|
|
| Line 1: |
Line 1: |
| '''Slater-type orbitals''' (STOs) are functions used as [[atomic orbital]]s in the [[linear combination of atomic orbitals molecular orbital method]]. They are named after the physicist [[John C. Slater]], who introduced them in 1930.<ref>
| |
| {{cite journal
| |
| |last1=Slater |first1=J. C.
| |
| |year=1930
| |
| |title=Atomic Shielding Constants
| |
| |journal=[[Physical Review]]
| |
| |volume=36 |issue= |page=57
| |
| |bibcode=1930PhRv...36...57S
| |
| |doi=10.1103/PhysRev.36.57
| |
| }}</ref>
| |
|
| |
|
| They possess exponential decay at long range and [[Kato theorem|Kato's cusp condition]] at short range (when combined as [[hydrogen-like atom]] functions, i.e. the analytical solutions of the stationary Schrödinger for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do [[Gaussian orbitals|Gaussian-type orbitals]]).
| |
|
| |
|
| ==Definition==
| | This can a strategy and conjointly battle activation where your entire family must manage your hold tribe and also protect it. You have on build constructions which is likely to provide protection for your soldiers along with each of our instruction. First concentrate on your protection and simply after its recently long been taken treatment. You might need to move forward which has the criminal offense arrange. As well as Military facilities, you likewise require to keep in thought [https://Www.Vocabulary.com/dictionary/processes processes] the way your tribe is certainly going. For instance, collecting tactics as well as raising your own tribe will be the key to good findings.<br><br>The amend delivers a information of notable enhancements, arch of which could quite possibly be the new Dynasty Struggle Manner. In this specific mode, you can asserting combating dynasties and reduce utter [http://en.search.wordpress.com/?q=rewards rewards] aloft their beat.<br><br>If you have had little ones who delight in video games, then you are aware how challenging it really can be always to pull them out among the t. v.. Their eye can be stuck towards the monitor for hours as these kinds of products play their preferred exercises. If you want aid regulating your little ones clash of clans Hack time, your own pursuing article has some suggestions for you.<br><br>An outstanding method to please young children with a gaming circle and ensure they endure fit is to purchase a Wii. This video gaming system needs real task perform. Your children won't be sitting for hours on conclude playing clash of clans hack. They will need to be moving around as a way to play the games inside of this particular system.<br><br>Or even a looking Conflict of Families Jewels Free, or you're just buying a Skimp Conflict of Tribes, currently have the smartest choice while on the internet, absolutely free as well as only takes a little bit to get all many.<br><br>It's to select the required xbox game gaming product. In the beginning, you should think about your standard requirements like a video game player, accompanied by check out the additional features made available from each unit you are since. Consider investigating on-line. Check testimonials to ascertain if other useful gamers have discovered complications with the unit. Ahead of buying a game process, you should know nearly everything you are able to actually regarding it.<br><br>Await game of the year versions of major bands. These often come out a yr or maybe more when original title, but are made lots of the downloadable and extra content that has been released in stages after a first title. In case you loved this article and you would want to receive details regarding clash of clans hack tool - [http://circuspartypanama.com read full article] - kindly visit our own web page. Most games offer a lot more bang for the monetary. |
| STOs have the following radial part:
| |
| | |
| : <math>R(r) = N r^{n-1} e^{-\zeta r}\,</math>
| |
| where
| |
| : ''n'' is a [[natural number]] that plays the role of [[principal quantum number]], ''n'' = 1,2,...,
| |
| : ''N'' is a [[normalizing constant]],
| |
| : ''r'' is the distance of the electron from the [[atomic nucleus]], and | |
| : <math>\zeta</math> is a constant related to the effective [[electric charge|charge]] of the nucleus, the nuclear charge being partly shielded by electrons. Historically, the effective nuclear charge was estimated by [[Slater's rules]].
