Scaled second-order perturbation corrections to configuration interaction singles: efficient and reliable excitation energy methods

Two modifications of the perturbative doubles correction to configuration interaction with single substitutions (CIS(D)) are suggested, which are excited state analogues of ground state scaled second-order M?ller-Plesset (MP2) methods. The first approach employs two parameters to scale the two spin components of the direct term of CIS(D), starting from the two-parameter spin-component scaled (SCS) MP2 ground state, and is termed SCS-CIS(D). An efficient resolution-of-the-identity (RI) implementation of this approach is described. The second approach employs a single parameter to scale only the opposite-spin direct term of CIS(D), starting from the one-parameter scaled opposite-spin (SOS) MP2 ground state, and is called SOS-CIS(D). By utilizing auxiliary basis expansions and a Laplace transform, a fourth-order algorithm for SOS-CIS(D) is described and implemented. The parameters that describe SCS-CIS(D) and SOS-CIS(D) are optimized based on a training set that includes valence excitations of various organic molecules and Rydberg transitions of water and ammonia, and they significantly improve upon CIS(D) itself. The accuracy of the two methods is found to be comparable. This arises from a strong correlation between the same-spin and the opposite-spin portions of the excitation energy terms. The methods are successfully applied to the zincbacteriochlorin-bacteriochlorin charge-transfer transition, for which time-dependent density functional theory, with presently available exchange-correlation functionals, is known to fail. The methods are also successfully applied to describe various electronic transitions outside of the training set. The efficiency of the SOS-CIS(D) and the auxiliary basis implementation of CIS(D) and SCS-CIS(D) are confirmed with a series of timing tests.     

Singly and doubly excited states of butadiene, acrolein, and glyoxal: Geometries and electronic spectra

Excited-state geometries and electronic spectra of butadiene, acrolein, and glyoxal have been investigated by the symmetry adapted cluster configuration interaction (SAC-CI) method in their s-trans conformation. Valence and Rydberg states below the ionization threshold have been precisely calculated with sufficiently flexible basis sets. Vertical and adiabatic excitation energies were well reproduced and the detailed assignments were given taking account of the second moments. The deviations of the vertical excitation energies from the experiment were less than 0.3 eV for all cases. The SAC-CI geometry optimization has been applied to some valence and Rydberg excited states of these molecules in the planar structure. The optimized ground- and excited-state geometries agree well with the available experimental values; deviations lie within 0.03 A and 0.7 degrees for the bond lengths and angles, respectively. The force acting on the nuclei caused by the excitations has been discussed in detail by calculating the SAC-CI electron density difference between the ground and excited states; the geometry relaxation was well interpreted with the electrostatic force theory. In Rydberg excitations, geometry changes were also noticed. Doubly excited states (so-called 2 (1)A(g) states) were investigated by the SAC-CI general-R method considering up to quadruple excitations. The characteristic geometrical changes and large energetic relaxations were predicted for these states.     

Model for atomic multiphoton ionization via Rydberg states

A model to treat multiphoton ionization through Rydberg states, taking into account blackbody radiation and other processes which lead to ionization with transfer of excitation among the closely-spaced highly excited Rydberg states is presented. The basic idea is to make the rate equations tractable by compacting all of the Rydberg states that experience physically similar processes into a single “bath” in the equations. The model is applied to sodium Rydberg atoms and the predictions show excellent agreement with experiments.     

Femtosecond photoelectron imaging of transient electronic states and Rydberg atom emission from electronically excited he droplets

Ultrafast relaxation of electronically excited pure He droplets is investigated by femtosecond time-resolved photoelectron imaging. Droplets are excited by extreme ultraviolet (EUV) pulses with photon energies below 24 eV. Excited states and relaxation products are probed by ionization with an infrared (IR) pulse with 1.6 eV photon energy. An initially excited droplet state decays on a time scale of 220 fs, leading predominantly to the emission of unaligned 1s3d Rydberg atoms. In a second relaxation channel, electronically aligned 1s4p Rydberg atoms are emitted from the droplet within less than 120 fs. The experimental results are described within a model that approximates electronically excited droplet states by localized, atomic Rydberg states perturbed by the local droplet environment in which the atom is embedded. The model suggests that, below 24 eV, EUV excitation preferentially leads to states that are localized in the surface region of the droplet. Electronically aligned 1s4p Rydberg atoms are expected to originate from excitations in the outermost surface regions, while nonaligned 1s3d Rydberg atoms emerge from a deeper surface region with higher local densities. The model is used to simulate the He droplet EUV absorption spectrum in good agreement with previously reported fluorescence excitation measurements.     

