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.

Embracing local suppression and enhancement of dynamic correlation effects in a CASΠDFT method for efficient description of excited states.

The recently proposed CASΠDFT method combines the reliable description of nondynamic electron correlation with the complete active space (CAS) wavefunction and the efficient treatment of dynamic correlation by density functional theory (DFT). This marriage is accomplished by adopting the DFT correlation energy functional modified with the local correction function of the on-top pair density (Π). The role of the correction function is to sensitize the correlation functional to local effects of suppression and enhancement of dynamic correlation and to account for an adequate amount of dynamic correlation energy. In this work we show that the presence of covalent and ionic configurations in a wavefunction gives rise to spatial regions where the effects of suppression and enhancement of correlation energy, respectively, dominate. The results obtained for the potential energy curves of the excited states of the hydrogen molecule prove that CASΠDFT is reliable for states that change their character along the dissociation curve. The method is also applied to the lowest excited states of six-membered heterocyclic nitrogen compounds such as pyridine, pyrazine, pyrimidine, and pyridazine. The obtained excitation energies for the n → π* and π → π* excitations confirm the good performance of CASΠDFT for excited states. The absolute average error of the method is 0.1 eV lower than that of the CCSD method and higher by the same amount than that of the more expansive CC3 variant. Compared with the coupled cluster methods, this encouraging performance of CASΠDFT is achieved at the negligible computational cost of obtaining the correlation energy.

Local Enhancement of Dynamic Correlation in Excited States: Fresh Perspective on Ionicity and Development of Correlation Density Functional Approximation Based on the On-Top Pair Density.

We discuss the interplay between the nondynamic and dynamic electron correlation in excited states from the perspective of the suppression of dynamic correlation (SDC) and enhancement of dynamic correlation (EDC) effects. We reveal that there exists a connection between the ionic character of a wave function and EDC. Following this finding we introduce a quantitative measure of ionicity based solely on local functions without referring to valence bond models. The ability to recognize both the SDC and EDC regions underlies the presented method, named CASΠDFT, combining complete active space (CAS) wave function and density functional theory (DFT) via the on-top pair density (Π) function. We extend this approach to excited states by devising an improved representation of the EDC effect in the correlation functional. The generalized CASΠDFT uses different DFT functionals for ground and excited states. Numerical demonstration for singlet π → π* excitations shows that CASΠDFT offers satisfactory accuracy at a fraction of the cost of the ab initio approaches.

Molecular multibond dissociation with small complete active space augmented by correlation density functionals.

Molecular multibond dissociation displays a variety of electron correlation effects posing a challenge for theoretical description. We propose a CASΠ(M)DFT approach, which includes these effects in an efficient way by combining the complete active space self-consistent field method with density functional theory (DFT). Within CASΠ(M)DFT, a small complete active space (CAS) accounts for the long-range intrabond and middle-range interbond nondynamic correlation in the stretched bonds. The common short-range dynamic correlation is calculated with the Lee-Yang-Parr (LYP) correlation DFT functional corrected for the suppression of dynamic correlation with nondynamic correlation. The remaining middle-range interbond dynamic correlation is evaluated with the modified LYP functional of the bond densities. As a result, CASΠ(M)DFT potential energy curves (PECs) calculated in the relatively small triple-zeta basis closely reproduce the benchmark complete basis set PECs for the following prototype multibonded molecules: N2, CO, H2O, and C2.

Reproducing benchmark potential energy curves of molecular bond dissociation with small complete active space aided with density and density-matrix functional corrections.

Various effects of electron correlation accompany molecular bond dissociation, which makes the efficient calculation of potential energy curves a notoriously difficult problem. In an attempt to reliably reproduce both absolute energies and shapes of the benchmark dissociation curves, calculations with the combined CASΠDFT method are carried out for the prototype molecules H2, BH, F2, and N2. The complete active space (CAS) part of CASΠDFT accounts for long-range nondynamic correlation, while short-range dynamic correlation is accounted for with the corrected Lee-Yang-Parr correlation functional of density functional theory (DFT). The correction represents the suppression of dynamic correlation with nondynamic correlation, and it is a function of the ratio x(r) between the conditional and conventional densities obtained with the CAS on-top pair density Π(r). For the single-bonded molecules H2, BH, and F2, CASΠDFT succeeds in reproducing the shapes and absolute energies (for H2 and BH) of the benchmark curves, while for the triple-bonded N2 molecule, the addition to CASΠDFT of a multibond correction is required. It accounts for the middle-range dynamic correlation of the same-spin electrons in the (symmetrized) high-spin atomic electron configurations of the dissociating N2.

Complete active space and corrected density functional theories helping each other to describe vertical electronic π → π* excitations in prototype multiple-bonded molecules.

