Article pubs.acs.org/JPCA
Cation States of Ethane: HEAT Calculations and Vibronic Simulations of the Photoelectron Spectrum of Ethane Published as part of The Journal of Physical Chemistry virtual special issue “Mark S. Gordon Festschrift”. Kin Long Kelvin Lee,† Scott M. Rabidoux,‡ and John F. Stanton*,§ †
School of Chemistry, University of New South Wales, Kensington, NSW 2052, Australia Institute for Computational Engineering and Sciences and §Department of Chemistry, The University of Texas at Austin, Austin, Texas 78712, United States
‡
ABSTRACT: High-accuracy ab initio calculations have been carried out on ethane and its radical cation. With the HEAT-345(Q) scheme, adiabatic ionization potentials of 11.52 and 11.57 eV are determined for the X̃ 2Eg and à 2A1g states, respectively, with an uncertainty of ±0.015 eV. Also considered in this report are linear and quadratic vibronic coupling involving both states. With this simple vibronic model, the photoelectron spectrum of ethane was simulated in the 11−15 eV region using linear and full quadratic Jahn− Teller coupling Hamiltonians, and with up to 70 billion direct product basis functions in a high-performance computing environment. Although the linear vibronic coupling model adequately reproduces the spectral envelope, the quadratic vibronic treatment results in much better agreement with the observed spectrum.
■
INTRODUCTION
X̃ 2Eg
The study of small organic molecules serves as a benchmark for understanding some of the most ubiquitous phenomena in chemical physics. Among these, the simplest alkanes (closed shell saturated hydrocarbons) are within reach for high-accuracy theoretical methods as well as a host of gas phase experimental work. From a fundamental standpoint, the spectroscopy of ethane has seen a wealth of investigations into its microwave,1 infrared,2,3 and VUV/X-ray absorption,4−6 with accompanying theoretical work.7−10 In particular, the high symmetry of its staggered form (D3d) has attracted much attention into the internal rotation dynamics.11−14 From a more applied perspective, ethane is emitted from the combustion of fossil fuels15 as well as being the second most abundant component of natural gas.16 In astrochemistry, some examples of where ethane is found include extraterrestrial objects such as ices,17,18 in Jupiter’s atmosphere,19,20 and in lakes thought to dot the surface of Titan, one of Saturn’s moons.21 Another interesting aspect of ethane is its ionization: by removing an electron, the result is an open shell radical cation whose reactivity and character is dramatically different from that of the closed-shell neutral. The orbital configuration of ethane in its electronic ground state is
à 2 A1g =(1a 2u)2 ...(3a1g)1(1eg )4
Previous work on the radical cation has investigated the structure and spectroscopy of these states. Calculations with both wave function and density functional methods found minimum energy geometries comprising interesting structural features. The lowest energy forms are a “diborane-like” structure in C2h symmetry and a “long-bond” structure of D3d symmetry,22−25 deriving from the X̃ 2Eg and à 2A1g states, respectively. Photoelectron spectroscopy by Turner and workers4,26 observed significant vibronic activity around 11−13 eV, ionization out of the 1eg and 3a1g orbitals producing the cation states as mentioned above. The sharp structure seen in this region was primarily attributed to the symmetric C−H deformation progression, where the Jahn− Teller distorted X̃ 2Eg state was thought to be responsible for the lower energy end of the spectrum and with the à 2A1g state at higher energy. The removal of an electron from ethane appears deceptively simple: in the diabatic picture it is simply production of an electron configuration as appropriately modified from the ones shown above. In this framework, there is no interaction between the X̃ 2Eg and à 2A1g states. However, coupling nuclear and electronic motion complicates this picture. First, nuclear displacements are able to lift the degeneracy of the components nominally associated with the X̃ 2Eg state (Jahn−Teller (JT)
2 X̃ 1A1g = (1a 2u)2 (1a1g)2 (2a1g )(2a u)2 (1e u)4 (3a1g)2 (1eg )4
To produce the lowest-lying states of the radical cation, the ejected electron may nominally originate from either the 1eg (HOMO) or the nearby 3a1g (HOMO−1) orbital: © XXXX American Chemical Society
=(1a 2u)2 ...(3a1g)2 (1eg )3
Received: July 26, 2016 Revised: September 5, 2016
A
