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Anharmonic Franck−Condon Factors for the X̃ 2B1 ← X̃ 1A1 Photoionization of Ketene Published as part of The Journal of Physical Chemistry A virtual special issue “Spectroscopy and Dynamics of Medium-Sized Molecules and Clusters: Theory, Experiment, and Applications”. Guntram Rauhut* University of Stuttgart, Institute for Theoretical Chemistry, Pfaffenwaldring 55, 70569 Stuttgart, Germany Downloaded via KAOHSIUNG MEDICAL UNIV on November 21, 2018 at 18:36:25 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

S Supporting Information *

ABSTRACT: The X̃ 2B1 ← X̃ 1A1 photoelectron spectra of ketene and its doubly deuterated isotopologue have been computed from correlated vibrational wave functions as determined from vibrational configuration interaction theory relying on multidimensional Born−Oppenheimer potential energy surfaces being obtained from explicitly correlated coupled-cluster calculations. Duschinsky effects were accounted for in all cases. Excellent agreement with available experimental data was achieved.

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state. (4) In 1992 Takeshita14 calculated selected harmonic vibrational frequencies for different ionized states of ketene and its deuterated analog by means of quantum-chemical CISD calculations. He also provided the first simulations for the photoelectron spectra of these species. (5) A few years later, Szalay et al.15 determined geometrical parameters and rotational constants for the ketene radical cation, and (6) Nooijen16 simulated the X̃ 2B1 ← X̃ 1A1 photoelectron spectrum by relying on the Hessian vertical Franck−Condon approximation. Consequently, a considerable amount of experimental and computational data is available for the lowest electronic state of the radical cations of these species. Although many data are available for the radical cation of ketene, accurate vibrational studies based on high-level anharmonic potential energy surfaces are still missing. In this study, ab initio calculations at the explicitly correlated coupledcluster level, i.e., UCCSD(T)-F12a,17 were used to determine multidimensional potential energy surfaces, which allow for the accurate calculation of vibrationally averaged molecular properties and the infrared and photoelectron spectra of these species. As the potential energy surfaces are spanned by mass weighted normal coordinates, the calculation for the deuterated molecule would in principle request an independent determination of the potential energy surface (PES) being represented by a multimode expansion.18 However, this would double the computational effort, which is quite substantial at this level of theory. Therefore, the generalized Duschinsky transformation has been used for converting the PES into the correct set of normal coordinates. For details see ref 19. To the best of my

INTRODUCTION Certainly, ketene is among the best studied molecules. Comprehensive experimental and theoretical surveys summarize a vast amount of data for this small molecule.1−5 For example, as early as in 1956 Quadbeck6 wrote an overview article about the importance of ketene in synthetic organic chemistry. From the theoretical point of view, East et al.2 presented high-level ab initio calculations about the vibrational properties of this system, which are in excellent agreement with experimental results. However, it is not the intention of this study here to summarize all the work focusing on ketene in its electronic ground state. Concerning the radical cation of ketene, significantly less has been done. Nevertheless, even this system has been subject to several detailed investigations. For example, Hall et al.7 studied the electronic states of the radical cation of ketene. Bouma et al.8 investigated the potential energy surface of H2CCO+•, and Orlova et al.9 were interested in the water-catalyzed hydrolysis of the radical cation of ketene, whereas Heinrich et al.10 had a detailed look at the reaction of H 2 CCO +• with ethylene. For sure, this list is not comprehensive. As this contribution here focuses on the photoionization of ketene to its radical cation in the lowest electronic state, six papers are of particular importance for this study: (1) Shirley and co-workers11 measured vibrationally resolved photoionization spectra of H2CCO and D2CCO and assigned the most important progressions in these spectra. (2) Wang et al.12 used zero kinetic energy (ZEKE) photoelectron spectroscopy to determine the vibrational frequencies of H2CCO+• and D2CCO+• in the X̃ 2B1 state, and (3) Merkt and co-workers13 employed rotationally resolved pulsed-fieldionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectroscopy to determine the vibrational frequencies and the rotational constants of H2CCO+• and D2CCO+• in the X̃ 2B1 © 2015 American Chemical Society

Received: July 17, 2015 Revised: September 29, 2015 Published: September 29, 2015 10264

DOI: 10.1021/acs.jpca.5b06922 J. Phys. Chem. A 2015, 119, 10264−10271

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mode wave functions (modals) φi were determined from 20 mode-dependent distributed Gaussians χμi per degree of freedom, i.e., φi = ∑μiCiμi χμi. All calculations were performed in a state-specific configuration selective manner as described in detail elsewhere.31 Vibrational angular momentum terms have been accounted for on the basis of a constant μ-tensor; i.e., the inverse of the effective moment of inertia tensor within the Watson Hamiltonian has been truncated after the zeroth-order term, which was found to be sufficient for very many applications.32 Up to quadruple excitations have been considered in the correlation space of the VCI calculations. For the doubly deuterated isotopologues, the generalized Duschinsky transformation has been used for rotating the potentials of the parent compounds, i.e., H2CCO and H2CCO+•:

knowledge, this route has never been followed up for the calculation of photoelectron spectra. Vibration configuration interaction calculations (VCI) close to the full-CI limit were used to determine high-level vibrational wave functions and thus very accurate Franck−Condon factors.20 A comparison with avaliable experimental data is attempted whenever possible.

