2nd Order Perturbation Theory Research Articles

Effect of α-Substitution on the Reactivity of C(sp3)-H Bonds in Pd0-Catalyzed C-H Arylation.

We report mechanistic studies on the reactivity of different α-substituted C(sp3)-H bonds, -CHnR (R = H, Me, CO2Me, CONMe2, OMe, and Ph, as well as the cyclopropyl and isopropyl derivatives -CH(CH2)2 and -CHMe2) in the context of Pd0-catalyzed C(sp3)-H arylation. Primary kinetic isotope effects, kH/kD, were determined experimentally for R = H (3.2) and Me (3.5), and these, along with the determination of reaction orders and computational studies, indicate rate-limiting C-H activation for all substituents except when R = CO2Me. This last result was confirmed experimentally (kH/kD ∼ 1). A reactivity scale for C(sp3)-H activation was then determined: CH2CO2Me > CH(CH2)2 ≥ CH2CONMe2 > CH3 ≫ CH2Ph > CH2Me > CH2OMe ≫ CHMe2. C-H activation involves AMLA/CMD transition states featuring intramolecular O → H-C H-bonding assisted by C-H → Pd agostic bonding. The "AMLA coefficient", χ, is introduced to quantify the energies associated with these interactions via natural bond orbital 2nd order perturbation theory analysis. Higher barriers correlate with lower χ values, which in turn signal a greater agostic interaction in the transition state. We believe that this reactivity scale and the underlying factors that determine this will be of use for future studies in transition-metal-catalyzed C(sp3)-H activation proceeding via the AMLA/CMD mechanism.

Ab initio study of the biogenic amino acids.

Ten amino acids have been subjected to the quantum chemical calculations using the ab initio MO-LCAO-SCF calculations. When the geometry optimization started form the X-ray structure confirming the zwitterionic form, the ab initio calculations in vacuo result in the amino acid (canonical) form with the hydrogen atom attached not to the amine but to the carboxylate group. At the optimum geometry, a number of properties were evaluated: dipole moment, dipole polarizability, molecular surface, molecular volume, HOMO, LUMO, ionization energy, and electron affinity using the ΔSCF approach and their values corrected for electron correlation by the 2nd order perturbation theory (MP2). Also, the Mulliken electronegativity and Pearson hardness were evaluated. These properties have been mutually correlated by employing the statistical multivariate methods: the cluster analysis, the probabilistic neural network classifier, the principal component analysis, and the Pearson pair correlation. In addition, the molecular electrostatic potential mapped on the isovalue surface of charge density has been drawn. After the full vibrational analysis, thermodynamic properties at 300K were evaluated: internal energy, entropy, and the free energy.

Time-Dependent Density Functional Theory Study of Copper(II) Oxo Active Sites for Methane-to-Methanol Conversion in Zeolites.

Copper-exchanged zeolites are useful materials for step-wise methane-to-methanol conversion (MMC). However, methanol yields on copper-exchanged zeolites are often modest, spurring interest in the development of active-site species that are activated at moderate temperatures, afford greater yields, and provide excellent methanol selectivities. Ultraviolet-visible (UV-vis) spectroscopy is a major tool for characterizing the active-sites and their evolution during the step-wise MMC process. However, computation of the UV-vis spectra of the copper-oxo active sites using Tamm-Dancoff time-dependent density functional theory (TDA-DFT) can be quite problematic. This has led to utilization of expensive methods based on multireference approaches, Green functions, and the Bethe-Salpeter equation. In this work, we examined the optical spectra of [CuO]+, [Cu2O]2+, [Cu2O2]2+, and [Cu3O3]2+ species implicated in MMC in zeolites. For the larger species, we examined how agreement with experimental data is improved with increasingly larger cluster models. For [CuO]+, we compared TDA-DFT against restricted active space 2nd-order perturbation theory, RASPT2. We found that signature peaks for [CuO]+ have multireference behavior. The excited states have many configuration state functions with a double excitation character. These effects are likely responsible for the poor utility of conventional TDA-DFT methods. Indeed, we obtain good agreement with experimental data and RASPT2 after accounting for 2h/2p excitations within TDA-DFT with a previously described configuration interaction singles and doubles, CIS(D)-style scheme. This was the case for [CuO]+, [Cu2O]2+, as well as a [Cu2O2]2+ species. Using a long-range corrected double-hybrid, ωB2PLYP, we provide for the first time computational evidence for the experimental UV-vis spectrum of the [Cu3O3]2+ active site motif.

