Analog signal processing

2014-01-08T09:10:54Z

151.151.16.6: /* Tools used in analog signal processing */

In [[computational chemistry]] and [[molecular physics]], '''Gaussian orbitals''' (also known as '''Gaussian type orbitals''', '''GTOs''' or '''Gaussians''') are [[function (mathematics)|function]]s used as [[atomic orbital]]s in the [[linear combination of atomic orbitals molecular orbital method|LCAO method]] for the representation of [[Molecular orbital|electron orbitals]] in [[molecule]]s and numerous properties that depend on these.<ref>{{cite journal|last=Gill|first=Peter M.W.|title=Molecular integrals Over Gaussian Basis Functions|journal=Advances in Quantum Chemistry|year=1994|volume=25|pages=141–205|doi=10.1016/S0065-3276(08)60019-2|url=http://cs.anu.edu.au/~Alistair.Rendell/hons09/qc/GillReview.pdf|accessdate=17 June 2011|bibcode = 1994AdQC...25..141G }}</ref>

== Rationale ==

The use of Gaussian orbitals in electronic structure theory (instead of the more physical [[Slater-type orbital]]s) was first proposed by [[S. Francis Boys|Boys]]<ref>S.F. Boys, [http://www.jstor.org/stable/98423 Proc. R. Soc. London Ser. A 200, 542 (1950)]</ref> in 1950. The principal reason for the use of Gaussian [[Basis set (chemistry)|basis functions]] in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4—5 orders of magnitude compared to [[Slater orbitals]] more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.

For reasons of convenience, many quantum chemistry programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested, as integral evaluation is much easier in the cartesian basis, and the spherical functions can be simply expressed using the cartesian functions.<ref>{{cite journal|last=Schlegel|first=H.|coauthors=Frisch, M.|title=Transformation between Cartesian and pure spherical harmonic Gaussians|journal=International Journal of Quantum Chemistry|year=1990|volume=54|issue=2|pages=83–87|doi=10.1002/qua.560540202 }}</ref>

== Mathematical form ==

The Gaussian basis functions obey the usual radial-angular decomposition
: <math>\ \Phi(\mathbf{r}) = R_l(r) Y_{lm}(\theta,\phi)</math>,
where <math>Y_{lm}(\theta,\phi)</math> is a [[Spherical harmonics|spherical harmonic]], <math>l</math> and <math>m</math> are the angular momentum and its <math>z</math> component, and <math>r,\theta,\phi</math> are spherical coordinates.

While for Slater orbitals the radial part is

: <math>\ R(r) = A(l,\alpha) r^l e^{-\alpha r}, </math>

<math>A(l,\alpha)</math> being a normalisation constant, for Gaussian primitives the radial part is

: <math>\ R(r) = B(l,\alpha) r^l e^{-\alpha r^2},</math>

where <math>B(l,\alpha)</math> is the normalization constant corresponding to the Gaussian.

Because an individual primitive Gaussian function gives a rather poor description for the electronic wave function near the nucleus, Gaussian basis sets are almost always contracted:

<math>\ R_l(r) = r^l \sum_{p=1,P} c_p A(l,\alpha_p) \exp(-\alpha_p r^2)</math>,

where <math>c_p</math> is the contraction coefficient for the primitive with exponent <math>\alpha_p</math>. The coefficients are given with respect to normalized primitives, because coefficients for unnormalized primitives would differ by many orders of magnitude. The exponents are reported in [[atomic units]]. There is a large library of published Gaussian basis sets optimized for a variety of criteria available at the [https://bse.pnl.gov/bse/portal EMSL basis set exchange].

