Fourth-order Perturbation Theory Research Articles

Spin-Vibronic Dynamics in Open-Shell Systems beyond the Spin Hamiltonian Formalism.

Vibronic coupling has a dramatic influence over a large number of molecular processes, ranging from photochemistry to spin relaxation and electronic transport. The simulation of vibronic coupling with multireference wave function methods has been largely applied to organic compounds, and only early efforts are available for open-shell systems such as transition metal and lanthanide complexes. In this work, we derive a numerical strategy to differentiate the molecular electronic Hamiltonian in the context of multireference ab initio methods and inclusive of spin-orbit coupling effects. We then provide a formulation of open quantum system dynamics able to predict the time evolution of the electron density matrix under the influence of a Markovian phonon bath up to fourth-order perturbation theory. We apply our method to Co(II) and Dy(III) molecular complexes exhibiting long spin relaxation times and successfully validate our strategy against the use of an effective spin Hamiltonian. Our study sheds light on the nature of vibronic coupling, the importance of electronic excited states in spin relaxation, and the need for high-level computational chemistry to quantify it.

Vibrationally excited states of 1H- and 2H-1,2,3-triazole isotopologues analyzed by millimeter-wave and high-resolution infrared spectroscopy with approximate state-specific quartic distortion constants.

In this work, we present the spectral analysis of 1H- and 2H-1,2,3-triazole vibrationally excited states alongside provisional and practical computational predictions of the excited-state quartic centrifugal distortion constants. The low-energy fundamental vibrational states of 1H-1,2,3-triazole and five of its deuteriated isotopologues ([1-2H]-, [4-2H]-, [5-2H]-, [4,5-2H]-, and [1,4,5-2H]-1H-1,2,3-triazole), as well as those of 2H-1,2,3-triazole and five of its deuteriated isotopologues ([2-2H]-, [4-2H]-, [2,4-2H]-, [4,5-2H]-, and [2,4,5-2H]-2H-1,2,3-triazole), are studied using millimeter-wave spectroscopy in the 130-375GHz frequency region. The normal and [2-2H]-isotopologues of 2H-1,2,3-triazole are also analyzed using high-resolution infrared spectroscopy, determining the precise energies of three of their low-energy fundamental states. The resulting spectroscopic constants for each of the vibrationally excited states are reported for the first time. Coupled-cluster vibration-rotation interaction constants are compared with each of their experimentally determined values, often showing agreement within 500kHz. Newly available coupled-cluster predictions of the excited-state quartic centrifugal distortion constants based on fourth-order vibrational perturbation theory are benchmarked using a large number of the 1,2,3-triazole tautomer isotopologues and vibrationally excited states studied.

Two bridge-particle-mediated RET between chiral molecules.

The problem of resonance energy transfer between a pair of chiral molecules mediated by two electrically polarizable bridging particles is solved using molecular quantum electrodynamics theory. In this framework, a single virtual photon propagates between any two-coupled entities and is responsible for the conveyance of excitation energy from the emitter to the absorber. Electric dipole and quadrupole, and magnetic dipole couplings linear in the Maxwell fields are employed for donor and acceptor, while each mediator scatters two virtual photons and responds quadratically to the electric displacement field via its electric dipole polarizability. This enables fourth-order diagrammatic perturbation theory to be used to compute the probability amplitude for the process. Individual multipole moment contributions to the Fermi golden rule rate are then extracted for oriented and isotropic systems. Discriminatory transfer rates arise when either the donor or the acceptor are electric-magnetic dipole and the other has a pure multipole moment, or when both are chiral, with mixed electric dipole-quadrupole contributions vanishing in the fluid phase. The bridge-mediated transfer rate is found to be a maximum for a collinear geometry. Moreover, a multi-level model of the mediator is necessary for energy migration. Asymptotically limiting rates for arbitrary and collinear geometries are also obtained for one center purely electric dipolar and the other purely quadrupolar, or both donor and absorber purely quadrupolar. Understanding is gained of radiationless and radiative transfer mechanisms between chiral moieties in a dielectric medium.

