Search

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 the total linear or angular momentum. This lack of conservation arises not so much from the hopping direction (which is easily adjusted) but more generally from propagating adiabatic dynamics along surfaces that are not time reversible. We show that one solution to this problem is to run along eigenvalues of phase-space electronic Hamiltonians H(R, P) (i.e., electronic Hamiltonians that depend on both nuclear position and momentum) with an electronic–nuclear coupling Γ · P [see Eq. (25)], and we delineate the conditions that must be satisfied by the operator Γ. The present results should be extremely useful as far as developing new semiclassical approaches that can treat systems where the nuclear, electronic orbital, and electronic spin degrees of freedom altogether are all coupled together, hopefully including systems displaying the chiral-induced spin selectivity effect. [ABSTRACT FROM AUTHOR]

Discrete variable representation (DVR) basis sets on the radial half-line, based on Bessel functions, are presented. These are Hankel transforms of the eigenfunctions of the particle in a spherical box in k space, but there is no box or bound on the radial variable r. The grid points extend to infinity on the r axis. The DVR functions are exactly orthonormal and exactly satisfy the interpolation properties usually associated with DVR functions. The exact matrix elements of the kinetic energy are computed, and the use of the Bessel DVR functions in radial eigenvalue problems is illustrated. The phase space or semiclassical interpretation of the Bessel DVR functions is presented, and variations on these functions, corresponding to alternative boundary conditions in k space, are discussed. An interesting feature of Bessel DVR functions is that they are based on a finite basis representation that is continuously infinite. [ABSTRACT FROM AUTHOR]

A new discrete variable representation (DVR) basis is presented, one that is based on Airy functions. That is, the functions of the "finite basis representation" (actually infinite in this case) are energy eigenfunctions in a constant force field. The exact matrix elements of the kinetic energy are computed. The use of the Airy DVR functions in diagonalizing a one-dimensional Hamiltonian is illustrated. [ABSTRACT FROM AUTHOR]

This paper explores the difficulties of constructing multidimensional discrete variable representation (DVR) basis sets and the strategies that can be used to overcome them. A parameter count shows that the conditions on a DVR basis set cannot be satisfied on most spaces of wave functions. One-dimensional, orthogonal polynomials are an exception, but the Y[sub lm]'s on the sphere only go 3/4 of the way, in a certain sense, toward supplying enough parameters to satisfy the DVR conditions. It is shown that DVR sets involving rotationally invariant subspaces of wave functions on the sphere (consisting of complete subshells only) exist only for small values of the angular momentum cutoff. However, an exploration of DVR sets invariant under subgroups of the full rotation group leads to the discovery of a 12-point DVR set that is invariant under the tetrahedral group, whose grid points are the vertices of an icosahedron. [ABSTRACT FROM AUTHOR]

A framework for discrete variable representation (DVR) basis sets is developed that is suitable for multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators in a central role and integrates semiclassical and phase space concepts into the basic framework. Rates of convergence of basis set expansions are emphasized, and it is shown that the DVR method gives exponential convergence, assuming conditions of analyticity and boundary conditions are met. A discussion of nonorthogonal generalizations of DVR functions is presented, in which it is shown that projected δ-functions and interpolating functions form a biorthogonal basis. It is also shown that one of the generalized DVR proposals due to Szalay [J. Chem. Phys. 105, 6940 (1996)] gives exponential convergence.© 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Examples of nontrivial (that is, non-Cartesian product) multidimensional discrete variable representation (DVR) basis sets are presented that are generalizations of sinc functions in one dimension. Their use in solving quantum problems in two dimensions is illustrated. Unlike all standard (one-dimensional) examples of DVR bases, these bases cannot be created by dividing out the roots of a generating function. It is argued that the difficulty of constructing nontrivial, multidimensional DVR bases is due to the restrictive nature of the DVR conditions, which cannot be satisfied on most subspaces of wave functions. The bases considered in this paper, however, are invariant under translations on a lattice in n-dimensional space, and the properties of the Abelian group of lattice translations allow the DVR conditions to be satisfied. More generally, the question of the relation between group theory and the conditions necessary to qualify a set of basis functions as a DVR basis is considered. It is shown how to construct orthonormal states that are related by the action of some group, and, in the case of Abelian groups, the extra conditions required to qualify the basis as a DVR set are discussed. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove the angular component of the derivative coupling between states J and K . In a previous set of papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135–1140 (2020)] posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating a rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a one-electron operator O ̂ whose matrix element J O ̂ K was the angular component of the derivative coupling. Here, we show that a one-electron operator can, in fact, be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and are of immediate use for FSSH calculations. [ABSTRACT FROM AUTHOR]