==Molecular integrals==
Taketa et al. (1966) presented the necessary mathematical equations for obtaining matrix elements in the Gaussian basis.<ref>{{cite journal|last=Taketa|first=Hiroshi|coauthors=Huzinaga, Sigeru; O-ohata, Kiyosi|title=Gaussian-Expansion Methods for Molecular Integrals|journal=Journal of the Physical Society of Japan|year=1966|volume=21|issue=11|pages=2313–2324|doi=10.1143/JPSJ.21.2313}}</ref> Since then much work has been done to speed up the evaluation of these integrals which are the slowest part of many quantum chemical calculations. Živković and Maksić (1968) suggested using [[Hermite]] Gaussian functions,<ref>{{cite journal|last=Živković|first=T.|coauthors=Maksić, Z. B.|title=Explicit Formulas for Molecular Integrals over Hermite-Gaussian Functions|journal=Journal of Chemical Physics|year=1968|volume=49|issue=7|pages=3083–3087|doi=10.1063/1.1670551 |bibcode = 1968JChPh..49.3083Z }}</ref> as this simplifies the equations. McMurchie and Davidson (1978) introduced recursion relations,<ref>{{cite journal|last=McMurchie|first=Larry E.|coauthors=Davidson, Ernest R.|title=One- and two-electron integrals over Cartesian Gaussian functions|journal=Journal of Computational Physics|year=1978|volume=26|issue=2|pages=218–31|doi=10.1016/0021-9991(78)90092-X|bibcode = 1978JCoPh..26..218M }}</ref> which greatly reduces the amount of calculations. [[John Pople|Pople]] and Hehre (1978) developed a local coordinate method.<ref>{{cite journal|last=Pople|first=J. A.|coauthors=Hehre, W. J.|title=Computation of electron repulsion integrals involving contracted Gaussian basis functions.|journal=J. Comp. Phys.|year=1978|volume=27|issue=2|pages=161–168|doi=10.1016/0021-9991(78)90001-3|bibcode = 1978JCoPh..27..161P }}</ref> Obara and Saika introduced efficient recursion relations in 1985,<ref>{{cite journal|last=Obara|first=S.|coauthors=Saika, A.|title=Efficient recursive computation of molecular integrals over Cartesian Gaussian functions|journal=J. Chem. Phys.|year=1986|volume=84|issue=7|pages=3963–74|doi=10.1063/1.450106|bibcode = 1986JChPh..84.3963O }}</ref> which was followed by the development of other important recurrence relations. Gill and Pople (1990) introduced a 'PRISM' algorithm which allowed efficient use of 20 different calculation paths.<ref>{{cite journal|last=Gill|first=Peter M. W.|coauthors=Pople, John A.|title=The Prism Algorithm for Two-Electron Integrals|journal=International Journal of Quantum Chemistry|date=December 1991|volume=40|issue=6|pages=753–772|doi=10.1002/qua.560400605|url=http://rscweb.anu.edu.au/~pgill/papers/026PRISM.pdf|accessdate=17 June 2011}}</ref>

==The POLYATOM System==
The POLYATOM System<ref>I.G. Csizmadia, M.C. Harrison, J.W. Moskowitz and B.T. Sutcliffe, ''Nonempirical LCAO-MO-SCF-CI calculations on organic molecules with gaussian-type functions. Introductory review and mathematical formalism'', Theoretica Chimica Acta, 6, 191, 1966.</ref> was the first package for ''ab initio'' calculations using Gaussian orbitals that was applied to a wide variety of molecules.<ref>A.C. Wahl, ''Chemistry by computer'', Scientific American, pages 54-70, April, 1970.</ref> It was developed in [[John Clarke Slater|Slater's]] Solid State and Molecular Theory Group (SSMTG) at [[MIT]] using the resources of the Cooperative Computing Laboratory. The mathematical infrastructure and operational software were developed by Imre Csizmadia,<ref>[http://books.google.com/books?id=3xCzIQDYOvoC&pg=PA248&lpg=PA248&dq=Imre+Csizmadia+quantum&source=bl&ots=kLojlrMul2&sig=xAxDEzcVsnP11uI00W8rXiTfajI&hl=en&ei=cVZiTcyPJIO88gaUndTXCw&sa=X&oi=book_result&ct=result&resnum=3&sqi=2&ved=0CB4Q6AEwAg#v=onepage&q=Imre%20Csizmadia%20quantum&f=false ''Imre Csizmadia'', Professor Emeritus of Chemistry, University of Toronto, in Reviews in Computational Chemistry, vol.15, p.248]</ref> Malcolm Harrison,<ref>[http://cs.nyu.edu/cs/review95/node12.html ''Malcolm C. Harrison'', Professor of Computer Science, New York University]</ref> Jules Moskowitz<ref>[http://www.google.com/#hl=en&sugexp=ldymls&xhr=t&q=Jules+Moskowitz&cp=10&pf=p&sclient=psy&aq=0&aqi=&aql=&oq=Jules+Mosk&pbx=1&bav=on.1,or.&fp=5e2b21bd614e0a97 ''Jules W. Moskowitz'', Professor Emeritus of Chemistry, New York University]</ref> and Brian Sutcliffe.<ref>[http://www.chilton-computing.org.uk/acl/associates/users/sutcliffe.htm ''Brian T. Sutcliffe'', Professor of Chemistry, York University]</ref>

