Empirical Correlation Methods for Temporary Anions - The Journal of

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Empirical Correlation Methods for Temporary Anions Thomas Sommerfeld* and Rebecca J. Weber Department of Chemistry and Physics, Southeastern Louisiana University, SLU 10878, Hammond, Louisiana 70402, United States ABSTRACT: A temporary anion is a short-lived radical anion that decays through electron autodetachment into a neutral molecule and a free electron. The energies of these metastable species are often predicted using empirical correlation methods because ab initio predictions are computationally very expensive. Empirical correlation methods can be justified in the framework of Weisskopf
Fano
Feshbach theory but tend to work well only within closely related families of molecules or within a restricted energy range. The reason for this behavior can be understood using an alternative theoretical justification in the framework of the Hazi
Taylor stabilization method, which suggests that the empirical parameters do not so much correct for the coupling of the computed state to the continuum but for electron correlation effects and that therefore empirical correlation methods can be improved by using more accurate electronic structure methods to compute the energy of the confined electron. This idea is tested by choosing a heterogeneous reference set of temporary states and comparing empirical correlation schemes based on Hartree
Fock orbital energies, Kohn
Sham orbital energies, and attachment energies computed with the equation-of-motion coupled-cluster method. The results show that using more reliable energies for the confined electron indeed enhances the predictive power of empirical correlation schemes and that useful correlations can be established beyond closely related families of molecules. Certain types of σ* states are still problematic, and the reasons for this behavior are analyzed. On the other hand, preliminary results suggest that the new scheme can even be useful for predicting energies of bound anions at a fraction of the computational cost of reliable ab initio calculations. It is then used to make predictions for bound and temporary states of the furantrione and croconic acid radical anions.

1. INTRODUCTION When an electron is attached to a closed-shell molecule, as a rule, the radical anion initially formed is less stable than its neutral parent plus a free electron and has thus a finite lifetime. These so-called temporary anions are characterized by their energy above the neutral parent molecule, Er, also referred to as its resonance energy or as the vertical attachment energy of the neutral, and by their lifetimes τ or decay widths Γ = p/τ. Well-known examples include temporary anions formed by attachment into the π* orbitals of CO2 and benzene, and a large number of other temporary anions has been observed.1 Once a temporary anion is formed, it may simply decay back into the neutral and a free electron, but provided the lifetime τ is sufficiently long, there are several other possible reaction pathways, including formation of quasi-stable anions with very long lifetimes essentially controlled by internal vibrational energy redistribution, dissociation of bonds (dissociative attachment), rearrangements, and, if the excess energy can be dissipated into an environment, formation of stable anions. Experimentally, temporary anions can be characterized using different electron-scattering or charge-exchange techniques. One efficient method is electron transmission spectroscopy (ETS) combined with the derivate technique introduced by Sanche and Schulz.1,2 A large body of data has been established using this method, and a detailed bibliography of electron transmission spectra is available at Paul Burrow’s University of Nebraska Website.3 Characterizing temporary anions theoretically is, despite impressive progress in the electronic structure field, still a challenging r 2011 American Chemical Society

task because it combines a continuum with a correlation problem. Wave functions of temporary anion states are part of the electronscattering continuum and, therefore nonsquare-integrable (non L 2) and have at the same time to describe pronounced higherorder electron correlation effects.4,5 Ideally, one would like a method that (1) imposes the correct scattering boundary conditions, (2) treats electron correlation well beyond second-order perturbation theory, and (3) can still be applied to fairly large molecules. Unfortunately, at this time, we do not have any method that comes even close, and realistically, one can have only one out of the three. Computational methods for temporary anions are always a combination of a method to treat the continuum, an electronic structure method, and, in most cases, a basis set. For an overview, it is most convenient to divide all of these combinations into four broad classes according to the way the continuum is treated. Class I are electron-scattering methods, such as the R matrix or the complex Kohn methods (see, for example, refs 6
8) that compute the electron-scattering cross section, which can then be analyzed to characterize temporary anion states. Methods in this class have been combined with single-particle methods and with configuration interaction methods,9 but the configuration spaces are necessarily

