Hamiltonian structure

Since we have discovered the underlying Hamiltonian structure of the QCMD model we are able to apply methods commonly used to construct suitable numerical integrators for Hamiltonian systems. Therefore we transform the QCMD equations (1) into the Liouville formalism. To this end, we introduce a new state z in the phase space, z = and define the nonlinear... 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

In the context of this paper, the most important conservation property of QCMD is related to its canonical Hamiltonian structure which implies the symplecticncs.s of the solution operator [1]. There are different ways to... 

For a space of eigenvectors of matrices of the gaussian orthogonal ensemble (k = N) the distribution of values of matrix elements of electromagnetic transition operators is gaussian, as follows from the central limit theorem. The ensemble averaging of hamiltonians guarantees that no correlations exist between the hamiltonian structure and the particular transition operator that is considered. 

There is also another way the mesoscopic time evolution Equation (55) can be introduced. We collect a list of well-established (i.e., well tested with experimental observations) time evolution equations on many different levels of description and try to identify their common features. This is indeed the way the time evolution Equation (55) has been first introduced. The Hamiltonian structure of the nondissipative part has been discovered first in the context of hydrodynamics by Clebsch (1895). Equations of the type (55) have started to appear in Dzyaloshinskii and Volovick (1980) and later in... 

At least in the traditional domains of chemical engineering and in the traditional core of instructions that chemical engineers receive during their education, fluid mechanics (transport phenomena) has played a key role. Also one of the principal motivations for creating nonequilibrium thermodynamics was an attempt to make fluid mechanics manifestly compatible with equilibrium thermodynamics. Even the noncanonical Hamiltonian structures that play such an important role in the multiscale nonequilibrium thermodyna mics presented in Section 3 have been first discovered... 

S.K. Gray and J.M. Verosky, Classical Hamiltonian structures in wave packet dynamics, J. Chem. Phys., 100 (1994) 5011. 

In three-dimensional flows the velocity field cannot be defined through a streamfunction, therefore the advection of fluid elements does not have the simple Hamiltonian structure as in two dimensions. One significant result on mixing in three dimensions is related to the existence of invariant surfaces in steady inviscid flows (Arnold, 1965). The velocity field of such flows is a solution of the time-independent Euler equation... 

Let us emphasize that the issues arising in the design and analysis of numerical methods for molecular dynamics are slightly different than those confronted in other application areas. For one thing the systems involved are highly structured having conservation properties such as first integrals and Hamiltonian structure. We address the issues related to the inherent structure of the molecular N-body problem in both this and the next chapter wherein we shall learn that symplectic discretizations are typically the most appropriate methods. 

Describing the Hamiltonian structure for a constrained system is a little complicated to do formally. The simplifying concept that we exploit is that the symplectic 2-form in the ambient space can be projected to the co-tangent bundle to define an associated symplectic form on the manifold. 

This example illustrates how the RPA can be used to derive explicit, analytical expressions for Hamiltonian structure factors. The generalization to blends that also contain copolymers is straightforward. 

Adler, M. On a trace functional for formal pseudodifferential operators and the Hamiltonian structure of Korteweg-de Vries type equations. Lecture Notes in Math, v. 755 (1979), 1-16. 

Kupershmidt, B. A., and Wilson G. Modifying Lax equations and the second Hamiltonian structure. Invent Math, 62 (1981) 403-436. 

Aref (1984) indicated that the equations which describe the particle trajectories in a two-dimensional flow have a Hamiltonian structure, that is. 

Using the Hamiltonian in equation Al.3.1. the quantum mechanical equation known as the Scln-ddinger equation for the electronic structure of the system can be written as... 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

Consider collisions between two molecules A and B. For the moment, ignore the structure of the molecules, so that each is represented as a particle. After separating out the centre of mass motion, the classical Hamiltonian that describes tliis problem is... 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

The interaction of the electron spin s magnetic dipole moment with the magnetic dipole moments of nearby nuclear spins provides another contribution to the state energies and the number of energy levels, between which transitions may occur. This gives rise to the hyperfme structure in the EPR spectrum. The so-called hyperfme interaction (HFI) is described by the Hamiltonian... 

They unfold a connection between parts of time-dependent wave functions that arises from the structure of the defining equation (2) and some simple properties of the Hamiltonian. 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

The NDCPA seems to be a very reasonable way to treat the properties of both electrons and excitons interacting with phonons with dispersion. In principal, the NDCPA can be applied to a system of the Hamiltonian with the electron(exciton)-phonon coupling terms of arbitrary structure. The NDCPA results in an algorithm which can be effectively treated numerically (for example, iteratively). The application of the NDCPA is not restricted to the... 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

Semiempirical calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

---