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COMPUTATIONAL DETAILS In the present work, all calculations were performed with a development version of the MOLPRO 2099.9 package of ab initio programs.21 The equilibrium geometries of ketene and its radical cation were determined at the (U)CCSD(T)-F12a level using the cc-pVTZ-F12 basis set of Peterson et al.22 The frozen core approximation has been used in all calculations, except within single-point energy calculations for the equilibrium structures of both molecules, which were performed at the all electron (U)CCSD(T)-F12a/cc-pCVQZ-F12 level. The DFMP2-F12 calculations, which are the first step in (U)CCSD(T)-F12a calculations, were performed using the DF-MP2F12/3C(FIX) method described in detail in refs 23 and 24. In this method the numerous two-electron integrals are computed using robust density fitting (DF) approximations25,26 (DF was not used in the Hartree−Fock and (U)CCSD(T)-F12 calculations). The aug-cc-pVnZ/MP2FIT basis sets of Weigend et al.27 were used as auxiliary bases, where n corresponds to the cardinal number of the orbital basis sets. For evaluating the Fock matrix (which is needed in the AO and RI basis sets) the cc-pVnZ/JKFIT basis sets27 were used. Unless otherwise noted, the same basis sets27 were also used for the resolution of the identity (RI); previous work has shown that these are well suited for this purpose.23 The complementary auxiliary basis set (CABS) approach was employed; i.e., the union of the AO and RI basis sets was used to approximate the resolution of the identity. The Hartree−Fock reference energies were improved by addition of the complementary auxiliary basis set (CABS) singles correction. Potential energy surfaces were represented by multimode expansions being truncated after the three-mode (3D) coupling terms.18,28 A multilevel scheme has been used, in which the 1D and 2D terms were evaluated at the CCSD(T)-F12a/cc-pVTZF12 level, whereas a smaller cc-pVDZ-F12 basis has been used for the 3D terms.29 A total of 24 grid points per vibrational degree of freedom have been used within a grid representation of the potential, which has been transformed to a polynomial representation subsequently. Polynomials up to eighth order were found to be sufficient for representing the potential. Thus, the potential is given as 3N − 6

V (q ⃗ ) =

8

⎛

∑ ∑ qir ⎜⎜pr(i) + i

r

⎝

⎛ 1 × ⎜⎜prs(ij) + 3 ⎝

3N − 6

1 2 8

∑∑ k

t

3N − 6

t −1/2 q ⃗′ =  L′ M′1/2 TM L q ⃗ + L′t M′1/2 (T re⃗ − re′⃗ + t ⃗)    S

8 s

⎞⎞

(ijk) ⎟⎟ qkt prst ⎟⎟

⎠⎠

(2)

L and L′ are the eigenvectors of the Hessians of the deuterated and the parent compound, and M and M′ denote the diagonal matrices of the nuclear masses. T is the Eckart transformation matrix, which reduces to a unit matrix in this particular application. re⃗ and r′e⃗ denote the equilibrium structure of the molecules in the molecule fixed coordinate system, and t ⃗ denotes the shift of the centers of mass, which occurs only formally as it does not contribute in calculations for either vertical excitations or isotopic substitutions. The second term of this equation vanishes once it is used for the transformation between isotopologues, i.e., d⃗ = 0⃗, because the different masses have no impact on the relative positions of the atoms of the equilibrium structures re⃗ and r′e⃗ and thus (Tre⃗ − r′e⃗ ) = 0⃗. It is noted that the Duschinsky matrix S is not a unitary matrix when M ≠ M′. Moreover, unlike the case for vertical electronic excitations, the Duschinsky matrix formally contains a nonvanishing vibration−rotation coupling block, which has been neglected in the present study.19 Therefore, the Duschinsky transformation used here is just approximate.19 Another important approximation arises from the fact that in multimode expansions of the PES high-order coupling terms may be rotated into the important low-order couplings. As the potential has been truncated after the three-mode coupling terms (3D) already, this formally might lead to non-negligible deviations. However, as will be shown below, the agreement of the frequencies of the deuterated species with experimental data is excellent and thus the introduced approximations appear to be appropriate and the deviations negligible. The Duschinsky matrices according to eq 2 for the transformation of PESs concerning isotopic substitutions are provided in the Supporting Information. The determinants of these matrices are as low as 0.217. These matrices show that rather strong mixing occurs between the modes of the deuterated system and its parent compound. A detailed discussion of the transformation of multidimensional potential energy surfaces for isotopic substitutions is provided in ref 19. For the calculation of the Franck−Condon factors a standard Duschinsky rotation, i.e., a unitary S matrix and a nonvanishing shift-vector d⃗, has been used.33 It shall be noted here that Duschinsky effects have also been accounted for by other authors within the determination of Franck−Condon factors in many anharmonic calculations not only for triatomic systems but also for more extended systems.34,35 The Duschinsky matrices for the vertical excitations are also provided in the

