An elementary but rigorous derivation is given for a variational principle guiding centre motion. The equations of motion resulting from the (the drift equations) possess exact conservation laws phase volume, energy (for time-independent systems), and angular momentum azimuthally symmetric systems). results carrying to higher order in adiabatic parameter are displayed. behaviour fields discussed, role clarified. application principles solution gyrokinetic discussed.

A Hamiltonian theory of guiding center motion which uses rectangular coordinates in physical space and noncanonical phase is presented. The averaging methods preserve two important features systems, viz., conservation energy (for time-independent fields) Liouville’s theorem. These are sacrificed by the traditional methods. also relieve much burden higher order perturbation calculations, drift equations for fully electromagnetic fields extended to one than they have been known past. first...

The problem of separating rotations from internal motions in systems such as macroscopic flexible bodies, atoms, molecules, nuclei, and solar is an old one, with many applications physics, chemistry, engineering. A new element, however, which has not been appreciated until fairly recently, the existence certain gauge fields on reduced configuration space for systems. These (non-Abelian) arise ``falling cat'' problem, changes shape induce external orientation; but they also have a dynamical...

A Hamiltonian treatment of the guiding center problem is given which employs noncanonical coordinates in phase space. Separation unperturbed system from perturbation achieved by using a coordinate transformation suggested theorem Darboux. As model to illustrate method, motion magnetic field B=B (x,y) ? studied. Lie transforms are used carry out expansion.

The traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented a way which clearly displays their differential-geometric foundations. These then generalized to the case noncanonical phase space. In new method H is treated, not as scalar space, but one component fundamental form p dq−Hdt. analysis applied this entire form, all its components.

Traditional approaches to the asymptotic behavior of coupled wave equations have difficulties in formulation a consistent version Bohr-Sommerfeld quantization conditions. These can be circumvented by using Weyl calculus diagonalize matrix operators. In analyzing diagonalized equations, geometric phases enter an important way, especially development rules. It turns out that Berry's phase is incorporated into symplectic structure ray space, influencing classical Hamiltonian orbits,...

Modern electronic structure theory is built around the Born-Oppenheimer approximation and construction of an Hamiltonian Ĥel(X) that depends on nuclear position X (and not momentum P). In this article, using well-known electron translation (Γ') rotational (Γ″) factors to couple transitions motion, we construct a practical phase-space both momentum, ĤPS(X,P). While classical dynamics run along eigensurfaces operator can recover many properties correctly, present some evidence motion ĤPS(X,P)...

We show that in the trace formula of Gutzwiller [J. Math. Phys. 8, 1979 (1967); 10, 1004 (1969); 11, 1791 (1970); 12, 343 (1971)] and Balian Bloch [Ann. (N.Y.) 60, 401 63, 592 (1971); 69, 76 (1972); 85, 514 (1974)], applied to systems two degrees freedom, Maslov index arising contribution from each periodic orbit is equal twice number times stable unstable manifolds wind around orbit. As a consequence, we find defined by either its or manifold. In this way it becomes apparent occurring an...

A framework for discrete variable representation (DVR) basis sets is developed that suitable multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators a central role and integrates semiclassical phase space concepts into basic framework. Rates convergence set expansions are emphasized, it shown method gives exponential convergence, assuming conditions analyticity boundary met....

We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends dynamics into complex phase space. Although we were able to give an accurate formulation of the had at time writing that paper only intuitive, heuristic understanding deeper causes make method work. A more mathematical was needed. To close this gap show in equivalence with first order expansion ℏ Schrödinger equation. further prove is equivalent stationary approximation, using...

We demonstrate that, for systems with spin–orbit coupling and an odd number of electrons, the standard fewest switches surface hopping algorithm does not conserve total linear or angular momentum. This lack conservation arises so much from direction (which is easily adjusted) but more generally propagating adiabatic dynamics along surfaces that are time reversible. show one solution to this problem run eigenvalues phase-space electronic Hamiltonians H(R, P) (i.e., depend on both nuclear...

