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Kohn-Sham solutions are constructed from ab initio densities obtained with multireference configuration interaction (MRCI) calculations for the transition state (TS) and for the intermediate complex (IC) of a prototype symmetrical reaction. The generalized approximations (GGAs) overestimate the relative stability of the systems in the experiment.

In spite of the high quality of exchange-correlation energies Exc obtained with the generalized gradient approximations (GGAs) of density functional theory, their xc potentials vxc are strongly deficient, yielding upshifts of ca. 5 eV in the orbital energy spectrum (in the order of 50% of highlying valence orbital energies). The GGAs share this deficiency with the local density approximation (LDA). We argue that this error is not caused by the incorrect long-range asymptotics of vxc or by self-interaction error. It arises from incorrect density dependencies of LDA and GGA exchange functionals leading to incorrect (too repulsive) functional derivatives (i.e., response parts of the potentials). The vxc potential is partitioned into the potential of the xc hole vxchole (twice the xc energy density xc), which determines Exc, and the response potential vresp, which does not contribute to Exc explicitly. The substantial upshift of LDA/GGA orbital energies is due to a too repulsive LDA exchange response potential vLDA xresp in the bulk region. Retaining the LDA exchange hole potential plus the B88 gradient correction to it but replacing the response parts of these potentials by the model orbital-dependent response potential vGLLBxresp of Gritsenko et al. [Phys. Rev. A 51, 1944 (1995)], which has the proper step-wise form, improves the orbital energies by more than an order of magnitude. Examples are given for the prototype molecules: dihydrogen, dinitrogen, carbon monoxide, ethylene, formaldehyde, and formic acid. [ABSTRACT FROM AUTHOR]

For chemistry an accurate description of bond weakening and breaking is vital. The great advantage of density matrix functionals, as opposed to density functionals, is their ability to describe such processes since they naturally cover both nondynamical and dynamical correlation. This is obvious in the Löwdin-Shull functional, the exact natural orbital functional for two-electron systems. We present in this paper extensions of this functional for the breaking of a single electron pair bond in N-electron molecules, using LiH, BeH+, and Li2 molecules as prototypes. Attention is given to the proper formulation of the functional in terms of not just J and K integrals but also the two-electron L integrals (K integrals with a different distribution of the complex conjugation of the orbitals), which is crucial for the calculation of response functions. Accurate energy curves are obtained with extended Löwdin-Shull functionals along the complete dissociation coordinate using full CI calculations as benchmark. [ABSTRACT FROM AUTHOR]

Recently, we have demonstrated that the problems finding a suitable adiabatic approximation in time-dependent one-body reduced density matrix functional theory can be remedied by introducing an additional degree of freedom to describe the system: the phase of the natural orbitals [K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, Phys. Rev. Lett. 105, 013002 (2010); K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, J. Chem. Phys. 133, 174119 (2010)]. In this article we will show in detail how the frequency-dependent response equations give the proper static limit (ω → 0), including the perturbation in the chemical potential, which is required in static response theory to ensure the correct number of particles. Additionally we show results for the polarizability for H2 and compare the performance of two different two-electron functionals: the phase-including Löwdin–Shull functional and the density matrix form of the Löwdin–Shull functional. [ABSTRACT FROM AUTHOR]

Time dependent density matrix functional theory in its adiabatic linear response formulation delivers exact excitation energies ?a and oscillator strengths fa for two-electron systems if extended to the so-called phase including natural orbital (PINO) theory. The Löwdin-Shull expression for the energy of two-electron systems in terms of the natural orbitals and their phases affords in this case an exact phase-including natural orbital functional (PILS), which is non-primitive (contains other than just J and K integrals). In this paper, the extension of the PILS functional to N-electron systems is investigated. With the example of an elementary primitive NO functional (BBC1) it is shown that current density matrix functional theory ground state functionals, which were designed to produce decent approximations to the total energy, fail to deliver a qualitatively correct structure of the (inverse) response function, due to essential deficiencies in the reconstruction of the two-body reduced density matrix (2RDM). We now deduce essential features of an N-electron functional from a wavefunction Ansatz: The extension of the two-electron Löwdin-Shull wavefunction to the N-electron case informs about the phase information. In this paper, applications of this extended Löwdin-Shull (ELS) functional are considered for the simplest case, ELS(1): one (dissociating) two-electron bond in the field of occupied (including core) orbitals. ELS(1) produces high quality ωα(R) curves along the bond dissociation coordinate R for the molecules LiH, Li2, and BH with the two outer valence electrons correlated. All of these results indicate that response properties are much more sensitive to deficiencies in the reconstruction of the 2RDM than the ground state energy, since derivatives of the functional with respect to both the NOs and the occupation numbers need to be accurate. [ABSTRACT FROM AUTHOR]

