Quasi-degenerate Perturbation Theory Research Articles

Systematic Investigation of Electronic States and Bond Properties of LnO, LnO+, LnS, and LnS+ (Ln = La-Lu) by Spin-Orbit Multiconfiguration Perturbation Theory.

The electronic structures of lanthanide monoxides (LnO/LnO+) and monosulfides (LnS/LnS+) for all lanthanide series elements (Ln = La-Lu) have been systematically analyzed with sophisticated quantum chemical calculations. The ground electronic configuration has been determined to be Ln 4fn6s1 or 4fn+1 for the neutral molecules and Ln 4fn for the cations. The low-lying energy states resulting from spin-orbit coupling and ligand field effects have been resolved using spin-orbit multiconfiguration quasi-degenerate second-order perturbation theory calculations. The ionization energies of LnO (5.20-7.06 eV) are about 0.3-2.2 eV lower than those of LnS (5.54-9.22 eV) due to the difference in the Ln 6s and 4f orbital energies from which an electron is removed during the ionization process. The bond dissociation energies (BDEs) have been computed by the state-averaged general multiconfigurational perturbation theory and the completely renormalized coupled-cluster [CR-CC(2,3)] methods. The BDEs are highly dependent on the lanthanide elements as several factors of the lanthanides affect the bond dissociation. The calculated bond lengths and energies agree well with available experimental values and are systematically predicted for the series of lanthanide monoxides and monosulfides where experimental values are not available. Furthermore, the LS terms of low-lying energy states and their corresponding bond properties have been clarified in detail to systematize the similarities and differences of the lanthanide compounds.

Theoretical study on stacking distance dependence of one- and two-photon absorption properties of phenalenyl π-dimer models

Abstract Stacking distance (d) dependence of the peak positions and intensities of one-photon absorption (OPA) and two-photon absorption (TPA) spectra of phenalenyl π-dimer models was investigated theoretically. Excitation energies and transition dipole moments required to simulate the OPA and TPA spectra were evaluated at the extended multiconfigurational quasi-degenerate second-order perturbation theory (XMC-QDPT2) level. The first TPA peak intensity for the model with d = 3.0 Å, related to the electronic transition along the stacking direction, was enhanced by a two-order magnitude compared with that for the model with d = 6.0 Å. We also obtained diradical character (y)-dependent behavior of the TPA property for the first TPA peak predicted from the valence-bond configuration interaction method based on the two-site model.

Theoretical Study on One- and Two-Photon Absorption Properties of π-Stacked Multimer Models of Phenalenyl Radicals

Effects of the number of monomers (N) on the two-photon absorption (TPA) properties of π-stacked multimer models consisting of phenalenyl radicals were investigated theoretically. We conducted spectral simulations for the π-stacked N-mer models (N = 2, 4, and 6) with different stacking distances (d1) and their alternation patterns (d2/d1). Excitation energies and transition dipole moments were calculated at the extended multi-configurational quasi-degenerate second-order perturbation theory (XMC-QDPT2) level based on the complete active space self-consistent field (CASSCF) wavefunctions with the active space orbitals constructed from the singly occupied molecular orbitals (SOMOs) of monomers. The TPA cross-section value per dimer unit at the first peak, originating from the electronic transition along the stacking direction, was predicted to increase significantly as the d2/d1 approaches one, as the d1 decreases, and as the N increases from 2 to 6. These tendencies are similar to the calculation results for the static hyperpolarizabilities.

Photoionization of aziridine: Nonadiabatic dynamics of the first six low-lying electronic states of the aziridine radical cation.

In this article, the theoretical photoionization spectroscopy of the aziridine (C2H5N) molecule is investigated. To start with, we have optimized the geometry of this molecule at the neutral electronic ground state at the density functional theory/augmented correlation-consistent polarized valence triple zeta level of theory using the G09 program. The electronic structure calculations were restricted to the first six low-lying electronic states in order to account for the experimental photoelectron spectrum of the C2H5N molecule. The first six low-lying electronic states (X̃2A', Ã2A', B̃2A″, C̃2A″, D̃2A', and Ẽ2A') of the potential energy surfaces (PESs) are calculated by both equation of motion-ionization potential-coupled cluster singles and doubles and multi-configuration quasi-degenerate perturbation theory abinitio quantum chemistry methods along the dimensionless normal displacement coordinates in which multiple conical intersections were established among the considered electronic states. A (6 × 6) model vibronic Hamiltonian is constructed on a diabatic electronic basis, using the symmetry selection rules and Taylor series expansion. The Cs symmetry point group of the aziridine molecule leads to electronic states symmetry of either A' or A″, and these states are close in energy, due to which the same symmetry electronic states avoid each other. To get a smooth diabatic PES, a fourfold diabatization scheme is used, which is implemented in the General Atomic and Molecular Electronic Structure Systems suite of programs. All the parameters used in the diabatic vibronic coupling model Hamiltonian are calculated in terms of the normal modes of vibrational coordinates. Finally, the vibronic model Hamiltonian constructed for the coupled six electronic states is used to solve both time-independent and time-dependent Schrödinger equations using the multi-configuration time-dependent Hartree program module to obtain the dynamical observables. The theoretical vibronic band structure is found to be in good accord with the available experimental results.