Modern electronic structure theory is built around the Born–Oppenheimer approximation and the construction of an electronic Hamiltonian H ̂ e l (X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ′) and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, H ̂ P S (X , P). While classical Born–Oppenheimer dynamics that run along the eigensurfaces of the operator H ̂ e l (X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of H ̂ P S (X , P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general. [ABSTRACT FROM AUTHOR]

We show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also show that previously proposed effective Ehrenfest equations of motion [M. Amano and K. Takatsuka, "Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions," J. Chem. Phys. 122, 084113 (2005) and V. Krishna, "Path integral formulation for quantum nonadiabatic dynamics and the mixed quantum classical limit," J. Chem. Phys. 126, 134107 (2007)] involving the non-Abelian Berry force do maintain momentum conservation. As a numerical example, we investigate the Kramers doublet of the methoxy radical using generalized Hartree–Fock with spin–orbit coupling and confirm that angular momentum is conserved with the proper equations of motion. Our work makes clear some of the limitations of the Born–Oppenheimer approximation when using ab initio electronic structure theory to treat systems with unpaired electronic spin degrees of freedom, and we demonstrate that Ehrenfest dynamics can offer much improved, qualitatively correct results. [ABSTRACT FROM AUTHOR]

For a system without spin–orbit coupling, the (i) nuclear plus electronic linear momentum and (ii) nuclear plus orbital electronic angular momentum are good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic transition, there should be no change in the total linear or angular momentum. Now, the standard surface hopping algorithm ignores the electronic momentum and indirectly equates the momentum of the nuclear degrees of freedom to the total momentum. However, even with this simplification, the algorithm still does not conserve either the nuclear linear or the nuclear angular momenta. Here, we show that one way to address these failures is to dress the derivative couplings (i.e., the hopping directions) in two ways: (i) we disallow changes in the nuclear linear momentum by working in a translating basis (which is well known and leads to electron translation factors) and (ii) we disallow changes in the nuclear angular momentum by working in a basis that rotates around the center of mass [which is not well-known and leads to a novel, rotationally removable component of the derivative coupling that we will call electron rotation factors below, cf. Eq. (96)]. The present findings should be helpful in the short term as far as interpreting surface hopping calculations for singlet systems (without spin) and then developing the new surface hopping algorithm in the long term for systems where one cannot ignore the electronic orbital and/or spin angular momentum. [ABSTRACT FROM AUTHOR]

The Darboux-Christoffel formula is a closed-form expression for the kernel of the operator that projects onto the first N of a system of one-dimensional polynomials, orthonormal with respect to some weighting function. It is a key element in the theory of Gaussian integration and in the theory of discrete variable representation or Lagrangian mesh methods for diagonalizing quantum Hamiltonians of a few degrees of freedom. The one-dimensional Darboux-Christoffel formula turns out to have a generalization that is valid in a semiclassical or asymptotic sense for a wider class of orthonormal functions than orthonormal polynomials. This class consists of the bound eigenfunctions of one-dimensional Hamiltonians with time-reversal invariance, such as kinetic-plus-potential Hamiltonians. It also has certain generalizations involving the unbound eigenfunctions of such Hamiltonians. [ABSTRACT FROM AUTHOR]

The rovibrational kinetic energy for an arbitrary number of rigid molecules is computed. The result has the same general form as the kinetic energy in the molecular rovibrational Hamiltonian, although certain quantities are augmented to account for the rotational energy of the monomers. No specific choices of internal coordinates or body frame are made in order to accommodate the large variety of such conventions. However, special attention is paid to how key quantities transform when these conventions are changed. An example system is analysed explicitly as an illustration of the formalism. [ABSTRACT FROM AUTHOR]

