On the Topological Phase around Conical Intersections with Tamm-Dancoff Linear-Response Time-Dependent Density Functional Theory.

Regions of nuclear-configuration space away from the Franck-Condon geometry can prove problematic for some electronic structure methods, given the propensity of such regions to possess conical intersections, i.e., (highly connected) points of degeneracy between potential energy surfaces. With the likelihood (perhaps even inevitability) for nonadiabatic dynamics simulations to explore molecular geometries in close proximity to conical intersections, it is vital that the performance of electronic structure methods is routinely examined in this context. In a recent paper [Taylor, J. T. J. Chem. Phys. 2023, 159, 214115.], the ability of linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) to provide a proper description of conical intersections, in terms of their topology and topography, was investigated, with particular attention paid to conical intersections between two excited electronic states. For the same prototypical molecules, protonated formaldimine and pyrazine, we herein consider whether AA LR-TDDFT can correctly reproduce the topological phase accumulated by the adiabatic electronic wave function upon traversing a closed path around an excited-to-excited state conical intersection despite not using the appropriate quadratic-response nonadiabatic coupling vectors. Equally, we probe the ability of the ground-to-excited state intersection ring exhibited by AA LR-TDDFT in protonated formaldimine to give rise to a similar topological phase in spite of its incorrect dimensionality.

On the description of conical intersections between excited electronic states with LR-TDDFT and ADC(2).

Conical intersections constitute the conceptual bedrock of our working understanding of ultrafast, nonadiabatic processes within photochemistry (and photophysics). Accurate calculation of potential energy surfaces within the vicinity of conical intersections, however, still poses a serious challenge to many popular electronic structure methods. Multiple works have reported on the deficiency of methods like linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) or algebraic diagrammatic construction to second-order [ADC(2)]-approaches often used in excited-state molecular dynamics simulations-to describe conical intersections between the ground and excited electronic states. In the present study, we focus our attention on conical intersections between excited electronic states and probe the ability of AA LR-TDDFT and ADC(2) to describe their topology and topography, using protonated formaldimine and pyrazine as two exemplar molecules. We also take the opportunity to revisit the performance of these methods in describing conical intersections involving the ground electronic state in protonated formaldimine-highlighting in particular how the intersection ring exhibited by AA LR-TDDFT can be perceived either as a (near-to-linear) seam of intersection or two interpenetrating cones, depending on the magnitude of molecular distortions within the branching space.

Effective homogeneity of Fermi-Amaldi-containing exchange-correlation functionals.

Parr and Ghosh [Phys. Rev. A. 51 3564 (1995)] demonstrated that when near-exact electron densities and potentials are used, the exchange-correlation energies of first- and second-row atoms are well-described by a combination of the Fermi-Amaldi functional with a functional that is homogeneous of degree one under density scaling. Insight into this observation is provided by considering their work from the perspective of the effective homogeneity of the overall exchange-correlation functional. By considering a general form that combines the Fermi-Amaldi functional with a functional that is homogeneous of degree k, it is shown that for these atoms, the functional of Parr and Ghosh (k = 1) exhibits essentially optimal effective homogeneities on the electron-deficient side of the integer. Percentage errors in effective homogeneities are close to percentage errors in exchange-correlation energies.

Thermodynamic equilibrium between locally excited and charge-transfer states through thermally activated charge transfer in 1-(pyren-2'-yl)-o-carborane.

Reversible conversion between excited-states plays an important role in many photophysical phenomena. Using 1-(pyren-2'-yl)-o-carborane as a model, we studied the photoinduced reversible charge-transfer (CT) process and the thermodynamic equilibrium between the locally-excited (LE) state and CT state, by combining steady state, time-resolved, and temperature-dependent fluorescence spectroscopy, fs- and ns-transient absorption, and DFT and LR-TDDFT calculations. Our results show that the energy gaps and energy barriers between the LE, CT, and a non-emissive 'mixed' state of 1-(pyren-2'-yl)-o-carborane are very small, and all three excited states are accessible at room temperature. The internal-conversion and reverse internal-conversion between LE and CT states are significantly faster than the radiative decay, and the two states have the same lifetimes and are in thermodynamic equilibrium.

