A study of dynamic quadrupolar and octupolar excitations and calculations on excited d and f states of atom and ions: He-sequence

The quadrupolar and octupolar distortion of the ions in the He-sequence caused by an external electro–magnetic field has been studied by a variation–perturbation method in the Hartree–Fock scheme. For certain frequencies singularities appear in the response of the system to the perturbation. Approximate representations for the excited d and f states have been obtained from a study of these resonances. Such a perturbation calculation has the advantage that representations of the different excited states are obtained independently. The orthogonality to all the lower lying levels of the same symmetry is not required. The only source of inaccuracy implicit in the procedure lies in the improper consideration of the inter-electronic interaction. This is corrected for by an independent calculation, which is again formulated in terms of a perturbation treatment. The resulting wave functions for the excited states are accurate in the Hartree–Fock model. Expectation values of several operators have been calculated with these corrected wave functions.     

SCF MO CI perturbation calculations of inner shell and valence shell vertical ionization potentials

A CI method for calculating inner and valence shell vertical ionization potentials is presented. It is based on ab initio SCF MO calculations for the neutral closedshell ground state followed by CI perturbation calculations for the ground and ion states including all spin and symmetry adapted singly and doubly excited configurations with respect to the main configurations of the state of interest. The state energy is computed by performing a CI calculation for a set of selected configurations, and then adding the contributions of the remaining configurations as estimated by second order Brillouin-Wigner perturbation theory. The use of the same set of MO's for all states together with the CI perturbation method makes the method rather rapid. The numerical results are, in spite of the limited Gaussian basis sets used, in good agreement with experiment.     

A practical valence bond method: a configuration interaction method approach with perturbation theoretic facility

The previously developed valence bond configuration interaction (VBCI) method (Wu, W.; Song, L.; Cao, Z.; Zhang, Q.; Shaik, S., J. Phys. Chem. A, 2002, 105, 2721) that borrows the general CI philosophy of the MO theory, is further extended in this article, and its methodological features are improved, resulting in three accurate and cost-effective procedures: (a) the effect of quadruplet excitation is incorporated using the Davidson correction, such that the new procedure reduces size consistency problems, with due improvement in the quality of the computational results. (b) A cost-effective procedure, named VBCI(D, S), is introduced. It includes doubly excited structures for active electrons and singly excited structures for inactive pairs. The computational results of VBCI(D, S) match those of VBCISD with much less computational effort than VBCISD. (c) Finally, a second-order perturbation theory is utilized as a means of configuration selection, and lead to considerable reduction of the computational cost, with little or no loss in accuracy. Applications of the new procedures to bond energies and barriers of chemical reactions are presented and discussed.     

Excitation Energies of Canonical Nucleobases Computed by Multiconfigurational Perturbation Theories

In this computational work, we assessed the performance of ab initio multireference (MR) methods for the calculation of vertical excitation energies of five nucleobases: adenine, guanine, cytosine, thymine and uracil. In total, we have studied 38 singlet and 30 triplet excited states. Where possible we used the multireference configuration interaction (MRCI) method as a reference for various flavors of multireference perturbation theory to second order. In particular, we have benchmarked CASPT2, NEVPT2 and XMCQDPT2. For CASPT2, we have analyzed the single‐state, multistate (MS) and extended MS variants. In addition, we have assessed the effect of the ionization potential electron affinity (IPEA) shift. For NEVPT2, we have used the partially and the strongly contracted variants. Further, we have tested the commonly used RI‐CC2, RI‐ADC2 and EOM‐CCSD methods. Generally, we observe the following trends for singlet excited states: NEVPT2 is the closest MR method to MRCISD+Q, closely followed by CASPT2 with the default IPEA shift. The same trend is observed for triplet states, although NEVPT2 and CASPT2‐IPEA are getting closer. Interestingly, the n, π* singlet excited states were described more accurately than π, π* excited states, while for triplet states the trend is inverted except for NEVPT2. This work is an important benchmark for future photochemical investigations.     

Combined DFT and wave function theory approach to excited states of lanthanide luminescent materials: A case study of LaF3:Ce3+

Lanthanide luminescent materials play key roles in modern society, but their first-principles treatment remains a great challenge due to complex manifold of electronic excited states and the difficulty in performing excited state structural relaxations that is necessary to model luminescent properties. Herein, we propose a practical approach that combines embedded cluster model (ECM) based multi-configurational wave function theory (WFT) and occupancy constrained density-functional theory plus the Hubbard U correction (OC-DFT + U) to treat lanthanide doped luminescent materials, using LaF₃:Ce³⁺, a typical scintillator with low symmetry, as a case study. We show that the combined approach yields accurate absorption energies with an error on the order of 200 cm⁻¹, but the emission energies are significantly underestimated, the origin of which is further clarified by vibrationally resolved absorption and emission spectra calculation. This work demonstrates the possibility of combining ECM-based wave function theory and periodic DFT into a comprehensive computational scheme for lanthanide luminescent materials and highlights the limitations of the current implementation of OC-DFT + U for excited state structural optimization.     