∑ ∑ qjs j

d⃗

(1)

where q denotes the normal coordinates and p the polynomial coefficients. Vibrational frequencies and vibrationally averaged molecular properties were determined from vibrational selfconsistent field (VSCF) calculations and a correlation treatment by vibrational configuration interaction (VCI) calculations based on the Watson Hamiltonian for polyatomic nonrotating molecules.28,30,31 Within the VSCF program, one 10265

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calculations, Csaszar and co-workers36 have shown for the water molecule that VARCs can be estimated by the expectation values of the diagonal elements of the μ-tensor as occurring in the Watson Hamiltonian,30 i.e., the inverse of the effective moment of inertia tensor. However, this approximation neglects Coriolis coupling contributions entirely. Therefore, the standard perturbational correction has been added to this approximation to yield improved results. Thus, the VARCs are approximately given by

Supporting Information. Even these matrices show nonnegligible off-diagonal elements, although less pronounced than in the case of the isotopic substitution. Therefore, for accurate calculations of Franck−Condon factors Duschinsky effects may not be neglected (see below). Consequently a 2fold Duschinsky transformation has been performed within the calculation of the FCFs for the deuterated species. Important transitions were selected on the basis of initial FCFs at the VSCF level, which hardly needs any time.20 Excitations up to the sixth root in up to four modes at the same time have been considered, which led to a sum of FCFs being larger than 0.9999. From these the most important have been selected, which were subsequently computed by state-specific configuration-selective VCI calculations with configuration spaces including quadruple excitations and accounting for vibrational angular momentum terms.20

Bνα ≈

⟨μαα ⟩ν 2

+

∑ kl

(ζkl α)2 (3ωk 2 + ωl 2) ⎛ 1 ⎞⎟ ⎜ν + k 2 2 ⎝ 2⎠ ωk − ωl

(3)

The second approach is based on standard vibrational perturbation theory (VPT2). In these calculations the underlying semiquartic force field has been retrieved from the polynomial representation of the PES within the multimode expansion (see ref 37). Results for H2CCO+• and D2CCO+• obtained from both approaches are listed in Table 2. Both sets of calculations yield very similar results, which are in good agreement with the available experimental data. Please note that only averaged values were given for B0 and C0 in the work of Merkt and co-workers13 and Wang et al.12 The computed rotational constant for the equilibrium geometry of Szalay et al.15 are consistently lower than the values reported here, which is a result of their lower computational level. Vibrational Frequencies. As accurately computed anharmonic frequencies for closed-shell ketene have been provided by East et al.,2 this study will exclusively focus on the frequencies for the ionized species. However, it shall be noted that the agreement of the anharmonic vibrational frequencies for deuterated ketene, which have been retrieved from the PES of the parent compound, with experimental data is excellent (see the Supporting Information). A mean deviation of just 1.4 cm−1 shows that the generalized Duschinsky transformation being applied to the PES of H2CCO does not lead to any significant deviations. The A1 frequencies of H2CCO+• and D2CCO+• have also been computed by Takeshita14 within the harmonic approximation. However, due to the rather large deviations from the accurate UCCSD(T)-F12a values, these results will not be discussed in detail. VCI and VPT2 results for H2CCO+• and D2CCO+• are listed in Table 3. Within the VPT2 calculations the cubic and quartic force constants have been retrieved from the polynomial representation of the potential and not by differentiation of the electronic energy. Therefore, the differences between the VCI and VPT2 results arise solely from the truncation of the potential within the semiquartic force field and the differences in the variational and perturbational methodologies. For H2CCO+• the computed values are in nice agreement with the experimental MATI and PFI-ZEKE data except for the fundamental and overtone of ν5 (CH2 wagging) and the overtone of ν8. Most interestingly, the

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RESULTS AND DISCUSSION Geometrical Parameters. Photoionization of ketene leads to a singly occupied CC π-orbital, which is depicted in Figure 1. As a consequence, one must expect a markedly longer CC

Figure 1. Singly occupied molecular orbital of the X̃ 2B1 ketene radical cation.