Incorporation of the Fermi-Amaldi Term into Direct Energy Kohn-Sham Calculations.

In direct energy Kohn-Sham (DEKS) theory, the density functional theory electronic energy equals the sum of occupied orbital energies, obtained from Kohn-Sham-like orbital equations involving a shifted Hartree exchange-correlation potential, which must be approximated. In the present study, the Fermi-Amaldi term is incorporated into approximate DEKS calculations, introducing the required -1/r contribution to the exchange-correlation component of the shifted potential in asymptotic regions. It also provides a mechanism for eliminating one-electron self-interaction error, and it introduces a nonzero exchange-correlation component of the shift in the potential that is of appropriate magnitude. The resulting electronic energies are very sensitive to the methodologies considered, whereas the highest occupied molecular orbital energies and exchange-correlation potentials are much less sensitive and are similar to those obtained from DEKS calculations using a conventional exchange-correlation functional.

Low energy electron impact resonances of anthracene probed by 2D photoelectron imaging of its radical anion.

Electron-molecule resonances of anthracene were probed by 2D photoelectron imaging of the corresponding radical anion up to 3.7 eV in the continuum. A number of resonances were observed in both the photoelectron spectra and angular distributions, and most resonances showed clear autodetachment dynamics. The resonances were assigned using density functional theory calculations and are consistent with the available literature. Competition between direct and autodetachment, as well as signatures of internal conversion between resonances, was observed for some resonances. For the 12B2g resonance, a small fraction of population recovers the ground electronic state as evidenced by thermionic emission. Recovery of the ground electronic state offers a route of producing anions in an electron-molecule reaction; however, the energy at which this occurs suggests that anthracene anions cannot be formed in the interstellar medium by electron capture through this resonance.

Photoelectron spectroscopy of para-benzoquinone cluster anions.

The photoelectron spectra of para-benzoquinone radical cluster anions, (pBQ)n - (n = 2-4), taken at hv = 4.00 eV are presented and compared with the photoelectron spectrum of the monomer (n = 1). For all clusters, a direct detachment peak can be identified, and the incremental increase in the vertical detachment energy of ∼0.4 eV n-1 predominantly reflects the increase in cohesion energy as the cluster size increases. For all clusters, excitation also leads to low energy electrons that are produced by thermionic emission from ground electronic state anionic species, indicating that resonances are excited at this photon energy. For n = 3 and 4, photoelectron features at lower binding energy are observed which can be assigned to photodetachment from pBQ- for n = 3 and both pBQ- and (pBQ)2 - for n = 4. These observations indicate that the cluster dissociates on the time scale of the laser pulse (∼5 ns). The present results are discussed in the context of related quinone cluster anions.

Simple DFT Scheme for Estimating Negative Electron Affinities.

A simple density functional theory (DFT) scheme is proposed for estimating negative vertical electron affinities of neutral systems, based on a consideration of the integer discontinuity and density scaling homogeneity. The key feature is the derivation of two system-dependent exchange-correlation functionals, one appropriate for the electron deficient side of the integer and one appropriate for the electron abundant side. The electron affinity is evaluated as a linear combination of frontier orbital energies from self-consistent Kohn-Sham calculations on the neutral system using these functionals. For two assessments comprising a total of 43 molecules, the scheme provides electron affinities that are in good agreement with experimental values and which are an improvement over those from the DFT method of Tozer and De Proft [ J. Phys. Chem. A 2005 , 109 , 8923 ]. The scheme is trivial to implement in any Kohn-Sham program, and the computational cost is that of a series of generalized gradient approximation Kohn-Sham calculations. More generally, the study provides a prescription for performing low-cost, self-consistent Kohn-Sham calculations that yield frontier orbital energies that approximately satisfy the appropriate Koopmans conditions, without the need for exact exchange.

Photoelectron spectroscopic study of I-·ICF3: a frontside attack SN2 pre-reaction complex.

