Reconciling semiclassical and Bohmian mechanics. II. Scattering states for discontinuous potentials

In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition, psi = psi1 + psi2, was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step potential, and square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a companion paper [C. Trahan and B. Poirier, J. Chem. Phys. 124, 034116 (2006), following paper].     

Nonempirical calculations on diatomic transition metals. II. RHF investigation of lowest closed-shell states of homonuclear 3d transition-metal dimers

Potential-energy curves of the 3d dimer series Sc₂ through Cu₂ are calculated for the lowest closed-shell states within the nonempirical RHF formalism using limited basis sets of minimal to near-double-zeta–plus-polarization size. Calculated spectroscopic constants are compared to semiempirical results as well as to experimental estimates. The possibility for closed- or open-shell ground states is discussed for each dimer. For diatomic Sc and Cu a detailed study of basis set effects on calculated molecular constants is carried out.     

Reconciling semiclassical and Bohmian mechanics. I. Stationary states

The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill behaved at caustics. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by divergent forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eigenstates in one degree of freedom are the primary focus, but more general applications are also anticipated.     

Integrating quantum and molecular mechanics

A computer algorithm is developed for integrating density functional quantum mechanics into a molecular mechanics program. The computationally infeasible aspects of the standard LCAO-MO approach (the iterative calculation of eigenvectors and the requirement of orthogonal expansions for the orbitals) are replaced with an efficient use of optimization via the trace theorem of linear algebra. The construction of a basis is also described for expanding the electron density that transforms with the molecular geometry. The combination of the trace method and the basis allow the solution for one configuration of atoms and electrons to be tracked over a wide range of internal conformations. The approach is readily adaptable to being used in the context of an imposed classical field that allows it to be used on part of a macromolecular complex. The initial implementation in the program AMMP is described. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1618–1633, 1999     

Phase-space dynamics and quantum mechanics

Ann-dimensional system with a classical HamiltonianH(p, q, t) may be described by a phase-space distribution function D(q, p, t). The dynamical equation for D(q, p, t) is postulated to be

Phase-space dynamics and quantum mechanics

Summary In a recent paper Deal has postulated a new dynamical equation for quantum mechanical phase-space distribution functions. We analyze the new equation and show that it may be related to the traditional standard and antistandard phase-space representations of quantum mechanics. A brief review of these and other representations is also given.     

Calculation of wave‐functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. II. Application of the local basis equation

The application of the local basis equation (Ferenczy and Adams, J. Chem. Phys. 2009 , 130, 134108) in mixed quantum mechanics/molecular mechanics (QM/MM) and quantum mechanics/quantum mechanics (QM/QM) methods is investigated. This equation is suitable to derive local basis nonorthogonal orbitals that minimize the energy of the system and it exhibits good convergence properties in a self‐consistent field solution. These features make the equation appropriate to be used in mixed QM/MM and QM/QM methods to optimize orbitals in the field of frozen localized orbitals connecting the subsystems. Calculations performed for several properties in divers systems show that the method is robust with various choices of the frozen orbitals and frontier atom properties. With appropriate basis set assignment, it gives results equivalent with those of a related approach [G. G. Ferenczy previous paper in this issue] using the Huzinaga equation. Thus, the local basis equation can be used in mixed QM/MM methods with small size quantum subsystems to calculate properties in good agreement with reference Hartree–Fock–Roothaan results. It is shown that bond charges are not necessary when the local basis equation is applied, although they are required for the self‐consistent field solution of the Huzinaga equation based method. Conversely, the deformation of the wave‐function near to the boundary is observed without bond charges and this has a significant effect on deprotonation energies but a less pronounced effect when the total charge of the system is conserved. The local basis equation can also be used to define a two layer quantum system with nonorthogonal localized orbitals surrounding the central delocalized quantum subsystem. © 2013 Wiley Periodicals, Inc.     

The role of excited states in the photolysis of n-butyraldehyde, II. Intersystem crossing quantum yields

Singlet triplet intersystem crossing quantum yields were determined for n-butyraldehyde, using the photosensitized cis-trans isomerization of piperylene to monitor the process. At room temperature and 313 nm, 0.58±0.05 and 0.72±0.05 were obtained for the triplet yield in the vapor phase and in isooctane, respectively.

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- - - . 0,58±0,05 0,72±0,05 313 .

