New Version of the Rayleigh Schrödinger Perturbation Theory: Examples

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1 New Versio of the Rayleigh Schrödiger Perturbatio Theory: Examples MILOŠ KALHOUS, 1 L. SKÁLA, 1 J. ZAMASTIL, 1 J. ČÍŽEK 2 1 Charles Uiversity, Faculty of Mathematics Physics, Ke Karlovu 3, Prague 2, Czech Republic 2 Uiversity of Waterloo, Waterloo, Otario N2L 3G1, Caada Received 22 Jauary 23; accepted 1 October 23 Published olie 9 Jue 24 i Wiley IterSciece ( DOI 1.12/qua.241 ABSTRACT: It has bee show i our precedig papers that the liear depedece of the perturbatio wave fuctios o the perturbatio eergies makes possible to calculate the exact perturbatio eergies from the values of the perturbatio wave fuctios correspodig to arbitrarily chose trial perturbatio eergies. The resultig versio of the perturbatio theory is very simple ca be used at large orders. I this paper, this method is applied to a few problems its umerical properties are discussed. 24 Wiley Periodicals, Ic. It J Quatum Chem 99: , Itroductio I this paper, we are iterested i the perturbatio theory for the boud states of the Schrödiger equatio: H x E x. (1) As usual i the Rayleigh Schrödiger perturbatio theory, we assume the Hamiltoia, wave fuctio, eergy to be i the forms H H H 1, (2) Correspodece to: L. Skála; skala@karlov.mff.cui.cz 1 2 2, (3) E E E 1 2 E 2, (4) where is a perturbatio parameter. Usig these assumptios i the Schrödiger equatio (1), we obtai the well-kow equatios for E, H E (5) H H 1 1 E i i, 1, 2,.... i (6) Iteratioal Joural of Quatum Chemistry, Vol 99, (24) 24 Wiley Periodicals, Ic.

2 KALHOUS ET AL. We ote that deotes the uperturbed wave fuctio of the Hamiltoia H. Depedig o the problem i questio, it ca be the groud-state as well as excited-state wave fuctio. Despite the well-kow formulatios that ca be foud i ay textbook o quatum mechaics, there is oe property of the perturbatio theory that has bee oticed [1, 2] used [3 9] oly recetly. It has bee show that the value of the perturbatio wave fuctio (x) at a arbitrarily chose poit x depeds liearly o the perturbatio eergy E. This liear depedece makes it possible to determie the exact perturbatio eergies from the values of (x) for two arbitrarily chose trial perturbatio eergies E by simple calculatio. I this way, the fuctios which are ot quadratically itegrable ca be used to calculate the exact perturbatio eergies E, i the ext step, the correspodig exact perturbatio fuctios. This method has a few advatages. I cotrast to the usual formulatio of the perturbatio theory, this method based o the computatio of from Eq. (6) for a give eergy E ca easily be programmed for arbitrary large orders of the perturbatio theory. For example, 2 perturbatio eergies E ecessary for fidig their large-order behaviour were calculated i [7]. Further, by solvig Eq. (6) umerically, both the discrete the cotiuous parts of the eergy spectrum is take ito accout, the perturbatio eergies E ca be calculated eve i cases whe oly a few boud states exist. The liear depedece of (x) othe eergy E makes it possible to avoid the usual shootig method reduce the computatioal time substatially. Fially, we ote that oly the wave fuctios are eeded i this method o itegrals have to be calculated. The aim of this paper is to apply this method to a few problems test its umerical properties. 2. Summary of the Method First we discuss a odegeerate multidimesioal case. We assume that the perturbatio fuctios i perturbatio eergies E i are already computed for i,..., 1. Solutio of Eq. (6) ca be writte as E, x E F x f 1 x, 1, 2,..., (7) where F x H E 1 x (8) 1 1 f 1 x H E H 1 1 x E i i x. i1 (9) The geeral solutio of Eq. (6) ca cotai also a term c (x) at the right-h side of Eq. (7), where c is a arbitrary costat. For the sake of simplicity, we assume c here. As it is see from Eq. (7), the perturbatio fuctio (E, x) depeds o the eergy E which is ot yet kow the poit x [x 1,...,x N ]in-dimesioal space. Equatios (7) (9) show that the structure of the perturbatio fuctios is very simple. It follows from Eq. (7) that the fuctio (E, x) isaliear fuctio of the eergy E. Further, it is see that F(x) is a fuctio idepedet of. We ote also that, except for the case that E is the exact perturbatio eergy, (E, x) is ot quadratically itegrable has o physical meaig. The fuctios F(x) f 1 (x) are calculated from Eqs. (8) (9) umerically with the coditios F(x b ) f 1 (x b ), where x b are poits at the boudary regio sufficietly distat from the potetial miimum. The same boudary coditios are used for the fuctio (x). We ote that the fuctio F(x) diverges i the exact calculatio, however, it has large but fiite values i umerical calculatios. The fuctios (E, x) for the exact perturbatio eergy E are quadratically itegrable. Therefore, we ca assume they obey the coditio E, x F x. (1) It follows from Eqs. (7) (1) that the fuctios (E, x) also satisfy the coditio E, x f 1 x. (11) Therefore, we ca eglect (E, x) i Eq. (7). The formula for the eergy E the reads E f 1 x. (12) F x This equatio ca be used at a arbitrarily chose poit x iside the itegratio regio except for the 326 VOL. 99, NO. 4

