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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]

Although time-dependent density functional theory (TDDFT) has become the tool of choice for real-time propagation of the electron density ρN(t) of N-electron systems, it also encounters problems in this application. The first problem is the neglect of memory effects stemming from the, in TDDFT virtually unavoidable, adiabatic approximation, the second problem is the reliable evaluation of the probabilities Pn(t) of multiple photoinduced ionization, while the third problem (which TDDFT shares with other approaches) is the reliable description of continuum states of the electrons ejected in the process of ionization. In this paper time-dependent Dyson orbital theory (TDDOT) is proposed. Exact TDDOT equations of motion (EOMs) for time-dependent Dyson orbitals are derived, which are linear differential equations with just static, feasible potentials of the electron–electron interaction. No adiabatic approximation is used, which formally resolves the first TDDFT problem. TDDOT offers formally exact expressions for the complete evolution in time of the wavefunction of the outgoing electron. This leads to the correlated probability of single ionization P1(t) as well as the probabilities of no ionization (P0(t)) and multiple ionization of n electrons, Pn(t), which formally solves the second problem of TDDFT. For two-electron systems a proper description of the required continuum states appears to be rather straightforward, and both P1(t) and P2(t) can be calculated. Because of the exact formulation, TDDOT is expected to reproduce a notorious memory effect, the “knee structure” of the non-sequential double ionization of the He atom. [ABSTRACT FROM AUTHOR]

Almost all functionals that are currently used in density matrix functional theory have been created by some a priori ansatz that generates approximations to the second-order reduced density matrix (2RDM). In this paper, a more consistent approach is used: we analyze the 2RDMs (in the natural orbital basis) of rather accurate multi-reference configuration interaction expansions for several small molecules (CH4, NH3, H2O, FH, and N2) and use the knowledge gained to generate new functionals. The analysis shows that a geminal-like structure is present in the 2RDMs, even though no geminal theory has been applied from the onset. It is also shown that the leading non-geminal dynamical correlation contributions are generated by a specific set of double excitations. The corresponding determinants give rise to non-JKL (non Coulomb/Exchange like) multipole-multipole dispersive attractive terms between geminals. Due to the proximity of the geminals, these dispersion terms are large and cannot be omitted, proving pure JKL functionals to be essentially deficient. A second correction emerges from the observation that the “normal” geminal-like exchange between geminals breaks down when one breaks multiple bonds. This problem can be fixed by doubling the exchange between bond broken geminals, effectively restoring the often physically correct high-spin configurations on the bond broken fragments. Both of these corrections have been added to the commonly used antisymmetrized product of strongly orthogonal geminals functional. The resulting non-JKL functional Extended Löwdin-Shull Dynamical-Multibond is capable of reproducing complete active space self-consistent field curves, in which one active orbital is used for each valence electron. [ABSTRACT FROM AUTHOR]

A number of consequences of the presence of the exchange–correlation hole potential in the Kohn–Sham potential are elucidated. One consequence is that the HOMO–LUMO orbital energy difference in the KS-DFT model (the KS gap) is not “underestimated” or even “wrong”, but that it is physically expected to be an approximation to the excitation energy if electrons and holes are close, and numerically proves to be so rather accurately. It is physically not an approximation to the difference between ionization energy and electron affinity I−A (fundamental gap or chemical hardness) and also numerically differs considerably from this quantity. The KS virtual orbitals do not possess the notorious diffuseness of the Hartree–Fock virtual orbitals, they often describe excited states much more closely as simple orbital transitions. The Hartree–Fock model does yield an approximation to I−A as the HOMO–LUMO orbital energy difference (in Koopmans' frozen orbital approximation), if the anion is bound, which is often not the case. We stress the spurious nature of HF LUMOs if the orbital energy is positive. One may prefer Hartree–Fock, or mix Hartree–Fock and (approximate) KS operators to obtain a HOMO–LUMO gap as a Koopmans' approximation to I−A (in cases where A exists). That is a different one-electron model, which exists in its own right. But it is not an “improvement” of the KS model, it necessarily deteriorates the (approximate) excitation energy property of the KS gap in molecules, and deteriorates the good shape of the KS virtual orbitals. [ABSTRACT FROM AUTHOR]

