Fast calculation of excited-state potentials for rare-gas diatomic molecules: Ne2 and Ar2

A fast method for obtaining excited-state potentials of rare-gas diatomic molecules is described. Two types of excited orbitals are used: molecular orbitals calculated in the field of a singly charged molecular ion, and atomic orbitals (properly symmetrized) obtained in a similar atomic system. The RPA equations are solved within the manifold of excitations from the highest occupied orbital in each symmetry to the lowest excited orbital of either type in each symmetry. A simple model for estimating the dynamic correlation correction to excitation (and ionization) energies is given. Applications to excited states of Ne₂(^1,3Σ⁺_{g, u}, ^1,3Π_{g, u}) and Ar₂(^1,3Σ⁺_{g, u}) are described. Two-electron integral transformations involve only three orbitals of each symmetry, and the RPA matrices are four-dimensional. The computational effort required for all excited-state potentials adds less than one-tenths (in terms of computer time) to the effort involved in the preliminary ground state Hartree—Fock calculations. The resulting potentials compare favorably with more elaborate CI calculations and give good agreement with spectroscopic and scattering data. Potential curves for the molecular ions are also given.     

Local Hartree–Fock orbitals using a three‐level optimization strategy for the energy

Using the three‐level energy optimization procedure combined with a refined version of the least‐change strategy for the orbitals—where an explicit localization is performed at the valence basis level—it is shown how to more efficiently determine a set of local Hartree–Fock orbitals. Further, a core–valence separation of the least‐change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree–Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality. © 2013 Wiley Periodicals, Inc.     

Theoretical study of the electronic spectroscopy of CO adsorbed on Pt(111)

The excited states of CO adsorbed on the Pt(111) surface are studied using a time-dependent density functional theory formalism. To reduce the computational cost, electronic excitations are computed within a reduced single excitation space. Using cluster models of the surface, excitation energies are computed for CO in the on-top, threefold, and bridge binding sites. On adsorption, there is a lowering of the 5sigma orbital energy. This leads to a large blueshift in the 5sigma- -> pi(CO*) excitation energy for all adsorption sites. The 1pi and 4sigma orbital energies are lowered to a lesser extent, and smaller shifts in the corresponding excitation energies are predicted. For the larger clusters, pi* excitations at lower energies are observed. These transitions correspond to excitations to virtual orbitals of pi* character which lie below the pi* orbitals of gas phase CO. These orbitals are associated predominantly with the metal atoms of the cluster. The excitation energies are also found to be sensitive to changes in the adsorption geometry. The electronic spectrum of CO on Pt(111) is simulated and the assignment of the bands observed in experimental electron energy loss spectroscopy discussed.     

Molecular-orbital-free algorithm for excited states in time-dependent perturbation theory

A nonlinear conjugate gradient optimization scheme is used to obtain excitation energies within the random phase approximation (RPA). The solutions to the RPA eigenvalue equation are located through a variational characterization using a modified Thouless functional, which is based upon an asymmetric Rayleigh quotient, in an orthogonalized atomic orbital representation. In this way, the computational bottleneck of calculating molecular orbitals is avoided. The variational space is reduced to the physically-relevant transitions by projections. The feasibility of an RPA implementation scaling linearly with system size N is investigated by monitoring convergence behavior with respect to the quality of initial guess and sensitivity to noise under thresholding, both for well- and ill-conditioned problems. The molecular-orbital-free algorithm is found to be robust and computationally efficient, providing a first step toward large-scale, reduced complexity calculations of time-dependent optical properties and linear response. The algorithm is extensible to other forms of time-dependent perturbation theory including, but not limited to, time-dependent density functional theory.     

Bond orbital approach for optical rotatory strength calculations

Directly determined localized approximate molecular Orbitals are used in excitation energy and optical rotatory strength calculations within the CNDO/2 scheme. Using strictly localized bond orbitals one obtains qualitatively good excitation energies, but quantitative agreement can be found only by considering delocalization effects, which have been proved to be crucial in determining the optical rotatory strength. The delocalization interactions are classified as through space and through bond ones and even the latter is found to have significant contributions. The chiroptical properties of the lowest lying transitions in the twisted glyoxal molecule are analysed in terms of localized molecular orbital contributions.     

AB initio calculations of oscillator and rotatory strengths in the random-phase approximation: planar and twisted butadiene

Minimal basis set (STO-4G) ab initio calculations in the random-phase approximation (RPA) are presented for the ordinary and rotatory intensities of the low-lying electronic transitions of twisted cis-butadiene, and planar trans-butadiene. The formally equivalent intensities agree much better in the RPA than in either monoexcited CI or Hartree-Fock virtual orbital calculations. Comparisons with other work are given, and an explanation is suggested for the sensitivity of the rotatory strengths to substituents.     

