Temperature‐Dependent Behavior of the Dual Fluorescence of 2‐(3‐Fluorophenyl)‐2,3‐dihydro‐1H‐benzo[f]isoindole‐1,3‐dione

The fluorescence behavior of 2‐(3‐fluorophenyl)‐2,3‐dihydro‐1H‐benzo[f]isoindole‐1,3‐dione ( 1 ) was studied in solvents of different polarity and viscosity. Dual luminescence is observed and the short‐wavelength emission is found to increase considerably with the solvent polarity. The ratio of the fluorescence quantum yield of the two states emitting, the one (SW*) at short wavelength and the other (LW*) at long wavelength, shows a bell‐shaped dependence on the reciprocal of the temperature in diethyl ether, butyronitrile, and propane‐1,2,3‐triol triacetate (glycerol triacetate; GTA). This has been interpreted as the result of a reversible interconversion between the two states. The enthalpy difference between the SW* and LW* excited states, as deduced from the slope of the ln (Φ/Φ) vs. 1/T curves in the high temperature range, is found to be solvent polarity and solvent viscosity independent as the same value (−7.3 kJ/mol) is obtained in the three above‐mentioned solvents. The independence from polarity is the consequence of a similar difference in dipole moment between the ground‐state and the SW* and LW* excited states (4.5 and 4.9 D, respectively, derived from solvatochromy). The activation energy of the SW*→LW* step deduced from the low temperature measurements in the nonviscous solvents, increases with solvent polarity (11.6 and 17.5 kJ/mol for diethyl ether and butyronitrile, respectively); they are greater than the viscous‐flow activation energy of the solvents indicating that the resolvation of the excited dipole controls the kinetics. In the nonviscous solvents, the LW* state originates from the SW* state, while in the viscous GTA solution, both states are formed simultaneously within the 1‐ps laser pulse.     

Electron correlation methods based on the random phase approximation

In the past decade, the random phase approximation (RPA) has emerged as a promising post-Kohn–Sham method to treat electron correlation in molecules, surfaces, and solids. In this review, we explain how RPA arises naturally as a zero-order approximation from the adiabatic connection and the fluctuation-dissipation theorem in a density functional context. This is contrasted to RPA with exchange (RPAX) in a post-Hartree–Fock context. In both methods, RPA and RPAX, the correlation energy may be expressed as a sum over zero-point energies of harmonic oscillators representing collective electronic excitations, consistent with the physical picture originally proposed by Bohm and Pines. The extra factor 1/2 in the RPAX case is rigorously derived. Approaches beyond RPA are briefly summarized. We also review computational strategies implementing RPA. The combination of auxiliary expansions and imaginary frequency integration methods has lead to recent progress in this field, making RPA calculations affordable for systems with over 100 atoms. Finally, we summarize benchmark applications of RPA to various molecular and solid-state properties, including relative energies of conformers, reaction energies involving weak and covalent interactions, diatomic potential energy curves, ionization potentials and electron affinities, surface adsorption energies, bulk cohesive energies and lattice constants. RPA barrier heights for an extended benchmark set are presented. RPA is an order of magnitude more accurate than semi-local functionals such as B3LYP for non-covalent interactions rivaling the best empirically parametrized methods. Larger but systematic errors are observed for processes that do not conserve the number of electron pairs, such as atomization and ionization.     

Comparisons of the RPA,SCRPA, Tamm–Dancoff,and full CI methods by analysis of their transition density matrices,oscillator strengths,and energy moments of oscillator strengths for the electric dipole transitions from the ground state of the B+ ion (frozen K-shell model)

The RPA, SCRPA , Tamm–Dancoff, and full CI methods are compared by analyzing their transition density matrices, oscillator strengths, and energy moments of oscillator strengths for the ¹S_ground – ¹P_odd transitions of the 4-electron B⁺ ion in the frozen K-shell approximation. It is found that the RPA gives transition density matrices that are aligned nearly as well as possible along those of the full CI , but have vector lengths that are significantly too long. The corresponding transition energies are significantly too small. These errors compensate to give oscillator strengths for the dominant transition that, for all forms of the oscillator strength, are within 1.6% of the corresponding full CI values. The SCRPA gives better transition density matrices than the RPA , but poorer oscillator strengths. The Tamm-Dancoff approximation gives very good values for the mixed length-velocity form of the oscillator strength. The RPA gives a static electric dipole polarizability that is nearly 20% larger than that of the full CI . The SCRPA gives a value 15% smaller than—and the Tamm-Dancoff approximation gives a mixed length-velocity value that is 11% larger than—that of the full CI . Other energy moments of oscillator strengths are also reported. Certain other approximations related to the RPA and the SCRPA are reported as well.     