First-principles calculations of double resonance Raman spectra for monolayer MoTe2

Since double resonance Raman (DRR) spectra are laser-energy dependent, the first-principles calculations of DRR for two-dimensional materials are challenging. Here, the DRR spectrum of monolayer MoTe$_2$ is calculated by home-made program, in which we combine {\em ab-initio} density-functional-theory calculations with the electron-phonon Wannier (EPW) method. Within the fourth-order perturbation theory, we are able to quantify not only the electron-photon matrix elements within the dipole approximation, but also the electron-phonon matrix elements using the Wannier functions. The reasonable agreement between the calculated and experimental Raman spectra is achieved, in which we reproduce some distinctive features of transition metal dichalcogenides (TMDCs) from graphene (for example, the dominant intervalley process involving an electron or a hole). Furthermore, we perform an analysis of the possible DRR modes over the Brillouin zone, highlighting the role of low-symmetry points. Raman tensors for some DRR modes are given by first principles calculations from which laser polarization dependence is obtained.

Multicomponent MP4 and the inclusion of triple excitations in multicomponent many-body methods.

This study implements the full multicomponent third-order (MP3) and fourth-order (MP4) many-body perturbation theory methods for the first time. Previous multicomponent studies have only implemented a subset of the full contributions, and the present implementation is the first multicomponent many-body method to include any connected triples contribution to the electron-proton correlation energy. The multicomponent MP3 method is shown to be comparable in accuracy to the multicomponent coupled-cluster doubles method for the calculation of proton affinities, while the multicomponent MP4 method is of similar accuracy as the multicomponent coupled-cluster singles and doubles method. From the results in this study, it is hypothesized that the relative accuracy of multicomponent methods is more similar to their single-component counterparts than previously assumed. It is demonstrated that for multicomponent MP4, the fourth-order triple-excitation contributions can be split into electron-electron and electron-proton contributions and the electron-electron contributions ignored with very little loss of accuracy of protonic properties.

Investigation of the Structural and Thermochemical Properties of [2.2.2]-Bicyclooctadiene Photoswitches.

Molecular photoswitches can under certain conditions be used to store solar energy in the so-called molecular solar thermal storage systems, which is an interesting technology for renewable energy solutions. The current investigations focus on the performance of seven different density functional theory (DFT) methods (B3LYP, CAM-B3LYP, PBE0, M06-2X, ωB97X-D, B2PLYP, and PBE0DH) when predicting geometries and thermochemical properties of the [2.2.2]-bicyclooctadiene (BOD) photoswitch. We find that all of the investigated DFT methods provide geometries that are in good agreement with those obtained using coupled cluster singles and doubles (CCSD) calculations. The dependence on the employed basis set is not large when predicting geometries. With respect to the thermochemical properties, we find that the M06-2X, CAM-B3LYP, PBE0, and ωB97X-D functionals all predict thermochemical properties that are in good agreement with the results of the CCSD, the CCSD including perturbative triples (CCSD(T)), and the explicitly correlated CCSD-F12 and CCSD(T)-F12 models. Lastly, for energy calculations, we tested the newly developed fourth-order cluster perturbation theory singles and doubles CPS(D-4) model, which in this study provides energy differences that are of CCSD and sometimes also CCSD(T) quality at a relatively low cost. We find that the CPS(D-4) model is an excellent choice for further investigation of BOD derivatives because accurate energies can be obtained routinely using this methodology. From the results, we also note that the predicted storage energies and storage energy densities for the BOD photoswitch are very large compared to other molecular solar thermal storage systems and that these systems could be candidates for such applications.