Interaction of CH4 with Electronically Excited O2: Ab Initio Potential Energy Surfaces and Reaction Kinetics

Multireference quantum chemical research with the aid of complete active space self-consistent field approach was performed to study the elementary reactions of $${{\text {CH}}_4}$$ with $${\text {O}}_2$$ in $${a^1\varDelta _g}$$ , $${b^1\varSigma _g^+}$$ , $${c^1\varSigma _u^-}$$ , and $${A^{\prime 3} \varDelta _u}$$ electronically excited states highly relevant for plasma-assisted combustion and for plasma-chemical fuel reforming. The thermodynamically and kinetically favorable reaction pathways and likely intersystem crossings for the first step of the methane oxidation have been found out. The key energy values were refined based upon the extended multiconfiguration quasi-degenerate 2nd-order perturbation theory. It has been exhibited that the reaction of $${{\text {O}}_2(a^1\varDelta _g)}$$ and $${{\text {O}}_2(A^{\prime 3} \varDelta _u)}$$ with $${{\text {CH}}_4}$$ proceeds through the abstraction of hydrogen with fairly low energy barriers that led to the formation of the $$\hbox {HO}_2$$ molecule in $${^2A^{\prime \prime }}$$ and $${^2A^{\prime }}$$ electronic states, respectively. These results were compared with the findings of previous theoretical investigations. The oxygen molecule in singlet sigma b state was evinced to be nonreactive with regard to the methane. However, for $${c^1\varSigma _u^-}$$ state, the reactive interaction was nevertheless found possible due to the significant probability of the nonadiabatic transitions. Appropriate thermal rate constants for revealed channels have been calculated employing variational transition-state theory and capture approximation. Corresponding three-parameter Arrhenius expressions for the broad temperature range ( $$T=300$$ –3000 K) were reported.

Low-lying excited states of model proteins: Performances of the CC2 method versus multireference methods.

A benchmark set of relevant geometries of a model protein, the N-acetylphenylalanylamide, is presented to assess the validity of the approximate second-order coupled cluster (CC2) method in studying low-lying excited states of such bio-relevant systems. The studies comprise investigations of basis-set dependence as well as comparison with two multireference methods, the multistate complete active space 2nd order perturbation theory (MS-CASPT2) and the multireference difference dedicated configuration interaction (DDCI) methods. First of all, the applicability and the accuracy of the quasi-linear multireference difference dedicated configuration interaction method have been demonstrated on bio-relevant systems by comparison with the results obtained by the standard MS-CASPT2. Second, both the nature and excitation energy of the first low-lying excited state obtained at the CC2 level are very close to the Davidson corrected CAS+DDCI ones, the mean absolute deviation on the excitation energy being equal to 0.1 eV with a maximum of less than 0.2 eV. Finally, for the following low-lying excited states, if the nature is always well reproduced at the CC2 level, the differences on excitation energies become more important and can depend on the geometry.

The protonated water trimer and its giant Fermi resonances

The protonated water trimer is a prototype “proton wire” that can transport two protons nearly concertedly. This “proton transfer mode” (PTM) is an important contributor to the infrared spectrum of the isolated gas-phase cluster. We have simulated its infrared spectrum for both the hydrated and deuterated isotopologues, using vibrational 2nd order perturbation theory (VPT2) and ab initio molecular dynamics (AIMD) trajectories. VPT2 calculations explain quantitatively the experimental spectra at both high and low frequencies, provided that high-level quantum chemistry is utilized. In the D2-tagged hydrated cluster, the PTM undergoes giant Fermi resonances (FR’s) with two combination bands. In the deuterated analogue, one observes a single FR of “normal” intensity, manifested as the doublet recently reported experimentally. We provide band assignment for both isotopologues, with and without the D2 tag applied in experiment. We discuss possible manifestations of the giant resonance on proton transfer through water wires.

Theoretical spectroscopy study of the low-lying electronic states of UX and UX(+), X = F and Cl.