Third-order corrections to random-phase approximation correlation energies

Several random-phase approximation (RPA) correlation methods were compared in third order of perturbation theory. While all of the considered approaches are exact in second order of perturbation theory, it is found that their corresponding third-order correlation energy contributions strongly differ from the exact third-order correlation energy contribution due to missing interactions of the particle-particle-hole-hole type. Thus a simple correction method is derived which makes the different RPA methods also exact to third-order of perturbation theory. By studying the reaction energies of 16 chemical reactions for 21 small organic molecules and intermolecular interaction energies of 23 intermolecular complexes comprising weakly bound and hydrogen-bridged systems, it is found that the third-order correlation energy correction considerably improves the accuracy of RPA methods if compared to coupled-cluster singles doubles with perturbative triples as a reference.     

A simple but fully nonlocal correction to the random phase approximation

The random phase approximation (RPA) stands on the top rung of the ladder of ground-state density functional approximations. The simple or direct RPA has been found to predict accurately many isoelectronic energy differences. A nonempirical local or semilocal correction to this direct RPA leaves isoelectronic energy differences almost unchanged, while improving total energies, ionization energies, etc., but fails to correct the RPA underestimation of molecular atomization energies. Direct RPA and its semilocal correction may miss part of the middle-range multicenter nonlocality of the correlation energy in a molecule. Here we propose a fully nonlocal, hybrid-functional-like addition to the semilocal correction. The added full nonlocality is important in molecules, but not in atoms. Under uniform-density scaling, this fully nonlocal correction scales like the second-order-exchange contribution to the correlation energy, an important part of the correction to direct RPA, and like the semilocal correction itself. For the atomization energies of ten molecules, and with the help of one fit parameter, it performs much better than the elaborate second-order screened exchange correction.     

A comparative analysis of Hartree-Fock and Kohn-Sham orbital energies

Hartree-Fock and Kohn-Sham orbital energies, the latter computed with several different exchange/correlation functionals, are compared and analyzed for 12 molecules. The Kohn-Sham energies differ significantly from experimental ionization energies, but by amounts that are, for a given molecule and exchange/correlation functional, roughly the same for all of the valence orbitals. With the exchange/correlation functionals used, the energy of the highest occupied Kohn-Sham orbital does not approximate the corresponding ionization potential any better than do the other orbital energies. Received: 24 October 1997 / Accepted 31 October 1997     

Localized exchange-correlation potential from second-order self-energy for accurate Kohn-Sham energy gap

A local Kohn-Sham (KS) exchange-correlation potential is derived by localizing the second-order self-energy operator, using approximations to the linear response Sham-Schlüter equation. Thanks to the use of the resolution-of-identity technique for the calculation of the self-energy matrix elements, the method is very efficient and can be applied to large systems. The authors investigate the KS energy gaps and lowest excitation energies of atoms and small- and medium-size molecules. Reference KS energy gaps (from accurate densities) of atoms and small molecules can be reproduced with great accuracy. For larger systems they found that the KS energy gap is smaller than the one obtained from the local-density approximation, showing the importance of an ab initio correlation in the Kohn-Sham potential.     

Third-order energy derivative corrections to the Kohn-Sham orbital hardness tensor

The third term in the Taylor expansion of the total energy functional around the number of electronsN is evaluated as the second-order derivative of orbital Kohn-Sham energies with respect to orbital occupancy. Present approach is an extension of an efficient algorithm to compute densityfunctional based orbital reactivity indices. Various energy derivatives used to approximate orbital reactivity indices are defined within the space spanned by the orbital occupation numbers and the Kohn-Sham one-electron energies. The third-order energy functional derivative has to be considered for singular hardness tensor ([η]). On the contrary, this term has negligible influence on the reactivity index values for atomic or molecular systems with positively defined hardness tensors. In this context, stability of a system in equilibrium state estimated through the eigenvalues of [η] is discussed. Numerical illustration of the Kohn-Sham energy functional derivatives in orbital resolution up to the third order is shown for benchmark molecules such as H₂O, H₂S, and OH⁻.     

Increasing the applicability of density functional theory. II. Correlation potentials from the random phase approximation and beyond

Density functional theory (DFT) results are mistrusted at times due to the presence of an unknown exchange correlation functional, with no practical way to guarantee convergence to the right answer. The use of a known exchange correlation functional based on wave-function theory helps to alleviate such mistrust. The exchange correlation functionals can be written exactly in terms of the density-density response function using the adiabatic-connection and fluctuation-dissipation framework. The random phase approximation (RPA) is the simplest approximation for the density-density response function. Since the correlation functional obtained from RPA is equivalent to the direct ring coupled cluster doubles (ring-CCD) correlation functional, meaning only Coulomb interactions are included, one can bracket RPA between many body perturbation theory (MBPT)-2 and CCD with the latter having all ring, ladder, and exchange contributions. Using an optimized effective potential strategy, we obtain correlation potentials corresponding to MBPT-2, RPA (ring-CCD), linear-CCD, and CCD. Using the suitable choice of the unperturbed Hamiltonian, Kohn-Sham self-consistent calculations are performed. The spatial behavior of the resulting potentials, total energies, and the HOMO eigenvalues are compared with the exact values for spherical atoms. Further, we demonstrate that the self-consistent eigenvalues obtained from these consistent potentials used in ab initio dft approximate all principal ionization potentials as demanded by ionization potential theorem.     

