Fourth-order Perturbation Theory Research Articles

Thermodynamic perturbation theory of simple liquids: Monte Carlo simulation of a hard sphere system and the Helmholtz free energy of SW fluids

The Monte Carlo method is used to simulate similar sized hard sphere systems in a wide range of densities (from η 0.005 to 0.530 with a step of η = 0.005). The models are used to calculate the coefficients of the thermodynamic perturbation theory series for SW fluids up to the fourth order. The width of the attraction zone of the SW potential λ is varied from 1 to 2.5 sphere diameters. The analytical expressions approximating the obtained coefficients by polynomials with respect to the variables η and λ are determined. The absolute accuracy of the approximation is estimated to be better than ±0.001. All the necessary data for the calculation of the Helmholtz free energy of SW fluids up to the fourth-order perturbation theory are given.

Investigating the zero-field splitting parameters of 3d5 ions (Mn2+ and Fe3+) doped α-Al2O3 crystal

The characteristics of the local environment around paramagnetic centers formed by 3d5 ions (Mn2+ and Fe3+) in α-Al2O3 crystal are investigated using theoretical analysis. The zero-field splitting (ZFS) parameters (ZFSPs) for Mn2+ and Fe3+ ions located at the substitutional Al3+ sites and the interstitial sites are modeled using several modelling approaches through superposition model (SPM) and the fourth-order perturbation theory (PT) formula. The possible structural distortions resulting from Mn2+ and Fe3+ ions are determined around Al3+ sites with C3v symmetry and interstitial sites with D3d symmetry. The results are discussed in respect to the potential of the sites to be located by 3d5 ions and modelling approaches.

Formation and partial melting of two types of spin-cluster glass behavior in vanadate spinel

We report the doping effect on the various properties of spinels Co1−xZnxV2O4 (0 ≤ x ≤ 0.2). For the parent compounds, the rise in magnetization, the valley in thermal conductance, the transition from the ferromagnetic arrangement to non-collinear alignment indicated by the specific heat for the V sublattice, especially the frequency dependence of AC susceptibility around T1 = 59 K, verify the occurrence of the transition at T1 besides the ferrimagnetic transition at TC. The ferrimagnetic transition at TC induces the spin-cluster glass behavior and the transition at T1 yields the new spin-cluster glass (NSCG) behavior. As the Zn2+-doped content increases, the above phenomena are gradually weakening to vanishing, but the glassy behavior at TC still exists for all samples. Through the fourth-order perturbation theory, we discuss the reasons for the gradual vanishing of the transition at T1.

Spin exchange interaction with tunable range between graphene quantum dots

We study the spin exchange between two electrons localized in separate quantum dots in graphene. The electronic states in the conduction band are coupled indirectly by tunneling to a common continuum of delocalized states in the valence band. As a model, we use a two-impurity Anderson Hamiltonian which we subsequently transform into an effective spin Hamiltonian by way of a two-stage Schrieffer-Wolff transformation. We then compare our result to that from a Coqblin-Schrieffer approach as well as to fourth order 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.

Drawbacks of Applying Perturbative Schemes to Meson Spectroscopy

We study meson-meson scattering in a soluble model which describes asymptotically free mesons and confined quark-antiquark pairs via coupled channels. Concretely, the two scattered mesons are assumed to interact through s-channel meson-exchange diagrams. Furthermore, we develop a perturbative expansion of the model, and show that the thus found resonance pole positions, including contributions up to fourth order in perturbation theory, completely fail to reproduce the exact results. This shows that the resonance predictions based on perturbative approximations in quark models may be highly unreliable.