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For Part I, see Ref. /1/     

Pattern recognition in molecular quantum mechanics

It is shown that the classical concept of an open system does not encompass quantal systems but has to be replaced by the non-Boolean notion of an entangled system. Molecular, chemical, or biological phenomena can be considered to be reduced to a fundamental theory like quantum mechanics only if the fundamental and the phenomenological theories are formally and interpretatively connected, and if the classifications used in the empirical sciences are shown to follow from a single set of fundamental dynamical laws. These conditions enforce a non-statistical and ontic interpretation of quantum mechanics, hence a non-Boolean calculus of propositions. In this interpretation the notion of a world state is well-defined, its Schmidt-decomposition defines a background-dependent model state for molecular systems and creates the phenomena we can observe. To any molecular system there is associated in an objective way a nonnegative number which we call the integrity. The integrity measures the inherent fuzziness of the system concept in a holistic theory, and is used to define recognizable molecular patterns.     

Harmonic oscillators in relativistic quantum mechanics

Relativistic generalisations of the harmonic oscillator are analysed. Lévy–Leblond, Dirac and Klein–Gordon equations which in the limit of a non-relativistic and spinless particle transform into Schrödinger equation for the harmonic oscillator are constructed. Properties of their solutions, in particular the structure of their spectra, are analysed. Applications to modelling phenomena relevant in quantum chemistry are briefly discussed.     

A Bohmian total potential view to quantum effects. II: decay of temporarily trapped states

Formation, persistence and decay of temporarily trapped states, the time-dependent generalization of resonances, are analysed within the framework of Bohmian Mechanics. More specifically, the so-called Bohm’s total potential, the sum of classical plus Bohm’s quantum potential, is used. It is found that both formation and decay are triggered by the frequency in the oscillations of the total potential. These oscillations have been studied at the specific locations where the classical potential displays maxima, i.e. the ‘walls’ temporarily capturing the system’s density. Our main result is that the total potential oscillation frequency is solely dependent on the steepness of the classical potential ramp and, surprisingly, independent of the classical barrier height and width, well depth and width, collision energy or wavepacket width.     

On the validity of Hamilton-Jacobi mechanics in the mean in quantum mechanics

It is shown that under certain conditions the time-dependent Hamilton–Jacobi equation (TDHJE) is satisfied in the mean in quantum mechanics provided the classical action is replaced by Dirac's complex quantal action. This may be regarded as the equivalent of Ehrenfest's theorem for the TDHJE . Expectation values of operators are discussed in a similar manner.     

Structures and stabilities of diacetylene-expanded polyhedranes by quantum mechanics and molecular mechanics

The structures, heats of formation, and strain energies of diacetylene (buta-1,3-diynediyl) expanded molecules have been computed with ab initio and molecular mechanics calculations. Expanded cubane, prismane, tetrahedrane, and expanded monocyclics and bicyclics were optimized at the HF/6-31G(d) and B3LYP/6-31G(d) levels. The heats of formation of these systems were obtained from isodesmic equations at the HF/6-31G(d) level. Heats of formation were also calculated from Benson group equivalents. The strain energies of these expanded molecules were estimated by several independent methods. An adapted MM3 molecular mechanics force field, specifically parametrized to treat conjugated acetylene units, was employed for one measure of strain energy and as an additional method for structural analysis. Expanded dodecahedrane and icosahedrane were calculated by this method. Expanded molecules were considered structurally in the context of their potential material applications.     

Resonance states of atomic anions

We study destabilization of an atom in its ground state with decrease of its nuclear charge. By analytic continuation from bound to resonance states, we obtain complex energies of unstable atomic anions with nuclear charge that is less than the minimum “critical” charge necessary to bind N electrons. We use an extrapolating scheme with a simple model potential for the electron, which is loosely bound outside the atomic core. Results for O^2? and S^2? are in good agreement with earlier estimates. Alternatively, we use the Hylleraas basis variational technique with three complex nonlinear parameters to find accurately the energy of two‐electron atoms as the nuclear charge decreases. Results are used to check the less accurate one‐electron model. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 255–261, 2001     

The atomic states of nickel

The Ni 3d ⁹ 4s ¹(³ D)–3d ¹⁰(¹ S) and 3d ⁹ 4s ¹(³ D)-3d ⁸4s ²(³ F) atomic excitation energies have been computed using large multireference CI wave functions in conjunction with a large ANO basis set. Radial correlation effects in the 3d shell are found to be very important and are included using CASSCF wave functions having the 3d and correlating 3d orbitals in the active space. The previous discrepancy (0.5 eV) with experiment for the ³ D–¹ S excitation is reduced to 0.1 eV when the 3d3d references are included in the CI. For the ³ F state, the 4s-4p near degeneracy gives rise to important 4s ²4p ² excitations in addition to the 3d3d excitations which are important for the ³ D and ¹ S states. Inclusion of only 3d-3d correlation in the ³ F and ³ D CI reference spaces yields a ³ F–³ D separation which is in error by 0.12 eV. Addition of the 4s ²4p ² excitations to the ³ F reference space is estimated to increase the discrepancy with experiment by an additional 0.1 eV.