Direct Perturbation Theory Research Articles

Analytical evaluation of first-order electrical properties based on the spin-free Dirac-Coulomb Hamiltonian

We report an analytical scheme for the calculation of first-order electrical properties using the spin-free Dirac-Coulomb (SFDC) Hamiltonian, thereby exploiting the well-developed density-matrix formulations in nonrelativistic coupled-cluster (CC) derivative theory. Orbital relaxation effects are fully accounted for by including the relaxation of the correlated orbitals with respect to orbitals of all types, viz., frozen-core, occupied, virtual, and negative energy state orbitals. To demonstrate the applicability of the presented scheme, we report benchmark calculations for first-order electrical properties of the hydrogen halides, HX with X = F, Cl, Br, I, At, and a first application to the iodo(fluoro)methanes, CH(n)F(3 - n)I, n = 0-3. The results obtained from the SFDC calculations are compared to those from nonrelativistic calculations, those obtained via leading-order direct perturbation theory as well as those from full Dirac-Coulomb calculations. It is shown that the full inclusion of spin-free (SF) relativistic effects is necessary to obtain accurate first-order electrical properties in the presence of fifth-row elements. The SFDC scheme is also recommended for applications to systems containing lighter elements because it introduces no extra cost in the rate-determining steps of a CC calculation in comparison to the nonrelativistic case. On the other hand, spin-orbit contributions are generally small for first-order electrical properties of closed-shell molecules and may be handled efficiently by means of perturbation theory.

Fourth-order relativistic corrections to electrical first-order properties using direct perturbation theory

In this work, we present relativistic corrections to first-order electrical properties obtained using fourth-order direct perturbation theory (DPT4) at the Hartree-Fock level. The considered properties, i.e., dipole moments and electrical-field gradients, have been calculated using numerical differentiation techniques based on a recently reported DPT4 code for energies [S. Stopkowicz and J. Gauss, J. Chem. Phys. 134, 064114 (2011)]. For the hydrogen halides HX, X=F, Cl, Br, I, and At, we study the convergence of the scalar-relativistic contributions by comparing the computed DPT corrections to results from spin-free Dirac-Hartree-Fock calculations. Furthermore, since in the DPT series spin-orbit contributions first appear at fourth order, we investigate their magnitude and judge the performance of the DPT4 treatment by means of Dirac-Hartree-Fock benchmark calculations. Finally, motivated by experimental investigations of the molecules CH(2)FBr, CHF(2)Br, and CH(2)FI, we present theoretical results for their halogen quadrupole-coupling tensors and give recommendations concerning the importance of higher-order scalar-relativistic and spin-orbit corrections.

Spectroscopic investigation of fluoroiodomethane, CH2FI: Fourier-transform microwave and millimeter-/submillimeter-wave spectroscopy and quantum-chemical calculations

Guided by theoretical predictions, the rotational spectrum of fluoroiodomethane, CH(2)FI, has been recorded and assigned. Accurate values are reported for the ground-state rotational constants, all quartic, sextic, and two octic centrifugal-distortion constants. The hyperfine structure of the rotational spectrum was thoroughly investigated using a Fourier-transform microwave spectrometer and the Lamb-dip technique in the millimeter-/submillimeter-wave region, thus allowing the accurate determination of the complete iodine quadrupole-coupling tensor and of the diagonal elements of the iodine spin-rotation tensor. Relativistic effects turned out to be essential for the accurate theoretical prediction of the dipole moment and quadrupole-coupling constants and were accounted for by direct perturbation theory and a spin-free four-component treatment based on the Dirac-Coulomb Hamiltonian. The relativistic corrections to the dipole moment amount to up to 34% and to the iodine quadrupole-coupling tensor to about 15-16% of the total values.

Direct perturbation theory in terms of energy derivatives: Fourth-order relativistic corrections at the Hartree–Fock level

In this work, the quantum-chemical treatment of relativistic effects by means of direct perturbation theory is extended from its lowest order, DPT2, to the next higher order, DPT4. The required theory is given in terms of energy derivatives with the DPT4 energy correction defined as the corresponding second derivative with respect to the relativistic perturbation parameter λ(rel) = c(2) and c as the speed of light. To facilitate the implementation in standard quantum-chemical program packages, a general formulation of DPT starting from a nonrelativistic Lagrangian is developed, thereby expanding both wave function and operators in terms of λ(rel). The corresponding expressions, which incorporate in an additive manner scalar-relativistic and spin-orbit contributions, are given at the Hartree-Fock level and have been implemented in the CFOUR program package using the available analytic second-derivative techniques. The accuracy of the DPT4 corrections at the HF level is investigated by comparison with rigorous four-component calculations. Scalar-relativistic and spin-orbit contributions are analyzed individually and the importance of the various terms to those corrections is discussed. Furthermore, the basis-set dependence of the computed DPT4 corrections is investigated.