Polarization as a tuning parameter for Floquet engineering: Magnetism in the honeycomb, square, and triangular Mott insulators

Magnetic exchange couplings can be tuned by coupling to periodic light, where the frequency and amplitude are typically varied: a process known as Floquet engineering. The polarization of the light is also important, and in this paper, we show how different polarizations, including several types of unpolarized light, can tune the exchange couplings in distinct ways. Using unpolarized light, for example, it is possible to tune the material without breaking either time-reversal or any lattice symmetries. To illustrate these effects generically, we consider single-band Hubbard models at half-filling on the honeycomb, square and triangular lattices. We derive the effective Heisenberg spin models to fourth order in perturbation theory for arbitrary fixed polarizations, and several types of unpolarized light that preserve time-reversal and lattice symmetries. Coupling these models to periodic light tunes first, second and third neighbor exchange couplings, as well as ring exchange terms on the square and triangular lattices. Circularly polarized light induces chiral fields for the honeycomb and triangular lattices, which favors non-coplanar magnetism and potential chiral spin liquids. We discuss how to maximize the enhancement of the couplings without inducing heating.

Resonance energy transfer mediated by a chiral molecule.

The problem of resonant energy transfer (RET) between an electric dipole donor, D, and an electric dipole acceptor, A, mediated by a passive, chiral third-body, T, is considered within the framework of molecular quantum electrodynamics theory. To account for the optical activity of the mediator, magnetic dipole and electric quadrupole coupling terms are included in addition to the leading electric dipole interaction term. Fourth-order diagrammatic time-dependent perturbation theory is used to obtain the matrix element. It is found that the Fermi golden rule rate depends on pure multipole moment polarizabilities and susceptibilities of T, as well as on various mixed electric and magnetic multipole moment response functions. The handedness of T manifests through mixed electric-magnetic dipole and mixed electric dipole-quadrupole polarizabilities, which affect the rate and, respectively, require the use of fourth-rank and sixth-rank Cartesian tensor averages over T, yielding non-vanishing isotropic rate formulae applicable to a chiral fluid medium. Terms of a similar order of magnitude proportional to the product of electric dipole polarizability and either magnetic dipole susceptibility or electric quadrupole polarizability of T are also computed for oriented and freely tumbling molecules. Migration rates dependent upon the product of the pure electric dipole or magnetic dipole polarizability with the mixed electric-magnetic or electric dipole-quadrupole analogs, which require fourth- and fifth-rank Cartesian tensor averaging, vanish for randomly oriented systems. Asymptotically limiting rate expressions are also evaluated. Insight is gained into RET occurring in complex media.

Discriminatory Resonance Energy Transfer Mediated by a Third Molecule.

Relay of resonant excitation energy between two chiral molecules by an inert third particle is studied using molecular quantum electrodynamics theory. A single virtual photon propagates between each interacting pair. Fourth-order diagrammatic time-dependent perturbation theory is employed to compute the matrix element. Rate terms dependent upon the chirality of the donor and acceptor species are extracted using the Fermi golden rule. Interestingly, the mediated rate is discriminatory. For freely tumbling particles it exhibits an inverse-square dependence on each interparticle separation distance, indicating a purely radiative exchange mechanism. Furthermore, the isotropic rate is found to be a maximum for a collinear geometry and vanishes when the angle between the donor, mediator, and acceptor is 90°. The indirect rate is compared with direct transfer between two chiral molecules. Insight is gained into discriminatory migration of energy in a dielectric medium.

Non-degenerate fourth order perturbation theory

There exist several forms of perturbation theory in Quantum Mechanics, namely: non-degenerate perturbation theory, degenerate perturbation theory, time-dependent perturbation theory, time-constant perturbation theory and time-harmonic perturbation theory. This paper presents an extension of the perturbation theory for non-degenerate states. It has been observed that most texts and journal papers treat only the first –order and second-order non-degenerate perturbation theory. Therefore, this paper attempts to treat and present the third order and fourth order non-degenerate perturbation theory in quantum mechanics. It can thus be asserted that the higher the order of perturbations of quantum systems that we know, the more successful will be our efforts in suppressing or eliminating them.