| |
| | |
| The normalization constant is computed from the integral
| |
| :<math> \int_0^\infty x^n e^{-\alpha x} dx = \frac{n!}{\alpha^{n+1}}. </math>
| |
| Hence
| |
| :<math>N^2 \int_0^\infty \left(r^{n-1}e^{-\zeta r}\right)^2 r^2 dr =1 \Longrightarrow N = (2\zeta)^n \sqrt{\frac{2\zeta}{(2n)!}}. </math>
| |
| | |
| It is common to use the [[spherical harmonics]] <math>Y_l^m(\mathbf{r})</math> depending on the polar coordinates
| |
| of the position vector <math>\mathbf{r}</math> as the angular part of the Slater orbital. | |
| | |
| ==Differentials==
| |
| The first radial derivative of the radial part of a Slater-type orbital is
| |
| :<math> {\partial R(r)\over \partial r} = \left[\frac{(n - 1)}{r} - \zeta\right] R(r) </math> | |
| The radial Laplace operator is split in two differential operators
| |
| :<math> \nabla^2 = {1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial \over \partial r}\right) </math>
| |
| The first differential operator of the Laplace operator yields
| |
| :<math> \left(r^2 {\partial\over \partial r} \right) R(r) = \left[(n - 1) r - \zeta r^2 \right] R(r) </math>
| |
| The total Laplace operator yields after applying the second differential operator
| |
| :<math> \nabla^2 R(r) = \left({1 \over r^2} {\partial\over \partial r} \right) \left[(n - 1) r - \zeta r^2 \right] R(r) </math>
| |
| the result
| |
| :<math>\nabla^2 R(r) = \left[{n (n - 1) \over r^2} - {2 n \zeta \over r} + \zeta^2 \right] R(r) </math>
| |
| Angular dependent derivatives of the spherical harmonics don't depend on the radial function and have to be evaluated separately.
| |
| | |
| ==Integrals==
| |
| The fundamental mathematical properties are those associated with the kinetic energy, nuclear attraction and Coulomb repulsion integrals for placement of the orbital at the center of a single nucleus. Dropping the normalization factor ''N'', the representation of the orbitals below is
| |
| :<math>\chi_{nlm}({\mathbf{r}}) = r^{n-1}e^{-\zeta r}Y_l^m({\mathbf{r}}).</math>
| |
| | |
| The [[Fourier transform]] is<ref>
| |
| {{cite journal
| |
| |last1=Belkic |first1=D.
| |
| |last2=Taylor |first2=H. S.
| |
| |year=1989
| |
| |title=A unified formula for the Fourier transform of Slater-type orbitals
| |
| |journal=[[Physica Scripta]]
| |
| |volume=39 |issue=2 |pages=226–229
| |
| |bibcode=1989PhyS...39..226B
| |
| |doi=10.1088/0031-8949/39/2/004
| |
| }}</ref>
| |
| :<math>\chi_{nlm}({\mathbf{k}})= \int d^3r e^{i{\mathbf{k}}\cdot {\mathbf{r}}} \chi_{nlm}({\mathbf{ r}})</math>
| |
| :<math>=4\pi (n-l)! (2\zeta)^n (ik/\zeta)^l Y_l^m({\mathbf{k}}) \sum_{s=0}^{\lfloor(n-l)/2\rfloor} \frac{\omega_s^{nl}}{(k^2+\zeta^2)^{n+1-s}}</math>,
| |
| | |
| where the <math>\omega</math> are defined by
| |
| :<math>\omega_s^{nl}\equiv(-\frac{1}{4\zeta^2})^s\frac{(n-s)!}{s!(n-l-2s)!}</math>.
| |
| | |
| The overlap integral is
| |
| :<math> \int \chi^*_{nlm}(r)\chi_{n'l'm'}(r)d^3r = \delta_{ll'}\delta_{mm'}\frac{(n+n')!}{(\zeta+\zeta')^{n+n'+1}} </math>
| |
| | |
| of which the normalization integral is a special case. The starlet in the | |
| superscript denotes [[Complex conjugate|complex-conjugation]].
| |
| | |
| The [[Kinetic energy#Quantum mechanical kinetic energy of rigid bodies|kinetic energy]] integral is
| |
| | |
| :<math>
| |
| \int \chi^*_{nlm}(r)(-\frac{\nabla^2}{2})\chi_{n'l'm'}(r)d^3r
| |
| =
| |
| \frac{1}{2}\delta_{ll'}\delta_{mm'}
| |
| \int_0^\infty dr e^{-(\zeta+\zeta')r}
| |
| \left[
| |
| [l'(l'+1)-n'(n'-1)]r^{n+n'-2}+2\zeta'n'r^{n+n'-1}-\zeta'^2r^{n+n'}
| |
| \right],
| |
| </math>
| |
| a sum over three overlap integrals already computed above. | |
| | |
| The Coulomb repulsion integral can be evaluated using the Fourier representation
| |
| (see above)
| |
| | |
| :<math>
| |
| \chi^*_{nlm}({\mathbf{r}})=\int\frac{d^3k}{(2\pi)^3}e^{i{\mathbf{k}}\cdot {\mathbf{r}}}
| |
| \chi^*_{nml}({\mathbf{k}})
| |
| </math> | |
| | |
| which yields
| |
| | |
| :<math>
| |
| \int \chi^*_{nlm}({\mathbf{r}})\frac{1}{|{\mathbf{r}}-{\mathbf{r}}'|}\chi_{n'l'm'}({\mathbf{r}}')d^3r
| |
| =
| |
| 4\pi
| |
| \int
| |
| \frac{d^3k}{(2\pi)^3}
| |
| \chi^*_{nlm}({\mathbf{k}})\frac{1}{k^2}\chi_{n'l'm'}({\mathbf{k}})
| |
| </math>
| |
| :<math>
| |
| =
| |
| 8\delta_{ll'}
| |
| \delta_{mm'}
| |
| (n-l)!
