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.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

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 1²B_2g 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.

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.

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.

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.

Atomic electron affinities and the role of symmetry between electron addition and subtraction in a corrected Koopmans approach

The essential aspects of zero-temperature grand-canonical ensemble density-functional theory are reviewed in the context of spin-density-functional theory and are used to highlight the assumption of symmetry between electron addition and subtraction that underlies the corrected Koopmans approach of Tozer and De Proft (TDP) for computing electron affinities. The issue of symmetry is then investigated in a systematic study of atomic electron affinities, comparing TDP affinities with those from a conventional Koopmans evaluation and electronic energy differences. Although it cannot compete with affinities determined from energy differences, the TDP expression yields results that are a significant improvement over those from the conventional Koopmans expression. Key insight into the results from both expressions is provided by an analysis of plots of the electronic energy as a function of the number of electrons, which highlight the extent of symmetry between addition and subtraction. The accuracy of the TDP affinities is closely related to the nature of the orbitals involved in the electron addition and subtraction, being particularly poor in cases where there is a change in principal quantum number, but relatively accurate within a single manifold of orbitals. The analysis is then extended to a consideration of the ground state Mulliken electronegativity and chemical hardness. The findings further emphasize the key role of symmetry in determining the quality of the results.

Revisiting the density scaling of the non-interacting kinetic energy

Two different density scaling approaches are compared and their prospects for use in functional development are reviewed.

Sample sizes for lesion magnetisation transfer ratio outcomes in remyelination trials for multiple sclerosis

BACKGROUND

Enhancing remyelination in MS might improve function and protect axons from future damage. Lesion magnetisation transfer ratio (MTR) is sensitive to myelin content, and may be a useful measure for trials evaluating potential remyelinating agents.

OBJECTIVE

Estimating sample sizes required for a parallel group, placebo-controlled trial in MS using change in mean MTR of all T2lesions as a primary outcome measure.

METHODS

The primary sample size calculation was derived from data from a natural history study of relapsing remitting MS (n=18). The MTR values observed in demyelinated and remyelinated lesions in an ex vivo study were used to estimate the effect of remyelination on lesion MTR. The ex vivo data were also used to independently calculate sample sizes in order to inform the robustness of the in vivo estimates.

RESULTS

Calculations suggest that 30% remyelination of T2 lesions could be detected with 80% power in 38 (95% confidence interval 12-96) patients per arm based on the in vivo data, and in 66 per arm based on the ex vivo data.

CONCLUSION

The sample sizes derived are in a range that makes MTR a feasible outcome measure for proof-of-concept trials of putative therapies achieving remyelination in MS lesions.

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.

Methodology of diffusion-weighted, diffusion tensor and magnetisation transfer imaging

MRI offers a number of opportunities to examine characteristics of tissue well below the spatial resolution of the imaging technique. The best known of these is diffusion imaging, which allows the production of images whose contrast reflects the ability of water molecules to move through the extravascular extracellular space. Less well-known, but increasingly important, is magnetisation transfer imaging, which produces contrast based on the ability of protons to move between the free water pool and local macromolecules. Both of these techniques offer unique information about the microscopic and molecular structure of tumour tissue. This article will briefly review the underlying theory and technical aspects associated with these imaging techniques.

Influence of Triplet Instabilities in TDDFT

Singlet and triplet vertical excitation energies from time-dependent density functional theory (TDDFT) can be affected in different ways by the inclusion of exact exchange in hybrid or Coulomb-attenuated/range-separated exchange-correlation functionals; in particular, triplet excitation energies can become significantly too low. To investigate these issues, the explicit dependence of excitation energies on exact exchange is quantified for four representative molecules, paying attention to the effect of constant, short-range, and long-range contributions. A stability analysis is used to verify that the problematic TDDFT triplet excitations can be understood in terms of the ground state triplet instability problem, and it is proposed that a Hartree-Fock stability analysis should be used to identify triplet excitations for which the presence of exact exchange in the TDDFT functional is undesirable. The use of the Tamm-Dancoff approximation (TDA) significantly improves the problematic triplet excitation energies, recovering the correct state ordering in benzoquinone; it also affects the corresponding singlet states, recovering the correct state ordering in naphthalene. The impressive performance of the TDA is maintained for a wide range of molecules across representative functionals.

