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Hydrogen and helium atoms in strong magnetic fields Thirumalai, Anand 2007

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Hydrogen and Helium Atoms in Strong Magnetic Fields by A n a n d T h i r u m a l a i B . T e c h . , T h e I n d i a n Ins t i tu te of Techno logy - Roorkee , 2002 M . A . S c , T h e U n i v e r s i t y of B r i t i s h C o l u m b i a , 2005 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F • M a s t e r of Science i n T h e F a c u l t y of G r a d u a t e S tudies (Phys ics ) T h e U n i v e r s i t y O f B r i t i s h C o l u m b i a September , 2007 © A n a n d T h i r u m a l a i 2007 Abstract T h e energy levels of hyd rogen a n d h e l i u m a toms i n magne t i c fields of ar-b i t r a r y s t reng th are ca lcu la t ed i n this s tudy. T h e current work conta ins es t imates of the g r o u n d a n d first few exc i t ed Har t r ee -Fock states of these systems t ha t are improvement s u p o n previous est imates . T h e m e t h o d o l -ogy involves c o m p u t i n g the eigenvalues a n d eigenvectors of the genera l ized Ha r t r ee -Fock p a r t i a l different ial equat ions for these one- a n d two-e lec t ron systems i n a self-consistent manner . T h e m e t h o d desc r ibed here in is app l i ca -ble to ca lcu la t ions of a t o m i c s t ruc ture i n magne t i c fields of a r b i t r a r y s t reng th as i t exp lo i t s the n a t u r a l symmet r i e s of the p r o b l e m w i t h o u t assumpt ions of any basis funct ions . i i Table of Contents A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f T a b l e s v L i s t o f F i g u r e s vi A c k n o w l e d g e m e n t s x D e d i c a t i o n x i 1 I n t r o d u c t i o n 1 2 L i t e r a t u r e R e v i e w 4 2.1 H y d r o g e n a n d H y d r o g e n - l i k e species 5 2.2 H e l i u m a n d H e l i u m - l i k e species 7 2.3 M a n y - e l e c t r o n Sys tems 12 3 T h e o r e t i c a l B a c k g r o u n d 13 3.1 T h e Ha r t r ee -Fock E q u a t i o n s - a de r iva t i on 13 iii Table of Contents 3.2 T h e H e l i u m A t o m a n d H e l i u m - l i k e species 16 ' 3.2.1 D e r i v a t i o n of the genera l ized H a r t r e e - F o c k equat ions i n p a r t i a l differential fo rm 17 3.2.2 T h e di rec t i n t e r ac t ion 22 3.2.3 T h e exchange in t e rac t ion 24 4 Numer i ca l Detai ls 28 4.1 N u m e r i c a l procedures for the hydrogen a t o m 28 4.2 N u m e r i c a l procedures for he l i um- l ike species 29 5 Results &; Discussion 32 5.1 T h e H y d r o g e n A t o m 32 5.2 T h e H e l i u m A t o m 37 6 Summary &: Conclusions 53 6.1 F u t u r e W o r k 55 Bibl iography 56 Appendices A Energy Values 64 B Source Code 66 i v List of Tables 5.1 T a b l e l i s t i n g the coefficients of the different r a t i o n a l funct ions employed i n the fits for the three states of hydrogen discussed above. T h e coefficient values have confidence bounds of 99%. . 36 5.2 T a b l e l i s t i n g the coefficients of the different r a t i o n a l funct ions emp loyed i n the fits for the three states of hydrogen discussed above. T h e coefficient values have confidence bounds of 99%. . 42 A . l T a b l e l i s t i n g the b i n d i n g energies of the three states of the hy-drogen a t o m considered i n th is s tudy. T h e q u a n t u m numbers co r respond to m, vz a n d ixz. T h e energies are i n R y d b e r g un i t s . 65 A . 2 T a b l e l i s t i n g the b i n d i n g energies of the three states of the h e l i u m a t o m considered i n th is s tudy. T h e q u a n t u m numbers cor respond to m a n d vz. T h e energies are i n un i t s of Ez<x> • • 65 v List of Figures 5.1 F i g u r e showing the v a r i a t i o n i n the b i n d i n g energy of the g r o u n d state of hydrogen w i t h the magne t i c field s t r eng th pa-rameter i n the range 1 0 - 3 < (5 < 1. T h e energy is r epor ted i n un i t s of R y d b e r g energy. T h e d a t a po in t s are results of the nu -m e r i c a l c a l c u l a t i o n w h i l e the l ine represents a fit to the da ta . T h e dashed l ine represents the results f rom first order pe r tu r -b a t i o n theory. T h e purpose be ing to i l l u s t r a t e the fact t ha t p e r t u r b a t i o n theory breaks d o w n w i t h increas ing the magne t i f f ield s t reng th 43 5.2 F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of three lowest m - states of hydrogen co r re spond ing to m=0,-l,-2, w i t h 7 T = + 1 w i t h the magne t i c f ield s t reng th pa ramete r i n the range 10~3 < (3 < 1. T h e energy is r epor ted i n un i t s of R y d b e r g energy. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u a t i o n wh i l e the l ines represent fits to the d a t a 44 v i List of Figures 5.3 F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of the state of hyd rogen co r re spond ing to m = 0 w i t h w i t h the magne t i c f ield s t r eng th paramete r i n the range 1 0 - 3 < (5 < 1. Resu l t s f rom b o t h the present s t udy as we l l as those f rom R e f [1] are p lo t t ed herein . T h e energy is repor ted i n uni ts of R y d b e r g energy. T h e d a t a poin ts are results of the n u m e r i c a l c a l c u a t i o n w h i l e the l ines represent fits to the d a t a 45 5.4 F i g u r e showing a p lo t of the eigenvalues ca l cu l a t ed as a func-t i o n of the are per finite element i n the mesh for a c a l c u l a t i o n for the g r o u n d state of hyd rogen at / ? = 1 0 - 1 . W i t h increas-i n g l y finer meshes for a sufficient d o m a i n size the eigenvalue ob t a ined approaches an a sympto te co r re spond ing to the l i m i t of in f in i t e ly fine mesh. T h e paramete r desc r ib ing the mesh size is the average area per f ini te element i n the mesh 46 5.5 F i g u r e showing a p lo t of the rea l pa r t of the wave funct ions , i.e. ip(p,z), of the g r o u n d state of the hydrogen a t o m w i t h chang ing magne t i c f ield s t rength . T h e wave funct ions are for (a) / ? = 1 0 - 3 , (b) /?