Relationship between interference pattern and molecular orbital shape in (e, 2e) electron momentum profiles of SF6

Journal of Electron Spectroscopy and Related Phenomena 209 (2016) 78–86

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Relationship between interference pattern and molecular orbital shape in (e, 2e) electron momentum profiles of SF6 Noboru Watanabe, Masakazu Yamazaki, Masahiko Takahashi ∗ Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan

a r t i c l e

i n f o

Article history: Received 5 February 2016 Received in revised form 2 April 2016 Accepted 4 April 2016 Available online 6 April 2016 PACS number: 34.80.Gs Keywords: Bond oscillation Momentum space wave function Electron momentum spectroscopy (e, 2e) spectroscopy SF6

a b s t r a c t We report an (e, 2e) study on interference effects in the spherically-averaged electron momentum density distributions of the five outermost molecular orbitals (MOs) of SF6 , which are each constructed from the 2p atomic orbitals (AOs) on the F atoms. The (e, 2e) experiment was performed in the symmetric noncoplanar geometry and at an incident electron energy of 1.2 keV. Interference patterns for each MO have then been obtained by dividing the experimental data by distorted-wave-Born-approximation cross section for an isolated F 2p AO. A detailed analysis of the results shows that the interference patterns afford a rare opportunity to experimentally clarify symmetries of the MOs. Furthermore, it is demonstrated that they provide information about not only the molecular geometry but also the spatial orientation of the constituent AOs even for this relatively complicated molecule. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Electron momentum spectroscopy (EMS), also known as binary (e, 2e) spectroscopy, is a powerful technique for investigating the electronic structures of atoms and molecules [1–7]. It is now well documented that EMS cross sections are directly related to the electron momentum density distribution of the ionized orbital. EMS thus enables one to look at individual electron orbitals in momentum-space (p-space). The advantage of this over doing in position-space (r-space) has long been demonstrated in terms of the inverse spatial reversal property [1–9]. Namely, an expansion in the r-space wave function corresponds to a contraction in the p-space wave function and vice versa, due to the nature of the DiracFourier transform. This is the material reason why EMS is sensitive particularly to the behavior of outer, loosely bound electrons that are of central importance in various chemical properties of atoms and molecules. Another notable feature of p-space wave functions is that for molecules, the information about the equilibrium nuclear positions Rj appears only in phase factors, which are introduced by the Dirac-Fourier transform. Hence, the electron momentum density distribution of a molecular orbital (MO) inherently possesses cosi-

∗ Corresponding author. E-mail address: [email protected] (M. Takahashi). http://dx.doi.org/10.1016/j.elspec.2016.04.004 0368-2048/© 2016 Elsevier B.V. All rights reserved.

nusoidal modulations, for instance, that with periodicity of 2/Rjk along the direction of the line connecting atoms j and k, separated by the distance, Rjk [3–6]. To put this simply, consider the following two wave functions for a homonuclear diatomic molecule that are given by a linear combination of atomic orbitals (AOs): ±

(r) =

1



2 (1 ± S)





 (r − R 1 ) ±  (r − R 2 ) .

Here (r − Rj )’s (j = 1 or 2) are the constituent AOs and S is their overlap integral. The plus sign corresponds to the bonding orbital and the minus sign to the antibonding orbital. The Dirac-Fourier transform of ± (r) provides the p-space MOs, ± (p), as ±

(p) =



1

2 (1 ± S)



 (p) exp (ip · R 1 ) ±  (p) exp (ip · R 2 )



with (p) being the p-space AO. Then interference terms arise in the electron momentum densities, | ± (p)|2 , from the phase factors, exp(ip·Rj ) (j = 1 or 2): |

±

2

(p) | =

| (p) |2 [1 ± cos p · (R 1 − R 2 )] . 1±S

From this equation one can see that | ± (p)|2 should exhibit their own cosinusoidal modulations, which are completely out of phase from each other. Phenomena of this kind are clearly characteristic of p-space wave functions and they are called bond oscillation

