Time-dependent four-component relativistic density-functional theory for excitation energies. II. The exchange-correlation kernel

We extend our previous formulation of time-dependent four-component relativistic density-functional theory [J. Gao, W. Liu, B. Song, and C. Liu, J. Chem. Phys. 121, 6658 (2004)] by using a noncollinear form for the exchange-correlation kernel. The new formalism can deal with excited states involving moment (spin)-flipped configurations which are otherwise not accessible with ordinary exchange-correlation functionals. As a first application, the global potential-energy curves of 16 low-lying omega omega-coupled electronic states of the AuH molecule have been investigated. The derived spectroscopic parameters, including the adiabatic and vertical excitation energies, equilibrium bond lengths, harmonic and anharmonic vibrational constants, fundamental frequencies, and dissociation energies, are grossly in good agreement with those of ab initio multireference second-order perturbation theory and the available experimental data.     

Analysis of environment response effects on excitation energies within subsystem-based time-dependent density-functional theory

We analyze the ability of subsystem time-dependent density-functional theory (sTDDFT) to describe environmental response effects. To this end, we utilize the recently proposed “exact” version of sTDDFT relying on projection-based embedding (PbE), which so far was applied only for the special case of two subsystems. We confirm that PbE-sTDDFT in combination with supersystem bases yields results equivalent to those of supermolecular TDDFT calculations for systems solvated by many solvent molecules, using the previously studied system of methylene-cyclopropene⋯(H₂O)₁₇ as an example. By means of this exact reference embedding framework, we are able to disentangle solvent effects introduced in terms of the embedding potential from those caused by solvent response couplings, both for the PbE variant and for sTDDFT with approximate non-additive kinetic energy functionals. Furthermore, we show that the use of a monomer basis introduces significant errors for the environmental response contribution. Employing a virtual-orbital localization strategy on top of PbE-sTDDFT, we can also directly assess the impact of inter-subsystem charge-transfer excitations on the entire solvent effect, which turn out to play a significant role for the environmental response. Finally, we analyze the response effects introduced by the individual solvent molecules and their interdependence, and show that a simple, pair-wise additive correction for solvent response yields excellent results in the present example.     

Intermolecular potentials based on symmetry-adapted perturbation theory with dispersion energies from time-dependent density-functional calculations

Recently, three of us have proposed a method [Phys. Rev. Lett. 91, 33201 (2003)] for an accurate calculation of the dispersion energy utilizing frequency-dependent density susceptibilities of monomers obtained from time-dependent density-functional theory (DFT). In the present paper, we report numerical calculations for the helium, neon, water, and carbon dioxide dimers and show that for a wide range of intermonomer separations, including the van der Waals and short-range repulsion regions, the method provides dispersion energies with accuracies comparable to those that can be achieved using the current most sophisticated wave-function methods. If the dispersion energy is combined with (i) the electrostatic and first-order exchange interaction energies as defined in symmetry-adapted perturbation theory (SAPT) but computed using monomer Kohn-Sham (KS) determinants, and (ii) the induction energy computed using the coupled KS static response theory, (iii) the exchange-induction and exchange-dispersion energies computed using KS orbitals and orbital energies, the resulting method, denoted by SAPT(DFT), produces very accurate total interaction potentials. For the helium dimer, the only system with nearly exact benchmark values, SAPT(DFT) reproduces the interaction energy to within about 2% at the minimum and to a similar accuracy for all other distances ranging from the strongly repulsive to the asymptotic region. For the remaining systems investigated by us, the quality of the SAPT(DFT) interaction energies is so high that these energies may actually be more accurate than the best available results obtained with wave-function techniques. At the same time, SAPT(DFT) is much more computationally efficient than any method previously used for calculating the dispersion and other interaction energy components at this level of accuracy.     

Excitation energies from time-dependent density-functional theory beyond the adiabatic approximation

Time-dependent density-functional theory in the adiabatic approximation has been very successful for calculating excitation energies in molecular systems. This paper studies nonadiabatic effects for excitation energies, using the current-density functional of Vignale and Kohn [Phys. Rev. Lett. 77, 2037 (1996)]. We derive a general analytic expression for nonadiabatic corrections to excitation energies of finite systems and calculate singlet s-->s and s-->p excitations of closed-shell atoms. The approach works well for s-->s excitations, giving a small improvement over the adiabatic local-density approximation, but tends to overcorrect s-->p excitations. We find that the observed problems with the nonadiabatic correction have two main sources: (1) the currents associated with the s-->p excitations are highly nonuniform and, in particular, change direction between atomic shells, (2) the so-called exchange-correlation kernels of the homogeneous electron gas, f(xc) (L) and f(xc) (T), are incompletely known, in particular in the high-density atomic core regions.     