=lCr2, (c) / ^ l c r 1 and (d) 0=1. T h e ef-fect of the increas ing b i n d i n g energy is c lea r ly v i s ib le i n the s h r i n k i n g of the wave func t ion w i t h increas ing 0. These p lo ts are not to be confused w i t h e lec t ron dens i ty 47 List of Figures 5.6 F i g u r e showing the v a r i a t i o n i n the b i n d i n g energy of the g r o u n d state of h e l i u m w i t h the magne t i c f ield s t r eng th pa-rameter i n the range 1 0 ~ 3 < fiz < 1- T h e energy is r epor t ed i n un i t s of R y d b e r g energy for nuclear charge Ze. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u l a t i o n w h i l e the l ine represents a fit to the d a t a 48 5.7 F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of three lowest M - states of h e l i u m cor respond ing to M = 0 , - l , - 2 . w i t h TT=+1 w i t h the magne t i c field s t reng th parameter i n the range 1 0 ~ 3 < Pz < 1- T h e energy is repor ted i n un i t s of R y d b e r g energy for nuclear charge Ze. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u a t i o n w h i l e the l ines represent fits to the d a t a 49 5.8 F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of the state of h e l i u m cor respond ing to M = - l w i t h 7 r=+ l w i t h the magne t i c f ield s t reng th paramete r i n the range 1 C T 3 < (3Z < 1. Resu l t s f rom b o t h the present s t u d y as w e l l as those f rom R e f [1] a n d Jones et a l [2, 3] are p l o t t e d here in . T h e energy is r epor ted i n un i t s of R y d b e r g energy for nuclear charge Ze. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u a t i o n w h i l e the l ines represent fits to the d a t a 50 List of Figures 5.9 F i g u r e showing a p lo t of the eigenvalues ca l cu la t ed as a func-t i o n of the are per finite element i n the mesh for a c a l c u l a t i o n for the g r o u n d state of hyd rogen at /?^=1. W i t h inc reas ing ly finer meshes for a sufficient d o m a i n size the eigenvalue ob-t a i n e d approaches a n a sympto t e co r re spond ing to the l i m i t of in f in i t e ly fine mesh. T h e paramete r desc r ib ing the mesh size is the average area per f ini te element i n the mesh 51 5.10 F i g u r e showing a p lo t of the rea l pa r t of the wave funct ions , i.e. ip(p,z), of one of the electrons c o m p r i s i n g the g r o u n d state of the h e l i u m a t o m w i t h chang ing magne t i c f ield s t rength . T h e q u a n t u m numbers cor respond to m 2 = - l , s ^ = - l / 2 , 7 r = + l . T h e wave funct ions are for (a) 0Z=1O~'\ (b) 0Z = IO~2, (c) ,5Z = 10" 1 a n d (d) / 3 z = l . T h e effect of the increas ing b i n d i n g energy is c lea r ly v i s ib le i n the s h r i n k i n g of the wave func t ion w i t h inc reas ing 13z- These p lo t s are not to be confused w i t h e lec t ron dens i ty 52 ix Acknowledgements I w o u l d l ike to t h a n k m y superv isor a n d mentor D r . J e r e m y S. H e y l for his gu idance a n d suppor t t h roughou t the course m y M a s t e r ' s degree. I w o u l d also l ike to t h a n k m y g o o d fr iend M r . A n d r e w J a s o n Penner for the m a n y f ru i t fu l scientif ic discussions. x Dedication To my parents. Chapter 1 Introduction T h e p r o b l e m of a toms i n magne t i c fields i n q u a n t u m mechanics has been inves t iga ted over m a n y decades. F o r mos t p h y s i c a l p rob lems encountered i n a t e r res t r i a l context the magne t i c fields are low enough i n s t reng th tha t t yp -i c a l l y a Zeeman- type pe r tu rba t i ve analys is [4] suffices for gaug ing the effects of the f ield o n the energy levels of the different e lec t ronic states. However th is p r o b l e m is i n t r i c a t e ly t i ed w i t h the number of electrons i n the a t o m i n ques t ion . A s the n u m b e r of electrons i n a n a t o m increases the p r o b l e m be-comes a n a l y t i c a l l y in t r ac t ab le and one has to resort to o ther a p p r o x i m a t i o n methods . T h e p r o b l e m is further c o m p l i c a t e d by the fact tha t i f the magne t i c field s t r eng th exceeds « 109C7, pe r tu rba t ive t r ea tment cannot be employed . B e -y o n d a p p r o x i m a t e l y th is f ield s t r eng th the m a g n i t u d e of the i n t e r ac t i on of an a t o m i c e lec t ron w i t h the magne t i c f ield becomes comparab l e to the m a g -n i t u d e of i ts i n t e r ac t i on w i t h the nucleus. T h u s one has to resort to a fu l l t r ea tment i n order to gauge the effect of the magne t i c f ield o n the energy levels of the different states of the a t o m . T h e m o t i v a t i o n to s t udy a toms i n magne t i c fields of s t rength b e y o n d the 1 Chapter 1. Introduction pe r tu rba t i ve regime was i n a large par t due to the d iscovery of such fields b e i n g present i n w h i t e d w a r f stars a n d n e u t r o n stars. These a s t rophys ica l ob-jects are ra ther c o m m o n i n our universe a n d they represent the f inal stages of s tel lar evo lu t ion . T h e former are fo rmed w h e n a red giant s tar sheds i ts outer envelope l eav ing b e h i n d a s m a l l star tha t is m o s t l y made up of h e l i u m (as wel l as some c a r b o n a n d oxygen) , w i t h s table the rmonuc lea r h e l i u m b u r n i n g . T h e la t te r , n e u t r o n stars, are formed i n d r a m a t i c supernova explosions; the s te l lar ma t t e r is c rushed under g r a v i t a t i o n a l forces to the po in t where deep w i t h i n the n e u t r o n star the dens i ty is expec ted to exceed nuclear densi ty. T h e mos t c o m m o n l y observed n e u t r o n