Electronic structure of the ground state of the benzyl radical

Results from the -electron approximation are given for the total energy, spin-density distribution, and electron density, which have been derived by configuration interaction via self-consistent orbitals for closed and open shells and with allowance for all singly excited configurations and some doubly excited ones. Some singly excited configurations do not mix with the ground-state configuration in the first order of perturbation theory but make a contribution to the latter greater than do some of the configurations that do mix. Incorporation of all singly excited configurations results in a lower ground-state energy if orbitals for a closed shell are used instead of those for an open one. Calculations via closed-shell orbitals lead to an inhomogeneous electron-density distribution, which is gradually smoothed out as the set of configurations is expanded. The spin-density distribution is very much dependent on the number of configurations used and on the orbitals employed in them.     

Hydrocarbon clusters from a foil diffusion source

Rydberg states and clusters of Rydberg states have been reported in several cases from high temperature sources, so-called diffusion sources. The present study uses the same technique as the one used for the study of excited Cs clusters (Åmanet al., 1990), and is aimed at highly excited Rydberg cluster formation from hydrogen containing molecules. Ionized clusters from a diffusion source with the graphite foil emitter at 1400 K are studied by time-of-flight mass spectrometry. The excited clusters are shown to be ionized by field ionization. The best results are found using ethylene in the source, and the flux from the source probably contains both hydrogen and hydrocarbon species. Typical clusters have a mass of 10,000–200,000 a.u., assuming singly charged clusters. The formation and stabilization (cooling) of such large clusters under the present conditions is only possible since excited states can form a condensed phase, of so called Rydberg matter (Manykinet al., 1981, 1982, Petterssonet al., 1992, Svenssonet al., 1991, 1992). The importance of excited hydrogen containing clusters for the chemistry and physics of interstellar space is pointed out.     

A multi-configuration LCAO–MO study for complex unsaturated molecules. II. Application to the benzene cation

The method of the MC –LCAO –MO approach, described in the preceding paper, is further applied to the benzene cation. Through the iteration process the π-electron energies and the molecular shapes are computed for the ground and two lowest excited states of the cation in both D_6h and D_2h geometries. A remarkable fact obtained is that a comparatively small variation of the geometrical structure (c. 0.010 – 0.013 Å bond length difference) brings about a considerable change of the energy value (c. 0.85 – 1.25 eV). The π-electronic excitation energies obtained from the iteration process are compared with the transition energies calculated from the usual method in which the structures of the excited states are assumed to be the same as the corresponding ground state structures. The difference in the excitation energy between the cation and the anion, and the CI effect on the excited states, are discussed. It is found that the doubly excited configurations play an important role in CI , which is somewhat different from that of the singly excited configurations. The stabilization energy due to the Jahn–Teller distortion is estimated for the ground state of the cation.     

Full configuration interaction for the benzyl radical

The electronic structure of the benzyl radical in its ground state has been computed using a model Hamiltonian due to Pariser–Parr with full configuration interaction as well as with different truncated configurational sets built on SCF open-shell orbitals. The correlation energy corresponding to this model was found to be equal to –0.929722 eV. With the singly excited configurations only 18% of this energy is taken into account. By extending the basis to include the doubly excited configurations one can account for 94% of the correlation energy. An analysis of the accuracy of the proton hyperfine splitting calculation caused by inaccurate computation of the wave function is given. If only singly and even doubly excited configurations are taken into account one cannot hope to obtain splittings with an accuracy of more than 0.5 g. Inclusion of triply excited configurations lowers this error by one order. In addition, the use of the simple McConnell relation may lead to an error in splitting calculations of no less than 1.5 g.     

Photophysics of Schiff Bases: Theoretical Study of Salicylidene Methylamine

The proton‐transfer reaction in a model aromatic Schiff base, salicylidene methylamine (SMA), in the ground and in the lowest electronically‐excited singlet states, is theoretically analyzed with the aid of second‐order approximate coupled‐cluster model CC2, time‐dependent density functional theory (TD‐DFT) using the Becke, three‐parameter Lee–Yang–Parr (B3LYP) functional, and complete active space perturbation theory CASPT2 electronic structure methods. Computed vertical‐absorption spectra for the stable ground‐state isomers of SMA fully confirm the photochromism of SMA. The potential‐energy profiles of the ground and the lowest excited singlet state are calculated and four photophysically relevant isomeric forms of SMA; α, β, γ, and δ are discussed. The calculations indicate two S₁/S₀ conical intersections which provide non‐adiabatic gates for a radiationless decay to the ground state. The photophysical scheme which emerges from the theoretical study is related to recent experimental results obtained for SMA and its derivatives in the low‐temperature argon matrices (J. Grzegorzek, A. Filarowski, Z. Mielke, Phys. Chem. Chem. Phys. 2011 , 13, 16596–16605). Our results suggest that aromatic Schiff bases are potential candidates for optically driven molecular switches.     