Full configuration interaction calculation of the low lying valence and Rydberg states of BeH

The all-electron full configuration interaction (FCI) vertical excitation energies for some low lying valence and Rydberg excited states of BeH are presented in this article. A basis set of valence atomic natural orbitals has been augmented with a series of Rydberg orbitals that have been generated as centered onto the Be atom. The resulting basis set can be described as 4s2p1d/2s1p (Be/H) + 4s4p3d. It allows to calculate Rydberg states up to n= {3,4,5} of the s, p, and d series of Rydberg states. The FCI vertical ionization potential for the same basis set and geometry amounts to 8.298 eV. Other properties such as FCI electric dipole and quadrupole moments and FCI transition dipole and quadrupole moments have also been calculated. The results provide a set of benchmark values for energies, wave functions, properties, and transition properties for the five electron BeH molecule. Most of the states have large multiconfigurational character in spite of their essentially single excited nature and a number of them present an important Rydberg-valence mixing that is achieved through the mixed nature of the particle MO of the single excitations.     

Comments about the representation of Rydberg and ionic excited states and their photochemistry

This paper discusses the conditions for representing Rydberg and ionic excited states of molecules. It especially shows the intrinsic difficulties of MO methods to treat the weak resonances between strongly polarized situations in highly polarizable symmetric systems; such situations occur in the long distance region for Rydberg excited states of homonuclear molecules, and for the 90 ° twisted singlet excited states of polyenes. The valence/Rydberg mixing is discussed, and some principles for the understanding of Rydberg photochemistry are proprosed, based on a few examples. The present knowledge of the photochemistry of zwitterionic excited states of polyenes is summarized.     

Observation of new Rydberg series in the many-electron transition region of N(2)

Fluorescence excitation spectra produced through photoexcitation of N(2) using synchrotron radiation in the spectral region between 50 and 62.5 nm have been obtained with a resolution of 0.004?nm. A broadband detector (in the 115-180 nm region) was employed to monitor fluorescence originated from neutral excited atomic nitrogen fragments which are produced through direct dissociation processes and predissociation from the well-known many-electron excited Rydberg states. We have identified a new Rydberg series (2 (2)Π(g)) 4sσ, a better resolved Rydberg (D (2)Π(g)) npσ series, and also the prominent Codling series converging to the D (2)Π(g), and C (2)Σ(u) (+) states of N(2) (+), respectively. By normalizing our relative fluorescence intensities to previously measured absolute fluorescence cross-section data we obtain the cross-section data of undispersed fluorescence in the 115-180?nm region. The fluorescence quantum yields for the present photodissociative excitation processes are found to be less than 0.05. The present results may provide important data for our understanding of competitions among the various decay channels of the many-electron transition states of N(2).     

Electronic states of the benzene dimer: a simple case of complexity

Electronic structure calculations of the excited states of the benzene dimer using equation-of-motion coupled-cluster method are reported. The calculations reveal large density of electronic states, including multiple valence, Rydberg, and mixed Rydberg-valence states. The calculations of the oscillator strengths for the transitions between the excimer state (i.e., the lowest excited state of the dimer, 1(1)B(1g)) and other excited states allowed us to identify the target state responsible for the excimer absorption as the E(1u) state of a mixed Rydberg-valence character at 3.04 eV above the excimer (1(1)B(1g)). Although at D(6h) the 1(1)B(1g) → E(1u) transition is symmetry-forbidden, small geometric displacements (to D(2h)) that have a negligible effect on the excitation energy split this degenerate state into the dark (4B(3u)) and bright (4B(2u)) components (oscillator strength of 0.3 au). The excitation energy for this transition depends strongly on the dimer structure, which explains the broad character of the experimentally observed excimer absorption spectrum.     