Whereas a density functional that incorporates dispersion interaction has remained elusive to date, we demonstrate that in principle the dispersion energy can be obtained from a density matrix functional. In density matrix functional theory one tries to find suitable approximations to the two-particle reduced density matrix (2RDM) in terms of natural orbitals (NOs) and natural orbital occupation numbers (ONs). The total energy is then given as a function(al) of the NOs and ONs, i.e., as an implicit functional of the 1RDM. The left-right correlation in a (dissociating) bond, as well as various types of dynamical correlation, can be described accurately with a NO functional employing only J and K integrals (JK-only functional). We give a detailed analysis of the full CI wavefunction of the He2 dimer, from which the dispersion part of the two-particle density matrix is obtained. It emerges that the entirely different physics embodied in the dispersion interaction leads to an essentially different type of exchange-correlation orbital functional for the dispersion energy (non-JK). The distinct NO functionals for the different types of correlation imply that they can be used in conjunction without problems of double counting. Requirements on the (primitive) basis set for Van der Waals bonding appear to be more modest than for other types of correlation. [ABSTRACT FROM AUTHOR]

Adiabatic response time-dependent density functional theory (TDDFT) suffers from the restriction to basically an occupied → virtual single excitation formulation. Adiabatic time-dependent density matrix functional theory allows to break away from this restriction. Problematic excitations for TDDFT, viz. bonding-antibonding, double, charge transfer, and higher excitations, are calculated along the bond-dissociation coordinate of the prototype molecules H2 and HeH+ using the recently developed adiabatic linear response phase-including (PI) natural orbital theory (PINO). The possibility to systematically increase the scope of the calculation from excitations out of (strongly) occupied into weakly occupied ('virtual') natural orbitals to larger ranges of excitations is explored. The quality of the PINO response calculations is already much improved over TDDFT even when the severest restriction is made, to virtually the size of the TDDFT diagonalization problem (only single excitation out of occupied orbitals plus all diagonal doubles). Further marked improvement is obtained with moderate extension to allow for excitation out of the lumo and lumo+1, which become fractionally occupied in particular at longer distances due to left-right correlation effects. In the second place the interpretation of density matrix response calculations is elucidated. The one-particle reduced density matrix response for an excitation is related to the transition density matrix to the corresponding excited state. The interpretation of the transition density matrix in terms of the familiar excitation character (single excitations, double excitations of various types, etc.) is detailed. The adiabatic PINO theory is shown to successfully resolve the problematic cases of adiabatic TDDFT when it uses a proper PI orbital functional such as the PILS functional. [ABSTRACT FROM AUTHOR]

The adiabatic approximation is problematic in time-dependent density matrix functional theory. With pure density matrix functionals (invariant under phase change of the natural orbitals) it leads to lack of response in the occupation numbers, hence wrong frequency dependent responses, in particular α(ω→0)≠α0 (the static polarizability). We propose to relinquish the requirement that the functional must be a pure one-body reduced density matrix (1RDM) functional, and to introduce additional variables which can be interpreted as phases of the one-particle states of the independent particle reference system formed with the natural orbitals, thus obtaining so-called phase-including natural orbital (PINO) functionals. We also stress the importance of the correct choice of the complex conjugation in the two-electron integrals in the commonly used functionals (they should not be of exchange type). We demonstrate with the Löwdin-Shull energy expression for two-electron systems, which is an example of a PINO functional, that for two-electron systems exact responses (polarizabilities, excitation energies) are obtained, while writing this energy expression in the usual way as a 1RDM functional yields erroneous responses. [ABSTRACT FROM AUTHOR]