Efficient generation of Heisenberg Hamiltonian matrices for VB calculations of potential energy surfaces

The spin‐Hamiltonian valence bond theory relies upon covalent configurations formed by singly occupied orbitals differing by their spin counterparts. This theory has been proven to be successful in studying potential energy surfaces of the ground and lowest excited states in organic molecules when used as a part of the hybrid molecular mechanics—valence bond method. The method allows one to consider systems with large active spaces formed by n electrons in n orbitals and relies upon a specially proposed graphical unitary group approach. At the same time, the restriction of the equality of the numbers of electrons and orbitals in the active space is too severe: it excludes from the consideration a lot of interesting applications. We can mention here carbocations and systems with heteroatoms. Moreover, the structure of the method makes it difficult to study charge‐transfer excited states because they are formed by ionic configurations. In the present work we tackle these problems by significant extension of the spin‐Hamiltonian approach. We consider (i) more general active space formed by n ± m electrons in n orbitals and (ii) states with the charge transfer. The main problem addressed is the generation of Hamiltonian matrices for these general cases. We propose a scheme combining operators of electron exchange and hopping, generating all nonzero matrix elements step‐by‐step. This scheme provides a very efficient way to generate the Hamiltonians, thus extending the applicability of spin‐Hamiltonian valence bond theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Analytic energy derivatives for the equation-of-motion coupled-cluster method: Algebraic expressions,implementation and application to theS 1 state of HFCO

Summary The general theory of analytic derivatives for the equation-of-motion coupled cluster (EOM-CC) method is reviewed. Special attention is paid to the EOM-CC singles and doubles (EOM-CCSD) approximation, which has the same computational scaling properties as the coupled-cluster singles doubles (CCSD) ground state method and is therefore applicable to a wide range of molecular systems. The detailed spin orbital equations that must be solved in EOM-CCSD gradient calculations are presented for the first time, and some guidelines are discussed regarding their computational implementation. Finally, use of the EOM-CCSD gradient method is illustrated by determining the structure, dipole moment components, harmonic frequencies and infrared intensities of formyl fluoride (HFCO) in its singlet excited (n, *) state.     

Characterizing the excited states of large photoactive systems by exciton models

The theoretical investigation of excited state for large photoactive systems plays the fundamental role in understanding various optical processes in material and biological system. Frenkel exciton (FE) model describing the excitation of the whole system as a collective effect of quasi-particles of excitons, that is, bound electron–hole pairs, is well-known as a simple but powerful theoretical scheme to present a clear and insightful physical picture for complicated excited state problems. In this mini-review, we summarize our recent developments of quantum chemical methods based on exciton models and their related applications for large photoactive systems. It is shown that our developed ab initio renormalized exciton model (REM) and block interaction product state (BIPS) schemes provide new efficient and automatic low-scaling excited state methods for both localized and delocalized excited states in large systems. Illustrative examples including simulations of both absorption and emission spectrum in large sized molecular aggregates, indicate the exciton model based methods provide promising computational tools for unravel the mechanism of photophysical and photochemical processes in large photoactive systems.     

Assessment and acceleration of binding energy calculations for protein–ligand complexes by the fragment molecular orbital method

In the field of drug discovery, it is important to accurately predict the binding affinities between target proteins and drug applicant molecules. Many of the computational methods available for evaluating binding affinities have adopted molecular mechanics‐based force fields, although they cannot fully describe protein–ligand interactions. A noteworthy computational method in development involves large‐scale electronic structure calculations. Fragment molecular orbital (FMO) method, which is one of such large‐scale calculation techniques, is applied in this study for calculating the binding energies between proteins and ligands. By testing the effects of specific FMO calculation conditions (including fragmentation size, basis sets, electron correlation, exchange‐correlation functionals, and solvation effects) on the binding energies of the FK506‐binding protein and 10 ligand complex molecule, we have found that the standard FMO calculation condition, FMO2‐MP2/6‐31G(d), is suitable for evaluating the protein–ligand interactions. The correlation coefficient between the binding energies calculated with this FMO calculation condition and experimental values is determined to be R = 0.77. Based on these results, we also propose a practical scheme for predicting binding affinities by combining the FMO method with the quantitative structure–activity relationship (QSAR) model. The results of this combined method can be directly compared with experimental binding affinities. The FMO and QSAR combined scheme shows a higher correlation with experimental data (R = 0.91). Furthermore, we propose an acceleration scheme for the binding energy calculations using a multilayer FMO method focusing on the protein–ligand interaction distance. Our acceleration scheme, which uses FMO2‐HF/STO‐3G:MP2/6‐31G(d) at R_int = 7.0 Å, reduces computational costs, while maintaining accuracy in the evaluation of binding energy. © 2015 Wiley Periodicals, Inc.     

