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Negative Electron Affinities from DFT: Fluorination of Ethylene Michael J. G. Peach,† Frank De Proft,‡ and David J. Tozer*,† †
Department of Chemistry, University of Durham, South Road, Durham, DH1 3LE United Kingdom, and Eenheid Algemene Chemie (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium
‡
ABSTRACT Four simple density functional theory methods are investigated to assess how well they can describe subtle variations in negative vertical electron affinities arising due to the successive fluorination of ethylene, where the magnitude of the variations is well below the absolute accuracy of the methods. Three of the approaches (two based on Koopmans' theorem and one using a potential wall) are able to describe the trends in the affinities using both compact and very diffuse basis sets. The best results are obtained using the potential wall. The fourth approach uses a conventional energy difference evaluation, which is only applicable when the basis set is compact; the breakdown upon addition of diffuse functions is strikingly illustrated. The results are interpreted in terms of the spatial extent of the relevant orbitals and the signs and values of the orbital energies. The study highlights the potential predictive value of simple DFT methods for describing trends in negative affinities. SECTION Molecular Structure, Quantum Chemistry, General Theory
T
Kohn-Sham density functional theory (DFT) calculation; the values are often empirically corrected,3,4 and stabilization approaches5,6 must be used to distinguish resonance solutions from discretized continuum solutions when the orbital energy is positive. (There has recently been renewed interest in the use of Koopmans' theorem in DFT,7,8 which, unlike Hartree-Fock, formally provides exact ground-state first ionization potentials and electron affinities, providing appropriate account is taken of the integer discontinuity.9,10) Alternatively, eq 1 can be used in conjunction with a scheme that artificially binds the excess electron, for example, using a compact basis set,11 point charge,12 or dielectric.13 We previously proposed14 that the negative electron affinity of a neutral N-electron system can be approximated in DFT using the simple expression
he vertical electron affinity of a neutral, N-electron system is defined as A ¼ EN - EN þ 1
ð1Þ
where EN and ENþ1 are the electronic energies of the neutral and anion, respectively, determined at the geometry of the neutral. In many cases, the anion is energetically unstable with respect to electron loss (neglecting dipole-bound or similar weakly bound states), and so the ground-state energy of the anion equals that of the neutral and A=0. Despite this, the experimental A in such cases is commonly stated to be negative, as determined from electron transmission spectroscopy1 (ETS). This corresponds not to the ground-state anion but rather to the short-lived temporary anion (shape resonance) associated with electron capture in an unoccupied orbital of the neutral, which is readily observed with ETS. The ab initio calculation of negative electron affinities that reproduce ETS values is a challenge. Straightforward application of eq 1 with any variational electronic structure method will yield a negative affinity if the basis set used to expand the one-electron orbitals is sufficiently compact; the excess electron in the anion will be artificially bound by the basis. As the basis set becomes increasingly diffuse, however, the electron will tend to leave the system, and the affinity will approach the ground-state value of A=0. A common way forward is to avoid the anion calculation completely and instead use a Koopmans expression2 A ¼ - εLUMO
A ¼ - ðεLUMO þ εHOMO Þ - I
where εHOMO and εLUMO are the highest occupied molecular orbital (HOMO) and LUMO energies of the neutral, determined from a conventional DFT calculation using a local exchange-correlation functional (e.g., the local density approximation or a generalized gradient approximation (GGA)). The quantity I is the standard vertical ionization potential of the neutral determined from the total electronic energies of the cation and neutral using the same functional I ¼ EN - 1 - EN
ð2Þ
ð4Þ
Received Date: July 30, 2010 Accepted Date: September 7, 2010 Published on Web Date: September 13, 2010
where εLUMO is the lowest unoccupied molecular orbital (LUMO) energy of the neutral, from a Hartree-Fock or
