Vibrational Effects on Electron Momentum Distributions of Outer

Aug 5, 2016 - where fee is the e−e collision factor and M(p) is the relevant structure factor .... three-parameter Lee−Yang−Parr (B3LYP)56,57 fu...
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Vibrational Effects on Electron Momentum Distributions of OuterValence Orbitals of Oxetane Yaguo Tang,† Xu Shan,† Jing Yang,† Shanshan Niu,† Zhe Zhang,† Noboru Watanabe,‡ Masakazu Yamazaki,‡ Masahiko Takahashi,‡ and Xiangjun Chen*,† †

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 ‡ 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 turnup at low momentum in the experimental momentum profile of the 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 EMDs of outer-valence orbitals of oxetane. The case of oxetane exhibits the significance of checking vibrational effects when performing electron momentum spectroscopy measurements.

1. 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 wave function of the ionized orbital in momentum space, that is, the 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 attract researchers’ attention until recently. EMS studies on conformational isomers8−25 have already shown that EMDs are remarkably affected by 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 the 1b3u orbital of C2H445 showed that EMS results seem to not be 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 © 2016 American Chemical Society

momentum profiles for specific orbitals of polyatomic molecules. 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, for exmaple, the effects of the ν5 mode (∼3000 cm−1) on the 1b3g orbital for ethylene.36 Oxetane or trimethylene oxide, a four-membered ring molecule of flexible structure, is one of the simplest molecules having ring-puckering motion.46−50 The unusual one-dimensional ring-puckering potential curve is proved to be a doubleminimum 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 excited vibrational levels in this mode are appreciably populated at room temperature because 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 Received: July 5, 2016 Revised: August 1, 2016 Published: August 5, 2016 6855

DOI: 10.1021/acs.jpca.6b06706 J. Phys. Chem. A 2016, 120, 6855−6863

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The Journal of Physical Chemistry A pp d3σ = (2π )4 1 2 fee Σav p f i dΩ1 dΩ 2 dE1 p0 pp = (2π )4 1 2 fee M(p) p0

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 reproduced the experiment. On the other hand, for other outervalence 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 inspired 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.

σEMS =

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 wave function of a molecular target can be described as a product of separate electronic, vibrational, and rotational wave functions. Ignoring rotational motions and employing the closure relation of the final vibrational eigenstates, the relevant structure factor can be simplified as35,36 M (p) =

∑ pν (T ) ∫ |χiν (Q )|2 ρf (p; Q ) dQ ν

1 Sf (Q ) 4π

ρf (p ; Q ) =

EMS is a coincidence experiment based on the kinematically complete (e, 2e) 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 (1)

p = p1 + p2 − p0

(2)

(4)

where χiν(Q) is the vibrational wave function of the eigenstate ν in the ground state and pν(T) is the related population at temperature T. ρf(p;Q) is the spherically averaged EMD at a given geometry with a coordinate Q, which reads

2. THEORETICAL METHODS

εf = E0 − E1 − E2

(3)

∫ |φf (p; Q )|2 dΩp

(5)

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 crossterms between vibrational modes are neglected χiν (Q ) =

∏ ξν (Q L) L

(6)

L

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

M(p) = ρf (p ; Q 0) +

∑ ΔML(p) L

= ρf (p ; Q 0) +

∑ (∑ pνL (T ) ∫ |ξνL(Q L)|2 L

νL

× ρf (p ; Q 0 + Q LqL̂ ) dQ L − ρf (p ; Q 0))

(7)

Figure 1. Main nuclear displacement vectors for vibrational normal modes of oxetane calculated by the B3LYP/aug-cc-pVTZ method. The red sphere represents oxygen, the yellow one represents carbon, and the gray one is the hydrogen atom. 6856