Mediation of resonance energy transfer by two polarisable particles.

The molecular quantum electrodynamics theory is employed to calculate the matrix element and Fermi golden rule rate for resonant transfer of electronic excitation energy between a donor and an acceptor in the vicinity of two neutral electric dipole polarizable particles, which play the role of bridging species. The emitter and absorber couple linearly to the electric displacement field via their electric dipole moments, while each mediator interacts quadratically with this field through its dynamic polarizability. This form of interaction Hamiltonian enables fourth-order perturbation theory to be used to compute the probability amplitude together with summation over 24 time-ordered diagrams representing a single virtual photon exchange between each pair of coupled particles. Expressions for the migration rate mediated by two inert molecules are obtained for an arbitrary arrangement of the four species that are in fixed mutual orientation or are freely tumbling. These formulae are valid for all interparticle separation distances outside the orbital overlap region. From the general result, rate equations applicable to an equidistant collinear configuration of the four bodies are evaluated. Near- and far-zone limiting forms of the transfer rate for the relay pathway are also calculated and exhibit inverse sixth and inverse square dependences on relative separation distances between pairs of particles, confirming the short-range (radiationless) and long-range (radiative) energy transfer mechanisms associated with two-body theory. The distance behavior of interference terms between two-, three-, and four-body terms is also examined, and the relative importance of each contribution to the total transfer rate is discussed.

An equation of state for Stockmayer fluids based on a perturbation theory for dipolar hard spheres.

We develop a perturbation theory for the difference between the Helmholtz energy of a Stockmayer fluid, i.e., a fluid interacting by a Lennard-Jones plus point-dipole potential, and a Lennard-Jones fluid. We show that the difference can be approximated by the perturbational Helmholtz energy contribution of a dipolar hard-sphere fluid with a suitably chosen effective hard-sphere diameter, relative to a hard-sphere fluid with the same effective diameter. We analyze both a third and fourth order perturbation theory, both written as Padé approximations. Several recipes for calculating the hard-sphere diameter are investigated; we find that the Weeks-Chandler-Andersen diameter is most suitable. Results of the perturbation theory are shown to be in good agreement with reference data for the Helmholtz energy, internal energy, and isochoric heat capacity as obtained from molecular simulations performed in this work and to vapor-liquid equilibrium data from the literature. Theoretical predictions of the proposed model are compared to results from the perturbation theory of Gubbins and Twu [Chem. Eng. Sci. 33, 863 (1978)], which is a theory based on a Lennard-Jones reference fluid. We find the theories are in good agreement. Our approach can easily be applied to van der Waals potentials, other than Lennard-Jones potentials. If a dipolar Mie fluid is considered, the approach merely requires calculation of the effective hard-sphere diameter for a Mie potential. We further note that the approach has a reduction in the variable space of the underlying correlation integrals, i.e., the correlation functions of a hard-sphere fluid depend on density only, whereas the Lennard-Jones reference correlation functions depend on density and temperature.

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures.

The time evolution in open quantum systems, such as a molecular aggregate in contact with a thermal bath, still poses a complex and challenging problem. The influence of the thermal noise can be treated using a plethora of schemes, several of which decompose the corresponding correlation functions in terms of weighted sums of exponential functions. One such scheme is based on the hierarchical equations of motion (HEOM), which is built using only certain forms of bath correlation functions. In the case where the environment is described by a complex spectral density or is at a very low temperature, approaches utilizing the exponential decomposition become very inefficient. Here, we utilize an alternative decomposition scheme for the bath correlation function based on Chebyshev polynomials and Bessel functions to derive a HEOM approach up to an arbitrary order in the environmental coupling. These hierarchical equations are similar in structure to the popular exponential HEOM scheme, but are formulated using the derivatives of the Bessel functions. The proposed scheme is tested up to the fourth order in perturbation theory for a two-level system and compared to benchmark calculations for the case of zero-temperature quantum Ohmic and super-Ohmic noise. Furthermore, the benefits and shortcomings of the present Chebyshev-based hierarchical equations are discussed.