Microwave, High-Resolution Infrared, and Quantum Chemical Investigations of CHBrF2: Ground and v4 = 1 States

A combined microwave, infrared, and computational investigation of CHBrF(2) is reported. For the vibrational ground state, measurements in the millimeter- and sub-millimeter-wave regions for CH(79)BrF(2) and CH(81)BrF(2) provided rotational and centrifugal-distortion constants up to the sextic terms as well as the hyperfine parameters (quadrupole-coupling and spin-rotation interaction constants) of the bromine nucleus. The determination of the latter was made possible by recording of spectra at sub-Doppler resolution, achieved by means of the Lamb-dip technique, and supporting the spectra analysis by high-level quantum chemical calculations at the coupled-cluster level. In this context, the importance of relativistic effects, which are of the order of 6.5% and included in the present work using second-order direct perturbation theory, needs to be emphasized for accurate predictions of the bromine quadrupole-coupling constants. The infrared measurements focused on the ν(4) fundamental band of CH(79)BrF(2). Fourier transform investigations using a synchrotron radiation source provided the necessary resolution for the observation and analysis of the rotational structure. The spectroscopic parameters of the v(4) = 1 state were found to be close to those of the vibrational ground state, indicating that the ν(4) band is essentially unaffected by perturbations.

Relativistic electron confined by isotropic parabolic potential

Energies and wave functions for quasibound states of a relativistic particle interacting with the field of a quadratic isotropic potential $V(r)=m{\ensuremath{\omega}}^{2}{r}^{2}/2$ are obtained using a power expansion method. The stabilization of autoionizing resonances is performed by introducing the spherical confinement of finite radius ${R}_{b}$. For comparison, the problem is investigated also in the framework of direct perturbation theory (DPT). It has been shown that the substantial degeneracy of nonrelativistic levels of the isotropic harmonic oscillator is completely removed. The relativistic effects, spin effects, fine-structure splittings, and radial densities of probability for several states are studied.

Near-field heat transfer between a nanoparticle and a rough surface

In this work we focus on the surface roughness correction to the near-field radiative heat transfer between a nanoparticle and a material with a rough surface utilizing a direct perturbation theory up to second order in the surface profile. We discuss the different distance regimes for the local density of states above the rough material and the heat flux analytically and numerically. We show that the heat transfer rate is larger than that corresponding to a flat surface at short distances. At larger distances it can become smaller due to surface polariton scattering by the rough surface. For distances much smaller than the correlation length of the surface profile, we show that the results converge to a proximity approximation, whereas in the opposite limit the rough surface can be replaced by an equivalent surface layer.

Relativistic corrections to electrical first-order properties using direct perturbation theory

Direct perturbation theory (DPT) is applied to compute relativistic corrections to electrical properties such as dipole moment, quadrupole moment, and electric-field gradient. The corrections are obtained as second derivatives of the energy and are given via method-independent expressions that involve the first derivative of the density matrix with respect to the relativistic perturbation as well as property integrals with additional momentum operators. Computational results obtained using Hartree-Fock (HF), second-order Moller-Plesset (MP2) perturbation theory, and the coupled-cluster singles and doubles approach augmented by a perturbative treatment of triple excitations are presented for the hydrogen halides HX with X=F, Cl, Br, (I, At) and the magnitude of relativistic effects, their basis-set dependence, and the limitations of DPT are discussed. We compare our results to those obtained using the second-order Douglas-Kroll method and benchmark them using four-component HF (Dirac-HF) and MP2 calculations. Relativistic effects are shown to be already important for elements of the third row (Na-Ar) when aiming at a high-accuracy quantum-chemical treatment. DPT provides reliable results for compounds containing elements up to the fourth period (K-Kr) and only breaks down when applied in lowest order to heavier elements. As a first application of the present DPT treatment for electrical properties, we report calculations for bromofluoromethane (CH(2)FBr) which was investigated using rotational spectroscopy by Cazzoli et al. [Mol. Phys. 106, 1181 (2008)] and for which consideration of relativistic effects turns out to be essential for good agreement between theory and experiment in the case of the bromine quadrupole-coupling constant.