Spectroscopic constants (Te, re, B0, ωe, and ωexe) have been calculated for the low-lying electronic states of UF, UF(+), UCl, and UCl(+) using complete active space 2nd-order perturbation theory (CASPT2), with a series of correlation consistent basis sets. The latter included those based on both pseudopotential (PP) and all-electron Douglas-Kroll-Hess Hamiltonians for the U atom. Spin orbit (SO) effects were included a posteriori using the state interacting method using both PP and Breit Pauli (BP) operators, as well as from exact two-component methods for U(+) and UF(+). Complete basis set (CBS) limits were obtained by extrapolation where possible and the PP and BP calculations were compared at their respective CBS limits. The PP-based method was shown to be reliable in calculating spectroscopic constants, in particular when using the state interacting method with CASPT2 energies (SO-CASPT2). The two component calculations were limited by computational resources and could not include electron correlation from the nominally closed shell 6s and 6p orbitals of U. UF and UCl were both calculated to have Ω = 9/2 ground states. The first excited state of UCl was calculated to be an Ω = 7/2 state at 78 cm(-1) as opposed to the same state at 435 cm(-1) in UF, and the other low-lying states of UCl showed a similar compression relative to UF. Likewise, UF(+) and UCl(+) both have Ω = 4 ground states and the manifold of low-lying excited Ω = 3, 2, 1, 0 states was energetically closer together in UCl(+) than in UF(+), ranging up to 776 cm(-1) in UF(+) and only 438 cm(-1) in UCl(+). As in previous studies, the final PP-based SO-CASPT2 results for UF(+) and UF agree well with experiment and are expected to be predictive for UCl and UCl(+), which are reported here for the first time.

NONLINEAR BEHAVIOR OF BARYON ACOUSTIC OSCILLATIONS IN REDSHIFT SPACE FROM THE ZEL'DOVICH APPROXIMATION

Baryon acoustic oscillations (BAO) are a powerful probe of the expansion history of the universe, which can tell us about the nature of dark energy. In order to accurately characterize the dark energy equation of state using BAO, we must understand the effects of both nonlinearities and redshift space distortions on the location and shape of the acoustic peak. In a previous paper we introduced a novel approach to 2nd order perturbation theory in configuration space using the Zel'dovich approximation, and presented a simple result for the first nonlinear term of the correlation function. In this paper, we extend this approach to redshift space. We show how perform the computation, and present the analytic result for the first nonlinear term in the correlation function. Finally, we validate our result through comparison to numerical simulations.

Power system multi-parameter small signal stability analysis based on 2nd order perturbation theory

• A novel method to analyze low-frequency oscillation modes is proposed. • Estimate accurately even several uncertain parameters in the system change largely. • The methods avoids the burdensome derivation calculation of 2nd order sensitivity. • Saving calculating time by avoiding solving complicated high-dimensional state matrix. • More accurate compared with the 1st order perturbation theory. A novel method based on multi-parameter 2nd order perturbation sensitivity is proposed to analyze the low-frequency oscillation modes in large-scale interconnected power system, since the low-frequency oscillation mode change is hard to determine due to the violent fluctuation of multiple parameters during operation. Firstly, the multi-parameter 2nd order perturbation sensitivity matrices of eigenvalues and eigenvectors are deduced. Then, their multi-parameter 2nd order estimated values are calculated. On the basis of this, the changing system oscillation modes under multiple parameters variation are estimated. The simulation results of WECC (Western Electricity Coordinating Council) system verify that this method is able to assess the small signal stability of the system relatively accurately even several parameters of the system change. Then it can adjust appropriate dispatching method accordingly to improve the damping of dominant oscillation mode. Also, this method makes the solving process direct and clear since it avoids the burdensome derivation calculation of 2nd order sensitivity, and it is time-saving by avoiding solving complicated high-dimensional state matrices.

Thermal Expansion of Some Fcc Transition Metals

Volume thermal expansion of some fcc transition metals have been studied using improved lattice dynamical model. In this approach, the contribution of s like electron is calculated in 2nd order perturbation theory for the local model pseudopotential (Heine - Abrenkov) while that of the d electrons is taken into account by introduction of repulsive potential. The present study confirms that the use of improved model to study such anharmonic property yields satisfactory results. Looking to the success of present study, the present lattice mechanical model may be used to study thermophysical properties in high temperature and high pressure regions.

