Fourth-order Perturbation Theory Research Articles

Multiphoton classification of prethreshold structures in measured microwave ionization curves of hydrogen Rydberg atoms

We present a complete multiphoton classification of prethreshold structures in measured microwave ionization data of hydrogen Rydberg atoms. We show that fourth-order quasienergy perturbation theory reproduces the locations of prethreshold structures, thus proving that prethreshold structures are a perturbative effect. Additional prethreshold structures are predicted to occur at field and frequency values, to our knowledge, not yet explored experimentally.

Trends in Stability for N18 Cages

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Trends in Stability for N18 Cages

Nitrogen molecules Nx have been extensively studied for their potential as high energy density materials (HEDM). Cages of three-coordinate nitrogen have been studied to determine the structural features that result in the most stable isomers. N12 cages, for example, have been shown to follow a general trend that molecules with more pentagons in the network are more stable than other molecules. Larger Nx cages (24 or more atoms) do not follow this trend, favoring cylindrical structures over spherical ones with larger numbers of pentagons. To determine which trends intermediate-sized Nx cages follow, a series of N18 cages are studied by theoretical calculations of their stability. Geometries are optimized by using Hartree−Fock theory and perturbation theory (MP3), with single energies calculated with the fourth-order perturbation theory (MP4). The correlation-consistent CC-PVDZ basis of Dunning is employed. The major result is a loose trend favoring pentagons, with the most stable molecules being the ones with an optimal combination of pentagons and triangles.

Magnetic dipole transitions in crystals

The magnetic dipole transitions in rare earth ions in crystals are described in terms of a model based on the fourth order perturbation theory. An analysis of the perturbing influence of crystal field potential and spin-orbit interaction is made. In addition to intra-shell interactions, inter-shell interactions that include the influence of the excited configurations are analysed. The transition amplitude is defined in terms of effective operators expressed by double unit tensor operators. The radial integrals of all effective operators are defined within the perturbed function approach, therefore the complete radial basis sets of one-electron states of given symmetry are taken into account. Conclusions mainly focus on the possible importance of magnetic dipole transitions in the description of electric dipole 0 ⟷ 0 transition due to the so-called borrowing mechanism.

Fourth-order perturbation theory for the half-filled Hubbard model in infinite dimensions

We calculate the zero-temperature self-energy to fourth-order perturbation theory in the Hubbard interaction $U$ for the half-filled Hubbard model in infinite dimensions. For the Bethe lattice with bare bandwidth $W$, we compare our perturbative results for the self-energy, the single-particle density of states, and the momentum distribution to those from approximate analytical and numerical studies of the model. Results for the density of states from perturbation theory at $U/W=0.4$ agree very well with those from the Dynamical Mean-Field Theory treated with the Fixed-Energy Exact Diagonalization and with the Dynamical Density-Matrix Renormalization Group. In contrast, our results reveal the limited resolution of the Numerical Renormalization Group approach in treating the Hubbard bands. The momentum distributions from all approximate studies of the model are very similar in the regime where perturbation theory is applicable, $U/W \le 0.6$. Iterated Perturbation Theory overestimates the quasiparticle weight above such moderate interaction strengths.

Comparative analysis of the microscopic spin-Hamiltonian expressions used for the non-Kramers Fe2+(3d6) ions with spin S=2 in reduced rubredoxin, desulforedoxin, and related systems

The predictions based on the crystal-field (CF) and spin-Hamiltonian (SH) theory play important role in the studies of the spectroscopic properties of the non-Kramers Fe2+ (S=2) ions in reduced rubredoxin, desulforedoxin, and related mononuclear synthetic analogues. The complexity of such systems as well as the various approximations used in analysis of experimental data contribute to inconclusive results derived from electron magnetic resonance, Mössbauer, and optical spectroscopy studies as well as theoretical calculations. In this paper various microscopic spin-Hamiltonian (MSH) approaches used in the pertinent literature and their limitations are critically examined. The CF and SH theory pertinent for the non-Kramers (3d6) Fe2+ ion with spin S=2 at tetragonal and the first- and second-kind orthorhombic symmetry sites are presented in the nutshell. As a benchmark we utilize the MSH expressions within the 5D approximation derived by computer up to fourth order of perturbation theory. This theoretical framework enables us to carry out a comparative analysis of the various, often inconsistent, second-order MSH expressions existing in the literature. The fourth-rank zero-field splitting terms are considered for the first time. The results of this study provide a starting point for detailed modeling of the spectroscopic properties of Fe2+ ion in complex biological systems and their synthetic analogues.