Electronic and vibrational spectroscopy of benzonitrile cation for astrochemical consideration.

The electronic and vibrational spectra of benzonitrile cation, C6H5CN+ (BZN+), in the gas phase at low temperatures are reported. Measurements were carried out using a cryogenic ion trapping apparatus. The mid-infrared spectrum shows a strong CN stretch at 2130 ± 1 cm-1 (4.694 ± 0.002 µm). The electronic spectrum is reported in the range 5040-5750Å. This covers the forbidden B2B2 ← X2B1 and allowed C2B1 ← X2B1 transitions. The spectrum is dominated by a broad absorption feature at wavelengths shorter than 5250Å, with the strongest absorption located at 5140Å. Experimental data are complemented with quantum chemical calculations carried out at the density functional theory and extended multi-configurational quasi-degenerate perturbation theory level. The spectroscopic results are discussed in the context of astronomical observations in the infrared and visible.

Spectroscopic and computational characterization of lanthanide-mediated bond activation of ethylamine

Ln (Ln = La and Ce) atom reactions with ethylamine are conducted in a pulsed laser vaporization supersonic molecular beam source. Dehydrogenation and association metal-containing species are observed with time-of-flight mass spectrometry, and the dehydrogenated Ln ethylimido species in the formula Ln(NC2H5) are characterized by single-photon mass-analyzed threshold ionization (MATI) spectroscopy and quantum chemical calculations. The theoretical calculations include density functional theory for both Ln species and a scalar relativity correction, electron correlation, and spin-orbit coupling through multiconfiguration quasi-degenerate second-order perturbation theory for the Ce species. The MATI spectrum of lanthanum ethylimido La(NCH2CH3) has a single vibronic band system from the ionization of the doublet ground state with the La 6s1 configuration, whereas that of the cerium ethylimido Ce(NCH2CH3) displays two vibronic band system from the ionization of the two lowest-energy spin-orbit coupling states with the Ce 4f16s1 configuration. Both Ln ethylimido complexes are formed by the thermodynamically and kinetically favorable concerted dehydrogenation of the amino group. Two additional isomers of Ln(NC2H5) include a four-membered metallacycle Ln(NHCH2CH2) from the dehydrogenation of the amino and methyl groups and a three-membered cycle Ln(NHCHCH3) from the dehydrogenation of the amino and methylene groups. The cyclic isomers are not observed experimentally as they are not populated under the experimental conditions.

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory.

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

Quasi-degenerate extension of local N-electron valence state perturbation theory with pair-natural orbital method based on localized virtual molecular orbitals.

Chemical phenomena involving near-degenerate electronic states, such as conical intersections or avoided crossing, can be properly described using quasi-degenerate perturbation theory. This study proposed a highly scalable quasi-degenerate second-order N-electron valence state perturbation theory (QD-NEVPT2) using the local pair-natural orbital (PNO) method. Our recent study showed an efficient implementation of the PNO-based state-specific NEVPT2 method using orthonormal localized virtual molecular orbitals (LVMOs) as an intermediate local basis. This study derived the state-coupling (or off-diagonal) terms to implement QD-NEVPT2 in an alternative manner to enhance efficiency based on the internally contracted basis and PNO overlap matrices between different references. To facilitate further acceleration, a local resolution-of-the-identity (RI) three-index integral generation algorithm was developed using LMOs and LVMOs. Although the NEVPT2 theory is considered to be less susceptible to the intruder-state problem (ISP), this study revealed that it can easily suffer from ISP when calculating high-lying excited states. We ameliorated this instability using the imaginary level shift technique. The PNO-QD-NEVPT2 calculations were performed on small organic molecules for the 30 lowest-lying states, as well as photoisomerization involving the conical intersection of 1,1-dimethyldibenzo[b,f] silepin with a cis-stilbene skeleton. These calculations revealed that the PNO-QD-NEVPT2 method yielded negligible errors compared to the canonical QD-NEVPT2 results. Furthermore, we tested its applicability to a large photoisomerization system using the green fluorescent protein model and the ten-state calculation of the large transition metal complex, showcasing that off-diagonal elements can be evaluated at a relatively low cost.