N. Watanabe et al. / Journal of Electron Spectroscopy and Related Phenomena 209 (2016) 78–86

[9,10]. Note that the interference pattern due to bond oscillation may provide a wealth of information about not only the molecular geometry but also the spatial distribution and symmetry of the MO in interest. In spite of its remarkable property, however, bond oscillation has scarcely been investigated. In fact, the first experimental evidence of its presence was given in 2012, by a joint study of the Sendai (e, 2e) group and the Hefei (e, 2e) group [11], for the three outermost MOs of CF4 , demonstrating the sensitivity of bond oscillation to the spatial orientation of the constituent AOs. Later, the Hefei (e, 2e) group conducted a high energy-resolution EMS study on H2 [12], in which the observed deviation from the Franck–Condon principle in vibrational ratio was ascribed to interference effects. We the Sendai (e, 2e) group also performed an EMS study on the same compound [13]. Here, special attention was paid to observing a full period of the interference pattern for H2 by covering a much wider range of the electron momentum, and the result was in support of the chemical bond theory in that shrinkage of the H 1s AOs occurs upon molecular formation. Under the circumstances mentioned above, we have carried out an EMS experiment for the five outermost MOs of SF6 [14], which are each constructed from equivalent 2p AOs located on the F atoms. Although it has happened that very recently an EMS measurement of interference patterns in SF6 was independently conducted and reported for the first four of the five MOs [15], any attempt beyond the observation of the interference patterns has not yet been made for this molecule. The principal aim of the present work is thus to elucidate the relationship between interference pattern and molecular orbital shape for SF6 , by extending our earlier study on CF4 [11]. The reason for this extension is threefold. Firstly, SF6 is a more complicated compound, belonging to the Oh point group. Secondly, although the three outermost MOs of CF4 and the five MOs of SF6 are all formed from the nonbonding F 2p AOs, they have a difference in the number of the type of internuclear distances to be considered. Namely, SF6 is an octahedral molecule and has two different F-F internuclear distances at the equilibrium geometry, whilst CF4 has only one F-F distance. Lastly, the five MOs of SF6 are each constructed from the F 2p AOs with various spatial orientations and belong to different irreducible representations of the Oh point group, so they offer an ideal opportunity to examine how information about the spatial orientations of the constituent AOs and symmetries of the MOs are manifested in the interference patterns. For this purpose, the present experiment on SF6 has been carried out by using one of our highly sensitive EMS spectrometers [16]. Considerable improvement in the momentum resolution and/or statistical accuracy of data have been achieved as compared with those of the EMS studies on SF6 in literature [15,17,18]. Furthermore, associated theoretical calculations have been made with effects of molecular vibration on the electron momentum densities [19–27] being involved. These calculations are motivated by an open question; how molecular vibration does affect bond oscillation phenomena.

2. Experiment EMS is a high-energy electron-impact ionization experiment that involves coincident detection of the inelastically scattered and ejected electrons [1–7]. 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. The EMS experiment on SF6 was carried out at E0 = 1.2 keV and in the symmetric noncoplanar geometry where the two outgoing electrons having equal energies and making equal scattering angles of 45◦ with respect to the incident electron beam axis are detected. In this kinematics, the magnitude of the target electron momentum,

79

p = |p|, can be determined from the out-of-plane azimuthal angle difference between the two outgoing electrons ( = 2 − 1 − ). Details of the spectrometer used have been given elsewhere [16]. Briefly, it 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 Ebind and p in parallel. The experimental result was obtained by accumulating data at an ambient sample gas pressure of 3.0 × 10−4 Pa for two weeks runtime. Commercially available SF6 gas (Japan Fine Products, >99.999%) was used in the experiment. During the measurement the electron gun produced an electron beam of typically 30 ␮A 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. No detectable impurities were observed in the binding energy spectra. 3. Theory 3.1. Theoretical momentum profiles Within the plane wave impulse approximation (PWIA), the EMS cross section is given by EMS (p) = (2)4

p1 p2 fee M (p) , p0

(1)

where fee is the electron–electron collision factor and М(p) is the quantity called electron momentum profile [4]. By using the Born–Oppenheimer approximation and by neglecting the influence of molecular vibration, М(p) is expressed as follows [3–7]: 1 M (p) = S˛ (Q 0 ) d˛ 4



|

˛

2

(p; Q 0 ) | dp .

