First-order Perturbation Correction Research Articles

Quasi-local stress-tensor formalism and the Casimir effect

We apply the quasi-local stress–energy tensor formalism to the Casimir effect of a scalar field confined between conducting planes located in a static spacetime. We show that the surface energy vanishes for both Neumann and Dirichlet boundary conditions and consequently the volume Casimir energy reduces to the famous zero point energy of the quantum field, i.e. [Formula: see text]. This enables us to reinforce previous results in the literature and extend the calculations to the case of massive and arbitrarily coupled scalar field. We found that there exists a first-order perturbation correction to the Casimir energy contrary to previous claims which state that it vanishes. This shows many orders of magnitude greater than previous estimations for the energy corrections and makes it detectable by near future experiments.

Photoelectron spectra of early 3d-transition metal dioxide molecular anions from GW calculations.

Photoelectron spectra of early 3d-transition metal dioxide anions, ScO2 -, TiO2 -, VO2 -, CrO2 -, and MnO2 -, are calculated using semilocal and hybrid density functional theory (DFT) and many-body perturbation theory within the GW approximation using one-shot perturbative and eigenvalue self-consistent formalisms. Different levels of theory are compared with each other and with available photoelectron spectra. We show that one-shot GW with a PBE0 starting point (G0W0@PBE0) consistently provides very good agreement for all experimentally measured binding energies (within 0.1 eV-0.2 eV or less). We attribute this to the success of PBE0 in mitigating self-interaction error and providing good quasiparticle wave functions, which renders a first-order perturbative GW correction effective. One-shot GW calculations with a Perdew-Burke-Ernzerhof (PBE) starting point do poorly in predicting electron removal energies by underbinding orbitals with typical errors near 1.5 eV. A higher exact exchange amount of 50% in the DFT starting point of one-shot GW does not provide very good agreement with experiment by overbinding orbitals with typical errors near 0.5 eV. While not as accurate as G0W0@PBE0, the G-only eigenvalue self-consistent GW scheme with W fixed to the PBE level provides a reasonably predictive level of theory (typical errors near 0.3 eV) to describe photoelectron spectra of these 3d-transition metal dioxide anions. Adding eigenvalue self-consistency also in W, on the other hand, worsens the agreement with experiment overall. Our findings on the performance of various GW methods are discussed in the context of our previous studies on other transition metal oxide molecular systems.

Effect of Perturbative Vibronic Correction for Weak Fluorescence in Thermally Activated Delayed Fluorescence Systems.

Minimizing the energy difference between the lowest singlet (S1) and the lowest triplet states, ΔEST, is the main strategy to design thermally activated delayed fluorescence (TADF) molecules, and spatially separating the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is the general method in the design. However, such a separation also tends to reduce the oscillator strength of the S1 state. In real systems, vibrations change the S1 oscillator strength, and thus one needs to consider the vibronic coupling toward searching for TADF candidate molecules. Here, we evaluate the importance of vibronic coupling by including the first-order perturbative correction to the transition dipole moments of carbazolyl-phthalonitrile derivatives. Indeed, some molecules display large enhancements in their oscillator strengths, with their fluorescence lifetimes reduced by 2 orders of magnitude. The twisting mode between the carbazole groups and phthalonitrile is the most important mode in inducing the perturbations. Thus, performing the perturbative correction is crucial in attaining more reliable predictions on their fluorescence propensities. We also observe that some other molecules, whose zeroth-order predicted fluorescence rates are much slower than the actual experimental data, are affected little by the same first-order correction. For these molecules, we deduce that the geometry-dependent excited-state switching kicks in. Our results demonstrate the significance of vibronic coupling in TADF molecules and the importance of adopting correction schemes as the guidelines for screening of useful TADF molecules.

Dynamical correction to the Bethe-Salpeter equation beyond the plasmon-pole approximation.

The Bethe-Salpeter equation (BSE) formalism is a computationally affordable method for the calculation of accurate optical excitation energies in molecular systems. Similar to the ubiquitous adiabatic approximation of time-dependent density-functional theory, the static approximation, which substitutes a dynamical (i.e., frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the BSE formalism. Here, going beyond the static approximation, we compute the dynamical correction of the electron-hole screening for molecular excitation energies, thanks to a renormalized first-order perturbative correction to the static BSE excitation energies. The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random-phase approximation. Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing one to assess the clear improvement brought by the dynamical correction for both singlet and triplet optical transitions.

Electrostatics and Dispersion in X-H···Y (X = C, N, O; Y = N, O) Hydrogen Bonds and Their Role in X-H Vibrational Frequency Shifts.