The CASΠDFT method, which combines the complete active space (CAS) wave function approach and density functional theory (DFT), offers an efficient description of important excitations to the lowest excited states. CASΠDFT employs a correlation DFT functional corrected with a function P[x] of the ratio xr of the conditional and conventional electron densities obtained with the CAS on-top pair density Π(r). The sectors of P[x] for x(r) ≤ 1 and x(r) > 1 represent the opposite effects of the suppression of dynamic correlation with nondynamic correlation and its enhancement due to the ionic-type excitation. The present combination of the self-consistent-field CAS and the corrected Lee-Yang-Parr correlation functional closely reproduces in the relatively small double-zeta basis the benchmark experimental lowest singlet vertical π → π* excitations in the prototype multiple-bonded molecules N2, CO, C2H2, and C2H4.

Nanostructured organosilicon luminophores and their application in highly efficient plastic scintillators.

Organic luminophores are widely used in various optoelectronic devices, which serve for photonics, nuclear and particle physics, quantum electronics, medical diagnostics and many other fields of science and technology. Improving their spectral-luminescent characteristics for particular technical requirements of the devices is a challenging task. Here we show a new concept to universal solution of this problem by creation of nanostructured organosilicon luminophores (NOLs), which are a particular type of dendritic molecular antennas. They combine the best properties of organic luminophores and inorganic quantum dots: high absorption cross-section, excellent photoluminescence quantum yield, fast luminescence decay time and good processability. A NOL consists of two types of covalently bonded via silicon atoms organic luminophores with efficient Förster energy transfer between them. Using NOLs in plastic scintillators, widely utilized for radiation detection and in elementary particles discoveries, led to a breakthrough in their efficiency, which combines both high light output and fast decay time. Moreover, for the first time plastic scintillators, which emit light in the desired wavelength region ranging from 370 to 700 nm, have been created. We anticipate further applications of NOLs as working elements of pulsed dye lasers in photonics, optoelectronics and as fluorescent labels in biology and medical diagnostics.

Comparative study of genome divergence in salmonids with various rates of genetic isolation.

The aim of the study is a comparative investigation of changes that certain genome parts undergo during speciation. The research was focused on divergence of coding and noncoding sequences in different groups of salmonid fishes of the Salmonidae (Salmo, Parasalmo, Oncorhynchus, and Salvelinus genera) and the Coregonidae families under different levels of reproductive isolation. Two basic approaches were used: (1) PCR-RAPD with a 20-22 nt primer design with subsequent cloning and sequencing of the products and (2) a modified endonuclease restriction analysis. The restriction fragments were shown with sequencing to represent satellite DNA. Effects of speciation are found in repetitive sequences. The revelation of expressed sequences in the majority of the employed anonymous loci allows for assuming the adaptive selection during allopatric speciation in isolated char forms.

Counterpoise correction is not useful for short and Van der Waals distances but may be useful at long range.

This article investigates the errors in supermolecule calculations for the helium dimer. In a full CI calculation, there are two errors. One is the basis set superposition error (BSSE), the other is the basis set convergence error (BSCE). Both of the errors arise from the incompleteness of the basis set. These two errors make opposite contributions to the interaction energies. The BSCE is by far the largest error in the short range and larger than (but much closer to) BSSE around the Van der Waals minimum. Only at the long range, the BSSE becomes the larger error. The BSCE and BSSE largely cancel each other over the Van der Waals well. Accordingly, it may be recommended to not include the BSSE for the calculation of the potential energy curve from short distance till well beyond the Van der Waals minimum, but it may be recommended to include the BSSE correction if an accurate tail behavior is required. Only if the calculation has used a very large basis set, one can refrain from including the counterpoise correction in the full potential range. These results are based on full CI calculations with the aug-cc-pVXZ (X = D, T, Q, 5) basis sets.

Double excitation effect in non-adiabatic time-dependent density functional theory with an analytic construction of the exchange-correlation kernel in the common energy denominator approximation.

Time-dependent density functional (response) theory (TDDF(R)T) is applied almost exclusively in its adiabatic approximation (ATDDFT), which is restricted to predominantly single electronic excitations and neglects additional roots of the TDDFT eigenvalue problem stemming from the interaction between single and double excitations. We incorporate the effect of the latter interaction into a non-adiabatic frequency-dependent and spatially non-local Hartree-exchange-correlation (Hxc) kernel fCEDAHxc (r1, r2, omega), the explicit analytical expression of which is derived for interacting single and double excitations well separated from the other excitations, within the common energy denominator approximation (CEDA) for the Kohn-Sham (KS) and interacting density response functions, chis and chi, respectively. The kernel fCEDAHxc (r1, r2, omega) obtained from the direct analytical inverse of chiCEDAs and chiCEDA is a sum of the delta-function and non-local orbital-dependent spatial terms with frequency-dependent factors, with which fCEDAHxc acquires a modulated quadratic dependence on omega. The effective incorporation in fCEDAHxc of the complete manifold of excited states (through the delta function term) represents an extension of the kernel reported by Maitra, Zhang, Cave, and Burke [J. Chem. Phys., 2004, 120, 5932]. In the TDDFT eigenvalue equations considered in the diagonal approximation, fCEDAHxc generates two excitation energies omegaq and omegaq+1, which both correspond to the same single KS excitation omegasq, thus producing the effect of the single-double excitation interaction.

