Second-order Calculations Research Articles

Analytic first-order derivatives of partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2)

A balanced treatment of dynamic and static electron correlation is important in computational chemistry, and multireference perturbation theory (MRPT) is able to do this at a reasonable computational cost. In this paper, analytic first-order derivatives, specifically gradients and dipole moments, are developed for a particular MRPT method, state-specific partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2). Only one linear equation needs to be solved for the derivative calculation if the Z-vector method is employed, which facilitates the practical application of this approach. A comparison of the calculated results with experimental geometrical parameters of O3 indicates excellent agreement although the calculated results for O3 - are slightly outside the experimental error bars. The 0-0 transition energies of various methylpyrimidines and trans-polyacetylene are calculated by performing geometry optimizations and seminumerical second-order geometrical derivative calculations. In particular, the deviations of 0-0 transition energies of trans-polyacetylene from experimental values are consistently less than 0.1 eV with PC-NEVPT2, indicating the reliability of the method. These results demonstrate the importance of adding dynamic electron correlation on top of methods dominated by static electron correlation and of developing analytic derivatives for highly accurate methods.

High-order calculation of triboelectric charge density using a transition matrix element in a triboelectric nanogenerator system

The second-order calculation for triboelectric charge density (σtribo) between a metal and polymer is performed analytically based on the molecular-ion-state model. The transition matrix element ∣MAB∣ is approximately calculated for two cases where the distance between metal and polymer is very small (d ≪ 1) and the electronic energy is near the Fermi energy level (EF) because electron transfer between metal and polymer is dominant at this level. The σtribo approaches zero-order approximation as d approaches zero. Thus, although d is very small (d ≪ 1), its effect as a second-order approximation, such as dielectric constant and dielectric thickness, is shown here.

Q y and Q x Absorption Bands for Bacteriochlorophyll a Molecules from LH2 and LH3.

Light-harvesting systems 2 and 3 (LH2 and LH3) act as antennas for the initial light capture by photosynthetic purple bacteria, thus initiating the conversion of solar energy into chemical energy. The main absorbers are carotenoids and bacteriochlorophylls (BChls), which harvest different parts of the solar spectrum. The first two optical transitions in BChl produce the Q y and Q x absorption bands. The large size of BChl molecules has prevented accurate computational determination of the electronic structures for the relevant states, until we recently succeeded in obtaining the excitation energies and transition dipole moments of the first (Q y) transition for all BChls in LH2 and LH3 using multi-state multiconfigurational second-order perturbation theory calculations. In this work, we go one step further, compute the corresponding values for the Q x transition, in line with previous work [ J. Am. Chem. Soc . 2017 , 139 , 7558 - 7567 ], and compare and assess our data against excitation energies obtained through time-dependent density functional theory methods. Interestingly, we find that the two transitions respond differently to BChls' geometrical factors, such as the macrocycle ring curvature and the dihedral torsion of the acetyl moiety. These findings will aid the unraveling of structure-function relationships for absorption and energy transfer processes in purple bacteria, and once again this demonstrates the viability of multireference quantum chemical methods as computational tools for the photophysics of biomolecules.

Microscopic structure of coexisting 0+ states in Ni68 probed via two-neutron transfer

The structure of low-spin states originating from shape-coexisting configurations in $^{68}_{40}$Ni$_{28}$ was directly probed via the two-neutron transfer reaction $^{66}$Ni$(t,p)^{68}$Ni in inverse kinematics using a radioactive ion beam on a radioactive target. The direct feeding to the first excited 0$^+$ state was measured for center-of-mass angles 4°–16° and amounts to an integral of 4.2(16)% relative to the ground state. The observed difference in feeding of the 0$^+$ states is explained by the transfer of neutrons, mainly in the $pf$ shell below $N=40$ for the ground state, and across $N=40$ in the $g_{9/2}$ orbital for the 0$_2^+$, based on second-order distorted-wave Born approximation calculations combined with state-of-the-art shell-model two-nucleon amplitudes. However, the direct feeding to the 2$_1^+$ state [29(3)%] is incompatible with these calculations.

Energy levels, weighted oscillator strengths, transition rates, lifetimes, hyperfine structures and isotope shifts of He-like-Sn deduced from the relativistic multiconfiguration Dirac–Hartree–Fock and second-order many-body perturbation theory calculations

We disclose relativistic multiconfiguration Dirac–Hartree–Fock (MCDHF) spectrum calculations for SnXLIX. Energy levels, weighted oscillator strengths, isotope shifts, hyperfine structure and Landé gJ factors are calculated for 127 odd- and even-parity states as well as lifetimes and transition rates between these states. To scrutinize the accuracy of our results, we have implemented parallel calculations using a Flexible Atomic Code (FAC) by introducing the Second-Order Many-Body Perturbation Theory (MBPT) method. Additionally, the Breit interaction and leading quantum electrodynamic effects (QED) are included as perturbations in extensive relativistic configuration interaction (RCI) calculations. We signal that, our calculations for SnXLIX are made for the first time and they provide to date the most accurate and complete atomic data related to this ion.