Density-operator approaches to transport through interacting quantum dots: Simplifications in fourth-order perturbation theory

Various theoretical methods address transport effects in quantum dots beyond single-electron tunneling while accounting for the strong interactions in such systems. In this paper we report a detailed comparison between three prominent approaches to quantum transport: the fourth-order Bloch-Redfield quantum master equation (BR), the real-time diagrammatic technique (RT), and the scattering rate approach based on the T-matrix (TM). Central to the BR and RT is the generalized master equation for the reduced density matrix. We demonstrate the exact equivalence of these two techniques. By accounting for coherences (nondiagonal elements of the density matrix) between nonsecular states, we show how contributions to the transport kernels can be grouped in a physically meaningful way. This not only significantly reduces the numerical cost of evaluating the kernels but also yields expressions similar to those obtained in the TM approach, allowing for a detailed comparison. However, in the TM approach an ad hoc regularization procedure is required to cure spurious divergences in the expressions for the transition rates in the stationary (zero-frequency) limit. We show that these problems derive from incomplete cancellation of reducible contributions and do not occur in the BR and RT techniques, resulting in well-behaved expressions in the latter two cases. Additionally, we show that a standard regularization procedure of the TM rates employed in the literature does not correctly reproduce the BR and RT expressions. All the results apply to general quantum dot models and we present explicit rules for the simplified calculation of the zero-frequency kernels. Although we focus on fourth-order perturbation theory only, the results and implications generalize to higher orders. We illustrate our findings for the single impurity Anderson model with finite Coulomb interaction in a magnetic field.

MASS OPERATOR OF THE INTERACTING ELECTRONS IN FORMALISM OF PERMUTATION GROUP THEORY

Double permutation (DP) method is developed here for designing Feynman diagrams for mass operator (MO) of interacting electrons in high orders of perturbation theory (PT). The derived expression allows finding the Young diagrams for the class of permutations corresponding to disconnect Feynman diagrams. The classification of DPs, carried out before, allows to identify the permutations corresponding to disconnected, singly connected (improper) diagrams and to derive expressions for intolerant cycles of permutations. Ordering the nonprimed digits in natural order in the cycles of DP, we avoid the permutations, corresponding to the Feynman diagrams of the same topology because of other numbering of nodes. Thus, the numbers of considered permutations is sufficiently reduced: (from 24 to 6 and from 720 to 42) in the second and the third orders of PT. All 414 expressions (diagrams) for MO in the fourth order of PT were designed using this method. The use of group theory allows us to conclude that no more Feynman diagrams can be designed. The developed method can be used as algorithm for Feynman diagrams designing for MO of interacting electrons (one sort fermions) in higher orders of PT.

Molecular QED of coherent and incoherent sum-frequency and second-harmonic generation in chiral liquids in the presence of a static electric field

Coherent second-order nonlinear optical processes are symmetry forbidden in centrosymmetric environments in the electric-dipole approximation. In liquids that contain chiral molecules, however, and which therefore lack mirror image symmetry, coherent sum-frequency generation is possible, whereas second-harmonic generation remains forbidden. Here we apply the theory of molecular quantum electrodynamics to the calculation of the matrix element, transition rate, and integrated signal intensity for sum-frequency and second-harmonic generation taking place in a chiral liquid in the presence and absence of a static electric field, to examine which coherent and incoherent processes exist in the electric-dipole approximation in liquids. Third- and fourth-order time-dependent perturbation theory is employed in combination with single-sided Feynman diagrams to evaluate two contributions arising from static field-free and field-induced processes. It is found that, in addition to the coherent term, an incoherent process exists for sum-frequency generation in liquids. Surprisingly, in the case of dc-field-induced second-harmonic generation, the incoherent contribution is found to always vanish for isotropic chiral liquids even though hyper-Rayleigh second-harmonic generation and electric-field-induced second-harmonic generation are both independently symmetry allowed in any liquid.

Padé-resummed fourth-order many-body perturbation theory: Applications to the description of Au–Au, Au–Hg, and Hg–Hg bonds

Pilot applications of Padé-resummed fourth-order many-body perturbation theory to the description of chemical bonds with strong contributions from the interactions of filled d-shells (Au–Au, Au–Hg, and Hg–Hg) are reported. The employed Padé approximants, especially the constrained [0/1,1] quadratic one, ensure radical improvements of accuracy with respect to the partial summation. The non-separability problem is discussed and bypassed through the direct resummation of perturbative expansions for differential correlation energies.

