Harmonic Hamiltonian Research Articles

Algebraic methods applied to the study of energy transfer in anharmonic systems

We make use of two different methodologies to study the transition probabilities in a molecular anharmonic system in the presence of an external perturbation. For the first method, we use a series expansion of the displacement coordinate keeping up to fourth order terms; for the second method we use a deformed algebra to approximate the anharmonic Hamiltonian via a harmonic oscillator's Hamiltonian written in terms of deformed operators. We evaluate vibrational transition probabilities as a function of the collision energy and compare the results obtained with the two approaches.

Time-averaged quantum dynamics and the validity of the effective Hamiltonian model

We develop a technique for finding the dynamical evolution in time of an averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms in Lindblad form. Applying the general equation to harmonic Hamiltonians, we confirm a previous formula for the Effective Hamiltonian together with a new decoherence term which should in general be included, and whose vanishing provides the criteria for validity of the Effective Hamiltonian approach. Finally, we apply the theory to examples of the AC Stark Shift and Three- Level Raman Transitions, recovering a new decoherence effect in the latter.

FRACTIONAL QUANTUM EULER–CAUCHY EQUATION IN THE SCHRÖDINGER PICTURE, COMPLEXIFIED HARMONIC OSCILLATORS AND EMERGENCE OF COMPLEXIFIED LAGRANGIAN AND HAMILTONIAN DYNAMICS

Fractional quantum Euler–Cauchy equation in the Schrödinger picture is derived from the fractional action-like variational approach recently introduced by the author. Many interesting consequences are revealed and explored, in particular the emergence of complexified harmonic oscillators, complexified Lagrangian and Hamiltonian and complexified fractional action integral.

Excess of low frequency vibrational modes and glass transition: A molecular dynamics study for soft spheres at constant pressure

Using molecular dynamics at constant pressure, the relationship between the excess of low frequency vibrational modes (known as the boson peak) and the glass transition is investigated for a truncated Lennard-Jones potential. It is observed that the quadratic mean displacement is enhanced by such modes, as predicted using a harmonic Hamiltonian for metastable states. As a result, glasses loose mechanical stability at lower temperatures than the corresponding crystal, since the Lindemann criteria are observed, as is also deduced from density functional theory. Finally, we found that the average force and elastic constant are reduced in the glass due to such excess of modes. The ratio between average elastic constants can be approximated using the 2/3 rule between melting and glass transition temperatures.

Interference Dynamics Simulation of Anisotropic Condensates with Multiple Vortices in Alkali Bose Gases

We performed simulations of interference where two anisotropic condensates have multiple vortices. The problem of how the multiple zipper pattern (MZP) appears is investigated on the basis of the time-dependent Gross–Pitaevskii equation. The anisotropic multiple-vortex state, which is a linear combination of eigen states of anisotropic harmonic potential Hamiltonian is used to reproduce the observed MZP. Multiple vortices are expressed by high winding numbers in a calculation. In the isotropic case, the MZP can be found only in the central region with a low density due to the large vortex core. In the anisotropic and high winding number case, the MZP with several dislocations appears. In fact, the MZP is clearly seen because of high density. Each wave function of the anisotropic multiple-vortex state is divided into two parts. In short, four wave packets form the MZP. Both the large anisotropy and high winding number are required in reproducing the MZP.

Phonon diffusion in harmonic and anharmonic one-dimensional quasiperiodic lattices

The phonon diffusivity in one-dimensional quasiperiodic lattices is studied using harmonic and inharmonic Hamiltonians. This study is performed by solving the equations of motion using a time discretization and the leap-frog algorithm. For the case of harmonic Hamiltonians, the results show that the variance of a wave packet in quasiperiodic systems is proportional to the time, as in a periodic lattice, but their diffusion constant is lower. This behaviour is qualitatively different from the electronic case, in which the variance increases as a power law of the time, with an exponent that depends upon the strength of the quasiperiodic potential. The difference between the electronic and phonon problems seems to be related to the localization degree of their long wavelength modes. In this limit, we present the time evolution of the phonon wave extension, showing a finite sound velocity given by averaged lattice parameters. Finally, for the inharmonic case, we found that the phonon diffusivity decreases as the nonlinear perturbation grows.