(2)

Here ˛ (p;Q0 ) is the p-space representation of the normalized Dyson orbital for the molecule at the equilibrium geometry, Q0 , and S␣ is the spectroscopic factor. (4␲)−1 ʃd˝p represents the spherical averaging due to the random orientation of gaseous molecular targets and d˛ denotes the degeneracy of the orbital. Electron momentum profile thus corresponds to the spherically-averaged electron momentum density distribution of the orbital. Recently, we have developed a theoretical method of calculating electron momentum profiles with vibrational effects being involved [21,24,26]. According to this harmonic analytical quantum mechanical approach, М(p) can be written as, M (p) = +

 v

1 S˛ (Q 0 ) d˛ 4 Pv (T )





|ϕ˛ (p, Q 0 ) |2 dp





ˆs − ˛ (p, Q 0 ) | vs (Qs )  vs (Qs ) | ˛ p, Q 0 + Qs q

s

(3) with ˛ (p,Q) being a spherically-averaged one-electron momentum density distribution at a given molecular geometry, Q = (Q1 , Q2 , . . ., Qs , . . .). Here Pv (T) is the Boltzmann’s statistical population at temperature T. vs (Qs ) is the harmonic oscillator function of the ˆs denotes a unit v vibrational level for the s-th normal mode, and q vector that points along the s-th normal coordinate. The first term of Eq. (3) is equivalent to the electron momentum profile at the equilibrium molecular geometry and it is the second term that is in charge of vibrational effects.

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In this study we calculated two kinds of theoretical momentum profiles for SF6 . One is electron momentum profiles calculated at the equilibrium geometry and the other is those calculated with vibrational effects being involved. The vibrational effects calculations were made as follows. First, the vibrational normal coordinates and frequencies of SF6 were obtained by means of the second-order Møller–Plesset (MP2) treatment with the augmented correlation-consistent polarization valence-triple-zeta (aug-cc-pVTZ) basis set [28]. In the calculation f-type diffuse functions were omitted from the basis set to save computational time. The normal coordinate analysis was carried out using the General Atomic Molecular Electronic Structure System (GAMESS) program [29]. ˛ (p,Q)’s were then computed at several molecular geometries distorted along each normal coordinate from the equilibrium. These calculations were conducted within the target Kohn–Sham approximation [30]. Briefly, for each distorted structures, the Kohn–Sham orbitals of SF6 were obtained by means of density functional theory (DFT) along with the Becke-3-parameters-LeeYang-Parr (B3LYP) functional [31] and aug-cc-pVTZ basis, and they were used as normalized Dyson orbitals. The DFT calculations were performed using the Gaussian03 program [32]. The Kohn–Sham orbitals were subsequently converted to ˛ (p,Q)’s with the aid of HEMS program developed by Bawagan et al. [33], assuming all the spectroscopic factor values for the relevant MOs are equal to each other. Finally, we obtained electron momentum profiles of the target at room temperature (298 K), according to Eq. (3). On the other hand, conventional theoretical momentum profiles, which do not involve vibrational effects, were computed by dropping the contribution of the second term of Eq. (3). All the theoretical momentum profiles were folded with the instrumental momentum resolution according to the procedure of Migdall et al. [34] in order to make comparisons with the experiment. The procedure has been shown to satisfactorily account for the influence of momentum resolution (see, for example, Ref. [21]).

An exponential, exp(ip·rj ), is subsequently expanded in terms of spherical harmonics as





exp ip · r j = 4







 

ˆ iL jL prj Y∗L,M rˆj YL,M (p),

(6)

L,M

where jL (prj ) is the spherical Bessel function of order L. Substitution of Eq. (6) into Eq. (5) yields

˛

(p) = 2p (p)

6 

1 



cj exp ip · R j

with



2p (p) = i

2 ␲

ˆ aj,m Y1,m (p),

(7)

m=−1

j=1

 j1 (pr) 2p (r) r 2 dr.

(8)

From Eqs. (1), (2), and (7), it follows that EMS (p) = 2p (p) S˛ d˛









exp ip · R jk

cj ck

j,k



 

 





∗ ˆ Y1,m ˆ aj,m a∗k,m Y1,m p  p

dp ,

(9)

m,m

where Rjk = Rj − Rk and  2p (p) = 4␲3 (p1 p2 /p0 )fee |2p (p)|2 , which corresponds to the EMS cross section for the 2p AO of an isolated F atom. After expanding the exponential, exp(ip·Rjk ), in terms of spherical harmonics, the integration over ˝p is performed using the following relations:

 

 

ˆ = (−1)m Yl,−m p ˆ , Y∗l,m p

(10)

 3.2. Interference factor

Yl1 ,m1 (ˆ p) Yl2 ,m2 (ˆ p) Yl3 ,m3 (ˆ p) dp



In analysis of bond oscillation we use the following model, which was originally developed to examine interference patterns in CF4 [11]. Since the five outermost MOs of SF6 consist of the F 2p AOs, they each can approximately be described as

˛

(r) =

6 



1   

cj 2p rj

 

aj,m Y1,m rˆ j

,

(4)

=

(2l1 + 1) (2l2 + 1) (2l3 + 1) 4

l1

l2

l3

0

0

0



l1

l2

l3

m1

m2

m3

 .