Calculation of K-edge circular dichroism of amino acids: comparison of random phase approximation with other methods

Soft x-ray natural circular dichroism of amino acids is studied by means of ab initio methods. Several approaches to evaluate the oscillator and rotary strengths of core-to-valence excitations are compared from the viewpoint of basis set dependence: ground-state Hartree-Fock (HF) orbital set employed in (i) random phase approximation (RPA), (ii) static exchange approach (STEX) (unrelaxed), (iii) core-ionized state HF orbital set applied in STEX (relaxed), and (iv) HF excited state orbital set for each core-to-valence excited state. Furthermore in (i) the PRA in the framework of the density functional method (DFT) is compared with the RPA where the ab initio HF orbital set is used. In (iv), the oscillator and rotary strengths evaluated by different orbital sets for the initial and final states, namely, nonorthogonal ground-state and core-excited HF orbitals, are compared with those evaluated by using the core-excited HF orbital set to describe the initial (ground) state. It was shown that, among considered methods, the RPA provides most consistent and less time-consuming results for circular dichroism core excitation spectra. Discussion of the low energy part of K edge circular dichroism spectra of five common amino acids obtained with the help of RPA is presented.     

On a definition of bond energies based on localized molecular orbitals

A theoretical model is presented for defining bond energies based on localized molecular Orbitals. These bond energies are obtained by rearranging the total SCF energy including the nuclear repulsion term to a sum over orbital and orbital interaction terms and then to total orbital terms, which can be interpreted as the energies of localized orbitals in a molecule. A scaling procedure is used to obtain a direct connection with experimental bond dissociation energies. Two scale parameters are employed, the C-C and the C-H bond dissociation energy in C₂H₆ for A-B and C-H type bonds, respectively. The implications of this scaling procedure are discussed. Numerical applications to a number of organic molecules containing no conjugated bonds gives in general a very satisfactory agreement between experimental and theoretical bond energies.     

Electronic spectrum of F<Subscript>2</Subscript>CO: theoretical calculations of vertical excitation energies and intensities

In this work, the linear response formalism with a triples-corrected CCSD reference wave function, LR-CCSDR(3), is applied to the calculation of vertical excitation energies of singlet states of the F₂CO molecule. A basis set of atomic natural orbitals augmented with a series of Rydberg functions has been used in the calculations. A large number of electronically excited states were calculated, and the valence, Rydberg, or mixed character of the states were investigated. In addition, the molecular quantum defect orbital (MQDO) method has been used to determine transition intensities involving Rydberg states. Excitation energies and transition intensities for Rydberg states with n > 3 are reported for the first time.     

Quasiparticle virtual orbitals in electron propagator calculations

The computational limits of accurate electron propagator methods for the calculation of electron binding energies of large molecules are usually determined by the rank of the virtual orbital space. Electron density difference matrices that correspond to these transition energies in the second-order quasiparticle approximation may be used to obtain a virtual orbital space of reduced rank that introduces only minor deviations with respect to the results produced with the full, original set of virtual orbitals. Numerical tests show the superior accuracy and efficiency of this approach compared to the usual practice of omission of virtual orbitals with the highest energies.     

Basis set effects on frontier molecular orbital energies and energy gaps: a comparative study between plane waves and localized basis functions in molecular systems

In order to study the Kohn-Sham frontier molecular orbital energies in the complete basis limit, a comparative study between localized functions and plane waves, obtained with the local density approximation exchange-correlation functional is made. The analyzed systems are ethylene and butadiene, since they are theoretical and experimentally well characterized. The localized basis sets used are those developed by Dunning. For the plane-waves method, the pseudopotential approximation is employed. The results obtained by the localized basis sets suggest that it is possible to get an estimation of the orbital energies in the limit of the complete basis set, when the basis set size is large. It is shown that the frontier molecular orbital energies and the energy gaps obtained with plane waves are similar to those obtained with a large localized basis set, when the size of the supercell and the plane-wave expansion have been appropriately calibrated.     

The generalized valence bond π orbitals of ethylene and allyl cation

Using a fixed sigma core obtained from full electron ab initio Hartree-Fock calculations, the spatially projected GVB orbitals for the pi electron systems of ethylene and allyl cation are reported. The GVB(SP) method generates wavefunctions possessing the correct spatial and spin symmetry without restricting the nature of the individual orbitals. The GVB(SP) wavefunction provides a simple interpretation of the molecule in terms of orbitals each containing a single electron. The resulting total energies and excitation energies agree very well with full configuration interaction calculations.     

A quantitative analysis of the through-space and the through-bond interactions between lone-pairs in azines: pyridazine,pyrimidine and pyrazine

The INDO calculations were performed on the three azines: pyridazine, pyrimidine, and pyrazine. The cannonical molecular orbitais obtained by these calculations were then transformed into the localized molecular orbitals. With the use of the localized molecular orbitals, the variation in the lone-pair orbital energies of these molecules were pursued in the light of the through-space and/or the through-bond interactions between the specified localized molecular orbitals in a molecule selectively. The interactions were expressed by the summation of several terms: through-space, through-bond, through-virtuals and coupling terms.     