DOI: 10.1021/acs.jpca.6b06706 J. Phys. Chem. A 2016, 120, 6855−6863

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The Journal of Physical Chemistry A where ξνL(QL) is the harmonic function describing molecular vibrations and ΔML(p) is the contribution to vibrational effects from the Lth normal mode. QL is the displacement from the equilibrium geometry Q0, and q̂L is the unit vector along the normal coordinate. eq 7 is easily understood because it can be regarded as a first-order approximation of eq 4 when the normal coordinate is introduced. On the basis of 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 ringpuckering 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 property of the harmonic functions was used.36 Due to the same odd−even property of the ring-puckering eigenfunctions that resulted from the symmetry of the potential curve, eq 7 is still valid for oxetane even if the nonharmonic ring-puckering wave fuction is included. As a result, vibrational effects can be computed according to eq 7, where the ringpuckering mode is described by the wave function 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 work,35 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 three-parameter Lee−Yang−Parr (B3LYP)56,57 functional and Dunning’s augmented correlation-consistent polarization valence basis set of triple-ζ quality (aug-cc-pVTZ)58,59 using the Gaussian 03 program.60 Then, ρf(p;Q0 + QLq̂L) is calculated

at the B3LYP/aug-cc-pVTZ level where the target 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 the University of Science and Technology Table 1. Vibrational Frequencies of Oxetane modea

mode characterb

calculated (cm−1)

experimentc (cm−1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

β-CH2 stretch (A1) α-CH2 stretch (A1) α-CH2 scissor (A1) β-CH2 scissor (A1) α-CH2 wag (A1) ring breathing (A1) ring deformation (A1) ring deformation (A1) α-CH2 stretch (A2) β-CH2 twist (A2) α-CH2 twist (A2) α-CH2 rock (A2) β-CH2 stretch (B1) α-CH2 stretch (B1) α-CH2 twist (B1) α-CH2 rock (B1) β-CH2 rock (B1) ring puckering (B1) α-CH2 stretch (B2) α-CH2 scissor (B2) α-CH2 wag (B2) β-CH2 wag (B2) C−O stretch (B2) C−C stretch (B2)

3074 3023 1541 1495 1370 1039 918 810 3055 1234 1159 837 3123 3053 1197 1147 768 48 3014 1514 1311 1261 1022 939

2979.0 2893.9 1505.0 1452.0 1343.1 1032.7 908.7 784.5 1230d 1096d 842 3006 2938.7 1183d 1137d 703 2887.1 1480 1289.0 1230 1008.3 937.3

The Herzberg’s notation66 is adopted here. bThe notation of α, β-CH2 is in line with that of Chan et al.47 cReference 67. dReference 68. a

Figure 2. Electron momentum profiles of the 3b1 orbital of oxetane. (a) Experimental momentum profile35 with theoretical ones considering vibrational effects. (b) Contributions from each vibrational mode on the 3b1 momentum profile. (c) Orbital map of the 3b1 orbital with C2v geometry. (d) Orbital map of the 3b1 orbital distorted along the ring-puckering mode. (e) Orbital map of the 3b1 orbital distorted along the ν14 α-CH2 stretch mode |ξνL(QL)|2/|ξνL(0)|2 = 1/4). The two different colors in the orbital maps mean opposite phases of the position−space wave functions. 6857

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The Journal of Physical Chemistry A of China using the Gaussian 09 program65 exhibits that the 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 orbitals and geometries. The displacement QL is chosen so that |ξνL(QL)|2/|ξνL(0)|2 = 3/4, 1/2, 1/4, 1/8, and 1/16. A fourth-order polynomial function is used to fit ρf(p;Q0 + QLq̂L) as a function of QL. Finally, the momentum profiles are obtained according to eq 7 considering thermally induced nuclear motions using the Boltzmann distribution at room temperature (298 K).

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 the scaling factor of 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 Figures 2−5. As a comparison, the EMS experimental results on the first five MOs of Yang et al.35 are also plotted in the figures.

3. RESULTS AND DISCUSSION There are 24 vibrational modes (8A1 + 4A2 + 6B1 + 6B2) for the oxetane molecule, as illustrated in Figure 1. The vibrational mode characters and the corresponding vibrational frequencies

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 wave functions. 6858

DOI: 10.1021/acs.jpca.6b06706 J. Phys. Chem. A 2016, 120, 6855−6863

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The Journal of Physical Chemistry A

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 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 |ξνL(QL)|2/|ξνL(0)|2 = 1/4. The two different colors in the orbital maps mean opposite phases of the position−space wave functions.

