Benchmarking DFT Results against ab Initio Calculations

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Vibrational Frequencies of Fractionally Charged Molecular Species: Benchmarking DFT Results against Ab Initio Calculations Ioan Bâldea J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b12946 • Publication Date (Web): 03 Mar 2017 Downloaded from http://pubs.acs.org on March 6, 2017

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The Journal of Physical Chemistry

Vibrational Frequencies of Fractionally Charged Molecular Species: Benchmarking DFT Results against Ab Initio Calculations Ioan Bâldea∗,†,‡ Theoretische Chemie, Universit¨ at Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany, and Institute of Space Sciences, National Institute of Lasers, Plasma and Radiation Physics, RO 077125, Bucharest-M˘agurele, Romania E-mail: [email protected]

Abstract Recent advances in nano-/molecular electronics and electrochemistry made it possible to continuously tune the fractional charge q of single molecules and to use vibrational spectroscopic methods to monitor such changes. Approaches to compute vibrational frequencies ω(q) of fractionally charged species based on the density functional theory (DFT) are faced with an important issue: the basic quantity used in these calculations, the total energy, should exhibit piecewise linearity with respect to the fractional charge but approximate, commonly utilized exchange correlation functionals do not obey this condition. In this paper, with the aid of a simple and representative example, we benchmark results for ω(q) obtained within the DFT against ab initio methods: CCSD (coupled cluster singles and doubles), and MP2 and MP3 (second ∗

To whom correspondence should be addressed Theoretische Chemie, Universit¨ at Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany ‡ Institute of Space Sciences, National Institute of Lasers, Plasma and Radiation Physics, RO 077125, Bucharest-M˘agurele, Romania †

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and third-order Møller-Plesset perturbation) expansions. These results indicate that, in spite of missing the aforementioned piecewise linearity, DFT-based values ω(q) can reasonably be trusted.

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Introduction

Understanding interactions between electron distribution and molecular vibrations is essential for chemical and molecular physics. However important are the results reported over decades in studies on native molecules or easily prepared via ionization or electron attachment, the insight that can be gained into this problem is inherently limited: such investigations can only sample molecular species possessing an integral charge. Significantly richer information on a given molecular species can be obtained if its (fractional) charge can be continuously tuned. Recent advances in nano-/molecular electronics 1–5 and electrochemistry 6–10 made it possible to conduct such investigations experimentally. The average number of electrons of molecules adsorbed on biased electrodes can be continuously tuned by varying the bias. 6,11 Particularly promising in this direction are molecular junctions under electrochemical gating, 7,8,12 wherein an almost complete redox process can be achieved. 13,14 This is considerably more than electron transfers of up to ∼ 10% as inferred in earlier SERS (surface enhanced Raman spectroscopy) 6 experiments or using two terminal platforms. 11 The fractional molecular charge q is not directly accessible in experiments. Therefore, to get information on this quantity, existing works in the field 6,11,13–18 attempted to investigate its impact on molecular vibrational frequencies. Computing vibrational frequencies ω = ω(q) of molecules with fractional charges q within the density functional theory (DFT) is confronted with an important issue. At frozen molecular geometry R (see below), the total electronic energy Eq (R), which is the basic quantity 2


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needed in such calculations, should exhibit piecewise linearity with respect to the fractional molecular charge q (say,) added (0 ≤ q ≤ 1) to a (neutral) molecule possessing N electrons 19

Eq (R) = (1 − q)EN (R) + qEN +1 (R)

(1)

Approximate, commonly used exchange-correlation functionals fail to comply with this linearity. This deviation from linearity is usually assigned to the unphysical interaction of an electron with itself (“self-interaction errors” 20–25 ) or the tendency of approximate functionals to spread out the electron density artificially (“delocalization errors” 26 ). Given the fact that vibrational frequencies are determined from (second-order) derivatives of the total ground state energy with respect to atomic coordinates, a natural question arises. Does or to what extent does this spurious behavior of the total energy as a function of q affect the vibrational frequencies computed for fractionally charged species. This is the question that motivated the present paper, wherein DFT results for ω = ω(q) will be benchmarked against results of ab initio quantum chemical calculations.

