Spurious Oscillations Caused by Density Functional Approximations: Who is to Blame? Exchange or Correlation?

We analyze the varying susceptibilities of different density functional approximations (DFAs) to present spurious oscillations on the profiles of several vibrational properties. Among other problems, these spurious oscillations cause significant errors in harmonic and anharmonic IR and Raman frequencies and intensities. This work hinges on a judicious strategy to dissect the exchange and correlation components of DFAs and pinpoint the origins of these oscillations. We identify spurious oscillations in derivatives of all energy components with respect to nuclear displacements, including those energy terms that do not involve numerical integrations. These indirect spurious oscillations are attributed to suboptimal electron densities resulting from a self-consistent field procedure using a DFA that exhibits direct spurious oscillations. Direct oscillations stem from inaccurate numerical integration of the exchange and correlation energy density functionals. A thorough analysis of direct spurious oscillations reveals that only a handful of exchange and correlation components are insensitive to spurious oscillations, giving rise to three families of functionals, BH&H, LSDA, and BLYP. Among the functionals in these families, we encounter four widespread DFAs: BLYP, B3LYP, LC-BLYP, and CAM-B3LYP. Certain DFAs like PBE appear less sensitive to spurious oscillations due to compensatory cancellations between their energy components. Additionally, we found non-negligible but small oscillations in PBE and TPSS, which could be safely employed provided a sufficiently large integration grid is used in the calculations. These findings hint at the key components of current approximations to be improved and emphasize the necessity to develop accurate DFAs suitable for studying molecular spectroscopies.

Pitfall in simulations of vibronic TD-DFT spectra: diagnosis and assessment.

In this Communication, we study the effect of spurious oscillations in the profiles of energy derivatives with respect to nuclear coordinates calculated with density functional approximations (DFAs) for formaldehyde, pyridine, and furan in their ground and electronic excited states. These spurious oscillations, which can only be removed using extensive integration grids that increase enormously the CPU cost of DFA calculations, are significant in the case of third- and fourth-order energy derivatives of the ground and excited states computed by M06-2X and ωB97X functionals. The errors in question propagate to anharmonic vibronic spectra computed under the Franck-Condon approximation, i.e., positions and intensities of vibronic transitions are affected to a large extent (shifts as significant as hundreds of cm-1 were observed). On the other hand, the LC-BLYP and CAM-B3LYP functionals show a much less pronounced effect due to spurious oscillations. Based on the results presented herein, we recommend either LC-BLYP or CAM-B3LYP with integration grids (250, 974) (or larger) for numerically stable simulations of vibronic spectra including anharmonic effects.

How Reliable Are Modern Density Functional Approximations to Simulate Vibrational Spectroscopies?

We show that properties of molecules with low-frequency modes calculated with density functional approximations (DFAs) suffer from spurious oscillations along the nuclear displacement coordinate due to numerical integration errors. Occasionally, the problem can be alleviated using extensive integration grids that compromise the favorable cost-accuracy ratio of DFAs. Since spurious oscillations are difficult to predict or identify, DFAs are exposed to severe performance errors in IR and Raman intensities and frequencies or vibrational contributions to any molecular property. Using Fourier spectral analysis and digital signal processing techniques, we identify and quantify the error due to these oscillations for 45 widely used DFAs. LC-BLYP and BH&H are revealed as the only functionals showing robustness against the spurious oscillations of various energy, dipole moment, and polarizability derivatives with respect to a nuclear displacement coordinate. Given the ubiquitous nature of molecules with low-frequency modes, we warrant caution in using modern DFAs to simulate vibrational spectroscopies.

Choosing Bad versus Worse: Predictions of Two-Photon-Absorption Strengths Based on Popular Density Functional Approximations.