how it works, the contribution of each vibrational mode is plotted in Figure 2b. It can be seen that the high intensity of the HOMO momentum profile at small momentum clearly originates from the ring-puckering and α-CH2 stretching motion of B1 symmetry. From the orbital map in Figure 2c, one can see that the 3b1 orbital of C2v geometry is constituted by an oxygen 2p lone pair (no) and α-CH bonds and is completely antisymmetric about the ring plane. Keep in mind that the momentum−space wave function φf(p;Q) is the Dirac− Fourier transform of the corresponding position−space wave function φf(r;Q), and its value at p = 0 reads φf(p;Q)|p=0 = (2π)−3/2 ∫ φf(r;Q) dr. Therefore, it is reasonable to have zero intensity at zero momentum for the 3b1 orbital with antisymmetry character about the ring plane. Yang et al.’s natural bond orbital analysis35 indicated that the different charge transfer from no to

For the 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 expected to show 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 2a, the present calculation with a comprehensive consideration of all vibrational modes (solid curve) reproduces the experiment quite well, which indicates that other vibrational motions (dashed−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 of different kinds of vibrational motions and 6859

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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 coordinate QL meeting |ξνL(QL)|2/|ξνL(0)|2 = 1/4. The two different colors in the orbital maps mean opposite phases of the position−space wave functions.

4b2 orbital, the intensity in the region of p < 1.0 au is increased to some extent. Obvious discrepancies between theory and experiment still exist in Figure 3a1,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 with the experimental result well except in the low-momentum region. It suggests that the uncertainty from the deconvolution process should be responsible for the remaining discrepancy in the momentum region of p > 0.25 au in Figure 3a1,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 line with the experimental results (Figure 4). It is easily understood with the

the nearby C−H bonds due to hyperconjugative interactions breaks the symmetry (Figure 2d) when the molecule is puckered and consequently contributes 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 2e), which also understandably results in 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 the 8a1 orbital, vibrational motions increase the intensity at p = 0.5−1.3 au, improving the agreement between theoretical and experimental momentum profiles, while for the 6860

DOI: 10.1021/acs.jpca.6b06706 J. Phys. Chem. A 2016, 120, 6855−6863

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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 outervalence 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.

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 the 2b1 orbital exhibits antisymmetry about the ring plane. The ν13 mode of the β-CH2 stretch and ν14 mode of the α-CH2 stretch of B1 symmetry are mainly responsible for the high intensity at zero momentum for the 2b1 orbital. The 1a2 orbital consisting of α-CH bonds displays d orbital character. The ν9 mode of the 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 improved agreement is achieved with experiment, a distinct underestimation of the turn-up still exists. Similar to the case of the 1b3g orbital of ethylene6 and the π* orbital of oxygen,7 the d-type 1a2 orbital of oxetane also likely suffers a distorted wave effect, resulting in a high turn-up and accounting for the remaining discrepancy between experiment and theoretical calculation considering vibrational effects. Because 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 can then certainly be attributed to the distorted wave effect in ionization from the 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. As for the final three outer-valence orbitals, 3b2, 7a1, and 1b1, where no experimental results are available, theoretical calculations (Figure 5) also show that the vibrational motions significantly affect momentum profiles. The 3b2 orbital of C2v geometry is antisymmetric about the β-CH2−O plane, thus yielding zero intensity at p ≈ 0. The vibrational modes like ν21 and ν23 will break this symmetry and make a positive contribution to the momentum intensity at p ≈ 0. Theoretical calculation considering vibrational effects predicts an evidently higher intensity in the low-momentum region. It is interesting that 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 C2v geometry gives the binding energies of the two orbitals at 16.5 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 (ν18, ν13) and B1 (1b1)⊗ B1 (ν18, ν13) create B1 and A1 symmetries, respectively. As a consequence, a coupling between 7a1 and 1b1 orbitals can easily occur through vibrational modes of B1 symmetry. As Figure 5b,c shows, the related vibronic coupling lowers the intensity of the momentum profile of the 7a1 orbital while notably increasing that of the 1b1 orbital in the low-momentum region.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: +86-551-63601170. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grants 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 the Gaussian 09 program was performed on the supercomputing system in the Supercomputing Center of the University of Science and Technology of China.

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REFERENCES

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4. CONCLUSIONS 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 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 gives a rational explanation of the discrepancy between experimental and theoretical 6861

DOI: 10.1021/acs.jpca.6b06706 J. Phys. Chem. A 2016, 120, 6855−6863

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