stars, pulsars , have been observed to have magne t i c fields o n the order of 10 1 1 - 1 0 1 3 G [1]. M a g n e t a r s , w h i c h are s t rong ly magne t i zed neu t ron stars [5], c an have magne t i c field s t rengths we l l i n excess of 1 0 1 3 G . W h i t e d w a r f stars on the other h a n d have somewha t less ex t reme fields, a lbe i t s t i l l h igh , ~ 106 - 10 8 G. T h e s t ruc tu re of a toms i n b o t h instances however, is cons iderab ly a l te red f rom the low field case. In order to fac i l i ta te a p roper u n d e r s t a n d i n g of the spec t r a of n e u t r o n stars a n d w h i t e d w a r f stars one mus t necessar i ly have accura te d a t a of the s t ruc ture of a toms i n the a tmospheres of these compac t objects . T h e work desc r ibed here in extends prev ious w o r k [6] and presents a n u m e r i c a l treat-ment of hyd rogen a n d h e l i u m a toms i n magne t i c fields of a r b i t r a r y s t rength , y i e l d i n g accurate results for the eigenvalues a n d eigenvectors of the first few l o w - l y i n g states over a w ide range of f ield s t rengths ( ~ 105 - 1 0 1 3 G ) . T h e ca l cu l a t ed energy eigenvalues are seen to be improvemen t s u p o n p rev ious 2 Chapter 1. Introduction estimates. 3 Chapter 2 Literature Review T h e p r o b l e m of a toms i n magne t i c fields has been inves t iga ted b o t h ana ly t -i c a l l y and n u m e r i c a l l y by var ious researchers i n the past . B r o a d l y speak ing , treatises i n the pas t have concerned themselves w i t h a n a l y z i n g the p r o b l e m i n three different regimes depend ing u p o n the s t r eng th of the magne t i c field; low, in te rmedia te a n d intense. In the presence of b o t h low-s t reng th a n d i n -tense magne t i c fields the i n t e r ac t ion of electrons i n a toms w i t h the magne t i c field a n d w i t h the nucleus c a n be t rea ted w i t h p e r t u r b a t i o n theory, t r ea t ing as a p e r t u r b a t i o n first the one a n d then the other respec t ive ly i n each case. However , i n the in te rmedia te field regime, the m a g n i t u d e of i n t e r ac t i on of an e lec t ron w i t h the f ield is on the same order of m a g n i t u d e as i ts i n t e r ac t i on w i t h the nucleus. A s a resul t , we c a n n o longer e m p l o y p e r t u r b a t i o n theo ry i n th i s regime a n d mus t resort to a more r igorous a n d comple te t rea tment . In the presence of a s t rong magne t i c f ield the spher ica l s y m m e t r y of the a t o m is b r o k e n a n d the a t o m is s t re tched a long the f ield [1]. T h e m a g n i t u d e of i n t e r a c t i o n o f the e lec t ron w i t h the magne t i c field c a n exceed t ha t w i t h the nucleus . T h i s t y p i c a l l y occurs w h e n the e lec t ron c y c l o t r o n energy becomes greater t h a n the C o u l o m b po t en t i a l energy, i.e. ku>B > Ze2/r [1]. Here , U>B is 4 Chapter 2. Literature Review the c y c l o t r o n frequency. A t h i g h field s t rengths a z i m u t h a l s y m m e t r y remains in tac t a n d thus i t has been observed t ha t i t is more convenient to m o d e l the a t o m i n c y l i n d r i c a l coordinates . It is convenient to classify the field s t rength [7] as low (Pz < 1 0 - 3 ) , in te rmedia te , also ca l l ed s t rong ( 1 0 - 3 < /3z < 1), a n d intense or h i g h (1 < /?z < oo) . It is to be no ted tha t the paramete r (3Z represents the magne t i c f ield s t r eng th as a d imensionless n u m b e r i n a t o m i c un i t s for an a t o m w i t h Z n u m b e r of pro tons ; the reader is referred to §3 .1 for def ini t ions of the same. T h e present chapter is ded ica ted to a short ch rono log ica l r ev iew of the l i t e ra ture u p o n the t o p i c of a toms i n magne t i c fields d i scuss ing the key results of different researchers. Thereaf ter , theore t i ca l b a c k g r o u n d is p r o v i d e d on the H a r t r e e - F o c k m e t h o d [8] i n the fo l lowing chapter . Present ly , i t is i n s t ruc t ive to beg in w i t h past w o r k done o n the hyd rogen a t o m i n a magne t i c field. A s u sua l m o d e l i n g th is s ingle e lec t ron sys t em one gains va luab le ins ight . 2.1 Hydrogen and Hydrogen-like species Since the 1970's the subject of a toms i n magne t i c fields has been a top ic of cons iderable interest sparked m a i n l y because of the d iscovery of intense magne t i c fields i n the v i c i n i t y of w h i t e dwar f stars [9-11] a n d n e u t r o n stars [12, 13]. T h u s there was a need to re -examine m u c h of the conven t iona l a t o m i c phys ics k n o w n at the t i m e to ex tend i t to encompass the behav iour of a toms i n such intense magne t i c fields. A s usua l , hyd rogen was e x a m i n e d 5 Chapter 2. Literature Review first; th i s b e i n g an excel lent s t a r t i ng po in t for deve lop ing the theory. T h e first of the key papers desc r ib ing a comprehens ive t r ea tmen t was by C a n u t o a n d K e l l y i n 1972 [14]. T h e y deve loped a s e m i - a n a l y t i c a l ap-p roach w h e r e i n they solved the Schrodinger equa t ion for h y d r o g e n i n a m a g -net ic f ield by r educ ing i t to a one-d imens iona l wave equa t i on p r o b l e m . A t the t ime , these ca lcu la t ions were sufficiently accura te for c a l c u l a t i n g m a n y a t o m i c proper t ies . L a t e r the same year, P r a d d a u d e [15] developed a new expans ion basis for the hydrogen ic wave func t ion i n Lague r re P o l y n o m i a l s . T h e advantage be ing tha t i t reduced the Schrodinger equa t ion to a set of inf in i te a lgebraic equat ions m a k i n g possible exped i t ed ca lcu la t ions of b o u n d states of hydrogen ic species. S i m o l a and V i r t a m o i n 1978 [16] presented a t r ea