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ð3Þ
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Equation 3 represents a simple correction to the DFT Koopmans value in eq 2, based on a consideration of how local functionals treat the integer discontinuity. The method exploits an approximate symmetry between electron addition and subtraction, using information about the cation and neutral to predict the energy of the (problematic) anion, without the need for an explicit calculation on the anion. See ref 9 for a detailed discussion of the theoretical arguments behind the method. In related work, we proposed15 an explicit Kohn-Sham approach where a potential wall is used to bind the excess electron in the anion, thus allowing electron densities, molecular orbitals, and properties of temporary anions to be determined. In the approach, a sphere of radius λRA is constructed around each atom A, where λ is a scaling parameter and RA is the Bragg-Slater radius. Within each sphere, the exchange-correlation potential in the anion Kohn-Sham equation is set equal to the conventional potential. Outside the spheres, the potential is set equal to a spin- and orbital-energy-dependent, r-independent value, chosen to recover the appropriate spin-density asymptotic decay of the anion from a knowledge of the neutral orbital energies. The parameter λ was originally chosen to ensure that the affinity of the neutral determined using A ¼ EN - ENλ þ 1
Table 1. Vertical Electron Affinities, in electronvolts eq 1
eq 3
eq 5
ETSa
cc-pVTZ C2H4
-2.45
0.93
-2.98
-2.71
-1.78 ( 0.05
C2H3F
-2.63
0.82
-3.07
-2.84
-1.91 ( 0.05
trans-C2H2F2
-2.70
0.82
-3.06
-2.85
-1.84 ( 0.05
cis-C2H2F2 1,1-C2H2F2
-2.72 -2.94
0.69 0.61
-3.18 -3.24
-2.99 -3.09
-2.18 ( 0.05 -2.39 ( 0.05
C2HF3
-3.07
0.57
-3.29
-3.17
-2.45 ( 0.05
C2F4
-3.39
0.37
-3.47
-3.45
-3.00 ( 0.05
aug-cc-pVTZ C2H4
-1.44
1.09
-2.79
-2.68
-1.78 ( 0.05
C2H3F
-1.46
1.02
-2.82
-2.76
-1.91 ( 0.05
trans-C2H2F2
-1.54
1.07
-2.76
-2.71
-1.84 ( 0.05
cis-C2H2F2 1,1-C2H2F2
-1.65 -1.38
0.89 0.85
-2.92 -2.95
-2.93 -2.99
-2.18 ( 0.05 -2.39 ( 0.05
C2HF3
-1.42
0.79
-3.00
-3.07
-2.45 ( 0.05
C2F4
-1.96
0.54
-3.24
-3.44
-3.00 ( 0.05 -1.78 ( 0.05
aug-cc-pVTZþp
ð5Þ
is close to the value from eq 3; EN is the conventional energy λ of the neutral, whereas ENþ1 is the energy of the anion determined using the potential wall for a given λ. More recently, it was shown16 that a system independent value of λ = 3.4 yields very similar results, and this will be used in the present study. Given the non-ground-state nature of negative affinities, it is unsurprising that eqs 1-3 and 5 do not provide values in quantitative agreement with ETS values. Equation 1 can break down completely, as discussed above, whereas values from eqs 3 and 5 are typically in error by 0.5 eV.17 Values from eq 2 can even have the incorrect sign!17 However, in many circumstances, particularly in the field of conceptual DFT,18 it is more important to be able to reproduce variations or trends, as opposed to absolute values, and we have illustrated that these simple methods can be successful at doing this.16,17 The aim of the present Letter is to consider a much more demanding test of the methods; we shall assess how well they can describe subtle variations in negative affinities arising because of changes in molecular structure, where the magnitude of the variations is well below the absolute accuracy of the methods. As the specific test, we consider the successive fluorination of ethylene, from C2H4 to C2F4, paying attention to the variation in the affinity with the number of fluorine atoms and with isomerism in the difluoro-substituted systems. The influence of diffuse basis functions is critically assessed. We consider the seven systems C2H4, C2H3F, 1,1-C2H2F2, cis-1,2-C2H2F2, trans-1,2-C2H2F2, C2HF3, and C2F4. Experimental ETS affinities from ref 19 are listed in Table 1. The error bars are such that the order of the experimental affinities cannot be unambiguously assigned across all seven systems.
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eq 2
C2H4
0.01
1.09
-2.79
-2.68
C2H3F
0.02
1.02
-2.82
-2.76
-1.91 ( 0.05
trans-C2H2F2
0.01
1.07
-2.76
-2.71
-1.84 ( 0.05
cis-C2H2F2
0.00
0.89
-2.92
-2.93
-2.18 ( 0.05
1,1-C2H2F2 C2HF3
0.01 0.02
0.86 0.80
-2.94 -3.00
-2.99 -3.07
-2.39 ( 0.05 -2.45 ( 0.05
C2F4
0.01
0.54
-3.23
-3.44
-3.00 ( 0.05
a
Ref 19.
