Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction

Using a new method it is possible to derive mean field equations from the microscopic $N$ body Schr\"odinger evolution of interacting particles without using BBGKY hierarchies. In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii equation which is usually derived assuming positivity of the interaction. The new method for dealing with mean field limits presented in [6] allows us to relax this condition. The price we have to pay for this relaxation is however that we have to restrict the scaling behavior to $\beta<1/3$ and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function $\mu^{\Psi_0}$ to a pure state $|\phi_0><\phi_0|$

Restorative practice and behaviour management in schools: discipline meets care.

The history of restorative practices in New Zealand schools is directly related to projects such as the Suspension Reduction Initiative (SRI) and the more recent Student Engagement Initiative (SEI); thus the origins of restorative practices in schools are linked with behaviour management and school discipline. During the same period, teachers' work has become more complex: They are working with an increasingly diverse range of students, which in turn requires epistemologically diverse teaching and relationship-building approaches to ensure maximum participation for all. Teachers are looking for new and better ways to interact with students in their classrooms, and those responsible for disciplinary systems are looking to restorative practice for new ways to resolve the increasing range and number of difficulties between teachers and students, students and other students, and between the school and parents. Restorative practices (RP) are currently seen as a way of achieving all this, so they carry a huge burden of hope. Relationship skills are a key competency in the new curriculum, and the philosophy of restoration offers both a basis for understanding and a process for putting this agenda into practice. In effect, it means educating for citizenship in a diverse world, including teaching the skills of conflict resolution. If we accept this philosophy, the curriculum for teacher education will require significant changes in what students are taught about behaviour and classroom management

Statistical analysis of activation and reaction energies with quasi-variational coupled-cluster theory

The performance of quasi-variational coupled-cluster (QV) theory applied to the calculation of activation and reaction energies has been investigated. A statistical analysis of results obtained for six different sets of reactions has been carried out, and the results have been compared to those from standard single-reference methods. In general, the QV methods lead to increased activation energies and larger absolute reaction energies compared to those obtained with traditional coupled-cluster theory

The role of spin-orbit effects in the mobility of N+ ions moving in a helium gas at low temperature

The mobility of N+ ions in ground-state helium gas at very low temperature is examined with explicit inclusion of spin–orbit coupling effects. The ionic kinetics is treated theoretically with the three-temperature model. The N+–He interaction potentials, including spin–orbit coupling, are determined using high-level ab initio calculations. Then, the classical and quantal transport cross sections, both needed in the computation of the mobility coefficients, are calculated in terms of the collisional energy of the N+–He system. The numerical results, at temperature 4.3 K, show the spin–orbit interactions have negligible effect on the mobility coefficients

Perturbation-adapted perturbation theory

A new general approach is introduced for defining an optimum zero-order Hamiltonian for Rayleigh–Schrödinger perturbation theory. Instead of taking the operator directly from a model problem, it is constructed to be a best fit to the exact Hamiltonian within any desired functional form. When applied to many-body perturbation theory for electrons, strongly improved convergence is observed in cases where the conventional Fock Hamiltonian leads to divergence or slow convergence

The determination of point groups from imprecise molecular geometries

We present a new approach for the assignment of a point group to a molecule when the structure conforms only approximately to the symmetry. It proceeds by choosing a coordinate frame that minimises a measure of symmetry breaking that is computed efficiently as a simple function of the molecular coordinates and point group specification

One-particle many-body Green's function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms

A thorough analytical and numerical characterization of the whole perturbation series of one-particle many-body Green’s function (MBGF) theory is presented in a pedagogical manner. Three distinct but equivalent algebraic (first-quantized) recursive definitions of the perturbation series of the Green’s function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green’s function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 but converges at the full-configuration-interaction (FCI) limit at n=∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approximation. The diagrammatic linkedness and thus size-consistency of the one-particle Green’s function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green’s function to expose a self-energy diagram and the removal of reducible diagrams are also justified mathematically using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams

Second-order MCSCF optimization revisited. I. Improved algorithms for fast and robust second-order CASSCF convergence

A new improved implementation of the second-order multiconfiguration self-consistent field optimization method of Werner and Knowles [J. Chem. Phys. 82, 5053 (1985)] is presented. It differs from the original method by more stable and efficient algorithms for minimizing the second-order energy approximation in the so-called microiterations. Conventionally, this proceeds by alternating optimizations of the orbitals and configuration (CI) coefficients and is linearly convergent. The most difficult part is the orbital optimization, which requires solving a system of nonlinear equations that are often strongly coupled. We present a much improved algorithm for solving this problem, using an iterative subspace method that includes part of the orbital Hessian explicitly, and discuss different strategies for performing the uncoupled optimization in a most efficient manner. Second, we present a new solver in which the orbital-CI coupling is treated explicitly. This leads to quadratic convergence of the microiterations but requires many additional evaluations of reduced (transition) density matrices. In difficult optimization problems with a strong coupling of the orbitals and CI coefficients, it leads to much improved convergence of both the macroiterations and the microiterations. Third, the orbital-CI coupling is treated approximately using a quasi-Newton approach with Broyden–Fletcher–Goldfarb–Shanno updates of the orbital Hessian. It is demonstrated that this converges almost as well as the explicitly coupled method but avoids the additional effort for computing many transition density matrices. The performance of the three methods is compared for a set of 21 aromatic molecules, an Fe(ii)-porphine transition metal complex, as well as for the [Cu2O2(NH3) 6]2+, FeCl3, Co2(CO)6C2H2, and Al4O2 complexes. In all cases, faster and more stable convergence than with the original implementation is achieved

Molecular second-quantized Hamiltonian:Electron correlation and non-adiabatic coupling treated on an equal footing

We introduce a new theoretical and computational framework for treating molecular quantum mechanics without the Born–Oppenheimer approximation. The molecular wavefunction is represented in a tensor-product space of electronic and vibrational basis functions, with electronic basis chosen to reproduce the mean-field electronic structure at all geometries. We show how to transform the Hamiltonian to a fully second-quantized form with creation/annihilation operators for electronic and vibrational quantum particles, paving the way for polynomial-scaling approximations to the tensor-product space formalism. In addition, we make a proof-of-principle application of the new Ansatz to the vibronic spectrum of C2