We show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also previously proposed effective equations motion [M. Amano K. Takatsuka, “Quantum fluctuation electronic wave-packet coupled with classical nuclear motions,” J. Chem. Phys. 122, 084113 (2005) V. Krishna, “Path integral formulation for quantum nonadiabatic the mixed limit,” 126, 134107 (2007)] involving non-Abelian Berry force do maintain...

Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove angular component derivative coupling between states J and K. In a previous set papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135–1140 (2020)] posited approach for such removal based on direct projection, while we isolated second by constructing differentiating rotationally invariant basis. Unfortunately, neither was able demonstrate one-electron operatorÔ whose matrix element JÔK coupling....

We derive generalizations of the semiclassical trace formula Gutzwiller [J. Math. Phys. 12, 343 (1971)] and Balian Bloch [Ann. 69, 76 (1972)] that are valid for systems exhibiting continuous symmetries. In particular, we consider symmetries which associated set conserved quantities Poisson-commute. For these systems, periodic orbits a given energy occur in families usual formula, is only when isolated, does not apply. formulas derive, density states determined by sum over rather than...

We consider the effect of a continuous family neutral (bouncing ball) orbits on energy spectrum quantized stadium billiard. Using semiclassical approximation we derive analytic expressions for standard two-point spectral measures. The corrections due to bouncing ball account some non-generic features observed in analysis cavity which was recently measured. Once contributions are subtracted, is shown be well reproduced by semi-classical trace formula based unstable periodic orbits. also study...

A new formula is presented for computing Maslov indices in integrable and near-integrable Hamiltonian systems. For several kinds of applications the particularly easy to use. It does not rely on counting caustics or other discontinuities. Its theoretical justification calls wave-packet concepts topological properties group symplectic matrices. Techniques are also manipulating index analytical expressions.

Trace formulas provide the only general relations known connecting quantum mechanics with classical in case that motion is chaotic. In particular, they connect quantal objects such as density of states periodic orbits. this paper, several trace formulas, including those Gutzwiller, Balian and Bloch, Tabor, Berry, are examined from a geometrical standpoint. New forms amplitude determinant asymptotic theory developed tools for examination. The meaning caustics these revealed terms...

This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The uses Wigner-Weyl correspondence to map nuclear operators into functions on classical phase space Moyal star product represent operator multiplication those functions. These are explained body paper. result is power series κ2, where κ = (m/M)1/4 usual parameter. lowest term approximation, while terms corrections....

We show that following the standard mantra of quantum chemistry and diagonalizing Born-Oppenheimer (BO) Hamiltonian ĤBO(R) is not optimal means to construct potential energy surfaces. A better approach diagonalize a phase-space electronic Hamiltonian, ĤPS(R, P), which parameterized by both nuclear position R momentum P. Such nonperturbative can be constructed using partial Wigner transform method has exactly same cost as BO for semiclassical calculation (and only slight increase in...

The semiclassical mechanics of the Wigner 6j-symbol is examined from standpoint WKB theory for multidimensional, integrable systems, to explore geometrical issues surrounding Ponzano-Regge formula. relations among methods Roberts and others deriving formula are discussed, a new approach, based on recoupling four angular momenta, presented. A generalization Yutsis-type spin network developed this purpose. Special attention devoted symplectic reduction, reduced phase space (the 2-sphere...

Nuclear Berry curvature effects emerge from electronic spin degeneracy and canlead to non-trivial spin-dependent (nonadiabatic) nuclear dynamics. However, such are completely neglected in all current mixed quantum-classical methods as fewest switches surface-hopping. In this work, we present a phase-space surface-hopping (PSSH) approach simulate singlet-triplet intersystem crossing We show that with simple pseudo-diabatic ansatz, PSSH algorithm can capture the relevant make predictions...

This paper concerns the representation of angular momentum operators in Born–Oppenheimer theory polyatomic molecules and various forms associated conservation laws. Topics addressed include question whether these laws are exactly equivalent or only to some order parameter κ = (m/M)1/4 what correlation is between quantum numbers representations. These questions both problems involving a single potential energy surface those with multiple, strongly coupled surfaces electrostatic model for...