Here the expressions in large parentheses denote 3j symbols [35]. By recognizing that the integral in Eq. (11) is necessarily zero unless l1 + l2 + l3 = even and m1 + m2 + m3 = 0, the EMS cross section can be written as

m=−1

j=1

EMS (p) = 2p (p) S˛ I˛ (p) , where cj ’s are coefficients of the linear combination of AOs expansion, 2p (rj ) is the radial part of the F 2p AO centered at the j-th F atom position Rj , and rj = r − Rj . Y,m denotes spherical harmonics with the total angular momentum quantum number of  and magnetic quantum number of m. Spatial orientation, in the molecular frame, of the j-th 2p AO is defined by a set of coefficients aj ,m ’s, which satisfy |aj,−1 |2 + |aj,0 |2 + |aj,1 |2 = 1. Here aj ,m ’s are taken so that the values of cj ’s become positive. The Dirac-Fourier transform is performed to generate the pspace representations of the molecular orbitals:

with

⎡

I˛ (p) = d˛ h˛ ⎣1 +

6 

cj (2)

3⁄2

 (0)

j=1



exp ip · R j









1   

exp ip · r j 2p rj

 

aj,m Y1,m rˆj

m=−1



(0)





(2)

Cjk j0 pRjk + Cjk j2



⎤  pRjk ⎦ .

(13)

Here h˛ = j cj 2 and Rjk = |Rjk |. The coefficients of the spherical Bessel functions of order 0 and 2 are given by

Cjk =

−



(12)

j= / k

4 cj ck ˛ (p) =

(11)



aj,m a∗k,m 3

 1/4

0 1 1 0

0



0

m 6 

0

1

1

0

m −m



 

m ˆ (−1) Y0,0 R jk

, cj2

j=1

(14) dr. (5)

N. Watanabe et al. / Journal of Electron Spectroscopy and Related Phenomena 209 (2016) 78–86

−4 cj ck





aj,m a∗k,m 3

5/4

m,m

(2)

Cjk =

2 1 1 0

0

 m

0 6 

2 −m

1

1

m

−m

Eqs. (14) and (15) are simplified to =



,



(2) Cjk

=

(16)



ˆ ˆj · R cj ck cos jk − 3 u jk





ˆ ˆk · R u jk

. Then experimental momentum profile for each ionization band has been produced by plotting the area under the corresponding Gaussian curve as a function of p. 4.2. Momentum profiles of SF6

cj 2

j



cj2

j=1

cj ck cos jk



ˆ (−1)m Y2,m−m R jk

,(15)

ˆ = R /|R |. By substituting the expressions of spherical with R jk jk jk ˆ jk and the numerical values of 3j symbols, harmonics in terms of R

(0) Cjk



81

 .

(17)

cj2

j

Here jk is the angle between the orientations of the constituent ˆj and u ˆk are unit vectors 2p AOs located on the F atoms j and k. u codirectional with the orientations of the 2p AOs. Since the function I˛ (p) governs the oscillatory feature of the interference pattern, it is henceforth referred to as the interference factor. 4. Results and discussion 4.1. Binding energy spectra The electronic structure of SF6 has extensively been investigated so far [36–42]. One of the reasons for the abundance of the studies is that the assignments of the valence ionization bands to MOs were somewhat controversial; several theoretical and experimental studies proposed different electronic configurations concerning the 5t1u , 1t2u , and 3eg orbitals [36,37,39]. In the present work, we use the following valence configuration, which has widely been used in recent experimental and theoretical studies [41,42]:

The experimental momentum profiles of the 1t1g , (1t2u + 5t1u ), 3eg , and 1t2g orbitals are shown in Figs. 2(a)–(d). Also depicted in the figures are the above-mentioned, two theoretical calculations. The theoretical momentum profiles calculated at the equilibrium geometry and those with vibrational effects are shown by solid and dashed lines, respectively. For making a comparison between experiment and theory, the sum of the experimental momentum profiles of the four lowest-lying bands (of the five outermost MOs) has been constructed and then height-normalized to the corresponding calculations with vibrational effects at p = 1.3 a.u. The scaling factor thus obtained has subsequently been applied to the individual experimental results so that all the experimental and theoretical momentum profiles share a common intensity scale. It can be seen from Fig. 2 that the equilibrium geometry calculations considerably underestimate the experiment at p < ∼1.0 a.u. for the 1t1g , (1t2u + 5t1u ), and 3eg orbitals. A similar tendency has also been observed by the Hefei (e, 2e) group [15], but higher momentum resolution achieved in the present study allows us to make a more detailed comparison of experiment with theory. It should be noted that the disagreement with experiment is unlikely to