This paper gives a natural orbital (NO) based analysis of the van der Waals interaction in (singlet) H2 at long distance. The van der Waals interaction, even if not leading to a distinct van der Waals well, affects the shape of the interaction potential in the van der Waals distance range of 5-9 bohrs and can be clearly distinguished from chemical bonding effects. In the NO basis the van der Waals interaction can be quantitatively covered with, apart from the ground state configurations (1σg)2 and (1σu)2, just the 4 configurations (2σg)2 and (2σu)2, and (1πu)2 and (1πg)2. The physics of the dispersion interaction requires and explains the peculiar relatively large positive CI coefficients of the doubly excited electron configurations (2σu)2 and (1πg)2 (the occupancy amplitudes of the 2σu and 1πgx, y NOs) in the distance range 5-9 bohrs, which have been observed before by Cioslowski and Pernal [Chem. Phys. Lett. 430, 188 (2006)]. We show that such positive occupancy amplitudes do not necessarily lead to the existence of zero occupation numbers at some H-H distances. [ABSTRACT FROM AUTHOR]

The key characteristics of electronic excitations of many-electron systems, the excitation energies ωα and the oscillator strengths fα, can be obtained from linear response theory. In one-electron models and within the adiabatic approximation, the zeros of the inverse response matrix, which occur at the excitation energies, can be obtained from a simple diagonalization. Particular cases are the eigenvalue equations of time-dependent density functional theory (TDDFT), time-dependent density matrix functional theory, and the recently developed phase-including natural orbital (PINO) functional theory. In this paper, an expression for the oscillator strengths fα of the electronic excitations is derived within adiabatic response PINO theory. The fα are expressed through the eigenvectors of the PINO inverse response matrix and the dipole integrals. They are calculated with the phase-including natural orbital functional for two-electron systems adapted from the work of Lowdin and Shull on two-electron systems (the phase-including Löwdin-Shull functional). The PINO calculations reproduce the reference fα values for all considered excitations and bond distances R of the prototype molecules H2 and HeH+ very well (perfectly, if the correct choice of the phases in the functional is made). Remarkably, the quality is still very good when the response matrices are severely restricted to almost TDDFT size, i.e., involving in addition to the occupied-virtual orbital pairs just (HOMO+1)-virtual pairs (R1) and possibly (HOMO+2)-virtual pairs (R2). The shape of the curves fα(R) is rationalized with a decomposition analysis of the transition dipole moments. [ABSTRACT FROM AUTHOR]

Shape corrections to the standard approximate Kohn-Sham exchange-correlation (xc) potentials are considered with the aim to improve the excitation energies (especially for higher excitations) calculated with time-dependent density functional perturbation theory. A scheme of gradient-regulated connection (GRAC) of inner to outer parts of a model potential is developed. Asymptotic corrections based either on the potential of Fermi and Amaldi or van Leeuwen and Baerends (LB) are seamlessly connected to the (shifted) xc potential of Becke and Perdew (BP) with the GRAC procedure, and are employed to calculate the vertical excitation energies of the prototype molecules N[sub 2], CO, CH[sub 2]O, C[sub 2]H[sub 4], C[sub 5]NH[sub 5], C[sub 6]H[sub 6], Li[sub 2], Na[sub 2], K[sub 2]. The results are compared with those of the alternative interpolation scheme of Tozer and Handy as well as with the results of the potential obtained with the statistical averaging of (model) orbital potentials. Various asymptotically corrected potentials produce high quality excitation energies, which in quite a few cases approach the benchmark accuracy of 0.1 eV for the electronic spectra. Based on these results, the potential BP-GRAC-LB is proposed for molecular response calculations, which is a smooth potential and a genuine "local" density functional with an analytical representation. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Time-dependent density functional theory (TDDFT) is applied for calculation of the excitation energies of the dissociating H[sub 2] molecule. The standard TDDFT method of adiabatic local density approximation (ALDA) totally fails to reproduce the potential curve for the lowest excited singlet [sup 1]Σ[sub u][sup +] state of H[sub 2]. Analysis of the eigenvalue problem for the excitation energies as well as direct derivation of the exchange-correlation (xc) kernel f[sub xc](r,r[sup ′],ω) shows that ALDA fails due to breakdown of its simple spatially local approximation for the kernel. The analysis indicates a complex structure of the function f[sub xc](r,r[sup ′],ω), which is revealed in a different behavior of the various matrix elements K[sub 1c,1c][sup xc] (between the highest occupied Kohn-Sham molecular orbital ψ[sub 1] and virtual MOs ψ[sub c]) as a function of the bond distance R(H-H). The effect of nonlocality of f[sub xc](r,r[sup ′]) is modeled by using different expressions for the corresponding matrix elements of different orbitals. Asymptotically corrected ALDA (ALDA-AC) expressions for the matrix elements K[sub 12,12][sup xc(στ)] are proposed, while for other matrix elements the standard ALDA expressions are retained. This approach provides substantial improvement over the standard ALDA. In particular, the ALDA-AC curve for the lowest singlet excitation qualitatively reproduces the shape of the exact curve. It displays a minimum and approaches a relatively large positive energy at large R(H-H). ALDA-AC also produces a substantial improvement for the calculated lowest triplet excitation, which is known to suffer from the triplet instability problem of the restricted KS ground state. Failure of the ALDA for the excitation energies is related to the failure of the local density as well as generalized gradient approximations to reproduce correctly the polarizability of dissociating H[sub 2]. The... [ABSTRACT FROM AUTHOR]