| |
| (n'-l)!
| |
| \frac{(2\zeta)^n}{\zeta^l}
| |
| \frac{(2\zeta')^{n'}}{\zeta'^l}
| |
| \int_0^\infty
| |
| dk k^{2l}
| |
| \sum_{s=0}^{\lfloor (n-l)/2\rfloor}
| |
| \frac{\omega_s^{nl}}{(k^2+\zeta^2)^{n+1-s}}
| |
| \sum_{s'=0}^{\lfloor (n'-l)/2\rfloor}
| |
| \frac{\omega_{s'}^{n'l'}}{(k^2+\zeta'^2)^{n'+1-s'}}
| |
| </math>
| |
| These are either individually calculated with the [[Methods of contour integration|law of residues]] or recursively
| |
| as proposed by Cruz et al. (1978).<ref>
| |
| {{cite journal
| |
| |last1=Cruz |first1=S. A.
| |
| |last2=Cisneros |first2=C.
| |
| |last3=Alvarez |first3=I.
| |
| |year=1978
| |
| |title=Individual orbit contribution to the electron stopping cross section in the low-velocity region
| |
| |journal=[[Physical Review A]]
| |
| |volume=17 |issue=1 |pages=132–140
| |
| |bibcode=1978PhRvA..17..132C
| |
| |doi=10.1103/PhysRevA.17.132
| |
| }}</ref>
| |
| | |
| ==STO Software==
| |
| Some quantum chemistry software uses sets of [[1s Slater-type function|Slater-type functions]] (STF) analogous to Slater type orbitals, but with variable exponents chosen to minimize the total molecular energy (rather than by Slater's rules as above). The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which give a displaced Gaussian) has led many to expand them in terms of Gaussians.<ref>
| |
| {{cite journal
| |
| |last1=Guseinov |first1=I. I.
| |
| |year=2002
| |
| |title=New complete orthonormal sets of exponential-type orbitals and their application to translation of Slater Orbitals
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=90 |issue=1 |pages=114–118
| |
| |doi=10.1002/qua.927
| |
| }}</ref>
| |
| | |
| Analytical ab initio software for poly-atomic molecules has been developed e.g. STOP: a Slater Type Orbital Package in 1996.<ref>
| |
| {{cite journal
| |
| |last1=Bouferguene |first1=A.
| |
| |last2=Fares |first2=M.
| |
| |last3=Hoggan |first3=P. E.
| |
| |year=1996
| |
| |title=STOP: Slater Type Orbital Package for general molecular electronic structure calculations
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=57 |issue=4 |pages=801–810
| |
| |doi=10.1002/(SICI)1097-461X(1996)57:4<801::AID-QUA27>3.0.CO;2-0
| |
| }}</ref>
| |
| | |
| SMILES uses analytical expressions when available and Gaussian expansions otherwise. It was first released in 2000.
| |
| | |
| Various grid integration schemes have been developed, sometimes after analytical work for quadrature (Scrocco). Most famously in the ADF suite of DFT codes.
| |
| | |
| ==References==
| |
| <references />
| |
| *{{cite journal
| |
| |last1=Harris |first1=F. E.
| |
| |last2=Michels |first2=H. H.
| |
| |year=1966
| |
| |title=Multicenter integrals in quantum mechanics. 2. Evaluation of electron-repulsion integrals for Slater-type orbitals
| |
| |journal=[[Journal of Chemical Physics]]
| |
| |volume=45 |issue=1 |pages=116
| |
| |bibcode=1966JChPh..45..116H
| |
| |doi=10.1063/1.1727293
| |
| }}
| |
| *{{cite journal
| |
| |last1=Filter |first1=E.
| |
| |last2=Steinborn |first2=E. O.
