Exact Hamiltonian Research Articles

The Vibrational Level Spectrum of H2O(X̃1A′) on a Partridge–Schwenke Potential up to the Dissociation Limit

We report accurate vibrational energy levels of H2O(X̃1A′) up to the dissociation limit on a recent ab initio potential energy surface of Partridge and Schwenke. Eigenvalues of the exact vibrational Hamiltonian in the Radau coordinates were obtained using the Lanczos algorithm. To improve computational efficiency, we utilized the extended symmetry-adapted discrete variable representation for the angular kinetic energy operator and adapted the hydrogen exchange symmetry in the radial coordinates using an index-swapping scheme. Normal mode assignments have been made to vibrational levels up to 20 000 cm−1, and several (local mode) stretching overtones up to the dissociation limit have been identified. Agreement with available experimental data is excellent.

A reduced-dimensionality quantum model which incorporates the full-dimensional energy of the system: Application to the vibrational predissociation of Cl2–Ne2

A reduced-dimensionality quantum model is proposed which incorporates the zero-point energy of the neglected modes in a systematic, natural way. In this model the reduced-dimensionality Hamiltonian is obtained by averaging the exact Hamiltonian over the dependence of the full-dimensional initial state of the neglected modes. The reduced Hamiltonian conserves all the terms of the full Hamiltonian, providing a more flexible description of the couplings between the modes considered explicitly in the model. The model is applied to simulate the vibrational predissociation dynamics of Cl2–Ne2, considering the three stretching modes of the complex. The results are compared to experimental data and to previous calculations using a reduced-dimensionality quantum model and a full-dimensional quantum–classical approach. The Cl2–Ne2 resonance lifetimes obtained agree only qualitatively with the experimental and previously calculated ones. By contrast, the present model predicts more correctly than previous calculations the behavior of the Cl2 fragment vibrational distributions observed experimentally. The applicability of the model is discussed and further refinements are suggested.

Quantum phases in artificial molecules

The few-particle state of carriers confined in a quantum dot is controlled by the balance between their kinetic energy and their Coulomb correlation. In coupled quantum dots, both can be tuned by varying the inter-dot tunneling and interactions. Using a theoretical approach based on the diagonalization of the exact Hamiltonian, we show that the transitions between different quantum phases can be induced through the inter-dot coupling both for a system of few electrons (or holes) and for aggregates of electrons and holes. We discuss their manifestations, in addition energy spectra (accessible through capacitance or transport experiments) and optical spectra.

Quantum dynamics in condensed phases via extended modes and exact interaction propagator relations

This paper presents a new approach to the study of quantum dynamics in condensed phases. The methodology is comprised of two main components. First, a formally exact method is described which allows the description of the liquid as a collection of coupled (through kinetic and potential coupling) harmonic modes. The modes are related to the Fourier modes of the component particle densities. Once the modes have been defined, a canonical transformation from the standard classical interparticle Hamilton function describes a new Hamilton function, which is exactly equivalent and defined on these harmonic coordinates. The final step in this section is the transformation of this Hamilton function into a quantum Hamiltonian operator. The second step in the process is the derivation of a new quantum mechanical evolution operator which is exact and allows the correction from a reference evolution operator, which is formed by adiabatic evolution on an approximate potential. A particular approximate potential which we suggest will be useful, is the collection of harmonic modes given in the Zwanzig Hamiltonian, weighted by the spectral density. Application of the reference interaction propagator methodology can then correct the approximate adiabatic evolution operator based on the approximate potential to the exact Hamiltonian of Fourier modes described above. A test problem of a double well nonlinearly coupled to a harmonic oscillator shows that the methodology obtains rapid numerical convergence. The paper closes with a description of how the methodology would be applied to a many-dimensional (hundreds of degrees of freedom) picture of reaction in a condensed phase.