An approximate Kohn-Sham exchange-correlation potential ν[sub xc][sup SAOP] is developed with the method of statistical averaging of (model) orbital potentials (SAOP) and is applied to the calculation of excitation energies as well as of static and frequency-dependent multipole polarizabilities and hyperpolarizabilities within time-dependent density functional theory (TDDFT). ν[sub xc][sup SAOP] provides high quality results for all calculated response properties and a substantial improvement upon the local density approximation (LDA) and the van Leeuwen-Baerends (LB) potentials for the prototype molecules CO, N[sub 2], CH[sub 2]O, and C[sub 2]H[sub 4]. For the first three molecules and the lower excitations of the C[sub 2]H[sub 4] the average error of the vertical excitation energies calculated with ν[sub xc][sup SAOP] approaches the benchmark accuracy of 0.1 eV for the electronic spectra. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Aim: To study the effect of epicardial adipose tissue on risk of left ventricular (LV) diastolic dysfunction (DD) in patients with visceral obesity., Material and Methods: Obesity leads to the development of LV DD and is a major cause of heart failure with preserved LV ejection fraction (HFpEF). However, the contribution of epicardial adipose tissue to DD is understudied. This study included 101 men with general obesity (body weight index, 32.9±3.6 kg /m2). Based on severity of epicardial obesity (EO), two groups were formed: group 1, patients with an epicardial adipose tissue thickness (EATt) >7 mm (n=70), and group 2, patients with EATt <7 mm (n=31). Arterial hypertension, diabetes mellitus, coronary atherosclerosis, and disorders of LV diastolic function according to echocardiography (EchoCG) were the exclusion criteria. Diastolic function and LV mechanics were evaluated by speckle-tracking EchoCG for all patients at the start of the study and again at 4.7±0.3 years., Results: At baseline, none of the patients of either group had significant differences in EchoCG characteristics of LV diastolic function (left atrial volume index, LV early diastolic longitudinal lengthening velocity, peak tricuspid regurgitation velocity, and the ratio of diastolic transmitral flow velocity to mean mitral annular velocity (E / e'). However, there were significant increases in the LV untwisting velocity to -122.11 [-142.0; -116.0 degrees /s -1] degrees/s and the time to LV peak untwisting velocity to 472.3 ms. Repeated EchoCG showed an increase in left atrial volume index in group 1 to 35.04 [33.0; 39.7] ml /m2. Repeated evaluation of the LV mechanics revealed increases in the times to LV peak untwisting and twisting and decreases in the LV twisting and untwisting velocities. The logistic regression analysis showed that EATt was a risk factor for LV DD in obesity. Furthermore, the ROC analysis determined the optimal EATt cut-off threshold of ≥9 mm as a predictor for LV DD development., Conclusion: EO facilitates the development of LV DD and, thus, represents a major cause for HFpEF. An EATt value of ≥9 mm can be considered as a risk factor for LV DD development in patients with EO.

Neurohumoral changes have recently attracted much attention as a part of the pathogenesis of heart failure. Activation of neurohumoral factors triggers processes resulting in changes of extracellular matrix composition and, thus, development of myocardial fibrosis. This article addresses a number of factors that directly contribute to the development of myocardial fibrosis.