We present a benchmark study of density functional approximation (DFA) performances in predicting the two-photon-absorption strengths in π-conjugated molecules containing electron-donating/-accepting moieties. A set of 48 organic molecules is chosen for this purpose, for which the two-photon-absorption (2PA) parameters are evaluated using different DFAs, including BLYP, PBE, B3LYP, PBE0, CAM-B3LYP, LC-BLYP, and optimally tuned LC-BLYP. Minnesota functionals and ωB97X-D are also used, applying the two-state approximation, for a subset of molecules. The efficient resolution-of-identity implementation of the coupled-cluster CC2 model (RI-CC2) is used as a reference for the assessment of the DFAs. Two-state models within the framework of both DFAs and RI-CC2 are used to gain a deeper insight into the performance of different DFAs. Our results give a clear picture of the performance of the density functionals in describing the two-photon activity in dipolar π-conjugated systems. The results show that global hybrids are best suited to reproduce the absolute values of 2PA strengths of donor-acceptor molecules. The range-separated functionals CAM-B3LYP and optimally tuned LC-BLYP, however, show the highest linear correlations with the reference RI-CC2 results. Hence, we recommend the latter DFAs for structure-property studies across large series of dipolar compounds.

How Many Electrons Does a Molecular Electride Hold?

Electrides are very peculiar ionic compounds where electrons occupy the anionic positions. In a crystal lattice, these isolated electrons often form channels or surfaces, furnishing electrides with many traits with promising technological applications. Despite their huge potential, thus far, only a few stable electrides have been produced because of the intricate synthesis they entail. Due to the difficulty in assessing the presence of isolated electrons, the characterization of electrides also poses some serious challenges. In fact, their properties are expected to depend on the arrangement of these electrons in the molecule. Among the criteria that we can use to characterize electrides, the presence of a non-nuclear attractor (NNA) of the electron density is both the rarest and the most salient feature. Therefore, a correct description of the NNA is crucial to determine the properties of electrides. In this paper, we analyze the NNA and the surrounding region of nine molecular electrides to determine the number of isolated electrons held in the electride. We have seen that the correct description of a molecular electride hinges on the electronic structure method employed for the analyses. In particular, one should employ a basis set with sufficient flexibility to describe the region close to the NNA and a density functional approximation that does not suffer from large delocalization errors. Finally, we have classified these nine molecular electrides according to the most likely number of electrons that we can find in the NNA. We believe this classification highlights the strength of the electride character and will prove useful in designing new electrides.

Performance of DFT functionals for calculating the second-order nonlinear optical properties of dipolar merocyanines.

The second-order nonlinear optical responses of a series of recently designed dipolar merocyanines are investigated using the 2006 Minnesota family of hybrid exchange-correlation functionals (XCFs), as well as the LC-BLYP, ωB97XD and CAM-B3LYP long-range (LR) corrected XCFs. The performance of these different levels of approximation is discussed in regard to reference second-order Møller-Plesset calculations and experimental data obtained from Hyper-Rayleigh Scattering (HRS) measurements. Particular focus is given to the influence of the amount of exact Hartree-Fock exchange included in the XCF on the magnitude of the static HRS responses, as well as to the impact of tuning the range-separation parameter in LR-XCFs, according to a system-specific nonempirical procedure. Frequency dispersion effects are also investigated, as well as their crucial role in the comparison between theoretical and experimental data.

A new tuned range-separated density functional for the accurate calculation of second hyperpolarizabilities.