Third-order S-matrix study of electron-electron correlation in nonsequential double ionization

Correlated electron momentum distribution of nonsequential double ionization of a Ne atom in intense laser fields is calculated using a third-order S-matrix formalism which takes into account the interaction between the Coulomb potential of the parent ion and the photoelectrons. Compared to the second-order S-matrix calculation, the correlation distributions of the two processes which take into account the effect of the ionic Coulomb potential on either of the two ionized electrons after their collision tend to be concentrated off the diagonal of the first and third quadrants, which are closer to the experimental observations. Our further analysis shows that this modification to the second-order calculation is due to the fact that, scattering of the emitted electron upon the parent ion will alter the photoelectron angular distribution from concentration in the laser direction to focus in a direction deviating from the laser field. And this will significantly reduce the momentum of the electron in the direction of the laser field. Our results are in accord with previous studies which reveal that electron-nuclear Coulomb interaction, electron-electron collision and the final-state electron repulsion all play significant roles in the formation of the fingerlike structure.

Relativistic two-dimensional hydrogen-like atom in a weak magnetic field

A two-dimensional (2D) hydrogen-like atom with a relativistic Dirac electron, placed in a weak, static, uniform magnetic field perpendicular to the atomic plane, is considered. Closed forms of the first- and second-order Zeeman corrections to energy levels are calculated analytically, within the framework of the Rayleigh–Schrödinger perturbation theory, for an arbitrary electronic bound state. The second-order calculations are carried out with the use of the Sturmian expansion of the two-dimensional generalized radial Dirac–Coulomb Green function derived in the paper. It is found that, in contrast to the case of the three-dimensional atom (Stefańska, 2015), in two spatial dimensions atomic magnetizabilities (magnetic susceptibilities) are expressible in terms of elementary algebraic functions of a nuclear charge and electron quantum numbers. The problem considered here is related to the Coulomb impurity problem for graphene in a weak magnetic field.

Non-linear general relativistic effects in the observed redshift

We present the second-order expression for the observed redshift, accounting for all the relativistic effects from the light propagation and from the frame change at the observer and the source positions. We derive the generic gauge-transformation law that any observable quantities should satisfy, and we verify our second-order expression for the observed redshift by explicitly checking its gauge transformation property. This is the first time an explicit verification is made for the second-order calculations of observable quantities. We present our results in popular gauge choices for easy use and discuss the origin of disagreements in previous calculations.

Even numbered carbon clusters: cost-effective wavefunction-based method for calculation and automated location of most structural isomers

Using second-order Moller-Plesset perturbation-theoretic calculations with extrapolation of the energy from the lowest steps of the hierarchical staircase to the complete basis set limit, a wave function-based approach emerges that rivals density functional theory in accuracy and cost-effectiveness. Tested on a large set of reactions, the method is now applied to the carbon clusters. Combined with variable-scaling opposite spin theory, the results approximate couple-cluster quality at no additional cost. Jointly with a stimulated breakup of the molecule by choosing a (simple or composite) driving coordinate at an adequate level of theory, the approach still offers a near automated tool for locating structural isomers along the optimized reaction coordinate for stimulated evolution so obtained. Adaptations are also suggested.

Finite Element Analysis of Second-order Effect of Plane High Strength Steel Frame Structures

High-strength structural steels have been gradually used in building structures and bridge structures because of their good mechanical properties and machinability. The mechanical behavior of high strength steel members is significantly different from that of ordinary steel members. Based on the approximate second-order calculation formula in the “Code for Design of Steel Structures” GB50017-2003, the non-linear increase coefficient of the material are introduced. The finite element analysis method is used to check the second-order nonlinear effect of typical high-strength steel frame. The results of the approximation second-order analysis and the second-order elastic-plastic analysis are compared. The second-order effect and the applicability of approximation method to high strength steel frame is analysed. The result of the calculation and analysis is valuable to engineering application.

N3LO corrections to jet production in deep inelastic scattering using the Projection-to-Born method

Computations of higher-order QCD corrections for processes with exclusive final states require a subtraction method for real-radiation contributions. We present the first-ever generalisation of a subtraction method for third-order (N3LO) QCD corrections. The Projection-to-Born method is used to combine inclusive N3LO coefficient functions with an exclusive second-order (NNLO) calculation for a final state with an extra jet. The input requirements, advantages, and potential applications of the method are discussed, and validations at lower orders are performed. As a test case, we compute the N3LO corrections to kinematical distributions and production rates for single-jet production in deep inelastic scattering in the laboratory frame, and compare them with data from the ZEUS experiment at HERA. The corrections are small in the central rapidity region, where they stabilize the predictions to sub per-cent level. The corrections increase substantially towards forward rapidity where large logarithmic effects are expected, thereby yielding an improved description of the data in this region.