Straightforward interpretation of excitations is possible if they can be described as simple single orbital-to-orbital (or double, etc.) transitions. In linear response time-dependent density functional theory (LR-TDDFT), the (ground state) Kohn-Sham orbitals prove to be such an orbital basis. In contrast, in a basis of natural orbitals (NOs) or Hartree-Fock orbitals, excitations often employ many orbitals and are accordingly hard to characterize. We demonstrate that it is possible in these cases to transform to natural excitation orbitals (NEOs) which resemble very closely the KS orbitals and afford the same simple description of excitations. The desired transformation has been obtained by diagonalization of a submatrix in the equations of linear response time-dependent 1-particle reduced density matrix functional theory (LR-TDDMFT) for the NO transformation, and that of a submatrix in the linear response time-dependent Hartree-Fock (LR-TDHF) equations for the transformation of HF orbitals. The corresponding submatrix is already diagonal in the KS basis in the LR-TDDFT equations. While the orbital shapes of the NEOs afford the characterization of the excitations as (mostly) simple orbital-to-orbital transitions, the orbital energies provide a fair estimate of excitation energies.

In recent years, several benchmark studies on the performance of large sets of functionals in time-dependent density functional theory (TDDFT) calculations of excitation energies have been performed. The tested functionals do not approximate exact Kohn-Sham orbitals and orbital energies closely. We highlight the advantages of (close to) exact Kohn-Sham orbitals and orbital energies for a simple description, very often as just a single orbital-to-orbital transition, of molecular excitations. Benchmark calculations are performed for the statistical average of orbital potentials (SAOP) functional for the potential [J. Chem. Phys. 2000, 112, 1344; 2001, 114, 652], which approximates the true Kohn-Sham potential much better than LDA, GGA, mGGA, and hybrid potentials do. An accurate Kohn-Sham potential does not only perform satisfactorily for calculated vertical excitation energies of both valence and Rydberg transitions but also exhibits appealing properties of the KS orbitals including occupied orbital energies close to ionization energies, virtual-occupied orbital energy gaps very close to excitation energies, realistic shapes of virtual orbitals, leading to straightforward interpretation of most excitations as single orbital transitions. We stress that such advantages are completely lost in time-dependent Hartree-Fock and partly in hybrid approaches. Many excitations and excitation energies calculated with local density, generalized gradient, and hybrid functionals are spurious. There is, with an accurate KS, or even the LDA or GGA potentials, nothing problematic about the "band gap" in molecules: the HOMO-LUMO gap is close to the first excitation energy (the optical gap).

Aim: To study effect of epicardial adiposity on risk of development and severity of coronary atherosclerosis., Material and Methods: We registered classical metabolic risk factors (RF) and additional factors of cardiovascular risk in 138 men aged 55.47+/-9.07 years with ischemic heart disease (IHD), functional class (FC) II-III angina, and I-III degree obesity. Diagnostic coronary angiography (CAG) was performed in all patients. Thickness of epicardial fat (tEF) in millimeters was measured at transthoracic echocardiography (ECG)., Results: Average tEF indexes were highest (10 [8; 10] mm) in the group of patients with multiple stenoses in coronary arteries (CA). According to ROC-analysis tEF appeared to be a predictor of significant coronary atherosclerosis. Its sensitivity was 80.4%, specificity - 67.6 % (cut-off value 6 mm). Epicardial adiposity was among factors associated with presence of coronary atherosclerosis (odds ratio [OR] 4.44, 95% confidence interval [CI] 2,06 to 9.59; p<0,001) along with age (OR 6.56; 95% CI 2.59 to 16.60; p=0,001), leptin (OR 3.50; 95% CI 1.46 to 8.37; p <0,001), resistin (OR 3.13; 95% CI 1.32 to 7.42; p <0,001) and waist circumference (OR 1.65; 95% CI 0.72 to 3.80; p=0.018).

We use the natural orbitals to define an independent particle system, from which the exact one-particle density matrix can be obtained with an ensemble of degenerate determinantal ground states. Also defining explicit phases for the orbitals, and admitting functionals that are dependent on those phases, time-dependent equations for the orbitals and occupation numbers are obtained from an action principle. The wrong polarizability and lack of double excitations of straightforward adiabatic time-dependent density matrix functional theory are then corrected, and the important symmetry χ(ω)=χ{*}(-ω), lost in previous ad hoc improvements, is restored. The extension of the response calculations beyond the occupied-virtual pairs, which are the only ones admitted in time-dependent density functional theory, leads to greatly improved response properties.

Time-dependent density functional theory (TDDFT) in its current adiabatic implementations exhibits three remarkable failures: (a) completely wrong behavior of the excited state surface along a bond-breaking coordinate; (b) lack of doubly excited configurations; (c) much too low charge transfer excitation energies. These TDDFT failure cases are all strikingly exhibited by prototype two-electron systems such as dissociating H2 and HeH+. We find for these systems with time-dependent density matrix functional theory that: (a) 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; (b) An adiabatic approximation is formulated in which also the double excitations are fully accounted for.