Bias loop corrections to the galaxy bispectrum

Combination of the power spectrum and bispectrum is a powerful way of breaking degeneracies between galaxy bias and cosmological parameters, enabling us to maximize the constraining power from galaxy surveys. Recent cosmological constraints treat the power spectrum and bispectrum on an uneven footing: they include one-loop bias corrections for the power spectrum but not the bispectrum. To bridge this gap, we develop the galaxy bias description up to fourth order in perturbation theory, conveniently expressed through a basis of Galilean invariants that clearly split contributions that are local and nonlocal in the second derivatives of the linear gravitational potential. In addition, we consider relevant contributions from short-range nonlocality (higher-derivative terms), stress-tensor corrections and stochasticity. To sidestep the usual renormalization of bias parameters that complicates predictions beyond leading order, we recast the bias expansion in terms of multipoint propagators, which take a simple form in our split-basis with loop corrections depending only on bias parameters corresponding to nonlocal operators. We show how to take advantage of Galilean invariance to compute the time evolution of bias and present results for the fourth-order parameters for the first time. We also discuss the possibilities of reducing the bias parameter space by using the evolution of bias and exploiting degeneracies between various bias contributions in the large-scale limit. Our baseline model allows to verify these simplifications for any application to large-scale structure data sets.

The Unified Theory of Resonance Energy Transfer According to Molecular Quantum Electrodynamics

An overview is given of the molecular quantum electrodynamical (QED) theory of resonance energy transfer (RET). In this quantized radiation field description, RET arises from the exchange of a single virtual photon between excited donor and unexcited acceptor species. Diagrammatic time-dependent perturbation theory is employed to calculate the transfer matrix element, from which the migration rate is obtained via the Fermi golden rule. Rate formulae for oriented and isotropic systems hold for all pair separation distances, R, beyond wave function overlap. The two well-known mechanisms associated with migration of energy, namely the R−6 radiationless transfer rate due to Förster and the R−2 radiative exchange, correspond to near- and far-zone asymptotes of the general result. Discriminatory pair transfer rates are also presented. The influence of an environment is accounted for by invoking the polariton, which mediates exchange and by introducing a complex refractive index to describe local field and screening effects. This macroscopic treatment is compared and contrasted with a microscopic analysis in which the role of a neutral, polarizable and passive third-particle in mediating transfer of energy is considered. Three possible coupling mechanisms arise, each requiring summation over 24 time-ordered diagrams at fourth-order of perturbation theory with the total rate being a sum of two- and various three-body terms.

Using fluorescence detected two-dimensional spectroscopy to investigate initial exciton delocalization between coupled chromophores.

Förster theory describes electronic exciton energy migration in molecular assemblies as an incoherent hopping process between donor and acceptor molecules. The rate is expressed in terms of the overlap integral between donor fluorescence and acceptor absorption spectra. Typical time scales for systems like photosynthetic antennae are on the order of a few picoseconds. Prior to transfer, it is assumed that the initially excited donor molecule has equilibrated with respect to the local environment. However, upon excitation and during the equilibration phase, the state of the system needs to be described by the full density matrix, including coherences between donor and acceptor states. While being intuitively clear, addressing this regime experimentally has been a challenge until the recently reported advances in fluorescence detected two-dimensional spectroscopy. Here, we demonstrate using fourth order perturbation theory the conditions for the presence of donor-acceptor coherence induced cross-peaks at zero waiting time between the first and the second pair of pulses. The approach is illustrated for a heterodimer model which facilitates an analytical solution.

Fourth-order vibrational perturbation theory with the Watson Hamiltonian: Report of working equations and preliminary results.