3 NEW VERSION OF THE PERTURBATION THEORY poits where the coditios (1) (11) are ot obeyed. If the perturbatio eergy E is calculated from Eq. (12), the correspodig perturbatio fuctio (E, x) ca be foud from Eqs. (7) (9). More detailed discussio of the method is give i [6, 9]. The eergy E 1 computed umerically from the equatio E 1 x f x F x (13) depeds slightly o the choice of the poit x. Calculatig 1 from Eq. (7) for 1, we obtai the fuctio Now we discuss the first-order perturbatio correctio to a degeerate eigevalue E. Assumig that the eergy E is -times degeerate the correspodig zero-order fuctio i Eq. (6) is replaced by the liear combiatio a j j, (16) j1 it follows from Eq. (6) that Eq. (7) ca be geeralized as 1 E 1, x E 1 j1 a j F j x a j f j x, (17) j1 1 E 1, x f x F x f x F x. (14) F x It shows that the fuctio 1 calculated i this way equals zero at the poit x : 1 E 1, x. (15) where F j x H E 1 j x (18) f j x H E 1 H 1 j x. (19) Therefore, the usual orthogoality coditio, 1, is ot fulfilled i umerical calculatios. It ca easily be show that this result ca be exteded to all fuctios. As show i [6], such fuctios ca have i some cases more simple form tha the usual perturbatio fuctios. If ecessary, the fuctios ca be made orthogoal to by the usual orthogoalizatio procedure. Therefore, the poit x used i the calculatio of the eergy (12) should be sufficietly distat from the poits where the fuctio (x) equals zero. Our method is a remarkable example of calculatig the perturbatio eergies E from the values of the fuctios F(x) f 1 (x) which are ot quadratically itegrable. Comparig with the stard formulatio of the perturbatio theory, large-order calculatios are simple i our method. To determie E 1, the values of F(x) f (x) at oly oe poit x are sufficiet. To determie E for 2, 3,...,oly the value of f 1 (x) at the poit x is to be computed. Therefore, this method of calculatig E is much faster tha the usual shootig method. We ote that the zero-order fuctio has to be foud oly for the state for which the perturbatio correctios are calculated. I cotrast to the usual perturbatio theory, other zero-order eergies wave fuctios are ot eeded i the calculatio. It is see from Eq. (17) that 1 (E 1, x) depeds o E 1 liearly as i the odegeerate case. Therefore, by aalogy with the odegeerate case, we ca derive the formula for the perturbatio eergy E 1 : E 1 j1 a j f j x a j F j x. (2) j1 Here, f (j) (x) F (j) (j) (x) are kow, E 1 a are to be foud. To fid E 1 a (j), we exploit the fact that the eergy E 1 is a costat use Eq. (2) at differet poits x x 1,..., x d iside the itegratio regio. The solutio of the equatios d j1 d j1 a j f j x 1 a j F j x 1 j1 a j f j x 2 a j F j x 2 j1 j1 d j1 a j f j x d a j F j x d (21) the yields sets of the coefficiets a (j). The values of the eergy E 1 are give by Eq. (2), the correspodig perturbatio fuctios equal 1 a j j. (22) j1 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 327