2

Methods

DFT calculations for non-integer charges of the fluorine molecule studied in this paper have been performed running the SIESTA 27 trunk-462 package 28 in parallel. 29 Unlike most quantum chemical packages available, like the GAUSSIAN 09 package 30 also employed in this study, SIESTA 27 allows to compute molecular vibrational properties for non-integer charge states. For calculations to fractional charge states SIESTA employs a DFT (density functional theory) implementation developed in Ref. 19. Similar to our recent studies, 14,18,31 we used the generalized gradient exchange correlation functional of Perdew, Burke, and Ernzerhof (GGA-PBE) 32 and split-valence TZP (split triple zeta polarized) basis sets. The methodology and computational issues in approaching vibrational properties of fractionally charged molecules utilized in the present work are similar to those of ref. 18, 14, and 31. To 3



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make the paper self-contained, relevant details will be presented below. SIESTA calculations employ a simulation box. To study a molecule one should choose a periodic unit cell that is sufficiently large. This should make spurious interactions between the molecule and its periodically repeated replicas negligible. Still, the unit cell needs not be very large in studies on neutral molecules. The key point here is that SIESTA utilizes stricly localized basis functions. Negligible interactions between the translated units are basically ensured by basis functions having vanishing overlaps across the unit cell boundaries. Having fulfilled this condition, the shape of the unit cell is not important. By contrast, for charged molecules, the total energy converges very slowly with respect to the cell size. To make convergence faster, SIESTA applies a Madelung correction term to energy, which is possible only for cubic cells. To comply with the aforementioned, similar to our previous works, 14,18,31 3

we used a cubic 30×30×30 ˚ A unit cell. As noted earlier, 14,18,31 we found no notable changes when varying the mesh energy cutoff between 300 Ry and 1000 Ry. This quantity defines the maximum kinetic energy of the electronic plane wave basis set utilized and is a characteristic of the three-dimensional real-space grid. SIESTA’s approach to vibrational properties is based on the dynamical matrix obtained via a finite difference calculation of the forces 33 based on the VIBRA utiliy, which is part of SIESTA distrbution. 28 This method represents an alternative to the linear response DFT schemes, 34 which are more popular approaches in dealing with phonons. For phonon studies, constraints must be much more tight than for standard geometry optimization. Similar to refs. 17 and 18, forces smaller than 0.1 meV/˚ A were imposed for all fractional charges. This is a much more severe restriction that typical default force tolerances (0.04 eV/˚ A in SIESTA and ∼ 0.023 eV/˚ A in GAUSSIAN 09). Similar to refs. 17 and 18, we also examined the effect of the finite difference procedure. We found no notable impact by varying the displacement used for computing the force constant matrix from 0.01 ˚ A to 0.02 ˚ A. Ab initio results presented here comprise Møller-Plesset (MP) at second- (MP2) and third- (MP3) order level of theory as well as at CCSD (coupled-cluster singles and doubles)

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level. GAUSSIAN 09 package 30 was employed to perform these calculations.