The frequency shifts of donor stretching vibration in X–H···Y (X = C, N, O; Y = N, O) hydrogen-bonded complexes of phenylacetylene, indole, and phenol are linearly correlated with the electrostatic component of the interaction energy. This linear correlation suggests that the electrostatic component, which is the first-order perturbative correction to the stabilization energy, is essentially localized on the X–H group. The linear correlation suggests that the electrostatic tuning rate, which is a measure of the X–H oscillator to undergo shifts upon hydrogen bonding per unit increase in the electrostatic component of the stabilization energy, was found to be in the order of O–H > N–H > C–H. Interestingly, for each of the donor groups, viz., C–H, N–H, and O–H, the vibrational frequency shifts were inversely correlated to the dipole moment of the acceptor separately, which is counterintuitive vis-à-vis the electrostatic component. This implies that extrapolation to zero dipole moment of the acceptor will yield very large shifts in the hydrogen-bonded X–H stretching frequencies. The trends in the variation of the dispersion and exchange-repulsion components and the total interaction energy vis-à-vis frequency shifts of donor stretching vibration are similar for hydrogen-bonded complexes of phenylacetylene, indole, and phenol. Furthermore, it was observed that the vibrational frequency shifts of all of the complexes are linearly correlated with the charge transfer from the filled orbital of the hydrogen acceptor to the vacant antibonding (σ*) orbital of the X–H donor group on the basis of natural bonding orbital calculations.

On certain series involving Laguerre polynomials

In this paper, we find closed-form sums for the series ∑n=1∞(α2)nn1n!1F1(−n,γ,x2), where α>0. This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H=−d2dx2+Bx2+Ax2+λxα, where 0≤x<∞, α,λ>0, A≥0.

Drag of buoyant vortex rings.

Extending from the model proposed by Vasel-Be-Hagh et al. [J. Fluid Mech. 769, 522 (2015)], a perturbation analysis is performed to modify Turner's radius by taking into account the viscous effect. The modified radius includes two terms; the zeroth-order solution representing the effect of buoyancy, and the first-order perturbation correction describing the influence of viscosity. The zeroth-order solution is explicit Turner's radius; the first-order perturbation modification, however, includes the drag coefficient, which is unknown and of interest. Fitting the photographically measured radius into the modified equation yields the time history of the drag coefficient of the corresponding buoyant vortex ring. To give further clarification, the proposed model is applied to calculate the drag coefficient of a buoyant vortex ring at a Bond number of approximately 85; a similar procedure can be applied at other Bond numbers.

First-Order Dyson Coordinates and Geometry

The mathematical constructs of the Dyson coordinates and geometry are introduced. The former are a unitary transformation of the normal coordinates and the anharmonic vibrational counterpart of the Dyson orbitals in electronic structure theory. The first-order Dyson coordinates bring the sums of the harmonic force constants and their first-order diagrammatic perturbation corrections (the first-order Dyson self-energy) to a diagonal form. The first-order Dyson geometry has no counterpart in electronic structure theory. It is the point on the potential energy surface at which the sums of the energy gradients and their first-order diagrammatic perturbation corrections vanish. It agrees with the vibrationally averaged geometry of vibrational self-consistent field (VSCF) theory in the bulk limit. These constructs provide a unified view of the relationship of VSCF and its diagrammatically size-consistent modifications as well as the self-consistent phonon method widely used in solid-state physics.

Stationary rotating black holes in theories with gravitational Chern-Simons Lagrangian term

We study the effects of introducing purely gravitational Chern-Simons Lagrangian terms in ordinary Einstein gravity on stationary rotating black hole solutions and on the associated thermodynamical properties, in a generic number of dimensions which support these terms (i.e. in D = 4k-1). We analyze the conditions, namely the number of vanishing angular momenta, under which the contributions of the Chern-Simons term to the equations of motion and the black hole entropy vanish. The particular case of a 7-dimensional theory in which a purely gravitational Chern-Simons term is added to the Einstein-Hilbert Lagrangian in D=7 dimensions is investigated in some detail. As we have not been able to find exact analytic solutions in nontrivial cases, we turn to perturbation theory and calculate the first-order perturbative correction to the Myers-Perry metric in the case where all angular momenta are equal. The expansion parameter is a dimensionless combination linear in the Chern-Simons coupling constant and the angular momentum. Corrections to horizon and ergosurface properties, as well as black hole entropy and temperature, are presented.