ExROPPP: Fast, accurate, and spin-pure calculation of the electronically excited states of organic hydrocarbon radicals.

Recent years have seen an explosion of interest in organic radicals due to their promise for highly efficient organic light-emitting diodes and molecular qubits. However, accurately and inexpensively computing their electronic structure has been challenging, especially for excited states, due to the spin-contamination problem. Furthermore, while alternacy or "pseudoparity" rules have guided the interpretation and prediction of the excited states of closed-shell hydrocarbons since the 1950s, similar general rules for hydrocarbon radicals have not to our knowledge been found yet. In this article, we present solutions to both of these challenges. First, we combine the extended configuration interaction singles method with Pariser-Parr-Pople (PPP) theory to obtain a method that we call ExROPPP (Extended Restricted Open-shell PPP) theory. We find that ExROPPP computes spin-pure excited states of hydrocarbon radicals with comparable accuracy to experiment as high-level general multi-configurational quasi-degenerate perturbation theory calculations but at a computational cost that is at least two orders of magnitude lower. We then use ExROPPP to derive widely applicable rules for the spectra of alternant hydrocarbon radicals, which are completely consistent with our computed results. These findings pave the way for highly accurate and efficient computation and prediction of the excited states of organic radicals.

Lagrangian of extended multiconfigurational self-consistent field second-order quasidegenerate perturbation theory combined with reference interaction site model self-consistent field constraint spatial electron density.

Lagrangians of the state-averaged multiconfigurational self-consistent field (SA-MCSCF) and multistate extended second-order quasidegenerate perturbation theory (MS-XMCQDPT2) coupled with the reference interaction site model self-consistent field constraint spatial electron density are defined. In addition, variational equations were derived to calculate the excitation energies of the target molecules dissolved in various solvents. The theory was applied to a phenol molecule in various solutions, and the gradients and Hessian matrices were calculated to evaluate the absorption spectral lines, including the broadening bandwidth. Numerical calculations revealed fine structures in any solvent surroundings. The main intramolecular vibrational modes related to such fine structures were stretching vibrations of the aromatic ring and the oxygen atom of the phenol molecule. The present theory plays an important role in predicting the structure of potential energy surfaces, such as Hessian matrices for various solvent types, during the photoexcitation process.

Unveiling the quantum secrets of triel metal triangles: a tale of stability, aromaticity, and relativistic effects.

Low lying electronic states of Al3-, Ga3-, In3-, and Tl3- have been characterized using high level multiconfigurational quasi degenerate perturbation theory on the multiconfigurational self-consistent field. Among these species, the singlet states emerge as the predominant energy minima, displaying remarkable stability. However, within the Tl3- series, our investigation leads to the identification of the high-spin , as the most stable spin state, a result corroborated by previous experimental detection via photoelectron spectroscopy. Similarly, we have also identified the singlet state of In3- as the signal detected previously experimentally. By applying Mandado's rules and an array of aromaticity indicators, it is conclusively demonstrated that both the singlet and quintet states exhibit multiple-fold aromaticity, while the triplets exhibit conflicting aromaticity. Furthermore, this investigation highlights the significant impact of relativistic effects, as they enhance the stability of the state relative to its singlet counterpart. These findings shed new light on the electronic structures and properties of these ions, offering valuable insights into their chemical behavior and potential applications.

Spin-orbit coupling of electrons on separate lanthanide atoms of Pr2O2 and its singly charged cation.

Although it plays a critical role in the photophysics and catalysis of lanthanides, spin-orbit coupling of electrons on individual lanthanide atoms in small clusters is not well understood. The major objective of this work is to probe such coupling of the praseodymium (Pr) 4f and 6s electrons in Pr2O2 and Pr2O2+. The approach combines mass-analyzed threshold ionization spectroscopy and spin-orbit multiconfiguration second-order quasi-degenerate perturbation theory. The energies of six ionization transitions are precisely measured; the adiabatic ionization energy of the neutral cluster is 38 045 (5) cm-1. Most of the electronic states involved in these transitions are identified as spin-orbit coupled states consisting of two or more electron spins. The electron configurations of these states are 4f46s2 for the neutral cluster and 4f46s for the singly charged cation, both in planar rhombus-type structures. The spin-orbit splitting due to the coupling of the electrons on the separate Pratoms is on the order of hundreds of wavenumbers.