tment of the hyd rogen a t o m where they deve loped further some of the ideas presented i n the s emina l work by C a n u t o a n d K e l l y i n 1972 [14]. T h e y adop ted a basis of L a n d a u orb i ta l s for e x p a n d i n g the wave func t ion for hy-drogen at finite fields. These o rb i ta l s descr ibe the m o t i o n of the e lec t ron t ransverse to the magne t i c f ield. T h e C o u l o m b p o t e n t i a l was t rea ted as a p e r t u r b a t i o n , th i s was jus t i f i ed by app roach ing the in te rmedia te field region by decreas ing the magne t i c f ield s t rength f rom inf in i ty . T h e r e su l t ing cou-p led o r d i n a r y differential equat ions were solved i t e ra t ive ly us ing a coup led channels m e t h o d to p roduce accurate eigenenergies and eigenfunct ions for the wave funct ions . F r i e d r i c h i n 1982 [17] ob t a ined b i n d i n g energies for the hyd rogen a t o m i n an intense magne t i c f ield by d i a g o n a l i z i n g the H a m i l t o -n i a n m a t r i x i n a non-o r thogona l basis. L a n d a u o rb i t a l s were aga in assumed 6 Chapter 2. Literature Review to descr ibe m o t i o n transverse to the field. These results were seen to be i n close agreement w i t h c o n t e m p o r a r y w o r k of W u n n e r a n d R u d e r [18]. L a t e r Rosne r et a l [19, 20] o b t a i n e d n u m e r i c a l so lu t ions of the Schrod inger equa t ion for hyd rogen i n a magne t i c f ield by r ep lac ing the C o u l o m b in t e rac t ion by an effective po t en t i a l , g i v i n g the extent of m i x i n g be tween the different states by the C o u l o m b po ten t i a l . Ivanov [21] i n 1988 so lved the p r o b l e m of the hydro -gen a t o m i n a n intense magne t i c f ield b y s o l v i n g the two d i m e n s i o n a l p a r t i a l different ia l equa t ion represent ing the t ime- independen t Schrod inger equa t ion u s ing f ini te difference me thods a n d R i c h a r d s o n e x t r a p o l a t i o n to increase the ob t a ined accuracy of eigenvalues. F o r a comprehens ive rev iew u p o n the top ic the reader is referred to R u d e r et a l [1]. 2.2 Helium and Helium-like species T h e h y d r o g e n a t o m is more r ead i ly solvable i n c o m p a r i s o n to the h e l i u m a t o m i n a magne t i c f ield as compl i ca t i ons due to o ther electrons are absent. However , the va luab le ins ight ga ined f rom s o l v i n g the hyd rogen a t o m was i n s t r u m e n t a l i n the so lu t i on of the h e l i u m a t o m i n s t rong magne t i c fields i n the years of research fo l lowing the s emina l w o r k by C a n u t o a n d K e l l y i n 1972 [14]. W e sha l l descr ibe here in once aga in a short ch rono log ica l rev iew of the t o p i c of h e l i u m a n d he l i um- l i ke species i n s t rong magne t i c fields a n d the key research done i n th i s area over the past 4 decades. A s i n the case of hyd rogen research in to the behav iour of h e l i u m and he l ium- l ike species i n 7 Chapter 2. Literature Review intense magne t i c field was spur red by the d iscovery of s t rong magne t i c fields i n c o m p a c t a s t rophys ica l objects; w h i t e dwar f a n d neu t ron stars. T h e reader is referred to R u d e r et a l [1] for a more comprehens ive rev iew of the top ic . T h e p r o b l e m was first t a c k l e d by K a d o m t s e v i n 1971 [22] us ing the T h o m a s - F e r m i m o d e l a n d la ter i m p r o v e d u p o n by M u e l l e r et a l i n 1971 [23]. I n i t i a l w o r k o n the t o p i c emp loyed pu re ly v a r i a t i o n a l techniques . T h e first of such ca lcu la t ions were per formed by C o h e n et a l i n 1970 [24]. L a t e r , improvemen t s were made by H e n r y et a l i n 1974 [25] w h o ca l cu la t ed the b i n d i n g energy of the negat ive hydrogen ion us ing a v a r i a t i o n a l technique w i t h t r i a l wave funct ions . A g a i n i n 1975, M u e l l e r et a l [26] o b t a i n e d bet-ter cons t ra in ts for the uppe r -bounds of energies for some l o w - l y i n g states of he l i um- l i ke species; H~, He a n d Li+. T h e y assumed a t r i a l wave func t ion c o m p r i s i n g of g r o u n d L a n d a u o rb i t a l s desc r ib ing m o t i o n of the e lec t ron per-pend ic l a r to the magne t i c field and gaussians desc r ib ing the m o t i o n pa ra l l e l to the field. W i t h such a wave func t ion chosen ab initio t hey were able to ca l -cu la te be t ter bounds for the eigenenergies us ing the c lass ica l R a y l e i g h - R i t z m e t h o d . Banerjee et a l i n 1974 [27] i m p r o v e d u p o n such est imates us ing s i m i l a r techniques , whe re in aga in , a one-parameter v a r i a t i o n a l fo rm was as-s u m e d for desc r ib ing the e lec t ron wave funct ions pa ra l l e l to the f ie ld. L a r s e n i n 1979 [28] ca l cu la t ed the b i n d i n g energies of the negat ive hyd rogen ion i n an intense magne t i c f ield us ing a v a r i a t i o n a l technique w h e r e i n the t r i a l wave funct ions were different for the different regimes of magne t i c field in tens i ty ; l ow to modera t e a n d the h i g h field case. In 1982, G a d i y a k et a l [29] were able 8 Chapter 2. Literature Review to o b t a i n suff iciently accurate results for hyd rogen v i a a v a r i a t i o n a l app roach a n d for h e l i u m v i a a Z-dependent p e r t u r b a t i o n theory. V i n c k e a n d B a y e i n 1989 [30] o b t a i n e d improvemen t s of es t imates of the energies of the negat ive h y d r o g e n i o n a n d h e l i u m i n s t r ong magne t i c fields e m p l o y i n g a v a r i a t i o n a l t echnique w h e r e i n the wave func t ion was seperable in to L a n d a u - l i k e o rb i ta l s a n d the a x i a l par t of the wave funct ions were t a k e