DOI: 10.1021/acs.jpca.6b07516 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A distortion27,28). Though degenerate at high-symmetry (D3d) geometries, motion along eg vibrational modes splits the X̃ 2Eg state into a pair of states; in first order, one raised and the other lowered in energy. A second interaction comes from the pseudoJahn−Teller (PJT) effect,29 which involves the nondegenerate à 2A1g electronic state coupling with X̃ 2Eg, also through eg coordinates. These interactions strongly mix the simple diabatic picture of the ionization and the actual vibronic states probed experimentally are quite complicated. It is easily seen that the radical cation states of ethane are exemplary of complex vibronic phenomena where a manifold of strongly coupled states contributes to a spectrum. Previous work by Mahapatra and workers30,31 has attempted to simulate the vibronic coupling by introducing activity from the totally symmetric modes and the JT/PJT modes separately. Furthermore, the treatment of vibronic coupling was separated into linear and quadratic contributions, and the results were convolved afterward. The assumption is that the linear and quadratic contributions to the vibronic Hamiltonian are separable and interaction between these terms is negligible. However, because the degenerate 2Eg state is relatively close in energy to the 2A1g, the interstate coupling can be large. The aim of the current paper is 2-fold. The first involves an accurate determination of the adiabatic ionization potential (IP) of the X̃ 2Eg and à 2A1g states. Although there have been numerous experimental and theoretical studies on the adiabatic IP, there is still a surprisingly large uncertainty attached to this very fundamental property of what is a small, prototypical molecule. Here, the HEAT procedure32−34 is used to calculate the IPs. Second, vibronic coupling in the low-lying states of the radical cation of ethane is studied with coupled-cluster methods. We simulate the photoelectron spectrum of ethane in the 11−15 eV region by utilizing a quadratic vibronic coupling Hamiltonian, including the aforementioned coupling mechanisms: JT coupling within the X̃ state and PJT coupling between the X̃ and à states. The model includes totally symmetric, JT, and PJT active modes without simplifying approximations made in the calculations. We show that the linear model is only qualitatively faithful to the observed spectrum, and that considerable improvement is achieved when quadratic coupling terms are included.
functions, chosen in line with the HEAT-345(Q) strategy. This means the HF contribution is considered using aug-cc-pCVXZ (X = T, Q, 5) and the correlation energy from CCSD(T) with aug-cc-pCVXZ (X = Q, 5) followed by extrapolation to the complete basis set (CBS) limit. The ZPE and DBOC were calculated using CCSD(T)/cc-pVQZ and HF/aug-cc-pVTZ respectively. Scalar relativistic corrections were calculated as the sum of the mass-velocity, one- and two-electron Darwin contributions. The iterative triples contribution, EΔT−(T), to the CCSD energy was estimated with the cc-pVXZ (X = T, Q) basis sets as the extrapolated difference between fc-CCSD(T) and fcCCSDT correlation energies. Finally, the CCSDT(Q) term (EΔHLC) was approximated at the cc-pVDZ level using the MRCC code36 interfaced with CFOUR. Vibronic Coupling Hamiltonian. Regarding ionization of ethane, the simple diabatic picture produces a cation by removing an electron from either the eg or a1g orbital of the X̃ 1A1g ground state configuration, resulting in either the X̃ 2Eg or the à 2A1g cation states, respectively. However, experimentally the photoelectron spectrum is seen to be complicated by significant vibronic coupling. To model this process, we consider the diabatic interaction within and between the two cation electronic states. Within the 2Eg manifold, certain doubly degenerate nuclear displacements (of eg symmetry) are JT active. In a similar fashion, the 2A1g state will couple with the neighboring 2Eg state, also through eg modes (PJT coupling). To address this problem computationally, one must compute coupling terms with electronic structure methods. Several methods for treating vibronic coupling exist, but we have chosen to use the Köppel, Domcke, and Cederbaum (KDC) model37 in its parallel computer implementation by Rabidoux et al.38 It will become apparent in the following sections that the latter choice is motivated by the relatively large size of the vibronic basis set required to achieve converged results. Because the method has been outlined in many places in the literature, we discuss it only briefly here. The problem at hand is to diagonalize the molecular Hamiltonian in a quasidiabatic electronic basis where the nuclear kinetic energy operator (T̂ N) is assumed diagonal. We begin with the Schrödinger equation:
COMPUTATIONAL METHODS HEAT Calculations. All of the calculations in this report were done with a development version of the CFOUR program.35 The HEAT protocol has been covered in previous publications,32−34 and we only briefly outline the details here. It involves a systematic treatment of dynamical correlation as well as including smaller contributions such as those due to scalar relativity and nuclear motion effects on the potential energy surface (PES); the latter treated adiabatically by the diagonal Born−Oppenheimer correction (EDBOC). Following geometry optimization at the ae-CCSD(T)/cc-pVQZ level, each effect is included individually, resulting in the following expression for the HEAT energy of a species:
The usual approach is to express the molecular wave function in terms of the Born−Huang expansion:
(Ĥ − E)Ψ(r,q) = 0