Received: March 25, 2011 Revised: May 17, 2011 Published: May 17, 2011 6675

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much smaller than those used in configuration interaction calculations for ground and excited states of systems of similar size. Class II are methods that reformulate the scattering problem in terms of calculations that explicitely use only square-integrable wave functions such as the complex scaling,10,11 complex absorbing potential,12,13 and stabilization methods.14,15 All of these methods use a parametrized Hamiltonian, and in a first step, several eigenvalues of the Hamiltonian are computed as a function of the parameter. Energies and lifetimes of temporary states can then be extracted from the accumulated data. Because squareintegrable wave functions are used, standard bound-state techniques can be applied, but because several eigenvalues are needed, Class II continuum methods can only be combined with electronic structure methods that yield several states. Class III are extrapolation methods. Here, a stabilizing potential is added to the Hamiltonian such that the temporary state sought becomes bound. The energy of this bound state is then computed as a function of the strength of the stabilizing potential and extrapolated to vanishing stabilization. Examples include increasing the nuclear charges and extrapolating back to the physical charge,16,17 as well as putting the system into a solvent and extrapolating to zero permittivity.18,19 As one needs to compute only a single bound state, extrapolation methods can be combined with any electronic structure method. However, it is by no means clear whether the energy of a bound state can be smoothly extrapolated through a continuum threshold, and extrapolation methods work best if the state in question is not too high in the continuum and not totally symmetric. Finally, Class IV are methods that establish an empirical correlation between experimental energies of temporary anions and computed energies of an excess electron confined into the same space occupied by the molecule. These methods are the focus of the present paper and are discussed in more detail in the next section. Here, we note that empirical correlation methods can be set up with any electronic structure method; however, if higher temporary anion states are to be included (for example, benzene has two temporary anion states (2E2u and 2B2g), electronic structure methods suitable for computing excited states are needed. In the next section, empirical correlation methods are briefly reviewed, including the electronic structure methods and basis sets combined with them. The linear fit parameters used in this context can be understood as to “correct” the computed energy for (1) the interaction of the computed discrete state with the continuum and (2) all errors resulting from missing electron correlation effects and incomplete basis sets. The focus of this study is to combine the empirical correlation method with coupled-cluster methods and various basis sets and to investigate whether the predictive power of empirical schemes can be enhanced by decreasing the empirical corrections due to (2). All empirical methods need a set of reference data, and in section 3, the set used here is briefly discussed. In section 4, results from combining empirical correlation techniques with the equation-of-motion coupled-cluster method are presented, and the performance is compared with schemes that use Hartree
Fock and Kohn
Sham orbital energies. The new method is used to predict resonance states of croconic acid and furantrione, and section 5 concludes.

2. EMPIRICAL CORRELATION METHODS Empirical correlation, or Class IV methods, aim at establishing a linear relationship Eexp ¼ mEcomp þ b