A density matrix functional with occupation number driven treatment of dynamical and nondynamical correlation.

A recently proposed series of corrections to the earliest JK-only functionals has considerably improved the prospects of density matrix functional theory (DMFT). Still, the most advanced of these functionals (correction C3) requires a preselection of the terms in the pair density Gamma(r(1),r(2)) involving the bonding and antibonding natural orbitals (NOs) belonging to an electron pair bond. Ideally, a DMFT functional should only depend on the NOs and their occupation numbers, and we propose a functional with an occupation number driven weighing of terms in the pair density. These are formulated as "damping" for certain ranges of occupation numbers of the two-electron cumulant that arises in the expansion of the two-particle density matrix of the paradigmatic two-electron system. This automatic version of C3, which we denote AC3, provides the correct dissociation limit for electron pair bonds and it excellently reproduces the potential energy curves of the multireference configuration interaction (MRCI) method for the dissociation of the electron pair bond in the series of the ten-electron hydrides CH(4), NH(3), H(2)O, and HF. AC3 reproduces closely the experimental equilibrium distances and at R(e) it yields correlation energies of the ten-electron systems with an average error in the absolute values of only 3.3% compared to the MRCI values. We stress the importance of treatment of strong correlation cases (NO occupation numbers differing significantly from 2.0 and 0.0) by appropriate terms in the cumulant.

Adiabatic approximation of time-dependent density matrix functional response theory.

Time-dependent density matrix functional theory can be formulated in terms of coupled-perturbed response equations, in which a coupling matrix K(omega) features, analogous to the well-known time-dependent density functional theory (TDDFT) case. An adiabatic approximation is needed to solve these equations, but the adiabatic approximation is much more critical since there is not a good "zero order" as in TDDFT, in which the virtual-occupied Kohn-Sham orbital energy differences serve this purpose. We discuss a simple approximation proposed earlier which uses only results from static calculations, called the static approximation (SA), and show that it is deficient, since it leads to zero response of the natural orbital occupation numbers. This leads to wrong behavior in the omega-->0 limit. An improved adiabatic approximation (AA) is formulated. The two-electron system affords a derivation of exact coupled-perturbed equations for the density matrix response, permitting analytical comparison of the adiabatic approximation with the exact equations. For the two-electron system also, the exact density matrix functional (2-matrix in terms of 1-matrix) is known, enabling testing of the static and adiabatic approximations unobscured by approximations in the functional. The two-electron HeH(+) molecule shows that at the equilibrium distance, SA consistently underestimates the frequency-dependent polarizability alpha(omega), the adiabatic TDDFT overestimates alpha(omega), while AA improves upon SA and, indeed, AA produces the correct alpha(0). For stretched HeH(+), adiabatic density matrix functional theory corrects the too low first excitation energy and overpolarization of adiabatic TDDFT methods and exhibits excellent agreement with high-quality CCSD ("exact") results over a large omega range.

Assessment of a simple correction for the long-range charge-transfer problem in time-dependent density-functional theory.

The failure of the time-dependent density-functional theory to describe long-range charge-transfer (CT) excitations correctly is a serious problem for calculations of electronic transitions in large systems, especially if they are composed of several weakly interacting units. The problem is particularly severe for molecules in solution, either modeled by periodic boundary calculations with large box sizes or by cluster calculations employing extended solvent shells. In the present study we describe the implementation and assessment of a simple physically motivated correction to the exchange-correlation kernel suggested in a previous study [O. Gritsenko and E. J. Baerends J. Chem. Phys. 121, 655 (2004)]. It introduces the required divergence in the kernel when the transition density goes to zero due to a large spatial distance between the "electron" (in the virtual orbital) and the "hole" (in the occupied orbital). A major benefit arises for solvated molecules, for which many CT excitations occur from solvent to solute or vice versa. In these cases, the correction of the exchange-correlation kernel can be used to automatically "clean up" the spectrum and significantly reduce the computational effort to determine low-lying transitions of the solute. This correction uses a phenomenological parameter, which is needed to identify a CT excitation in terms of the orbital density overlap of the occupied and virtual orbitals involved. Another quantity needed in this approach is the magnitude of the correction in the asymptotic limit. Although this can, in principle, be calculated rigorously for a given CT transition, we assess a simple approximation to it that can automatically be applied to a number of low-energy CT excitations without additional computational effort. We show that the method is robust and correctly shifts long-range CT excitations, while other excitations remain unaffected. We discuss problems arising from a strong delocalization of orbitals, which leads to a breakdown of the correction criterion.