Green’s functions and strength functions for stationary quantum systems

By using the analytic properties of the retarded Green’s function for a stationary quantum system, the strength function that coincides with the energy distribution of an unperturbed state of the system over its exact states in a perturbing field is constructed. It is shown that, in general, this strength function has the form of a Breit-Wigner distribution with energy-dependent parameters and that its moments are determined by the expectation values of various powers of the exact Hamiltonian for the wave function of the unperturbed state. The strength function averaged over a certain energy interval is calculated, and its properties are investigated for a global regime of averaging. The resulting strength functions are used to determine the mean field and the optical potential for nucleons in nuclei and to investigate conditions under which quantum chaos emerges in various systems.

Quantum description of five-particle systems in principal-axis hyperspherical coordinates

To describe a 5-particle system, ‘principal-axis hyperspherical’ coordinates (made up of one hyperradius and eight angles as internal coordinates) and three Euler angles as external (rotational) coordinates are used. A mathematical procedure to derive, in a systematic manner, the exact quantum mechanical Hamiltonian of the system in terms of these coordinates is presented. A generalized angular momentum vector operator, which allows the generation of a profitable standard representation for the angular part of the problem, is introduced.

Electrostatic energy calculation for the interpretation of scanning probe microscopy experiments

We discuss the correct expression for the classical electrostatic energy used while analysing scanning probe microscopy (SPM) experiments if either a conducting tip or a substrate or both are used in the experiment. For this purpose a general system consisting of an arbitrary arrangement of finite metallic conductors at fixed potentials (maintained by external sources) and a distribution of point charges in free space are considered using classical electrostatics. We stress the crucial importance of incorporating into the energy the contribution coming from the external sources (the `battery'). Using the Green function of the Laplace equation, we show in a very general case that the potential energy of point charges which are far away from metals is equally shared by their direct interaction and the polarization interaction due to charge induced in metals by the remote charges (the image interaction). When the charges are located close to the metals, there is an additional negative term in the energy entirely due to image interaction. The exact Hamiltonian of a quantum system interacting classically with polarized metal conductors is derived and its application in the Hartree-Fock and the density functional theories is briefly discussed. As an illustration of the theory, we consider an interaction of several point charges with a metal plane and a spherical tip, based on the set-up of a real SPM experiment. We show the significance of the image interaction for the force imposed on the tip.

A potential energy surface for the electronic ground state of CO2

The potential energy surface for the electronic ground state of CO2 is refined by means of a two-step variational procedure using the exact rovibrational Hamiltonian in the bond length–bond angle coordinates. In the refinement, the observed rovibrational energy levels for J=0–4 below 16,000 cm−1, obtained from the effective spectroscopic constants of CO2 given by Rothman et al. (J Quant Spectrosc Radiat Transfer 1992, 48, 537) in the HITRAN data base, are used as the input data points. The accurate ab initio force constants of Martin et al. (Chem Phys Lett 1993, 205, 535) are taken as the initial guess for the potential. The root-mean-square error of this fit to the 431 observed rovibrational energy levels is 0.05 cm−1. With the optimized potential energy surface, we also calculate the rovibrational energy levels of 13C16O2 and 12C18O2. The results are in good agreement with experimental data. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 269–280, 2000

Quantum calculations of highly excited vibrational spectrum of sulfur dioxide. III. Emission spectra from the C̃ 1B2 state

We report quantum mechanical calculations of the X̃←C̃ emission spectra of SO2 at the red wing of the C̃ absorption band. The near equilibrium potential energy surface of the C̃ 1B2 state is deduced by fitting experimental vibrational frequencies using an exact quantum mechanical Hamiltonian. Low-lying vibrational eigenenergies on this double minimum potential agree well with experimental frequencies and the corresponding eigenstates show some interesting features. Both spectral positions and intensities of the X̃←C̃ transitions are obtained up to 16 000 cm−1 using a Chebyshev based spectral method, which does not explicitly construct vibrational eigenfunctions in the X̃ state. The emission spectra are in reasonably good agreement with experimental measurements.

Hankel Singular Values and Vectors of a Class of Infinite-Dimensional Systems: Exact Hamiltonian Formulas for Control and Approximation Problems

This paper derives exact formulas for singular values and vectors of Hankel operators whose symbol is a product of a single-input single-output inner function and a multi-input multi-output rational function. This class of Hankel operators arises from the sensitivity minimization H∞ control problem with a rational weight function and the approximation problem of transfer functions having rational outer parts. It is shown that there is a Hamiltonian transcendental equation characterizing singular values which leads to a matrix function formula for singular vectors.