bond in the electronic ground state of the radical cation than in the neutral closed-shell species. A comparison of the values obtained here, which are listed in Table 1, shows that this indeed can be observed. On the contrary, the CO bond in H2CCO+• is slightly shorter than in the parent compound. The results for neutral ketene are in excellent agreement with the very accurate re results of East et al.2 Vibrational averaging based on VCI wave functions leads to lengthening of all bonds (r0 vs re). Our re parameters are in qualitative agreement with the results of Szalay et al.15 and Takeshita,14 who obtained an even longer CC bond of 1.408 and 1.406 Å, respectively, at lower computational levels. Isotopic substitution has little impact on the vibrationally averaged parameters except for the CH bond lengths. Rotational Constants. For calculating vibrationally averaged rotational constants (VARCs), two strategies have been followed up. Within the framework of variational VCI

Table 1. Geometrical Parameters (Å and deg) of H2CCO, D2CCO, H2CCO+•, and D2CCO+• in the X̃ 1A1 and X̃ 2B1 States Obtained from (U)CCSD(T)-F12a/cc-pVTZ-F12 and VCI Calculations H2CCO r(CC) r(CO) r(CH) ∠(CCH)

H2CCO+•

D2CCO

D2CCO+•

re

r0

r0

re

r0

r0

1.3154 1.1629 1.0773 119.02

1.3195 1.1644 1.0800 118.80

1.3193 1.1645 1.0793 118.87

1.3906 1.1249 1.0847 117.99

1.3951 1.1259 1.0909 117.81

1.3952 1.1258 1.0884 117.81

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Table 2. VCI and VPT2 Rotational Constants (cm−1) of H2CCO+• and D2CCO+• in the X̃ 2B1 State Obtained from VCI and VPT2 Calculations Relying on UCCSD(T)-F12a/cc-pVTZ-F12 Calculations H2CCO+•

a

RC

VCI

VPT2

Ae Be Ce

9.116 0.333 0.322

9.116 0.333 0.322

A0 B0 C0

9.057 0.333 0.320

9.060 0.333 0.320

exp

D2CCO+• a

b

exp

9.14(4) 0.339(6)c 0.339(6)c

9.10(4) 0.335(9)c 0.335(9)c

VCI

VPT2

4.561 0.295 0.277

4.561 0.295 0.277

4.539 0.295 0.276

4.542 0.295 0.276

expa

expb

4.60(8) 0.270(18)c 0.270(18)c

4.61(6) 0.293(7)c 0.293(7)c

Experimental data taken from ref 12. bExperimental data taken from ref 13. c(B0 + C0)/2.

Table 3. Vibrational VCI and VPT2 Frequencies of H2CCO+• and D2CCO+• (All Quantities in cm−1) H2CCO+•

a

νi

sym

harm.

VCI

VPT2

ν11 ν12 ν13 ν14 ν15 ν16 ν17 ν18 ν19

A1 A1 A1 A1 B1 B1 B2 B2 B2

3123.5 2264.2 1391.7 1024.8 729.2 466.4 3261.3 1025.5 403.4

2994.3 2219.8 1347.2 1017.0 726.5 468.8 3119.0 1009.5 406.1

2998.4 2224.7 1343.5 1022.1 724.0 468.6 3113.3 1008.8 407.2

ν22 ν23 ν24 ν25 ν26 ν28 ν29

A1 A1 A1 A1 A1 A1 A1

4528.4 2783.4 2049.6 1458.4 932.8 2051.0 806.8

4419.4 2694.8 1971.7 1471.3 926.9 2017.2 812.8

4420.1 2676.8 1990.3 1462.9 926.7 2015.6 812.0

D2CCO+• harm.

VCI

VPT2

expa

expb

2232.1 2200.2 1094.2 914.6 594.5 437.3 2358.9 850.2 356.3

2241.5 2202.4 1097.3 913.2 592.8 437.3 2356.4 849.0 356.8

2243(2)

2241.3(1.0)

1094(2) 916(2) 586(5) 447(5)

1096.1(1.5) 916.3(2.0)

403(3)

2268.5 2251.6 1110.6 910.8 595.8 439.1 2438.1 858.7 355.3

1477(3) 930(10) 2029(3) 819(10)

4503.2 2221.2 1821.6 1191.6 878.2 1717.4 710.6

4390.3 2126.0 1775.9 1196.8 852.2 1701.0 713.1

4379.4 2132.4 1770.7 1190.1 851.9 1698.1 712.4

exp

a

2227(2) 1350(3) 1020(2) 745(3) 470(3)

exp

b

2226.5(1.2) 1351.4(2.0) 1021.4(1.2)

866(2) 359(10)

[2207(2)] 1190(10) 852(2)

Experimental data taken from ref 12. bExperimental data taken from ref 13.