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Results

The fluorine molecule F2 was chosen as a specific example for the present study for several reasons. The first reason is of technical nature. Choosing a diatomic molecule obviates the need of normal mode analysis needed for larger polyatomic molecules, which is not implemented in available quantum chemistry packages to allow phonon calculations to fractionally charged molecular species. In the present case, the vibrational frequency ω(q) of a molecular state possessing a fractional charge q can be simply obtained from the curvature of the total energy Eq (R) plotted as a function of the F-F distance R (see Figure 4 below). Second, we chose F2 because of the large change in frequency upon adding charge; upon complete reduction the change in frequency amounts to more than 500 cm−1 (cf. Tables S1 and S2 of the Supporting Information (SI)), which is roughly twice the change encountered in medium size molecules like bipyridine, 35 benzene 18 or benzene derivatives; 17 adding an electron to the diatomic light molecule is a process much more invasive than adding an electron to a heavy polyatomic molecule. Third and most importantly in the present context, we chose the F2 molecule just because the total energy computed as a function of electron addition q within the DFT notoriously deviates from the linearity expressed by Eq. (1). The corresponding curve computed at the distance (R) between the two F nuclei frozen at the value R0 in the ground state of the neutral molecule is convex; 23 it is presented in Figure 1. Parenthetically, concave-shaped curves were also reported. 36 The energy curve of Figure 1 is not only highly nonlinear, it even exhibits a minimum, which is nothing but an unphysical “prediction” of a most stable molecular species characterized by a fractional charge. In fact, even if computed within a method free of the DFT drawback noted above, the lowest energy Eq of the fractionally charged molecule needed for calculations of vibrational

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Figure 1: The dependence of the total energy Eq of the F2 molecule frozen at the geometry of the neutral species computed as a function of the fractional charge q at the PBE/TZP level of theory drastically deviates from the piecewise linearity predicted for exact functionals. 19 properties deviates to some extent from the linear dependence on q expressed by Eq. 1; see Figure 2. The reason is that, unlike in the situation presented in Figure 1, the geometry

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Figure 2: The dependence of the total energy Eq for the molecular fractional charge q at F-F distances optimized at the actual q value computed within various methods and the two basis sets indicated in the legend: (a) CCSD, (b) MP2, and (c) MP3. Notice that the nonlinear dependence on q is consequence of the fact that the molecular geometry was optimized by using Eq. 1 at each q value. R(q) is not frozen at the value that optimizes the energy of the neutral molecule R(q) 6= R(q)|q=0 ≡ R0 ; rather, R(q) represents the F-F distance which minimizes the RHS of Eq. (1) for a given value q of the fractional charge. This fact is illustrated in Figure 3. Noteworthy, to calculate vibration frequencies of a state with fractional charge q one needs calculations similar to those done in conjunction with the results presented in Figure 3 rather 6


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Figure 3: The dependence of the total energy Eq for the F2 molecule with a fractional charge q on the F-F distance R obtained at (a) CCSD/TZVP level, (b) MP2/TZVP, and (c) MP3/TZVP. Notice that the minimum location indicated by the dashed vertical bars changes with q. In both panels, the minimum energy of the neutral molecule (q = 0) at CCSD/TZP level was taken as zero energy. In panels a and b, values in angstrom of the F-F distances at the minima are (from left to right) 1.410, 1.520, 1.690, 1.851, 1.9357 and 1.401, 1.520, 1.681, 1.842, 1.930, respectively. than those of Figure 1; vibrations imply to consider slight departures from the optimum geometry R(q) at a given q. Ideally, ab initio quantum chemical calculations for molecular states with fractional charges should resort to the ensemble averaging over the neutral and (in our case) anionic states weighted in accordance with the fractional charge q within an approach based on the density matrix rather on the wave function. Still, ab initio quantum chemical packages available (or at least of our disposal) do not allow such computation for fractionally charged molecules. Therefore, the determination of the vibrational frequencies ω(q) was done in the following manner. Starting from the values EN (R) and EN +1 (R) computed ab initio, we built curves for the energy Eq (R) for various fractional charges q according to Eq. 1. These curves possess minima located at R = R(q) (cf. Figure 3 and Figure 4). By fitting the energy curves for various fractional charges q similar to those of Figure 3 around the minima R = R(q) we determined the (harmonic) vibrational frequencies. The procedure is illustrated in Figure 4. The curves for the frequencies ω(q) obtained in this way within the CCSD, MP2, and MP3 methods are depicted in Figure 5a, Figure 5b, and Figure 5c, respectively. The DFT-based 7



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ab initio results fit around minimum o R(q)=1.5196A

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Figure 4: By fitting the ab initio curve Eq versus R around the minimum R(q), Eq (R) = µ ω(q)2 [R − R(q)]2 + c [R − R(q)]3 , the value of the corresponding (harmonical) vibrational 2 frequency ω(q) can be computed. Here, µ denotes the reduced mass of the molecule. Computational details are indicated in the legend. curve of ω(q) computed by means of SIESTA is depicted in Figure 5d.

