Article pubs.acs.org/JPCA
Assessment of Various Electronic Structure Methods for Characterizing Temporary Anion States: Application to the Ground State Anions of N2, C2H2, C2H4, and C6H6 Michael F. Falcetta,*,† Laura A. DiFalco,† Daniel S. Ackerman,† John C. Barlow,† and Kenneth D. Jordan‡ †
Grove City College, Grove City, Pennsylvania 16127, United States University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
‡
ABSTRACT: The theoretical characterization of temporary anions is an especially challenging problem. In the present study we assess the performance of several electronic structure methods when used in conjunction with the stabilization method to characterize temporary anion states. The ground state anions of N2, C2H2, C2H4, and C6H6 are used as the test systems, with the most extensive testing being done for N2. For the 2Πg anion state of N2− the ADC(2), EOM-MP2, and EOM-CCSD methods give values of the resonance parameters in excellent agreement with the results of prior high-level calculations. For the hydrocarbon systems, the EOM-MP2 method consistently provides excellent agreement with the EOM-CCSD results for the test systems, whereas the ADC(2) considerably underestimates the widths for ethylene and benzene. Several density functional theory (DFT) approaches are tested and, although none performs as well as the EOM-MP2 method, it is found that inclusion of Hartree−Fock exchange greatly improves the results. Of the DFT-based methods, time-dependent DFT with standard correlation functionals and use of Hartree−Fock exchange provides the best performance for N2− over the range of bond lengths considered and is also found to give reasonable values of the resonance parameters of the three hydrocarbon molecules.
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INTRODUCTION The addition of an electron to a closed-shell neutral molecule results in the formation of a radical anion. Radical anions are important in multiple areas, including biology,1,2 atmospheric chemistry,3 plasma processing,4 and mass spectrometry.5 If the anion is less stable than the neutral molecule (that is, the electron affinity is negative), the anion has a finite lifetime with respect to loss of the extra electron. Although temporary anions (TA’s) may survive for as little as a few femtoseconds, they are important in DNA damage due to secondary electron formation upon exposure to radiation6 and in the pumping mechanism in the N2/CO2 laser,7−9 as well as in several other applications.10 They are readily detected as resonances in various electron scattering cross sections.7,8,11,12 TAs may be divided into two classes, those for which the excess electron is captured into an unfilled valence orbital without electronic excitation of the molecule and those for which electron capture is accompanied by electronic excitation.7,8 The former are the focus of the present study. Such TAs (also called shape resonances) are particularly challenging to treat theoretically as they require flexible basis sets containing diffuse functions. With such basis sets, the lowest energy eigenvalues of the N + 1 electron problem, where N is the number of electrons of the neutral molecule, correspond to approximations to the continuum rather than to the temporary anion. Special theoretical methods are required to avoid the collapse onto the continuum. In this work, we make use of the stabilization method,13,14 described below, to deal with this problem. For the immediate discussion © 2014 American Chemical Society
we assume that a method that avoids the collapse onto a continuum is used. Koopmans’ theorem15 (KT) equates the negative of the energies of the occupied and unoccupied Hartree−Fock (HF) orbital energies of an atom or molecule with the ionization potentials (IPs) and electron affinities (EAs), respectively. The KT approximation neglects both relaxation and correlation effects to the IP’s and EAs. In the case of IPs, these two effects tend to cancel, whereas for EAs they are generally of the same sign. As a result, KT estimates of EAs associated with temporary anions tend to be too high by 1−2 eV16−19 (provided one has avoided the collapse onto the continuum problem) and obtaining accurate EAs from computational methods requires the inclusion of relaxation and correlation corrections.20 In a diabatic picture, a temporary anion may be viewed as a discrete state, in which the excess electron is localized in a normally unfilled valence orbital, embedded in the autoionization continuum, with the finite lifetime resulting from the coupling of the discrete state to the continuum. A TA may be characterized by a complex energy.21 Eres = E R − iΓ/2
(1)
Special Issue: Kenneth D. Jordan Festschrift Received: January 11, 2014 Revised: February 10, 2014 Published: February 10, 2014 7489
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approximants (GPA).65 The simplest GPA that incorporates the correct branch structure for two interacting roots is