ν5 fundamental deviates at the VCI level by 18.5 cm−1 from the experimental value, whereas the corresponding overtone is in nice agreement with the MATI result. Because the 2-fold of the VCI fundamental would be below the computed overtone, both the VCI and the VPT2 result for this particular mode are not considered to be reliable. This might arise from regions of the PES with strong multireference character, which have been encountered within the generation of the PES. However, in all cases, the Hartree−Fock and the coupled-cluster iterations converged smoothly. Likewise, as the harmonic frequency of ν5 is already far below the experimental value (cf. Table 3), the origin of this problem appears to be in the electronic structure calculations rather than in the vibrational parts. For D2CCO+• all vibrational frequencies are in nice agreement with the experimental data, except the fundamental of ν8. This might arise from the rotation of the potential as the ν5 mode is now in good agreement with the experimental value. Note that the harmonic frequency of ν8 again is below the experimental value, which indicates problems in the PES arising from the electronic structure calculations. According to the MATI results the overtone of ν3 of D2CCO+• is higher in energy than the 2-fold of its fundamental. However, its signal in the MATI spectrum is quite strong. An analysis of the VCI results for the overtone of ν3 reveals strong Fermi resonances between this overtone and the fundamentals of ν1 and ν2. This leads to transition energies of 2126.0, 2200.2, and 2232.1 cm−1. The first one shows a

contribution of 42% of the VSCF configuration to the ν3 overtone, whereas the other two have contributions of just 16% and 18%, respectively. Therefore, the computed VCI overtone has been assigned to the first band, which, in addition, is in accord with the VPT2 results. Most likely, the band seen at 2207 cm−1 by Wang et al. is not the overtone of ν3, but the fundamental of ν2, which would not only explain its strong intensity but also resolve the dilemma of a positive anharmonicity for the overtone of ν3. Strong resonances were also found for the overtone of ν2, for which the leading VCI coefficient is as low as 0.57, i.e., a contribution of just 33%. All this indicates that the calculation of the anharmonic frequencies for the X̃ 2B1 state of the ketene radical cation is significantly more complex than for the neutral H2CCO or D2CCO. Franck−Condon Factors and Photoelectron Spectra. The computed photoelectron spectrum for the X̃ 2B1 ← X̃ 1A1 transition of ketene along with a copy of the spectrum recorded by Shirley and co-workers11 is shown in Figure 2. A comparison of both spectra reveals excellent agreement. It shall be noted that the computed spectrum has not been shifted to match the 0−0′ transition. Concerning the 0−0′ transition, Wang et al.12 determined the ionization potential from ZEKE spectra to be 77539 ± 2 cm−1, whereas Shirley and co-workers provide a value for the 0−0′ transition of 77583 ± 4 cm−1.11 The VCI calculations performed here yield a slightly higher value of 77618 cm−1. The remaining difference must be addressed to 10267

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Table 4. Selected VCI Franck−Condon Factors of the X̃ 2B1 ← X̃ 1A1 Photoionization of Ketene state 0̃ 62 41 31 52 42 21 3162 3141 43 2141 3142 2131 3143 22 2241 2231 2341

%a

FCF

ν̃ − ν̃0b

ν̃− ν0c

99 84 83 87 87 48 92 51 67 53 73 42 79 33 84 53 34 33

4.062e−01 6.669e−03 1.357e−01 6.704e−03 6.597e−03 4.419e−03 1.987e−01 5.161e−03 8.805e−03 5.856e−05 7.280e−02 3.336e−03 3.720e−03 2.010e−04 4.469e−02 1.390e−02 5.265e−04 1.327e−03

0.0 926.9 1017.0 1347.2 1471.3 1971.7 2219.8 2287.6 2370.4 2964.0 3246.7 3332.8 3566.5 4327.5 4419.4 5450.5 5763.0 7631.8

9.62 9.74 9.75 9.79 9.81 9.87 9.90 9.91 9.92 9.99 10.03 10.04 10.07 10.16 10.17 10.30 10.34 10.57

a

Contribution of the leading configuration to the VCI state. bRelative energy in cm−1 in the ionized state. cExcitation energy in eV with respect to the vibrational ground state of the singlet state.

Figure 2. Experimental (top) and computed (bottom) photoelectron spectrum for the X̃ 2B1 ← X̃ 1A1 transition of ketene, the latter being obtained from vibrational configuration interaction calculations (convolution by Lorentzians with a fwhm of 8 meV). The experimental spectrum has been reprinted with permission from B. Niu, Y. Bai, and D. A. Shirley, J. Chem. Phys. 1993, 99, 2520. Copyright 2015, AIP Publishing LCC.