n to be a s imp le p r o d u c t of an exponen t i a l a n d a power l aw func t ion . T h u s the a s y m p t o t o t i c behav iour i n the l i m i t s of z —> oo a n d z —> 0 respec t ive ly was we l l mode l ed . F o r o ther references o n the t o p i c of a v a r i a t i o n a l t r ea tment the reader is referred to C h e n , R u d e r m a n and S u t h e r l a n d [31], Glasser and K a p l a n [32] and Glasser [33] and references there in . T h e above men t ioned studies were s teady i m -provements u p o n the uppe r b o u n d s for the b i n d i n g energies of h e l i u m - l i k e species. T h e m a j o r i t y of these studies however were a n a l y t i c a l a n d were therefore l i m i t e d i n the i r a p p l i c a b i l i t y to a p r o b l e m tha t was inhe ren t ly more t r ac t ab le as a n u m e r i c a l p r o b l e m . O n e of the first a t t empt s at a p p l y i n g the Ha r t r ee -Fock ( H F ) technique [8] to the so lu t i on of two-e lec t ron systems was by V i r t a m o i n 1976 [34]. V i r t a m o ca luca ted the g r o u n d state energies of he l i um- l ike species; H~, He a n d Li+ us ing the self-consistent field procedure ( S C F ) due to Har t r ee [8]. T h e pro-cedures o u t l i n e d the re in fo rmed the basis for m u c h fur ther work . E l sewhere the H F m e t h o d for c a l c u l a t i n g the spec t ra of a toms h a d been successfully a p p l i e d to many-e l ec t ron systems [35, 36] i n a comprehens ive manner . T h e concepts of s ingle- a n d m u l t i - conf igura t ion H F methods [35] were success-9 Chapter 2. Literature Review fu l ly adap ted by P r o s c h e l et a l [37] and followed afterwards again by T h u r n e r et a l i n 1993 [38] to compu te accura te ly and sys t ema t i ca l l y the energy levels of the g r o u n d a n d first few l o w - l y i n g states respect ively, of hydrogen , h e l i u m a n d he l i um- l i ke species i n magne t i c f ield s t rengths r a n g i n g f rom low to s t rong . T h i s work , la ter c o m p i l e d b y R u d e r et a l [1], forms a b e n c h m a r k for t e s t ing results for two-e lec t ron systems. T h e y employed L a n d a u o rb i t a l s to descr ibe m o t i o n pe rpend icu l a r to the f ield a n d solved the one -d imens iona l coup led sys t em of equat ions thus a r i s ing to o b t a i n eigenenergies a n d eigenfunct ions. In 1992, L a i et a l [39] m o d e l e d hydrogen molecules a n d chains i n magne t i c fields B » 10 9C7 us ing a H F m e t h o d w i t h L a n d a u o rb i t a l s a cco rd ing to the s t a n d a r d p r e s c r i p t i o n i n R u d e r et a l [1] w i t h success. It is to be observed t ha t w i t h the H F me thods as the workhorse for such ca lcu la t ions s teady progress was made i n the 1980's a n d 1990's i n c a l c u l a t i n g the energies of the g r o u n d a n d first few exc i ted states of two-elec t ron systems. These me thods have been seen to y i e l d accura te results over a w ide range of magne t i c f ield s t rengths. In 1994 Ivanov [40] o b t a i n e d results for the b i n d i n g energies of a few l o w - l y i n g states by a d o p t i n g an unres t r i c t ed H F m e t h o d . N o a s s u m p t i o n of basis was made for e x p a n d i n g the wave func t ion of the e lec t ron a n d the re su l t ing t w o - d i m e n s i o n a l p a r t i a l different ia l equat ions represent ing the H F equat ions for the electrons were solved u s ing a finite difference m e t h o d , i te ra t ive ly , o n a mesh w i t h agreeable results . D e p a r t i n g f rom the n o r m of i t e ra t ive me thods for so lv ing the H F equat ions , Jones et a l [2, 3, 41]employed a released-phase quantum Monte-Carlo method for 10 Chapter 2. Literature Review d e t e r m i n i n g the g r o u n d and first few exc i ted states of the h e l i u m a t o m . T h e i r resul ts were seen to agree w i t h those r e su l t ing f rom conven t iona l H F methods . H e y l a n d He rnqu i s t i n 1997 [42] descr ibed b o t h an a n a l y t i c a l as wel l a nu-m e r i c a l a p p r o x i m a t i o n for eva lua t ing the effective in ter -e lec t ronic potent ia l s . T h e i r n u m e r i c a l technique ex tended the idea of a d o p t i n g a basis of funct ions for d i rec t ions b o t h the t ransverse a n d pa ra l l e l to the field. T h e y cons t ruc ted the wave func t ion i n the a x i a l d i r e c t i o n w i t h the he lp of h a r m o n i c osc i l l a to r H e r m i t e p o l y n o m i a l s . T h i s m e t h o d was seen to y i e l d accura te results for h i g h magne t i c fields j3z > 1 0 3 for hydrogen and h e l i u m . T h e y employed a m e t h o d of c a l c u l a t i n g the t o t a l energy of the sys t em g iven an assumed set of wave funct ions a n d then proceeded to m i n i m i s e th is energy by v a r y i n g parameters . T h i s was seen to y i e l d accura te results consis tent w i t h other w o r k [1]. O n e of the advantages over o ther me thods was observed to be the s ign i f ican t ly lesser amoun t of c o m p u t a t i o n invo lved . M o r e recently, M o r i a n d H a i l e y [43] a n d M o r i a n d H o [44] adop ted a pe r tu rba t i ve app roach to t reat the exchange te rms a n d higher L a n d a u states w i t h cons iderable success for f ind ing uppe r bounds for the energies for the g r o u n d and first few exc i t ed states of h e l i u m a n d other m i d - Z a toms. T h e au thors observed the re in tha t th i s cons ide rab ly exped i t ed the H F i te ra t ions . 