Exact solutions of relativistic two-body motion in lineal gravity

We develop the canonical formalism for a system of N bodies in lineal gravity and obtain exact solutions to the equations of motion for $N=2.$ The determining equation of the Hamiltonian is derived in the form of a transcendental equation, which leads to the exact Hamiltonian to infinite order in the gravitational coupling constant. In the equal mass case, explicit expressions of the trajectories of the particles are given as functions of the proper time, which show characteristic features of the motion depending on the strength of gravity (mass) and the magnitude and sign of the cosmological constant. As expected, we find that a positive cosmological constant has a repulsive effect on the motion, while a negative one has an attractive effect. However, some surprising features emerge that are absent for a vanishing cosmological constant. For a certain range of the negative cosmological constant, the motion shows a double maximum behavior as a combined result of an induced momentum-dependent cosmological potential and the gravitational attraction between the particles. For a positive cosmological constant, not only bounded motions but also unbounded ones are realized. The change of the metric along the movement of the particles is also exactly derived.

Principal-axis hyperspherical description of six-particle systems: quantum-mechanical treatment

Principal-axis hyperspherical coordinates (made up of one hyperradius and 11 angles as internal coordinates) and three Euler angles as external (rotational) coordinates are used to describe six-particle systems. The exact quantum mechanical Hamiltonian of the system in terms of these coordinates is established and is very simple. Properties of quasimomenta and their algebra are given.

Majorana fermion formulation of the two channel Kondo model

We show that a Majorana fermion description of the two channel Kondo model can emerge quite naturally as a representation of the algebra associated with the spin currents in the two channels. Using this representation we derive an exact equivalent Hamiltonian for the two channel model expressed entirely in terms of Majorana fermions. The part of the Hamiltonian that is coupled to the impurity spin corresponds to the vector of the σ–τ model (compactified two channel model). Consequently all the thermodynamic properties associated with the impurity spin can be calculated from the σ–τ model alone. The equivalent model can be used to confirm the interpretation of the many-body excitation spectrum of the low energy fixed point of the two-channel model as due to free Majorana fermions with appropriate boundary conditions.

N-chlorodifluoromethanimine, CF2=NCl: Quadrupole coupling and r structure

The microwave spectra of CF212=15N35Cl, CF212=15N37Cl, and CF213=14N35Cl have been recorded and analyzed. Rotational, quartic centrifugal distortion and quadrupole coupling constants were determined. The off-diagonal element of the chlorine quadrupole coupling tensor, χab, and the principal element of that tensor, χz, were obtained as 63.7 and −103.5 MHz, respectively, for Cl35 (average), and 51 and −82 MHz, respectively, for Cl37 in CF212=15N37Cl. During the analysis of the spectrum of the C13 species, it was necessary to diagonalize the complete exact Hamiltonian of the asymmetric rotor with two quadrupole nuclei because of irregular hyperfine patterns of some transitions caused by previously neglected matrix elements from the N14 quadrupole. Earlier data for the normal isotopic species were re-evaluated because neglected off-diagonal matrix elements of the Hamiltonian led to data for χab and χz inconsistent with the results obtained during this investigation. The rotational constants of all five isotopic species were used to determine the r0 structure. All structural parameters for the planar molecule except for the difference between the CF distances could be determined. The results for the distances r(C=N) and r(NCl) and the angle α(CNCl) were 1.254(27) Å, 1.713(17) Å, and 113.62(26)°, respectively. The experimental data for the structure, the inertial defect, and the centrifugal distortion constants are compared to results from ab initio calculations.

Exact Relativistic Two-Body Motion in Lineal Gravity

We consider the N-body problem in (1+1) dimensional lineal gravity. For 2 point masses (N=2) we obtain an exact solution for the relativistic motion. In the equal mass case we obtain an explicit expression for their proper separation as a function of their mutual proper time. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant.