■
Ψ(r,q) =
∑ ψμ(r;q) χμ (q) (3)
μ 37
Here the quasidiabatic electronic states of interest (ψ(r;q)) and vibrational (χ(q)) wave functions form a direct-product basis for Ψ(r,q). The electronic basis is truncated to only include the three lowest lying electronic states of the cation; the degenerate 2 (A,B) Eg pair and the 2A1g states. The next state above these is the B̃ 2Eu state; whereas it also experiences vibronic (JT and PJT) interactions, it is considerably higher in energy (∼2.7 eV from the X̃ 2Eg state30) and is ignored here. Projection of H onto the electronic basis yields a nuclear Schrödinger equation of the form
∞ ∞ E HEAT = E HF + ECCSD(T) + EΔT − (T) + EΔHLC + EZPE
+ E DBOC + Erel
(2)
⎡⎛T AA 0 ⎛ χA ⎞ 0 ⎞ ⎛ V AA V AB V AC ⎞⎤⎥⎛ χA ⎞ ⎢⎜ N ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ χ Ĥ = ⎢⎜ 0 TNBB 0 ⎟ + ⎜ V BA V BB V BC ⎟⎥⎜ χB ⎟ = E⎜ B ⎟ ⎢⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ χ ⎜ χC ⎟ ⎟ CA ⎢⎜ V CB V CC ⎠⎥⎦⎝ C ⎠ 0 TNCC ⎠ ⎝ V ⎝ ⎠ ⎣⎝ 0
(1)
∞ where E∞ HF and ECCSD(T) are the extrapolated HF energy and CCSD(T) correlation contribution, respectively. In this work, the bulk of these calculations are based on all-electron (ae) energies with relatively large basis sets containing augmented
(4) B
DOI: 10.1021/acs.jpca.6b07516 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A Because TN is assumed diagonal, the necessary computations are those needed to parametrize the diagonal and off-diagonal elements of V, where the superscripts A, B, C correspond to the 2 A,B Eg and 2A1g states, respectively. The coupling between states is described by a second-order Taylor expansion about the equilibrium position of the ground state neutral molecule in dimensionless normal coordinates q, which takes the form V ii = Eivert +
∑ Fkiiqk + k
1 2
∑ Fkliiqkql + ...
where n = [n1, n2, ..., nm] and m is the number of vibrational modes. In the present case, a total of nine modes are considered corresponding to the three a1g modes and the three eg pairs. The parametrized quasidiabatic electronic Hamiltonian is then projected onto this basis. With all the necessary parameters, we obtain energy levels and wave functions from the eigenvalues and eigenvectors of H obtained with the Lanczos procedure. The result can be visualized as a “stick spectrum”, where the positions (transition energy) of each stick are the eigenvalues of H, and the height (transition probability) of each stick is determined by its corresponding eigenvector. A spectrum comparable to the experimental spectrum then requires convolving the stick spectrum with some empirical broadening function. In the present report, the spectra shown are stick spectra computed as discussed above and convolved with a Lorentzian function of 40 meV line width.
(5)
kl
and V ij =
∑ Fkijqk + k
1 2
∑ Fklijqkql + ...
(6)
kl
Evert i
■
Here is the vertical energy difference between the electronic ground state of ethane and the ith cation state at the ground state geometry, Fii and Fij are force and coupling constants, respectively, that couple modes qk within (intrastate) and between (interstate) quasidiabatic electronic states. In a linear vibronic coupling model, the off-diagonal potential expansions Vij are truncated to include first derivatives (Fkqk), and the second derivatives in eq 5 are assumed to be unchanged from the ground state. The quadratic vibronic coupling model relaxes this assumption and includes second derivatives (Fklqkql) of the cation states, with k and l corresponding here to the coupling modes of eg symmetry. In this report, quadratic force constants are calculated in the diagonal blocks of Ĥ for the totally symmetric modes as well as for the JT active modes. The parameters of V can be computed using electronic structure theory. Evert and Fii are easily i obtained from the standard methods of adiabatic ab initio quantum chemistry, whereas the coupling terms Fij are based on the quasidiabatic generalization of coupled-cluster theory due to Ichino et al.39 All of these calculations were done at the EOMIPCCSD/ANO0 level, using the wave function of neutral ethane as the reference state. We shall differentiate the force and coupling constants by their respective activities: the totally symmetric modes are referred to as κk (k ∈ a1g), JT constants are λk (k ∈ eg), and the PJT coupling constants are λ′k (k ∈ eg). Our model Hamiltonian then takes the form of the 3 × 3 matrix ⎛ E vert +κ−λ − λ′ λ A ⎜ Eg ⎜ +κ+λ λ E Evert λ′ B V=⎜ g ⎜ ⎜ − λ′ + λ′ E Avert 1g ⎝
⎞ ⎟ ⎟ ⎟ ⎟ κ ⎟⎠
RESULTS AND DISCUSSION Ethane and Its Radical Cation States. The X̃ 1A1g neutral and à 2A1g radical cation states were optimized with D3d symmetry, whereas the X̃ 2Eg state of C2H6+ was treated in the reduced C2h symmetry that arises due to JT distortion. Harmonic frequency analysis was carried out with each structure at the CCSD(T)/cc-pVQZ level, which verified that all structures are minima on the corresponding adiabatic potential energy surfaces. The optimized geometries for ground state ethane and the two lowest-lying cation electronic states are shown in Figure 1.