11 Chapter 2. Literature Review 2.3 Many-electron Systems F o r a toms w i t h more t h a n two e lec t ron there is however not m u c h work i n the l i t e ra ture . F lowers et a l i n 1977 [45] a t t e m p t e d a v a r i a t i o n a l s t u d y of the i r o n a t o m i n a s t rong magne t i c field. T h e a p p r o x i m a t i o n s were seen to y i e l d reasonably g o o d results . B o t h one-d imens iona l me thods of Neuhauser et a l [46, 47] a n d two-d imens iona l Har t ree- fock me thods of Jones et a l [2] a n d Ivanov 1998 [48] and Ivanov and Schmelcher [49-52] have had success w i t h v a r y i n g degrees of n u m e r i c a l accuracy for many-e l ec t ron systems up to car-b o n . T h e c o m p l i c a t i o n i n such procedures arises f rom the c a l c u l a t i o n of the in ter -e lec t ron po ten t ia l s a n d the i n t e r ac t i on w i t h the nucleus w h i c h involve c a l c u l a t i n g m i x i n g be tween the different states. M o r e w o r k is necessary i n th is area to o b t a i n more accura te a n d s t r ingent bounds for the energies of these systems i n s t rong magne t i c fields. T h i s concludes our survey of the l i te ra ture . T h e next chapter is devoted to d e s c r i b i n g the H a r t r e e - F o c k m e t h o d a n d a d e r i v a t i o n is g iven for the u n -res t r ic ted two d i m e n s i o n a l m e t h o d used i n th is s tudy. 12 Chapter 3 Theoretical Background T h i s chapter conta ins a desc r ip t i on of the Ha r t r ee -Fock f o r m a l i s m . T h e me thods desc r ibed concern themselves w i t h a toms w i t h a r b i t r a r y a t o m i c n u m b e r Z, a n d a r b i t r a r y magne t i c field s t rengths. 3.1 The Hartree-Fock Equations - a derivation W e sha l l beg in w i t h the H a m i l t o n i a n for the hyd rogen a t o m i n a magne t i c f ie ld. T h e single e lec t ron sys t em is perhaps the s imples t a n d yet the most i n s t ruc t ive to m o d e l . T h e m a i n in te rac t ions be ing those w i t h the nucleus a n d the magne t i c field respect ively. A p p e a l i n g to the n a t u r a l symmet r i e s of the p r o b l e m , the H a m i l t o n i a n for the single e lec t ron of the hyd rogen a t o m i n c y l i n d r i c a l coord ina tes is g iven by 13 Chapter 3. Theoretical Background „„ e2B2 2 Ze2 1 +pBB (lz + 2 s z + — - p 2 - - r 3.1 8 m e 47re0 | r | where me is the mass of the e lect ron and the B the magne t i c field s t rength ; the vector B is or ien ted a long- the pos i t ive z- axis . T h e r e m a i n i n g s y m b o l s have the i r usua l meanings . It is of course i m p l i c i t l y a ssumed here in tha t the nucleus is in f in i t e ly massive. L e t us assume a ce r t a in fo rm for the wave func t ion of the single e lec t ron tt = V>(p,z)e i T n*x(s) (3.2) It c a n be seen i m m e d i a t e l y tha t such a choice precludes the use of a basis of funct ions for desc r ib ing the behav iour of the e lec t ron b o t h pa ra l l e l a n d pe rpend icu l a r to the magne t i c field. T h u s , the t ime- independen t Schrod inger equa t ion i n un i t s of B o h r r a d i i can be w r i t t e n '1 8 ( d \ d2\ m2 n o . ;PcTP{%) + dr2) + ^+2^m-1)+ 2J2 2Z ib (p, z) eib (p, z) (3.3) where m is the a z i m u t h a l q u a n t u m number . In def ining E q . (3.3) i t has been assumed tha t the e lec t ron sp in is an t i - a l igned w i t h the magne t i c field. 14 Chapter 3. Theoretical Background A d d i t i o n a l l y the paramete r e is defined as 2E E 2 2 - IT ( 3 ' 4 ) Eoo is the R y d b e r g energy. T h e q u a n t i t y a RS 1/137 is the fine s t ruc ture constant . T h e paramete r (3 is defined i n the usua l way as (3-5) where B0 is the c r i t i c a l f ield s t r eng th at w h i c h po in t the t r a n s i t i o n to the intense magne t i c field regime occurs [1]. A t th is f ield s t r eng th the L a r m o r a n d the B o h r r a d i i are equa l i n magn i tude . B0 = (3.6) en In general , the r e l a t i on aH = OBP~1^2 holds . W h e r e aH a n d as are the L a r m o r a n d B o h r r a d i i respect ively. T h u s , b e y o n d a value of (3 ~ 1 the t r a n s i t i o n to the intense magne t i c field regime occurs a n d the in t e r ac t i on of the e lec t ron w i t h the nucleus becomes progress ively less d o m i n a n t as (3 increases. T h e L a r m o r rad ius is g iven by a# = \Jeh/2me wh i l e the B o h r r ad ius is g iven by the express ion as = h/amec. It can be seen tha t E q . (3.3) is a l inear second order p a r t i a l different ia l equa t ion , a n d i n the present s t u d y i t was so lved n u m e r i c a l l y o n a c o m p u t e r us ing f ini te-element techniques . Fo r detai ls o n the n u m e r i c a l t r ea tment of E q . (3.3) see C h a p t e r 4. 15 Chapter 3. Theoretical Background 3.2 The Helium Atom and Helium-like species F o r c a l c u l a t i n g the a t o m i c s t ruc ture of two-e lec t ron systems we adopt an i t e ra t ive m e t h o d , the so ca l l ed self-consistent field method [8] w h i c h essen-t i a l l y solves the Ha r t r ee -Fock equat ions for the electrons. It is to be no ted at th i s j u n c t u r e tha t the equat ions to be der ived are o b t a i n e d by e x t r e m i s i n g an energy func t iona l ; i.e. e m p l o y i n g the v a r i a t i o n a l p r i n c i p l e a n d therefore the e igensolut ions are guaranteed to be g o o d a p p r o x i m a t i o n s to the t rue so lu t ions , p r o v i d e d the choice of wave funct ions are reasonab ly g o o d approx-ima t ions . T h e detai ls for the v a r i a t i o n a l t rea tment sha l l not be g iven here a n d the reader is referred to M e s s i a h [53] for the same. O n e s tar ts w i t h a s ingle e lect ron, i m a g i n i n g tha t i t in teracts indepen-d e n t l y w i t h the nucleus a n d moves i n a po t en t i a l governed b y the nucleus a n d the average repuls ive effect of the other electrons i n the a t o m . (The magne t i c f ield is added as par t of the conjugate m o m e n t u m . ) T h u s , we as-sume tha t the e lec t rons ' ' in the a t o m can be represented