Time-dependent four-component relativistic density functional theory for excitation energies

Time-dependent four-component relativistic density functional theory within the linear response regime is developed for calculating excitation energies of heavy element containing systems. Since spin is no longer a good quantum number in this context, we resort to time-reversal adapted Kramers basis when deriving the coupled Dirac-Kohn-Sham equation. The particular implementation of the formalism into the Beijing density functional program package utilizes the multipolar expansion of the induced density to facilitate the construction of the induced Coulomb potential. As the first application, pilot calculations on the valence excitation energies and fine structures of the rare gas (Ne to Rn) and Group 12 (Zn to Hg) atoms are reported. To the best of our knowledge, it is the first time to be able to account for spin-orbit coupling within time-dependent density functional theory for excitation energies.     

The calculation of excitation energies based on the relativistic two-component zeroth-order regular approximation and time-dependent density-functional with full use of symmetry

In the present work, we propose a relativistic time-dependent density-functional theory (TDDFT) based on the two-component zeroth-order regular approximation and a noncollinear exchange-correlation (XC) functional. This two-component TDDFT formalism has the correct nonrelativistic limit and affords the correct threefold degeneracy of triplet excitations. The relativistic TDDFT formalism is implemented into the AMSTERDAM DENSITY FUNCTIONAL program package for closed-shell systems with full use of double-group symmetry to reduce the computational effort and facilitate the assignments. The performance of the formalism is tested on some closed-shell atoms, ions, and a few diatomic molecules containing heavy elements. The results show that the fine structure of the excited states for most atoms and ions studied here can be accurately accounted for with a proper XC potential. For the excitation energies of the molecules studied here, the present formalism shows promise and the error encountered is comparable to that of nonrelativistic TDDFT calculations on light elements.     

The electronic spectrum of AgCl2: ab initio benchmark versus density-functional theory calculations on the lowest ligand-field states including spin-orbit effects

The X 2pi(g), 2sigma(g)+, and 2delta(g) states of AgCl2 have been studied through benchmark ab initio complete active space self-consistent field plus second-order complete active space multireference Moller-Plesset algorithm (CASSCF+CASPT2) and complete active space self-consistent field plus averaged coupled pair functional (CASSCF+ACPF) and density-functional theory (DFT) calculations using especially developed basis sets to study the transition energies, geometries, vibrational frequencies, Mulliken charges, and spin densities. The spin-orbit (SO) effects were included through the effective Hamiltonian formalism using the LambdaSSigma ACPF energies as diagonal elements. At the ACPF level, the ground state is 2pi(g) in contradiction with ligand-field theory, SCF, and large CASSCF; the adiabatic excitation energies for the 2sigma(g)+ and 2delta(g) states are 1640 and 18,230 cm(-1), respectively. The inclusion of the SO effects leads to a pure omega = 32(2pi(g)) ground state, a omega = 12 (66%2pi(g) and 34%2sigma(g)+) A state, a omega = 12 (34%2pi(g) and 66%2sigma(g)+) B state, a omega = 52(2delta(g))C state, and a omega = 32(99%2delta(g))D state. The X-A, X-B, X-C, and X-D transition energies are 485, 3715, 17 246, and 20 110 cm(-1), respectively. The B97-2, B3LYP, and PBE0 functionals overestimate by approximately 100% the X 2pi(g)-2sigma(g)+T(e) but provide a qualitative energetic ordering in good agreement with ACPF results. B3LYP with variable exchange leads to a 42% optimal Hartree-Fock exchange for transition energies but all equilibrium geometries get worsened. Asymptotic corrections to B3LYP do not provide improved values. The nature of the bonding in the X 2pi(g) state is very different from that of CuCl2 since the Mulliken charge on the metal is 1.1 while the spin density is only 0.35. DFT strongly delocalizes the spin density providing even smaller values of around 0.18 on Ag not only for the ground state, but also for the 2sigma(g)+ state.     

The performance of time-dependent density functional theory based on a noncollinear exchange-correlation potential in the calculations of excitation energies

In the present work we have studied the accuracy of excitation energies calculated from spin-flip transitions with a formulation of time-dependent density functional theory based on a noncollinear exchange-correlation potential proposed in a previous study. We compared the doublet-doublet excitation energies from spin-flip transitions and ordinary transitions, calculated the multiplets splitting of some atoms, the singlet-triplet gaps of some diradicals, the energies of excited quartet states with a doublet ground state. In addition, we attempted to calculate transition energies with excited states as reference. We compared the triplet excitation energies and singlet-triplet separations of the excited state from spin-flip and ordinary transitions. As an application, we show that using excited quartet state as reference can help us fully resolve excited states spin multiplets. In total the obtained excitation energies calculated from spin-flip transitions agree quite well with other theoretical results or experimental data.     