where ER is the resonance position, defined relative to the energy of the neutral at the given geometry (essentially the energy of the peak of the relevant feature in the total cross section) and Γ is the associated decay width. Neglecting the impact of vibrational structure, Γ can be associated with the width of the feature in the scattering cross section. The lifetime of the resonance state, τ, is defined as
τ = ℏ/Γ
P(α)E2 + Q (α)E + R(α) = 0
(3)
where P, Q, and R are polynomials in α whose coefficients are determined by fitting to the calculated energies for real values of α. Equation 3 is appropriate for stabilization graphs with well separated avoided crossings. If there are two or more close avoided crossings, the GPA can be generalized by including higher order terms in E with associated polynomials in α.20 The stabilization method described above has been used with configuration interaction calculations (CISD) to determine the complex energy of the lowest energy temporary anions of N2 and Mg and in the KT approximation to characterize qualitatively the temporary anions of several polyatomic molecules.16,17,19,20 Recently a series of studies have employed the stabilization method with the unfilled orbitals from DFT calculations, both without56−58 and with long-range corrected (LC) functionals59 to characterize temporary anion states. The goal of the present paper is to investigate how several commonly used electronic structure methods fare at predicting ER and Γ values of temporary anion states when used in conjunction with the stabilization method. Our main test system is N2−/N2 at various bond lengths. The approaches that worked well for the N2− system were then applied to the ground state anions of C2H2, C2H4, and C6H6. The calculations were performed using the Gaussian 09,66 FHI-aims,67 cfour,68 and PSI469 codes.
(2)
This characterization of the resonance state is a local approach, in contrast to more complete treatments in which Γ may be energy dependent and nonlocal and ER includes an energydependent level shift.21 Although these additional dependencies are of vital importance in modeling threshold phenomena and bond-breaking processes, the local approach suffices for the characterization of TAs far from threshold, and only the local approach is considered here. To simulate the vibrational structure in the vicinity of resonances, it is important to determine ER and Γ as functions of the molecular geometry.22 A variety of computational approaches to calculating the properties of TA’s have been developed. These include methods that explicitly solve the scattering problem23,24 as well as approaches that involve modifications of standard quantum chemical methods. Examples of the latter include the complex scaling,25−30 complex absorbing potential (CAP),31−38 and stabilization methods.16,17,19,55−63 In addition, one can add a perturbation to the system to convert the TA to a bound state and then extrapolate to zero perturbation.39−54 In this study we adopt the stabilization method,13,14 in which one calculates, using an L2 basis set, the energies of the system of interest as a function of an adjustable parameter. The spectrum of eigenvalues from such a calculation contains two types of solutions: resonances and discretized approximations to the continuum (DC). The wave functions associated with the former have much of their amplitude in the region of the atom or molecule, whereas the wave functions for the latter have little weight in this region. The finite lifetime derives from the coupling of the localized resonance state to the DC solutions. A common method of implementing the stabilization method is to multiply the exponents of the most diffuse basis functions used in the calculation by a scaling factor, α.16,17,55−63 The eigenvalues of the N + 1 electron system are then calculated at several values of α and the resulting data are displayed as a stabilization graph. The interpretation of stabilization graphs has been covered in detail in earlier publications.16,17,55−63 Here we note that the mixing between a resonance and a DC solution is usually readily apparent as an avoided crossing. The challenge then is to determine the real and imaginary parts of the resonance energy from the data encapsulated in the stabilization graph. A commonly employed method for extracting ER and Γ from stabilization graphs involves analytically continuing the energies, E(α), into the complex plane, and using the stationary condition, dE/dα = 0, to determine the complex resonance energy.20 The resulting complex, stationary value of the scaling factor (αstat) is then substituted into the functional form for E, giving the complex stationary energy. Several functional forms for E(α) have been employed, but it is now generally recognized that, because stabilization graphs possess avoided crossings, it is important to adopt functional forms that incorporate square-root (or higher order) branch singularities.20,64 This can be accomplished by use of generalized Padé