ð1Þ

where Ecomp is a computed energy, Eexp is the experimentally determined resonance energy, and m and b are empirical parameters, which are typically fitted to a training set and can then be used to predict unknown resonance energies. The key point of these methods is that, in principle, any electronic structure method may be used to obtain Ecomp and that, again in principle, good correlations may result even with methods that are computationally orders of magnitude less expensive than applying Class I
III methods. In this section, we briefly review Class IV method combinations that have been used in the past and discuss physical justifications for this type of approach. Empirical correlation methods have initially been used (and are still used) as qualitative tools in contexts such as assignment of electron transmission spectra1 and extrapolation of adiabatic electron affinities in closely related families of molecules (see, e.g., refs 20 and 21). Later, in the early 1990s, two landmark contributions22,23 established empirical correlation methods as quantitative tools; Chen and Gallup22 put empirical correlations on a firm theoretical basis within the framework of Weisskopf
Feshbach
Fano theory,24
26 and Staley and Strnad23 performed the first systematic study using several basis sets and a large data set of vertical electron affinities. More recent developments focus on combining correlation methods with different electronic structure methods. The methods established in refs 22 and 23 and practically all earlier work rely on Koopmans’s Theorem (KT),27 that is, a Hartree
Fock or semiempirical MO calculation is performed for the neutral molecule, and the orbital energies of the relevant unoccupied orbitals are used for Ecomp (owing to the sign convention of scattering theory, the usual change of sign in KT is not needed here). Over the past decade, three new schemes based on density functional theory (DFT) were introduced. Modelli’s method still uses an orbital energy, but the Hartree
Fock orbital is replaced by a Kohn
Sham orbital,28 Tozer and De Proft use an expression derived from the Fukui function that can be understood as the Kohn
Sham orbital energy plus a correction term,29
31 and Vera and Pierini introduced a somewhat more expensive method computing Ecomp from the difference of two independent DFT calculations for the neutral and the anion (ΔDFT).32 However, the latter two groups focused on the lowest attachment state only and therefore on the narrower energy range of up to 3 eV. All empirical correlation methods are based on the idea that the energy of an excess electron spatially confined to the molecular valence region provides a measure for the resonance energy. One way of connecting the spatially confined excess electron to the scattering wave function is Weisskopf
Feshbach
Fano theory,24
26 where the temporary anion state is described as a discrete state |dæ that is embedded in and interacting with a continuum. This physical justification for empirical correlation approaches is discussed in great depth in ref 22. Another argument has been made more recently in the context of conceptual DFT in refs 29 and 30. Here, we present a justification using the language of the Hazi
Taylor stabilization method,14 which lets the empirical parameters b and m appear in a new light. In the stabilization method, the molecule is put into a “box” that can be represented explicitly by a box-like potential33,34 or implicitly through a localized basis set14,35 (cf., ref 36). Owing to the box potential, the continuum of the original Hamiltonian is replaced with a discretized pseudocontinuum, and when the size of the box is systematically varied, so are the energies of the pseudocontinuum states. The coupling of the discrete state |dæ with the pseudocontinuum then becomes visible as a series of avoided crossings 6676

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Figure 1. Schematic representation of a stabilization plot. A temporary state is characterized by a series of stabilization plateaus and avoided crossings. The innermost stabilization plateau is indicated by an arrow.

in a so-called stabilization plot (Figure 1), from which the energy of the temporary anion can be obtained from the stabilization plateaus between the crossings, and the lifetime can be extracted from the crossings themselves.15,33 In the present context, the relevant feature of the stabilization plot is the innermost stabilization plateau indicated on the left-hand side of Figure 1. If it was possible to choose a basis set corresponding to the box size associated with the innermost stabilization plateau, one had a method to compute the energy of a temporary anions state with a single calculation. Clearly, this perfect basis set does not exist because the precise position of the stabilization plateaus will depend on the molecule as well as on the energy of the resonance state itself, and thus even for the same molecule, different resonances will have plateaus at different box sizes. However, established empirical correlation schemes suggest that while no basis set or explicit confinement is perfect, some are close enough to be useful. The difference between the stabilization and the Weisskopf
Feshbach
Fano justification22 is that in the latter, the coupling to the continuum is already included in the computed value Ecomp. This suggests that the role of the empirical parameters m and b in eq 1 is not so much to account for the coupling with the continuum but rather to correct for shortcomings of the electronic structure method, and using more reliable electronic structure methods to compute Ecomp should yield better correlations as outlined in the Introduction. Before testing this assumption with numerical results, let us briefly discuss two technical aspects of empirical correlation methods. First, as mentioned above, empirical correlations can, in principle, be combined with any electronic structure method, yet many molecules form more than one temporary anion state (e.g., butadiene forms two) or form both bound and temporary anions (e.g., p-benzoquinone), and unless only the lowest state is of interest, a method to compute several attachment states is needed. This requirement rules out many standard ground-state methods and makes methods attractive that, similar to KT, start from the closed-shell neutral, and directly yield several attachment states. Here, the equation-of-motion coupled-cluster with single and double substitutions method (EOM-CCSD)37 is employed. Alternatives are methods based on the one-particle Green’s function and multireference configuration interaction methods.

