Evaluation of Molecular Integrals in a Mixed Gaussian and Plane-Wave Basis by Means of the Faddeeva Function and Its Derivatives

1998 ◽  
Vol 63 (8) ◽  
pp. 1264-1284 ◽  
Author(s):  
Petr Čársky ◽  
Tomáš Reschel
Keyword(s):  
Wave Function ◽  
Plane Wave ◽  
Electron Scattering ◽  
Polyatomic Molecules ◽  
Explicit Formulas ◽  
Molecular Integrals

We present formulas for two-electron free-free exchange integrals and integrals with three Gaussians and one plane-wave function that are required in calculations of electron scattering by polyatomic molecules. The formulas of integrals with p- and d-type Gaussians were obtained by differentiation of the fundamental integrals (sk'|sk) and (ss|sk) that contain s-type Gaussians only and that may be evaluated by means of the Faddeeva function w. Explicit formulas are given for the two types of integrals in the spd Gaussian basis.

Atomic Imaging of Crystals using Large-Angle Electron Scattering in STEM

1990 ◽  
Vol 48 (1) ◽  
pp. 74-75
Author(s):  
D.E. Jesson ◽  
S. J. Pennycook
Keyword(s):  
Wave Function ◽  
Electron Scattering ◽  
Numerical Solutions ◽  
Large Angle ◽  
Real Space ◽  
High Angle ◽  
Analytical Treatment ◽  
Order Zero ◽  
Experimental Conditions ◽  
Ewald Sphere

It is well known that conventional atomic resolution electron microscopy is a coherent imaging process best interpreted in reciprocal space using contrast transfer function theory. This is because the equivalent real space interpretation involving a convolution between the exit face wave function and the instrumental response is difficult to visualize. Furthermore, the crystal wave function is not simply related to the projected crystal potential, except under a very restrictive set of experimental conditions, making image simulation an essential part of image interpretation. In this paper we present a different conceptual approach to the atomic imaging of crystals based on incoherent imaging theory. Using a real-space analysis of electron scattering to a high-angle annular detector, it is shown how the STEM imaging process can be partitioned into components parallel and perpendicular to the relevant low index zone-axis.It has become customary to describe STEM imaging using the analytical treatment developed by Cowley. However, the convenient assumption of a phase object (which neglects the curvature of the Ewald sphere) fails rapidly for large scattering angles, even in very thin crystals. Thus, to avoid unpredictive numerical solutions, it would seem more appropriate to apply pseudo-kinematic theory to the treatment of the weak high angle signal. Diffraction to medium order zero-layer reflections is most important compared with thermal diffuse scattering in very thin crystals (<5nm). The electron wave function ψ(R,z) at a depth z and transverse coordinate R due to a phase aberrated surface probe function P(R-RO) located at RO is then well described by the channeling approximation;

An Empirical Formula Approach to Total Cross Sections for Electron Scattering on Polyatomic Molecules

2005 ◽  
Vol 22 (11) ◽  
pp. 2801-2804
Author(s):  
Du Chao-Ling ◽  
Sun Jin-Feng ◽  
Liu Yu-Fang ◽  
Cheng Guang-Xu ◽  
Chen Yan-Feng
Keyword(s):  
Electron Scattering ◽  
Cross Sections ◽  
Empirical Formula ◽  
Polyatomic Molecules ◽  
Total Cross Sections

Second quantization

2003 ◽  
Keyword(s):  
Wave Function ◽  
Vibrational Excitation ◽  
Formal Solution ◽  
Algebraic Approach ◽  
Polyatomic Molecules ◽  
Algebraic Equations ◽  
Second Quantization ◽  
Closed Set ◽  
Quantum Operators ◽  
General Tool

Second quantization (SQ) concepts were introduced in chapter 2 as a general tool to treat excitations in molecular collisions for which the dynamics were described in cartesian coordinates. This SQ-formulation, which was derived from the TDGH representation of the wave function, could be introduced if the potential was expanded locally to second order around the position defined by a trajectory. It is, however, possible to use the SQ approach in a number of other dynamical situations, as for instance when dealing with the vibrational excitation of diatomic and polyatomic molecules, or with energy transfer to solids and chemical reactions in the socalled reaction path formulation. Since the formal expressions in the operators are the same, irrespective of the system or dynamical situation, the algebraic manipulations are also identical, and, hence, the formal solution the same. But the dynamical input to the scheme is of course different from case to case. In the second quantization formulation of the dynamical problems, one solves the operator algebraic equations formally. Once the formal solution is obtained, we can compute the dynamical quantities which enter the expressions. The advantage over state or grid expansion methods is significant since (at least for bosons) the number of dynamical operators is much less than the number of states. In order to solve the problem to infinite order, that is, also the TDSE for the system, the operators have to form a closed set with respect to commutations. This makes it necessary to drop some two-quantum operators. Historically, the M = 1 quantum problem, namely that of a linearly forced harmonic oscillator, was solved using the operator algebraic approach by Pechukas and Light in 1966 [131]. In 1972, Kelley [128] solved the two-oscillator (M = 2) problem and the author solved the M = 3 and the general problem in 1978 [129] and 1980 [147], respectively. The general case was solved using graph theory designed for the problem and it will not be repeated here. But the formulas are given in this chapter and in the appendices B and C.

To the theory of electron scattering by polyatomic molecules

1994 ◽  
Vol 35 (2) ◽  
pp. 182-187
Author(s):  
M. D. Él'kin ◽  
É. K. Kosterina
Keyword(s):  
Electron Scattering ◽  
Polyatomic Molecules

R-matrix calculations for polyatomic molecules: electron scattering by

1997 ◽  
Vol 30 (18) ◽  
pp. 4087-4096 ◽  
Author(s):  
L A Morgan ◽  
C J Gillan ◽  
J Tennyson ◽  
Xiushan Chen
Keyword(s):  
Electron Scattering ◽  
Polyatomic Molecules ◽  
R Matrix
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