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Discussion

The inspection of curves for energy of Figure 2 reveals a very weak dependence on the ab initio method employed (CCSD, MP2, or MP3). This is an indication that electron correlation effects are adequately accounted for within the aforementioned ab initio methods. The impact of the basis set on the total energy appears to be more significant. In general, because the extra electron of an anion is weakly bound to the molecule, basis sets including diffuse functions are needed for an adequate description of anionic states. 35,37 In the present case of the F2 molecule this effect can be seen by comparing the curves computed using basis sets without (TZVP) and with (aug-cc-pVTZ 30 ) diffuse functions. The larger the fractional charge q added to the neutral molecule, the larger is the difference between the total energy computed by using the TZVP basis set and that computed by means of the aug-cc-pVTZ basis set; inclusion of diffuse basis functions acts to stabilize the anionic species (cf. Figure 2 as well as Figure 6). With the basis set TZVP that does

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Figure 5: Dependence of the vibrational frequency ω = ω(q) on the fractional charge q. The method utilized as well as computational details are indicated in the legend. Notice that for each pair of the curves depicted in panels a to c the ab initio method is the same; the difference between the mates of each pair of curves quantifies the the impact of the diffuse basis functions, which are included in the aug-cc-pVTZ basis set but not in the ccpVTZ basis set. Because the SIESTA curve for frequency (panel d) is slightly affected by numerical noise 17,18 (see the main text for detail) we also present the linear fit (green line) of the computed values (red points). Notice that in all panels of this figure the x- and y-ranges are identical. not include diffuse basis functions, the theoretical values for the adiabatic electroaffinity are EA ≡ E|q=0 − E|q=−1 ≃ 2.5 eV. They are significantly smaller that the values EA ≈ 3 eV computed with the basis set aug-cc-pVTZ that contains diffuse functions and are in reasonable agreement with the experimental value EAexp = 3.005 ± 0.071 eV; 38 see also Tables S1 and S2 of the SI. As intuitively expected, the frequency of the molecular vibration is red-shifted when moving from the neutral to the anionic species. Adding charge to the molecule acts to weaken 9



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the chemical bond between the F atoms; see Figure 5. Nevertheless, the impact of the diffuse basis functions on the vibrational frequencies is relatively weak. After inspecting Figure 5 one can by no means conclude that a larger amount of the extra charge added yields a larger difference between the frequencies computed by using the TVZP and aug-cc-pVTZ basis sets; a behavior different from that displayed by the curves for the total energy (Figure 2). From the point of view of the present investigation on molecular vibrational frequencies this somewhat surprising result, which confirms a similar finding reported earlier, 35 represents a technical advantage. Diffuse basis sets are not included with SIESTA standard distributions, 28 and combining basis sets and pseudopotentials may be presumably responsible for non negligible albeit not dramatic numerical noise encountered in SIESTA studies on vibrational effects of fractionally charged species. 14,17,18 Although weaker than in case of larger molecules 14,17,18 this numerical noise is still visible in Figure 5d, where the dependence on the fractional charge q of the vibrational frequency ω(q) is presented. As suggested by Figure 5d, the dependence of ω(q) on q is practically linear. By and larger, the agreement between the results for ω(q) versus q obtained within DFT calculations (Figure 5d) and the other ab initio methods presented in Figure 5a, b and c is reasonable; notice that, in order to facilitate the comparison, we have chosen identical x- and y-ranges in all panels of Figure 5. Figure 5 reveals that the larger differences between the DFT-based values and those based > 0.8) is transferred on ab initio approaches occur in cases where an almost entire electron (q ∼

to the molecule. It is tempting to assign this behavior at larger q to a poorer description of the anionic species by using the GGA-PBE exchange correlation functional. The fact that deviations between the F-F distance R(q) computed by SIESTA and the ab initio values are more substantial at larger q values (cf. Figure 6) gives further support to this idea. The significantly larger R(q)-values of the F-F distance predicted by SIESTA imply a weaker chemical bond and are in accord with the SIESTA values of ω(q) that are significantly lower than the ab initio values of ω(q) at larger q.