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Second is the choice of basis set. It is well-established that if the basis set itself is used as a confinement, correlation with experimental attachment energies strongly degrades when diffuse basis functions, that is, functions with exponents substantially smaller than those used in standard valence basis sets, are included.23,38 This may seem counterintuitive as including diffuse functions in the form of “þ” or “aug” functions is essential when investigating bound anions and is even necessary to reliably compute certain properties of neutral systems such as intermolecular interactions. Yet, augmenting the basis set with diffuse functions substantially increases the size of the basis set box and, therefore, substantially decreases the energy of the lowest pseudocontinuum state. Thus, if temporary anions are studied with diffuse basis sets, the lowest state is likely to have considerable pseudocontinuum character, or in the language of the stabilization method, the box size has been increased beyond the innermost stabilization plateau, and consequently the computed energy is no longer associated with the state of interest. For certain types of resonances, this may even happen when only standard valence bases are used,39 a point that we will come back to several times in the discussion of our numerical results.

3. REFERENCE DATA The goal of the paper is to investigate empirical methods for predicting resonance positions, that is, vertical electron attachment energies, a property directly measured in electron-scattering techniques. In particular, by using electron transmission spectroscopy (ETS), a large body of experimental data has been accumulated, and ETS is used here as the primary data source. The reference molecules considered were selected based on several criteria. First, we were aiming for a heterogeneous set in the sense that resonances associated with antibonding orbitals of double bonds, triple bonds, aromatic systems, and sigma bonds were included and that, in addition, for each of these bond types, molecules with different heteroatoms participating in the relevant functional groups were chosen. Therefore, for example, our set representing π*-type resonances associated with double bonds contains a variety of molecules with CdC, CdN, and CdO bonds. This choice practically neglects the more subtle variations within a given family, that is, different substitution patters in benzene, which empirical methods reproduce rather well,23,40 but rather focuses on the coarse trends with which empirical methods are struggling.28 Clearly, the availability of a published ETS value for the resonance energies of a given molecule is a necessary condition, and from the available data, fairly small, symmetrical molecules were selected because this choice minimizes issues with the assignment of the computed states to the observed signals as well as issues with conformer mixtures in the experiments. Table 1 lists the 35 selected molecules having a total of 66 resonance states. 4. AB INITIO RESULTS The first step of the computational study was to find minimal energy geometries for all 35 molecules. In view of a consistent method that is applicable for larger molecules, all optimizations were performed using many-body second-order perturbation theory (MP2) and the 6-31G* basis set.41 This method is expected to be a reasonable trade-off between practicability and the need to compute rather accurate gas-phase conformations. The quality of the employed equilibrium geometry impacts the empirical fit as the computed energy, Ecomp, does depend on it. For π* resonances, 6677

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Table 1. Reference Molecules and Their Resonance Statesa π* Resonances of Double Bonds

π* Resonances of Triple Bonds

1.78 (b2g)49

ethylene (D2h)

00 50

propene (Cs)

1.99 (a )

49

buta-1,3-diene (C2h)

cyanogen (D¥h)

0.58 (πu), 5.37 (πg)46

00 1

1.51 (b1)

carbon dioxide (D¥h)

2.84 (e)51 1.75 (π)52

hydrogen cyanide (C¥v)

2.26 (π)51

1

dinitrogen (D¥h)

2.2 (πg)54

55

butadiyne (D¥h)

1.00 (πu), 5.60 (πg)46

56

hexa-2,4-diyne (C2h)

1.40 (au/bu), 5.15 (ag/bg)46

1.19 (a )

acetone (C2v)

acetonitrile (C3v) carbon monoxide (C¥v)

2.05 (a )

acetaldehyde (Cs)

2.60 (π1g)1

00 53

0.62 (au), 2.8 (bg)

formamide (Cs)

acetylene (D¥h)

3.58 (πu)

formaldehyde (C2v) formic acid (Cs)

0.86 (b1) 1.8 (a00 )55

glyoxal (C2h)

1.9 (bg)57

p-benzoquinone (D2h)

0.69 (b2g), 1.41 (au), 4.37 (b2u)58

dimethyl diazine (C2h)

0.83 (bg)59 Systems with σ* Resonances

chloromethane (Cs) dichloromethane (C2v)

π* Resonances of Aromatic Systems 60

3.45 (a)

60

1.23 (a1), 3.38 (b2) 0

00 60

benzene (D6h)

1.12 (e2u), 4.82 (b2g)61

furan (C2v)

1.73 (b1), 3.15 (a2)62

dichlorofluoromethane (Cs) dichlorodifluoromethane (C2v)

