Figure 5. Theoretical electron momentum profiles considering vibrational effects for (a1) 3b2, (b1) 7a1, and (c1) 1b1 orbitals and contributions from each vibrational mode on (a2) 3b2, (b2) 7a1, and (c2) 1b1 momentum profiles. The position space orbital maps calculated with C2v geometry are shown as insets in (a1)-(c1) and the position space orbital maps for the distorted structures along specific vibrational modes are shown as insets in (a2)-(c2) with the 2

2

coordinate QL meeting ξ vL (QL ) / ξ vL (0) = 1/ 4 . The two different colors in the

orbital maps mean opposite phases of the position-space wavefunctions.


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Figure 1. submitted by Y. G. Tang et al.



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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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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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

Table of Contents graphic

TOC. submitted by Y. G. Tang et al.


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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


v1

v2

v4

v3

v7

v8

v9

v13

v14

v15

v16

v19

v20

v21

v22

v10

ACS Paragon Plus Environment

v5

v11

v6

v12

v17

v23

v18

v24


0.06

Relative Intenstity

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

Experiment C2v Ring-puckering mode

Other vibrational modes All vibrational modes

0.012

Page 34 of 38

Contributions from each vibrational mode on the 3b1 orbital ring-puckering v14 D-CH2 stretch of B1 symmetry

3b1

0.008

0.04

v9 D-CH2 stretch of A2 symmetry sum of other modes

0.004

0.02 0.000

0.00 0.0

0.5

1.0

1.5

2.0

0.0

0.5

Momentum(a.u.)

1.0

(a)

(b)

C2v

v14

ring puckering

(c)

1.5

Momentum(a.u.)

(d) ACS Paragon Plus Environment

(e)

2.0

Page 35 of 38

0.08

(a1) 8a1 Relative Intenstity

0.06

Experiment C2v Ring-puckering mode Other vibrational modes All vibrational modes

0.002

(a2) 8a1

0.001

0.04 0.000

0.02

v1

-0.001

v4

v13

v21

v22

v23

v24 0.00 0.0 0.08

0.5

(b1) 4b2 Relative Intenstity

0.06

1.0

1.5

2.0

sum of other modes

-0.002 0.0

0.5

1.0

0.003


(b2) 4b2 0.002

1.5

2.0

v5

v10

v14

v19

v20

v23

v24 sum of other modes

0.001

0.04

0.000

0.02 -0.001

0.00 0.0 0.14 0.12

Relative Intenstity

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


0.5

(c) 8a1+4b2

0.10

1.0

1.5

2.0

0.0

0.5

1.0

1.5

Momentum(a.u.)


0.08 0.06 0.04 0.02 0.00 0.0

8a1 0.5

1.0

1.5

2.0

Momentum(a.u.)


4b2

2.0


0.12


Relative Intenstity

(a1) 2b1 0.08

0.006

v10 v14

(a2) 2b1

v13

0.002

0.04

0.00 0.0

Relative Intenstity

0.12

v13 v15

sum of other modes

0.004

0.000

v14

-0.002

0.5

1.0

1.5

H1 2.0


(b1) 1a2

0.0

0.012

0.5

(b2) 1a2

1.0

v9 v10

1.5

2.0

v12 v15

sum of other modes

0.008

0.08 0.004

v9

0.04 0.000

0.00 0.0

0.5

1.0

1.5

2.0

0.0

0.5

(c) 2b1+1a2 0.20

1.0

1.5

Momentum(a.u.)


0.25

Relative Intenstity

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

Page 36 of 38

0.15

0.10

0.05

2b1 0.00 0.0

0.5

1.0

1.5

2.0

Momentum(a.u.)


1a2

2.0

Page 37 of 38

Relative Intensity

0.06

(a1) 3b2

C2v Ring-puckering mode Other vibrational modes All vibrational modes

0.004

(a2) 3b2

v15

v16

v19

v20

v21

v22

v23

0.04

ring-puckering sum of other modes

0.002

v21 0.02 0.000

C2v 0.00 0.0

0.5

1.0

1.5

2.0

2.5

v23 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

2.5

3.0

2.5

3.0

0.8

(b1) 7a1


Relative Intensity

0.6

0.02

(b2) 7a1

0.00

-0.02

0.4

ring puckering

v9 v13 v14

-0.04 0.2


C2v 0.0 0.0 0.20

Relative Intensity

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


0.5

1.0

(c1) 1b1

1.5

2.0

2.5

3.0

0.0 0.08


0.15

0.5

1.0

1.5

ring puckering

0.06 v13 0.04

v14

0.02


v13

v17

0.05

0.00

C2v 0.5

1.0

1.5

2.0

2.5

2.0

(c2) 1b1

0.10

0.00 0.0

v13

-0.06

3.0

-0.02 0.0

0.5

Momentum (a.u.)


1.0

1.5

2.0

Momentum (a.u.)


0.06

ring puckering

C2v

Relative Intenstity

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

0.04

HOMO 3b1


0.02

0.00 0.0

v14

0.5

1.0

1.5

2.0

Momentum(a.u.)


Page 38 of 38

Vibrational Effects on Electron Momentum Distributions of Outer

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