The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: a large sample of mock galaxy catalogues

We present a fast method of producing mock galaxy catalogues that can be used to compute covariance matrices of large-scale clustering measurements and test the methods of analysis. Our method populates a 2nd-order Lagrangian Perturbation Theory (2LPT) matter field, where we calibrate masses of dark matter halos by detailed comparisons with N-body simulations. We demonstrate the clustering of halos is recovered at ~10 per cent accuracy. We populate halos with mock galaxies using a Halo Occupation Distribution (HOD) prescription, which has been calibrated to reproduce the clustering measurements on scales between 30 and 80 Mpc/h. We compare the sample covariance matrix from our mocks with analytic estimates, and discuss differences. We have used this method to make catalogues corresponding to Data Release 9 of the Baryon Oscillation Spectroscopic Survey (BOSS),producing 600 mock catalogues of the "CMASS" galaxy sample. These mocks enabled detailed tests of methods and errors that formed an integral part of companion analyses of these galaxy data.

NONLINEAR BEHAVIOR OF BARYON ACOUSTIC OSCILLATIONS FROM THE ZEL'DOVICH APPROXIMATION USING A NON-FOURIER PERTURBATION APPROACH

Baryon acoustic oscillations are an excellent technique to constrain the properties of dark energy in the Universe. In order to accurately characterize the dark energy equation of state, we must understand the effects of both the nonlinearities and redshift space distortions on the location and shape of the acoustic peak. In this paper, we consider these effects using the Zel'dovich approximation and a novel approach to 2nd order perturbation theory. The second order term of the Zel'dovich power spectrum is built from convolutions of the linear power spectrum with polynomial kernels in Fourier space, suggesting that the corresponding term of the the Zel'dovich correlation function can be written as a sum of quadratic products of a broader class of correlation functions, expressed through simple spherical Bessel transforms of the linear power spectrum. We show how to systematically perform such a computation. We explicitly prove that our result is the Fourier transform of the Zel'dovich power spectrum, and compare our expressions to numerical simulations. Finally, we highlight the advantages of writing the nonlinear expansion in configuration space, as this calculation is easily extended to redshift space, and the higher order terms are mathematically simpler than their Fourier counterparts.

Ab initio investigation of the ground X2A′ [ X2A1] and low-lying excited electronic states of C 2B

Ab initio single- and multi-reference approaches, i.e. restricted open-shell Hartree–Fock (ROHF), coupled-cluster singles-and-doubles with approximate triples correction [CCSD(T)], complete active space self-consistent field (CASSCF), 2nd-order CAS perturbation theory (CASPT2), multi-reference singles-and-doubles configuration-interaction (MRSDCI), and density-functional B3LYP – in the cc-pVQZ basis set – have been employed to investigate the structure of the ground X 2 A′ [ X 2 A 1] state of the triatomic C 2B. We found that the ground state of the cyclic C 2B is prone to symmetry breaking along the asymmetric stretching coordinate. Single-reference highly correlated CCSD(T)/aug-cc-pVQZ method (as well as ‘static’-correlated CASSCF, and B3LYP) predicts two equivalent global minima of the adiabatic potential energy function, with the barrier between the minima of about 20 cm −1 at C 2v geometry. Wavefunctions almost free from SB are obtained using the MRSDCI approach based on the CASSCF and state-averaged (SA) CASSCF wavefunctions. Highly correlated multi-reference full-valence (FV) CASPT2 model exactly predicts a C 2v equilibrium geometry; however, the highest level FV-CAS-MRSDCI/aug-cc-pVQZ method predicts lower-symmetry structure. We found that this effect is not artifactual as stated in Leonard et al. (2000) [5]. SB arises from a pseudo-Jahn–Teller effect (PJTE), the non-adiabatic interaction with the first excited 1 2 B 2 state, which has the same symmetry as the asymmetric stretching coordinate in the ground state. Several low-lying excited electronic states have been optimized using SA-CAS-MRSDCI/aug-cc-pVTZ procedure. The first excited doublet is 1 2 B 2 (0.95 eV), and the lowest-lying quartet state 1 4 B 2 (1.74 eV). The vertical excitation spectra and transition matrix elements for six low-lying doublet and four quartet states were calculated using full valence SA-CAS-MRSDCI/aug-cc-pVTZ procedure.