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COMPUTATIONAL METHODS
Except where explicitly noted otherwise, a modified aug-ccpVTZ basis set was used on second row atoms. Specifically, the most diffuse p function of the aug-cc-pVTZ basis set was replaced with four p functions with exponents of 0.08, 0.04, 0.02, and 0.01, which were scaled by a factor of α, which ranged from 0.36 to 1.96. Addition of still more diffuse p functions had no effect on the calculated resonance energies. In the Results and Discussion section it is demonstrated that the inclusion of additional polarization functions in the basis set does not significantly alter the resonance parameters of N2−. The aug-ccpVDZ basis set70 with the diffuse p functions removed was used for the hydrogen atoms. Test calculations demonstrated that these functions had no effect on the resonance parameters of benzene or acetylene. The TDDFT calculations on benzene employed only a single diffuse p function (exponent of 0.04) on the carbon atom. The exponent of this function is scaled to construct the stabilization graph. The reduction in the basis set size for benzene was done because of numerical instabilities encountered when the default basis set was used. Some of the methods employed to characterize the anion state actually calculate the excitation energies of the N + 1 electron system. These can be “tricked” into giving EAs by initially placing the excess electron into a diffuse continuum-like orbital belonging to the totally symmetric representation, described by adding an s function of exponent of 0.001 on each atom or at the center of the molecule. This method has been used in the past to characterize bound anions.71 In the present study the equation-of-motion coupled-clustersingles-plus-doubles (EOM-CCSD) method72 is used to obtain the benchmark resonance parameters against which the results obtained by other methods are compared. In addition to EOMCCSD, the methods considered include EOM-MP2,73 diagonal 7490
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algebraic diagrammatic construction (d-ADC(2)), 74 full ADC(2),75 KT (HF),15 configuration interaction with single excitations (CIS),76 and CIS with perturbative doubles corrections (CIS(D)).77 Several density functional methods are considered, both in the KT approximation and in the timedependent DFT (TDDFT) approximation.78 The functionals considered include standard generalized gradient approximation functionals (GGAs), LC functionals that employ exact exchange at long electron−electron distances, and functionals that use HF exchange at all electron−electron distances (designated as “HFE”). The calculations using exact exchange are particularly relevant in light of a recent study of Lee et al.,79 who found that the DFT calculations using Hartree−Fock densities provide a good description of bound anion states. The functionals used include CAMB3LYP,80 ωB97XD,81 LCBLYP,82 LC-ωPBE,83 HFE_PBE,84 and HFE_BLYP.82c,d Finally, stabilization calculations on the N2−/N2 system are carried out using both the nonself-consistent G0W074 and selfconsistent GW074 Green’s function methods employing either HF or PBE orbitals. For all theoretical methods considered, calculations were carried out at the experimental value of the equilibrium bond length (Req) 1.09767 Å of N2.85 For the KT, EOM, and selected DFT methods, calculations were also performed at other values of R spanning a range of Req ± 0.12 Å. The 1s orbital of the nitrogen or carbon atoms were frozen in the CIS and CIS(D) calculations. Except for the results in Table 1, the complex resonance energies from analytic continuation represent the average of at least 12 different calculations, using slightly different data sets as input and using various orders for the polynomials in eq 3. The standard deviations about the averages give a measure of the uncertainty in the resonance parameters resulting from the analytic continuation procedure.
roots were employed. This is not necessary, but it leads to greater numerical stability. Table 1 reports the resonance parameters of N2− obtained from analytic continuation of the EOM-CCSD data reported in Table 1. Sensitivity of the Resonance Parameters of N2− to the Details of the Analytic Continuation Procedurea polynomial order
ER (eV)
Γ/2 (eV)
9 9 9 9 10 10 10 10 11 11 11 11 12 12 12 12 average std dev
2.5683 2.5760 2.5802 2.5798 2.5820 2.5797 2.5792 2.5792 2.5807 2.5854 2.5769 2.5839 2.5804 2.5786 2.5843 2.5835 2.580 0.004
0.2902 0.2832 0.2852 0.2838 0.2883 0.2835 0.2830 0.2846 0.2865 0.2951 0.2780 0.2839 0.2865 0.2805 0.2870 0.2832 0.285 0.004
a
The results were obtained from the data points used to construct the stabilization graph shown in Figure 1. bThe integer in the first column gives the order of the polynomials used in the fit, and the different entries to the use of different sets of input data.