Conserving approximations in direct perturbation theory: new semianalytical impuritysolvers and their application to general lattice problems

For the treatment of interacting electrons in crystal lattices, approximations based on thepicture of effective sites, coupled in a self-consistent fashion, have proven very useful.Particularly in the presence of strong local correlations, a local approach to theproblem, combining a powerful method for the short-ranged interactions with thelattice propagation part of the dynamics, determines the quality of results to alarge extent. For a considerable time the noncrossing approximation (NCA) indirect perturbation theory, an approach originally developed by Keiter for theAnderson impurity model, was a standard for the description of the local dynamics ofinteracting electrons. In the last couple of years exact methods like the numericalrenormalization group (NRG), as pioneered by Wilson, have surpassed this approximationas regarding the description of the low-energy regime. We present an improvedapproximation level of direct perturbation theory for finite Coulomb repulsionU, the crossing approximation 1 (CA1), and discuss its connections with other generalizations ofNCA. CA1 incorporates all processes up to fourth order in the hybridization strengthV in a self-consistent skeleton expansion, retaining the full energy dependence of the vertexfunctions. We reconstruct the local approach to the lattice problem from the point of viewof cumulant perturbation theory in a very general way and discuss the proper use ofimpurity solvers for this purpose. Their reliability can be tested in applications to, forexample, the Hubbard model and the Anderson-lattice model. We point out shortcomingsof existing impurity solvers and improvements gained with CA1 in this context.

Matrix formulation of direct perturbation theory of relativistic effects in a kinetically balanced basis

Direct perturbation theory (DPT) and its quasi-degenerate version (QD-DPT) in a matrix formulation, i.e. DPT-mat and QD-DPT-mat are derived from the matrix representation of the Dirac operator in a kinetically balanced basis, both in the intermediate and the unitary normalization. The results are compared with those of an earlier formulation in terms of operators and wave functions. In the wave function formulation it is imperative to describe the weak singularities of the wave function at the position of a point nucleus correctly and to satisfy the key relation between large and small components locally. This formulation is incompatible with an expansion in a regular basis. In a matrix formulation in a kinetically balanced basis both the large and the small component are expanded in regular basis sets and the key relation is only satisfied in the mean. DPT is essentially a theory at bispinor level. Although it is possible to eliminate the small component to arrive at a quasi-relativistic theory, this requires some care. A both compact and numerically stable formulation is in terms of the large and the small component. The generalization from a theory for one state to a quasi-degenerate formulation for a set of states, is very simple in the matrix formulation, but rather complicated and somewhat indirect at wave function level, where an intermediate quasi-relativistic step is needed. The advantage of the matrix formulation is particularly pronounced in the unitary normalization.

Direct perturbation theory for the dark soliton solution to the nonlinear Schrödinger equation with normal dispersion

After finding the basic solutions of the linearized nonlinear Schrödinger equation by the method of separation of variables, the perturbation theory for the dark soliton solution is constructed by linear Green's function theory. In application to the self-induced Raman scattering, the adiabatic corrections to the soliton's parameters are obtained and the remaining correction term is given as a pure integral with respect to the continuous spectral parameter.

Influence of Spectral Filtering and Nonlinear Gain on anOptical Soliton

Using the direct soliton perturbation theory, we investigate theevolution of soliton parameters and the first-order correction of brightsoliton in a system with linear and nonlinear gain (or loss) andspectral filtering in a comprehensive way. The results obtained by meansof our analytic method are consistent with numerical simulations. It isalso found that the stable soliton propagation which has beeninvestigated in a previous report by others is the limit case of ourresults.

Stability of dark soliton solutions of the quintic complex Ginzburg–Landau equation in the case of normal dispersion

Dark soliton solutions of the one-dimensional complex Ginzburg–Landau equation (CGLE) are analysed for the case of normal group-velocity dispersion. The CGLE can be transformed to the nonlinear Schrödinger equation (NLSE) with perturbation terms under some practical conditions. The main properties of dark solitons are analysed by applying the direct perturbation theory of the NLSE. The results obtained may be helpful for the research on the optical soliton transmission system.

Dissipative dynamics of dark solitons in elongated Bose–Einstein condensates

The dissipative dynamic stability is investigated of dark solitons in elongated Bose–Einstein condensates that can be described by the Gross–Pitaevskii equation including an additional term. Based on the direct perturbation theory for the nonlinear Schrodinger equation, the dependence of the soliton velocity on time is explicitly given, and the shape of dark solitons remaining unchanged under the dissipative condition is confirmed theoretically for the first time. It is found that the dynamically stable dark solitons turn out to be thermodynamically unstable.

Direct perturbation theory for a nearly integrable nonlinear equation with application to dark-soliton solutions

Since the linearized equation for a nearly integrable nonlinear equation is essentially a linear equation, the direct perturbation theory is now developed in the frame of Green's function theory of linear differential equations. The perturbed NLS equation with normal dispersion is treated as an example. Some obscure points in previous works about this equation are clarified.

