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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]

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]

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]

A complete and explicit expression for the exchange kernel f[sub xσ] of density functional response theory (DFRT) is derived in terms of the occupied Kohn-Sham (KS) orbitals ψ[sub iσ]. It is based on the common energy denominator approximation (CEDA) for the KS Green's function (O. V. Gritsenko and E. J. Baerends, Phys. Rev. A 64, 042506 (2001)). The kernel f[sub xσ][sup CEDA] is naturally subdivided into the Slater f[sub Sσ][sup CEDA] and the 'response' f[sub respσ][sup CEDA] parts, which are the derivatives of the Slater ν[sub Sσ] and response ν[sub respσ] potentials, respectively. While f[sub Sσ][sup CEDA] is obtained with a straightforward differentiation of ν[sub Sσ], some terms of f[sub respσ][sup CEDA] are obtained from the solution of linear equations for the corresponding derivatives. All components of f[sub xσ][sup CEDA] are explicitly expressed in terms of the products ψ[sub iσ][sup *]ψ[sub jσ] of the occupied KS orbitals taken at the positions r[sub 1] and r[sub 2], as well as the potentials of these products at r[sub 3]. The coefficients in these expressions are obtained by inversion of the matrix, associated with the overlap matrix of the products ψ[sub iσ][sup *]ψ[sub jσ] and ψ[sub kσ][sup *]ψ[sub lσ]. Terms are indicated, which generate in an external electric field an ultra-nonlocal potential δν[sub xσ], counteracting an external field, and possible approximations to f[sub xσ][sup CEDA] are considered. [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]