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 Optics and Photonics Journal, 2012, 2, 332-337 http://dx.doi.org/10.4236/opj.2012.24041 Published Online December 2012 (http://www.SciRP.org/journal/opj) Simple Method of the Formation of the Hamiltonian Matrix for Some Schrödinger Equations Describing the Molecules with Large Amplitude Motions George А. Pitsevich, Alex E. Malevich Belarusian State University, Мinsk, Belarus Email: pitsevich@bsu.by Received September 8, 2012; revised October 7, 2012; accepted October 18, 2012 ABSTRACT A simple approach to the formation of a Hamiltonian matrix for some Schrödinger equations describing the molecules with large amplitude motions has been proposed. The algorithm involving one or several variables has been concretely defined for the basis functions represented by Fourier series and orthogonal polynomials, taking Hermitian polynomials as an example. Keywords: Schrödinger Equation; Large Amplitude Motions; Hamiltonian Matrix 1. Introduction Algebraic approaches to solving of Schrödinger equa-tions have several advantages compared to other methods. Provided the basis functions used for expansion of the wave functions and the potential energy of a system are adequately selected, the Schrödinger equation takes the matrix form, in the end its solution being reduced to the derivation of eigenfunctions and eigenvalues for the Hamiltonian matrix. When studying the molecules whose variables are changing with a large amplitude, the Ham-iltonian matrix derivation is a nontrivial problem. This work presents an algorithm to form the Hamiltonian ma-trix for some Schrödinger equations describing mole-cules and molecular systems with several variables of this type. Such equations may be illustrated by the fol-lowing: 22ddFUE (1)   2222,,,,,st stABU stststEst  (2)   222222,,,, ,,,, ,,,,xyzxyz xyzFxyzU xyzxyzExyz   (3) In all cases it is assumed that with a change in the vari-ables the kinematic parameters remain invariable. This may be attained by an adequate selection of a coordinate system strongly related to the molecule, both in the case of symmetric [1-4] and low-symmetry [5-9] molecules. Equation (1), in particular, describes internal vibrations in a molecule of methanol, taking the effective vibra-tional constant as F [10-12]. In some cases an invariable character of the molecular kinematic parameters for large amplitude motions may be considered as a physically valid approximation. Specifically, Equation (3) may be used for the description of motion of a hydrogen atom in the process of hydrogen bonding if we neglect motion of the oxygen atoms, the amplitude of which is in fact con-siderably smaller than that of H motion as a mass ratio of these atoms is 1:16. This paper presents “quick” ap-proaches to construct the Hamiltonian matrix for some basis functions. 2. Using of Fourier Series Since the use of Fourier series for solving of equations of the form given in (1) is frequently described in the lit-erature and in some works the formation algorithm for the Hamiltonian matrix is given in detail, we begin our analysis from Equation (2). Let the potential energy be given in the form: ,i,,,e;,ab ks ltklkla bUstu ab  (4) Then a wave function is derived as: Copyright © 2012 SciRes. OPJ G. А. PITSEVICH, A. E. MALEVICH 333i,,ens mtnmnmst b (5) Substituting (4) and (5) into (2), we obtain: i22,,i,,,ee0ns mtnmnmab nks mltkl nmnmkla bnA mBEbub  (6) Next we define coefficients for the exponential . In the second term the following condition must be fulfilled: iensmt;nk nk n nml ml mm   (7) Instead of (6), we have: i22,i,,,,ee0ns mtnmab ns mtnnmmnmnmn nmma bnA mBEbub  N (8) Then we construct the finite matrix with the dimen-sions . This means that n and m are varying within the limits from to c per unity. From (8) we derive: 2221 21;ccc ci22,i,,,,ee0ns mtnmabcns mtnnmmnmnmcnnm ma bnA mBEbub  (9) Now we take (9) as a matrix equation of the form ij jj jHbEb, where jb—column vector that, ac- cording to (5), gives the wave function corresponding to the energy jE. It is clear that a pair of the indices ,nm numbers rows of the Hamiltonian matrix and a pair of the indices ,nmb—its columns. Next, to derive the Hamiltonian matrix from (9), first we have to fix an order of the coefficients nm in the column vector of the wave function defined by Equation (5). For example, if , the transposed column vector may be of the form: 1c1, 11,01,10,10,00,11,11,01,1;;;;;;;;b bbbbbbbbb  (10) Let us assume that in the same order from top to bot-tom there is a change in the index pair ,nm,ab num-bering rows of the Hamiltonian matrix. Then a matrix element of H is numbered by two index pairs, ,,,nmnm. Considering that usually, for the diagonal element H c;nnmm we can write: 2200,,,nm nmHnAmB uE  (11) and for nondiagional elements we can write: ,,,,if andnnmmnm nmHunna mmb  (12) ,,, 0if ornm nmHnna mmb  (13) Numbering matrix elements of H by the ordinary indices ,ij2 each of which is varying from 1 to , we should establish for each of them a one- to-one correspondence to a pair of numbers by the prin-ciple: 21c,iniim; ,jjjnm3i. Specifically, in the case given by (10) for  we have ; 33 1n 1m; and for 6j we have 6; 6. Now an algo-rithm for the formation of the matrix 0n1mHtakes the fol-lowing form: 2200ii iiHnA mBuE (14) ,if andijijijnn mmiji jHunnamm b (15) 0if oriji jijHnnamm b (16) Let us write the Hamiltonian matrix in the explicit form with the use of (14 - 16) for . Besides, we assume that the index order is determined by the relation of (10), and 1c1ab. Then we have: 0, 11,01,1000,10, 11,11,01, 1000,11,11,0001,01,10, 11,01,1001,11,01, 10,10, 11,11,01, 1001,1 1,00,11,1 1,0001,0 10000000000000000000uuuABuuuuuuAuuuuABuuuu uuBuuuu uuuuuuuu uuuBuuu,1 0,1001,11,01, 10,10, 1001,1 1,00,100000000000 0uABuuuu uuAuuu u000ABu Next we consider the case of three variables. Let the Schrödinger equation be of the form: Copyright © 2012 SciRes. OPJ G. А. PITSEVICH, A. E. MALEVICH 334   22222,, ,,,,,, ,,,,2str str strABCstU strstrEstr r (17) Then we define an algorithm to form the Hamiltonian matrix when using three-dimensional Fourier series. Let the potential energy be given as: ,, i,,, ,,,e ;,,abc hs kt lrhklhkl abcUstr uabc N (18) A wave function takes the form: i,,,, ens mt qrnmqnmqstr b (19) Substituting (18) and (19) into (17), we obtain:  i222,,,, i,,,,,,ee0ns mt qrnmqnmqabc nhsmktlqrhkl nmqnmqhklabcnA mB qCEbub (20) Let us find coefficients for the exponential iens mt qr . The following condition must be fulfilled: ;;;nhnhn nmk mk m mlqlql l    (21) Instead of (20), we have: i222,, i,,,,,,, ,ee0ns mt qrnmqabc nsmt qrnnmmllnmqnmqn nm ml labcnA mB qCEbub N (22) We construct the finite matrix with the dimensions , i.e. n, m, and q are varying within the limits from to d per unity. From Equa-tion (22) we get: 3321 21;ddd di222,, i,,,,,,, ,ee0ns mt qrnmqabcdns mt lrnnmmllnmqnmqdnnmmllabcnA mB qCEbub  (23) Now three indices ,,nml number rows and three indices ,,nml1c number columns of the Hamiltonian matrix. To derive a Hamiltonian matrix from (23), we again fix an order of the coefficients nmq in the column vector for the wave function defined by (19). For exam-ple, if , the transposed column vector may be of the form: b1, 1, 11, 1,01, 1,11,0, 11,0,01,0,11,1, 11,1,01,1,10, 1, 10, 1,00, 1,10,0, 10,0,00,0,10,1, 10,1,00,1,11, 1, 11, 1,01, 1,11,0, 11,0,01,0,11,;;;;;;;;;;;;;;; ;;;;;;;;bb bbbbbb bbbbbbbbbbbbbbbbbb    ;1, 11,1,01,1,1;;bb (24) We assume that a change of three indices ,,nmq, numbering rows for the Hamiltonian matrix is in the same order from top to bottom. Then a matrix ele- ment of H is numbered by two pairs of three indices, ,, . Considering that, as previously, we have , then for the diagonal elements ,,,nmq nmqHd,,abcq;;mqnnm we can write: 222000,,,,,nmq nmqHnAmBqCu E (25) And for nondiagonal elements we can write: ,,,, ,,,if ;andnnmmq qnmq nmqHunnammb qqc   (26) ,, ,,,0ifor ornmq nmqHnnammb qqc  (27) when numbering the matrix elements of H by the ordi-nary indices ,ij each of which is varying from 1 to 321c, we should establish for each of them a one-to-one correspondence to three numbers by the prin-ciple: q,,iiinmi; ,,jjjjnmq3i. Specifically, in the case given by (24) for we have 31n; 31m; 31q, and for we have 1919j1n; 19 1m; 19 1q. Now an algorithm to form the ma-trix H takes the form: 222000ii iiiHnA mB qCuE (28) ,,if ;andiji jijijnn mm qqijij ijHunnmm bqq c a  (29) 0ifor oriji jij ijHnnammb qqc (30) 3. Using of Orthogonal Polynomials Let us consider the Schrödinger equation with one vari-able (31), taking orthogonal Hermitian polynomials nH as an example. 22ddxRUxxExx (31) Let the potential energy be given as:  0mkkkUxuH x (32) Copyright © 2012 SciRes. OPJ G. А. PITSEVICH, A. E. MALEVICH 335And we are looking for a wave function of the form:  2120exnnnxbHx (33) We substitute (32) and (33) into (31):   020200012140nnnnnnnnnmkn nknkRnEbH xRnnb HxRbH xubHx Hx  (34) Using the orthogonality of Hermitian polynomials, we can write:   2,,,212,,;ednknk lnkllnkxlnknk lHxHxc HxcHxHxH xx (35) As a result, Equation (34) takes the form:   ,,2002000,211204lnkk nlnn nnnnmnknnnnklnkcubHRnEbH xRnnbHxRbH xx  (36) Taking the coefficients for nH, we construct a matrix with the dimensions 1h1,h. In the second term of Equation (36) we must assume 22nnnn  in the third term we assume , and in the fourth –. Instead of (36), we get: 22nnnnln  220122104nnnnhnnnnkn nknkRnEbHxRnnb H xRbHx cHxbu    (37) As previously, the index numbers rows of the Hamiltonian matrix, whereas the index n numbers its columns. Let an order of indices in the column vector of the wave function be so that a form of the transposed vector is given by: n01;; hbbbb (38) In a similar way we will number rows of the Hamilto-nian matrix from top to bottom from 0 to h per unity. According to Equation (37), at the first stage we can fill the Hamiltonian matrix with the elements existing for representation of the potential energy in the form by the following principle: nnk kcunnnnk kkHcu (39) Summation in (39) is over all the existing indices k for the specified index pair ,nn. Next, to every diagonal element nnH we add 12nR and to every element of the diagonal, parallel to the main diagonal and posi-tioned above it as a next nearest ,2nnH, we add 21nn R. And in the case of a similar diagonal positioned as a next nearest below  we add ,2nnH14R. So, diagonal elements take the form: 12nnnnkkkHnRcu   (40) Nondiagonal elements are of the form: 2,21nnn nkkk2,HnnRc  u (41) 2,14nnn nkkk2,HRcu   (42) The remaining nondiagonal elements are as (39). Us-ing the ordinary indices varying from 1 to ,ij 1h, we can rewrite this algorithm as: 1, 1,12iiiik kkHiRcu  (43) ,21, 1,1iiiik kkHiiR cu  (44) ,2 1,3,14iiiikkkHRc u  (45) 1, 1,ijijk kkHcu (46) Finally, we consider Equation (3), trying to construct the Hamiltonian matrix with the use of Hermitian poly-nomials as basis functions. Let the potential energy be represented as:  ,,,, 0,, ;,,abcklm klmklmUxyzu HxH yHzabc N (47) A wave function is derived as follows:  2202222,,e ;rnts ntsntsxyzb HxHyHzrxyz (48) Substituting (47) and (48) into (3), we obtain: Copyright © 2012 