| |
| |year=1978
| |
| |title=Extremely compact formulas for molecular two-center and one-electron integrals and Coulomb integrals over Slater-type atomic orbitals
| |
| |journal=[[Physical Review A]]
| |
| |volume=18 |issue=1 |pages=1–11
| |
| |bibcode=1978PhRvA..18....1F
| |
| |doi=10.1103/PhysRevA.18.1
| |
| }}
| |
| *{{cite journal
| |
| |last1=McLean |first1=A. D.
| |
| |last2=McLean |first2=R. S.
| |
| |year=1981
| |
| |title=Roothaan-Hartree-Fock Atomic Wave Functions, Slater Basis-Set Expansions for Z = 55–92
| |
| |journal=[[Atomical Data and Nuclear Data Tables]]
| |
| |volume=26 |issue=3–4 |pages=197–381
| |
| |bibcode=1981ADNDT..26..197M
| |
| |doi=10.1016/0092-640X(81)90012-7
| |
| }}
| |
| *{{cite journal
| |
| |last1=Datta |first1=S.
| |
| |year=1985
| |
| |title=Evaluation of Coulomb integrals with hydrogenic and Slater-type orbitals
| |
| |journal=[[Journal of Physics B]]
| |
| |volume=18 |issue=5 |pages=853–857
| |
| |bibcode=1985JPhB...18..853D
| |
| |doi=10.1088/0022-3700/18/5/006
| |
| }}
| |
| *{{cite journal
| |
| |last1=Grotendorst |first1=J.
| |
| |last2=Steinborn |first2=E. O.
| |
| |year=1985
| |
| |journal=[[Journal of Computational Physics]]
| |
| |title=The Fourier transform of a two-center product of exponential-type functions and its efficient evaluation
| |
| |volume=61 |issue=2 |pages=195–217
| |
| |bibcode=1985JCoPh..61..195G
| |
| |doi=10.1016/0021-9991(85)90082-8
| |
| }}
| |
| *{{cite journal
| |
| |last1=Tai |first1=H.
| |
| |year=1986
| |
| |title=Analytic evaluation of two-center molecular integrals
| |
| |journal=[[Physical Review A]]
| |
| |volume=33 |issue=6 |pages=3657–3666
| |
| |bibcode=1986PhRvA..33.3657T
| |
| |doi=10.1103/PhysRevA.33.3657
| |
| |pmid=9897107
| |
| }}
| |
| *{{cite journal
| |
| |last1=Grotendorst |first1=J.
| |
| |last2=Weniger |first2=E. J.
| |
| |last3=Steinborn |first3=E. O.
| |
| |year=1986
| |
| |title=Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators
| |
| |journal=[[Physical Review A]]
| |
| |volume=33 |issue=6 |pages=3706–3726
| |
| |bibcode=1986PhRvA..33.3706G
| |
| |doi=10.1103/PhysRevA.33.3706 | |
| |pmid=9897112
| |
| }}
| |
| *{{cite journal
| |
| |last1=Grotendorst |first1=J.
| |
| |last2=Steinborn |first2=E. O.
| |
| |year=1988
| |
| |title=Numerical evaluation of molecular one- and two-electron multicenter integrals with exponential-type orbitals via the Fourier-transform method
| |
| |journal=[[Physical Review A]]
| |
| |volume=38 |issue=8 |pages=3857–3876
| |
| |bibcode=1988PhRvA..38.3857G
| |
| |doi=10.1103/PhysRevA.38.3857
| |
| |pmid=9900838
| |
| }}
| |
| *{{cite journal
| |
| |last1=Bunge |first1=C. F.
| |
| |last2=Barrientos |first2=J. A.
| |
| |last3=Bunge |first3=A. V.
| |
| |year=1993
| |
| |title=Roothaan-Hartree-Fock Ground-State Atomic Wave Functions: Slater-Type Orbital Expansions and Expectation Values for Z=2–54
| |
| |journal=[[Atomic Data and Nuclear Data Tables]]
| |
| |volume=53 |issue=1 |pages=113–162
| |
| |bibcode=1993ADNDT..53..113B
| |
| |doi=10.1006/adnd.1993.1003
| |
| }}
| |
| *{{cite journal
| |
| |last1=Harris |first1=F. E.
| |
| |year=1997
| |
| |title=Analytic evaluation of three-electron atomic integrals with Slater wave functions
| |
| |journal=[[Physical Review A]]
| |
| |volume=55 |issue=3 |pages=1820–1831
| |
| |bibcode=1997PhRvA..55.1820H
| |
| |doi=10.1103/PhysRevA.55.1820
| |
| }}
| |
| *{{cite journal
| |
| |last1=Ema |first1=I.