Theoretical spectroscopy for unraveling the intensity mechanism of the optical and photoluminescent spectra of chiral Re(I) transition metal complexes.

In this work, we present a computational study that is able to predict the optical absorption and photoluminescent properties of the chiral Re(I) family of complexes [fac-ReX(CO)3L], where X is either Cl or I and L is N-heterocyclic carbene extended with π-conjugated [5]-helicenic unit. The computational strategy is based on carefully calibrated time dependent density functional theory calculations and operates in conjunction with an excited state dynamics approach to treat in addition to absorption (ABS) and photoluminescence (PL), electronic circular dichroism (ECD), and circularly polarized luminescence (CPL) spectroscopies, respectively. The employed computational approach provides, an addition, access to the computation of phosphorescence rates in terms of radiative and non-radiative relaxation processes. The chosen molecules consist of representative examples of non-helicenic (NHC) and helicenic diastereomers. The agreement between theoretical and experimental spectra, including absorption (ABS, ECD) and emission (PL, CPL), is excellent, validating a quantitative interpretation of the spectral features on the basis of natural transition orbitals and TheoDore analyses. It is demonstrated that across the set of studied Re(I) diastereomers, the emission process in the case of NHC diastereomers is metal to ligand charge transfer in nature and is dominated by the easy-axis anisotropy of the emissive excited multiplet. On the contrary, in the cases of the helicenic diastereomers, the emission process is intra ligand charge transfer in nature and is dominated by the respective easy-plane anisotropy of the emissive excited multiplet. This affects remarkably the photoluminescent properties of the molecules in terms of PL and CPL spectral band shapes, spin-vibronic coupling, relaxation times, and the respective quantum yields. Spin-vibronic coupling effects are investigated at the level of the state-average complete active space self-consistent field in conjunction with quasi-degenerate second order perturbation theory. It is in fact demonstrated that a spin-vibronic coupling mechanism controls the observed photophysics of this class of Re(I) complexes.

Low-energy hole subband dispersions in a cylindrical Ge nanowire: the effects of the nanowire growth direction

We examine the validity of the spherical approximation in the Luttinger–Kohn Hamiltonian in calculating the subband dispersions of the hole gas. We calculate the realistic hole subband dispersions (without the spherical approximation) in a cylindrical Ge nanowire by using quasi-degenerate perturbation theory. The realistic low-energy hole subband dispersions have a double-well anticrossing structure, that consists with the spherical approximation prediction. However, the realistic subband dispersions are also nanowire growth direction dependent. When the nanowire growth direction is restricted in the (100) crystal plane, the detailed growth direction dependences of the subband parameters are given. We find the spherical approximation is good approximation, it can nicely reproduce the real result in some special growth directions.

Iron(II) Complexes with Porphyrin and Tetrabenzoporphyrin: CASSCF/MCQDPT2 Study of the Electronic Structures and UV-Vis Spectra by sTD-DFT.

The geometry and electronic structures of iron(II) complexes with porphyrin (FeP) and tetrabenzoporphyrin (FeTBP) in ground and low-lying excited electronic states are determined by DFT (PBE0/def2-TZVP) calculations and the complete active space self-consistent field (CASSCF) method, followed by the multiconfigurational quasi-degenerate second-order perturbation theory (MCQDPT2) approach to determine the dynamic electron correlation. The minima on the potential energy surfaces (PESs) of the ground (3A2g) and low-lying, high-spin (5A1g) electronic states correspond to the planar structures of FeP and FeTBP with D4h symmetry. According to the results of the MCQDPT2 calculations, the wave functions of the 3A2g and 5A1g electronic states are single determinant. The electronic absorption (UV-Vis) spectra of FeP and FeTBP are simulated within the framework of the simplified time-dependent density functional theory (sTDDFT) approach with the use of the long-range corrected CAM-B3LYP function. The most intensive bands of the UV-Vis spectra of FeP and FeTBP occur in the Soret near-UV region of 370-390 nm.