4 KALHOUS ET AL. The usual formulatio of the first-order degeerate perturbatio theory ca be derived from Eq. (2) as follows. Substitute Eqs. (18) (19) ito Eq. (2), we obtai for a poit x E 1 j1 d j1 a j H E 1 H 1 j x a j H E 1. (23) j x By multiplyig the umerator deomiator by (H E ), we obtai E 1 j1 E 1 j1 d j1 a j H 1 j x a j j x (24) a j j x a j H 1 j x. (25) j1 Further, by multiplyig this equatio from the left side by the complex cojugate fuctio (i)* (x), itegratig over x, assumig orthoormality of the fuctios (j) (x), a stard secular problem is obtaied: where W ij E 1 ija j, i 1,...,, (26) j1 3. Examples W ij i H 1 j. (27) For the oe-dimesioal problems, this method has already bee used with very good results (see [3 8]). For this reaso, several two-dimesioal problems with icreasig complexity are ivestigated here COUPLED ANHARMONIC OSCILLATORS As the first example, we discuss the problem havig oly boud discrete states. We calculated the perturbatio eergies for the groud state of two oliearly coupled harmoic oscillators [9 11]: TABLE I Perturbatio eergies E for the groud state of two coupled harmoic oscillators (28); E is the exact zero-order eergy. H 2 x x 2 2 x 1 2 x 2 2 x 1 2 x 2 2 x 1 2 x 2 2. (28) To compute from Eq. (6) i the regio x 1 [11, 11], x 2 [11, 11], we used a grid of poits , 16 16, , , assumed that the fuctios equal zero at the border of this regio, solved the correspodig system of differece equatios i double-precisio accuracy i Fortra. The same grid of poits was used i all examples give below, oly the itegratio regio was differet. To elimiate the effect of a o-zero step of the grid the perturbatio eergies were extrapolated to a ifiitely dese grid by meas of the Richardso extrapolatio. This extrapolatio is substatial for icreasig the accuracy of the results. The groudstate perturbatio eergies are show i Table I. Oly the digits which agree i the calculatios for the poits x [, ] x [1, 1] are show. It is see that the eergies deped o the choice of the poit x oly very little. All digits show i Table I also agree with a idepedet calculatio made E VOL. 99, NO. 4

5 NEW VERSION OF THE PERTURBATION THEORY TABLE II Perturbatio eergies E for the groud state of two coupled Morse oscillators (29); E is the exact zero-order eergy. by meas of the differece equatio method suggested i [11]. It is also see that the method ca be used at large orders. Therefore, these results are quite satisfactory COUPLED MORSE OSCILLATORS As a secod test, we chose a more difficult problem with oly oe boud state of the zero-order Hamiltoia whe the stard perturbatio theory yields the first-order correctio E 1 oly. The perturbatio eergies were calculated for the groud state of two coupled Morse oscillators: H 2 x x e x1 2 1 e x2 2 1 e x1 2 1 e x2 2. (29) The itegratio regio x 1 [12, 2] x 2 [12, 2] was used. The groud-state perturbatio eergies are show i Table II. Oly the digits that agree i the calculatios for the poits x [4, 4] x [5, 6] are show. The value of E 1.25 was verified by aalytic calculatio. Decreasig accuracy of the eergies E with icreasig is due to the fact that the fuctios spread with icreasig rapidly get out of the itegratio regio. These results show that, i cotrast to the stard perturbatio theory, our method ca be used for calculatig large-order perturbatio eergies eve if there are oly a few zero-order boud states. E 3/ BARBANIS HAMILTONIAN As aother example of usig our method we calculated the perturbatio eergies for the groud state of the Barbais Hamiltoia: H 2 x 2 2 y 2 x 2 x 2 y 2 y 2 xy 2. (3) Here, we used the frequecies x 1 y 1. This Hamiltoia has bee ofte studied as a simple model for systems with the Fermi resposes such as the CO 2 stretch bed resoace. Potetial i this Hamiltoia is ot bouded from below. The groud-state perturbatio eergies were calculated both umerically aalytically (see Table III). I the umerical calculatio, the itegratio regio x [11, 11] y [11, 11] was used. Similarly to the precedig examples, oly the digits that agree i the calculatios for the poits x [.2,.5] x [.6,.8] are show. Agai, the eergies deped o the choice of the poit x oly little. The umerical results agree with a idepedet aalytic calculatio made i Maple [12] with a accuracy to 6 9 digits. Due to the atisymmetry of the perturbatio potetial i Eq. (3), the odd-order perturbatio eergies E 1, E 3,...equal zero HÉNON-HEILES HAMILTONIAN As the last test of our method, the Héo Heiles Hamiltoia was used. The perturbatio eergies were calculated for the groud state the first excited state of the zero-order Hamiltoia 2 H 1 2 x x 2 x y y 2 y 2 (31) with the perturbatio potetial 2 H 1 yx 2 y 3. (32) The frequecies x 2, y 1.3, the parameter 1 correspodig to a very flat shallow potetial miimum were used. Similarly to the Barbais potetial, the perturbatio potetial is ot bouded from below. The aalytic eergies wave fuctios of the Héo Heiles potetial ca be calculated as follows. First, we express the eigefuctios of H as the products of the eigefuctios of the harmoic os- INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 329

