International Journal of Pure and Applied Sciences and Technology

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1 Int. J. Pure Appl. Sc. Technol., 4() (03), pp Internatonal Journal of Pure and Appled Scences and Technology ISSN Avalable onlne at Research Paper Schrödnger State Space Matrx Parametres Estmaton J. J. Medel, * and R. Palma Computng Research Centre, Real-tme and Automatc Control Department. Computng School, Natonal Polytechnc Insttute, Mexco Cty, Mexco * Correspondng author, e-mal: (jjmedelj@yahoo.com.mx) (Receved: 3-9-; Accepted: -0-) Abstract: The Schrödnger equaton as a quantum superposton descrbes the atomc orbtal as a lnear dfference functons bass set. Another alternatve s that ths equaton could be wrtten as a m-dmensonal state space lnear stochastc system wth unnown matrx coeffcents wth smooth movements around the statonary regon. Thus, the matrx s estmated n a probablty sense consderng the dstrbuton functon and ts stochastc moments. The matrx estmaton result s an alternatve soluton wthout losng the general system propertes. The estmaton s based on a gradent stochastc technque wth an nstrumental varable such that the contrbuton matrx s optmal n a probablty sense. Therefore, the advantages wth respect to tradtonal solutons are focused on estmatng the matrx contrbuton on-lne wth a lnear complexty. Keywords: Schrödnger equaton, Mathematcal methods, Fnte dfference methods, Flters, Control theory.. Introducton The Schrödnger equaton H E, s governed by physcs, chemstry and bology prncples and has an approxmate predctve power [9]. However, for over 80 years snce ts ncepton; t was thought to be nsoluble except for a few specal cases. Snce 000, Naatsuj et al., n [9] formulated a general method solvng the Schrödnger equaton appled to atoms and molecules n analytcal expanson form, havng a hgh numercal and computatonal complexty [5]. In a physcal sense, the Schrödnger equaton as a quantum superposton descrbes the atomc orbtal as a bass functon set. Ths s expanded as a lnear functonal combnaton. It s descrbed as a matrces array n fnte dfferences wth unnown coeffcents [6], [7], [8].

2 Int. J. Pure Appl. Sc. Technol., 4() (03), An ntal assumpton s that the orbtal number s ncluded n the lnear expanson. In a sense, m- atomc orbtal can be numbered from to m, wth m Z. The expresson for the -th orbtal s gven n (). c χ () r Where s an orbtal represented as the sum of m-atomc orbtals and ts correspondng contrbuton coeffcent c r, and r,m, m Z, where the Hartree-Foc procedure s used obtanng the expanson coeffcents [].. Prelmnares Quantum calculatons are typcally performed wthn a fnte bass functons set. In these cases, the wave functons under consderaton are represented as vectors. The components correspond to coeffcents n the bass set lnear functons combnaton. The operatons are represented as matrces n fnte dfferences. A mnmum bass set s one n whch each atom s represented by a basc Hartree-Foc calculaton functon [4, 7]. The orbtal s expressed as a lnear electron functons combnaton bass, centred on atomc components. The basc atomc orbtal s the hydrogen atom and s nown as Slater-type orbtal [4]. An atomc orbtal s a mathematcal functon that descrbes the wave-le behavour of ether one or a par of electrons n an atom [7]. Atomc orbtal s typcally descrbed as hydrogen scheme (meanng one-electron) wave functons over space. The wave functon for a complex atom electrons cloud may be seen as beng bult up n an electron confguraton that s a sum of smpler hydrogen atomc orbtals. The orbtal approxmaton s symbolcally represented for the atomc wave functon, and s descrbed n ().... () r r 3 Where each, functon wth, m, m Z, s defned as 3-dmensonal ndvdual electron property. The spn s ncluded n the wave functon as a spn-orbtal product. The wave functon has to be antsymmetrc wth respect to electron nterchange wrtten as a dagonal matrx [7]. The goal of most quantum calculatons s the producton of a wave functon. Ths s acheved f all or are nown. consttuent orbtals are descrbed by { } 3. Results and Dscusson, m Quantum super-poston s a fundamental prncple of quantum mechancs. Mathematcally, t refers to Schrödnger equaton solutons property havng a lnear form n state space of a nown m-th-partcle quantum object. Then, the vector contans the poston coordnates assocated wth quantum objects. The stochastc gradent of a wave functon s symbolzed by. Thus, the state space s defned n (3). (,, ) (,, ) W ( ) (3),, Where (,, ) s the Schrödnger equaton, and W s a stochastc system n Hlbert space wth L norm,,, ) s descrbed n a probablty space. ( Wth states as x, x, then x ɺ x, xɺ, where x y z 4π 4π and 4π 4π 0 wth c.