Figure 1. Optimized structures for ethane and its cation states. Values shown are in Å and deg. Values in brackets are literature values where available. For the X̃ 1A1g state, ref 40. For the X̃ 2Eg state, ref 42.
(7)
1
where κ = ∑k κkqk + 2 ∑kl κklqkl , λ = ∑kλkqk and λ′ = ∑kλk′qk. With the electronic part of the Hamiltonian parametrized, what remains is its diagonalization; although being less involved and intrinsically complicated than the parametrization, this is actually the most computationally intensive aspect of the problem. The molecular eigenstates in eq 3 are solved for variationally in a direct product basis of diabatic electronic states and m-dimensional harmonic oscillator wave functions (νn, taken as the product of m one-dimensional harmonic oscillators) centered about the equilibrium geometry of ground state ethane χμ (q) =
The structure of the ground state of ethane has been reproduced well, with parameters in good agreement with those derived from microwave measurements on ethane and its deuterated isotopologues.40 For the 2Eg state, our calculated structure possesses a longer C−C bond than those from previously reported calculations: 1.579 Å at the MP2/6-31G** level24 (even shorter with B3LYP/6-311G(2d,p), 1.436 Å).23 This is likely due to the additional dynamic correlation retrieved in our ae-CCSD(T)/cc-pVQZ optimization. Meanwhile, the “partially dissociated” 2A1g state, with a long C−C distance and near-planar methyl groups agrees well with a previous calculation by Lunell and Huang.24 In general, however, these structures are consistent with what is known in the literature to be “diborane-like” and “long bond”,23 with the former (and lower energy) of these two
∑ cnμνn(q) n
=
∑ ∑ ... ∑ cnμνn (q1) νn (q2)... νn (qm) 1
n1
n2
nm
2
m
(8) C
DOI: 10.1021/acs.jpca.6b07516 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A states being the C2h structure. As later sections of this paper are concerned with simulating the vibronic activity, the notation that will be used to refer to the relevant normal modes is shown in Table 1. Not included in this table are vibrations of ungerade
Table 3. Magnitude of the Linear Force and Coupling Constants Calculated at the EOMIP-CCSD/ANO0 Level for the Three Cation Statesa Totally Symmetric Modes (κk)
mode no./symmetry
frequency/cm−1
assignments
ν1/a1g ν2/a1g ν3/a1g ν4, ν5/eg ν6, ν7/eg ν8, ν9/eg
3067.6 1445.4 1019.4 3133.8 1520.8 1237.6
symmetric C−H stretches symmetric C−H deformation C−C stretch antisymmetric C−H stretches antisymmetric C−H deformation C−H rocking
E∞ HF E∞ CCSD(T) EΔT−(T) ECCSDT(Q) DBOC Erel ZPE adiabatic IP (eV)
à 2A1g
−79.266951 −0.559594 0.000120 −0.000390 0.004728 −0.031977 0.075074
−78.877089 −0.519924 0.000090 −0.000288 0.004704 −0.032058 0.068980 11.52
−78.892118 −0.502778 0.000105 −0.000378 0.004746 −0.032104 0.068597 11.57
A1g
2106.8 (2342.3) 1268.3 (1347.8) 1082.2 (1091.3) JT and PJT Modes
109.2 (107.3) 5037 (5183.0) 1851.1 (2065.6)
vibrational mode (eg)
λkb
λ′k
ν4, ν5 ν6, ν7 ν8, ν9
1695.6 (1877.7) 2677.2 (2444.7) 1505.0 (1561.5)
2953.4 (298.4) 2285.5 (2194.7) 1437.4 (2675.4)
a
Values in the parentheses are constants by Venkatesan and Mahapatra31 calculated at the MP2/cc-pVDZ level with finite differences. Constants are given in cm−1. bPhase for constants supplied are for 2EAg .