by a single slater de te rminan t w h i c h an t i - symmet r i zes the i n d i v i d u a l e lec t ron wave funct ions . A short d e r i v a t i o n of the key equat ions are g iven be low a s suming a single-conf igura t ion f o r m for the a t o m as descr ibed above. I t is t o be no t ed t ha t no res t r ic t ions are i m p o s e d u p o n the a t o m i c elec-t rons such as the c o m m o n l y employed adiabatic approximation [1] however, th i s is achieved i n prac t ice by d e m a n d i n g tha t the appropr i a t e b o u n d a r y con-16 Chapter 3. Theoretical Background d i t ions be met . In neu t ron star a n d w h i t e d w a r f star a tmospheres where the magne t i c fields are ra ther s t rong , i t is seen tha t w o r k i n g w i t h i n the ad i aba t i c a p p r o x i m a t i o n y ie lds g o o d d iv idends [37]. T h e technique can , however, be emp loyed for c a l c u l a t i n g energy eigenvalues a n d wave funct ions for configu-ra t ions other t h a n the ad iaba t i c case. 3.2.1 Derivation of the generalized Hartree-Fock equations in partial differential form L e t us b e g i n w i t h the H a m i l t o n i a n of an N - e l e c t r o n a t o m sp l i t in to one- a n d t w o - b o d y te rms T h e first pa r t of the H a m i l t o n i a n cons i s t ing of one -body in te rac t ions is g iven b y the s t a n d a r d p re sc r ip t ion (us ing po la r c y l i n d r i c a l coordinates) where i = 1 , 2 , N A n d the t w o - b o d y t e r m i n the H a m i l t o n i a n is s i m p l y the C o u l o m b inter-a c t i o n be tween the ith a n d j t h e lectrons. (3.7) (3.8) 17 Chapter 3. Theoretical Background w{ri,rj) = (3.9) L e t us assume tha t the wave func t ion of a g iven conf igura t ion of electrons is g iven by where, AN is the a n t i - s y m m e t r i z a t i o n opera tor . T h u s , it can be seen that a s ingle slater de te rminan t is assumed to represent the a t o m i c conf igura t ion of a l l electrons. T h e single pa r t i c l e wave funct ions are assumed to be of the same f o r m as assumed for the case of the hydrogen a t o m i n E q . (3.2); exp l i c i t l y , where i labels the electrons for each of the N electrons. T h e single pa r t i c l e wave funct ions ipi(pi, 2 j ) are t aken to be rea l funct ions . W e sha l l res t r ic t ourselves for the t i m e be ing to the ad i aba t i c a p p r o x i m a -t i o n a n d also require tha t a l l the electrons i n the a t o m have the same sp in ; we sha l l look at o n l y the fu l ly sp in -po la r i sed states w i t h each electron 's sp in b e i n g an t i -a l igned w i t h the magne t i c f ield. E x t e n s i o n to other non -ad iaba t i c conf igura t ions is a ma t t e r of l o o k i n g at exc i t ed states; e m p l o y i n g . t h e appro-* = AN ( ^ 1 , ^ 2 , ^ 3 , ...,1pN-l,tpN) (3.10) *i = Mpi,Zi)eirrul,iXi(si) (3-11) 18 Chapter 3. Theoretical Background pr ia te b o u n d a r y cond i t ions . W r i t i n g out the genera l ized Ha r t r ee -Fock equat ions for d e t e r m i n i n g the single pa r t i c l e wave funct ions tpi we have - (ipj (rj) | w (n, r5) | ipi (r,-)} ipj (r*) ] = Eiipi(ri) (3.12) where i = 1 , 2 , 3 , N S u b s t i t u t i n g the ansatz g iven i n E q . (3.11), the assumed i n d i v i d u a l elec-t r o n wave funct ions, in to E q . (3.12) we o b t a i n after r ea r rang ing some terms f l d f d \ d2\ m? n o , 1 N o 2 2 ^ P i +zi I I F i - r3 = e l ^ ( P l , z i ) e i m ^ (3.13) where i,j = 1, 2, 3 , A / 19 Chapter 3. Theoretical Background It is to be no ted t ha t the c o n t r i b u t i o n due to e lec t ron sp in has been aver-aged out a priori. W e have chosen to work i n un i t s of B o h r r a d i i a long w i t h the def ini t ions g iven be low. T h e B o h r rad ius for an a t o m of nuclear charge Z is g iven by CIB/Z, w i t h a s as defined earl ier . T h e magne t i c field s t r eng th pa ramete r is defined as Bz — B/Bz = B/Z2. T h e reference magne t i c f ield s t r eng th for nuc lear charge Z, at w h i c h the t r a n s i t i o n to the intense m a g n e t i c field regime occurs is g iven b y Bz = Z2BQ w i t h B0 as defined i n E q . (3.6). F i n a l l y , the energy paramete r is defined as e; — Ei/(Z2E00), w i t h E^ as defined i n E q . (3.4). T h e above w r i t t e n E q . (3.13) represents the N - c o u p l e d H a r t r e e - F o c k equat ions i n p a r t i a l different ial fo rm for a n N - e l e c t r o n sy s t em w i t h nuclear charge Z . It is i n s t ruc t ive at th is stage to q u a l i t a t i v e l y examine the N-equa t ions w r i t t e n above a n d ascr ibe p h y s i c a l s ignif icance to each t e r m appea r ing there in . It c a n be seen i m m e d i a t e l y t ha t these equat ions resemble the Scrod inger eqau t ion for each of the i n d i v i d u a l states occup ied by the N - electrons. T h e first of the m i x i n g te rms represents the di rec t i n t e rac t ion , wh i l e the second represents the exchange in t e r ac t ion between the electrons i n the a t o m . T h e H a m i l t o n i a n w r i t t e n thus represents energy o f the e lec t ron i n a g iven state, where the t o t a l p o t e n t i a l i n w h i c h i t is cons t r a ined to move consists of the c u m u l a t i v e effects of the nucleus, an average field due to the presence of the other electrons a n d the magne t i c field. I n th is sense the theory is a m e a n field theory. T h e sys t em of equat ions is solved i t e ra t ive ly ; see C h a p t e r 4 for n u m e r i c a l deta i ls . If one neglects the exchange i n t e r ac t i on te rms i n E q . 