The configuration interaction method,and the triplet-singlet splitting in CH2

A configuration-interaction (CI) method in which the interaction matrix is never constructed has been investigated, following the original suggestion of Roos. Two methods have been used (1) for singlet states, which can be represented by a one determinant configuration of doubly occupied orbitals, CI with all singly and doubly excited configurations, (2) for states for which the restricted self-consistent field approximation is a single determinant, CI with all singly and doubly excited determinants. In case (2), the wavefunction may not be exactly an eigenfunction of S ². The methods were investigated using a double-zeta plus polarisation basis for CH₂. Both methods must give the same result for the lowest singlet ground state. Keeping the bond length fixed at 2.10 and 2.04 bohr respectively the bond angle for the singlet and triplet were found to be 100.8 ° and 132.0 °, with energies ?39.0312 a.u. and ?39.0563 a.u. respectively. These are the lowest variational energies obtained for these systems; the singlet-triplet splitting is thus predicted to be 15.4 kcal/mol.     

Ab initio calculations for the potential curves and spin–orbit coupling of Mg2

 The ground state and several low-lying excited states of the Mg₂ dimer have been studied by means of a combination of the complete-active-space multiconfiguration self-consistent-field (CASSCF)/CAS multireference second-order perturbation theory (CASPT2) method and coupled-cluster with single and double excitations and perturbative contribution of connected triple excitations [CCSD(T)] scheme. Reasonably good agreement with experiment has been obtained for the CCSD(T) ground-state potential curve but the dissociation energy of the only experimentally known A¹Σ _u ⁺ excited state of Mg₂ is somewhat overestimated at the CASSCF/CASPT2 level. The spectroscopic constants D _e, R _e and ω_e deduced from the calculated potential curves for other states are also reported. In addition, some spin–orbit matrix elements between the excited singlet and triplet states of Mg₂ have been evaluated as a function of internuclear separation. Received: 10 May 2001 / Accepted: 15 August 2001 / Published online: 30 October 2001     

The shifted scheme in the general-model-space diagrammatic perturbation theory

The shifted scheme of many-body perturbation theory is applied to open-shell states within the framework of the general-model-space theory. Rules for shifting the denominators of folded diagrams. which appear in open-shell perturbation expansions, are given. The finite-order energies in the shifted scheme obtained in two equivalent representations may differ. This happens, for instance, in the case of triplet states. For ³Σ_u⁺ states of the He₂, differences up to 0.07 mhartree have been found in third order. A similar phenomenon is the size inconsistency of the shifted scheme observed by Silver in the ground state of He₂. A possible advantage of the shifted scheme is its faster convergence for excited states.     

Doubly excited3Se,3De and3Ge states of two-electron atomic systems

Summary Time-dependent perturbation theory has been applied to calculate the doubly excited triplet statesNsns:³S^e,Npnp:³D^e andNdnd:³G^e (N=2, 3, 4,n=N+1, ... ,5) for He, Li⁺, Be²⁺ and B³⁺. A time-dependent harmonic perturbation causes simulataneous excitation of both the electrons with a change of spin state. The doubly excited energy levels have been identified as the poles of an appropriately constructed linearized variational functional with respect to the driving frequency. In addition to the transition energies, effective quantum numbers of these doubly excited states have been calculated and analytic representations of their wave functions are obtained. These are utilized to estimate the Coulomb repulsion term for these states which checks the consistency of the wave functions. These wave functions may also be used for calculating other physical properties of the systems.     

An application of second-order n-electron valence state perturbation theory to the calculation of excited states

n–electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory where all the zero-order wave functions are of multireference nature, being generated as eigenfunctions of a two–electron model Hamiltonian. The absence of intruder states makes NEVPT an interesting choice for the calculation of electronically excited states. Test calculations have been performed on several valence and Rydberg transitions for the formaldehyde and acetone molecules; the results are in good accordance with the best calculations and with the existing experimental data.Contribution to the Jacopo Tomasi Honorary Issue