A derivation of fourth-order vibrational perturbation theory (VPT4) based on the Watson Hamiltonian in dimensionless rectilinear normal coordinates is presented. Terms that are linear and cubic in the (nk + ), with nk being the zeroth-order harmonic oscillator quantum numbers, appear in fourth order and extend the much simpler second-order vibrational perturbation theory model. The rather involved expressions for the fourth-order terms are derived with Rayleigh-Schrödinger perturbation theory, the process of verifying their correctness is described, and a computer code to generate the VPT4 constants from the potential energy surface derivatives is provided. The paper concludes with numerical examples featuring the H2O, Si2C, and cyclic-C3H2 molecules.

A third and fourth order perturbation theory for dipolar hard spheres

In this work, we develop and analyze a third order and a fourth order perturbation theory for non-polarizable dipolar hard spheres. The electrostatic potential is split into a short-ranged and a long-ranged part. A perturbation expansion is applied for the short-ranged potential where the contribution of higher order terms is approximated through a [2,1]-Padé approximation for the third order theory and by a [2,2]-Padé approximation for the fourth order theory. Rodgers and Weeks [J. Chem. Phys. 131, 244108 (2010)] developed the Local Molecular Field (LMF) theory for the long-ranged contribution of the electrostatic energy. The LMF theory requires the relative permittivity of the considered fluid. We apply a perturbation theory for the relative permittivity and show that the LMF is then in very good agreement with the results for the long-ranged energy from molecular simulations. The short-ranged contribution to the Helmholtz energy from the third order and from the fourth order perturbation theory is, for densities ρ* ≤ 1 and for dipole moments of μ*2 ≤ 1, in very good agreement with the data from molecular simulations. For larger dipole densities (up to μ*2 ≤ 4 for ρ* ≲ 1), we observe significantly improved results for the fourth order perturbation theory compared to the third order perturbation theory.

Geometric Energy Derivatives at the Complete Basis Set Limit: Application to the Equilibrium Structure and Molecular Force Field of Formaldehyde.

Geometric energy derivatives which rely on core-corrected focal-point energies extrapolated to the complete basis set (CBS) limit of coupled cluster theory with iterative and noniterative quadruple excitations, CCSDTQ and CCSDT(Q), are used as elements of molecular gradients and, in the case of CCSDT(Q), expansion coefficients of an anharmonic force field. These gradients are used to determine the CCSDTQ/CBS and CCSDT(Q)/CBS equilibrium structure of the S0 ground state of H2CO where excellent agreement is observed with previous work and experimentally derived results. A fourth-order expansion about this CCSDT(Q)/CBS reference geometry using the same level of theory produces an exceptional level of agreement to spectroscopically observed vibrational band origins with a MAE of 0.57 cm-1. Second-order vibrational perturbation theory (VPT2) and variational discrete variable representation (DVR) results are contrasted and discussed. Vibration-rotation, anharmonicity, and centrifugal distortion constants from the VPT2 analysis are reported and compared to previous work. Additionally, an initial application of a sum-over-states fourth-order vibrational perturbation theory (VPT4) formalism is employed herein, utilizing quintic and sextic derivatives obtained with a recursive algorithmic approach for response theory.

Origin of Biquadratic Exchange Interactions in a Mott Insulator as a Driving Force of Spin Nematic Order

We consider a series of Mott insulators in unit of two orbitals each hosting spin-1/2 electron, and by pairing two spin-1/2 into spin-1 triplet, derive the effective exchange interaction between the adjacent units via fourth order perturbation theory. It turns out that the biquadratic exchange interaction between spin-1, which is one of the origins of the nematic order, arises only in processes where the four different electrons exchange cyclically along the twisted loop, which we call "twisted ring exchange" processes. We show that the term becomes the same order with the Heisenberg exchange interactions when the on-orbital Coulomb interaction is not too large. Whereas, the inter-orbital Coulomb interactions give rise to additional processes that cancel the twisted ring exchange, and strongly suppresses the biquadratic term. The Mott insulator with two electrons on degenerate two orbitals is thus not an ideal platform to study such nematic orders.