6 KALHOUS ET AL. TABLE III Perturbatio eergies E for the groud state of the Barbais Hamiltoia (3); E is the exact zero-order eergy.* E um E a * The eergies calculated umerically from Eq. (12) are deoted as E um. The aalytic eergies deoted as E a were calculated for,2,...,2.odd-order perturbatio eergies equal zero. cillator. For example, the groud-state zero-order wave fuctio equals x, y /4 5 3/4 e x2 13/ 2 y2 x y. (33) Aalogously, orthoormal excited state wave fuctios correspodig to low quatum umbers ca be writte i the form 1 x, y /4 5 1/4 ye x2 13/ 2 y2 x 1 y, (34) 3 x, y /4 5 1/4 y15 13y 2 e x2 13/ 2 y 2 x 3 y, (35) 21 x, y /4 5 1/4 1 4x 2 ye x2 13/ 2 y2 2 x 1 y. (36) Now we discuss the groud-state perturbatio problem with the zero-order wave fuctio 33 VOL. 99, NO. 4

7 NEW VERSION OF THE PERTURBATION THEORY x, y x, y. (37) After a simple but tedious calculatio, H 1 ca be expressed as a liear combiatio of the eigefuctios of the uperturbed Hamiltoia H : H The first-order perturbatio eergy equals zero: (38) E 1 H 1. (39) The correspodig eigefuctio ca be calculated from the equatio with the result 1 H E 1 E 1 H 1, (4) /4 5 3/4 y12 689y 2 57x 2 e x2 13/ 2 y 2. (41) Higher order perturbatio eergies wave fuctios ca be obtaied i a similar way. For example, the secod- third-order perturbatio wave fuctios equal /4 5 3/ y y y x x 2 y x 2 y x x 4 y 2 e x2 13/ 2 y 2 (42) /4 5 3/4 y y y y y x x 2 y x 2 y x 2 y x x 4 y x 4 y x x 6 y 2 e x2 13/ 2 y 2. (43) These fuctios agree very well with the umerically calculated oes are show i Figures 1 4. It is see that the groud-state perturbatio fuctios go to zero at the boudaries of the itegratio regio x [11, 11] y [11, 11]. With icreasig, the fuctios spread out from the cetre of the itegratio regio their absolute value goes up. I umerical calculatios, the itegratio regio x [11, 11], y [11, 11] was used. The umerical aalytical groud-state first excited state perturbatio eergies are show i Tables IV VII. Oly the digits which agree i the calculatios for the poits x [.2,.5] x [.6,.8] are show. Depedece of the results o the choice of the poit x is small, as i the precedig examples. The umerical results agree with the aalytic oes with a accuracy of 6 9 digits. Because of the atisymmetry of the perturbatio potetial (32), the odd-order perturbatio eergies FIGURE 1. Groud-state zero-order wave fuctio for the Héo Heiles Hamiltoia (31, 32). INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 331