Photodetachment and 2D photoelectron spectra of the mass-selected I-·CF3I complex are presented together with electronic structure calculations. Calculations show that the I- is located at the iodine side of CF3I. Vertical and adiabatic detachment energies were measured at 4.03 and approximately 3.8 eV, respectively. The photoelectron spectra and molecular orbitals show a significant covalent bonding character in the cluster. The presence of electronic excited states is observed. Below threshold, iodide is generated which can be assigned to the photoexcitation of degenerate charge-transfer bands from the off-axis p-orbitals localised on iodide. Near the onset of two spin-orbit thresholds, bright excited states are seen in the experiment and calculations. Excitation of these leads to the formation of slow electrons. The spectroscopy of I-·CF3I is compared to the well-studied I-·CH3I cluster, a pre-reaction complex in the text-book I- + CH3I SN2 reaction. Despite the reversed stereodynamics (i.e. inversion of the CX3 between X = H and F) of the SN2 reaction, striking similarities are seen. Both complexes possess charge transfer excited states near their respective vertical detachment energies and exhibit vibrational structure in their photoelectron spectra. The strong binding is consistent with observations in crossed molecular beam studies and molecular dynamics simulations that suggest that iodine as a leaving group in an SN2 reaction affects the reaction dynamics.

Approximating the Shifted Hartree-Exchange-Correlation Potential in Direct Energy Kohn-Sham Theory.

Levy and Zahariev [Phys. Rev. Lett. 113 113002 (2014)] have proposed a new approach for performing density functional theory calculations, termed direct energy Kohn-Sham (DEKS) theory. In this approach, the electronic energy equals the sum of orbital energies, obtained from Kohn-Sham-like orbital equations involving a shifted Hartree-exchange-correlation potential, which must be approximated. In the present study, density scaling homogeneity considerations are used to facilitate DEKS calculations on a series of atoms and molecules, leading to three nonlocal approximations to the shifted potential. The first two rely on preliminary Kohn-Sham calculations using a standard generalized gradient approximation (GGA) exchange-correlation functional and the results illustrate the benefit of describing the dominant Hartree component of the shift exactly. A uniform electron gas analysis is used to eliminate the need for these preliminary Kohn-Sham calculations, leading to a potential with an unconventional form that yields encouraging results, providing strong motivation for further research in DEKS theory.

Range-Separation Parameter in Tuned Exchange-Correlation Functionals: Successive Ionizations and the Fukui Function.

The range-separation parameter in tuned, range-separated exchange-correlation functionals is investigated in two contexts. First, the system dependence of the parameter is investigated for a series of systems obtained by successively ionizing a single species, paying particular attention to the degree of linearity in the energy versus electron number curve. The parameter exhibits significant system dependence and, therefore, achieving near-linearity in one segment of the curve leads to strong nonlinearity in other segments. This provides a challenging test case for the development of new functionals designed to overcome the known problems of this class of functional. Next, the study considers whether a range-separation parameter tuned to a Koopmans energy condition is also applicable for the analogous density condition. This is tested by comparing two formulations of the Fukui function of conceptual density functional theory, for three representative systems. Both formulations yield the same general features and are not highly sensitive to the range-separation parameter. However, the agreement between the two is near-optimal when the energy-tuned parameter is used, indicating that this parameter is applicable for the analogous density condition.

Fractional Electron Loss in Approximate DFT and Hartree-Fock Theory.

Plots of electronic energy vs electron number, determined using approximate density functional theory (DFT) and Hartree-Fock theory, are typically piecewise convex and piecewise concave, respectively. The curves also commonly exhibit a minimum and maximum, respectively, in the neutral → anion segment, which lead to positive DFT anion HOMO energies and positive Hartree-Fock neutral LUMO energies. These minima/maxima are a consequence of using basis sets that are local to the system, preventing fractional electron loss. Ground-state curves are presented that illustrate the idealized behavior that would occur if the basis set were to be modified to enable fractional electron loss without changing the description in the vicinity of the system. The key feature is that the energy cannot increase when the electron number increases, so the slope cannot be anywhere positive, meaning frontier orbital energies cannot be positive. For the convex (DFT) case, the idealized curve is flat beyond a critical electron number such that any additional fraction of an electron added to the system is unbound. The anion HOMO energy is zero. For the concave (Hartree-Fock) case, the idealized curve is flat up to some critical electron number, beyond which it curves down to the anion energy. A minimum fraction of an electron is required before any binding occurs, but beyond that, the full fraction abruptly binds. The neutral LUMO energy is zero. Approximate DFT and Hartree-Fock results are presented for the F → F(-) segment, and results approaching the idealized behavior are recovered for highly diffuse basis sets. It is noted that if a DFT calculation using a highly diffuse basis set yields a negative LUMO energy then a fraction of an electron must bind and the electron affinity must be positive, irrespective of whether an electron binds experimentally. This is illustrated by calculations on Ne → Ne(-).