Effects of observation on quantum interference in a laterally coupled double quantum dot using a quantum dot charge sensor

We theoretically investigated the effects of the observation by a quantum dot charge sensor on an Aharonov–Bohm effect in a laterally coupled double quantum dot using the interpolative 2nd-order perturbation theory. In particular, we introduce the notion of the coherent indirect coupling, which characterizes the strength of the indirect coupling between two quantum dots via the reservoir. As the Coulomb interaction V S for the charge sensing increases, the linear conductance through a double quantum dot decreases because of the many-body correlation effect. The visibility of Aharonov–Bohm oscillation in the linear conductance behaves super-linearly for weak sensing interaction regime and sub-linearly for strong sensing interaction regime.

Spin–orbit ab initio investigation of the UV photoinduced bond cleavage in iodotrimethylstannane

The photoinduced homolytic cleavage of the Sn–I bond in iodotrimethylstannane (CH3)3SnI, observed after UV irradiation, is investigated by means of spin–orbit ab initio calculations based on CASSCF (complete active space self-consistent field) and MS-CASPT2 (multi-state complete active space 2nd order perturbation theory) methods. The absorption electronic spectrum is characterized by ten low-lying 1,3A′ and 1,3A″ spin eigenstates corresponding to py(I),px(I) → σ*SnI; σSnI → σ*SnI and py(Sn), px(Sn) → σ*SnI, where σSnI and σ*SnI are the bonding and anti-bonding orbitals of the Sn–I bond along the pz axis. From the 1A′ electronic ground state and these ten spin eigenstates, 21 spin–orbit states are generated leading to various deactivation channels of (CH3)3SnI, corresponding to the formation of radicals (CH3)3Sn• and •I and to the ionic species (CH3)3Sn+ and I–. Irradiation into the upper band at 175 nm should lead to the heterolytic cleavage of the Sn–I bond to form the ionic primary products exclusively, whereas absorption into the shoulder at 250 nm induces the homolytic breaking with formation of the radical products.

MOLCAS 7: The Next Generation

Some of the new unique features of the MOLCAS quantum chemistry package version 7 are presented in this report. In particular, the Cholesky decomposition method applied to some quantum chemical methods is described. This approach is used both in the context of a straight forward approximation of the two-electron integrals and in the generation of so-called auxiliary basis sets. The article describes how the method is implemented for most known wave functions models: self-consistent field, density functional theory, 2nd order perturbation theory, complete-active space self-consistent field multiconfigurational reference 2nd order perturbation theory, and coupled-cluster methods. The report further elaborates on the implementation of a restricted-active space self-consistent field reference function in conjunction with 2nd order perturbation theory. The average atomic natural orbital basis for relativistic calculations, covering the whole periodic table, are described and associated unique properties are demonstrated. Furthermore, the use of the arbitrary order Douglas-Kroll-Hess transformation for one-component relativistic calculations and its implementation are discussed. This section especially focuses on the implementation of the so-called picture-change-free atomic orbital property integrals. Moreover, the ElectroStatic Potential Fitted scheme, a version of a quantum mechanics/molecular mechanics hybrid method implemented in MOLCAS, is described and discussed. Finally, the report discusses the use of the MOLCAS package for advanced studies of photo chemical phenomena and the usefulness of the algorithms for constrained geometry optimization in MOLCAS in association with such studies.

NMR shielding as a probe of intermolecular interactions: ab initio and density functional theory studies

Ab initio and density functional theory (DFT) calculations of nuclear magnetic resonance shielding tensors in benzene-methane and two isomers of the benzene dimer are reported, with the aim of probing the changes in shielding induced by the formation of supramolecular complexes from isolated molecules. It is shown that the changes in shielding (and hence of chemical shift) for hydrogen nuclei are broadly in line with expectations from "shielding cones" based on aromatic ring current, but that changes for carbon nuclei are rather more subtle. More detailed analysis indicates that the change in isotropic shielding results from much larger changes in individual components of the shielding tensor and in diamagnetic/paramagnetic shielding contributions. Benchmark data were obtained using Møller-Plesset 2nd order perturbation theory with a medium-sized basis set, but it is shown that Hartree-Fock and most density functional theory methods reproduce all essential changes in shielding, and do so in a reasonably basis set independent fashion. The chosen method is then applied to a DNA-intercalator complex.