Time-dependent density functional theory in its current adiabatic implementations exhibits three striking failures: (a) Totally wrong behavior of the excited state surface along a bond-breaking coordinate, (b) lack of doubly excited configurations, affecting again excited state surfaces, and (c) much too low charge transfer excitation energies. We address these problems with time-dependent density matrix functional theory (TDDMFT). For two-electron systems the exact exchange-correlation functional is known in DMFT, hence exact response equations can be formulated. This affords a study of the performance of TDDMFT in the TDDFT failure cases mentioned (which are all strikingly exhibited by prototype two-electron systems such as dissociating H2 and HeH+). At the same time, adiabatic approximations, which will eventually be necessary, can be tested without being obscured by approximations in the functional. We find the following: (a) In the fully nonadiabatic (ω-dependent, exact) formulation of linear response TDDMFT, it can be shown that linear response (LR)-TDDMFT is able to provide exact excitation energies, in particular, the first order (linear response) formulation does not prohibit the correct representation of doubly excited states; (b) within previously formulated simple adiabatic approximations the bonding-to-antibonding excited state surface as well as charge transfer excitations are described without problems, but not the double excitations; (c) an adiabatic approximation is formulated in which also the double excitations are fully accounted for. [ABSTRACT FROM AUTHOR]

The local-density approximation of density functional theory (DFT) is remarkably accurate, for instance, for geometries and frequencies, and the generalized gradient approximations have also made bond energies quite reliable. Sometimes, however, one meets with failure in individual cases. One of the possible routes towards better functionals would be the incorporation of orbital dependence (which is an implicit density dependency) in the functionals. We discuss this approach both for energies and for response properties. One possibility is the use of the Hartree–Fock-type exchange energy expression as orbital-dependent functional. We will argue that in spite of the increasing popularity of this approach, it does not offer any advantage over Hartree–Fock for energies. We will advocate not to apply the separation of exchange and correlation, which is so ingrained in quantum chemistry, but to model both simultaneously. For response properties the energies and shapes of the virtual orbitals are crucial. We will discuss the benefits that Kohn–Sham potentials can offer which are derived from either an orbital-dependent energy functional, including the exact-exchange functional, or which can be obtained directly as orbital-dependent functional. We highlight the similarity of the Hartree–Fock and Kohn–Sham occupied orbitals and orbital energies, and the essentially different meanings the virtual orbitals and orbital energies have in these two models. We will show that these differences are beneficial for DFT in the case of localized excitations (in a small molecule or in a fragment), but are detrimental for charge-transfer excitations. Again, orbital dependency, in this case in the exchange-correlation kernel, offers a solution. [ABSTRACT FROM AUTHOR]

Spin-unrestricted Kohn–Sham (KS) solutions are constructed from accurate ab initio spin densities for the prototype doublet molecules NO2, ClO2, and NF2 with the iterative local updating procedure of van Leeuwen and Baerends (LB). A qualitative justification of the LB procedure is given with a “strong” form of the Hohenberg–Kohn theorem. The calculated energies [variant_greek_epsilon]iσ of the occupied KS spin orbitals provide numerical support to the analogue of Koopmans’ theorem in spin-density functional theory. In particular, the energies -[variant_greek_epsilon]iβ of the minor spin (β) valence orbitals of the considered doublet molecules correspond fairly well to the experimental vertical ionization potentials (VIPs) Ii1 to the triplet cationic states. The energy -[variant_greek_epsilon]Hα of the highest occupied (spin-unpaired) α orbital is equal to the first VIP IH0 to the singlet cationic state. In turn, the energies -[variant_greek_epsilon]iα of the major spin (α) valence orbitals of the closed subshells correspond to a fifty-fifty average of the experimental VIPs Ii1 and Ii0 to the triplet and singlet states. For the Li atom we find that the exact spin densities are represented by a spin-polarized Kohn–Sham system which is not in its ground state, i.e., the orbital energy of the lowest unoccupied β spin orbital is lower than that of the highest occupied α spin orbital (“a hole below the Fermi level”). The addition of a magnetic field in the -z direction will shift the β levels up so as to restore the Aufbau principle. This is an example of the nonuniqueness of the mapping of the spin density on the KS spin-dependent potentials discussed recently in the literature. The KS potentials may no longer go to zero at infinity, and it is in general the differences νsσ(∞)-[variant_greek_epsilon]iσ that can be interpreted as (averages of) ionization energies. In total, the present results suggest the spin-unrestricted KS theory as a natural one-electron independent-particle model for interpretation and assignment of the experimental photoelectron spectra of open-shell molecules. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Theoretical and numerical insight is gained into the ∈-I relations between the Kohn-Sham orbital energies ∈[sub i] and relaxed vertical ionization potentials (VIPs) I[sub j], which provide an analog of Koopmans' theorem for density functional theory. The Kohn-Sham orbital energy ∈[sub i] has as leading term -n[sub i]I[sub i]- Σ[sub j ∈ Ω[sub s](i)]n[sub j]I[sub j], where I[sub i is the primary VIP for ionization (φ[sub i])[sup -1] with spectroscopic factor (proportional to the intensity in the photoelectron spectrum) n[sub i] close to 1, and the set Ω[sub s](i) contains the VIPs I[sub j] that are satellites to the (φ[sub i])[sup -1] ionization, with small but non-negligible n[sub j]. In addition to this "average spectroscopic structure" of the ∈[sub i] there is an electron-shell step structure in ∈[sub i] from the contribution of the response potential υ[sub resp]. Accurate KS calculations for prototype second- and third-row closed-shell molecules yield valence orbital energies - ∈[sub i], which correspond closely to the experimental VIPs, with an average deviation of 0.08 eV. The theoretical relations are numerically investigated in calculations of the components of the ∈ -I relations for the H[sub 2] molecule, and for the molecules CO, HF, H[sub 2]O, HCN. The derivation of the ∈ -I relations employs the Dyson orbitals (the n[sub i] are their norms). A connection is made between the KS and Dyson orbital theories, allowing the spin-unrestricted KS xc potential to be expressed with a statistical average of individual xc potentials for the Dyson spin-orbitals as leading term. Additional terms are the correction υ[sub c,kin,σ] due to the correlation kinetic effect, and the "response" υ[sub resp,σ], related to the correction to the energy of (N-1) electrons due to the correlation with the reference electron. [ABSTRACT FROM AUTHOR]