System-dependent exchange-correlation functional with exact asymptotic potential and εHOMO ≈ - I.

Density scaling considerations are used to derive an exchange-correlation explicit density functional that is appropriate for the electron deficient side of the integer and which recovers the exact r → ∞ asymptotic behaviour of the exchange-correlation potential. The functional has an unconventional mathematical form with parameters that are system-dependent; the parameters for an N-electron system are determined in advance from generalised gradient approximation (GGA) calculations on the N- and (N - 1)-electron systems. Compared to GGA results, the functional yields similar exchange-correlation energies, but HOMO energies that are an order of magnitude closer to the negative of the vertical ionisation potential; for anions, the HOMO energies are negative, as required. Rydberg excitation energies are also notably improved and the exchange-correlation potential is visibly lowered towards the near-exact potential. Further development is required to improve valence excitations, static isotropic polarisabilities, and the shape of the potential in non-asymptotic regions. The functional is fundamentally different to conventional approximations.

Revisiting the density scaling of the non-interacting kinetic energy.

Scaling relations play an important role in the understanding and development of approximate functionals in density functional theory. Recently, a number of these relationships have been redefined in terms of the Kohn-Sham orbitals [Calderín, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 86, 032510]. For density scaling the author proposed a procedure involving a multiplicative scaling of the Kohn-Sham orbitals whilst keeping their occupation numbers fixed. In the present work, the differences between this scaling with fixed occupation numbers and that of previous studies, where the particle number change implied by the scaling was accommodated through the use of the grand canonical ensemble, are examined. We introduce the terms orbital and ensemble density scaling for these approaches, respectively. The natural ambiguity of the density scaling of the non-interacting kinetic energy functional is examined and the ancillary definitions implicit in each approach are highlighted and compared. As a consequence of these differences, Calderín recovered a homogeneity of degree 1 for the non-interacting kinetic energy functional under orbital scaling, contrasting recent work by the present authors [J. Chem. Phys., 2012, 136, 034101] where the functional was found to be inhomogeneous under ensemble density scaling. Furthermore, we show that the orbital scaling result follows directly from the linearity and the single-particle nature of the kinetic energy operator. The inhomogeneity of the non-interacting kinetic energy functional under ensemble density scaling can be quantified by defining an effective homogeneity. This quantity is shown to recover the homogeneity values for important approximate forms that are exact for limiting cases such as the uniform electron gas and one-electron systems. We argue that the ensemble density scaling provides more insight into the development of new functional forms.

Molecular Binding in Post-Kohn-Sham Orbital-Free DFT.

Molecular binding in post-Kohn-Sham orbital-free DFT is investigated, using noninteracting kinetic energy functionals that satisfy the uniform electron gas condition and which are inhomogeneous under density scaling. A parameter is introduced that quantifies binding, and a series of functionals are determined from fits to near-exact effective homogeneities and/or Kohn-Sham noninteracting kinetic energies. These are then used to investigate the relationship between binding and the accuracy of the effective homogeneity and noninteracting kinetic energy at the equilibrium geometry. For a series of 11 molecules, the binding broadly improves as the effective homogeneity improves, although the extent to which it improves is dependent on the accuracy of the noninteracting kinetic energy; optimal binding appears to require both to be accurate simultaneously. The use of a Thomas-Fermi-von Weizsäcker form, augmented with a second gradient correction, goes some way toward achieving this, exhibiting molecular binding on average. The findings are discussed in terms of the noninteracting kinetic potential and the Hellmann-Feynman theorem. The extent to which the functionals can reproduce the system-dependence of the near-exact effective homogeneity is quantified, and potential energy curves are presented for selected molecules. The study provides impetus for including density scaling homogeneity considerations in the design of noninteracting kinetic energy functionals.