High-resolution IR absorption spectroscopy of polycyclic aromatic hydrocarbons in the 3 μm region: role of hydrogenation and alkylation

Aim. We aim to elucidate the spectral changes in the 3 μm region that result from chemical changes in the molecular periphery of polycyclic aromatic hydrocarbons (PAHs) with extra hydrogens (H-PAHs) and methyl groups (Me-PAHs). Methods. Advanced laser spectroscopic techniques combined with mass spectrometry were applied on supersonically cooled 1,2,3,4-tetrahydronaphthalene, 9,10-dihydroanthracene, 9,10-dihydrophenanthrene, 1,2,3,6,7,8-hexahydropyrene, 9-methylanthracene, and 9,10-dimethylanthracene, allowing us to record mass-selective and conformationally selective absorption spectra of the aromatic, aliphatic, and alkyl CH-stretches in the 3.175 − 3.636 µm region with laser-limited resolution. We compared the experimental absorption spectra with standard harmonic calculations and with second-order vibrational perturbation theory anharmonic calculations that use the SPECTRO program for treating resonances. Results. We show that anharmonicity plays an important if not dominant role, affecting not only aromatic, but also aliphatic and alkyl CH-stretch vibrations. The experimental high-resolution data lead to the conclusion that the variation in Me- and H-PAHs composition might well account for the observed variations in the 3 μm emission spectra of carbon-rich and star-forming regions. Our laboratory studies also suggest that heavily hydrogenated PAHs form a significant fraction of the carriers of IR emission in regions in which an anomalously strong 3 μm plateau is observed.

Reassigning the CaH+ 11Σ → 21Σ vibronic transition with CaD.

We observe vibronic transitions in CaD+ between the 11Σ and 21Σ electronic states by resonance enhanced multiphoton photodissociation spectroscopy in a Coulomb crystal. The vibronic transitions are compared with previous measurements on CaH+. The result is a revised assignment of the CaH+ vibronic levels and a disagreement with multi-state-complete-active-space second-order perturbation theory theoretical calculations by approximately 700 cm-1. Updated high-level coupled-cluster calculations that include core-valence correlations reduce the disagreement between theory and experiment to 300 cm-1.

Ground state of an exciton in a three-dimensional parabolic quantum dot: Convergent perturbative calculation

Working in the effective-mass approximation, we apply a powerful convergent perturbative technique of Turbiner's to the calculation of the ground state energy and the wave function of an exciton confined to a three-dimensional parabolic quantum dot. Unlike the usual Rayleigh–Schrödinger perturbation theory, Turbiner's approach works well even in the regime of strong coupling and does not require the knowledge of the full solution to the undisturbed problem. The second-order convergent calculation presented below is in excellent agreement with the results of exact numerical simulations for a wide range of system's confinement parameters.

Investigating neutron-proton pairing in sd -shell nuclei via (p,He3) and (He3,p) transfer reactions

Neutron-proton pairing correlations are investigated in detail via $np$ transfer reactions in $N=Z\phantom{\rule{4pt}{0ex}}sd$-shell nuclei. In particular, we study the cross-section ratio of the lowest ${0}^{+}$ and ${1}^{+}$ states as an observable to quantify the interplay between $T=0$ (isoscalar) and $T=1$ (isovector) pairing strengths. The experimental results are compared to second-order distorted-wave Born approximation calculations with proton-neutron amplitudes obtained in the shell-model formalism using the universal $sd$-shell interaction B. Our results suggest underestimation of the nonneglible isoscalar pairing strength in the shell-model descriptions at the expense of the isovector channel.

Structure and binding energy of the H2S dimer at the CCSD(T) complete basis set limit.