Energy levels of 1s2n d (n ≤ 9) states for lithium-like ions

The full-core plus correlation method with multi-configuration interaction wave functions is extended to the calculation of the non-relativistic energies of 1s2n d (n ≤ 9) states for the lithium isoelectronic sequence from Z = 11 to 20. Relativistic and mass-polarization effects on the energy are calculated as the first-order perturbation correction. The quantum-electrodynamics correction is also included. The fine structure splittings are determined from the expectation values of spin—orbit and spin—other-orbit interaction operators in the Pauli-Breit approximation. Combining the term energies of lowly excited states obtained with the quantum defects calculated by the single channel quantum defect theory, each of which is a smooth function of energy and approximated by a weakly varying function of energy, the ion potentials of highly excited states (n ≥ 6) are obtained with the semi-empirical iteration method. The results are compared with experimental data in the literature and found to be closely consistent with the regularity.

Perturbative correction to the ground-state properties of one-dimensional strongly interacting bosons in a harmonic trap

We calculate the first-order perturbation correction to the ground-state energy and chemical potential of a harmonically trapped boson gas with contact interactions about the infinite repulsion Tonks-Girardeau limit. With $c$ denoting the interaction strength, we find that, for a large number of particles $N$, the $1/c$ correction to the ground-state energy increases as ${N}^{5/2}$, in contrast to the unperturbed Tonks-Girardeau value that is proportional to ${N}^{2}$. We describe a thermodynamic scaling limit for the trapping frequency that yields an extensive ground-state energy and reproduces the zero temperature thermodynamics obtained by a local-density approximation.

Oscillator strengths for 2 2S–n2 P transitions of the lithium isoelectronic sequence from Z = 11 to 20

The dipole-length, dipole-velocity and dipole-acceleration absorption oscillator strengths for the 1s22s–1s2 np (3 ≤ n ≤ 9) transitions of lithium-like systems from Z = 11 to 20 are calculated by using the energies and the multi-configuration interaction wave functions obtained from a full core plus correlation method, in which relativistic and mass-polarization effects on the energy, as the first-order perturbation corrections, are included. The results of three forms are in good agreement with each other, and closely agree with the experimental data available in the literature. Based on the quantum defects obtained with quantum defect theory (QDT), the discrete oscillator strengths for the transitions from the ground state to highly excited states 1s2 np (n ≥ 10) and oscillator strength densities corresponding to the bound-free transitions are obtained for these ions.

Suppression of Photoionization by a Static Field

The dc field Stark effect is studied theoretically for atoms in high intensity laser fields. We prove that the first-order perturbation corrections for the energy and photoionization rate vanish when the dc field strength serves as a perturbational strength parameter. Our calculations show that by applying a dc field in the same direction as the polarization direction of the ac field, the photoinduced ionization rate is almost entirely suppressed. This suppression is attributed to changes in the phase shift of the continuum atomic wave functions which can be controlled by the dc field.

A rigorous derivation of gravitational self-force

There is general agreement that the MiSaTaQuWa equations should describe the motion of a ‘small body’ in general relativity, taking into account the leading order self-force effects. However, previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features. For example, all previous derivations have invoked, without proper justification, the step of ‘Lorenz gauge relaxation’, wherein the linearized Einstein equation is written in the form appropriate to the Lorenz gauge, but the Lorenz gauge condition is then not imposed—thereby making the resulting equations for the metric perturbation inequivalent to the linearized Einstein equations. (Such a ‘relaxation’ of the linearized Einstein equations is essential in order to avoid the conclusion that ‘point particles’ move on geodesics.) In this paper, we analyze the issue of ‘particle motion’ in general relativity in a systematic and rigorous way by considering a one-parameter family of metrics, gab(λ), corresponding to having a body (or black hole) that is ‘scaled down’ to zero size and mass in an appropriate manner. We prove that the limiting worldline of such a one-parameter family must be a geodesic of the background metric, gab(λ = 0). Gravitational self-force—as well as the force due to coupling of the spin of the body to curvature—then arises as a first-order perturbative correction in λ to this worldline. No assumptions are made in our analysis apart from the smoothness and limit properties of the one-parameter family of metrics, gab(λ). Our approach should provide a framework for systematically calculating higher order corrections to gravitational self-force, including higher multipole effects, although we do not attempt to go beyond first-order calculations here. The status of the MiSaTaQuWa equations is explained.

Effects of Zero-Point and Thermal Vibrational Averaging on Computed NMR Properties of a Model Compound for Purine Nucleosides.