Figure 1. Several different analytic continuation calculations using different order polynomials and different sets of data from the stabilization graph were carried out. Nearly the same resonance parameters are obtained from the different calculations. Table 2 reports the resonance parameters from EOM-CCSD stabilization calculations using basis sets derived from aug-cc-
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RESULTS AND DISCUSSION Figure 1 shows the stabilization graph for N2− at Req calculated using the EOM-CCSD method with the default basis set. The
Table 2. Comparison of the EOM-CCSD Values of Resonance Parameters of N2− with the Results of Experiment and Prior Calculations method EOM-CCSD aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z Boomerang modela MRCI/stab.b MPPTc EOM-CCSD/CAPd experimente
Figure 1. Stabilization graph for the 2Πg state of N2− at Req obtained using the EOM-CCSD method.
a e
figure reports the energies for the first four 2Πg eigenvalues. The second and third roots undergo a pronounced avoided crossing near α = 1.1, indicating the presence of a resonance state. At small α values, there is also a weak interaction between the third and fourth roots. Similarly, the second root does not level off for large values of α due to a weak interaction with the first root. In doing the analytic continuation, data from all four
Reference 22. Reference 87.
b
Reference 20.
c
ER (eV)
Γ/2 (eV)
2.580 2.49 2.49 2.20 2.62 2.36 2.44 2.32
0.285 0.251 0.248 0.270 0.224 0.215 0.195 0.205
Reference 42.
d
Reference 38.
pVTZ, aug-cc-pVQZ,70 and aug-cc-pV5Z70 replacing, in each case, the most diffuse p function with the four p functions used in the default basis set. The imaginary part of the resonance energy changes only slightly when proceeding from the aug-ccpVTZ basis set, and the real part of the resonance energy drops by only 0.1 eV with the expansion of the basis set. 7491
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KT(HF) method gives a value of ER about 1 eV too high and a value of Γ/2 about a factor of 2 larger than the EOM-CCSD result. As expected, these results are very close to those from static-exchange scattering calculations which give86 ER = 3.7 eV and Γ/2 = 0.58 eV. The next two entries in Table 3 are from stabilization calculations using the CIS and CIS(D) methods. The CIS method, which allows for relaxation but does not account for electron correlation, gives resonance parameters very close to the KT(HF) results. The CIS(D) method, which includes correlation effects perturbatively, gives a value for ER only 0.2 eV above the EOM-CCSD result but gives an imaginary part about a factor 10 too small. The reason for the severe underestimation of the width by the CIS(D) and by some of other theoretical methods will be discussed later. The next four entries in Table 3 present the results of stabilization calculations using KT in conjunction with various LC functionals. Both the KT(CAMB3LYP) and KT(ωB97XD) methods yield values of ER that are significantly lower than the EOM-CCSD result and values for Γ that are about an order of magnitude smaller than the corresponding EOM-CCSD results. The KT(LC-BLYP) and KT(LC-ωPBE) fare somewhat better, giving ER values about 0.7 eV lower than the EOM-CCSD result and Γ values about a factor of 2 too small. Thus, none of the range-separated DFT methods, when used