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.

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.

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.

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 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.

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.

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.

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.

Experimental and theoretical studies on organic D-π-A systems containing three-coordinate boron moieties as both π-donor and π-acceptor.

Four linear π-conjugated systems with 1,3-diethyl-1,3,2-benzodiazaborolyl [C(6)H(4)(NEt)(2)B] as a π-donor at one end and dimesitylboryl (BMes(2)) as a π-acceptor at the other end were synthesized. These unusual push-pull systems contain phenylene (-1,4-C(6)H(4)-; 1), biphenylene (-4,4'-(1,1'-C(6)H(4))(2)-; 2), thiophene (-2,5-C(4)H(2)S-; 3), and dithiophene (-5,5'-(2,2'-C(4)H(2)S)(2)-; 4) as π-conjugated bridges and different types of three-coordinate boron moieties serving as both π-donor and π-acceptor. Molecular structures of 2, 3, and 4 were determined by single-crystal X-ray diffraction. Photophysical studies on these systems reveal blue-green fluorescence in all compounds. The Stokes shifts for 1, 2, and 3 are notably large at 7820-9760 cm(-1) in THF and 5430-6210 cm(-1) in cyclohexane, whereas the Stokes shift for 4 is significantly smaller at 5510 cm(-1) in THF and 2450 cm(-1) in cyclohexane. Calculations on model systems 1'-4' show the HOMO to be mainly diazaborolyl in character and the LUMO to be dominated by the empty p orbital at the boron atom of the BMes(2) group. However, there are considerable dithiophene bridge contributions to both orbitals in 4'. From the experimental data and MO calculations, the π-electron-donating strength of the 1,3-diethyl-1,3,2-benzodiazaborolyl group was found to lie between that of methoxy and dimethylamino groups. TD-DFT calculations on 1'-4', using B3LYP and CAM-B3LYP functionals, provide insight into the absorption and emission processes. B3LYP predicts that both the absorption and emission processes have strong charge-transfer character. CAM-B3LYP which, unlike B3LYP, contains the physics necessary to describe charge-transfer excitations, predicts only a limited amount of charge transfer upon absorption, but somewhat more upon emission. The excited-state (S(1)) geometries show the borolyl group to be significantly altered compared to the ground-state (S(0)) geometries. This borolyl group reorganization in the excited state is believed to be responsible for the large Stokes shifts in organic systems containing benzodiazaborolyl groups in these and related compounds.

Negative electron affinities from DFT: influence of asymptotic exchange-correlation potential and effective homogeneity under density scaling.

The influence of the asymptotic exchange-correlation potential and density-scaling homogeneity on negative electron affinities determined using the approach of Tozer and De Proft [J. Phys. Chem. A2005, 109, 8923] is investigated. Application of an asymptotic correction to the potential improves the accuracy for several of the systems with the most negative affinities, reflecting their diffuse lowest unoccupied orbitals. For systems with modest affinities, it reduces the accuracy marginally. Enforcing a near-exact effective homogeneity through a simple shift in the potential leads to improved correlation with experimental values but significantly overestimated affinities. Optimal effective homogeneities are therefore determined, and a simple scheme is proposed for enforcing an average optimal value. Application of the scheme to a series of organic molecules maintains the excellent correlation with the experimental values while significantly reducing the absolute errors.

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.

Effective homogeneity of the exchange-correlation and non-interacting kinetic energy functionals under density scaling.

Correlated electron densities, experimental ionisation potentials, and experimental electron affinities are used to investigate the homogeneity of the exchange-correlation and non-interacting kinetic energy functionals of Kohn-Sham density functional theory under density scaling. Results are presented for atoms and small molecules, paying attention to the influence of the integer discontinuity and the choice of the electron affinity. For the exchange-correlation functional, effective homogeneities are highly system-dependent on either side of the integer discontinuity. By contrast, the average homogeneity-associated with the potential that averages over the discontinuity-is generally close to 4/3 when the discontinuity is computed using positive affinities for systems that do bind an excess electron and negative affinities for those that do not. The proximity to 4/3 becomes increasingly pronounced with increasing atomic number. Evaluating the discontinuity using a zero affinity in systems that do not bind an excess electron instead leads to effective homogeneities on the electron abundant side that are close to 4/3. For the non-interacting kinetic energy functional, the effective homogeneities are less system-dependent and the effect of the integer discontinuity is less pronounced. Average values are uniformly below 5/3. The study provides information that may aid the development of improved exchange-correlation and non-interacting kinetic energy functionals.

On the evaluation of the non-interacting kinetic energy in density functional theory.

The utility of both an orbital-free and a single-orbital expression for computing the non-interacting kinetic energy in density functional theory is investigated for simple atomic systems. The accuracy of both expressions is governed by the extent to which the Kohn-Sham equation is solved for the given exchange-correlation functional and so special attention is paid to the influence of finite Gaussian basis sets. The orbital-free expression is a statement of the virial theorem and its accuracy is quantified. The accuracy of the single-orbital expression is sensitive to the choice of Kohn-Sham orbital. The use of particularly compact orbitals is problematic because the failure to solve the Kohn-Sham equation exactly in regions where the orbital has decayed to near-zero leads to unphysical behaviour in regions that contribute to the kinetic energy, rendering it inaccurate. This problem is particularly severe for core orbitals, which would otherwise appear attractive due to their formally nodeless nature. The most accurate results from the single-orbital expression are obtained using the relatively diffuse, highest occupied orbitals, although special care is required at orbital nodes.

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.

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.