First-Principles Calculation of 17O and 25Mg NMR Shieldings in MgO at Finite Temperature: Rovibrational Effect in Solids

The temperature dependence of (17)O and (25)Mg NMR chemical shifts in solid MgO have been calculated using a first-principles approach. Density functional theory, pseudopotentials, a plane-wave basis set, and periodic boundary conditions were used both to describe the motion of the nuclei and to compute the NMR chemical shifts. The chemical shifts were obtained using the gauge-including projector augmented wave method. In a crystalline solid, the temperature dependence is due to both (i) the variation of the averaged equilibrium structure and (ii) the fluctuation of the atoms around this structure. In MgO, the equilibrium structure at each temperature is uniquely defined by the cubic lattice parameters, which we take from experiment. We evaluate the effect of the fluctuations within a quasiharmonic approximation. In particular, the dynamical matrix, defining the harmonic Hamiltonian, has been computed for each equilibrium volume. This harmonic Hamiltonian was used to generate nuclear configurations that obey quantum statistical mechanics. The chemical shifts were averaged over these nuclear configurations. The results reproduce the previously published experimental NMR data measured on MgO between room temperature and 1000 degrees C. It is shown that the chemical shift behavior with temperature cannot be explained by thermal expansion alone. Vibrational corrections due to the fluctuations of atoms around their equilibrium position are crucial to reproduce the experimental results.

Entropy, Entanglement, and Area: Analytical Results for Harmonic Lattice Systems

We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures, and for classical correlations in classical harmonic lattice systems. These findings further suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible.

Harmonic Theory of Thermal Two-Photon Absorption in Benzene

A correlation function formalism is applied to compute the two-photon absorption spectrum of benzene. Using harmonic Hamiltonians for the ground and excited electronic states, we find that the theory agrees qualitatively with the experimentally observed sparsity of the thermal two-photon absorption spectrum as compared with the single-photon absorption spectrum. An expression for the average vibrational energy in the excited state is derived. We find that cooling of the nascent vibrational energy in the electronically excited state is not as extensive in the two-photon absorption process as compared to the single-photon case.

Harmonic lattice behavior of two-dimensional colloidal crystals.

Using positional data from videomicroscopy and applying the equipartition theorem for harmonic Hamiltonians, we determine the wave-vector-dependent normal mode spring constants of a two-dimensional colloidal model crystal and compare the measured band structure to predictions of the harmonic lattice theory. We find good agreement for both the transversal and the longitudinal modes. For q-->0, the measured spring constants are consistent with the elastic moduli of the crystal.

Ab initio calculations of mode selective tunneling dynamics in CH312OH and CH313OH

A simplified formulation of the harmonic reaction path Hamiltonian (RPH) approach is used to calculate mode specific tunneling splittings and stereomutation times in CH312OH and CH313OH. The experimental torsional spectrum is very well reproduced, as well as the few known isotope shifts. The mode specific changes in tunneling splitting are investigated for the excitation of fundamentals and OH stretching overtones. Good agreement between experiment and the RPH model is obtained, except for excitations of modes which are perturbed by anharmonic resonances. The inverted tunneling splittings (E level below A) experimentally observed for the fundamental transitions of the CH-stretching modes ν2 and ν9 and of the CH-rocking mode ν11 are shown to result from a pure symmetry effect and not from a breakdown of vibrational adiabaticity. Introducing a proper geometrical phase factor but retaining the adiabatic separation of the torsional dynamics yields calculated values of Δν̃2=−3.6 cm−1, Δν̃9=−3.2 cm−1, and Δν̃11=−8.2 cm−1 that are in satisfactory agreement with experimental data. Negative tunneling splittings are also predicted for the asymmetric CH-bending modes ν4 and ν10 and the CH3-rocking mode ν7. A smooth decrease of the tunneling splitting is calculated for increasing OH stretching excitation [Δν̃(ν1)=6.2 cm−1,…,Δν̃(6ν1)=1.5 cm−1] in quantitative agreement with experiment [Δν̃(ν1)=6.3 cm−1,…,Δν̃(6ν1)=1.6 cm−1]. The effect is shown to result in about equal parts from the increase of the effective torsional barrier and the effective lengthening of the OH bond.

