Intricate behavior of one-electron potentials from the Euler equation for electron density and corresponding gradient force fields in crystals was studied. Bosonic and fermionic quantum potentials were utilized in bonding analysis as descriptors of the localization of electrons and electron pairs. Channels of locally enhanced kinetic potential and the corresponding saddle Lagrange points were found between chemically bonded atoms linked by the bond paths. Superposition of electrostatic φ_es (r) and kinetic φ_k (r) potentials and electron density ρ(r) allowed partitioning any molecules and crystals into atomic ρ- and potential-based φ-basins; the φ_k-basins explicitly account for electron exchange effect, which is missed for φ_es-ones. Phenomena of interatomic charge transfer and related electron exchange were explained in terms of space gaps between ρ- and φ-zero-flux surfaces. The gap between φ_es- and ρ-basins represents the charge transfer, while the gap between φ_k- and ρ-basins is proposed to be a real-space manifestation of sharing the transferred electrons. The position of φ_k-boundary between φ_es- and ρ-ones within an electron occupier atom determines the extent of electron sharing. The stronger an H‧‧‧O hydrogen bond is, the deeper hydrogen atom’s φ_k-basin penetrates oxygen atom’s ρ-basin. For covalent bonds, a φ_k-boundary closely approaches a φ_es-one indicating almost complete sharing the transferred electrons, while for ionic bonds, the same region corresponds to electron pairing within the ρ-basin of an electron occupier atom.
In this paper, the (3+1)-dimensional constant coefficient of Date–Jimbo–Kashiwara–Miwa (CCDJKM) equation is studied. All of the vector fields, infinitesimal generators, Lie symmetry reductions and different similarity reduction solutions are constructed. Due to the arbitrary functions in the infinitesimal generators, the (3+1)-dimensional CCDJKM equation can further be reduced to many (2+1)-dimensional partial differential equations. The explicit solutions of the similarity reduction equations, which include the quasi-periodic wave solution, the interaction solution between the periodic wave and a kink soliton and the trigonometric function solutions, are constructed with proper arbitrary function selection, and these new exact solutions are given out analytically and graphically.
AbstractWe study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.
A scheme is introduced to quantitatively analyze the magnetically induced molecular current density vector field $\mathbf{J}$. After determining the set of zero points of $\mathbf{J}$, which is called its {\em stagnation graph} (SG), the line integrals $\Phi_{\ell_i}=-\frac{1}{\mu_0} \int_{\ell_i} \mathbf{B}_\mathrm{ind}\cdot\mathrm{d}\mathbf{l}$ along all edges $\ell_i$ of the connected subset of the SG are determined. The edges $\ell_i$ are oriented such that all $\Phi_{\ell_i}$ are non-negative and they are weighted with $\Phi_{\ell_i}$. An oriented flux-weighted (current density) stagnation graph (OFW-SG) is obtained. Since $\mathbf{J}$ is in the exact theoretical limit divergence free and due to the topological characteristics of such vector fields the flux of all separate vortices and neighbouring vortex combinations can be determined by adding the weights of cyclic subsets of edges of the OFW-SG. The procedure is exemplified by the case of LiH for a perpendicular and weak homogeneous external magnetic field $\mathbf{B}$}
Abstract
Given an open and bounded set
Ω
⊆
ℝ
n
{\Omega\subseteq\mathbb{R}^{n}}
and a family
𝐗
=
(
X
1
,
…
,
X
m
)
{\mathbf{X}=(X_{1},\ldots,X_{m})}
of Lipschitz vector fields on Ω, with
m
≤
n
{m\leq n}
, we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e.
F
(
u
,
A
)
=
∫
A
f
(
x
,
u
(
x
)
,
X
u
(
x
)
)
𝑑
x
,
F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx,
with f being a Carathéodory integrand.
Real-Space Interpretation of Interatomic Charge Transfer and Electron Exchange Effects by Combining Static and Kinetic Potentials and Associated Vector Fields