0.96 (a ), 2.81 (a ) 0.98 (a1), 2.35 (b2), 3.88 (b1)60

pyrazine (D2h) pyridine (C2v)

0.07 (b3u), 0.87 (au), 4.10 (b2g)63 0.62 (b1), 1.16 (a2), 4.58 (b1)64

dichloroethane (C2h)

1.70 (bu), 2.40 (ag)65

thiophene (C2v)

1.15 (b1), 2.63 (a2)62

dimethyl disulfide (C2h)

66

1.04 (bu), 2.72 (bu)

Systems with More than One Type of Resonance State chlorobenzene (C2v)

0.75 (a2), 0.75 (b1), 2.42 (a1), 4.39 (b1)67

phenyl acetylene (C2v)

0.35 (b1), 1.03 (a2), 2.41 (b2), 3.28 (b1), 4.83 (b1)68

benzonitrile (C2v)

0.57 (a2), 2.57 (b2), 3.19 (b1), 4.62 (b1)51

nitrobenzene (C2v)

0.55 (a2), 1.36 (b1), 4.69 (b1)69

a

For each molecule, the experimentally observed resonance positions (in eV), the symmetry of the conformation used in the present calculations, and the symmetry label of the relevant virtual orbitals are listed. For some of these molecules, one may expect that several conformations are populated in the experiments; dimethyl disulfide is one example. However, at least in this case, the resonance energy does not depend strongly on the conformation, and only the listed conformations are included in the computational study.

this dependence is expected to be rather weak; however, for σ* resonances, the change in bond-order upon anion formation is drastic, and, in particular, for σ* states associated with a particular bond, that is, for strongly localized σ* states, the resonance position is known to depend strongly on the geometry. This behavior contributes to the disappointing performance of empirical methods for σ* resonances, a point we will come back to below. Electron attachment energies were then computed with KT and the EOM-CCSD method using four different Dunning basis sets, the double-ζ and triple-ζ sets established in the early 1970s (DZP and TZVP)42,43 and the more recent correlation consistent valence double-ζ and triple-ζ sets (cc-pVDZ and cc-pVTZ).44,45 Let us briefly note that replacing Hartree
Fock with Kohn
Sham orbital energies as suggested in ref 28 leads, for the reference set considered here, to a slightly worse correlation with experiment and is therefore not reported in detail. A plot of the experimental reference data versus the KT and EOM-CCSD results obtained with the DZP set is shown in Figure 2. The figure shows in the first place the well-understood trend4 that, owing to the missing correlation effects, the KT attachment energies are generally higher than the EOM-CCSD ones. Moreover, even without detailed analysis, a substantially stronger scatter of the KT values can be seen. Results obtained with the other three basis sets are different for each molecule, yet the overall picture is very similar for all four basis sets used.

Figure 2. Comparison of the correlations (eq 1) obtained using Koopmans’s Theorem and the EOM-CCSD methods with the DZP basis set. For KT, the rms is 0.45 eV; for EOM-CCSD, it is 0.34 eV (cf. Table 2).

The visual impression of a stronger scatter of the KT results is substantiated by the linear fit results listed in Table 2. For the DZP basis set, the average deviation from the linear relationship of eq 1 decreases from 0.45 eV for KT-based results to 0.34 eV for EOM-CCSD electron attachment energies. In comparison with the DZP set, the overall fit to the results from the other basis sets 6678

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Table 2. Fitted Parameters, m and b, for Use in Equation 1a KT

EOM-CCSD

Ndf

rms

m

b

DZP

63

0.45

0.691


1.19

TZVP cc-pVDZ

58 61

0.51 0.48

0.793 0.659


1.27
1.17

cc-pVTZ

59

0.53

0.701


1.06

DZP

63

0.34

0.838


0.802

TZVP

61

0.37

0.909


0.620

cc-pVDZ

63

0.37

0.816


0.810

cc-pVTZ

62

0.36

0.857


0.397

a Fits based on computed energies, Ecomp, obtained from KT or EOMCCSD calculations with different basis sets are shown, including the number of degrees of freedom of the fit, Ndf, and the root-mean-square deviation of the data from the fitted relationship. The Ndf values are different for different basis sets because with some basis sets, certain σ*resonances cannot be identified (see text). Root-mean-square deviations and values for the parameter b are given in eV.

