A general toolbox for the calculation of higher-order molecular properties using SCF wave functions at the one-, two- and four-component levels of theory

2012 ◽  
Author(s):  
Kenneth Ruud ◽  
Radovan Bast ◽  
Bin Gao ◽  
Andreas J. Thorvaldsen ◽  
Ulf Ekström ◽  
...  
Keyword(s):  
Higher Order ◽  
Wave Functions ◽  
Molecular Properties ◽  
The One

scholarly journals Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Applied Sciences ◽  
Degree Of Approximation ◽  
Higher Order ◽  
Second Order ◽  
Dimensional Case ◽  
Moduli Of Continuity ◽  
One Dimensional ◽  
Multidimensional Generalization ◽  
The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

General coalescence conditions for the exact wave functions. II. Higher-order relations for many-particle systems

2014 ◽  
Vol 140 (21) ◽  
pp. 214103 ◽  
Author(s):  
Yusaku I. Kurokawa ◽  
Hiroyuki Nakashima ◽  
Hiroshi Nakatsuji
Keyword(s):  
Higher Order ◽  
Particle Systems ◽  
Wave Functions ◽  
Order Relations

scholarly journals Intercombination Transitions between Levels X1Σg+ and A 3Πu in C2

1987 ◽  
Vol 120 ◽  
pp. 103-105
Author(s):  
J. Le Bourlot ◽  
E. Roueff
Keyword(s):  
Transition Probabilities ◽  
Energy Levels ◽  
Wave Functions ◽  
Spin Orbit Coupling ◽  
Energy Matrix ◽  
First Order ◽  
Emission Transition ◽  
Intercombination Transition ◽  
The One ◽  
Potential Curves

We present a new calculation of intercombination transition probabilities between levels X1Σg+ and a 3Πu of the C2 molecule. Starting from experimental energy levels, we calculate RKR potential curves using Leroy's Near Dissociation Expansion (NDE) method; these curves give us wave functions for all levels of interest. We then compute the energy matrix for the four lowest states of C2, taking into account Spin-Orbit coupling between a 3Πu and A 1Πu on the one hand and X 1Σ+g and b 3Σg− on the other. First order wave functions are then derived by diagonalization. Einstein emission transition probabilities of the Intercombination lines are finally obtained.

Semi-empirical Molecular Orbital Energy Levels of the Hexammine and Chloroammine Complexes of Co(IV)

1967 ◽  
Vol 22 (2) ◽  
pp. 170-175 ◽  
Author(s):  
Walter A. Yeranos ◽  
David A. Hasman
Keyword(s):  
Molecular Orbital ◽  
Energy Levels ◽  
Empirical Evaluation ◽  
Wave Functions ◽  
Electronic Transitions ◽  
Orbital Energy ◽  
Molecular Orbital Energy ◽  
The One ◽  
Semi Empirical

Using the recently proposed reciprocal mean for the semi-empirical evaluation of resonance integrals, as well as approximate SCF wave functions for Co3+, the one-electron molecular energy levels of Co (NH3) 3+, Co (NH3) 5Cl2+, and Co (NH3) 4Cl21+ have been redetermined within the WOLFSBERG–HELMHOLZ approximation. The outcome of the study fits remarkably well with the observed electronic transitions in the u.v. spectra of these complexes and prompts different band assignments than previously suggested.

scholarly journals Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

2019 ◽  
Vol 12 (3) ◽  
pp. 303-332 ◽  
Author(s):  
Alexandra Convent ◽  
Jean Van Schaftingen
Keyword(s):  
Sobolev Space ◽  
Isometric Embedding ◽  
Riemannian Metrics ◽  
Higher Order ◽  
Necessary Condition ◽  
Smooth Maps ◽  
Homogeneous Sobolev Space ◽  
The One ◽  
Intrinsic Definition ◽  
Sobolev Spaces Between Manifolds

AbstractWe define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N. When M and N are endowed with Riemannian metrics, {p\geq 1} and {k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space {\dot{W}^{k,p}(M,N)}. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space {\dot{W}^{k,p}(M,N)} is that {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.

scholarly journals Exact statistical properties of the Burgers equation

2000 ◽  
Vol 417 ◽  
pp. 323-349 ◽  
Author(s):  
L. FRACHEBOURG ◽  
Ph. A. MARTIN
Keyword(s):  
White Noise ◽  
Large Distance ◽  
Burgers Equation ◽  
Higher Order ◽  
Statistical Properties ◽  
Inviscid Limit ◽  
Initial Condition ◽  
Spatial Correlations ◽  
One Dimensional ◽  
The One

The one-dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behaviour of spatial correlations of the field is determined. Since higher-order distributions factorize in terms of the one- and two- point functions, our analysis provides an explicit and complete statistical description of this problem.

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