We therefore choose to split the assessment into two, within each of which there is no ambiguity in the experimental order. Specifically, we consider (1) the variation in affinity with increasing number of fluorine atoms in C2H4, C2H3F, cis-1,2C2H2F2, C2HF3, and C2F4 and (2) the variation in affinity between the three difluoro isomers 1,1-C2H2F2, cis-1,2-C2H2F2, and trans-1,2-C2H2F2. Vertical electron affinities were calculated using the Perdew-Burke-Ernzerhof20 (PBE) GGAexchange-correlation functional using a range of basis sets, from relatively compact to very diffuse. All calculations were performed at optimized PBE/aug-cc-pVQZ geometries. Affinities were determined using the conventional expression in eq 1, the Koopmans expression in eq 2, the corrected Koopmans expression in eq 3, and the potential wall approach in eq 5 using λ = 3.4. The experimental ETS affinities all correspond to electron capture in the π* orbital, and so to be directly comparable, the calculated values must correspond to the same chemical process. For affinities determined using eqs 1 and 5, this requires the singly occupied molecular orbital (SOMO) of the anion to be a π* orbital, whereas for eqs 2 and 3 it requires the LUMO of the neutral to be a π*. Table 1 lists the calculated electron affinities, and Figures 1-3 present correlation plots between calculated and experimental ETS affinities. Where appropriate, values of R2 and the slope of the linear best-fit line are discussed; values near
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Figure 2. Correlation plots for assessment 2: variation with difluoro isomerism.
Figure 1. Correlation plots for assessment 1: variation with number of fluorine atoms.
respectively. Absolute errors are also reasonably small. The Koopmans expression in eq 2 yields an improved R2 value of 0.982, but the slope is slightly larger at 2.18. The most prominent observation, however, is that these affinities have the incorrect sign. The corrected Koopmans expression in eq 3 recovers the correct sign, yielding an excellent R2 at 0.991 but a slight degradation in the slope at 2.53. (Note that the difference between affinities computed using eqs 2 and 3 is almost independent of the system, indicating that they all exhibit similar size integer discontinuities.9,14) The potential wall approach in eq 5 maintains the excellent R2 value of 0.992, but the slope is now notably improved to 1.69. Overall, therefore, the best results are obtained using eq 1 or 5. For assessment 2, the same general observations are made, although now the use of eq 1 yields a poor R2 value of 0.689, with a slope of 1.71. Of the remaining methods, the best results are obtained using the potential wall approach in eq 5, for which R2 = 0.999 and the slope is 2.30.
unity indicate a method that accurately reproduces the variation. Calculations were performed with the Gaussian 0921 and CADPAC22 programs, using an unrestricted formalism for the open-shell systems. First, consider the results determined using the relatively compact cc-pVTZ basis set. Conventional calculations on the anions (i.e., no potential wall) yield π* SOMOs for all systems except cis-1,2-C2H2F2; for this system, π* occupation was obtained by swapping the orbital occupancy and reconverging, with a minimal rise in the total energy. Such an approach is theoretically valid because it yields the lowest energy of the given space-spin symmetry. Calculations on the neutral systems yield π* LUMOs in all cases, whereas potential wall calculations on the anions yield π* SOMOs throughout. Results for assessments 1 and 2 are shown in Figures 1a and 2a, respectively. For assessment 1, the conventional approach in eq 1 works well, with R2 and slope values of 0.974 and 1.27,
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which the R2 and slope values are 0.998 and 1.61 for assessment 1 and 0.973 and 1.88 for assessment 2. To illustrate the acute effect of diffuse functions, we subsequently augmented the aug-cc-pVTZ basis set with two additional sets of diffuse p functions on all atoms, with exponents determined from the geometric progression. We denote the resultant basis set aug-cc-pVTZþp, for which the minimum p exponent is just 0.0031 (on the carbon atoms). Results determined using this basis set are presented in Figures 1c and 2c. Equation 1 now breaks down completely, giving affinities that are essentially zero. (In fact, they are marginally positive, although further investigation showed that the values are sensitive to exchange-correlation functional, being marginally negative for the Becke-Lee-YangParr (BLYP)23,24 functional but even more positive using the local density approximation (LDA).25,26 This increase in affinity from BLYP to PBE to LDA is consistent with that found for the positive affinities of light, main-group molecules.) aug-ccpVTZþp affinities determined using the other three methods are indistinguishable from those obtained using aug-cc-pVTZ. The overall conclusion from Figures 1 and 2 is therefore that eqs 2, 3, and 5 are all able to describe the trends (i.e., ordering) in the affinities using both compact and very diffuse basis sets, with R2 values close to unity. The slope values are further from unity, indicating a modest accuracy in the absolute variations, which is unsurprising given the challenging nature of negative electron affinities. Notwithstanding the previous comment about experimental error bars, it is instructive to combine the data in Figures 1 and 2 into single correlation plots. See Figure 3. The basis set dependence of the affinities is readily understood by considering the variation in the anion SOMOs and neutral LUMO with the diffuseness of the basis set. We illustrate this in Figure 4 for the representative molecule, C2H4, but analogous arguments will apply to all the systems. A fixed molecule plot size and contour of 0.001 au is used throughout. The first row shows the anion SOMO for the conventional (no potential wall) calculations. As the basis set becomes more diffuse, the SOMO expands enormously; the electron tends to leave the system, the anion energy drops, and the affinity becomes close to zero. The second row of Figure 4 shows the neutral LUMO. It expands slightly from cc-pVTZ to aug-ccpVTZ but is then unaffected by the addition of further diffuse functions. The LUMO energy (see below) shows a similar variation, and this is reflected in the modest change in affinities from eqs 2 and 3 from cc-pVTZ to aug-cc-pVTZ but insensitivity to the addition of further diffuse functions. (Note that for eq 3, the effect of diffuse functions on the HOMO energy and ionization potential must also be considered, but this is small.) The third row of Figure 4 shows the anion SOMO for the potential wall calculations. It is essentially unaffected by the change in basis set due to the confining nature of the potential wall, and this is reflected in the near-constant, negative affinities. Figure 4 shows that the anion SOMO determined using the potential wall is smaller than the neutral LUMO when diffuse basis sets are used. This nonintuitive observation arises because the position of the potential wall is too close to the molecule, as evidenced by the affinities being too negative.