Quasirelativistic theory I. Theory in terms of a quasi-relativistic operator¶

In this first part of a general analysis of a quasi-relativistic theory, i.e. a relativistic theory for electrons only, the transformation of the Dirac operator to an operator with two-component spinor solutions is studied, while a forthcoming second part will be devoted to a transformation at matrix level. We start with a simple derivation of the key relation between the upper (ϕ) and lower (χ) components of the Dirac bispinor (ψ) for both electrons and positrons. The three possible choices of a non-hermitian quasi-relativistic Hamiltonian L, a hermitian one with non-unit metric, and a hermitian one L + with unit metric are compared. The eigenfunctions of the first two are the upper components ϕ of ψ, while those of L + are the Foldy–Wouthuysen-type spinorsφ. Some general properties of quasi-relativistic Hamiltonians and their eigenfunctions are discussed, especially the behaviour near the position of a nucleus. Exact solutions, and even variational ones are only obtained if the orbital basis describes the weak singularities at the positions of the nuclei correctly. A new derivation of the quasi-relativistic version of direct perturbation theory (DPT) is given, followed by the theory of quasi-relativistic effective Hamiltonians, both non-hermitian and hermitian ones. The classical Foldy–Wouthuysen transformation is then presented as the singular limit of quasi-relativistic effective Hamiltonians with the model space extended to the entire space of positive-energy states. Finally, the problems that arise for a Douglas–Kroll transformation or the regular approximation at operator level are studied in detail. In part 2, it will be shown that everything becomes much simpler if one performs the transformation from relativistic to quasi-relativistic theory at matrix level. ¶This paper is dedicated to Andrzej Sadlej on the occasion of his 65th birthday.

Frequency-dependent hyperpolarizabilities of the Ne, Ar, and Kr atoms using the approximate coupled cluster triples model CC3

The frequency-dependent electric field-induced second harmonic generation (ESHG) second hyperpolarizabilities gamma of neon, argon, and krypton are calculated using the approximate coupled cluster triples model CC3. Systematic basis set investigations are carried out to establish basis set limits, and scalar relativistic effects are accounted for by direct perturbation theory. To estimate higher-order correlation effects, full configuration-interaction results are used to benchmark the accuracy of CC3. The best theoretical estimates obtained thereby for the static second hyperpolarizabilities gamma(0) are 107.4, 1159, and 2589 a.u. for neon, argon, and krypton, respectively. These values as well as the results for the dispersion curve of the parallel component gamma( parallel) agree well with the latest experimental values from electric field-induced second harmonic generation. In addition, the dispersion of the perpendicular component gamma( perpendicular) and the hyperpolarizability ratios gamma( parallel)gamma( perpendicular) has been studied for the first time on a consistently correlated ab initio level. The analysis of the results indicates that, in particular for neon and krypton, the presently available experimental values are flawed.

A novel analytic technique for the disintegration of coherent structure solutions of non-integrable and integrable nonlinear PDEs under forcing

We consider a technique for deriving exact analytic coherent structure (pulse/front/domain wall) solutions of general NLPDEs via the use of truncated invariant Painlevé expansions. Coupling this to Melnikov theory, we then consider the breakdown to chaos of such analytic coherent structure solutions of the Zakharov–Kuznetsov and reaction-diffusion equations under forcing. We also demonstrate that similar treatments are possible for integrable systems (using the well-studied forced KdV equation as an example) where the soliton/kink solutions represent the homoclinic/heteroclinic structures of the reduced ODEs. A method of treating the dynamics of the system prior to the onset of chaos by the use of intrinsic harmonic balance, multiscale or direct soliton perturbation theory is briefly discussed. It is conceivable that resummation of such perturbation series via the use of Fade approximants or other techniques may enable one to analytically follow the homoclinic or heteroclinic tangling beyond the first transversal intersection of the stable and unstable manifolds and into the chaotic regime.

Green's function method for perturbed DNLS equation with the squared spectrum parameter

A general direct perturbation theory for the perturbed DNLS equation is developed. The completeness relation is given by the Green's function method similar to that used for the linear equations. The effect of damping on a single soliton is calculated as an example, and the result is good with the squared spectrum parameter κ .

Calculating adiabatic evolution of the perturbed DNLS/MNLS solitons

Adiabatic evolution equations for parameters of the perturbed DNLS/MNLS solitons obtained by recently developed direct perturbation theory [Phys. Rev. E 65 (2002) 066608] are reformulated in simpler forms for practical use. A symbolic computation technique is developed to calculate them, based on the residue theorem. Effects of the third-order dispersion on the MNLS solitons are studied as an example.