Theoretical proposal for superconducting spin qubits

We propose and theoretically investigate spin superconducting qubits. Spin superconducting qubit consists of a single spin confined in a Josephson junction. We show that owing to spin-orbit interaction, superconducting difference across the junction can polarize this spin. We demonstrate that this enables single qubit operations and more complicated quantum gates, where spins of different qubits interact via a mutual inductance of superconducting loop where the junctions are embedded. Recent experimental realizations of Josephson junctions made of semiconductor quantum dots in contact with superconducting leads have shown that the number of electrons in the quantum dot can be tuned by a gate voltage. Spin superconducting qubit is realized when the number of electrons is odd. We discuss the qubit properties at phenomenological level. We present a microscopic theory that enables us to make accurate estimations of the qubit parameters by evaluating the spin-dependent Josephson energy in the framework of fourth-order perturbation theory.

Weak-coupling quantum Monte Carlo calculations on the Keldysh contour: Theory and application to the current-voltage characteristics of the Anderson model

We present optimized implementations of the weak-coupling continuous-time Monte Carlo method defined for nonequilibrium problems on the Keldysh contour. We describe and compare two methods of preparing the system before beginning the real-time calculation: the "interaction quench" and the "voltage quench", which are found to be suitable for large and small voltage biasses, respectively. We also discuss technical optimizations which increase the efficiency of the real-time measurements. The methods allow the accurate simulation of transport through quantum dots over wider interaction ranges and longer times than have heretofore been possible. The current-voltage characteristics of the particle-hole symmetric Anderson impurity model is presented for interactions U up to 10 times the intrinsic level width Gamma. We compare the Monte Carlo results to fourth order perturbation theory, finding that perturbation theory begins to fail at U/Gamma>4. Within the parameter range studied we find no evidence for a splitting of the Kondo resonance due to the applied voltage. The interplay of voltage and temperature and the Coulomb blockade conductance regime are studied.

Fourth-order constant denominator perturbation theory

A constant denominator perturbation theory based on a zeroth-order Hamiltonian characterized by degenerate subsets of orbitals is developed to fourth order. This formulation allows a decoupling of the numerators of the perturbation sequence, allowing for a much more rapid evaluation of the resulting sums. Although the theory is general, a constant denominator is chosen as a scaled difference between the average occupied and average virtual orbital energies. The correction for this choice is folded back into the perturbation. The scale can be chosen such that the first-order wave-function yields the lowest possible variational bound, or it can be chosen based upon the sequence of energy terms and the size of the wave function corrections. Results are presented within the localized bond model utilizing both the Pariser-Parr-Pople and CNDO model Hamiltonians. When the scale is chosen variational the second-order theory yields a useful bound, usually containing most of the basis set correlation. The third order appears as a scaled Langhoff-Davidson correction. The fourth-order theory displays remarkable stability, even in those cases in which Nesbet-Epstein partitioning seems poorly converged and Moller-Plesset theory uncertain. Other possible scalings are introduced and discussed, but the utility of these must await the more accurate predictions of fifth order for comparison.

Electron correlation and properties of many-electron systems

Different computational methods of quantum chemistry, which are currently employed in accurate calculations of atomic and molecular properties, are discussed and their advantages and disadvantages are investigated in calculations of electric properties of Be, FH, and H2O. A complete fourth-order many-body perturbation theory (MBPT) approach based on the coupled Hartree–Fock (CHF) solution for the perturbed problem reveals that in several cases the fourth-order correlation correction to the CHF result is dominated by those terms which involve triply substituted intermediate states. Moreover, in some cases a complete fourth-order MBPT result appears to be inferior to those obtained in approximate MBPT treatments which involve only certain classes of intermediate states. Hence, a complete fourth-order MBPT approach may not be sufficient for accurate predictions of atomic and molecular properties. The applicability of the complete active space (CAS) SCF method is studied in calculations of electric properties of FH. Although this method offers some advantages over the MBPT approach, its usefulness may be limited by the requirement of rather large active orbital spaces which have to be used in order to achieve sufficiently high accuracy. The CAS SCF results can be improved by using multireference CI methods with reference functions built of the CAS SCF natural orbitals. A good performance of the MBPT method limited to singly and doubly substituted intermediate states suggests that reasonable results can also be obtained in similar configuration interaction (CI) calculations, provided the latter are corrected for the erratic treatment of unlinked clusters. It is shown that the unlinked cluster corrections due to Davidson and Siegbahn result in reasonable estimates of the total correlation contribution to different properties. Because of its fairly routine character the CI method limited to single and double substitutions may in this way provide useful data with relatively small computational effort.