Figure 3. Deviations of the resonance energies predicted using eq 1 from the experimental reference data. Predictions are based on Ecomp from EOM-CCSD/DZP and the parameters listed in Table2.

is slightly worse, yet the improvements in going from KT to EOM-CCSD are somewhat larger, and at the EOM-CCSD level, the performance of all four basis sets is similar (Table 2). Another aspect of the fitting results are the parameters m and b themselves. In the sense that fully converged stabilization calculations may be expected to reproduce the experimental results, “better” methods should approach m = 1 and b = 0. Parameters derived from EOM-CCSD attachment energies are indeed much closer to this ideal than the parameters from fitting KT results with slopes, m, increased to about 0.85, and typical shifts, b, decreased by about 0.5 eV (Table 2). Nevertheless, EOM-CCSD combined with any of the four basis sets used here is still a fair bit from the unit-slope zero-shift ideal with the cc-pVTZ set performing best in this sense, m = 0.857 and b = 0.397 eV. Despite the better performance of empirical correlation with EOM-CCSD derived attachment energies, the well-documented finding that empirical correlations work best for closely related families of molecules is still apparent in the data. The errors of the resonance energies predicted by eq 1 are plotted versus the experimental resonance position in Figure 3, with a color coding breaking the reference set down into four broad families, π* resonances associated with aromatic systems, double bonds, and

Figure 4. Contour plots of the first three virtual orbitals of dichloromethane obtained with the TZVP basis set. Cuts through the Cl
C
Cl plane are shown. The difference between two contours is 0.025 Bohr
1.5.

triple bonds, as well as σ* resonances. (The σ* resonances are further separated into totally symmetric (a) and not totally symmetric states (b); see the discussion below.) Positions of σ* resonances are by far the most problematic to predict with errors as large as 0.8 eV. Positions of π* resonances of double and triple bonds are, with the exception of a few states, far more reliable, and π* resonances associated with aromatic systems can be predicted fairly accurately as all aromatic π* states show small errors of less than 0.3 eV. 6679

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Table 3. Comparison of Empirically Predicted Vertical Electron Attachment Energies with Ab Initio Results for Bound Anionsa ΔCCSD(T)

EOM-CCSD DZP

Out of the molecules with double and triple bonds, two systems stand out. For the 2Πu resonance of CO2 and for both π* resonances of NCCN, the errors are consistently large. Part of this problem may be the challenging assignment of the NCCN resonance close to the threshold,46 but for CO2 and the second resonance state of NCCN, there seems to be no simple explanation, and only studying more basis sets will show whether the observed trend is really systematic. For σ* states, the situation is somewhat clearer. In the first place, as noted before,39 there can be pronounced mixing with continuum states even if valence basis sets without any diffuse functions are used. One example for this behavior is shown in Figure 4. If the TZVP basis set is used for CH2Cl2, there are two low-energy virtual orbitals with a1 symmetry that both show the correct nodal structure with respect to the carbon
chlorine bonds, indicating that the relevant σ* orbital is strongly mixing with the lowest pseudocontinuum state in that symmetry. In the language of the stabilization method, the TZVP set provides a confinement such that the system is already beyond the first stabilization plateau in the region of the first crossing. In contrast, in b2 symmetry, there is only one orbital that has the correct antibonding character (lowest panel of Figure 4). As one would expect from particle-in-the-box energies, strong mixing with the lowest pseudocontinuum state occurs only for totally symmetric states and more often with triple-ζ than with double-ζ basis sets. Only for much larger molecules will other symmetries be affected. This mixing is far less problematic when the EOM-CCSD method is used because the computed resonance positions are much lower in energy, whereas the energies of pseudocontinuum states remain essentially the same. Moreover, when EOM-CCSD is used, there are two criteria that one can use to judge the extent of pseudocontinuum mixing. First, in all cases considered here, a single natural orbital with an occupation number near unity can be identified, and the nodal structure of this natural orbital can be analyzed. Second, the different attachment states can be distinguished by their single-particle character, with pseudocontinuum states typically showing a single particle character close to 100%, while states describing resonance show values in the 90
95% range. The second issue with σ* states is that, as discussed above, the resonance energy depends strongly on the relevant bond length. Thus, energies of σ* are much more sensitive to the geometries employed in the calculations, and it might be necessary to use well-converged geometries to improve the reliability of correlation methods for σ* resonance. More research is needed to investigate this effect further. One important test for empirical correlation methods is their performance in the threshold region, that is, whether bound anions are indeed predicted to be bound and whether resonances are indeed predicted to be metastable. The former can be tested using three molecules in the test set, p-benzoquinone, nitrobenzene, and glyoxal, which are able to bind an excess electron at their equilibrium geometries, and the same is true for the two molecules, croconic acid and furantrione (Figure 5), for which resonance energies will be predicted below. The vertical electron