Trans-[CrX2([15]aneN4)]+(X=F,CI)의 전자분광학과 리간드장 해석

trans-<TEX>$[CrX_2([15]aneN_4)]ClO_4 ([15]aneN_4=1,4,8,12-tetraazacyclopentadecane; X=F, Cl)$</TEX>의 전자흡수 스펙트럼을 리간드장 이론으로 해석하였다. AOMX 프로그램을 사용하여 관측한 스핀허용 전이와 계산값을 최적화시켰다. 결정한 CFT 파라미터를 AOM, NSH 및 여러 가지 다른 파라미터와 관련시켜 구하고, 이를 화학적 견지에서 논의하였다. 리간드장 해석으로부터 이들 착물에서는 F 리간드가 강한<TEX>${\sigma}-$</TEX> 및 <TEX>${\pi}-$</TEX>주개인 반면에 Cl 원자는 Cr(III) 이온에 약한 <TEX>${\sigma}-$</TEX> 및 <TEX>${\pi}-$</TEX>주개 성질이 있음을 확인할 수 있었다. The electronic absorption spectra of trans-<TEX>$[CrX_2([15]aneN_4)]ClO_4\;([15]aneN_4$</TEX>=1,4,8,12-tetraazacyclopentadecane; X=F, Cl) complexes have been interpreted using the ligand field theory. An AOMX program is used to optimize the differences between the calculated and the observed positions for the spin-allowed transition bands. The crystal field theory(CFT) parameter is directly related to the angular overlap model(AOM), normalized spherical harmonic hamiltonians(NSH), and semiempirical parameters. The various ligand field parameters are discussed in terms of their chemical significances. According to the ligand field analysis, we can confirm that the fluoride ligand is a very strong <TEX>${\sigma}-$</TEX> and <TEX>${\pi}-$</TEX>donor while the chloride has weak <TEX>${\sigma}-$</TEX> and <TEX>${\pi}-$</TEX>donor properties toward chromium(III) ion.

Mean field Ehrenfest quantum/classical simulation of vibrational energy relaxation in a simple liquid.

We give a detailed account of the statistical mechanical properties of the mean field Ehrenfest quantum/classical method as applied to liquid phase vibrational energy transfer using a simple harmonic oscillator model Hamiltonian. Depending on the shape of the initial quantum wave packet, a (partial) breakdown of detailed balance is observed, where the frictional response of the classical bath is only correlated to quasiclassical features of the evolving quantum state, i.e., a classical-like fluctuation-dissipation theorem holds. Only in the case of a coherent initial state (minimum uncertainty wave packet) does the mean field method produce physically meaningful results, namely, exponential relaxation (tau=tau(cl)) towards a quasiclassical equilibrium.

Third harmonic generation as a third-order quasi-exactly solvable system

The two degrees of freedom in the usual third harmonic generation effective Hamiltonian can be separated, both classically and quantum mechanically, into a harmonic oscillator and a nonlinear oscillator. In turn, the quantum Hamiltonian of the nonlinear oscillator can be written as a cubic polynomial in the generators of a quasi-exactly solvable Lie algebra, and the physically relevant eigenstates are precisely those that can be determined exactly (although, in general, not explicitly). Since the standard Jeffreys–Wentzel–Kramers–Brillouin methods are not easily applicable to general third-order differential equations, we use a Bohr–Sommerfeld quantization of the classical orbits to obtain approximate explicit formulas for the corresponding quantum eigenvalues.

Supersymmetric dynamical invariants

We address the problem of identifying the (nonstationary) quantum systems that admit supersymmetric dynamical invariants. In particular, we give a general expression for the bosonic and fermionic partner Hamiltonians. Due to the supersymmetric nature of the dynamical invariant the solutions of the time-dependent Schr\"odinger equation for the partner Hamiltonians can be easily mapped to one another. We use this observation to obtain a class of exactly solvable time-dependent Schr\"odinger equations. As applications of our method, we construct classes of exactly solvable time-dependent generalized harmonic oscillators and spin Hamiltonians.