The experimental and computed photoelectron spectra of deuterated ketene are shown in Figure 3. As outlined above, the spectrum of the deuterated compound relies on a 2-fold Duschinsky rotation and has been obtained from the potential energy surfaces of the nondeuterated parent compound. The first Duschinsky-like rotation accounts for the different masses,

deficiencies in the electronic structure calculations mainly arising from missing high-order coupled-cluster terms, relativistic effects, spin−orbit coupling in the calculation of the ionized molecule, and diagonal Born−Oppenheimer corrections. The reported value for the adiabatic excitation energy of Szalay et al. of 78070 cm−1 is significantly higher, and the 0−0′ value of Nooijen16 as deduced from his simulated photoelectron spectrum differs by more than 1500 cm−1. The most important Franck−Condon factors are summarized in Table 4. Most interestingly, Shirley and co-workers11 identified two progressions with strong intensity, namely 2n and 2n41, and two progressions with weak intensities, 2n31 and 2n42. Consequently, all intensities arise from progressions involving the CO stretching mode ν2. Nooijen16 confirmed the assignment of the first three progressions, whereas Takeshita did not identify the 2n31 progression, but all others. The last one of Shirley progressions, i.e., 2n42, was computed to be very low in intensity and it decays rather rapidly. Another progression, 314n was found to be stronger in intensity (cf. Table 4) . Besides that, the 4n progression, which has also been identified by Takeshita, shows non-negligible contributions to the spectrum. Consequently, not all progressions include the CO stretching mode. According to the data presented in Table 4 the very weak peaks in the spectrum, which arise from FCFs being smaller than 0.01, are due to the first elements of a multitude of weak and/or fast decaying progressions arising from the modes mentioned above rather than further progressions of mode ν2. Luckily, mode ν5, the CH2 wagging mode, which showed a slightly larger deviation from the experimentally determined frequency (cf. 3), has no impact on the photoelectron spectrum.

Figure 3. Experimental (top) and computed (bottom) photoelectron spectrum for the X̃ 2B1 ← X̃ 1A1 transition of doubly deuterated ketene, the latter being obtained from vibrational configuration interaction calculations (convolution by Lorentzians with a fwhm of 8 meV). The experimental spectrum has been reprinted with permission from B. Niu, Y. Bai, and D. A. Shirley, J. Chem. Phys. 1993, 99, 2520. Copyright 2015, AIP Publishing LCC. 10268

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the two spectra not only arise from a shift of the frequencies but also mainly from altered intensities of the individual transitions. A comparison of the computed and the experimental spectra for deuterated ketene shows that the intensities are not as nicely reproduced as for nondeuterated ketene. The two split peaks are slightly too low in intensity and the peaks at high excitation energies (i.e., >10.2 eV) can hardly be seen in the calculated spectrum. This is due to the fact that obviously not all important transitions have been included in the calculations. As a consequence, the sum over all FCFs at the VCI level being considered in the simulation of the spectrum is as low as 0.896, whereas it is 0.967 for the nondeuterated species. Note that the neglect of Duschinsky effects would even lower these numbers. This is due to the very low contributions of the leading configurations in the VCI vectors (as shown in Table 6). As noted in the Computational Details, the selection of important FCFs is based on the underlying VSCF FCFs. However, important VSCF configurations might result in unimportant VCI FCFs once the corresponding VCI state is dominated by many other configurations rather than the leading VSCF configuration. As a consequence, important VCI FCFs might be missing and, in addition, the picking of the correct state within the VCI calculations becomes an errorprone task and will lead to a further deterioration of the spectrum. Note that the information about the leading configuration given in the first columns of Tables 4 and 6 is of limited value once its contribution is very low. As the leading configurations for deuterated ketene decrease rapidly for the high-lying states, a lower quality of the spectrum must be anticipated in the high-energy region in comparison to the nondeuterated species. Nevertheless, even for deuterated molecule, the agreement with the experimental spectrum must still be considered very good.

whereas the second one pays respect to the different sets of normal coordinates in the neutral and the ionized systems. The account of the second Duschinsky transformation has nonnegligible impact on the magnitude of the FCFs. Table 5 shows Table 5. Franck−Condon Factors of the X̃ 2B1 ← X̃ 1A1 Photoionization of Doubly Deuterated Ketene with and without the Inclusion of Duschinsky Effects state

FCF (with DE)a

FCF (w/o DE)b

ν̃ − ν̃0b

31 32 33

8.694e−02 3.396e−04 1.181e−03

6.883e−02 6.062e−06 1.431e−03

1094.2 2126.2 3165.0

a

VCI FCFs including Duschinsky effects. bVCI FCFs without Duschinsky effects. bRelative energy in cm−1 in the ionized state.