Axonal damage in the making: neurofilament phosphorylation, proton mobility and magnetisation transfer in multiple sclerosis normal appearing white matter

Aims

Multiple sclerosis (MS) leaves a signature on the phosphorylation and thus proton binding capacity of axonal neurofilament (Nf) proteins. The proton binding capacity in a tissue is the major determinant for exchange between bound and free protons and thus the magnetisation transfer ratio (MTR). This study investigated whether the MTR of non-lesional white matter (NLWM) was related to the brain tissue concentration of neurofilament phosphoforms.

Methods

Unfixed post-mortem brain slices of 12 MS patients were analysed using MTR, T1 at 1.5 T. Blocks containing NLWM were processed for embedding in paraffin and inspected microscopically. Adjacent tissue was microdissected, homogenised and specific protein levels were quantified by ELISA for the Nf heavy chain (NfH) phosphoforms, glial fibrillary acidic protein (GFAP), S100B and ferritin.

Results

Averaged hyperphosphorylated NfH (SMI34) but not phosphorylated NfH (SMI35) levels were different between individual patients NLWM. The concentration of hyperphosphorylated NfH-SMI34 correlated with T1 (R = 0.70, p = 0.0114) and — inversely — with MTR (R =−0.73, p = 0.0065). NfH-SMI35 was not correlated to any of the MR indices.

Conclusions

Post-translational modifications of axonal proteins such as phosphorylation of neurofilaments occur in NLWM and may precede demyelination. The resulting change of proton mobility influences MTR and T1. This permits the in vivo detection of these subtle tissue changes on a proteomic level in patients with MS.

Longitudinal changes in magnetisation transfer ratio in secondary progressive multiple sclerosis: data from a randomised placebo controlled trial of lamotrigine

Sodium blockade with lamotrigine is neuroprotective in animal models of central nervous system demyelination. This study evaluated the effect of lamotrigine on magnetisation transfer ratio (MTR), a putative magnetic resonance imaging measure of intact brain tissue, in a group of subjects with secondary progressive multiple sclerosis (MS). In addition, the utility of MTR measures for detecting change in clinically relevant pathology was evaluated. One hundred seventeen people attending the National Hospital for Neurology and Neurosurgery or the Royal Free Hospital, London, UK, were recruited into a double-blind, parallel-group trial. Subjects were randomly assigned by minimisation to receive lamotrigine (target dose 400 mg/day) or placebo for 2 years. Treating and assessing physicians and patients were masked to treatment allocation. Results of the primary endpoint, central cerebral volume, have been published elsewhere. Significant differences between the verum and placebo arms were seen in only two measures [normal appearing grey matter (NAGM) p = 0.036 and lesion peak height (PH) p = 0.004], and in both cases there was a greater reduction in MTR in the verum arm. Significant correlations were found of change in MS functional composite with all MTR measures except lesion and normal appearing white matter (NAWM) PH. However, the change in MTR measures over 2 years were small, with only NAGM mean (p = 0.001), lesion peak location (p = 0.11) and mean (p < 0.0001) changing significantly from baseline. These data did not show that lamotrigine was neuroprotective. The clinical correlation of MTR measures was consistent, but the responsiveness to change was limited.

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.

Magnetization transfer ratio abnormalities reflect clinically relevant grey matter damage in multiple sclerosis

BACKGROUND

In multiple sclerosis, grey matter (GM) damage appears more clinically relevant than either white matter damage or lesion load.

OBJECTIVE

We investigated if normal-appearing white matter (NAWM) and grey matter tissue changes assessed by magnetization transfer ratio were associated with long-term disability.

METHODS

Sixty-nine people were assessed 20 years after presentation with a clinically isolated syndrome (CIS) [28 still CIS, 31 relapsing-remitting multiple sclerosis, 10 secondary progressive multiple sclerosis], along with 19 healthy subjects. Mean magnetization transfer ratio, peak height (PH) and peak location of the normalized magnetization transfer ratio histograms were determined in NAWM and grey matter, as well as, white matter and GM Fraction (GMF) and T(2)-weighted lesion load.