The equations-of-motion method for F2: Transition energies,oscillator strengths and born cross sections

The equations-of-motion method has been used to study various electronic states of F₂. The transition energies have been found in both the random phase approximation (RPA) and higher random approximation (HRPA) using single particle—hole components in the excitation operators. We have also computed generalized oscillator strengths (Born cross sections) for the scattering of high energy electrons by F₂.     

Noniterative Doubles Corrections to the Random Phase and Higher Random Phase Approximations: Singlet and Triplet Excitation Energies

The second-order noniterative doubles-corrected random phase approximation (RPA) method has been extended to triplet excitation energies and the doubles-corrected higher RPA method as well as a shifted version for calculating singlet and triplet excitation energies are presented here for the first time. A benchmark set consisting of 20 molecules with a total of 117 singlet and 71 triplet excited states has been used to test the performance of the new methods by comparison with previous results obtained with the second-order polarization propagator approximation (SOPPA) and the third order approximate coupled cluster singles, doubles and triples model CC3. In general, the second-order doubles corrections to RPA and HRPA significantly reduce both the mean deviation as well as the standard deviation of the errors compared to the CC3 results. The accuracy of the new methods approaches the accuracy of the SOPPA method while using only 10–60% of the calculation time. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.     

Linear response and quasiparticle calculations as probes of the Kohn-Sham eigenvalues in metals

The Kohn-Sham eigenvalues were formally introduced into density functional theory as Lagrange multipliers in the implementation of the minimum principle for the total energy of a many-electron system. No general results are available concerning the physical significance of these one-electron eigenvalues (with the exception of the highest occupied level, which equals the Fermi energy). Recent ab initio calculations of dynamical response in metals make explicit use of the Kohn-Sham band structure, and associated wave functions, through the use of spectral representations. This opens up the possibility of examining the significance of the eigenvalues at an “empirical” level, i.e., through direct comparison with the results of spectroscopic measurements. A particularly interesting example is afforded by new inelastic x-ray scattering experiments on A1. For a special wave vector transfer, q_o ≈︂ 1.5k_F, the measured spectrum provides a direct mapping of the Kohn-Sham noninteracting spectrum. For a range of wave vectors about q_o, the bare Kohn-Sham spectrum still reproduces all the main features of the measurements; this suggests that, in this metal, the Kohn-Sham eigenvalues are good approximations to the quasiparticle energies. We also discuss the interplay between Kohn-Sham bands and the energy of the “anomalous” plasmon in Cs, whose dispersion bears a signature of the excited-state band structure. Finally, and in a more formal framework, we outline the results of a first-principles comparison between quasiparticle amplitudes and Kohn-Sham wave functions at a jellium surface; the latter turn out to be excellent approximations to the former. © 1996 John Wiley & Sons, Inc.     

Local exchange-correlation approximations and first-row molecular dissociation energies

Recent Xα calculations of bond energies and other related properties of first-row diatomic molecules show very encouraging agreement with experiment. In the worst cases, however, the Xα dissociation energies overestimate the experimental values by almost 2 eV. Therefore, we have examined several refinements of the Xα theory and their effects on molecular bond lengths, bond energies, and vibrational frequencies. Among them, gradient corrections to the Xα exchange energy and also some variations of the local spin-density correlation energy approximation are considered. We find that a local exchange-correlation functional with gradient corrections gives dissociation energies in significantly better agreement with experiment than the Xα approximation.     

On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions

The correlation energy in the direct random phase approximation (dRPA) can be written, among other possibilities, either in terms of the interaction strength averaged correlation density matrix, or in terms of the coupled cluster doubles amplitudes obtained in the direct ring approximation (drCCD). Although the corresponding dRPA correlation density matrix on the one hand, and the drCCD amplitude matrix on the other hand, differ significantly, they yield identical energies. Similarly, the analogous RPA and rCCD correlation energies calculated from antisymmetrized two-electron integrals are identical to each other despite very different underlying working equations. In the present communication, a direct correspondence between amplitudes and densities is established and investigated with perturbation theory arguments. Our analysis also sheds some light on the properties of recently proposed RPA/rCCD variants which use antisymmetrized integrals in part of the equations and nonantisymmetrized integrals in others.     