Atomic pseudopotentials for reproducing π-orbital electron behavior in sp2 carbon atoms

A pseudopotential system for an sp² carbon atom is built and tested as a building block for various pseudohydrocarbon polyenes and polycyclic aromatic hydrocarbons. This pseudosystem has a central charge of Z = 1; it contains only one electron. It is employed in ab-initio calculations in which several physical characteristics including the orbital energies and first ionization energy, as well as first excitation energy and UV spectra, are found to be well-reproduced by the pseudosystem. Remarkably, not only are the π excitation energies in good agreement with the reference calculations, but also transition densities and intensities, confirming that the virtual space obtained with the pseudopotentials is of excellent quality. Finally, this approach is capable of reproducing the π electron systems of small or large, planar or nonplanar hydrocarbons at low computational cost.     

Direct algorithm for the random-phase approximation

An algorithm for calculating excitation energies and transition moments in the random-phase approximation (RPA) of the polarization propagator is presented. The algorithm includes direct solution of the RPA eigenvalue problem and direct evaluation of products of superoperator Hamiltonian matrices with excitation vectors. Given sufficient memory, only one integral evaluation step per iteration is needed. Illustrative calculations on the excitation energies and oscillator strengths of ethylene are presented. © 1996 John Wiley & Sons, Inc.     

A simplified relativistic time-dependent density-functional theory formalism for the calculations of excitation energies including spin-orbit coupling effect

In the present work we have proposed an approximate time-dependent density-functional theory (TDDFT) formalism to deal with the influence of spin-orbit coupling effect on the excitation energies for closed-shell systems. In this formalism scalar relativistic TDDFT calculations are first performed to determine the lowest single-group excited states and the spin-orbit coupling operator is applied to these single-group excited states to obtain the excitation energies with spin-orbit coupling effects included. The computational effort of the present method is much smaller than that of the two-component TDDFT formalism and this method can be applied to medium-size systems containing heavy elements. The compositions of the double-group excited states in terms of single-group singlet and triplet excited states are obtained automatically from the calculations. The calculated excitation energies based on the present formalism show that this formalism affords reasonable excitation energies for transitions not involving 5p and 6p orbitals. For transitions involving 5p orbitals, one can still obtain acceptable results for excitations with a small truncation error, while the formalism will fail for transitions involving 6p orbitals, especially 6p1/2 spinors.     

Coupled cluster studies. IV. Analysis of the correlated wavefunction in canonical and localized orbital basis for ethylene,carbon monoxide,and carbon dioxide

An a posteriori analysis of the correlated wavefunctions of three small molecules using canonical and localized orbitals shows that, while more excitations are nearly zero for canonical orbitals than for localized ones, in the latter case a straightforward way exists for a priori selection of negligible excitations. In the case of the larger molecule cytosine the same observation is made. However, in this case 99% of the correlation energy is obtained already with ≈ 10% of the excitations when localized orbitals are used, while ≈ 36% of them are necessary in canonical basis. Furthermore it is shown that, using localized orbitals, the excitations can be split into subsets which can be calculated individually. The results suggest that 80–90% of the correlation energies given by MP2, CCL, or CCD can be obtained from the contributions of individual chemical bonds and their interactions. A simple derivation of the orbital invariant formalism of Pulay and Sæbø for the calculation of MP2 and MP3 correlation energies is given.     

Compact basis sets for LCAO-LSD calculations. Part II: Tests for Cr2 and Ni4

The compact orbital and auxiliary basis sets for LCAO-LSD calculations introduced in Part I are tested in molecular calculations on Cr₂ and Ni₄. The present results for spectroscopic constants and valence orbital energies obtained using medium size orbital expansions with a double-zeta representation for valence orbitals are in very good agreement with those previously calculated with very extended sets. Since the computational time of the present calculations is reduced severalfold compared with the extended basis set calculations, the present basis sets allow increased efficiency of the LCAO-LSD calculations and allow the method to be extended to larger systems.     

Local pair natural orbitals for excited states

We explore how in response calculations for excitation energies with wavefunction based (e.g., coupled cluster) methods the number of double excitation amplitudes can be reduced by means of truncated pair natural orbital (PNO) expansions and localized occupied orbitals. Using the CIS(D) approximation as a test model, we find that the number of double excitation amplitudes can be reduced dramatically with minor impact on the accuracy if the excited state wavefunction is expanded in state-specific PNOs generated from an approximate first-order guess wavefunction. As for ground states, the PNO truncation error can also for excitation energies be controlled by a single threshold related to generalized natural occupation numbers. The best performance is found with occupied orbitals which are localized by the Pipek-Mezey localization. For a large test set of excited states we find with this localization that already a PNO threshold of 10(-8)-10(-7), corresponding to an average of only 40-80 PNOs per pair, is sufficient to keep the PNO truncation error for vertical excitation energies below 0.01 eV. This is a significantly more rapid convergence with the number doubles amplitudes than in domain-based local response approaches. We demonstrate that the number of significant excited state PNOs scales asymptotically linearly with the system size in the worst case of completely delocalized excitations and sub-linearly whenever the chromophore does not increase with the system size. Moreover, we observe that the flexibility of state-specific PNOs to adapt to the character of an excitation allows for an almost unbiased treatment of local, delocalized and charge transfer excited states.