(core)22 (4a1g )2 (3t1u )6 (2eg )4 (5a1g )2 (4t1u )6 (1t2g )6 (3eg )4 {(5t1u )6 (1t2u )6 }(1t1g )6 , where the 5t1u and 1t2u MOs are almost degenerate in energy. Fig. 1 shows examples of experimental binding energy spectra of SF6 , which were constructed by plotting the number of true coincident events as a function of Ebind . The spectra at two specific azimuthal angle differences,  = 1◦ and 10◦ , are depicted to illustrate the momentum dependent variations of the band intensities, together with the -angle integrated spectrum obtained by summing up the spectra at each . Vertical bars indicate the ionization energies reported in the photoelectron spectroscopy (PES) study [41]. In this study we focus our investigation on the four lowest-lying bands at Ebind = 15.7, 17.2, 18.5, and 19.8 eV, all of which are ascribed to ionization from the F 2p nonbonding MOs, as mentioned earlier. For the second band, the associated 1t2u and 5t1u orbitals are too closely spaced in energy to be resolved experimentally, even with high resolution PES, and hence sum of their contributions, (1t2u + 5t1u ), is considered here. To extract the contributions from the 1t1g , (1t2u + 5t1u ), 3eg , and 1t2g ionization, a deconvolution procedure was used. In the procedure a Gaussian curve was assumed for each ionization band, whose energy-position and width were determined by taking into account Franck–Condon widths estimated from the PES study [41] in addition to the instrumental energy resolution. The obtained deconvoluted curves are shown as dashed curves and their sum as a solid curve. A similar fitting procedure has been repeated for a series of binding energy spectra at each

Fig. 1. Binding energy spectra of SF6 . Vertical bars indicate the ionization energies of the outer valence orbitals. The broken curves represent the Gaussian deconvolution functions of the data and the solid curve their sum.

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N. Watanabe et al. / Journal of Electron Spectroscopy and Related Phenomena 209 (2016) 78–86

Fig. 2. Comparison between experimental and theoretical momentum profiles for the (a) 1t1g , (b) (1t2u + 5t1u ), (c) 3eg , and (d) 1t2g orbitals of SF6 . The solid and dashed lines represent the equilibrium geometry and vibrational effects calculations, respectively. The inserted figures are the corresponding theoretical electron density distributions in r-space.

originate in insufficient flexibility of the basis set used, because extension of the basis set from aug-cc-pVTZ to the augmented correlation-consistent polarization valence-quadruple-zeta (augcc-pVQZ) [28] has been found not to appreciably change the equilibrium geometry calculations in Fig. 2. The influence of molecular vibration is a possible source of the observed disagreement. Indeed, it can be seen from Fig. 2(a) that for the 1t1g orbital, the vibrational effects calculation predicts higher intensity than the equilibrium geometry calculation at small p and significantly reduces the deviation from the experimental result. On the other hand, however, there is no noticeable difference between the two kinds of calculations for the (1t2u + 5t1u ), 3eg , and 1t2g orbitals, showing that influence of nuclear motion to their electron momentum profiles is quite small. These observations indicate that except the low p component of the 1t1g momentum profile, one can safely neglect effects of molecular vibration in analyzing the experimental data in terms of bond oscillation. A plausible explanation of the remaining differences between experiment and theory is distorted-wave effect [4,6,43], so the effect is considered in the data analysis for extracting interference patterns, as discussed below. 4.3. Bond oscillation Within our model described in Section 3.2, the electron momentum profiles of the nonbonding MOs can be expressed as a product of interference factor, I˛ (p), and the EMS cross section of the 2p AO,  2p (p). It is thus expected that interference factor can be seen by dividing the experimental momentum profiles by  2p (p). Here, to take into account distorted-wave effects,  2p (p) was calculated with the distorted-wave Born approximation (DWBA) [43] using the Hartree–Fock (HF) wave function reported by Clementi and Roetti [44]. This may be validated by the fact that DWBA calculations with the HF wave functions have shown excellent agreement with experimental momentum profiles for atomic orbitals [45,46]. Figs. 3(a)–(d) show the experimental interference factors of the individual MOs, thus obtained. The equilibrium geometry and