Spectroscopy of nS, nP, and nD Rydberg series of Cs atoms on helium nanodroplets

The preparation of an artificial superatom consisting of a positive charge inside a superfluid helium nanodroplet and an electron in an orbital surrounding the droplet is of fundamental interest and represents an experimental challenge. In this work, nanodroplets of several thousand helium atoms are doped with single caesium (Cs) atoms. While on the droplet, the Cs valence electron is excited in two steps through an intermediate state into nS, nP, and nD states. The excitation is monitored by laser induced fluorescence or, for high principal quantum numbers, by resonant three-photon-ionization. On-droplet Rydberg excitations are resolved up to about n = 20. The energies are compared with those of free Cs atom Rydberg states and quantum defects as well as the on-droplet ionization threshold are derived.     

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.     

Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3

A benchmark set of 28 medium-sized organic molecules is assembled that covers the most important classes of chromophores including polyenes and other unsaturated aliphatic compounds, aromatic hydrocarbons, heterocycles, carbonyl compounds, and nucleobases. Vertical excitation energies and one-electron properties are computed for the valence excited states of these molecules using both multiconfigurational second-order perturbation theory, CASPT2, and a hierarchy of coupled cluster methods, CC2, CCSD, and CC3. The calculations are done at identical geometries (MP26-31G*) and with the same basis set (TZVP). In most cases, the CC3 results are very close to the CASPT2 results, whereas there are larger deviations with CC2 and CCSD, especially in singlet excited states that are not dominated by single excitations. Statistical evaluations of the calculated vertical excitation energies for 223 states are presented and discussed in order to assess the relative merits of the applied methods. CC2 reproduces the CC3 reference data for the singlets better than CCSD. On the basis of the current computational results and an extensive survey of the literature, we propose best estimates for the energies of 104 singlet and 63 triplet excited states.     

Theoretical prediction of the electronic excited states and resonance Raman intensities in formamide from coupled cluster calculations

Electronic excitations and the resonance Raman spectrum of formamide were obtained from ab initio electron correlation calculations using the equation of motion coupled cluster (EOM-CCSD) method. Interpretation of the UV spectrum on the basis of calculated vertical excitation energies and oscillator strengths accounts for all experimental bands previously assigned. Our assignment, however, suggests an additional Rydberg band at about 7.4 eV which may be hidden under the main absorption. We also show that the Rydberg states appear pairwise, corresponding to n and π hole states, respectively. Using analytic derivative techniques, derivatives of the excited state energies with respect to normal coordinates of the ground state were calculated. Approximate resonance Raman intensities have been determined.     

The electronic states of 1,2,3-triazole studied by vacuum ultraviolet photoabsorption and ultraviolet photoelectron spectroscopy, and a comparison with ab initio configuration interaction methods

The Rydberg states in the vacuum ultraviolet photoabsorption spectrum of 1,2,3-triazole have been measured and analyzed with the aid of comparison to the UV valence photoelectron ionizations and the results of ab initio configuration interaction (CI) calculations. Calculated electronic ionization and excitation energies for singlet, triplet valence, and Rydberg states were obtained using multireference multiroot CI procedures with an aug-cc-pVTZ [5s3p3d1f] basis set and a set of Rydberg [4s3p3d3f] functions. Adiabatic excitation energies obtained for several electronic states using coupled-cluster (singles, doubles, and triples) and complete active space self-consistent field procedures agree well with experimental values. Variations in bond lengths with the electronic state are discussed. The lowest energy UV band (～5.5-6.5 eV) is assigned to three electronically excited states and demonstrates the occurrence of a nonplanar upper state on the low energy side. A UV photoelectron spectrum with an improved resolution yielded adiabatic and vertical ionization energies and reorganization energies for several of the lowest cationic states. As well as excitations to the s, p, d-Rydberg states are the excitations consistent with an f-series.     