Simulation Techniques in Parametric Hamiltonians

Using simulation techniques, a general definition of parametric Hamiltonians (Hpa), is presented. The minimal distance between a parametric and the exact Hamiltonian (Hexa) is defined in terms of the sum of the least distance between the corresponding energy functional components. A reference electronic energy functional for simulation of Hexa is proposed based on a CI energy functional. General properties of parametric Hamiltonians are analyzed considering reduction of the number of particles, integrals, and basis set, elimination of orthogonal transformations, and correlation and total energy functionals. Parametric Hamiltonians based on simulation of binding energy functionals and a simple parametrization scheme are proposed. Some preliminary tests have been performed with the parametrization of a characterized Hpa and calculations of organic (CH4, C2H2, C2H4, and C2H6) and transition metal (Ni2 and Ni5) systems.

Quantum mechanical study of the vibrational–rotational structure of [O2(3Σg-)]2. Part I

Starting with a previously defined potential energy surface for the singlet ground state of [O2(3Σg-)]2, we calculate the bound rovibrational states of the dimer for J⩽6 by solving the secular problem over the exact Hamiltonian, considering the two monomers as rigid. It follows the LC-RAMP treatment of Tennyson and van der Avoird, which includes centrifugal distortions and Coriolis interactions, using a product basis of analytical radial functions (Morse oscillators) and angular momentum eigenfunctions. The symmetries of the permutation-inversion group G16 are used to reduce the size of the calculations. We find that the (O2)2 dimer has 26 bound vibrational levels (J=0). The dissociation energy of the ground vibrational level, D0″=73.2 cm-1, is found to be in good agreement with previously predicted experimental values.

Many-body calculations for the relaxed (110) surface of GaAs

Quantum Monte Carlo methods are a stochastic approach to directly tackle the manybody problem in solids. They have proven to describe virutually exactly the ground state of correlated bulk systems, like the homogeneous electron gas or solids of C, Ge, Si and GaAs. Especially Variational quantum Monte Carlo calculations using nonlocal ab initio pseudopotentials offer a way to study systematically many-body effects at solid surfaces, safely founded on the variational principle “the lower the energy, the better the wave function”. Here we report on first attempts for the relaxed (110) surface of GaAs, serving as a prototype of semiconductor surfaces. A finite layer geometry is chosen as the boundary condition of the multidimensional stochastic integration scheme. The exact many-body Hamiltonian is cast in a form allowing for rapid evaluation. New parameters in the correlated trial wave function increase the variational freedom necessary to take into account the influence of the surface. Their physical meaning and their statistical significance are discussed in detail.

State-to-state three-atom reactive scattering using adiabatic rotation approximations

In this work we explore the applicability of the adiabatic rotation approximations to state-to-state quantum reactive scattering for three-body systems within the context of the coupled-channel hyperspherical coordinates method. A novel refinement to the adiabatic rotation approximation advocated by Bowman (Chem. Phys. Lett. 1994, 217, 36) has been obtained by the inclusion of the asymmetric rotational terms into the Hamiltonian. Within this approach, the total Hamiltonian is separated in an exact Hamiltonian for zero total angular momentum plus an effective rotational energy term that is added to the potential energy surface to form an effective potential on which the reactive dynamics evolves. Calculations of state-to-state reaction probabilities, integral and differential cross-sections for the prototypical reactions Cl+H2 and F+H2 are presented to test their accuracy and computational advantages in comparison with exact quantum mechanical calculations.

Non-Abelian density functional theory

Given a vector space of microscopic quantum observables, density functional theory is formulated on its dual space. A generalized Hohenberg-Kohn theorem and the existence of the universal energy functional in the dual space are proven. In this context ordinary density functional theory corresponds to the space of one-body multiplication operators. When the operators close under commutation to form a Lie algebra, the energy functional defines a Hamiltonian dynamical system on the coadjoint orbits in the algebra's dual space. The enhanced density functional theory provides a new method for deriving the group theoretic Hamiltonian on the coadjoint orbits from the exact microscopic Hamiltonian.