vibrational effects calculations are also presented in the figures in the form of the interference factor, which were generated by dividing their PWIA electron momentum profiles by the F 2p PWIA cross section. A glance at Fig. 3 shows that the experimental results clearly exhibit oscillations, as expected, and that the interference patterns are largely dependent upon the MO shape. For instance, the oscillation of the 1t1g orbital interference pattern is almost out of phase to that of the 1t2g orbital. These observations are supported by the associated PWIA calculations. The clue for understanding the observations lies in the analytical interference factor, I˛ (p), and its explicit expression is thus derived for each nonbonding MO. Here the spatial orientation of the constituent F 2p AOs is considered, since it determines the values of the coefficients involved in I˛ (p), Cjk (0) and Cjk (2) (see Eqs. (16) and (17)). On each F atom there are two 2p AOs which are directed perpendicularly to the corresponding F-S axis. Such 2p AOs have local ␲ symmetry with respect to the F-S axis and are referred to as ␲-type AOs. On the other hand, the remaining F 2p AOs are directed along the F-S axis and have local ␴ symmetry, so they are denoted as ␴-type AOs. Standard techniques of group theory tells one that the ␲-type AOs span T1g + T2g + T1u + T2u whilst the ␴-type AOs do A1g + Eg + T1u . Among them there is only one Eg component and the 3eg orbital can therefore be determined by simply taking a linear combination of the ␴-type AOs so that it belongs to the Eg irreducible representation. Similarly, the 1t1g , 1t2g , and 1t2u orbitals can be derived by constructing associated symmetry-adapted linear combinations of the ␲-type AOs. These symmetry-adapted bases provide the following interference factors: I1t1g (p) = 3h1t1g [1 − j0 (pR1 ) − j2 (pR1 ) − 3j2 (pR2 )] ,

(18a)

I1t2u (p) = 3h1t2u [1 + j0 (pR1 ) + j2 (pR1 ) − 2j0 (pR2 ) − 2j2 (pR2 )] , (18b) I3eg (p) = 2h3eg [1 − j0 (pR1 ) + 2j2 (pR1 ) − 3j2 (pR2 )] ,

(18c)

N. Watanabe et al. / Journal of Electron Spectroscopy and Related Phenomena 209 (2016) 78–86

83

Fig. 3. Comparison of experimental and theoretical interference patterns for the (a) 1t1g , (b) (1t2u + 5t1u ), (c) 3eg , and (d) 1t2g orbitals of SF6 . The inserted figures are the corresponding theoretical electron density distributions in r-space. The figures (e)–(h) depict the associated analytical interference factors with the F-F distances of R1 = 5.90 Bohr and R2 = 4.17 Bohr.

I1t2g (p) = 3h1t2g [1 − j0 (pR1 ) − j2 (pR1 ) + 3j2 (pR2 )] ,



(18d) I5t1u (p) = 3h5t1u



1 + j0 (pR1 ) +

2 1 − x2 2 + x2

 j2 (pR1 )



4 (1 + 2x) 4 (1 − x) + j0 (pR2 ) + j2 (pR2 ) . 2 + x2 2 + x2 with the F-F distances of R1 = 5.90 Bohr and R2 = 4.17 Bohr, which are obtained from the S-F bond length reported in an electron diffraction study, RSF = 2.95 Bohr [47]. On the other hand, the 5t1u orbital is expressed as a linear combination of two kinds of symmetry-adapted bases which consist of the ␲-type and ␴-type AOs respectively, and its interference factor can be written as,

(18e)

Here x (= C␴ /C␲ ) is the ratio of the coefficient of the ␴type component (C␴ ) to that of the ␲-type component (C␲ ). The value of x is estimated to be 0.72 from our DFT calculation.

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Momentum [a.u.] Fig. 5. Contributions of the j2 (pR1 ) and j2 (pR2 ) terms extracted by taking linear combinations of the experimental interference factors for the 1t1g , 3eg , and 1t2g orbitals. Solid and chain lines are fitting curves, 9h j2 (pR1 ) and 18h j2 (pR2 ), which are employed to reproduce the experimental results with R1 , R2 , and h being used as fitting parameters.

