Calculation of the vibrational wave function of polyatomic molecules

2000 ◽  
Vol 112 (6) ◽  
pp. 2655-2667 ◽  
Author(s):  
Per-Olof Åstrand ◽  
Kenneth Ruud ◽  
Peter R. Taylor
Keyword(s):  
Wave Function ◽  
Polyatomic Molecules ◽  
Vibrational Wave Function

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.

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.

Prospect of Retrieving Vibrational Wave Function by Single-Object Scattering Sampling

2013 ◽  
Vol 4 (19) ◽  
pp. 3345-3350 ◽  
Author(s):  
Hosung Ki ◽  
Kyung Hwan Kim ◽  
Jeongho Kim ◽  
Jae Hyuk Lee ◽  
Joonghan Kim ◽  
...  
Keyword(s):  
Wave Function ◽  
Single Object ◽  
Vibrational Wave Function ◽  
Object Scattering

scholarly journals Spatial Imaging of theH2+Vibrational Wave Function at the Quantum Limit

2012 ◽  
Vol 108 (7) ◽  
Author(s):  
L. Ph. H. Schmidt ◽  
T. Jahnke ◽  
A. Czasch ◽  
M. Schöffler ◽  
H. Schmidt-Böcking ◽  
...  
Keyword(s):  
Wave Function ◽  
Quantum Limit ◽  
Vibrational Wave Function

The intersection of potential energy surfaces in polyatomic molecules

1975 ◽  
Vol 344 (1637) ◽  
pp. 147-156 ◽  
Keyword(s):  
Wave Function ◽  
Potential Energy ◽  
Electronic State ◽  
Configuration Space ◽  
Potential Energy Surfaces ◽  
Polyatomic Molecules ◽  
The State ◽  
State Changes ◽  
Energy Surfaces ◽  
Recent Claim

It is proved that if the wave function of a given electronic state changes sign when transported adiabatically round a loop in nuclear configuration space, then the state must become degenerate with another one at some point within the loop. I t is further shown that this condition is satisfied by certain unsymmetrical triatomic systems, thereby disposing of a recent claim that the non-crossing rule for diatomic molecules applies also to polyatomic molecules.

Other Applications of Neural Networks to Quantum Mechanical Problems

2012 ◽  
Author(s):  
Lionel Raff ◽  
Ranga Komanduri ◽  
Martin Hagan ◽  
Satish Bukkapatnam
Keyword(s):  
Neural Networks ◽  
Wave Function ◽  
Linear Combination ◽  
Vibrational Spectra ◽  
Schrodinger Equation ◽  
Usual Method ◽  
Basis Functions ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Vibrational Wave Function

The usual method for solving the vibrational Schrödinger equation to obtain molecular vibrational spectra and the associated wave functions generally involves the expansion of the vibrational wave function, ψk(y), in terms of a linear combination of a set of basis functions.

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