AB initio effective core potentials including relativistic effects. A procedure for the inclusion of spin-orbit coupling in molecular wavefunctions

The first ab initio procedure for the treatment of spin-orbit coupling in molecules based on the use of relativistic effective potentials derived from Dirac-Fock atomic wavefunctions is presented. A rigorous definition for the spin-orbit operator is given and its use in molecular calculations discussed.     

A simplified approach for the coupling of excitation energy transfer

A simplified approach for computing the electronic coupling of nonradiative excitation-energy transfer is proposed by following Scholes et al.’s construction on the initial and final states [G.D. Scholes, R.D. Harcourt, K.P. Ghiggino, J. Chem. Phys. 102 (1995) 9574]. The simplification is realized through defining a set of orthogonalized localized MOs, which include the polarization effect of the charge densities. The method allows calculating the coupling of both the singlet-to-singlet and triplet-to-triplet energy transfer. Numerical tests are performed for a few of dimers with different intermolecular orientations, and the results demonstrate that Coulomb term are the major contribution to the coupling of singlet-to-singlet energy transfer whereas in the case of triplet-to-triplet energy transfer, the dominant effect is arisen from the intermolecular charge-transfer states. The present application is on the Hartree-Fock level. However, the correlated wavefunctions which are normally expanded in terms of the determinant wavefunctions can be employed in the similar way.     

Performance of the M11 and M11-L density functionals for calculations of electronic excitation energies by adiabatic time-dependent density functional theory

Adiabatic time-dependent density functional theory is a powerful method for calculating electronic excitation energies of complex systems, but the quality of the results depends on the choice of approximate density functional. In this article we test two promising new density functionals, M11 and M11-L, against databases of 214 diverse electronic excitation energies, and we compare the results to those for 16 other density functionals of various kinds and to time-dependent Hartree-Fock. Charge transfer excitations are well known to be the hardest challenge for TDDFT. M11 is a long-range-corrected hybrid meta-GGA, and it shows better performance for charge transfer excitations than any of the other functionals except M06-HF, which is a specialized functional that does not do well for valence excitations. Several other long-range-corrected hybrid functionals also do well, and we especially recommend M11, ωB97X, and M06-2X for general spectroscopic applications because they do exceptionally well on ground-state properties as well as excitation energies. Local functionals are preferred for many applications to extended systems because of their significant cost advantage for large systems. M11-L is a dual-range local functional and-unlike all previous local functionals-it has good performance for Rydberg states as well as for valence states. Thus it is highly recommended for excitation energy calculations on extended systems.     

Electron density fitting for the Coulomb problem in relativistic density-functional theory

A density fitting approach for the Coulomb matrix representation within the four-component formulation of relativistic density-functional theory is presented. Our implementation, which uses G-spinor basis sets, shares all the advantages of those found in nonrelativistic quantum chemistry. We show that very accurate Coulomb energies may be obtained using a modest number of scalar auxiliary basis functions for molecules containing heavy atoms. The efficiency of this new implementation is demonstrated in a detailed study of the spectroscopic properties of the gold dimer, and its scaling behavior has been tested by calculations of some closed-shell gold clusters (Au2, Au3+, Au4, Au5+). The algorithm is found to scale as O(N3), just as it does in the nonrelativistic case, and represents a dramatic improvement in efficiency over the conventional approach in the calculation of the Coulomb matrix, with computation times that are reduced to less than 3% for Au2 and up to 1% in the case of Au5+.     

Assessment of time-dependent density functional theory for predicting excitation energies of bichromophoric peptides: case of tryptophan-phenylalanine

The ability of applied time-dependent density functional theory to predict the near-ultraviolet absorption spectrum of bichomophoric peptides in the gas phase has been tested by calculating the vertical excitation energies of the Tryptophan-Phenylalanine (Trp-Phe) dipeptide. We show that the contamination of the low-frequency part of the spectrum by spurious charge-transfer excitations depends both on the conformation of the peptide chain and the exchange-correlation approximation. For the most stable structure investigated, a hybrid density functional appears to eliminate a large proportion of the spurious states.     

Existence of time-dependent density-functional theory for open electronic systems: time-dependent holographic electron density theorem

We present the time-dependent holographic electron density theorem (TD-HEDT), which lays the foundation of time-dependent density-functional theory (TDDFT) for open electronic systems. For any finite electronic system, the TD-HEDT formally establishes a one-to-one correspondence between the electron density inside any finite subsystem and the time-dependent external potential. As a result, any electronic property of an open system in principle can be determined uniquely by the electron density function inside the open region. Implications of the TD-HEDT on the practicality of TDDFT are also discussed.     