Trace formulas provide the only general relations known connecting quantum mechanics with classical mechanics in the case that the classical motion is chaotic. In particular, they connect quantal objects such as the density of states with classical periodic orbits. In this paper, several trace formulas, including those of Gutzwiller, Balian and Bloch, Tabor, and Berry, are examined from a geometrical standpoint. New forms of the amplitude determinant in asymptotic theory are developed as tools for this examination. The meaning of caustics in these formulas is revealed in terms of intersections of Lagrangian manifolds in phase space. The periodic orbits themselves appear as caustics of an unstable kind, lying on the intersection of two Lagrangian manifolds in the appropriate phase space. New insight is obtained into the Weyl correspondence and the Wigner function, especially their caustic structures. [ABSTRACT FROM AUTHOR]

Focuses on the application of phase integral or WKB theory to multicomponent wave equations. Examination of the case of coupled channel equations in atomic or molecular scattering theory in the Born-Oppenheimer approximation; Investigation of the problem of mode conversion, also called surface jumping or Landau-Zener-Stuckelberg transitions.

Chemical relaxation phenomena, including photochemistry and electron transfer processes, form a vigorous area of research in which nonadiabatic dynamics plays a fundamental role. However, for electronic systems with spin degrees of freedom, there are few if any applicable and practical quasiclassical methods. Here, we show that for nonadiabatic dynamics with two electronic states and a complex-valued Hamiltonian that does not obey time-reversal symmetry (as relevant to many coupled nuclear-electronic-spin systems), the optimal semiclassical approach is to generalize Tully's surface hopping dynamics from coordinate space to phase space. In order to generate the relevant phase-space adiabatic surfaces, one isolates a proper set of diabats, applies a phase gauge transformation, and then diagonalizes the total Hamiltonian (which is now parameterized by both R and P). The resulting algorithm is simple and valid in both the adiabatic and nonadiabatic limits, incorporating all Berry curvature effects. Most importantly, the resulting algorithm allows for the study of semiclassical nonadiabatic dynamics in the presence of spin–orbit coupling and/or external magnetic fields. One expects many simulations to follow as far as modeling cutting-edge experiments with entangled nuclear, electronic, and spin degrees of freedom, e.g., experiments displaying chiral-induced spin selectivity. [ABSTRACT FROM AUTHOR]

We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of h of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper. [ABSTRACT FROM AUTHOR]

Differential forms provide a simple and powerful computational tool for finding canonical variables for drift motion in toroidal geometry. The method is easily applied to relativistic drift motion. [ABSTRACT FROM AUTHOR]

We demonstrate a remarkable analogy between the measurement of radiance and the well-known van Cittert–Zernike Theorem. [Copyright &y& Elsevier]

An analysis of electron drift waves and drift-cyclotron waves has been carried out in cylindrical geometry using a k-space integral equation. The model assumes a low-β electrostatic plasma in a straight magnetic field with a Gaussian density profile. Above the threshold for the driftcyclotron instability, a complicated spectrum of modes is found, with the fundamental radial eigenmode not necessarily the most unstable. In solving the integral equation, improved computational efficiency is obtained by decomposing the eigenfunctions on an associated Laguerre polynomial basis, a method which works best for low-azimuthal (m < 25) and low-radial (n < 10) mode numbers. [ABSTRACT FROM AUTHOR]

This article describes a direct method for the exact computation of 3nj symbols from the defining series, and continues discussing properties and asymptotic formulas focusing on the most important case, the 6j symbols or Racah coefficients. Relationships with families of hypergeometric orthogonal polynomials are presented and the asymptotic behavior is studied to account for some of the most relevant features, both from the viewpoints of the basic geometrical significance and as a source of accurate approximation formulas, such as those due to Ponzano and Regge and Schulten and Gordon. Numerical aspects are specifically investigated in detail, regarding the relationship between functions of discrete and of continuous variables, exhibiting the transition in the limit of large angular momenta toward both Wigner's reduced rotation matrices (or Jacobi polynomials) and harmonic oscillators (or Hermite polynomials). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010 [ABSTRACT FROM AUTHOR]