SciRes. OPJ G. А. PITSEVICH, A. E. MALEVICH 336        020202020202032(1)11444nts ntsntsnts ntsntsnts ntsntsnts ntsntsnts ntsntsnts ntsntsnts ntsntsfnk htRntsEbHxHyHzRn nbHxHyHzRt tbHx Hy HzRs sbHx Hy HzRbHxH yH zRbHxHyH zRbHxH yHzcc  000abclrsmntsklm fhrnts klmfhrcbuHxHyHz (49) Suppose that we need to construct a matrix with the dimensions . We determine coeffi- 31dd31cients for the factor  :ntsHxH yH z       2,,,2,,, 22,,,2,,,232212121)444nts ntsntsn tsnts ntsnts ntsntsn tsnts ntsnt sRnt sEb H xHyHzRnnbHxH yH zRttbHxH yHzRssbH xH yH zRbHxHyHzRbHxHyHzRbH            00ntsdnnk ttlssm ntsklmntsnts klmxH yHzcccbuHxHyHz  (50) This expression may be rewritten as follows:  2, ,,2,,,22,,, 2,,, 203221 2121 4404nts nnts ntsntsntsntsdntsnnkt tls smntsklmnts klmRn tsEbHxRnn bRttbRRRssbbbRbcccbu          (51) We fix an order of the coefficients nts in the column vector of the wave function defined by Equation (48). For example, if , the transposed column vector may be of the form: b2d0,0,0 0,0,10,0,2 0,1,0 0,1,10,1,2 0,2,0 0,2,10,2,21,0,01,0,11,0,2 1,1,01,1,11,1,21,2,0 1,2,1 1,2,22,0,0 2,0,12,0,2 2,1,0 2,1,12,1,2 2,2,0 2,2,12,2,2;;;;;;;;;;;;;;;;;;; ;;;; ;;bbbbbbbbbbbbbbbbbbbbbbbbbbbb;(52) Let us assume that a change in three indices ,,nts numbering rows of the Hamiltonian matrix is in the same order from top to bottom. The matrix element His numbered by a pair of three indices ,,, ,,nts nts. As earlier, first we can fill the Hamiltonian matrix with the existing elements representing the potential energy of the form by the following principle: H n nkttls smklmcccu,,, ,,n nkt tls smklmnts ntskl mHcccu  (53) Summation in Equation (53) is performed over all the existing triples ,,klm for the pair of the specified triples ,,nts and ,,nts. For the main diagonal ,, ,,,nts ntsH  we must add 32nts R. To the nondiagonal elements of the form  ,, ,2,,;nts ntsH ,, ,2,,nts ntsH, and we must add ,, , ,,2nts ntsH 21Rn n;  21Rt t, and 2Rs s1. Finally, to the diagonal elements of the form ,, ,nts nH 2,,;tsH ,, ,,2,nts ntsH , and ,, , ,,2nts nts  we must add 1R. Thus, we have: 4,, ,,,32nts ntsnnk ttlssm klmklmHnts Rcccu  (54)  ,2,,, ,2,,21nnk ttl ssm klmnts ntskl mHcccuRn n   (55)  ,2,,, ,,2,21nnkt tlssmklmnts ntskl mHcc cuRt t    (56) ,2,,, ,,,221nnkttls smklmnts ntsklmHcccuRs s    (57)  2, ,,, ,2,,14nn kttlssmklmnts ntsklmHcccu R (58)  ,2,,, ,,2,14nnkt tlssmklmnts ntsklmHcc cuR   (59)  ,, , ,,214nnk ttl ssm klmnts ntsklmHcccu R  (60) In other cases, we have (53). Now numbering the ma-trix elements of H by the ordinary indices ,ij each of Copyright © 2012 SciRes. OPJ G. А. PITSEVICH, A. E. MALEVICH Copyright © 2012 SciRes. OPJ 337which is varying from 1 to , we have to estab-lish for them a one-to-one correspondence to three num-bers according to the principle: 321d,,ii iints; ,,jjjjnts5i17j. Specifically, in the case given by (52) for we have; ; , and for we have 17 ; 1750n1t51t251sn; 17 . Then an al-gorithm to construct the matrix under condition of (52) takes the following form: 1sijHiii iHnt,18 ,19;iiiin nklmHci,613;10 1iiii nnkklmHci ,21, 4, 7,10,13iiii nnkklmHci32is18 ,iik ttl s6, ,2;1921i it tlssc 2,16,19, 22iii ittls scc ;iissm klmcu21;inn21;tt21;ssii iinnk ttlklmcc m klmicuklmuR25;klmuRRiis;mc,,m (61) c R (62)  (63) , (64) 18 ,iinkttlc18, ,iiiinklmHc14klmu;19;Riisismc (65) 6, ,2;1921i iikttls6,13;101iiii nnklm14;ismklm ;Hcc c uRi  (66) 216,19,22iiiittl ss2,1, 4, 7,10,13iii innkklm14,,25;m klm;Hccc ij ijttl ssmuRi,ijij nnkklm (67) klmHcccu (68) 4. Conclusion In this way we have derived analytical expressions for elements of the Hamiltonian matrix describing the mole-cules characterized by motions with a large amplitude. 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