with the KT approximation, gives accurate values of the resonance parameters. In contrast, the KT(HFE_BPE) and KT(HFE_BLYP) methods give ER and Γ values in reasonable agreement with the EOM-CCSD results, with the ER value being too large by only 0.2−0.4 eV, and the Γ values too large by a factor of 1.4−1.6. In general, the TDDFT calculations give similar resonance parameters as obtained from the KT calculations using the same functional. The final set of results was obtained with various Green’s function approaches. The d-ADC(2), and G0W0(HF) methods give ER values near 2.8 eV, in reasonable agreement with the EOM-CCSD result. However, both of these methods give a width about 5 times smaller than the EOM-CCSD result. This is apparently caused by the fact that the ADC(2) and G0W0, like the CIS(D) method discussed above, do not allow the orbitals to relax in response to electron correlation effects. Table 3 also reports resonance parameters for three Green’s function methods, GW0(HF), GW0(PBE), and ADC(2), that
Consequently, the smaller basis set will be employed for all subsequent calculations. As indicated in Table 2, the resulting resonance parameters of N2− are in close agreement with those obtained from other high-level ab initio calculations including multireference configuration interaction (MRCI),20 multipartioning perturbation theory (MPPT),42 and EOM-CCSD using a complex absorbing potential (CAP/EOM-CCSD)38 as well as with the empirical parameters deduced from experimental data by Dubé and Herzenberg.22 Table 3 summarizes the resonance parameters of N2− at Req obtained using various electronic structure methods. The Table 3. Complex Resonance Energies of N2− at Reqa method
ER (eV)
Γ/2 (eV)
KT(HF) CIS CIS(D) KT(CAMB3LYP) KT(ωB97XD) KT(LC-BLYP) KT(LC-ωPBE) KT(HFE_PBE) KT(HFE_BLYP) TDDFT(CAMB3LYP) TDDFT(LC-BLYP) TDDFT(LC-ωPBE) TDDFT(HFE_BLYP) TDDFT(HFE_PBE) d-ADC(2) G0W0 (HF) SC_GW0 (HF) SC_GW0 (PBE) ADC(2) EOM-MP2 EOM-CCSD
3.72(5) 3.77(3) 2.760(1) 0.514(6) 1.285(9) 1.90(2) 1.88(1) 3.03(6) 2.80(3) 0.84(4) 1.920(2) 1.960(2) 2.86(2) 3.078(7) 2.760(1) 2.85(1) 2.53(8) 2.15(7) 2.417(3) 2.523(7) 2.580(4)
0.55(4) 0.57(2) 0.020(1) 0.0261(6) 0.047(5) 0.16(4) 0.12(1) 0.38(4) 0.41(5) 0.059(7) 0.179(3) 0.089(7) 0.38(1) 0.27(1) 0.037(3) 0.063(8) 0.20(4) 0.26(3) 0.245(3) 0.28(1) 0.285(4)
a
The numbers in parentheses give the uncertainties in the last decimal resulting from the analytic continuation procedure.
EOM-MP2 method gives resonance parameters essentially identical to those from the EOM-CCSD calculations. The entry labeled KT(HF) is from a stabilization calculation using the virtual orbital energies from HF calculations on neutral N2. The
Table 4. Real Part of the Resonance Energy (eV) for N2− as a Function of Bond Length (Å) bond length (Å) method
1.01767
1.05767
1.09767
1.13767
1.17767
EOM-CCSD EOM-MP2 ADC(2) KT(HF) CIS CIS(D) KT(LC-BLYP) KT(LC-ωPBE) KT(HFE_BLYP) KT(HFE_PBE) TDDFT(LC-BLYP) TDDFT(LC-ωPBE) TDDFT(HFE_BLYP) TDDFT(HFE_PBE)
3.48 3.44 3.37 4.75 4.72 3.74 2.76 2.87 3.73 3.99 2.88 2.96 3.89 4.12
3.01 2.96 2.88 4.26 4.22 3.26 2.39 2.37 3.33 3.50 2.40 2.44 3.60 3.59
2.58 2.52 2.42 3.72 3.77 2.76 1.90 1.88 2.80 2.95 1.92 1.96 2.86 3.08
2.15 2.11 1.99 3.30 3.29 2.35 1.47 1.45 2.39 2.49 1.47 1.52 2.43 2.60
1.75 1.72 1.58 2.85 2.84 1.98 1.08 1.03 1.96 2.03 1.09 1.09 1.98 2.13
7492
avg %RMSD 0.82 3.71 21.71 21.65 4.13 12.67 13.08 4.48 6.86 12.31 11.74 6.27 8.89
