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2018 | OriginalPaper | Chapter

2. Ab Initio Methods

Authors : Kaoru Ohno, Keivan Esfarjani, Yoshiyuki Kawazoe

Published in: Computational Materials Science

Publisher: Springer Berlin Heidelberg

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Abstract

In fields such as materials science, electronics, mechanical engineering and bioscience, not to mention physics and chemistry, the keywords “first principles” and “ab initio” have been widely used recently. The principal idea of these keywords is to regard many-atom systems as many-body systems composed of electrons and nuclei, and to treat everything on the basis of first principles of quantum mechanics, without introducing any empirical parameters.

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previous chapter Introduction
next chapter Tight-Binding Methods
Footnotes
1
Recall \(\partial \mathcal{{G}}/\partial t=\mathcal{{J}}\). The constant of integration may be absorbed in the definition of the total energy
 
2
For more complicated potentials involving many-body forces the definition of the local energy is not unique, but one can show that it does not affect the total thermal conductivity [382].
 
3
Ergodic means the whole volume of the phase space will eventually be “visited” provided the simulation is continued for a long enough time.
 
4
Note that the breakdown of the total energy between the local energies of the particles is not unique. Here we divided it into N equal parts, but for convenience one may adopt other choices.
 
5
At temperatures below the Debye Temperature, the number of Normal processes involve low frequency phonons for which the phonon number can be approximated by \(k_B T/\hbar \omega _{{\varvec{\scriptstyle k}}\lambda }\), while in an Umklapp process, which involves large wavenumbers, or modes near the Brillouin zone boundary, the phonon frequency is on the order or larger than the Debye frequency and thus is proportional to \(e^{-\omega _D/T}\).
 
6
The derivation is discussed in the section on phonons.
 
7
This is due to the central-limit theorem, which states that in the limit where N goes to infinity, the average of N random numbers \(x_i\) has a Gaussian distribution of mean \(\bar{x}\) and of variance \( \sigma _x/\sqrt{N} \), where \(\bar{x}\) is the true mean of the distribution and \(\sigma _x\) its variance.
 
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Metadata
Title
Ab Initio Methods
Authors
Kaoru Ohno
Keivan Esfarjani
Yoshiyuki Kawazoe
Copyright Year
2018
Publisher
Springer Berlin Heidelberg
Book
Computational Materials Science

Print ISBN: 978-3-662-56540-7

Electronic ISBN: 978-3-662-56542-1

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-662-56542-1_2