the FCFs of the 3n progression for the deuterated system with and without Duschinsky effects, respectively. This example shows that a proper account of Duschinsky effects is mandatory once quantitative agreement is aimed at. For qualitative agreement it might be sufficient to neglect Duschinsky effects, but this depends strongly on the system as has been shown recently.20 The most obvious difference between the spectra of deuterated and nondeuterated ketene is the occurrence of the two split peaks at ∼9.75 and ∼10.02 eV. An investigation of the FCFs, as listed in Table 6, shows that the transitions into the 62 Table 6. Selected VCI Franck−Condon Factors of the X̃ 2B1 ← X̃ 1A1 Photoionization of Doubly Deuterated Ketene state 0̃ 62 41 31 64 21 11 2141 1141 2131 1131 22 1121 12 1132 2241 2231 23

%a

FCF

ν̃ − ν̃0b

ν̃ − ν0c

99 58 58 78 34 67 67 40 33 44 44 33 19 18 21 9 10 7

4.006e−01 1.356e−02 4.459e−02 8.693e−02 1.320e−03 4.806e−02 1.656e−01 4.864e−03 1.619e−02 5.685e−03 2.867e−02 1.940e−03 1.152e−02 2.214e−02 7.126e−03 1.271e−04 3.176e−04 1.206e−04

0.0 852.2 914.6 1094.2 1856.2 2200.2 2232.1 3108.9 3137.9 3293.9 3339.7 4390.3 4402.7 4432.2 4471.0 5311.0 5492.1 6564.9

9.62 9.73 9.74 9.76 9.85 9.90 9.90 10.01 10.01 10.03 10.04 10.17 10.17 10.17 10.18 10.28 10.30 10.44

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SUMMARY AND CONCLUSIONS The X̃ 2B1 ← X̃ 1A1 photoelectron spectra of ketene and its doubly deuterated isotopologue have been computed from vibrational wave functions being determined from VCI calculations relying on anharmonic potential energy surfaces obtained from explicitly correlated coupled-cluster theory. Rather than performing independent calculations for the deuterated species, the corresponding multidimensional potential energy surfaces have been obtained by exploiting the generalized Duschinsky transformation. The agreement with experimental spectra is very good although small deviations still arise from deficiencies in the calculations as for example missing contributions in the electronic structure calculations, which lead to a slight shift of the entire spectrum, or the selection of important FCFs based on VSCF FCFs, which results in missing contributions to the spectrum in highenergy regions, which usually go along with high state densities. The assignment of the dominant peaks in the photoionization spectra agrees with those of Shirley. However, some of the progressions with very low intensities in the spectra as provided by Shirley and co-workers11 could not be found and alternative assignments for the weak peaks in the spectra has been given. The progressions identified for D2CCO+• differ to some extent from the analysis of Takeshita. Except for some very few outliers, the agreement of the fundamental modes for H2CCO+• and D2CCO+• are in nice agreement with the experimental values of Wang et al.12 and Merkt and coworkers.13 A number of experimentally unseen transitions has been provided.

a

Contribution of the leading configuration to the VCI state. bRelative energy in cm−1 in the ionized state. cExcitation energy in eV with respect to the vibrational ground state of the singlet state.

and 31 states have gained significantly more intensity. Shirley and co-workers identified five progressions, namely 2n, 2n41, 2231, 2n4131, and 2n32, from which the analysis provided here confirms the first three. According to the VCI FCFs listed in Table 6, further progressions are 62n, 1n, and 113n, and thus an overtone progression of the B1 mode ν6 also contributes to the spectrum. Most interestingly, Takeshita’s analysis did not provide progressions involving mode ν2. The peak at about 10.145 eV gains considerable intensity from transitions into states involving mode 1. Consequently, the differences between 10269

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The Journal of Physical Chemistry A