RESULTS

Median expanded disability status scale for multiple sclerosis patients was 2.5 (range 1-8). GM-PH, and less so, NAWM mean and peak location, were lower in multiple sclerosis patients (P = 0.009) versus controls, relapsing-remitting multiple sclerosis versus CIS (P = 0.008) and secondary progressive multiple sclerosis versus relapsing-remitting multiple sclerosis (P = 0.002). GM-PH (as well as GMF) correlated with expanded disability status scale (r(s) = -0.49; P = 0.001) and multiple sclerosis functional score (r(s) = 0.51; P = 0.001). GM-PH independently predicted disability with similar strength to the associations of GMF with clinical measures.

CONCLUSION

Grey matter damage was related to long-term disability in multiple sclerosis cohort with a relatively low median expanded disability status scale. Markers of intrinsic grey matter damage (magnetization transfer ratio) and tissue loss offer clinically relevant information in multiple sclerosis.

Grey matter magnetization transfer ratio independently correlates with neurological deficit in secondary progressive multiple sclerosis

Although there is substantial brain grey matter pathology in secondary progressive multiple sclerosis (MS), there has been limited investigation of its contribution to disability.This study investigated the correlation of magnetization transfer ratio (MTR) measures taken from brain grey matter, normal appearing white matter (NAWM) and lesions with neurological deficit and disability in 113 people with secondary progressive MS. In order to adjust for the potential effects of focal white matter lesions and global brain atrophy, T2 lesion volume and normalized brain volume (NBV) were also calculated for each subject. Clinical measures included the expanded disability status scale (EDSS) and the multiple sclerosis functional composite (MSFC) scores. Linear regression analysis was used to assess the age- and gender-adjusted correlation of MTR histogram mean, peak height and peak location with the MSFC and individual component measures. Logistic regression analysis was used to determine whether imaging measures could be used to predict if subjects were in the higher disability group (EDSS > or = 6.5).Significant correlations were detected between MSFC composite and mean MTR in (i) normal appearing white matter (NAWM; r = 0.327, p < 0.0001), (ii) grey matter (r = 0.460, p < 0.0001) and (iii) lesions (r = 0.394, p < 0.0001). Although NBV and T2 lesion volume correlated significantly with MSFC, grey matter histogram mean emerged as the best predictor of MSFC score. None of the MRI measures significantly predicted higher EDSS.These results suggest that brain grey matter pathology plays an important role in determining neurological impairment. The apparent paucity of correlation between MRI measures and EDSS is likely to represent the relative insensitivity of the latter measure in this study group.

TDDFT diagnostic testing and functional assessment for triazene chromophores

A simple diagnostic test based on orbital overlap [M. J. G. Peach et al., J. Chem. Phys., 2008, 128, 044118] may be used to help judge the reliability of excitation energies in time-dependent density functional theory (TDDFT) when using generalized gradient approximation (GGA) and hybrid functionals. Orbital plots are used to illustrate the test for a model tripeptide and for 4-(N,N-dimethylamino)benzonitrile, which are representative of systems containing low- and high-overlap charge-transfer excitations. The scheme is then applied to a series of triazene chromophores in solvent, highlighting the relationship between overlap and oscillator strength and its implications for theoretical absorption spectra. No low-overlap excitations are observed with a hybrid functional; a single one is identified using a GGA. To assess the diagnostic test and to judge functional performance, gas phase triazene TDDFT excitations are compared with correlated ab initio values. The diagnostic test correctly identifies two low-overlap problematic GGA excitations. However, it does not identify another problematic excitation where the electron is excited to a spatially extended orbital, which necessarily has reasonable overlap with the occupied orbital; an improved diagnostic quantity is required for such cases. The best agreement between TDDFT and correlated ab initio excitations is obtained using a Coulomb-attenuated functional; the errors are significantly smaller than from the GGA and hybrid functionals. The study provides further support for the high quality excitations from Coulomb-attenuated functionals, negating the need for diagnostic tests.