Kinetic equations for transport through single-molecule transistors

We present explicit kinetic equations for quantum transport through a general molecular quantum-dot, accounting for all contributions up to 4th order perturbation theory in the tunneling Hamiltonian and the complete molecular density matrix. Such a full treatment describes not only sequential, cotunneling and pair tunneling, but also contains terms contributing to renormalization of the molecular resonances as well as their broadening. Due to the latter all terms in the perturbation expansion are automatically well-defined for any set of system parameters, no divergences occur and no by-hand regularization is required. Additionally we show that, in contrast to 2nd order perturbation theory, in 4th order it is essential to account for quantum coherence between non-degenerate states, entering the theory through the non-diagonal elements of the density matrix. As a first application, we study a single-molecule transistor coupled to a localized vibrational mode (Anderson-Holstein model). We find that cotunneling-assisted sequential tunneling processes involving the vibration give rise to current peaks i.e. negative differential conductance in the Coulomb-blockade regime. Such peaks occur in the cross-over to strong electron-vibration coupling, where inelastic cotunneling competes with Franck-Condon suppressed sequential tunneling, and thereby indicate the strength of the electron-vibration coupling. The peaks depend sensitively on the coupling to a dissipative bath, thus providing also an experimental probe of the Q-factor of the vibrational motion.

Automatic fitting procedures for EPR spectra of disordered systems: Matrix diagonalization and perturbation methods applied to fluorocarbon radicals

Two types of automatic fitting procedures for EPR spectra of disordered systems have been developed, one based on matrix diagonalization of a general spin Hamiltonian, the other on 2nd order perturbation theory. The first program is based on a previous Fortran code complemented with a newly written interface in Java to provide user-friendly in and output. The second is intended for the special case of free radicals with several relatively weakly interacting nuclei, in which case the general method becomes slow. A least squares’ fitting procedure utilizing analytical or numerical derivatives of the theoretically calculated spectrum with respect to the g- and hyperfine structure (hfs) tensors was used to refine those parameters in both cases. ‘Rigid limit’ ESR spectra from radicals in organic matrices and in polymers, previously studied experimentally at low temperature, were analyzed by both methods. Fluorocarbon anion radicals could be simulated, quite accurately with the exact method, whereas automatic fitting on, e.g. the c-C 4F 8 − anion radical is only feasible with the 2nd order approximative treatment. Initial values for the 19F hfs tensors estimated by DFT calculations were quite close to the final. For neutral radicals of the type XCF 2CF 2 the refinement of the hfs tensors by the exact method worked better than the approximate. The reasons are discussed. The ability of the fitting procedures to recover the correct magnetic parameters of disordered systems was investigated by fittings to synthetic spectra with known hfs tensors. The exact and the approximate methods are concluded to be complementary, one being general, but limited to relatively small systems, the other being a special treatment, suited for S = 1/2 systems with several moderately large hfs.

Photodissociation dynamics of trimethyltin iodide

The photodissociation dynamics of (CH 3) 3SnI has been investigated in the semi-classical approximation and in the quasi-resonant approximation by URIMIR (Unimolecular Reactions Induced by Multiphoton or Infra-Red Radiation), as well as by direct numerical integration of the time dependent Schrödinger equation with a one-dimensional model describing the Sn–I bond elongation. Potential energy functions (PEF) and associated dipole moment functions (DMF) result from non-linear fits of the MS-CASPT2 (Multi-State Complete Active Space 2nd Order Perturbation Theory) potential energies and dipole transition moments including spin–orbit effects. Preliminary simulations of the excited states dynamics suggest that the ionic channel should be populated by irradiation at about 207 nm. The branching ratio for the formation of ionic products ((CH 3) 3Sn + + I −) to that of radical products ((CH 3) 3Sn + I) could be higher than 5:1 for pulse lengths on the order of 150–200 fs at 50 TW cm −2.