(Table 3, values in parentheses), with the exception of the PJT constants where discrepancies are greater. The quadratic force and coupling constants are divided into their respective activities: constants for totally symmetric modes are given in Table 4, JT and PJT constants for the eg modes are given in Tables 6 and 5, respectively.
Table 2. Summary of the Individual Contributions to the HEAT Energy of Ethane and Its Cation Statesa X̃ 2Eg
2
Eg
ν1 ν2 ν3
symmetry (a1u, a2u, and eu), which we have excluded in our vibronic coupling model because single-quantum excitations in these modes do not contribute to the spectrum. Determining the Adiabatic IPs. The results of HEAT calculations are shown in Table 2. Between the two cation states,
X̃ 1A1g
2
vibrational mode (a1g)
Table 1. Gerade Harmonic Vibrational Frequencies and Symmetries of the X̃ 1A1g State of the Neutral Calculated at the CCSD(T)/cc-pVQZ Level of Theory (Assignments Taken from NIST41)
Table 4. Quadratic Coupling Matrix for the Totally Symmetric Modes Calculated at the EOMIP-CCSD/ANO0 Levela 2
ν1 ν1 ν2 ν3 a
3033.7 −258.4 12.9
2
A1g ν2
1621.5 84.5
Eg
ν3
ν1
ν2
ν3
1040.3
3160.7 90.4 5.8
987.8 4.4
860.2
−1
Constants are given in cm .
Table 5. Quadratic PJT Coupling Constants for the 2A1g Statea
a ∞ ECCSD(T)
is the extrapolated ae-CCSD(T)/aug-cc-pVXZ contribution (X = Q, 5). Unless otherwise specified, values are given in Hartrees.
ν4 ν4 ν5 ν6 ν7 ν8 ν9
we see that the small corrections (i.e., relativistic and DBOC) are minor compared to the SCF and coupled-cluster contributions. The largest difference, apart from the HF contribution, is the extrapolated CCSD(T) term which stabilizes the X̃ 2Eg relative to the à 2A1g state. The CCSDT(Q) energy is also seen to contribute importantly to differentiating the two radical cation states. Earlier work on the adiabatic ionization to the X̃ 2Eg state by Zuilhof et al.23 using CBS-APNO put the value at 11.55 eV, which is slightly above our HEAT value. However, our result, which carries an uncertainty of no more than 0.015 eV, is in excellent agreement with the latest evaluated experimental IP (11.52 ± 0.04 eV41). Vibronic Coupling Matrix Elements. The objective of this section is to briefly describe parametrization of the linear and quadratic vibronic coupling Hamiltonians using quasidiabatic EOMIP-CCSD theory.39 First, the simple linear vibronic coupling model is considered, where we only include the first order terms κk, λk, and λ′k for the three lowest states of the ethane cation. The corresponding vertical ionization energies for the 2 Eg and the 2A1g states calculated at the EOMIP-CCSD/ANO0 level are 12.60 and 12.96 eV, respectively. Linear force and coupling constants are given in Table 3 and are in relatively good agreement with the constants of Venkatesan and Mahapatra31 calculated at the MP2/cc-pVDZ level using finite differences
a
ν5
ν6
ν7
ν8
ν9
3256.4 3256.4 −164.9
1520.4 −164.9
1520.4 −34.1
271.4 271.4
1159.6 −34.1
1159.6
Constants are given in cm−1.
Photoelectron Spectra Simulation. Having calculated the necessary coupling elements in the previous section, these parameters can be used to compute simulated photoelectron spectra to compare with the experimental measurement by Baker et al.4 The observed ionization onset is at ∼11.4 eV with significant vibronic activity up to 12.8 eV. Panel A of Figure 2 shows the result of the linear vibronic coupling model compared with the experimental spectrum, and the vibrational basis functions used are shown in Table 7. The plot was obtained with a direct product basis of dimension 3.6 billion, which was required to obtain a converged spectral envelope; this calculation required 86.4 GB of memory. The calculation utilized 256 cores on the Stampede supercomputer38 and required only 36 min for the 1000 Lanczos algorithm iterations. As we can see, this linear model offers only qualitatively satisfactory reproduction of the position and intensities observed in the experiment. D
DOI: 10.1021/acs.jpca.6b07516 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A Table 6. Quadratic JT Coupling Matrix for the Doubly Degenerate Electronic State (2Eg ⊗ eg)a 2 A Eg
ν4 ν5 ν6 ν7 ν8 ν9 a
ν4
ν5
3210.1 27.8 169.5 94.9 101.2 65.3
3265.7 94.9 −20.3 65.3 −29.5
ν6
2 B Eg
ν7
1344.6 −170.7 203.3 218.4
1003.3 218.4 233.5
ν8
772.6 44.3
ν9
ν4
ν5
ν6
ν7
ν8
ν9
861.1
3265.7 27.8 −20.3 94.9 −29.5 65.3
3210.1 94.9 169.5 65.3 101.2
1003.3 −170.7 −233.5 218.4
1344.6 218.4 203.3
861.1 44.3
772.6
Constants are given in cm−1. Values in black are intrastate force constants, whereas values in bold are interstate coupling constants (i.e., EAg ⊗ EBg ).