| |
| |last2=García de La Vega |first2=J. M.
| |
| |last3=Miguel |first3=B.
| |
| |last4=Dotterweich |first4=J.
| |
| |last5=Meißner |first5=H.
| |
| |last6=Steinborn |first6=E. O.
| |
| |year=1999
| |
| |title=Exponential-type basis functions: single- and double-zeta B function basis sets for the ground states of neutral atoms from Z=2 to Z=36
| |
| |journal=[[Atomic Data and Nuclear Data Tables]]
| |
| |volume=72 |issue=1 |pages=57–99
| |
| |bibcode=1999ADNDT..72...57E
| |
| |doi=10.1006/adnd.1999.0809
| |
| }}
| |
| *{{cite journal
| |
| |last1=Fernández Rico |first1=J.
| |
| |last2=Fernández |first2=J. J.
| |
| |last3=Ema |first3=I.
| |
| |last4=López |first4=R.
| |
| |last5=Ramírez |first5=G.
| |
| |year=2001
| |
| |title=Four-center integrals for Gaussian and Exponential Functions
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=81 |issue=1 |pages=16–28
| |
| |doi=10.1002/1097-461X(2001)81:1<16::AID-QUA5>3.0.CO;2-A
| |
| }}
| |
| *{{cite journal
| |
| |last1=Guseinov |first1=I. I.
| |
| |last2=Mamedov |first2=B. A.
| |
| |year=2001
| |
| |title=On the calculation of arbitrary multielectron molecular integrals over Slater-Type Orbitals using recurrence relations for overlap integrals: II. Two-center expansion method
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=81 |issue=2 |pages=117–125
| |
| |doi=10.1002/1097-461X(2001)81:2<117::AID-QUA1>3.0.CO;2-L
| |
| }}
| |
| *{{cite journal
| |
| |last1=Guseinov |first1=I. I.
| |
| |year=2001
| |
| |title=Evaluation of expansion coefficients for translation of Slater-Type orbitals using complete orthonormal sets of Exponential-Type functions
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=81 |issue=2 |pages=126–129
| |
| |doi=10.1002/1097-461X(2001)81:2<126::AID-QUA2>3.0.CO;2-K
| |
| }}
| |
| *{{cite journal
| |
| |last1=Guseinov |first1=I. I.
| |
| |last2=Mamedov |first2=B. A.
| |
| |year=2002
| |
| |title=On the calculation of arbitrary multielectron molecular integrals over Slater-Type Orbitals using recurrence relations for overlap integrals: III. auxiliary functions Q<sup>1</sup><sub>nn'</sub> and G<sup>q</sup><sub>-nn</sub>
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=86 |issue=5 |pages=440–449
| |
| |doi=10.1002/qua.10045
| |
| }}
| |
| *{{cite journal
| |
| |last1=Guseinov |first1=I. I.
| |
| |last2=Mamedov |first2=B. A.
| |
| |year=2002
| |
| |title=On the calculation of arbitrary multielectron molecular integrals over Slater-Type Orbitals using recurrence relations for overlap integrals: IV. Use of recurrence relations for basic two-center overlap and hybrid integrals
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=86 |issue=5 |pages=450–455
| |
| |doi=10.1002/qua.10044
| |
| }}
| |
| *{{cite journal
| |
| |last1=Özdogan |first1=T.
| |
| |last2=Orbay |first2=M.
| |
| |year=2002
| |
| |title=Evaluation of two-center overlap and nuclear attraction integrals over Slater-type orbitals with integer and non-integer principal quantum numbers
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=87 |issue=1 |pages=15–22
| |
| |doi=10.1002/qua.10052
| |
| }}
| |
| *{{cite journal
| |
| |last1=Harris |first1=F. E.
| |
| |year=2003
| |
| |title=Comment on ''Computation of Two-Center Coulomb integrals over Slater-Type orbitals using elliptical coordinates''
| |
| |journal=[[International Journal of Quantum Chemistry]]
| |
| |volume=93 |issue=5 |pages=332–334
| |
| |doi=10.1002/qua.10567
| |
| }}
| |
| | |
| ==See also==
| |
| | |
| [[Basis set (chemistry)|Basis sets used in computational chemistry]]
| |
| | |
| {{DEFAULTSORT:Slater-Type Orbital}}
| |
| [[Category:Quantum chemistry]]
| |
| [[Category:Computational chemistry]]
| |