1.88


1.82

nitrobenzene

b2

0.02180


0.784


0.59

trans-glyoxal

au

0.02366


0.782


0.70

croconic acid

b1 a2


0.9196
0.7188


1.57
1.40


1.43
1.22

furantrione

a00


1.429


2.00


2.07

benzoquinone b2g

Figure 5. Molecular structure of 2,3,4-furantrione and croconic acid.


1.286

predicted VAE eq 1 AUG-cc-pVTZ

a

Five of the investigated molecules form electronically stable anion states at the equilibrium geometries of the neutral, and for these stable states, the attachment energy computed with the EOM-CCSD/DZP method and the corresponding empirical prediction from eq 1 (m and b taken from Table 2) are compared with ΔCCSD(T)/AUG-cc-pVTZ results. Note that the electron-scattering sign convention is used and that negative values imply energies below the neutral. All values are in eV.

Table 4. Predicted Vertical Attachment Energies of Croconic Acid and Furantrione (cf., Figure 5)a EOM-CCSD/DZP croconic acid

furantrione

predicted VAE

1π*(b1)


0.9196


1.57

2π*(a2)


0.7188


1.40

3π*(a2)

4.558

4π*(b1)

4.948

3.34

1π*(a00 )


1.429


2.00

2π*(a00 )

1.803

0.71

3π*(a00 )

4.840

3.25

3.02

Attachment energies corresponding to antibonding π states have been computed using the EOM-CCSD method and the DZP basis set, and from these results, the resonance energies are predicted using eq 1 with m and b taken from Table 2.

a

attachment energy of these systems is not available from ETS (or any other direct experimental measurement), but as the anions are bound, the attachment energies can be calculated with standard methods. Table 3 compares the attachment energies predicted from the EOM-CCSD/DZP calculations through eq 1 with results from CCSD(T)/AUG-cc-pVTZ calculations. The differences between the empirically predicted attachment energies and the coupled-cluster results are surprisingly small, less than 0.2 eV in all cases, which is about the level of accuracy expected for the CCSD(T)/AUG-cc-pVTZ method in the first place. Thus, not only can the established empirical scheme be expected to reliably predict whether radical anions are stable or metastable, but it seems that it even can be used to predict binding energies of radical anions at a fraction of the computational cost normally required for reliable ab initio results. Last, the EOM-CCSD/DZP based empirical method is used to predict resonance energies for two molecules for which we have not been able to find any experimental results. Croconic acid is one of the rare organic molecules that form ferroelectric crystals at room temperature,47 and 2,3,4-furantrione is a well-known reducing agent48 and a model for the oxidized form of vitamin-C. Similar to p-benzoquinone, these two molecules can form stable radical anions at their respective equilibrium geometries, but they 6680

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The Journal of Physical Chemistry A also possess two metastable states each. The predictions listed in Table 4 show that furantrione can be expected to show two ETS signals, one below 1 eV and one somewhat above 3 eV. Croconic acid, on the other hand, is predicted to have two metastable states close in energy somewhat above 3 eV, and these two will probably merge into a single ETS signal.