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Table 5. Imaginary Part of the Resonance Energy (eV) for N2− as a Function of Bond Length (Å) bond length (Å) method
1.01767
1.05767
1.09767
1.13767
1.17767
EOM-CCSD EOM-MP2 ADC(2) KT(HF) CIS CIS(D) KT(LC-BLYP) KT(LC-ωPBE) KT(HFE_BLYP) KT(HFE_PBE) TDDFT(LC-BLYP) TDDFT(LC-ωPBE) TDDFT(HFE_BLYP) TDDFT(HFE_PBE)
0.52 0.51 0.49 1.05 1.05 0.21 0.35 0.28 0.78 0.75 0.37 0.21 0.76 0.51
0.38 0.37 0.35 0.74 0.81 0.10 0.27 0.21 0.59 0.58 0.27 0.14 0.57 0.38
0.29 0.28 0.24 0.55 0.57 0.02 0.16 0.12 0.41 0.45 0.18 0.09 0.38 0.27
0.20 0.20 0.17 0.43 0.45 0.00 0.10 0.06 0.27 0.31 0.10 0.04 0.27 0.19
0.13 0.14 0.11 0.34 0.34 0.00 0.05 0.04 0.21 0.17 0.05 0.03 0.18 0.13
do allow for orbital relaxation in the presence of correlation effects. These methods give values of the width in fairly close agreement with the EOM-CCSD result. For accurate calculation of properties such as vibrational excitation cross sections it is essential to have accurate values of both the real and imaginary parts of the resonance energy as a function of the relevant geometrical parameters. Thus it is important to establish that the electronic structure method used accurately describes how the resonance parameters change with geometrical distortion. Tables 4 and 5 report, respectively, the calculated real and imaginary parts of the resonance energy as a function of the N2 bond length. The sixth column of each table reports, for each theoretical method, the percent root-meansquare deviation calculated using the differences between the results of that method and the EOM-CCSD results for the five bond lengths used. We consider first the results for the real part of the energy. For the KT and CIS methods the %RMSD errors are about 22%, and for the various range-selected DFT methods the errors are about 12−13%, whereas for the DFT methods using HF exchange for all electron−electron distances they are 4−9%, with the smallest error being for the KT(HFE_BLYP) method. The TDDFT calculations yield results slightly poorer than the KT results obtained using the same functional. Figure 2 reports over a wider range of bond lengths, the real part of the resonance energy calculated using the EOM-CCSD, EOM-MP2, ADC(2), and TDDFT(HFE_PBE) methods. The real part of the resonance energy calculated using the ADC(2) and EOM-MP2 methods are essentially identical to the corresponding EOM-CCSD result over the entire range of bond lengths considered. Although the TDDFT(HFE_PBE) energies are consistently too high, the error in the slope of the ER vs R curve calculated using the TDDFT(HFE_PBE) method is quite small. Table 5 reports the calculated imaginary parts of the N2− energy as a function of the internuclear distance. The % RMSD errors are about 52% for the KT and CIS methods and between 17 and 39% for the CIS(D) and the various DFT methods, with the exception of TDDFT(HFE-PBE), for which it is only 1.7%. From Figure 3, it is seen that the EOM-CCSD, EOMMP2, ADC(2), and TDDFT(HFE_PBE) methods give essentially the same Γ vs R curves over a wide range of bond lengths.
avg %RMSD 1.56 7.61 51.36 53.25 38.86 19.60 26.36 21.95 21.04 19.50 31.65 17.47 1.67
Figure 2. Real part of the resonance energy of the 2Πg state of N2− as a function of bond length calculated using the EOM-CCSD, EOM-MP2, ADC(2), and TTDFT(HFE_PBE) methods.