A method for the incorporation of thermal averaging into the calculation of nuclear magnetic resonance properties is given. These properties are computed using density functional theory, anharmonic first-order perturbation corrections to a normal-mode analysis, and standard statistical mechanical averaging. The method is applied to the calculation of chemical shieldings and spin-spin coupling constants (J couplings) of 1'-imidazolyl-2'-deoxy-β-ribofuranose (IDR), a model compound for purine nucleosides, at the B3LYP/6-311++G(2d,p) level of theory. Thermal averaging causes substantial changes in the values of computed parameters. The calculated harmonic normal modes of IDR are also investigated; we find reasonable agreement with published results from vibrational spectroscopy on DNA fragments. Finally, the calculated magnetic and structural data regarding the reported hydrogen bond between H8 on the imidazole ring and O5' on the sugar ring are investigated; we find that such data do not strongly support the formation of a hydrogen bond between these two atoms.

Exact solutions for perturbed confined hydrogen atoms: Polarizabilities and nuclear shielding factors

We extend our recent treatment of a well-known model of a nonrelativistic hydrogen-like atom confined within an impenetrable spherical ``box'' of radius . Analytical closed-form solutions are presented for the first-order perturbation corrections appropriate to the dipole polarization of states. solutions are expressed conveniently in terms of Kummer (confluent hypergeometric) functions, exactly as for the field-free atom, by exploiting several well-known recurrence and differential relations for these functions. The functions and all necessary integrals are evaluated using standard routines, which exploit analytical properties of these functions. The accuracy of our procedures has been checked by a similar calculation of the dipole shielding factors. Our procedures are easily extended to states of nonzero angular momentum, as well as to higher-multipole perturbations. Our exact values may be of interest in assessing the utility of this simple model of atomic confinement.

Improving the accuracy of WKB and Langer eigenvalues by higher-order variational method

Higher-order variational method is adapted for improving the accuracy of WKB and Langer eigenvalues by using the WKB and Langer eigenfunctions as trial functions. The results are compared with the first-order variational method using WKB and Langer functions as trial functions and with other approximate methods (finite element method, Langer, first-, third- and fifth-order WKB, first- and third-order WKB variational, first-order perturbation correction).

Simple and Surprisingly Accurate Approach to the Chemical Bond Obtained from Dimensional Scaling

We present a new dimensional scaling transformation of the Schrödinger equation for the two electron bond. This yields, for the first time, a good description of the bond via D scaling. There also emerges, in the large-D limit, an intuitively appealing semiclassical picture, akin to a molecular model proposed by Bohr in 1913. In this limit, the electrons are confined to specific orbits in the scaled space, yet the uncertainty principle is maintained. A first-order perturbation correction, proportional to 1/D, substantially improves the agreement with the exact ground state potential energy curve. The present treatment is very simple mathematically, yet provides a strikingly accurate description of the potential curves for the lowest singlet, triplet, and excited states of H2. We find the modified D-scaling method also gives good results for other molecules. It can be combined advantageously with Hartree-Fock and other conventional methods.

Excitation Energies of 1s2ns (6 ⩽ n ⩽ 9) States for Lithium-Like Systems from Z =11 to 20

The non-relativistic energies of 1s2ns (6 ⩽ n ⩽ 9)states for the lithium-like systems from Z = 11 to 20 arecalculated by using a full-core-plus-correlation (FCPC) method. Therelativistic and mass-polarization effects on the energy are calculatedby the first-order perturbation corrections. The correction from thequantum-electrodynamics effect is also included using effective nuclearcharge. Based on these results and the quantum defect theory, thequantum defects of 1s2ns series for these ions, as afunction of energy, are determined. The comparisons between theionization potentials for 1s2ns states (6 ⩽ n ⩽ 9)obtained by the FCPC method and the semi-empirical method are carriedout. The results show that their agreement is very well and theenergies of all discrete states (n ⩾ 10) below the ionizationthreshold of this series for the ions can be predicted by using theirquantum defects.

Near-Elliptic Core Polarization-Maintaining Photonic Crystal Fiber: Modeling Birefringence Characteristics and Realization

We investigate the modal birefringence behavior of a near-elliptic core index-guiding photonic crystal fiber (PCF) by using the first-order perturbation correction technique of mode analysis combined with semivectorial mode-field convergence method. The algorithm requires simple formulation, yields reasonably accurate results, and is efficient as needed for a design analysis; it is also applicable to any index-guiding regular structure/PCFs. The efficacy of our design tool has been verified with otherwise known solutions. Using this as a design recipe, we have realized PCFs with high birefringence characteristics.