previous photoelectron spectrum without the need for any empirical corrections. The next logical step would be to consider a quartic treatment of the Hamiltonian, although to do so would require a significant leap in computational cost as well as a vastly more tedious and complicated parametrization. Another venture could involve a full mode treatment of vibronic coupling (in the current report, modes of ungerade symmetry are neglected with the assumption that they will not contribute significantly). However, such efforts could not be used to further analyze the experimental spectrum unless higher resolution measurements were to become available.
■
CONCLUSIONS This paper reports a theoretical investigation of the relative stability and spectroscopy of the two lowest-lying cation states of ethane. This is the second simplest alkane and is an excellent example to study because it exhibits prototypical features of vibronic coupling handled here by the KDC model. Using the HEAT-345(Q) procedure, we have determined the adiabatic IP for the X̃ 2Eg and à 2A1g states to be 11.52 and 11.57 eV, respectively, with the former comparing well with a recent evaluated experimental value. Dynamic correlation was found to be the dominant contribution to the difference; with increasing basis, the energy of 2Eg is lowered relative to the 2A1g state. Higher order effects make only minor contributions to the energy difference between the two states. Finally, using the vibronic coupling model we attempted to reproduce the experimental photoelectron spectrum by Baker et al.4 It was found that a quadratic vibronic coupling model provides a quite good reproduction of the energies and intensities in the spectrum, but the linear coupling model is only qualitatively satisfactory.
Figure 2. Comparison of the experimental photoelectron spectrum adapted from ref 4 and simulation results from the (A) linear vibronic coupling model and (B) quadratic vibronic coupling model. The simulated spectrum is the result from convolving the stick spectrum with a Lorentzian of line width 0.04 eV.
Table 7. Harmonic Oscillator Basis Functions Used for Each Vibrational Mode in the Respective Linear and Quadratic Vibronic Coupling Models ν1
ν2
ν3
4
24
12
5
24
ν4
ν5
ν6
ν7
Linear Vibronic Coupling 4 4 16 16 Quadratic Vibronic Coupling 12 4 4 28 28
ν8
ν9
16
16
36
36
■
An improvement to the linear vibronic coupling model is obtained by inclusion of quadratic coupling into the model Hamiltonian; that is, including quadratic coupling terms for the totally symmetric, JT, and PJT active modes. This calculation needed 70.2 billion basis functions (1.68 TB of memory) to obtain a converged spectrum and was run on 2048 cores of the Stampede supercomputer. The time required for 1000 Lanczos iterations was 4.4 h. To our knowledge, this is the largest timeindependent KDC model calculation yet performed. As seen in panel B of Figure 2, the low-energy end of the spectrum is now in good agreement with the experimental spectrum. In this case, not only the line positions but also intensities are reproduced well for a rather substantial (∼1 eV) range above the origin of the simulation. In this section, we have treated ionization of ethane in the 11−15 eV region with a quadratic vibronic coupling model from first-principles and shown rather good agreement with the
AUTHOR INFORMATION
Corresponding Author
*J. F. Stanton. E-mail:
[email protected]. Phone: (512) 471-5903. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