Figure 3. The imaginary part of the resonance energy of the 2Πg state of N2− as a function of bond length calculated using the EOM-CCSD, EOM-MP2, ADC(2), and TTDFT(HFE_PBE) methods.
To further test the EOM-CCSD, EOM-MP2, ADC(2), and DFT(HFE_PBE) methods for calculation of resonance parameters, the stabilization method was applied to the ground state anions of acetylene (2Πg), ethylene (2B2g), and benzene (2E2u). Results for each molecule are presented in Table 6. For each of these molecules the EOM-MP2 resonance parameters agree closely with those from the EOM-CCSD method, with 7493
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experimental method for characterizing temporary anions is electron transmission spectroscopy (ETS).12 However, in the case of ethylene and benzene, the ground state anions display vibrational structure,12 making it difficult to extract accurate values of the resonance widths as vibrational motion leads to an averaging of the width over a range of geometries. Although the ET spectrum of acetylene does not display vibrational structure, distortion of the molecule from a linear geometry is expected to affect the lifetime, again making it difficult to extract an accurate value of the resonance width from the ET spectrum. (See discussion in ref 12.) The entries for the experimental halfwidths in Table 5 are taken from the widths of the features in electron transmission. Although extensive experimental work has been done for each of the hydrocarbons considered, resonance parameters derived from ET spectra were chosen for comparison because most data on temporary anions of hydrocarbons has been obtained using ET spectroscopy. Both KT(HFE_BLYP) and KT(HFE_PBE) yield values that are too high for both the real and imaginary parts of the resonance energy, with the KT(HFE_PBE) results being consistently closer to those from the EOM-CCSD method. Use of the TDDFT method with the HFE_PBE functional improves agreement with the EOM-CCSD results. The TDDFT(HFE_PBE) method yields reasonable widths but differs from the EOM-CCSD in the predicted values of ER by several tenths of an eV. Similar performance was observed in the N2− system. Table 6 also gives resonance parameters of acetylene, ethylene, and benzene obtained from prior theoretical studies. In general, the resonance parameters from the present study are in better agreement with experiment than are the prior theoretical results. An exception is benzene, for which the finite element discrete model (FEDM) gives a resonance width nearly identical to the EOM-CCSD value but a real part of the resonance energy in much better agreement with experiment.92
Table 6. Complex Resonance Energies (eV) of Ground States of C2H2−, C2H4−, and C6H6− at the Equilibrium Geometry of the Neutral Moleculea ER (eV)
molecule/method −
EOM-CCSD EOM-MP2 ADC(2) KT(HFE_BLYP) KT(HFE_BPE) TDDFT(HFE_PBE) MRCI (ref 88) EOM-CCSD (ref 38) expt (ref 12) EOM-CCSD EOM-MP2 ADC(2) KT(HFE_BLYP) KT(HFE_BPE) TDDFT(HFE_PBE) Kohn var (ref 89)b expt (ref 12) EOM-CCSD EOM-MP2 ADC(2) KT(HFE_BLYP) KT(HFE_BPE) TDDFT(HFE_PBE) SECP (ref 90)c SEP (ref 91)d FEDM (ref 92)e expt (ref 12)
C2H2 ( Πg) 2.77(1) 2.65(3) 2.59(8) 3.0(1) 2.9(1) 2.4(2) 2.99 2.61 2.6 C2H4− (2B2g) 2.06(2) 1.91(3) 1.78(1) 2.58(6) 2.62(7) 2.49(3) 1.38 1.78 C6H6− (2E2u) 1.562(3) 1.546(1) 1.135(3) 2.241(7) 2.211(8) 2.26(3) 1.82 2.23 1.047 1.12
Γ/2 (eV)
2
0.75(6) 0.68(2) 0.71(8) 0.89(8) 0.81(8) 0.6(1) 0.55 0.38 ∼0.8 0.32(3) 0.30(2) 0.245(1) 0.66(3) 0.54(7) 0.31(3) 0.23 ∼0.3 0.045(8) 0.043(5) 0.016(3) 0.15(1) 0.100(8) 0.10(2) 0.09 ∼0.1 0.043 ∼0.06
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a
The numbers in parentheses give the uncertainties in the last decimal resulting from the analytic continuation procedure. bKohn variational method. cStatic exchange−correlation−polarization. dStatic exchange−polarization. eFinite element discrete model.