Fourth-Order Perturbation Expansion for Hubbard Model on a Two-Dimensional Square Lattice

We investigate the Hubbard model on a two-dimensional square lattice by the perturbation expansion to the fourth order in the on-site Coulomb repulsion U. Numerically calculating all diagrams up to the fourth order in self-energy, we examine the convergence of perturbation series in the lattice system. We indicate that the coefficient of each order term rapidly decreases as in the impurity Anderson model for T > 0.1t in the half-filled case, but it holds in the doped case even at lower temperatures. Thus, we can expect that the convergence of perturbation expansion in U is very good in a wide parameter region also in the lattice system, except for T < 0.1t in the half-filled case. We next calculate the density of states in the fourth-order perturbation. In the half-filled case, the shape in a moderate correlation regime is quite different from the three peak structure in the second-order perturbation. Remarkable upper and lower Hubbard bands locate at w = +(-)U/2, and a pseudogap appears at the Fermi level w=0. This is considered as the precursor of the Mott-Hubbard antiferromagnetic structure. In the doped case, quasiparticles with very heavy mass are formed at the Fermi level. Thus, we conclude that the fourth-order perturbation theory overall well explain the asymptotic behaviors in a strong correlation regime.

Interatomic van der Waals potential in the presence of a magnetoelectric sphere

On the basis of a general formula obtained earlier via fourth-order perturbation theory within the framework of macroscopic quantum electrodynamics, the van der Waals potential between two neutral, unpolarized, ground-state atoms in the presence of a homogeneous, dispersing and absorbing magnetoelectric sphere is studied. When the radius of the sphere becomes sufficiently large, the result asymptotically agrees with that for two atoms near a planar interface. In the opposite limit of a very small sphere, the sphere can effectively be regarded as being a third ground-state atom, and the nonadditive three-atom van der Waals potential is recovered. To illustrate the effect of a sphere of arbitrary radius, numerical results are presented for the triangular arrangement where the atoms are at equidistance from the sphere, and for the linear arrangement where the atoms and the sphere are aligned along a straight line. As demonstrated, the enhancement or reduction of the interaction potential in the presence of purely electric or magnetic spheres can be physically understood in terms of image charges.

Theoretical study of the potential energy surface for the interaction of cisplatin and their aquated species with water

In this work, a systematic study of the interaction of neutral cisplatin ([Pt(NH(3))(2)Cl(2)]) and their charged aquated species ([Pt(NH(3))(2)Cl(H(2)O)](+) and [Pt(NH(3))(2)(H(2)O)(2)](2+)) with water was carried out. The potential energy surface (PES) was analyzed by considering 35 spatial orientations for the interacting species. The calculations were performed at various levels of theory including Moller-Plesset fourth order perturbation theory and density functional theory (DFT-B3LYP) using extended basis sets. Lennard-Jones (12-6) plus Coulomb classical potential was also used to assess the repulsion-dispersion and electrostatic contributions. The effect of atomic charges on the interaction energies is discussed using Mulliken, charges from electrostatic potential grid method and natural bond orbital schemes. The outcomes show that the electrostatic term plays a primary role on the calculation of interaction energies, with the absolute values of atomic charges from different approaches significantly affecting the overall interaction. Unusual results were revealed by basis set superposition error calculations for the structures located on the platinum-water PES.