K.L.K.L. acknowledges funding from the Australian Postgraduate Award. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. We also acknowledge support from the U.S. National Science Foundation, through grants OCI-1148125 and CHE1361031. E
DOI: 10.1021/acs.jpca.6b07516 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A
■
(24) Lunell, S.; Huang, M.-B. Theoretical confirmation of the e.s.r. spectrum of the ethane cation. J. Chem. Soc., Chem. Commun. 1989, 1031−1033. (25) Toriyama, K. Structures and reactions of radical cations of some prototype alkanes in low temperature solids as studied by ESR spectroscopy. J. Chem. Phys. 1982, 77, 5891. (26) Al-Joboury, M. I.; Turner, D. W. Molecular photoelectron spectroscopy. Part VI. Water, methanol, methane, and ethane. J. Chem. Soc. B 1967, 373−376. (27) Jahn, H. A.; Teller, E. Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy. Proc. R. Soc. London, Ser. A 1937, 161, 220−235. (28) Longuet-Higgins, H. C.; Opik, U.; Pryce, M. H. L.; Sack, R. A. Studies of the Jahn-Teller Effect. II. The Dynamical Problem. Proc. R. Soc. London, Ser. A 1958, 244, 1−16. (29) Köppel, H.; Cederbaum, L. S.; Domcke, W. Interplay of Jahn− Teller and pseudo−Jahn−Teller vibronic dynamics in the benzene cation. J. Chem. Phys. 1988, 89, 2023. (30) Kumar, R. R.; Venkatesan, T. S.; Mahapatra, S. Multistate and multimode vibronic dynamics: The Jahn-Teller and pseudo-Jahn-Teller effects in the ethane radical cation. Chem. Phys. 2006, 329, 76−89. (31) Venkatesan, T. S.; Mahapatra, S. Exploring the Jahn-Teller and pseudo-Jahn-Teller conical intersections in the ethane radical cation. J. Chem. Phys. 2005, 123, 114308. (32) Harding, M. E.; Vázquez, J.; Ruscic, B.; Wilson, A. K.; Gauss, J.; Stanton, J. F. High-accuracy extrapolated ab initio thermochemistry. III. Additional improvements and overview. J. Chem. Phys. 2008, 128, 114111. (33) Bomble, Y. J.; Vázquez, J.; Kállay, M.; Michauk, C.; Szalay, P. G.; Császár, A. G.; Gauss, J.; Stanton, J. F. High-accuracy extrapolated ab initio thermochemistry. II. Minor improvements to the protocol and a vital simplification. J. Chem. Phys. 2006, 125, 064108. (34) Tajti, A.; Szalay, P. G.; Császár, A. G.; Kállay, M.; Gauss, J.; Valeev, E. F.; Flowers, B. A.; Vázquez, J.; Stanton, J. F. HEAT: High accuracy extrapolated ab initio thermochemistry. J. Chem. Phys. 2004, 121, 11599. (35) Stanton, J. F.; Gauss, J.; Harding, M. E.; Szalay, P. G.; Auer, A. A.; Bartlett, R. J.; Benedikt, U.; Berger, C.; Bernholdt, D. E.; Bomble, Y. J.; et al. CFOUR, Coupled-Cluster techniques for Computational Chemistry, 2015. For the current version, see http://www.cfour.de. (36) Kallay, M.; Rolik, Z.; Csontos, J.; Ladjánszki, I.; Szegedy, L.; Ladoczki, B.; Samu, G. MRCC. J. Chem. Phys. 2013, 139, 094105. (37) Köppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. Adv. Chem. Phys. 1984, 57, 59−246. (38) Rabidoux, S. M.; Eijkhout, V.; Stanton, J. F. Parallelization Strategy for Large-Scale Vibronic Coupling Calculations. J. Phys. Chem. A 2014, 118, 12059−12068. (39) Ichino, T.; Gauss, J.; Stanton, J. F. Quasidiabatic states described by coupled-cluster theory. J. Chem. Phys. 2009, 130, 174105. (40) Hirota, E.; Endo, Y.; Saito, S.; Duncan, J. L. Microwave spectra of deuterated ethanes: Internal rotation potential function and rz structure. J. Mol. Spectrosc. 1981, 89, 285−295. (41) Lias, S. G.; Bartmess, J. E.; Liebman, J. F.; Holmes, J. L.; Levin, R. D.; Mallard, W. G. In NIST Chemistry WebBook; Linstrom, P. J., Mallard, W. G., Eds.; NIST Standard Reference Database; NIST: Gaithersburg, MD. (42) Shchapina, I. Y.; Chuvylkin, N. D. A simple method for quantitative estimation of C-H bond lengths in hydrocarbon radical cations. Russ. Chem. Bull. 1996, 45, 306−311.