Figure 3. Combined correlation plots for assessments 1 and 2.
Next, consider results determined using the aug-cc-pVTZ basis set, a commonly used basis set that includes diffuse functions. Conventional calculations on the anions now yield σ* SOMOs in all cases rather than π*. The same occupancy swap procedure employed above was used to converge to higher energy π* SOMO states. Calculations on the neutral systems yield π* LUMOs for all systems except C2F4, for which a σ* orbital is obtained. For this system we simply replaced εLUMO in eqs 2 and 3 with the π* εLUMOþ1; the resultant affinities change by just 0.05 eV. Potential wall calculations on the anions again yield π* SOMOs throughout. Results for assessments 1 and 2 are presented in Figures 1b and 2b, respectively. As expected, eq 1 now starts to break down. The affinities become insufficiently negative, with a very poor R2 value of just 0.517 in Figure 1b and a negative slope in Figure 2b. By contrast, the affinities determined using eqs 2, 3, and 5 are less affected by the diffuse functions, being rather similar to the cc-pVTZ values, particularly for eq 5. The best results are again obtained using the potential wall approach in eq 5, for
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Figure 4. Basis set dependence of C2H4 conventional anion SOMO, neutral LUMO, potential wall anion SOMO, and orbital energies (below, in au).
Increasing the value of λ improves the affinities and causes the SOMO to expand to a size intermediate between the cc-pVTZ and aug-cc-pVTZ conventional SOMOs. It is also relevant to point out that when λ is increased in this manner, the first anion to switch to a ground-state σ* SOMO is C2F4. This is consistent with the comments in refs 19 and 27 that this is the ground-state configuration, together with our aforementioned observation that C2F4 is the one neutral system for which the LUMO is σ* when diffuse basis sets are used. For the case of positive affinities, the use of approximate DFT functionals has been controversial28,29 because the SOMO energy is often positive, implying an unbound orbital. The issue was recently reconsidered by Lee et al.30 It is therefore informative to consider the orbital energies for the anion SOMOs and neutral LUMO in C2H4 and the implications for orbital binding. The orbital energies are listed in Figure 4. For the conventional calculations, the asymptotic value of the exchange-correlation potential in the Kohn-Sham equations is zero, and so a negative orbital energy corresponds to a bound orbital (in the limit of a complete basis set). The anion SOMO energy for the conventional calculations is, unsurprisingly, positive, but it decreases toward zero as the electron leaves the system. This is exactly the expected behavior; see refs 30 and 31 for further discussion in the case of partial electron loss. The neutral LUMO energy is negative for all three basis sets, and so the orbital is bound. By contrast, the anion SOMO energy for the potential wall calculations is positive for the three basis sets, with a value similar to conventional cc-pVTZ. For these potential wall
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calculations, however, the asymptotic value of the exchangecorrelation potential is also positive, having a value above the SOMO energy by an amount equal to the negative of the neutral LUMO energy; see eq 15 of ref 15. Therefore, despite being positive, the SOMO energy is still below the asymptote, meaning the electron is bound. In conclusion, we have assessed how well four simple DFT methods can describe subtle variations in negative vertical electron affinities arising due to the successive fluorination of ethylene, where the magnitude of the variations is well below the absolute accuracy of the methods. Three of the approaches (two based on Koopmans' theorem and one using a potential wall) are able to describe the trends in the affinities using both compact and very diffuse basis sets. The best results are obtained using the potential wall. The fourth approach uses a conventional energy difference evaluation, which is only applicable when the basis set is compact; the breakdown upon addition of diffuse functions has been strikingly illustrated. The results have been interpreted in terms of the spatial extent of the relevant orbitals and the signs and values of the orbital energies. The study highlights the potential predictive value of simple DFT methods for describing trends in negative affinities.
AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed. E-mail: D.J.Tozer@ Durham.ac.uk. Fax: þ44 191 384 4737.
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