Direct approximation of the long- and short-range components of the exchange-correlation Kohn-Sham potential

An approximation scheme was developed for the Kohn-Sham exchange-correlation potential v_xcσ, making use of a partitioning of v_xcσ into a long-range screening v_scrσ and a short-range response v_resp component. For the response part, a model v^mod_respσ was used, which represents v_resp as weighted orbital density contributions, the weights being determined by the orbital energies. v^mod_respσ possesses the proper short-range behavior and the atomic-shell stepped structure characteristic for v_resp. For the screening part, two model potentials v^mod_scrσ were used, one with the accurate Slater potential; the other one with the generalized gradient approximation (GGA) for the exchange part. Both use the GGA for the Coulomb correlation contribution to v_scrσ. The scheme provides an adequate approximation to v_xcσ in the outer-valence region with both the proper asymptotics and a rather accurate estimate of the ionization potential from the highest one-electron energy and a reasonable estimate of atomic E_xc and total energies E_tot. © 1997 John Wiley & Sons, Inc.     

Basis set convergence of molecular correlation energy differences within the random phase approximation

The basis set convergence of energy differences obtained from the random phase approximation (RPA) to the correlation energy is investigated for a wide range of molecular interactions. For dispersion bound systems the basis set incompleteness error is most pronounced, as shown for the S22 benchmark [P. Jurecka et al., Phys. Chem. Chem. Phys. 8, 1985 (2006)]. The use of very large basis sets (> quintuple-zeta) or extrapolation to the complete basis set (CBS) limit is necessary to obtain a reliable estimate of the binding energy for these systems. Counterpoise corrected results converge to the same CBS limit, but counterpoise correction without extrapolation is insufficient. Core-valence correlations do not play a significant role. For medium- and short-range correlation, quadruple-zeta results are essentially converged, as demonstrated for relative alkane conformer energies, reaction energies dominated by intramolecular dispersion, isomerization energies, and reaction energies of small organic molecules. Except for weakly bound systems, diffuse augmentation almost universally slows down basis set convergence. For most RPA applications, quadruple-zeta valence basis sets offer a good balance between accuracy and efficiency.     

A new approach to intermolecular forces in non-bonding regions

A new method to compute intermolecular energies in non-bonding regions is presented. It is based on the assumption that in such regions molecules can be reviewed as the sum of distorted, possibly overlapping, and electron exchanging atoms. The intermolecular energy change at a given distance is due to the sum of the atomic energy changes caused by these distortions. The energy change of any particular atom is computed in a Hartree—Fock model in which the effect of the other atom is represented by an effective potential. This potential in turn is computed from a calculation at a slightly larger intermolecular distance of the potential seen by an external electron in the field of the “other” atom. This potential computed in the RPA approximation and involves the distorted Hartree—Fock orbitals of the other atom (computed in a similar manner to the above) and the RPA response function of the other distorted atom.     

Density-functional theory with effective potential expressed as a direct mapping of the external potential: applications to atomization energies and ionization potentials

In this paper the authors further develop and apply the direct-mapping density functional theory to calculations of the atomization energies and ionization potentials. Single-particle orbitals are determined by solving the Kohn-Sham [Phys. Rev. A. 140, 1133 (1965)] equations with a local effective potential expressed in terms of the external potential. A two-parametric form of the effective potential for molecules is proposed and equations for optimization of the parameters are derived using the exchange-only approximation. Orbital-dependent correlation functional is derived from the second-order perturbation theory in its Moller-Plesset-type zeroth-order approximation based on the Kohn-Sham orbitals and orbital energies. The total atomization energies and ionization potentials computed with the second-order perturbation theory were found to be in agreement with experimental values and benchmark results obtained with ab initio wave mechanics methods.     

On the Stability of Cyclophane Derivates Using a Molecular Fragmentation Method

A molecular fragmentation method is used to study the stability of cyclophane derivates by decomposing the molecular energy into the molecular strain and intramolecular interaction energies. The molecular strain energies obtained by utilising the fragmentation method are in good agreement with existing experimental data. The intramolecular interaction energies calculated as the difference between the supermolecular energy and the bonded fragment energies are repulsive in the cyclophanes studied. The nature of this interaction is studied for groups of systematically extended doubled layered paracyclophane systems using the random‐phase approximation (RPA), two recently developed extensions to the RPA and standard density functional theory (DFT) methods including dispersion corrections. Upon a systematic increase in conjugation the strongly repulsive intramolecular interaction energy reduces and thus leads to an increase in the stability. Finally, existing experimental and theoretical estimates of the molecular strain are compared with the results of this work.