8 KALHOUS ET AL. FIGURE 2. First-order perturbatio fuctio 1 for the groud state of the Héo Heiles Hamiltoia (31, 32). FIGURE 4. Third-order perturbatio fuctio 3 for the groud state of the Héo Heiles Hamiltoia (31, 32). E 1, E 3,... equal zero. I umerical calculatios, these eergies differ from zero with a accuracy of 7 9 digits. The eve-perturbatio eergies E 2, E 4,..., are egative, their absolute value icreases rapidly with the order of the perturbatio theory. Results i Tables IV VII idicate that the absolute values of the perturbatio eergies E icrease with the eergy differece of the excited state the groud state. FIGURE 3. Secod-order perturbatio fuctio 2 for the groud state of the Héo Heiles Hamiltoia (31, 32). Cocludig this sectio, our method of solvig the perturbatio problem gives good results i the all ivestigated cases icludig the most difficult Héo Heiles Hamiltoia. The absolute values of the perturbatio eergies E icrease rapidly for large. 4. Coclusios I summary, the method described i this paper is simple efficiet alterative to the usual formulatio of the perturbatio theory. It ca be used for oe-dimesioal as well as multidimesioal problems for odegeerate as well as degeerate eigevalues. Its mai advatages are easy calculatio of the large-order perturbatios the possibility to fid the perturbatio correctios, eve i cases whe oly a few zero-order boud states exist. The aalytic umerical results for the ivestigated examples show that our versio of the perturbatio theory yields results with good accuracy ca be used at large orders. The umerical results show i all ivestigated examples that the absolute values of the perturbatio eergies E icrease rapidly for large. They also idicate that the perturbatio series (4) are diverget, asymptotic series i these cases. They are obviously related to differet asymptotic behavior of the wave fuctios correspodig to the zeroorder Hamiltoia H H the full Hamiltoia H H H 1 for x 3. Therefore, oe should be very careful whe trucatig such perturbatio series at low orders. 332 VOL. 99, NO. 4

9 TABLE IV Perturbatio eergies E for the groud state of the Héo Heiles Hamiltoia (31, 32); E is the exact zeroorder eergy.* E um E a * The eergies calculated from Eq. (12) are deoted as E um. The aalytic eergies deoted as E a were calculated for,2,..., 16. For lack of space, the aalytic eergies E a 1, E a 12, E a 14, E a 16 are writte as a decimal umber oly. Odd-order perturbatio eergies equal zero. TABLE V Perturbatio eergies E for the first excited [, 1] state of the Héo Heiles Hamiltoia (31, 32); E is the exact zero-order eergy.* E um E a * The eergies calculated from Eq. (12) are deoted as E um. The aalytic eergies deoted as E a were calculated for,2,..., 12. For lack of space, the aalytic eergies E a 8, E a 1, E a 12 are writte as a decimal umber oly. Odd-order perturbatio eergies equal zero.

10 TABLE VI Perturbatio eergies E for the secod excited [1, ] state of the Héo Heiles Hamiltoia (31, 32); E is the exact zero-order eergy.* E um E a * The eergies calculated from Eq. (12) are deoted as E um. The aalytic eergies deoted as E a were calculated for,2,..., 12. For lack of space, the aalytic eergies E a 1 E a 12 are writte as a decimal umber oly. Odd-order perturbatio eergies equal zero. TABLE VII Perturbatio eergies E for the third excited [1, 1] state of the Héo Heiles Hamiltoia (31, 32); E is the exact zero-order eergy.* E um E a * The eergies calculated from Eq. (12) are deoted as E um. The aalytic eergies deoted as E a were calculated for,2,..., 12. For lack of space, the aalytic eergies E a 1 E a 12 are writte as a decimal umber oly. Odd-order perturbatio eergies equal zero.

11 NEW VERSION OF THE PERTURBATION THEORY ACKNOWLEDGMENTS The authors thak the GA U.K. (grat 166/), the GA C.R. (grat 22//126), the MS C.R. (grat 1-1/2653) of Czech Republic NSERC of Caada (J. Z. is a NATO Sciece Fellow) for support. Refereces 1. Skála, L.; Čížek, J. J Phys A 1996, 29, L129 L Skála, L.; Čížek, J. J Phys A 1996, 29, Guardiola, R.; Ros, J. J Phys A 1996, 29, Zojil, M. J Phys A 1996, 29, Au, C. K.; Chow, C. K.; Chu, C. S. J Phys A 1997, 3, Skála, L.; Čížek, J.; Zamastil, J. J Phys A 1999, 32, Skála, L.; Čížek, J.; Kapsa, V.; Weiger, E. J. Phys Rev A 1997, 56, Skála, L.; Čížek, J.; Weiger, E. J.; Zamastil, J. Phys Rev A 1999, 59, Skála, L.; Kalhous, M.; Zamastil, J.; Čížek, J. J Phys A 22, 35, L167 L Vrscay, E. R.; Hy, C. R. J Phys A 1989, 22, Baks, T.; Beder, C. M.; Wu, T. T. Phys Rev D 1973, 8, Moaga, M. B.; Geddes, K. O.; Heal, K. M.; Labah, G.; Vorkoetter, S. M.; McCarro, J.; DeMarco, P. Maple 9 Itroductory Programmig Guide, ISBN , Maplesoft, Caada 23. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 335