5. SUMMARY AND CONCLUSIONS The equation-of-motion coupled-cluster method has been used to establish empirical correlation schemes for predicting the energies of temporary anion states. This study was initiated on the basis of an alternative theoretical justification of empirical correlation methods that suggests that the role of the empirical parameters is not so much to account for the coupling of the computed state to the electron-scattering continuum but to correct for the approximate treatment of electron correlation and use of an incomplete basis set. Numerical results substantiate this notion as empirical correlations based on computed energies from equation-of-motion coupled-cluster calculations considerably improve schemes based on orbital energies from Hartree
Fock or Kohn
Sham calculations. In contrast to the latter schemes, which have been used as a work horse for assigning ETS spectra, coupled-cluster-based correlations are far less sensitive to the employed basis set and allow one to establish correlation methods beyond families of closely related molecules. In the present work, a heterogeneous set of reference molecules was studied that was comprised of π* resonances associated with double bounds, triple bonds, and aromatic systems, as well as σ* resonances, and showed in each of these classes a considerable variety of heterosubstituted systems. Empirical correlations were investigated using orbital energies from Hartree
Fock and density functional calculation as well as attachment energies computed with the equation-of-motion coupled-cluster method. The EOM-CCSD results do not only yield a better correlations in the sense of a reduced root-mean-square deviation but also predict lower energies for the confined electrons such that especially higher lying resonances can be untangled more easily from the pseudocontinuum. As a byproduct of these lower energies, EOM-CCSD correlations also yield empirical parameters closer to the ideal m = 1 and b = 0 that is expected from fully converged computations. There are still problematic cases, in particular, for σ* resonances, root-mean-square deviations are large. An analysis of the associated orbitals shows that, in particular, for carbon
chlorine bonds, the lowest σ* resonances can already be mixed into the pseudocontinuum even if valence basis sets without additional diffuse functions are used (see also ref 39). In these cases, σ* orbital energies can obviously not be used to establish a meaningful correlation. Again, due to the lower energies predicted for confined electrons at correlated levels of theory, EOM-CCSD states are less prone to mixing with the lowest pseudocontinuum states. Moreover, the EOM-CCSD states can be analyzed in terms of the nodal pattern of the natural orbitals and in terms of their one-particle character, enabling a more reliable judgment of the extent of pseudocontinuum character in attachment states. Finally, it is worth noting that the energies of σ* resonances will vary much more strongly across the Franck
Condon zone of the neutral than the energies of π* resonances and that a part of the problem with establishing a satisfactory correlation for σ* resonances may be the use of rather approximate MP2/6-31G* geometries. More research is needed to settle this issue.

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In the empirical correlation schemes investigated, the basis set has two tasks. It needs to be sufficiently flexible to describe the electronic structure of the radical anions at a correlated level, and at the same time, it is needed to provide the confinement of the excess electron. In this context, it has been pointed out repeatedly that adding diffuse functions to the basis set can destroy the correlation between the experimental and computed orbital energies as the calculations increasingly describe a free electron,23,38 and the same conclusion was reached more recently for density functional approaches.31 Yet, other authors focusing more on the low-energy resonances advocated diffuse basis functions in a density functional context.32 Our results show that the empirical correlation indeed degrades from above, as one would expect based on the position of the lowest pseudocontinuum state, and that the EOM-CCSD derived methods are far less sensitive than orbital energy correlations, again, owing to the electron correlation stabilization of anions in the sense that EOM-CCSD electron attachment energies are far lower than the corresponding KT values while the position of the lowest pseudocontinuum state is practically independent of electron correlation. Thus, if the energy range of up to 6 or 7 eV is of interest, diffuse functions cannot be used. On the other hand, if only states in the energy range of about up to 3 eV are of interest, standard diffuse functions can be added to EOM-CCSD (or presumably density functional methods) without destroying the correlation. However, diffuse functions do not markedly improve the correlation in a lower root-mean-square deviation sense, and the same is true for higher-quality valence basis sets. The empirical parameters, on the other hand, are closer to the converged m = 1, b = 0 if better valence sets or diffuse functions are used. The established EOM-CCSD/DZP correlation is computationally more expensive than approaches using orbital energies, but only very slightly so. The main reason is the small DZP basis set, which tends to make the initial MP2 geometry optimization more time-consuming than the EOM-CCSD calculation itself. Thus, for molecules in the size range considered here, EOM-CCSD calculations can be done with the same computational resources needed for less reliable orbital-energy-based approaches.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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