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In this vein, one should note that calculations using GAUSSIAN 09 with the hybrid B3LYP functional yield frequencies closer to those obtained within the ab initio methods CCSD, MP2, and MP3: At B3LYP/TZVP and B3LYP/aug-cc-pVTZ levels, the values are ω(q)|q=−1 = 355.26 cm−1 and ω(q)|q=−1 = 357.85 cm−1 , respectively. Unfortunately, genuine hybrid functionals like B3LYP are not implemented in SIESTA; 28 in SIESTA’s nomenclature, “hybrid” functionals means admixtures of GGA and LDA functionals, and the latter is known to provide a too crude description of molecules; therefore, one cannot expect improvements using such “hybrid” functionals.

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Figure 6: Dependence of the F-F distance R(q) on the fractional charge q. The methods (CCSD, MP2, MP3, and DFT/SIESTA) utilized as well as computational details are indicated in the legend. Notice that in all panels of this figure the x- and y-ranges are identical. To end this section, the agreement between the SIESTA/DFT estimates for vibrational frequencies of fractionally charged molecular states and those based on the ab initio methods CCSD, MP2, and MP3 is reasonable. In assessing the quality of this agreement we should be aware that, whatever the theoretical method employed, certain differences between state11



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of-the-art theoretical estimates and experiments 38–41 continue to be significant even for such a small molecule like F2 ; see Tables S1 and S2 of the SI. In particular, one should note that applying semiempirical scaling factors ∼ 100 ± 10% in order to bring computed vibrational frequencies in good quantitative agreement with experiment represents current practice in the field. 42 Furthermore, as previously pointed out, 17,18,35 the values of these semiempirical factors for neutral and anionic species may be different.

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Conclusion

In this paper, we have considered the impact of adding a fractional charge q to a simple diatomic molecule (F2 ) on its vibrational frequency ω(q). The results reported here indicate that, in spite of a well-known drawback of the DFT — namely the failure to observe the piecewise linearity with respect to the fractional charge —, the DFT-based vibrational frequencies can reasonably be trusted; the values obtained for ω(q) within the DFT were found to be in reasonable good agreement with those computed by means of well-established ab initio methods of quantum chemistry (CCSD, MP2, and MP3). This is important because DFT calculations for vibrational properties can be performed for molecular sizes considerably larger than ab initio calculations. The change in the vibrational frequency upon adding an electron to the neutral F2 molecule turned out to be substantially larger than that encountered in other, mediumsize molecules. 13,17,18,35 Therefore, one can expect a better agreement between DFT and ab initio estimates for vibrational frequencies of larger molecules, which are of interest, e.g., for fabricating molecular junctions; adding/removing an electron to/from a polyatomic molecule is a less invasive process than in a small molecule like F2 .

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Acknowledgments The author acknowledges with thanks financial support for this research provided by the Deutsche Forschungsgemeinschaft (grant BA 1799/3-1) and computational support by the State of Baden-W¨ urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 40/467-1 FUGG.

Supporting Information Supplementary details, tables and one figure. The Supporting Information is available free of charge via the Internet at http://pubs.acs.org.

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Molecular energy E(q)

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"Nonlinear Nonlinear DFT curve E=E(q)...?" " ... A known flaw of DFT!"

-1

Fractional charge q

0

"Does it affect phonon computations...?" "... Not very much!"

Figure 7: TOC

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Benchmarking DFT Results against ab Initio Calculations

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