Dipole-bound anions of highly polar molecules: ethylene carbonate and vinylene carbonate

Results of experimental and theoretical studies of dipole-bound negative ions of the highly polar molecules ethylene carbonate (EC, C3H4O3, mu=5.35 D) and vinylene carbonate (VC, C3H2O3, mu=4.55 D) are presented. These negative ions are prepared in Rydberg electron transfer (RET) reactions in which rubidium (Rb) atoms, excited to ns or nd Rydberg states, collide with EC or VC molecules to produce EC- or VC- ions. In both cases ions are produced only when the Rb atoms are excited to states described by a relatively narrow range of effective principal quantum numbers, n*; the greatest yields of EC- and VC- are obtained for n*(max)=9.0+/-0.5 and 11.6+/-0.5, respectively. Charge transfer from low-lying Rydberg states of Rb is characteristic of a large excess electron binding energy (Eb) of the neutral parent; employing the previously derived empirical relationship Eb=23/n*(max)(2.8) eV, the electron binding energies are estimated to be 49+/-8 meV for EC and 24+/-3 meV for VC. Electron photodetachment studies of EC- show that the excess electron is bound by 49+/-5 meV, in excellent agreement with the RET results, lending credibility to the empirical relationship between Eb and n*(max). Vertical electron affinities for EC and VC are computed employing aug-cc-pVDZ atom-centered basis sets supplemented with a (5s5p) set of diffuse Gaussian primitives to support the dipole-bound electron; at the CCSD(T) level of theory the computed electron affinities are 40.9 and 20.1 meV for EC and VC, respectively.     

Alignment, vibronic level splitting, and coherent coupling effects on the pump-probe polarization anisotropy

The pump-probe polarization anisotropy is computed for molecules with a nondegenerate ground state, two degenerate or nearly degenerate excited states with perpendicular transition dipoles, and no resonant excited-state absorption. Including finite pulse effects, the initial polarization anisotropy at zero pump-probe delay is predicted to be r(0) = 3/10 with coherent excitation. During pulse overlap, it is shown that the four-wave mixing classification of signal pathways as ground or excited state is not useful for pump-probe signals. Therefore, a reclassification useful for pump-probe experiments is proposed, and the coherent anisotropy is discussed in terms of a more general transition dipole and molecular axis alignment instead of experiment-dependent ground- versus excited-state pathways. Although coherent excitation enhances alignment of the transition dipole, the molecular axes are less aligned than for a single dipole transition, lowering the initial anisotropy. As the splitting between excited states increases beyond the laser bandwidth and absorption line width, the initial anisotropy increases from 3/10 to 4/10. Asymmetric vibrational coordinates that lift the degeneracy control the electronic energy gap and off-diagonal coupling between electronic states. These vibrations dephase coherence and equilibrate the populations of the (nearly) degenerate states, causing the anisotropy to decay (possibly with oscillations) to 1/10. Small amounts of asymmetric inhomogeneity (2 cm(-1)) cause rapid (130 fs) suppression of both vibrational and electronic anisotropy beats on the excited state, but not vibrational beats on the ground electronic state. Recent measurements of conical intersection dynamics in a silicon napthalocyanine revealed anisotropic quantum beats that had to be assigned to asymmetric vibrations on the ground electronic state only [Farrow, D. A.; J. Chem. Phys. 2008, 128, 144510]. Small environmental asymmetries likely explain the observed absence of excited-state asymmetric vibrations in those experiments.     

A Hilbert space multi-reference coupled-cluster study of the H4 model system

Summary Employing the Hilbert space ansatz, a fully quadratic coupled-cluster method with a multidimensional reference space is applied to a DZP basis study of the model system, H₄. The reference space is described by two to four configurations at the level of single and double excitations, and single and double excitation operators are included in the expansions for the cluster and wave operator through quadratic terms. The performance of quadratic MRCCSD is investigated for the ground and three excited states of the H₄ system consisting of two stretched hydrogen molecules in a trapezoidal configuration where the degree of quasidegeneracy is varied from a nondegenerate situation to a completely degenerate one. Compared to full CI, in the highly degenerate region, the MRCCSD works quite well. In less degenerate regions, the accuracy is less satisfactory.     

Theoretical prediction of valence and Rydberg excited states: Minnesota exchange-correlation functionals vs symmetry adapted cluster-configuration interaction

During this contribution, we present a benchmark investigation on the applicability of several Minnesota functionals from various classes like local meta-generalized and meta-nonseparable gradient approximations, hybrids, and range-separated hybrids for describing the valence and Rydberg excitation energies of some organic compounds from different categories. Furthermore, the performances of Minnesota density functionals from density functional theory are also assessed against a wave function theory based approach in the context of excite states calculations, symmetry adapted cluster-configuration interaction (SAC-CI) method. Pragmatically, the singles and doubles linked excitation operators are considered in the SAC-CI wave functions. With more or less different accountabilities of the considered methods, it is shown that the M06-2X, M05-2X, and M11 functionals have the best performances for valence excited states. On the other hand, for Rydberg excited states although the SAC-CI method outperforms others, the statistical analyses reveal that the efficiency of some Minnesota functionals is also respectable.     