This study presents results for the binding energy and geometry of the H2S dimer which have been computed using Møller-Plesset perturbation theory (MP2, MP4) and coupled cluster (CCSD, CCSD(T)) calculations with basis sets up to aug-cc-pV5Z. Estimates of De, EZPE, Do, and dimer geometry have been obtained at each level of theory by taking advantage of the systematic convergence behavior toward the complete basis set (CBS) limit. The CBS limit binding energy values of De are 1.91 (MP2), 1.75 (MP4), 1.41 (CCSD), and 1.69 kcal/mol (CCSD[T]). The most accurate values for the equilibrium S-S distance rSS (without counterpoise correction) are 4.080 (MP2/aug-cc-pV5Z), 4.131 (MP4/aug-cc-pVQZ), 4.225 (CCSD/aug-cc-pVQZ), and 4.146 Å (CCSD(T)/aug-cc-pVQZ). This study also evaluates the effect of counterpoise correction on the H2S dimer geometry and binding energy. As regards the structure of (H2S)2, MPn, CCSD, and CCSD(T) level values of rSS, obtained by performing geometry optimizations on the counterpoise-corrected potential energy surface, converge systematically to CBS limit values of 4.099 (MP2), 4.146 (MP4), 4.233 (CCSD), and 4.167 Å (CCSD(T)). The corresponding CBS limit values of the equilibrium binding energy De are 1.88 (MP2), 1.76 (MP4), 1.41 (CCSD), and 1.69 kcal/mol (CCSD(T)), the latter in excellent agreement with the measured binding energy value of 1.68 ± 0.02 kcal/mol reported by Ciaffoni et al. [Appl. Phys. B 92, 627 (2008)]. Combining CBS electronic binding energies De with EZPE predicted by CCSD(T) vibrational second-order perturbation theory calculations yields Do = 1.08 kcal/mol, which is around 0.6 kcal/mol smaller than the measured value of 1.7 ± 0.3 kcal/mol. Overall, the results presented here demonstrate that the application of high level calculations, in particular CCSD(T), in combination with augmented correlation consistent basis sets provides valuable insight into the structure and energetics of the hydrogen sulfide dimer.

RETRACTED: Ab initio study of AlBui3 self-association

By the methods of the Density Functional Theory and second-order Moller-Plesset perturbation theory calculations the self-association process of triisobutylaluminum AlBui3 in the temperature range of 203–403 K is studied. The effects of solvation were studied using COSMO polarized continuum solvation model. Thermodynamic parameters of AlBui3 monomer and dimer were calculated and the data obtained were used to obtain their equilibrium concentrations. It has been shown for the first time that in the gas phase and non-polar solvents AlBui3 predominately exists as a dimer.

MP4 Study of the Anharmonic Coupling of the Shared Proton Stretching Vibration of the Protonated Water Dimer in Equilibrium and Transition States.

The structure and harmonic and anharmonic IR spectra of the protonated water dimer (PWD) were calculated in C1, C2, and Cs symmetry at the MP4/acc-pVTZ level of theory. We found that structure and IR spectra are practically identical in C2 and C1 symmetry, demonstrating that an equilibrium C1 configuration of the PWD is not realized. Anharmonic coupling of the shared proton stretching vibration with all other modes in the PWD in C2 and Cs symmetry was the focus of this investigation. For this purpose, 28 two-dimensional potential energy surfaces (2D PES) were built at the MP4/acc-pVTZ level of theory and the corresponding vibrational Schrödinger equations were solved using the DVR method. Differences in the coupling of the investigated mode with other modes in the C2 and Cs configurations, along with some factors that determine the red- or blue-shift of the stretching vibration frequency, were analyzed. We obtained a rather reasonable value of the stretching frequency of the bridging proton (1058.4 cm-1) unperturbed by Fermi resonance. The Fermi resonance between the fundamental vibration ν7 and the combined vibration ν2 + ν6 of the same symmetry was analyzed through anharmonic second-order perturbation theory calculations, as well as by 3D PES constructed using Q2, Q6, and Q7 as normal coordinates. A significant (up to 50%) transfer of intensity from the fundamental vibration to the combined one was found. We have estimated the frequency of the bridging proton stretching vibration in the Cs configuration of the PWD based on calculations of the intrinsic anharmonicity and anharmonic double modes interactions at the MP4/acc-pVTZ level of theory (1261 cm-1).

Explicit correlation factors

We analyze the performance of 17 different correlation factors in explicitly correlated second-order many-body perturbation calculations for correlation energies. Highly performing correlation factors are found to have near-universal shape and size in the short range of electron-electron distance (0<r12<1.5a.u.). The long-range behavior (r12>1.5a.u.) is insignificant insofar as the factor becomes near constant, leaving an orbital expansion to describe decoupled electrons. An analysis based on a low-rank Taylor expansion of the correlation factor seems limited, except that a negative second derivative with the value of around −1.3a.u. correlates with high performance.

Ground-state correlations within a nonperturbative approach

The contribution of the two-phonon configurations to the ground state of $^{4}\mathrm{He}$ and $^{16}\mathrm{O}$ is evaluated nonperturbatively using a Hartree-Fock basis within an equation-of-motion phonon method using a nucleon-nucleon optimized chiral potential. Convergence properties of energies and root-mean-square radii versus the harmonic oscillator frequency and space dimensions are investigated. The comparison with the second-order perturbation theory calculations shows that the higher-order terms have an appreciable repulsive effect and yield too-small binding energies and nuclear radii. It is argued that four-phonon configurations, through their strong coupling to two phonons, may provide most of the attractive contribution necessary for filling the gap between theoretical and experimental quantities. Possible strategies for accomplishing such a challenging task are discussed.