CONCLUSIONS The 2Πg state of N2− is a valuable benchmark system for evaluating theoretical methods for characterizing temporary anion systems. In this work, several electronic structure methods were used in conjunction with the stabilization method to characterize the real and imaginary pats of the energy of the temporary anion of N2. The EOM-CCSD results were used as the standard of comparison. The less computationally demanding EOM-MP2, ADC(2), and TDDFT(HFE_PBE) methods give results in fairly good agreement with those from the EOM-CCSD calculations. Both the KT(HFE_PBE) and TDDFT(HFE_PBE) methods are more reliable for predicting the resonance parameters than are DFT methods that use a range-separation strategy to provide a correct description of long-range exchange interactions. The TDDFT(HFE_PBE) method performs better than the KT(HF_PBE) method in predicting the resonance parameters of N2− as the bond length is changed. The EOM-CCSD, EOM-MP2, ADC(2), and TDDFT(HFE_PBE) methods were also used to calculate the complex resonance energies of the ground state anions of acetylene, ethylene, and benzene at the equilibrium geometries of the neutral molecules. The EOM-MP2 method gives the best agreement with EOM-CCSD results at considerably less computational cost and so is the most promising technique for application to calculation of the resonance parameters of larger molecules. The ADC(2) method gives far too small a
the real part of the resonance energy being slightly smaller for the EOM-MP2. The real part of the resonance energy from the ADC(2) calculations is 0.2−0.4 eV lower than the corresponding EOM-CCSD values. Although there is no appreciable difference between the imaginary part of the resonance energy of C2H2− derived from the ADC(2) method and that from the EOM-CCSD result, the ADC(2) values of the widths are noticeably smaller for C2H4− and C6H6−. As electron correlation effects are expected to be most important in benzene, it is not surprising that the ADC(2) method, with its relatively simple treatment of electron correlation, gives resonance parameters significantly different from the EOMCCSD results in this case. Both the EOM-CCSD and EOM-MP2 methods give resonance parameters of the three hydrocarbons in good agreement with experiment, although there is a noticeable tendency of the resulting real part of the resonance energy to be higher than the experimental value. This could reflect the need to employ more flexible basis sets or, more likely, the need to include electron correlation effects not recovered by these methods. Obtaining accurate resonance widths from experiment for comparison with the calculated results is more challenging. For polyatomic molecules, the most widely used 7494
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value of the width of the anion state of benzene. The KT(HFE(BLYP) and KT(HFE_PBE) methods give reasonable agreement with the EOM-CCSD results for acetylene and N2 but overestimate both the real and imaginary parts for ethylene and benzene. The TDDFT(HFE_PBE) method provides reasonable values of Γ for all the systems but has errors of several tenths of an electronvolt in ER, with the TDDFT(HFE_PBE) result being too high for all the systems except C2H2−. Nonetheless, our results show that DFT methods using Hartree−Fock exchange are viable methods for characterizing temporary anion states. The inclusion of electron correlation effects is essential for accurate prediction of the real and imaginary parts of the energies of temporary anion states. In addition, for characterizing the resonance widths it is important to use a method that allows for orbital relaxation in response to correlation effects. The EOM-MP2, EOM-CCSD, ADC(2), and SC_GW0 methods all allow for such coupling between correlation and relaxation effects. Even the most sophisticated wave functionbased method considered, EOM-CCSD, tends to produce resonance energies that are somewhat too high. This apparently reflects the need to include still higher-order correlations, e.g., through the EOM-CC3 method.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This research was supported by grants from the National Science Foundation, CHE1111235, and the Grove City College Sweezy Fund. Some of the calculations were carried out on the computers in the University of Pittsburgh Center for Simulation and Modeling. The authors thank Vamsee Voora for helpful discussions.
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