Jahn-Teller effect in Rydberg series: a multi-state vibronic coupling problem

Two simple limiting cases of Jahn-Teller (JT) coupling in Rydberg states of polyatomic molecules are considered, namely(i) JT coupling in Rydberg orbitals as well as in the ionization continuum (nondegenerate ion core, degenerate Rydberg series) and(ii) JT coupling in the ion core (degenerate ion core, nondegenerate Rydberg series). For both models simple and efficient algorithms for the computation of spectra (dynamical JT effect) are developed. The orbital JT effect is shown to represent a novel type of multi-state vibronic coupling, giving rise to interesting spectroscopic phenomena, among them resonant inter-Rydberg perturbations and JT induced autoionization. Particular attention is paid to the demonstration of the characteristic spectroscopic signatures of the two types of JT coupling in Rydberg states.     

Electronic excited states of Si(100) and organic molecules adsorbed on Si(100)

The electronically excited states of the Si(100) surface and acetylene, benzene, and 9,10-phenanthrenequinone adsorbed on Si(100) are studied with time-dependent density functional theory. The computational cost of these calculations can be reduced through truncation of the single excitation space. This allows larger cluster models of the surface in conjunction with large adsorbates to be studied. On clean Si(100), the low-lying excitations correspond to transitions between the pi orbitals of the silicon-silicon dimers. These excitations are predicted to occur in the range 0.4-2 eV. When organic molecules are adsorbed on the surface, surface --> molecule, molecule --> surface, and electronic excitations localized within the adsorbate are also observed at higher energies. For acetylene and benzene, the remaining pipi* excitations are found to lie at lower energies than in the corresponding gas-phase species. Even though the aromaticity of 9,10-phenanthrenequinone is retained, significant shifts in the pipi* excitations of the aromatic rings are predicted. This is in part due to structural changes that occur upon adsorption.     

Double-hybrid density functional theory for excited electronic states of molecules

Double-hybrid density functionals are based on a mixing of standard generalized gradient approximations (GGAs) for exchange and correlation with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional (dubbed B2PLYP) contains only two empirical parameters that describe the mixture of HF and GGA exchange (ax) and of the PT2 and GGA correlation (ac), respectively. Extensive testing has recently demonstrated the outstanding accuracy of this approach for various ground state problems in general chemistry applications. The method is extended here without any further empirical adjustments to electronically excited states in the framework of time-dependent density functional theory (TD-DFT) or the closely related Tamm-Dancoff approximation (TDA-DFT). In complete analogy to the ground state treatment, a scaled second-order perturbation correction to configuration interaction with singles (CIS(D)) wave functions developed some years ago by Head-Gordon et al. [Chem. Phys. Lett. 219, 21 (1994)] is computed on the basis of density functional data and added to the TD(A)-DFTGGA excitation energy. The method is implemented by applying the resolution of the identity approximation and the efficiency of the code is discussed. Extensive tests for a wide variety of molecules and excited states (of singlet, triplet, and doublet multiplicities) including electronic spectra are presented. In general, rather accurate excitation energies (deviations from reference data typically <0.2 eV) are obtained that are mostly better than those from standard functionals. Still, systematic errors are obtained for Rydberg (too low on average by about 0.3 eV) and charge-transfer transitions but due to the relatively large ax parameter (0.53), B2PLYP outperforms most other functionals in this respect. Compared to conventional HF-based CIS(D), the method is more robust in electronically complex situations due to the implicit account of static correlation effects by the GGA parts. The (D) correction often works in the right direction and compensates for the overestimation of the transition energy at the TD level due to the elevated fraction of HF exchange in the hybrid GGA part. Finally, the limitations of the method are discussed for challenging systems such as transition metal complexes, cyanine dyes, and multireference cases.