A global investigation of excited state surfaces within time-dependent density-functional response theory

This work investigates the capability of time-dependent density functional response theory to describe excited state potential energy surfaces of conjugated organic molecules. Applications to linear polyenes, aromatic systems, and the protonated Schiff base of retinal demonstrate the scope of currently used exchange-correlation functionals as local, adiabatic approximations to time-dependent Kohn-Sham theory. The results are compared to experimental and ab initio data of various kinds to attain a critical analysis of common problems concerning charge transfer and long range (nondynamic) correlation effects. This analysis goes beyond a local investigation of electronic properties and incorporates a global view of the excited state potential energy surfaces.     

Theoretical study of the electronic spectra of square-planar platinum (II) complexes based on the two-component relativistic time-dependent density-functional theory

In the present work the electronic spectra of [PtCl(4)](2-), [PtBr(4)](2-), and [Pt(CN)(4)](2-) are studied with a recently proposed relativistic time-dependent density-functional theory (TDDFT) based on the two-component zeroth-order regular approximation and a noncollinear exchange-correlation (XC) functional. The contribution to the double group excited states in terms of singlet and triplet single group excited states is estimated through the inner product of the transition density matrix obtained from two-component and scalar relativistic TDDFT calculations to better understand the double group excited states. Spin-orbital coupling effects are found to be very important in order to simulate the electronic spectra of these complexes. The results show that the two-component TDDFT formalism can afford excitation energies with high accuracy for the transition-metal systems studied here when use is made of a proper XC potential.     

Photoabsorption in sodium clusters on the basis of time-dependent density-functional theory

The photoabsorption spectra of a continuous series of Na(n) clusters (n相似文献   

Interpretation of indirect nuclear spin-spin coupling tensors for polyatomic xenon fluorides and group 17 fluorides: results from relativistic density-functional calculations

Significant improvements have been made recently in the calculation of NMR indirect nuclear spin-spin coupling tensors (J). In particular, the relativistic zeroth-order regular approximation density-functional theory (ZORA-DFT) approach holds great promise for the calculation of spin-spin coupling constants for a variety of chemical systems containing heavy nuclei. In the present work, the ZORA-DFT method is applied to the calculation of the complete reduced coupling tensors, K, for a range of chlorine-, bromine-, iodine-, and xenon-containing species: K(Cl,F) for ClF(2)(+), ClF(3), ClF(4)(+), ClF(5), ClF(6)(-), and ClF(6)(+); K(Br,F) for BrF(3), BrF(6)(-), and BrF(6)(+); K(I,F) for IF(4)(+) and IF(6)(+); K(Xe,F) for XeF(+), XeF(2), XeF(3)(+), XeF(4), XeF(5)(-), XeF(5)(+), and XeF(7)(+). These species represent a wide variety of geometrical bonding arrangements. Agreement between the calculated coupling constants and available experimental data is excellent, and the absolute sign of the coupling constants is provided. It is shown that (1)K(iso) may be positive or negative even within the same molecule, e.g., K(Cl,F)(iso) may be of either sign, depending on the local environment. Periodic trends in (1)K(iso) for isovalent and isostructural molecules are evident. The spin-spin coupling anisotropies, Delta K, and the orientations of the K tensors are also determined. The success of the calculations is a direct result of employing reliable geometries and considering both scalar and spin-orbit relativistic effects. The dependence of K(Cl,F)(iso) and K(Xe,F)(iso) on the local molecular and electronic structure is discussed in terms of the paramagnetic spin-orbit (PSO) and combined Fermi-contact spin-dipolar (FC+SD) coupling mechanisms. The PSO term depends strongly on the number of valence shell electron lone pairs on the central heavy atom, and the FC+SD contribution increases with the Cl[bond]F or Xe[bond]F bond length for a given series of compounds. This interpretation allows for the successful rationalization of the existing experimental data.     

Performance of time-dependent density functional and Green functions methods for calculations of excitation energies in radicals and for Rydberg electronic states

Time-dependent density functional (TD-DFT) and perturbation theory-based outer valence Green functions (OVGF) methods have been tested for calculations of excitation energies for a set of radicals, molecules, and model clusters simulating points defects in silica. The results show that the TD-DFT approach may give unreliable results not only for diffuse Rydberg states, but also for electronic states involving transitions between MOs localized in two remote from each other spatial regions, for example, for charge-transfer excitations. For the. O-SiX(3) clusters, where X is a single-valence group, TD-DFT predicts reasonable excitation energies but incorrect sequence of electronic transitions. For a number of cases where TD-DFT is shown to be unreliable, the OVGF approach can provide better estimates of excitation energies, but this method also is not expected to perform universally well. The OVGF performance is demonstrated to be satisfactory for excitations with predominantly single-determinant wave functions where the deviations of the calculated energies from experiment should not exceed 0.1-0.3 eV. However, for more complicated transitions involving multiple bonds or for excited states with multireference wave functions the OVGF approach is less reliable and error in the computed energies can reach 0.5-1 eV.