3 Int. J. Pure Appl. Sc. Technol., 4() (03), Therefore c consdered the wor Palma et al., n whch the propertes of the m-dmensonal stochastc estmator for a blac-box model were descrbed n [0], the Schrödnger equaton state space n a matrcal form s gven n (4). xɺ xɺ 0 c x 0 x dw dw (4) A lnear relatonshp for a mult-electron system s bult as follows. Let the Schrödnger equaton as (4) n an affne state space gven by (5) t was necessary to change the rows n (4). Where m x, m and[ dw, ] T [ w, ] T are the nnovaton stochastc processes for (4) and (5), respectvely. x dw w mɺ mɺ c 0 0 m m w w (5) From (), each molecular orbtal,, mɺ mɺ n, n Z,, has an assocated affne state space (6). c 0 0 m m w w (6) And the total state space s gven by (7). Mɺ M ɺ C 0 0 I M M W W (7) m Where C, I R are constant vectors, defned by C : { c }, I : {, m m } and, m C 0 M W A,, W. The lnear relatonshp for mult-electron model s (8). 0 I M W ɺ A W (8) After the Schrödnger state space model s formed, ts goal s to estmate the unnown parametres denoted by A ~. Frst, transformng (8) n dfferences havng as a stochastc sample set taen m Z sze from a random vector. Second, the matrx parametres correspondence has the form I A. Thrd, the whte nose has an approxmated dscrete descrpton. Then (8) n A dfferences has the form (9). A The matrx A s ntegrated by the contrbuton weghts set. It s a matrx wth unnown parametres []. Although the Schrödnger equaton was solved by [5, 9], numercal ntegraton methods are usually nsuffcent for accuracy and has a hgh computatonal cost, havng an exponental growth wth respect to the orbtals numbers consdered [9]. In spte of, the state space contrbuton of (9) s numercally reduced wth a lnear computatonal complexty observng n ths model the stablty condtons [3]. Unfortunately, the new system has unnown contrbuton weghts reflected n the matrx A. W W In a probablty sense, t has an optmal descrpton based on the nnovaton process W (9), data nformaton, was developed usng the stochastc gradent, symbolcally descrbed n (0) [6, 8]. ~ A E{ M } E{ W M } (0) Wth W, M are the process nose and the correlaton matrx, respectvely; and the nnovaton () s bult from (9). W A ()

4 Int. J. Pure Appl. Sc. Technol., 4() (03), Then () has a dagonal form (). M W 0 C 0 M 0 () 0 M W 0 I 0 M T M 0 ϑ be an nstrumental varable. Multplyng () by ϑ 0 M Let, wth second probablty moment havng the form (3). E{( ) } ~ W A E{ M } ϑ (3) M, ϑ det( M ) 0, andm ϑ M. Fnally, the optmal stochastc estmator s gven by (0). Schrödnger equaton numercally smulated n state space s shown n Fgures and. Fgure.a) depctng the C components, Fgure.b) depctng the I component. Fgures.c) and.d) show the thrd and fourth components. M l 0.5 M a) b) M M c) d) Fgure. Numercal smulaton: a) Frst, C5 component, b) Second, I component, c) Thrd, zero component, d) Fourth, zero component.

5 Int. J. Pure Appl. Sc. Technol., 4() (03), AA a) b) Fgure. Numercal results: a) Matrx parametres: Frst n red, second n green, thrd n blue, fourth n brown, b) Functonal error and ts nset. Fgure.a) shows matrx parametres numercal evoluton and, Fgure.b) shows the functonal error wth an accuracy order of Conclusons Due to Schrödnger beng transformed n state space and developed n dagonal form (), the optmal stochastc estmator for a mult-electron has the form (3). Therefore, the numercal complexty s lneal [0]. Fgure 3.a) shows the tradtonal computatonal complexty soluton method, Fgure 3.b) descrbes a lnear state space descrpton wth lnear computatonal complexty. Fgure 3. Computatonal complexty: a) Tradtonal Schrödnger soluton, b) State Space Schrödnger descrpton. The matrx parametres C s estmated usng the dagonalzaton process for m orbtal wthout calculatng the tradtonal potental terms. For the case m havng lowest tme executon as shown n Fgure 3. Ths has mportant mplcatons for data storage and computng when evaluatng very large complex systems. Acnowledgements Ths document was checed by Englsh Teacher Lawrence Whtehll, England.

6 Int. J. Pure Appl. Sc. Technol., 4() (03), References [] C.J. Cramer, Essentals of Computatonal Chemstry, John Wley and Sons Ltd., Chchester, England, 00. [] U.K. Dewangan, Structural parameter dentfcaton A new concept, Int. J. Pure Appl. Sc. Technol., 0() (0), -9. [3] B. Ghanbar, M.G. Porshoouh and B. Rahm, A numercal method for solvng systems of nonlnear ODE s, Int. J. Pure Appl. Sc. Technol., 3() (0), [4] F. Jensen, Introducton to Computatonal Chemstry, John Wley and Sons Ltd., Chchester, England, 999. [5] Y. Kuroawa, H. Naashma and H. Naatsuj, Free teratve-complement-nteracton calculatons of the hydrogen molecule, Phys. Rev., A7(005), [6] E.L. Lehmann and G. Casella, Theory of Pont Estmaton, Sprnger-Verlag, New Yor, 998. [7] I.N. Levne, Quantum Chemstry, Prentce Hall, New Jersey, 99. [8] J.J. Mortensen, K. Kaasbjerg, S.L. Fredersen, J.K. Norsov, J.P. Sethna and K.W. Jacobsen, Bayesan error estmaton n densty-functonal theory, Phys. Rev. Lett., 95(005), 640. [9] H. Naatsuj, H. Naashma, Y. Kuroawa and A. Ishawa, Solvng the Schrödnger equaton of atoms and molecules wthout analytcal ntegraton based on the free teratve-complementnteracton wave functon, Phys. Rev. Lett., 99(007), [0] R.O. Palma, J.J. Medel and G.A. Garrdo, An m-dmensonal stochastc estmator, Rev. Mex. Fís., 58() (0), 0069.