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits, respectively. The energy variation of the classical periodicity ( τ) is also dramatic, having the special limiting case of τ→∞ at the ‘top’ of the classical motion (i.e., the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to fourth order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the ‘top’ in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.

Distance Dependence of the Mn-Mn Exchange Interaction in IV-VI Semimagnetic Semiconductors

In the present paper we calculate the exchange interaction between two manganese ions in IV-VI semiconductors with the rocksalt structure. The method of calculations is based on the fourth order perturbation theory with respect to hybridization between band states and localized d orbitals of Mn ions. This hybridization is described by three Harrison integrals: V p d σ , V p d π , and V s d σ . The band states of IV-VI semiconductor are obtained from the semiempirical tight binding model built from s and p orbitals of cations and anions. The resulting exchange term in the Hamiltonian is of the form - Σ i , j = x y z J i j S 1 i S 2 j , however nondiagonal terms of the exchange integral tenso J i > not = j are very small. The dependence of J i j on the Mn-Mn distance is non-monotonic. We also discuss the influence of the local crystal deformations on the exchange integral.

Perturbative computation of the gluonic effective action via Polyakov’s world-line path integral

The Polyakov world-line path integral describing the propagation of gluon field quanta is constructed by employing the background gauge fixing method and is subsequently applied to analytically compute the divergent terms of the one (gluonic) loop effective action to fourth order in perturbation theory. The merits of the proposed approach is that, to a given order, it reduces to performing two integrations, one over a set of Grassmann and one over a set of Feynman-type parameters through which one manages to accomodate all Feynman diagrams entering the computation at once.

Computation of the conventional strain energy in oxaziridine

The conventional strain energy for oxaziridine is determined within the isodesmic, homodesmotic, and hyperhomodesmotic models. Using these models, the conventional strain energy for cyclopropane, cyclobutane, cyclopentane, and cyclohexane are also computed for comparison purposes. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies are computed for all pertinent molecular systems using SCF theory, second-order perturbation theory, and density functional theory and employing two basis sets of triple-zeta valence quality: 6-311G(d,p) and 6-311+G(2df,2pd). Single-point fourth-order perturbation theory and CCSD(T) calculations employing the larger basis set are also computed at the MP2/6-311+G(2df,2pd) optimized geometries to determine the effect of higher-order correlation effects on ring strain computation. Finally, the ring strain of oxaziridine is compared to other small ring systems.

Ab Initio Structure Investigation of the Enol Forms of β-Diketones RCOCH2COR (R = H, CH3, CF3)

The geometrical parameters, force field parameters, and vibrational frequencies of the enol forms of β-diketones RCOCH2COR (R = H, CH3, CF3) were calculated using the ab initio MO LCAO SCF method involving broad bases of Cartesian Gaussian functions. For acetylacetone (R = CH3), the potential surface of rotation of the two methyl groups was investigated in detail, and the minimal energy route of the transfer of the hydroxyl proton inside the chelate ring was calculated. For hexafluoroacetylacetone (R = CF3) and malonic dialdehyde (R = H), calculations were carried out only for stationary points on the potential energy surface. For all the geometrical configurations under study, the relative energies were refined with inclusion of electron correlation in terms of second-, third-, and fourth-order Moller–Plesset perturbation theory; for molecules with R = H, CH3, the configuration interaction procedure taking account of all single and double excitations was additionally employed. The experimental and theoretical data are compared with those available in the literature. The substituent effect on the geometrical and electronic structure of the chelate ring is studied.