==See also==
*[[Quantum chemistry computer programs]]

==References==
{{reflist}}

==External links==
* [http://www.shef.ac.uk/chemistry/orbitron/ A visualization of all common and uncommon atomic orbitals, from 1s to 7g] (''Note that the radial part of the expressions given corresponds to [[Slater orbitals]] rather than Gaussians. The angular parts, and hence their shapes as displayed in figures, are the same as those of spherical Gaussians.'')
* [http://www.ccl.net/cca/documents/basis-sets/basis.html Explanation of Gaussian basis set]
* [https://bse.pnl.gov/bse/portal Basis set exchange]
* [http://lnu.diva-portal.org/smash/get/diva2:282089/FULLTEXT01 Detailed derivation of Gaussian orbital based matrix elements in electron structure calculations]

[[Category:Molecular physics]]
[[Category:Quantum chemistry]]

Money flow index

2013-12-19T18:26:26Z

151.151.109.24: added lookup of money flow index patterns

In [[optimization (mathematics)|optimization]], a '''descent direction''' is a vector <math>\mathbf{p}\in\mathbb R^n</math> that, in the sense below, moves us closer towards a local minimum <math>\mathbf{x}^*</math> of our objective function <math>f:\mathbb R^n\to\mathbb R</math>.

Suppose we are computing <math>\mathbf{x}^*</math> by an iterative method, such as [[line search]]. We define a descent direction <math>\mathbf{p}_k\in\mathbb R^n</math> at the <math>k</math>th iterate to be any <math>\mathbf{p}_k</math> such that <math>\langle\mathbf{p}_k,\nabla f(\mathbf{x}_k)\rangle < 0</math>, where <math> \langle , \rangle </math> denotes the [[inner product]]. The motivation for such an approach is that small steps along <math>\mathbf{p}_k</math> guarantee that <math>\displaystyle f</math> is reduced, by [[Taylor's theorem]].

Using this definition, the negative of a non-zero gradient is always a
descent direction, as <math> \langle -\nabla f(\mathbf{x}_k), \nabla f(\mathbf{x}_k) \rangle = -\langle \nabla f(\mathbf{x}_k), \nabla f(\mathbf{x}_k) \rangle < 0 </math>.

Numerous methods exist to compute descent directions, all with differing merits. For example, one could use [[gradient descent]] or the [[conjugate gradient method]].

More generally, if <math>P</math> is a [[positive definite]] matrix, then
<math>d = -P \nabla f(x)</math>
is a descent direction
<ref name="?">{{cite book | author = J. M. Ortega and W. C. Rheinbold | title = Iterative Solution of Nonlinear Equations in Several Variables | pages = 243 | year = 1970 | doi = 10.1137/1.9780898719468
}}</ref>
at <math>x</math>.
This generality is used in [[preconditioned]] gradient descent methods.

{{DEFAULTSORT:Descent Direction}}
[[Category:Mathematical optimization]]

{{Reflist}}

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Analog signal processing

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