The calculation of nonlinear optical properties (NLOPs) using density functional theory (DFT) remains a challenge in computational chemistry. Although the existing range-separated functionals display the best performance for the calculation of this type of properties, their errors strongly depend on the family of molecules studied. Herein, we have explored a new strategy to empirically tune the range-separated LC-BLYP method to improve the accuracy of the calculation of the second hyperpolarizabilities (γ), which are poorly described by current density functional approximations. First, we benchmarked nine of the most accurate commonly used range-separated hybrid and optimally tuned functionals (i.e. B3LYP, PBE0, BH&HLYP, M06-2X, MN15, ωB97X-D, CAM-B3LYP, LC-BLYP and OT-LC-BLYP) for the calculation of γ using as a reference the CCSD(T) values of a chemically diverse set of 60 molecules. Among these nine functionals, LC-BLYP gives the lowest average errors. We determined the value of the range-separation parameter ω required to reproduce the CCSD(T) second hyperpolarizabilities with the LC-BLYP functional (ωCC) for the set of 60 molecules. Our new tuned range-separated functional, Tα-LC-BLYP, uses a quadratic correlation between ωCC and a molecular descriptor in terms of the linear polarizability and the number of electrons in the molecule. The average error of the γ values obtained with Tα-LC-BLYP is reduced by half or more as compared with the most accurate among the nine density functional approximations benchmarked.

Partition of optical properties into orbital contributions.

Nonlinear optical properties (NLOPs) play a major role in photonics, electro-optics and optoelectronics, and other fields of modern optics. The design of new NLO molecules and materials has benefited from the development of computational tools to analyze the relationship between the electronic structure of molecules and their optical response. In this paper, we present a new means to analyze the response property through the partition of NLOPs in terms of orbital contributions (PNOC). This tool can be used to obtain a real-space representation of the NLOPs, providing a powerful visualization aid to connect the magnitude of the optical property with some parts of the molecule. Unlike other methods to analyze NLOPs, the PNOC decomposes the optical property into orbitals of the unperturbed system, furnishing this method with the ability to assess the performance of single- and multi-determinant electronic structure methods. PNOC can be also used to design small basis sets for an accurate description of large systems, saving a substantial amount of computer time for the calculation of optical properties.

Can Density Functional Theory Be Trusted for High-Order Electric Properties? The Case of Hydrogen-Bonded Complexes.

This work reports on an extensive assessment of the performance of a wide palette of density functional approximations in predicting the (high-order) electric properties of hydrogen-bonded complexes. To this end, we compute the electronic and vibrational contributions to the electric polarizability and the first and second hyperpolarizabilities, using the CCSD(T)/aug-cc-pVTZ level of theory as reference. For all the studied properties, the average absolute errors below 20% can only be obtained using the CAM-B3LYP functional, while LC-BLYP and MN15 are shown to be only slightly less accurate (average absolute errors not exceeding 30%). Among Minnesota density functionals, i.e., M06, M06-2X, and MN15, we only recommend the latter one, which quite accurately predicts the electronic and vibrational (hyper)polarizabilities. We also analyze the optimal tuning of the range-separation parameter µ for the LC-BLYP functional, finding that this approach does not bring any systematic improvement in the predictions of electronic and vibrational (hyper)polarizabilities and the accuracy of computed properties is largely system-dependent. Finally, we report huge errors in predicting the vibrational second hyperpolarizability by ωB97X, M06, and M06-2X functionals. Based on the explicit evaluation of anharmonic terms contributing to the second hyperpolarizability, this failure is traced down to a poor determination of third- and fourth-order energy derivatives with respect to normal modes. These results reveal serious flaws of some density functional approximations and suggest caution in selecting the appropriate functional to calculate not only electronic and vibrational (hyper)polarizabilities but also other molecular properties that contain vibrational anharmonic contributions.

Gas-Phase Photolysis of Hg(I) Radical Species: A New Atmospheric Mercury Reduction Process.

The efficient gas-phase photoreduction of Hg(II) has recently been shown to change mercury cycling significantly in the atmosphere and its deposition to the Earth's surface. However, the photolysis of key Hg(I) species within that cycle is currently not considered. Here we present ultraviolet-visible absorption spectra and cross-sections of HgCl, HgBr, HgI, and HgOH radicals, computed by high-level quantum-chemical methods, and show for the first time that gas-phase Hg(I) photoreduction can occur at time scales that eventually would influence the mercury chemistry in the atmosphere. These results provide new fundamental understanding of the photobehavior of Hg(I) radicals and show that the photolysis of HgBr increases atmospheric mercury lifetime, contributing to its global distribution in a significant way.