REFERENCES
(1) Hirota, E.; Mastumura, C. Microwave Spectrum of Ethane. J. Chem. Phys. 1971, 55, 981. (2) Henry, L.; Valentin, A.; Lafferty, W. J.; Hougen, J. T.; Malathy Devi, V.; Das, P. P.; Narahari Rao, K. Analysis of high resolution fourier transform and diode laser spectra of the ν9 band of ethane. J. Mol. Spectrosc. 1983, 100, 260−289. (3) Hepp, M.; Herman, M. Vibration-rotation bands in ethane. Mol. Phys. 2000, 98, 57−61. (4) Baker, A. D.; Baker, C.; Brundle, C. R.; Turner, D. W. The electronic structures of methane, ethane, ethylene and formaldehyde studied by high-resolution molecular photoelectron spectroscopy. Int. J. Mass Spectrom. Ion Phys. 1968, 1, 285−301. (5) Au, J. W.; Cooper, G.; Burton, G. R.; Olney, T. N.; Brion, C. E. The valence shell photoabsorption of the linear alkanes, CnH2n+2 (n = 1−8): absolute oscillator strengths (7−220 eV). Chem. Phys. 1993, 173, 209− 239. (6) Au, J. W.; Cooper, G.; Brion, C. E. The molecular and dissociative photoionization of ethane, propane, and n-butane: absolute oscillator strengths (10−80 eV) and breakdown pathways. Chem. Phys. 1993, 173, 241−265. (7) Galasso, V. Ab initio study of multiphoton absorption properties of methane, ethane, propane, and butane. Chem. Phys. 1992, 161, 189− 197. (8) Caldwell, J. W.; Gordon, M. S. Excited states and photochemistry of saturated molecules. The vibrational structure in the electronic spectrum of ethane. J. Mol. Spectrosc. 1982, 96, 383−394. (9) Buenker, R. J.; Peyerimhoff, S. D. Ab initio calculations on the electronic spectrum of ethane. Chem. Phys. 1975, 8, 56−67. (10) Raymonda, J. W.; Simpson, W. T. Experimental and Theoretical Study of Sigma-Bond Electronic Transitions in Alkanes. J. Chem. Phys. 1967, 47, 430−448. (11) Hougen, J. T. Perturbations in the Vibration-Rotation-Torsion Energy Levels of an Ethane Molecule Exhibiting Internal Rotation Splittings. J. Mol. Spectrosc. 1980, 82, 92−116. (12) Wilson, E. B. Nuclear Spin and Symmetry Effects in the Heat Capacity of Ethane Gas. J. Chem. Phys. 1938, 6, 740. (13) Kemp, J. D.; Pitzer, K. S. Hindered Rotation of the Methyl Groups in Ethane. J. Chem. Phys. 1936, 4, 749−3. (14) Pitzer, K. S. Internal Rotation in Molecules with Two or More Methyl Groups. J. Chem. Phys. 1942, 10, 605−606. (15) Aydin, M.; Verhulst, K. R.; Saltzman, E. S.; Battle, M. O.; Montzka, S. A.; Blake, D. R.; Tang, Q.; Prather, M. J. Recent decreases in fossil-fuel emissions of ethane and methane derived from firn air. Nature 2011, 476, 198−201. (16) Hargreaves, R. J.; Buzan, E.; Dulick, M.; Bernath, P. F. Highresolution absorption cross sections of C2H6 at elevated temperatures. Molecular Astrophysics 2015, 1, 20. (17) Brown, M. E.; Barkume, K. M.; Blake, G. A.; Schaller, E. L.; Rabinowitz, D. L.; Roe, H. G.; Trujillo, C. A. Methane and Ethane on the Bright Kuiper Belt Object 2005 FY9. Astron. J. 2007, 133, 284−289. (18) Gibb, E. L.; Mumma, M. J.; Dello Russo, N.; DiSanti, M. A.; Magee-Sauer, K. Methane in Oort cloud comets. Icarus 2003, 165, 391− 406. (19) Bilger, C.; Rimmer, P.; Helling, C. Small hydrocarbon molecules in cloud-forming brown dwarf and giant gas planet atmospheres. Mon. Not. R. Astron. Soc. 2013, 435, 1888−1903. (20) Ridgway, S. T. Jupiter: Identification of Ethane and Acetylene. Astrophys. J. 1974, 187, L41−L43. (21) Mastrogiuseppe, M.; Poggiali, V.; Hayes, A.; Lorenz, R.; Lunine, J.; Picardi, G.; Seu, R.; Flamini, E.; Mitri, G.; Notarnicola, C.; et al. The bathymetry of a Titan sea. Geophys. Res. Lett. 2014, 41, 1432−1437. (22) Shubina, T. E.; Fokin, A. A. Hydrocarbon σ-radical cations. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2011, 1, 661−679. (23) Zuilhof, H.; Dinnocenzo, J. P.; Reddy, A. C.; Shaik, S. Comparative Study of Ethane and Propane Cation Radicals by B3LYP Density Functional and High-Level ab Initio Methods. J. Phys. Chem. 1996, 100, 15774−15784. F
DOI: 10.1021/acs.jpca.6b07516 J. Phys. Chem. A XXXX, XXX, XXX−XXX