Fig. 4. Analytical interference factors for the 3eg and 1t2u orbitals. The curve for the doubly degenerated 3eg orbital is multiplied by 3/2 to compare with that for the triply degenerated 1t2u orbital. The inserted figures are the theoretical electron density distributions in r-space.

Here j0 (x) = sin x/x and j2 (x) = {(3-x2 )sin x − 3x cos x}/x3 were used to derive the above equation. One may notice that at large p, Eq. (19) can be approximated to

⎡

For a comparison with the experiment, I˛ (p)’s are plotted in Figs. 3(e)–(h). Because of large separations in space of the F atoms, the overlap integrals between the 2p AOs should be small. We have thus approximated the proportional factor, h˛ = j cj 2 , as unity due to the normalization condition for each MO. It can be seen from Fig. 3 that I˛ (p)’s derived from group theory considerations are in good overall agreement with the experimental results. This observation indicates that the interference pattern exhibits a characteristic oscillatory structure depending upon the symmetry of the nonbonding MO, and further that by comparing the experimental and analytical interference factors, the symmetries of the MOs associated with the 1st, 2nd, and 4th ionization bands (1t1g , 1t2g , and 3eg orbitals) can unambiguously be determined without relying on implication of ab initio MO calculations. To get further insights into the bond oscillation, we examine how the interference patterns reflect the shapes of the nonbonding MOs. Firstly, we consider the intensity of the oscillatory structure at p = 0, I˛ (0), which is governed by Cjk (0) since j0 (0) = 1 and j2 (0) = 0. It can be seen from Eq. (16) that Cjk (0) is proportional to cos jk with

jk being the angle between the orientations of the j-th and k-th AOs. This implies that I˛ (0) should become larger as the directions of the F 2p AOs are closer to each other. Indeed, as can be seen from Fig. 3(b), the experimental interference factor of the (1t2u + 5t1u ) orbitals shows significant intensity at p ∼ 0, reflecting the fact that all the 2p AOs in the 5t1u orbital point to the same direction. It has been revealed that I˛ (0) carries information about the degree of orientation of the constituent AOs in the molecular frame. Next, we consider the amplitude of the oscillatory structures. As can be seen from Fig. 3, the oscillation amplitude of each MO varies with p in a different way. To illustrate this point more closely, the analytic interference factors of the 3eg and 1t2u orbitals are plotted in Fig. 4 over a wide p range. It is evident from the figure that the oscillation amplitude for the 1t2u orbital decreases with the increase in p much faster than that for the 3eg orbital does. For understanding this observation, we revisit Eq. (13). Substitution of Eqs. (16) and (17) into Eq. (13) yields

⎡



I˛ (p) = d˛ h˛ ⎣1 + 3

j= / k



cj ck  h˛

ˆ ˆj · R u jk



ˆ ˆk · R u jk

   sin pRjk pRjk

I˛ (p) ≈ d˛ h˛ ⎣1 + 3

j= / k

(2)

h˛

ˆ ˆj · R u jk



 ⎤  sin pRjk ˆ ⎦ . (20) ˆk · R u jk pRjk

It immediately tells us that the oscillation amplitude at large p is ˆ and u ˆ multiplied to sin (pR )/(pR ). Since ˆj · R ˆk · R governed by u jk jk jk jk ˆk ) is a unit vector codirectional with the orientation of the j-th ˆj (u u (k-th) AO, the oscillation amplitude at large p reflects the tilt angles of the F 2p AOs with respect to the associated F-F axis. For instance, as can be seen from its theoretical density distribution in r-space, the orientation of the j-th and k-th AOs in the 1t2u orbital is perpenˆ =u ˆ ˆj · R dicular to Rjk for all pairs of j and k, and thus u jk ˆk · R jk = 0. This rationally accounts for the rapid decrease of the oscillation amplitude with p for the 1t2u orbital. On the other hand, for the 3eg orbital, ˆ )’s arise from the ␴-type AOs and as a conˆj · R large values of (u jk sequence, the oscillatory structure exhibits a slow decrease in the amplitude. These arguments conclude that measurements of bond oscillation provide a rare opportunity to experimentally clarify the irreducible representation of the symmetry group of a nonbonding MO and further to experimentally get reliable information as to the spatial orientation of the constituent 2p AOs. Finally, it may be worthwhile to note that bond oscillation reflects the relevant internuclear distance and hence it can render information about the molecular structure also. Indeed, our earlier EMS study on CF4 [11] has demonstrated that the F-F internuclear distance can be determined from the interference pattern. In the present study a similar attempt has been made for a more complicated compound, SF6 . It can be seen from Eqs. (18a)–(18e) that each interference factor consists of R1 - and R2 -dependent terms. For simplicity, contributions from the j2 (pR1 ) and j2 (pR2 ) terms have, respectively, been extracted by taking linear combinations of I˛ (p)’s as follows:

 + Cjk

 cj ck 

−

3 I3e (p) − I1t1g (p) ≈ 9h j2 (pR1 ) , 2 g

(21a)

I1t2g (p) − I1t1g (p) ≈ 18h j2 (pR2 ) .