Benchmarking density-functional theory calculations of NMR shielding constants and spin-rotation constants using accurate coupled-cluster calculations.

Accurate sets of benchmark nuclear-magnetic-resonance shielding constants and spin-rotation constants are calculated using coupled-cluster singles-doubles (CCSD) theory and coupled-cluster singles-doubles-perturbative-triples [CCSD(T)] theory, in a variety of basis sets consisting of (rotational) London atomic orbitals. The accuracy of the calculated coupled-cluster constants is established by a careful comparison with experimental data, taking into account zero-point vibrational corrections. Coupled-cluster basis-set convergence is analyzed and extrapolation techniques are employed to estimate basis-set-limit quantities, thereby establishing an accurate benchmark data set. Together with the set provided for rotational g-tensors and magnetizabilities in our previous work [O. B. Lutnæs, A. M. Teale, T. Helgaker, D. J. Tozer, K. Ruud, and J. Gauss, J. Chem. Phys. 131, 144104 (2009)], it provides a substantial source of consistently calculated high-accuracy data on second-order magnetic response properties. The utility of this benchmark data set is demonstrated by examining a wide variety of Kohn-Sham exchange-correlation functionals for the calculation of these properties. None of the existing approximate functionals provide an accuracy competitive with that provided by CCSD or CCSD(T) theory. The need for a careful consideration of vibrational effects is clearly illustrated. Finally, the pure coupled-cluster results are compared with the results of Kohn-Sham calculations constrained to give the same electronic density. Routes to future improvements are discussed in light of this comparison.

Density Scaling of Noninteracting Kinetic Energy Functionals.

The influence of imposing an approximate density scaling condition on a noninteracting kinetic energy functional is investigated. A simple generalized gradient approximation (GGA) is presented, which satisfies both the density scaling condition and the usual coordinate scaling condition; the remaining multiplicative constant is determined from an energy criterion. In post-Kohn-Sham calculations, noninteracting kinetic energies of the closed-shell molecules of the G1 set determined using the GGA are a modest improvement over those determined using the corresponding local functional, which does not satisfy the density scaling condition. Potential energy curves of CO, F2, and P2 exhibit binding with the GGA, compared to purely repulsive curves with the local functional. Adjusting the exponent in the GGA form in order to optimize energy accuracy violates the density scaling condition, and two of the diatomics no longer exhibit binding. Results are compared with those from other local/GGA functionals in the literature.

Assessment of Tuning Methods for Enforcing Approximate Energy Linearity in Range-Separated Hybrid Functionals.

A range of tuning methods, for enforcing approximate energy linearity through a system-by-system optimization of a range-separated hybrid functional, are assessed. For a series of atoms, the accuracy of the frontier orbital energies, ionization potentials, electron affinities, and orbital energy gaps is quantified, and particular attention is paid to the extent to which approximate energy linearity is actually achieved. The tuning methods can yield significantly improved orbital energies and orbital energy gaps, compared to those from conventional functionals. For systems with integer M electrons, optimal results are obtained using a tuning norm based on the highest occupied orbital energy of the M and M + 1 electron systems, with deviations of just 0.1-0.2 eV in these quantities, compared to exact values. However, detailed examination for the carbon atom illustrates a subtle cancellation between errors arising from nonlinearity and errors in the computed ionization potentials and electron affinities used in the tuning.

Overcoming low orbital overlap and triplet instability problems in TDDFT.

Low orbital overlap and triplet instability problems in time-dependent density functional theory (TDDFT) are investigated for a new benchmark set, encompassing challenging singlet and triplet excitation energies of local, charge-transfer, and Rydberg character. The low orbital overlap problem is largely overcome for both singlet and triplet states by the use of a Coulomb-attenuated functional. For all the categories of functional considered, however, errors associated with triplet instability problems plague high overlap excitations, as exemplified by the excited states of acenes and polyacetylene oligomers. Application of the Tamm-Dancoff approximation reduces these errors for both singlet and triplet states, while leaving low-overlap excitations unaffected. The study illustrates the synergy between overlap and stability and highlights the success of a combined, Coulomb-attenuated Tamm-Dancoff approach.