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electron Investigation of the X̃ 2B1 Ground State of CH2CO+ and CD2CO+. J. Chem. Phys. 2002, 117, 6546−6555. (13) Willitsch, S.; Haldi, A.; Merkt, F. Rovibrational Energy Level Structure of the 2iB1 Ground Electronic State of CH2CO+ and CD2CO+. Chem. Phys. Lett. 2002, 353, 167−177. (14) Takeshita, K. A. Theoretical Study on the Ionic States with Analysis of Vibrational Levels of the Photoelectron Spectrum of Ketene (C2H2O and C2D2O). J. Chem. Phys. 1992, 96, 1199−1209. (15) Szalay, P. G.; Csaszar, A. G.; Nemes, L. Electronic States of Ketene. J. Chem. Phys. 1996, 105, 1034−1045. (16) Nooijen, M. First-Principles Simulation of the UV Absorption Spectrum of Ketene. Int. J. Quantum Chem. 2003, 95, 768−783. (17) Knizia, G.; Adler, T. B.; Werner, H.-J. Simplified CCSD(T)-F12 Methods: Theory and Benchmarks. J. Chem. Phys. 2009, 130, 054104. (18) Bowman, J. M.; Carrington, T.; Meyer, H.-D. Variational Quantum Approaches for Computing Vibrational Energies of Polyatomic Molecules. Mol. Phys. 2008, 106, 2145−2182. (19) Meier, P.; Oschetzki, D.; Berger, R.; Rauhut, G. Transformation of Potential Energy Surfaces for Estimating Isotopic Shifts in Anharmonic Vibrational Frequency Calculations. J. Chem. Phys. 2014, 140, 184111. (20) Meier, P.; Rauhut, G. Comparison of Methods for Calculating Franck-Condon Factors Beyond the Harmonic Approximation: How Important are Duschinsky Rotations? Mol. Phys. 2015, 10.1080/ 00268976.2015.1074740. (21) Werner, H.-J.; Knowles, P.; Lindh, R.; Manby, F.; Schütz, M. MOLPRO, Development Version 2099.9, a package of ab initio programs. 2015; see: http://www.molpro.net. (22) Peterson, K. A.; Adler, T. B.; Werner, H.-J. Systematically Convergent Basis Sets for Explicitly Correlated Wavefunctions: The Atoms H, He, B−Ne, and Al−Ar. J. Chem. Phys. 2008, 128, 084102. (23) Werner, H.-J.; Adler, T. B.; Manby, F. R. General Orbital Invariant MP2-F12 Theory. J. Chem. Phys. 2007, 126, 164102. (24) Knizia, G.; Werner, H.-J. Explicitly Correlated RMP2 for HighSpin Open-Shell Reference States. J. Chem. Phys. 2008, 128, 154103. (25) Manby, F. R. Density Fitting in Second-Order Linear-r12 Møller-Plesset Perturbation Theory. J. Chem. Phys. 2003, 119, 4607− 4613. (26) May, A. J.; Manby, F. R. An Explicitly Correlated Second Order Møller-Plesset Theory Using a Frozen Gaussian Geminal. J. Chem. Phys. 2004, 121, 4479−4485. (27) Weigend, F.; Köhn, A.; Hättig, C. Efficient Use of the Correlation Consistent Basis Sets in Resolution of the Identity MP2 Calculations. J. Chem. Phys. 2002, 116, 3175−3183. (28) Rauhut, G. Efficient Calculation of Potential Energy Surfaces for the Generation of Vibrational Wave Functions. J. Chem. Phys. 2004, 121, 9313−9322. (29) Pflüger, K.; Paulus, M.; Jagiella, S.; Burkert, T.; Rauhut, G. Multi-Level Vibrational SCF Calculations and FTIR Measurements on Furazan. Theor. Chem. Acc. 2005, 114, 327−332. (30) Watson, J. K. G. Simplification of the Molecular VibrationRotation Hamiltonian. Mol. Phys. 1968, 15, 479−490. (31) Neff, M.; Rauhut, G. Toward Large Scale Vibrational Configuration Interaction Calculations. J. Chem. Phys. 2009, 131, 124129. (32) Neff, M.; Hrenar, T.; Oschetzki, D.; Rauhut, G. Convergence of Vibrational Angular Momentum Terms within the Watson Hamiltonian. J. Chem. Phys. 2011, 134, 064105. (33) Duschinsky, F. The Importance of the Electron Spectrum in Multi Atomic Molecules. Concerning the Franck-Condon Principle. Acta Physiochim. URSS 1937, 7, 551. (34) Mok, D. K. W.; Lee, E. P. F.; Chau, F.-T.; Wang, D.; Dyke, J. M. A New Method of Calculation of Franck-Condon Factors Which Includes Allowance for Anharmonicity and the Duschinsky Effect: Simulation of the HeI Photoelectron Spectrum of ClO2. J. Chem. Phys. 2000, 113, 5791−5803. (35) Luis, J. M.; Bishop, D. M.; Kirtman, B. A Different Approach for Calculating Franck-Condon Factors Including Anharmonicity. J. Chem. Phys. 2004, 120, 813−822.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b06922. Tables containing anharmonic VCI frequencies and the Duschinsky matrices for ketene and its deuterated analog (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Phone: +49 (0)711 685-64405. Fax: +49 (0)711 685-64442. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaft (DFG, grant Ra 656/19-1) is kindly acknowledged. This research was supported in part by the bwHPC initiative and the bwHPC-C5 project provided through associated compute services of the JUSTUS HPC facility at the University of Ulm. bwHPC and bwHPC-C5 are funded by the Ministry of Science, Research and the Arts Baden-Württemberg (MWK) and the Deutsche Forschungsgemeinschaft (DFG). The author is very grateful to the unknown referees, who revealed a substantial error in the assignment of the FCFs in the original version of the manuscript.

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REFERENCES

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