(21b)



cos pRjk



pRjk





 ⎤ ⎦ .(19) 3

sin pRjk

2 + 

pRjk

N. Watanabe et al. / Journal of Electron Spectroscopy and Related Phenomena 209 (2016) 78–86

Here the proportional factor in I˛ (p) is assumed to be independent of the orbital index (h˛ = h ), because its value should be close to unity due to the reason explained above. The experimental R1 - and R2 -dependent components are presented in Fig. 5. It can be seen from the figure that reflecting the longer F-F internuclear distance, the R1 -dependent term exhibits a shorter period of oscillation than the R2 -dependent term. For a quantitative analysis, 9h j2 (pR1 ) and 18h j2 (pR2 ) are subsequently employed as fitting curves to reproduce the experimental results with R1 , R2 , and h being used as fitting parameters. The data below 0.4 a.u. were not taken into account in the fitting procedure because considerable vibrational effects have been observed for the 1t1g orbital at low momentum. The best fit to the experiment is presented in Fig. 5. The resulting R1 and R2 values of 5.30 Bohr and 3.93 Bohr are found to be in reasonable agreement with R1 = 5.90 Bohr and R2 = 4.17 Bohr reported by electron diffraction [47]. This strongly suggests that molecular structure information can be obtained from bond oscillation even for molecules having several different internuclear distances. 5. Conclusion In this study we have performed the EMS experiment for the five outermost MOs of SF6 , which are each constructed from the 2p AOs on the F atoms. The associated theoretical calculations have also been carried out, while vibrational effects being involved. Comparisons between the experiment and theory have shown that, except the low p component of the 1t1g electron momentum profile, influence of molecular vibration is quite small and hence it can be neglected in discussing bond oscillation. Interference patterns for each MO have then been obtained by dividing the measured electron momentum profiles by DWBA cross section for an isolated F 2p AO. It has been shown that the interference patterns are largely dependent upon the shapes of the nonbonding MOs and that their overall features are well accounted for by our model. Further analysis has shown that they allow one to experimentally clarify the symmetries of the nonbonding MOs and also that they provide information about the degree of orientation of the 2p AOs as well as about the tilt angles of the AOs with respect to the internuclear directions. In addition, it has been demonstrated that the molecular structure information can also be extracted from the interference patterns even for this relatively complicated molecule having two different F-F distances. Acknowledgements This research was supported by the Grant-in-Aids for Scientific Research (S) (No. 20225001) and (A) (No. 25248002), and Grant-inAid for Challenging Exploratory Research (No. 25620006) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] I.E. McCarthy, E. Weigold, (e, 2e) spectroscopy, Phys. Rep. 27 (1976) 275. [2] C.E. Brion, Looking at orbitals in the laboratory: the experimental investigation of molecular wavefunctions and binding energies by electron momentum spectroscopy, Int. J. Quantum Chem. 29 (1986) 1397. [3] K.T. Leung, Experimental momentum-space chemistry by (e, 2e) spectroscopy, in: Z.B. Maksic (Ed.), Theoretical Models of Chemical Bonding Part 3, Springer-Verlag, Berlin, 1991, pp. 339–386. [4] I.E. McCarthy, E. Weigold, Electron momentum spectroscopy of atoms and molecules, Rep. Prog. Phys. 54 (1991) 789. [5] M.A. Coplan, J.H. Moore, J.P. Doering, (e, 2e) spectroscopy, Rev. Mod. Phys. 66 (1994) 985. [6] E. Weigold, I.E. McCarthy, Electron Momentum Spectroscopy, Kluwer Academic/Plenum Publishers, New York, 1999. [7] M. Takahashi, Looking at molecular orbitals in three-dimensional form: from dream to reality, Bull. Chem. Soc. Jpn. 82 (2009) 751.

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