Ab initio quantum-chemical computations of the absorption cross sections of HgX2 and HgXY (X, Y = Cl, Br, and I): molecules of interest in the Earth's atmosphere.

The electronic-structure properties of the low-lying electronic states and the absorption cross sections (σ(E)) of mercury halides HgCl2, HgBr2, HgI2, HgBrCl, HgClI, and HgBrI have been determined within the UV-vis spectrum range (170 nm ≤ λphoton ≤ 600 nm) by means of the DKH3-MS-CASPT2/SO-RASSI quantum-chemical methodology (with the ANO-RCC basis set) and a semi-classical computational strategy based on nuclear sampling for simulating the band shapes. Computed band energies show a good agreement with the available experimental data for HgX2 with errors around 0.1-0.2 eV; theoretical and σ(E) are within the same order of magnitude. For the mixed HgXY compounds, the present computed data allow us to interpret previously proposed absorption bands estimated from the spectra of the parent molecules HgX2 and HgY2, measured in methanol solution. The analyses performed on the excited-state electronic structure and its changes around the Franck-Condon region provide a rationale on the singlet-triplet mixing of the absorption bands and the heavy-atom effect of the Hg compounds. Furthermore, the present benchmark of HgX2 and HgXY absorption σ values together with the previous benchmark of the electronic-structure properties of HgBr2 [see S. P. Sitkiewicz, et al., J. Chem. Phys., 2016, 145, 244304] has been helpful to set up a methodological and computational protocol which shall be used for predicting the atmospheric absorption and photolysis properties of several Hg compounds present in the atmospheric cycle of Hg.

Photoreduction of gaseous oxidized mercury changes global atmospheric mercury speciation, transport and deposition.

Anthropogenic mercury (Hg(0)) emissions oxidize to gaseous Hg(II) compounds, before deposition to Earth surface ecosystems. Atmospheric reduction of Hg(II) competes with deposition, thereby modifying the magnitude and pattern of Hg deposition. Global Hg models have postulated that Hg(II) reduction in the atmosphere occurs through aqueous-phase photoreduction that may take place in clouds. Here we report that experimental rainfall Hg(II) photoreduction rates are much slower than modelled rates. We compute absorption cross sections of Hg(II) compounds and show that fast gas-phase Hg(II) photolysis can dominate atmospheric mercury reduction and lead to a substantial increase in the modelled, global atmospheric Hg lifetime by a factor two. Models with Hg(II) photolysis show enhanced Hg(0) deposition to land, which may prolong recovery of aquatic ecosystems long after Hg emissions are lowered, due to the longer residence time of Hg in soils compared with the ocean. Fast Hg(II) photolysis substantially changes atmospheric Hg dynamics and requires further assessment at regional and local scales.

Ab initio quantum-chemical computations of the electronic states in HgBr2 and IBr: Molecules of interest on the Earth's atmosphere.

The electronic states of atmospheric relevant molecules IBr and HgBr2 are reported, within the UV-Vis spectrum range (170nm≤λphoton≤600 nm) by means of the complete-active-space self-consistent field/multi-state complete-active-space second-order perturbation theory/spin-orbit restricted-active-space state-interaction (CASSCF/MS-CASPT2/SO-RASSI) quantum-chemical approach and atomic-natural-orbital relativistic-correlation-consistent (ANO-RCC) basis sets. Several analyses of the methodology were carried out in order to reach converged results and therefore to establish a highly accurate level of theory. Good agreement is found with the experimental data with errors not higher than around 0.1 eV. The presented analyses shall allow upcoming studies aimed to accurately determine the absorption cross sections of interhalogen molecules and compounds with Hg that are relevant to better comprehend the photochemical processes taking place in the atmosphere.