Quantum electrodynamics of qubits

A systematic description of a spin one-half system endowed with magnetic moment or any other two-level system qubit interacting with the quantized electromagnetic field is developed. This description exploits a close analogy between a two-level system and the Dirac electron that comes to light when the two-level system is described within the formalism of second quantization in terms of fermionic creation and annihilation operators. The analogy enables one to introduce all the powerful tools of relativistic QED albeit in a greatly simplified form. The Feynman diagrams and the propagators turn out to be very useful. In particular, the QED concept of the vacuum polarization finds its close counterpart in the photon scattering off a two level system leading via the linear response theory to the general formulas for the atomic polarizability and the dynamic single spin susceptibility. To illustrate the usefulness of these methods, we calculate the polarizability and susceptibility up to the fourth order of perturbation theory. These ab initio calculations resolve some ambiguities concerning the sign prescription and the optical damping that arise in the phenomenological treatment. We also show that the methods used to study two-level systems qubits can be extended to many-level systems qudits. As an example, we describe the interaction with the quantized electromagnetic field of an atom with four relevant states: one S state and three degenerate P states. I. INTRODUCTION Two-level quantum systems, called qubits by Schumacher 1, play a fundamental role in quantum information theory. In this context they are usually treated as mathematical objects living in a two-dimensional Hilbert space. In reality, qubits always exist as material objects and we should not forget that they are endowed with concrete physical properties. In this paper we shall deal with two-level systems that interact directly with the electromagnetic field, such as spin one-half particles endowed with magnetic moment or twolevel atoms. Thus our results do not apply to qubits encoded in the polarization states of photons. We shall restrict ourselves in this paper to isolated qubits interacting only with the quantized electromagnetic field. Therefore the calculated decay rates will include only the spontaneous emission. A two-level system is the simplest model of a quantum system and yet in the presence of a coupling to the quantized electromagnetic field an exact solution has not been obtained. Even in the simplest case, when the electromagnetic field is restricted to just one mode, the model has been exactly solved only in the rotating-wave approximation by Jaynes and Cummings 2. Among the approximate solutions, perturbation theory is still the most universal and effective tool, especially in the world of electromagnetic phenomena. In the present paper we develop a systematic and complete theory based on an observation that a two-level system can be treated as a relativistic trapped electron. The translational degrees of freedom of such an electron are practically frozen. The only “degree of freedom” that remains is the electron’s ability to undergo transitions between two discrete energy states. In order to fully unfold the connection between the QED and the theory of two-level systems, we shall perform the second quantization of the standard theory of qubits. The description of two-level systems in terms of creation and annihilation operators has been introduced before cf., for example, 3 but no one has exploited the full potential of this formulation. The crucial element in our formulation is the systematic use of Feynman diagrams. To expose a close analogy with the relativistic theory, including the form of the propagators, we shall choose the energy scale in such a way that the energy levels of the two-level system have opposite signs. In this way, we arrive at a picture of a two-level system that coincides with the Dirac-sea view of quantum electrodynamics. The ground state of the two-level system corresponds to the occupation of the negative energy state, while the excited state corresponds to the occupation of the positive energy state accompanied by a hole in the negative energy sea. The transition between these two states due to the interaction with a photon can be represented by the two elementary Feynman diagrams shown in Fig. 1.

Two alternative derivations of the static contribution to the radiation-induced intermolecular energy shift

Two contrasting physical viewpoints and calculational approaches are adopted within the framework of molecular quantum electrodynamics for the evaluation of the static contribution to the change in mutual interaction energy between a pair of electric dipole polarizable molecules in an intense radiation field. This term arises when a real photon is scattered by the same molecular center with coupling between the two bodies occurring via exchange of a single virtual photon. In one method it is found that utilization of an effective three-photon interaction operator enables the energy shift to be obtained using second order perturbation theory with summation over only four time-ordered diagrams, each of which contain collapsed interaction vertices. The result is then shown to be obtained even more easily in a second approach that involves calculating the expectation values for both molecules in the ground electronic state and the field containing $N$ photons of mode $(\stackrel{P\vec}{k},\ensuremath{\lambda})$ of the electric dipole moments induced at each molecule by the incident field, which are coupled to the resonant dipole-dipole interaction tensor. The static contribution in question is shown to arise from the interaction of a permanent electric dipole moment in one species with the first hyperpolarizability of the other. Both methods are compared and contrasted with a previous computation in which contributions to the energy shift arising from 48 time-ordered diagrams were summed using fourth order perturbation theory.

Bond variations in the complexes [CuCl6]4−

An extended version of the angular overlap model developed from the fourth-order variational perturbation theory was used to explain why sd mixing for [CuCl6]4− and other pseudooctahedral complexes [CuL6]n− with the split ground state 2E is responsible for strong softening of the vibrational E mode, which results in lengthening and even cleavage of the bonds between the metal atom and the axial ligands.