A simple natural orbital mechanism of "pure" van der Waals interaction in the lowest excited triplet state of the hydrogen molecule.

A treatment of van der Waals (vdW) interaction by density-matrix functional theory requires a description of this interaction in terms of natural orbitals (NOs) and their occupation numbers. From an analysis of the configuration-interaction (CI) wave function of the 3Sigmau + state of H2 and the exact NO expansion of the two-electron triplet wave function, we demonstrate that the construction of such a functional is straightforward in this case. A quantitative description of the vdW interaction is already obtained with, in addition to the standard part arising from the Hartree-Fock determinant /1sigmag(r1)1sigmau(r2)/, only two additional terms in the two-electron density, one from the first "excited" determinant /2sigmag(r1)2sigmau(r2)/ and one from the state of 3Sigmau + symmetry belonging to the (1pig)1(1piu)1 configuration. The potential-energy curve of the 3Sigmau + state calculated around the vdW minimum with the exact density-matrix functional employing only these eight NOs and NO occupations is in excellent agreement with the full CI one and reproduces well the benchmark potential curve of Kolos and Wolniewicz [J. Chem. Phys. 43, 2429 (1965)]. The corresponding terms in the two-electron density rho2(r1,r2), containing specific products of NOs combined with prefactors that depend on the occupation numbers, can be shown to produce exchange-correlation holes that correspond precisely to the well-known intuitive picture of the dispersion interaction as an instantaneous dipole-induced dipole (higher multipole) effect. Indeed, (induced) higher multipoles account for almost 50% of the total vdW bond energy. These results serve as a basis for both a density-matrix functional theory of van der Waals bonding and for the construction of orbital-dependent functionals in density-functional theory that could be used for this type of bonding.

An improved density matrix functional by physically motivated repulsive corrections.

An improved density matrix functional [correction to Buijse and Baerends functional (BBC)] is proposed, in which a hierarchy of physically motivated repulsive corrections is employed to the strongly overbinding functional of Buijse and Baerends (BB). The first correction C1 restores the repulsive exchange-correlation (xc) interaction between electrons in weakly occupied natural orbitals (NOs) as it appears in the exact electron pair density rho(2) for the limiting two-electron case. The second correction C2 reduces the xc interaction of the BB functional between electrons in strongly occupied NOs to an exchange-type interaction. The third correction C3 employs a similar reduction for the interaction of the antibonding orbital of a dissociating molecular bond. In addition, C3 applies a selective cancellation of diagonal terms in the Coulomb and xc energies (not for the frontier orbitals). With these corrections, BBC still retains a correct description of strong nondynamical correlation for the dissociating electron pair bond. BBC greatly improves the quality of the BB potential energy curves for the prototype few-electron molecules and in several cases BBC reproduces very well the benchmark ab initio potential curves. The average error of the self-consistent correlation energies obtained with BBC3 for prototype atomic systems and molecular systems at the equilibrium geometry is only ca. 6%.

Asymptotic correction of the exchange-correlation kernel of time-dependent density functional theory for long-range charge-transfer excitations.

Time-dependent density functional theory (TDDFT) calculations of charge-transfer excitation energies omegaCT are significantly in error when the adiabatic local density approximation (ALDA) is employed for the exchange-correlation kernel fxc. We relate the error to the physical meaning of the orbital energy of the Kohn-Sham lowest unoccupied molecular orbital (LUMO). The LUMO orbital energy in Kohn-Sham DFT--in contrast to the Hartree-Fock model--approximates an excited electron, which is correct for excitations in compact molecules. In CT transitions the energy of the LUMO of the acceptor molecule should instead describe an added electron, i.e., approximate the electron affinity. To obtain a contribution that compensates for the difference, a specific divergence of fxc is required in rigorous TDDFT, and a suitable asymptotically correct form of the kernel fxc(asymp) is proposed. The importance of the asymptotic correction of fxc is demonstrated with the calculation of omegaCT(R) for the prototype diatomic system HeBe at various separations R(He-Be). The TDDFT-ALDA curve omegaCT(R) roughly resembles the benchmark ab initio curve omegaCT CISD(R) of a configuration interaction calculation with single and double excitations in the region R=1-1.5 A, where a sizable He-Be interaction exists, but exhibits the wrong behavior omegaCT(R)<