Calculating Vertical Ionization Energies of Hydrated Biological Chromophores Based on Multiconfigurational Perturbation Theory

Here we introduce a methodology for calculating vertical detachment energies (VDE) and vertical ionization energies (VIE) of anionic and neutral chromophores in aqueous environment. The proposed method is based on the extended multiconfigurational quasidegenerate perturbation theory coupled to the explicit treatment of solvent effects in the frame of the effective fragment potential method. We show that the solvent polarization contribution must be considered for getting accurate quantitative estimations of VDEs and VIEs. The calculated values of VDE for phenolate (7.3 eV) and VIE for phenol (7.9 eV) in aqueous environment are in good agreement with the experimental results obtained using X-ray and multiphoton UV photoelectron spectroscopy. Our approach will be useful for studying processes of photoinduced electron transfer from anionic as well as neutral biological chromophores in aqueous solution.

Calculating Vertical Ionization Energies of Hydrated Biological Chromophores Based on Multiconfigurational Perturbation Theory

Here we introduce a methodology for calculating vertical detachment energies (VDE) and vertical ionization energies (VIE) of anionic and neutral chromophores in aqueous environment. The proposed method is based on the extended multiconfigurational quasidegenerate perturbation theory coupled to the explicit treatment of solvent effects in the frame of the effective fragment potential method. We show that the solvent polarization contribution must be considered for getting accurate quantitative estimations of VDEs and VIEs. The calculated values of VDE for phenolate (7.3 eV) and VIE for phenol (7.9 eV) in aqueous environment are in good agreement with the experimental results obtained using X-ray and multiphoton UV photoelectron spectroscopy. Our approach will be useful for studying processes of photoinduced electron transfer from anionic as well as neutral biological chromophores in aqueous solution.

Electronic Excitation of ortho-Fluorothiophenol.

ortho-Fluorothiophenol (o-FTP) photodissociates through the well-known πσ* process. The fluorine atom of o-FTP introduces a feature in the photodissociation of o-FTP that does not occur in most other πσ* processes because the fluorine atom can form a hydrogen bond with the hydrogen atom of the SH group. Theoretical computations can serve as a good way to study these reactions because they usually proceed very quickly, and the current spectroscopies cannot probe the details of the processes as thoroughly as theory can. Here we use completely renormalized equation-of-motion coupled cluster theory with single and double excitations and a quasiperturbative treatment of connected triple excitations (CR-EOM-CCSD(T)) and quasidegenerate perturbation theory, in particular extended multistate complete-active-space second-order perturbation theory (XMS- CASPT2), to calculate the four lowest singlet states of o-FTP and hybrid density functional theory to optimize the geometries of the two lowest singlet states. We find that ten active electrons in nine active orbitals are sufficient to provide a good reference function for all four states. We find that the ground electronic state and the first excited singlet state both exhibit strongly bent hydrogen bonds. We also use density functional theory with the Tamm-Dancoff approximation and the SMD solvation model to successfully simulate the electronic spectrum of o-FTP in n-hexane solvent.

Simulating Spin–Orbit Coupling with Quasidegenerate N-Electron Valence Perturbation Theory

We present the first implementation of spin-orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QDNEVPT2). The SO-QDNEVPT2 approach enables the computations of ground- and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin-orbit coupling. In addition to SO-QDNEVPT2 with the full description of one- and two-body spin-orbit interactions at the level of two-component Breit-Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin-orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin-orbit coupling with a relatively low computational cost.

Light driven magnetic transitions in transition metal dichalcogenide heterobilayers

Motivated by the recent excitement around the physics of twisted transition metal dichalcogenide (TMD) multilayer systems, we study strongly correlated phases of TMD heterobilayers under the influence of light. We consider both waveguide light and circularly polarized light. The former allows for longitudinally polarized light, which in the high frequency limit can be used to selectively modify interlayer hoppings in a tight-binding model. We argue based on quasi-degenerate perturbation theory that changes to the interlayer hoppings can be captured as a modulation to the strength of the moiré potential in a continuum model. As a consequence, waveguide light can be used to drive transitions between a myriad of different magnetic phases, including a transition from a 120∘ Neel phase to a stripe ordered magnetic phase, or from a spin density wave phase to a paramagnetic phase, among others. When the system is subjected to circularly polarized light we find that the effective mass of the active TMD layer is modified by an applied electromagnetic field. By simultaneously applying waveguide light and circularly polarized light to a system, one has a high level of control in moving through the phase diagram in-situ. Lastly, we comment on the experimental feasibility of Floquet state preparation and argue that it is within reach of available techniques when the system is coupled to a judiciously chosen bath.