Experimental and theoretical studies of the photophysical properties of 2- and 2,7-functionalized pyrene derivatives.

Pyrene derivatives substituted at the 2- and 2,7-positions are shown to display a set of photophysical properties different from those of derivatives substituted at the 1-position. It was found that, in the 2- and 2,7-derivatives, there was little influence on the S(2) ← S(0) excitation, which is described as "pyrene-like", and a strong influence on the S(1) ← S(0) excitation, which is described as "substituent-influenced". In contrast, the 1-substituted derivatives display a strong influence on both the S(1) ← S(0) and the S(2) ← S(0) excitations. These observations are rationalized by considering the nature of the orbitals involved in the transitions. The existence of a nodal plane passing through the 2- and 7-positions, perpendicular to the molecular plane in the HOMO and LUMO of pyrene, largely accounts for the different behavior of derivatives substituted at the 2- and 2,7-positions. Herein, we report the photophysical properties of a series of 2-R-pyrenes {R = C(3)H(6)CO(2)H (1), Bpin (2; pin = OCMe(2)CMe(2)O), OC(3)H(6)CO(2)H (3), O(CH(2))(12)Br (4), C≡CPh (5), C(6)H(4)-4-CO(2)Me (6), C(6)H(4)-4-B(Mes)(2) (7), B(Mes)(2) (8)} and 2,7-R(2)-pyrenes {R = Bpin (9), OH (10), C≡C(TMS) (11), C≡CPh (12), C≡C-C(6)H(4)-4-B(Mes)(2) (13), C≡C-C(6)H(4)-4-NMe(2) (14), C(6)H(4)-4-CO(2)C(8)H(17) (15), N(Ph)-C(6)H(4)-4-OMe (16)} whose syntheses are reported elsewhere. Furthermore, we compare their properties to those of several related 1-R-pyrene derivatives {R = C(3)H(6)CO(2)H (17), Bpin (18), C≡CPh (19), C(6)H(4)-4-B(Mes)(2) (20), B(Mes)(2) (21)}. For all derivatives, modest (0.19) to high (0.93) fluorescence quantum yields were observed. For the 2- and 2,7-derivatives, fluorescence lifetimes exceeding 16 ns were measured, with most being ca. 50-80 ns. The 4-(pyren-2-yl)butyric acid derivative (1) has a long fluorescence lifetime of 622 ns, significantly longer than that of the commercially available 4-(pyren-1-yl)butyric acid (17). In addition to measurements of absorption and emission spectra and fluorescence quantum yields and lifetimes, time-dependent density functional theory calculations using the B3LYP and CAM-B3LYP functionals were also performed. A comparison of experimental and theoretically calculated wavelengths shows that both functionals were able to reproduce the trend in wavelengths observed experimentally.

Should negative electron affinities be used for evaluating the chemical hardness?

Despite recent advances in computing negative electron affinities using density-functional theory, it is an open issue as to whether it is appropriate to use negative electron affinities, instead of zero electron affinity, to compute the chemical hardness of atoms and molecules with metastable anions. We seek to answer this question using the accepted empirical rules linking the chemical hardness to the atomic size and the polarizability; we also propose a new correlation with the C6 London dispersion coefficient. For chemical reactivity in the gas phase, it seems to make no difference whether negative, or zero, electron affinities are used for systems with metastable anions. For reactions in solution the evidence that is presently available is insufficient to establish a preference. In addressing this issue, we noted that electron affinity data from which atomic chemical hardness values are computed are out of date; an update to Pearson's classic 1988 table [Inorg. Chem., 1988, 27, 734-740] is thus provided.

Bonding in negative ions: the role of d orbitals in the heavy analogues of pyridine and furan radical anions.

We have used a potential wall method to investigate the role of d orbitals in the a(2) singly-occupied molecular orbitals of (2)A(2) negative ion states of two molecular series: pyridine, phosphabenzene, arsabenzene, stibabenzene (C(5)H(5)X, X = {N, P, As, Sb}), and furan, thiophene, selenophene, tellurophene (C(4)H(4)X, X = {O, S, Se, Te}). Unlike for the lower lying doubly occupied orbitals, heteroatom d-carbon p in-phase (bonding) interactions in these a(2) orbitals are clearly identified and explain the 0.5 eV stabilization of the (2)A(2) radical anion state in those compounds where the heteroatoms have d orbitals in the valence shell, compared to compounds where d orbitals are missing in the valence shell of the heteroatoms. The performance of both the potential wall approach and the approximate expression of Tozer and De Proft for calculating negative electron affinities has been also investigated, through a comparison with results obtained using electron-transmission spectroscopy experiments.

Dispersion, static correlation, and delocalisation errors in density functional theory: an electrostatic theorem perspective.

Dispersion, static correlation, and delocalisation errors in density functional theory are considered from the unconventional perspective of the force on a nucleus in a stretched diatomic molecule. The electrostatic theorem of Feynman is used to relate errors in the forces to errors in the electron density distortions, which in turn are related to erroneous terms in the Kohn-Sham equations. For H(2), the exact dispersion force arises from a subtle density distortion; the static correlation error leads to an overestimated force due to an exaggerated distortion. For H(2)(+), the exact force arises from a delicate balance between attractive and repulsive components; the delocalisation error leads to an underestimated force due to an underestimated distortion. The net force in H(2)(+) can become repulsive, giving the characteristic barrier in the potential energy curve. Increasing the fraction of long-range exact orbital exchange increases the distortion, reducing delocalisation error but increasing static correlation error.