Josephson coupling through a quantum dot

We derive, via fourth-order perturbation theory, an expression for the Josephson current through a gated interacting quantum dot. We analyze our expression for two different models of the superconductor-dot-superconductor (SDS) system. When the matrix elements connecting dot and leads are featureless constants, we compute the Josephson coupling ${J}_{c}$ as a function of the gate voltage and Coulomb interaction. In the limit of a diffusive dot, we compute the probability distribution ${P(J}_{c})$ of Josephson couplings. In both cases, \ensuremath{\pi} junction behavior ${(J}_{c}<0)$ is possible, and is not simply dependent on the parity of the dot occupancy.

Orbital ordering and exchange interaction in the manganites

The microscopic origin of the exchange interaction in manganites is studied by solving an electronic model Hamiltonian for the Mn-O-Mn triad. It is shown that the magnetic structure of ${\mathrm{La}}_{1\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{MnO}}_{3}$ is correctly described within an electronic Hamiltonian model, provided that the appropriate orientation of the $\mathrm{Mn}{(e}_{g})$ orbitals induced by the Jahn-Teller effect is taken into account. The Jahn-Teller distortions of the ${\mathrm{MnO}}_{6}$ octahedra control the orientation of the ${e}_{g}$ orbitals in the crystal, which in turn is shown to determine the sign of the magnetic exchange. Electron hopping involving the $\mathrm{Mn}{(t}_{2g})$ orbitals is found to be important in certain situations, for instance, it can cause a sign change in the exchange interaction, from ferromagnetic to antiferromagnetic, as a function of the Mn-O-Mn bond angle. All our results are obtained by exact diagonalization of the model Hamiltonian, either by direct diagonalization or by diagonalization using the Lanczos method, if the Hamiltonian is too big, and are rationalized using results of the fourth-order perturbation theory. The exchange interactions (signs and magnitudes) of the end members ${\mathrm{LaMnO}}_{3}$ and ${\mathrm{CaMnO}}_{3}$ as well as of the half-doped compound, ${\mathrm{La}}_{1/2}{\mathrm{Ca}}_{1/2}{\mathrm{MnO}}_{3},$ are all described correctly within the model.

A summation procedure that improves the accuracy of the fourth-order Mo/ller–Plesset perturbation theory

A procedure is demonstrated for summing the Mo/ller–Plesset many-body perturbation expansion based on the ability of quadratic summation approximants to locate branch point singularities in the complex plane of the perturbation parameter. Accuracy comparable to that from CCSDT coupled-cluster calculations is obtained using fourth-order perturbation theory.

Higher-order anharmonic contributions to the Debye-Waller factor

Abstract We have presented a detailed analysis of the higher-order anharmonic contributions to the average square atomie displacement (u 2) or Debye-Waller factor by the diagrammatic method and the Zubarev type Green's function method. The Hamiltonian employed in the detailed analysis contains the anharmonic terms up to O(λ4)that is the cubic, quartic, quintic and sextic terms which arise from the Taylor expansion of the potential energy. Thus the various contributions to (u 2) of O(λ2) and O(λ4), presented in the diagrammatic language, arise from the first-, second-, third- and the fourth-order perturbation theory. This analysis establishing interrelationships between the Helmholtz free energy (F), self-energy and (u 2) reveals that a total of 16 diagrams contribute to (u 2) for a Bravais lattice or a lattice with a basis where every atom is at a site of inversion symmetry. A simple prescription is given for the derivation of (u 2) diagrams from the F diagrams. Out of 16 diagrams, two are of O(λ2) and 14...

Superexchange via cluster states: Calculations of spin-phonon coupling constants forCuGeO3

Calculations for spin-phonon coupling constants in ${\mathrm{CuGeO}}_{3}$ are presented, applying fourth-order superexchange perturbation theory to an extended Hubbard model. In our treatment, the intermediate oxygen ligand states are described by bandlike cluster states, due to the presence of significant $\mathrm{O}(p)\ensuremath{-}\mathrm{O}(p)$ hopping. We also include the effect of the Ge ions on the O ligand states. We find a considerable q dependence of the spin-phonon coupling. For the $\ensuremath{\pi}$ modes involved in the spin-Peierls transition, our results of the coupling constants are in fair agreement with the work of Werner, Gros, and Braden. Yet some discrepancies remain, especially concerning the coupling to the vibrations, which modulate the O(2)-O(2)-Ge angle $\ensuremath{\delta}.$ Our studies of the pressure dependence of the magnetic coupling constants ${J}_{1}$ and ${J}_{2}$ suggest that relatively large nonlinear effects are present, even at small pressure values, probably due to the existence of a ``soft'' van-der-Waals--type bond direction in ${\mathrm{CuGeO}}_{3}.$

