|
|
| Line 1: |
Line 1: |
| {{jargon|date=January 2014}}
| | She is known by the title of Myrtle Shryock. Bookkeeping is my occupation. The favorite hobby for my kids and me is to perform baseball and I'm attempting to make it a profession. Her family members lives in Minnesota.<br><br>Here is my blog - [http://www.1a-pornotube.com/blog/29575 http://www.1a-pornotube.com/blog/29575] |
| | |
| '''Zero differential overlap''' is an approximation in computational [[molecular orbital]] theory that is the central technique of [[semi-empirical quantum chemistry methods|semi-empirical methods]] in [[quantum chemistry]]. When computers were first used to calculate bonding in molecules, it was possible to only calculate diatomic molecules. As computers advanced, it became possible to study larger molecules, but the use of this approximation has always allowed the study of even larger molecules. Currently semi-empirical methods can be applied to molecules as large as whole proteins. The approximation involves ignoring certain integrals, usually two-electron repulsion integrals. If the number of orbitals used in the calculation is N, the number of two-electron repulsion integrals scales as N<sup>4</sup>. After the approximation is applied the number of such integrals scales as N<sup>2</sup>, a much smaller number, simplifying the calculation.
| |
| | |
| ==Details of approximation==
| |
| If the molecular orbitals <math>\mathbf{\Phi}_i \ </math> are expanded in terms of ''N'' basis functions, <math>\mathbf{\chi}_\mu^A \ </math> as:-
| |
| | |
| :<math>\mathbf{\Phi}_i \ = \sum_{\mu=1}^N \mathbf{C}_{i\mu} \ \mathbf{\chi}_\mu^A \, </math>
| |
| | |
| where ''A'' is the atom the basis function is centred on, and <math>\mathbf{C}_{i\mu} \ </math> are coefficients, the two-electron repulsion integrals are then defined as:-
| |
| | |
| :<math> \langle\mu\nu|\lambda\sigma\rangle = \iint \mathbf{\chi}_\mu^A (1) \mathbf{\chi}_\nu^B (1) \frac{1}{r_{12}} \mathbf{\chi}_\lambda^C (2) \mathbf{\chi}_\sigma^D (2) d\tau_1\,d\tau_2 \ </math> | |
| | |
| The zero differential overlap approximation ignores integrals that contain the product <math> \mathbf{\chi}_\mu^A (1) \mathbf{\chi}_\nu^B (1) </math> where ''μ'' is not equal to ''ν''. This leads to:-
| |
| | |
| :<math> \langle\mu\nu|\lambda\sigma\rangle = \delta_{\mu\nu} \delta_{\lambda\sigma} \langle\mu\mu|\lambda\lambda\rangle </math>
| |
| | |
| where <math> \delta_{\mu\nu} = \begin{cases}0 & \mu \ne \nu \\ 1 & \mu = \nu \ \end{cases} </math>
| |
| | |
| The total number of such integrals is reduced to ''N''(''N'' + 1) / 2 (approximately ''N''<sup>2</sup> / 2) from [''N''(''N'' + 1) / 2][''N''(''N'' + 1) / 2 + 1] / 2 (approximately ''N''<sup>4</sup> / 8), all of which are included in [[Ab initio quantum chemistry methods|ab initio]] [[Hartree–Fock]] and [[post-Hartree–Fock]] calculations.
| |
| | |
| ==Scope of approximation in semi-empirical methods==
| |
| Methods such as the [[Pariser–Parr–Pople method]] (PPP) and [[CNDO/2]] use the zero differential overlap approximation completely. Methods based on the intermediate neglect of differential overlap, such as [[INDO]], [[MINDO]], [[ZINDO]] and [[SINDO]] do not apply it when ''A'' = ''B'' = ''C'' = ''D'', i.e. when all four basis functions are on the same atom. Methods that use the neglect of diatomic differential overlap, such as [[MNDO]], [[PM3 (chemistry)|PM3]] and [[Austin Model 1|AM1]], also do not apply it when ''A'' = ''B'' and ''C'' = ''D'', i.e. when the basis functions for the first electron are on the same atom and the basis functions for the second electron are the same atom.
| |
| | |
| It is possible to partly justify this approximation, but generally it is used because it works reasonably well when the integrals that remain – <math>\langle\mu\mu|\lambda\lambda\rangle</math> – are parameterised.
| |
| | |
| ==References==
| |
| *{{cite book
| |
| | last = Jensen
| |
| | first = Frank
| |
| | authorlink =
| |
| | coauthors =
| |
| | title = Introduction to Computational Chemistry
| |
| | publisher = John Wiley and Sons
| |
| | year = 1999
| |
| | location = Chichester
| |
| | pages = 81–82
| |
| | url = http://hdl.handle.net/2027/uc1.31822026137414 <!-- HathiTrust -->
| |
| | isbn = 978-0-471-98085-8
| |
| | oclc = 466189317 }}
| |
| | |
| [[Category:Computational chemistry]]
| |
| | |
| | |
| {{chem-stub}}
| |
| {{quantum-stub}}
| |
She is known by the title of Myrtle Shryock. Bookkeeping is my occupation. The favorite hobby for my kids and me is to perform baseball and I'm attempting to make it a profession. Her family members lives in Minnesota.
Here is my blog - http://www.1a-pornotube.com/blog/29575