Spin-wave damping in the two-dimensional ferromagneticXYmodel

$〈{\mathrm{LS}}_{\mathbf{q}}^{\ensuremath{\perp}}{\mathrm{LS}}_{\ensuremath{-}\mathbf{q}}^{\ensuremath{\perp}}〉,$ which is related to the effect of damping of spin waves in a two-dimensional classical ferromagnetic $\mathrm{XY}$ model, is considered. The damping rate ${\ensuremath{\Gamma}}_{\mathbf{q}}$ is calculated using the leading diagrams due to the quartic-order deviations from the harmonic spin Hamiltonian. The resulting four-dimensional integrals are evaluated by extending the techniques developed by Gilat and others for spectral density types of integrals. ${\ensuremath{\Gamma}}_{\mathbf{q}}$ is included into the memory function formalism due to Reiter and Solander, and Menezes, to determine the dynamic structure function $S(\mathbf{q},\ensuremath{\omega}).$ For the infinite sized system, the memory function approach is found to give nondivergent spin-wave peaks, and a smooth nonzero background intensity (``plateau'' or distributed intensity) for the whole range of frequencies below the spin-wave peak. The background amplitude relative to the spin-wave peak rises with temperature, and eventually becomes higher than the spin-wave peak, where it appears as a central peak. For finite-sized systems, there are multiple sequences of weak peaks on both sides of the spin-wave peaks whose number and positions depend on the system size and wave vector in integer units of $2\ensuremath{\pi}/L.$ These dynamical finite-size effects are explained in the memory function analysis as due to either spin-wave difference processes below the spin-wave peak or sum processes above the spin-wave peak. These features are also found in classical Monte Carlo--spin-dynamics simulations.

Low-temperature dynamics of the one-dimensional quantum XY model

We calculate the spin-spin relaxation function for a one-dimensional quantum XY ferromagnet, of arbitrary spins, at low temperature, using a projection operator technique. The static correlation functions, that are needed for the calculation of the dynamics, are evaluated using a self-consistent harmonic theory that renormalizes the bare harmonic hamiltonian. In the classical limit we have obtained an analytical expression for the time-dependent memory function, that agrees with previous results.

Derivation of the Self-Consistent Phonon Theory from Zubarev Type Green's Function

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Vibrational energy relaxation in the condensed phases: Quantum vs classical bath for multiphonon processes

We develop a theory for relating quantum and classical time correlation functions in the context of vibrational energy relaxation. The treatment is based on the assumption that both the quantum and the classical systems are characterized by effective harmonic Hamiltonians with identical normal modes; and the solute-solvent interaction is taken to be linear in the solute vibrational coordinate, but nonlinear in the bath coordinates. We propose an approximate “quantum correction” which allows the determination of the quantum energy relaxation rates from the classical force-force time correlation functions in the limit of large solute’s vibrational frequency. We test the accuracy of this approximate correction against exact numerical results for two forms of the solute-solvent interaction (exponential and power law), and find it to be accurate for a wide range of solute vibrational frequencies and for different solvent thermodynamic states. A simple form of the “quantum correction” is proposed for the models based on Lennard-Jones interactions. In all cases it is found that the vibrational relaxation time in a fully quantum system is better approximated by a fully classical theory (classical oscillator in classical bath) than by a mixed quantum-classical theory (quantum oscillator in classical bath).

Semiclassical analysis of a quasi-exactly solvable system: second harmonic generation

The usual second harmonic generation effective Hamiltonian is shown to be equivalent to a one-dimensional Schrodinger operator with a sextic polynomial potential. This operator belongs to the class of quasi-exactly solvable models, for which a finite number of eigenvalues and eigenvectors can be determined exactly. A Jeffreys-Wentzel-Kramers-Brillouin analysis provides accurate asymptotic expansions for these eigenvalues in the limit of large unperturbed energies.