Barrier for the H2CO→H2+CO reaction: A discrepancy between high-level electronic structure calculations and experiment

The unimolecular dissociation of formaldehyde to H2+CO was studied using extended basis set calculations and a variety of medium-to-high accuracy correlation recovery techniques. These included second and fourth order perturbation theory, multireference configuration interaction wave functions, coupled cluster theory with perturbative triples and full iterative triples, and estimated full configuration interaction wave functions. The intrinsic error of the electronic structure methods was assessed by extrapolating total energies to the complete basis set limit. Our best estimate of the barrier height, including zero point vibrational effects, is 81.9±0.3 kcal/mol, almost 3 kcal/mol larger than the experimental value of 79.2±0.8 kcal/mol. This estimate includes corrections for the effects of finite basis set truncation (which is negligible at the quintuple zeta level), higher order correlation recovery, core/valence correlation, and scalar relativistic effects. Using the same theoretical approach, we estimate the exothermicity of the dissociation reaction to be −1.6 kcal/mol, compared to experimental values in the −0.4 to −2.2 kcal/mol range. New calculations of the unimolecular dissociation rate constants using a variety of techniques failed to reconcile theory and experiment.

MC-QCISD: Multi-Coefficient Correlation Method Based on Quadratic Configuration Interaction with Single and Double Excitations

This paper presents a multi-coefficient correlation method based on quadratic configuration interaction with single and double excitations (MC-QCISD) and basis sets using segmented contraction and having the same exponential parameters in the s and p spaces. The results are comparable to a previous multi-coefficient correlation method based on coupled cluster theory with less efficient correlation-consistent basis sets, and they are better than a previous multi-coefficient correlation method based on Møller−Plesset fourth order perturbation theory with single, double, and quadruple excitations with correlation-consistent basis functions. The mean unsigned error per bond of the MC-QCISD method is 0.72 kcal/mol. The new method should be very efficient for computing geometries of open-shell transition states.

Magnetic Exchange Across the Cyanide Bridge

The superexchange interaction in cyano-bridged transition metal dimers is analysed by a valence bond configuration interaction model in combination with fourth order perturbation theory. Ferro- and antiferromagnetic contributions to the exchange splitting are expressed in terms of metal-ligand hybridization matrix elements, metal-to-ligand, lig-and-to-metal, metal-to-metal charge transfer energies, and intra-atomic exhange integrals. The formalism is simplified and we arrive at a two-parameter model, with which it is possible to estimate magnetic ordering temperatures of cubic stoichiometric Prussian Blue like phases containing first row transition metals. The relevance and applicability of our theoretical results is demonstrated on critical temperatures reported in the literature. The model accounts for both ferri- and ferromagnetic ordering temperatures. It is found that cyanide is an efficient mediator of exchange due to the participation of the π and π* orbitals to equal extent.

MBPT and DFT Study of Hydrogen Cyanide Borane(1) Oligomers and Dehydrogenated Analogues

Geometry optimizations and fragmentation energies of the series of hydrogen cyanide borane(1) oligomers (up to hexamers) and their dehydrogenated analogues are presented. Structural parameters are examined at different computational levels: MBPT(2) (second-order many-body perturbation theory), SDQ-MBPT(4) (fourth-order many-body perturbation theory limited to singly-, doubly-, and quadruply excited configurations), and DFT-B3LYP (density functional theory with Becke's three-parameter hybrid functional using the Lee−Yang−Parr correlation functional). These oligomers seem to be suitable input for the models of the polymer chain because of their periodic structure emerging from successive HCNBH or HCNB addition. The stability of the oligomers in terms of initial “cracking” of the weakest bond is fairly high, ranging from 97 to 443 kJ/mol for the HCNBH series and from 329 to 589 kJ/mol for the HCNB series.