The standard local density approximation and generalized gradient approximations fail to properly describe the dissociation of an electron pair bond, yielding large errors (on the order of 50 kcal/mol) at long bond distances. To remedy this failure, a self-consistent Kohn-Sham (KS) method is proposed with the exchange-correlation (xc) energy and potential depending on both occupied and virtual KS orbitals. The xc energy functional of Buijse and Baerends [Mol. Phys. 100, 401 (2002); Phys. Rev. Lett. 87, 133004 (2001)] is employed, which, based on an ansatz for the xc-hole amplitude, is able to reproduce the important dynamical and nondynamical effects of Coulomb correlation through the efficient use of virtual orbitals. Self-consistent calculations require the corresponding xc potential to be obtained, to which end the optimized effective potential (OEP) method is used within the common energy denominator approximation for the static orbital Green's function. The problem of the asymptotic divergence of the xc potential of the OEP when a finite number of virtual orbitals is used is addressed. The self-consistent calculations reproduce very well the entire H[sub 2] potential curve, describing correctly the gradual buildup of strong left-right correlation in stretched H[sub 2]. [ABSTRACT FROM AUTHOR]

For spin-unrestricted Kohn-Sham (KS) calculations on systems with an open shell ground state with total spin quantum number S, we offer the analog of the Koopmans'-type relation between orbital energies and ionization energies familiar from the Hartree-Fock model. When (case I) the lowest ion state has spin S- 1/2 (typically when the neutral molecule has a (less than) half filled open shell), the orbital energy of the highest occupied orbital (ø[SUBH]), belonging to the open shell with majority spin (α) electrons, is equal to the ionization energy to this lowest ion state with spin S - 1/2: ε[SUPα][SUBH]-I[SUPS-1/2](ø[SUP-1][SUBH]). For lower (doubly occupied) orbitals the ionization ø[SUP-1][SUBH] leaves an unpaired electron that can couple to the open shell to S± 1/2 states: ε[SUPβ][SUBi]= -I[SUPS+l/2](ø[SUP-1][SUBi]) (exact identity for i = H- 1 ), ε[SUPα][SUBi] = - {[2 S/(2S + 1)]I[SUPS-1/2] (ø[SUP-1][SUBi]) + [ 1/(2S + 1 )]I[SUPS+1/2] (ø[SUP-1][SUBi])}, reducing to a simple average in the case of a doublet ground state (single electron outside closed shells). When the lowest ion state has spin S+ 1/2 (case II; typically for more than half filled open shells): ε[SUPα][SUBH] = ε[SUPβ][SUBH]=- I[SUPS- 1/2](ø[SUP-1][SUBH]; for i

An approximate Kohn–Sham (KS) exchange potential v[sub xσ][sup CEDA] is developed, based on the common energy denominator approximation (CEDA) for the static orbital Green’s function, which preserves the essential structure of the density response function. v[sub xσ][sup CEDA] is an explicit functional of the occupied KS orbitals, which has the Slater v[sub Sσ] and response v[sub respσ][sup CEDA] potentials as its components. The latter exhibits the characteristic step structure with “diagonal” contributions from the orbital densities |ψ[sub iσ]|[sup 2], as well as “off-diagonal” ones from the occupied–occupied orbital products ψ[sub iσ]ψ[sub j(≠1)σ][sup *]. Comparison of the results of atomic and molecular ground-state CEDA calculations with those of the Krieger–Li–Iafrate (KLI), exact exchange (EXX), and Hartree–Fock (HF) methods show, that both KLI and CEDA potentials can be considered as very good analytical “closure approximations” to the exact KS exchange potential. The total CEDA and KLI energies nearly coincide with the EXX ones and the corresponding orbital energies &Vegr;[sub iσ] are rather close to each other for the light atoms and small molecules considered. The CEDA, KLI, EXX–&Vegr;[sub iσ] values provide the qualitatively correct order of ionizations and they give an estimate of VIPs comparable to that of the HF Koopmans’ theorem. However, the additional off-diagonal orbital structure of v[sub xσ][sup CEDA] appears to be essential for the calculated response properties of molecular chains. KLI already considerably improves the calculated (hyper)polarizabilities of the prototype hydrogen chains H[sub n] over local density approximation (LDA) and standard generalized gradient approximations (GGAs), while the CEDA results are definitely an improvement over the KLI ones. The reasons... [ABSTRACT FROM AUTHOR]

Theoretical analysis and results of calculations are put forward to interpret the energies -ε[sub k] of the occupied Kohn–Sham (KS) orbitals as approximate but rather accurate relaxed vertical ionization potentials (VIPs) I[sub k]. Exact relations between ε[sub k] and I[sub k] are established with a set of linear equations for the ε[sub k], which are expressed through I[sub k] and the matrix elements ε[sub k][sup resp] of a component of the KS exchange-correlation (xc) potential v[sub xc], the response potential v[sub resp]. Although -I[sub k] will be a leading contribution to ε[sub k], other I[sub j≠k] do enter through coupling terms which are determined by the overlaps between the densities of the KS orbitals as well as by overlaps between the KS and Dyson orbital densities. The orbital energies obtained with “exact” KS potentials are compared with the experimental VIPs of the molecules N[sub 2], CO, HF, and H[sub 2]O. Very good agreement between the accurate -ε[sub k] of the outer valence KS orbitals and the corresponding VIPs is established. The average difference, approaching 0.1 eV, is about an order of magnitude smaller than for HF orbital energies. The lower valence KS levels are a few eV higher than the corresponding -I[sub k], and the core levels some 20 eV, in agreement with the theoretically deduced upshift of the KS levels compared to -I[sub k] by the response potential matrix elements. Calculations of 64 molecules are performed with the approximate v[sub xc] obtained with the statistical averaging of (model) orbitals potentials (SAOP) and the calculated ε[sub k] are compared with 406 experimental VIPs. Reasonable agreement between the SAOP -ε[sub k] and the outer valence VIPs is found with an average deviation of about 0.4 eV. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The performance of exchange and correlation (xc) functionals of the generalized gradient approximation (GGA) type and of the meta-GGA type in the calculation of chemical reactions is related to topological features of the electron density, which are connected to the orbital structure of chemical bonds within Kohn-Sham (KS) theory. Seventeen GGA and meta-GGA xc functionals are assessed for 15 hydrogen abstraction reactions and 3 symmetrical S(sub N)2 reactions.

Studies the benchmark calculations of chemical reactions in density functional theory by comparing the accurate Kohn-Sham (KS) solution with generalized gradient approximations for the hydrogen abstraction and symmetrical four-center exchange chemical reactions. Calculation of the KS quantities.

The exchange-correlation potentials vxc which are currently fashionable in density functional theory (DFT), such as those obtained from the local density approximation (LDA) or generalized gradient approximations (GGAs), all suffer from incorrect asymptotic behavior. In atomic calculations, this leads to substantial overestimations of both the static polarizability and the frequency dependence of this property. In the present paper, it is shown that the errors in atomic static dipole and quadrupole polarizabilities are reduced by almost an order of magnitude, if a recently proposed model potential with correct Coulombic long-range behavior is used. The frequency dependence is improved similarly. The model potential also removes the overestimation in molecular polarizabilities, leading to slight improvements for average molecular polarizabilities and their frequency dependence. For the polarizability anisotropy we find that the model potential results do not improve over the LDA and GGA results. Our method for calculating frequency-dependent molecular response properties within time-dependent DFT, which we described in more detail elsewhere, is summarized. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]