20 Chapter 3. Theoretical Background (3.13) t h e n one ob ta ins a s imple r set of equat ions p roposed due to Har t ree , the so-ca l led Hartree Equations. Ha r t r ee [8] p roposed these b y v i r t u e of i n -t u i t i v e a rguments . T h i s essent ia l ly amoun t s to a s suming t ha t the t o t a l wave func t ion of the N - electrons is a s imp le p r o d u c t of the N - i n d i v i d u a l wave funct ions ra ther t h a n an an t i - symmet r i s ed p roduc t . T h e Ha r t r ee equat ions c a n be so lved us ing i t e ra t ive techniques a n d owing to the absence of exchange t e rms the ca lcu la t ions are cons iderab ly shorter [8]. However , i f we choose to preserve the corre la t ions between the electrons by r e t a in ing the exchange te rms t h e n a n t i - s y m m e t r y of the t o t a l wave func t ion is m a i n t a i n e d a n d these equat ions are the so-cal led Hartree-Fock Equations. T h e Har t r ee equat ions suffer f rom the fact tha t neglec t ing the exchange between the electrons makes the sys t em of equat ions less s y m m e t r i c t h a n the fu l l H a r t r e e - F o c k E q u a t i o n s . T h i s is because the H a m i l t o n i a n hi + Wij is not the same for a l l of the i n -d i v i d u a l states. A s a resul t the e igensolut ions of the N - equat ions are not m u t u a l l y o r t hogona l l ead ing to erroneous es t imates of the energies [8]. T h e reader is referred to the the b o o k by Ha r t r ee [8] for deta i ls . T h e Har t r ee -F o c k E q u a t i o n s o n the other h a n d do not suffer f rom such a d r a w b a c k bu t are c o m p u t a t i o n a l l y more invo lved . O f key concern i n the c o m p u t a t i o n of the eigenvalues a n d eigenvectors is the d e t e r m i n a t i o n of the d i rec t a n d exchange in te rac t ions between the elec-t rons . In the current s t udy these have been deal t w i t h i n a manner ra ther different f rom earl ier treatises. In shor t , these con t r i bu t i ons are essent ia l ly e x t r a po ten t ia l s tha t a d d to the ex i s t i ng single pa r t i c l e H a m i l t o n i a n , except 21 Chapter 3. Theoretical Background they are coup led t h r o u g h the wave funct ions we igh t ing t h e m i n a g iven equa-t i o n i n E q . (3.13). T h e procedure employed for eva lua t ing these po ten t ia l s is desc r ibed be low. 3.2.2 The direct interaction T h e t r ea tmen t of the d i rec t a n d exchange po ten t ia l s cons idered here a n d i n the subsequent sec t ion is based u p o n r igorous ly d e t e r m i n i n g these po ten t ia l s by s o l v i n g the i r co r re spond ing P o i s s o n equat ions . T h i s is ca r r i ed out i n a nove l way deve loped i n th i s s t udy by e m p l o y i n g the square of the i n d i v i d u a l e lect rons ' m o m e n t u m opera tors —ihVi. T h e detai ls of m e t h o d deve loped i n th i s s t u d y are g iven be low. It is to be no ted tha t the exponen t i a l factors w i t h imcb are to be in te rpre ted w i t h the appropr i a t e s ign depend ing u p o n whe ther t hey are w r i t t e n i n the b r a or i n the co r r e spond ing ket; a m inus s ign for the former a n d a p lus s ign for the la t ter . L e t us first examine the in tegra l represent ing the d i rec t i n t e r ac t ion be-tween the electrons. = ( ^ ( p , , ^ ) e ^ ^ | - r ^ | ^ ( P i , ^ ) e ^ ^ ) (3.14) \'i rj\ L e t us act o n b o t h sides of E q . (3.14) w i t h the opera tor p\ = —h2Vf, we t h e n o b t a i n 22 Chapter 3. Theoretical Background - n2V2^D = - h 2 ^ 3 { P j ) zj)eh"^\ - 47r53(r1 - r ^ f o , Zj)eim^) (3.15) W h i c h t h e n i m m e d i a t e l y y ie lds V 1 2 $ D = - 4 7 T | ^ ( A , ^ ) | 2 (3.16) T h e R H S of E q . (3.16) is the square of the j t h e lect ron 's wave func t ion eva lua ted u s ing the coordina tes of the ith e lec t ron. N o t i n g t ha t E q . (3.16) is a P o i s s o n equa t ion w i t h a source t e r m , i t is observed t ha t i t is n u m e r i c a l l y t r ac tab le a n d solved us ing the appropr i a t e b o u n d a r y cond i t i ons to y i e l d the p o t e n t i a l <&D w h i c h is due to the d i rec t i n t e r ac t i on be tween the ith and j t h electrons; see C h a p t e r 4 for detai ls o n the numer ics . It is to be no ted however, t ha t i n contras t to prev ious work , the p r o b l e m i n E q . (3.16) is somewhat s imple r despi te h a v i n g to solve a p a r t i a l different ial equa t ion as one does not have to f ind a p p r o x i m a t e expressions for the m i x i n g t e rms a r i s ing f rom the i n t e r ac t i on between different e lec t ronic states a n d the C o u l o m b po ten-t i a l . T h e reader is referred to [1, 37] and references there in for the different a p p r o x i m a t i o n me thods emp loyed i n o b t a i n i n g est imates for E q . (3.14). W e now t u r n out a t t en t ion to the other t w o - b o d y t e r m i n E q . (3.13), the exchange in te rac t ion . 23 Chapter 3. Theoretical Background 3.2.3 The exchange interaction W e sha l l fo l low the same m e t h o d o l o g y as i n our t r ea tment of the d i rec t i n t e r a c t i o n t e r m . L e t us re -wr i te our t e r m i n E q . (3.13) tha t relates to the exchange in t e r ac t ion between the ith a n d j t h e lectrons S B = ( ^ . ( p j ^ ^ e ^ l — ^ l ^ ^ . ^ e ^ ) (3.17) i ' i rj\ A g a i n , as i n our p rev ious t rea tment , let us act o n b o t h sides of E q . (3.17) w i t h the opera tor V 2 , th i s t i m e d r o p p i n g the r edundan t factor of — h2, to o b t a i n V 2 ^ = (^{Pj,z3)e^\ - to&ifi - f-MfazJe*"*) (3.18) U p o n c a r r y i n g out the in tegra l we get V 2 $ £ = - 4 7 r ^ ( p i , ZiWpu zxy^~m^ (3.19) It is no ted tha t by the def in i t ion i n E q . (3.11) ip* = ip for the spa t i a l par t of the i n d i v i d u a l e lec t ron wave funct ions . A t th is stage, let us make the ansatz tha t $E = aE{pi)zi)e*mi-mt)*< (3.20) L e t us act o n b o t h sides of E q . (3.20) w i t h the L a p l a c i a n opera tor , V 2 , 24 Chapter 3. Theoretical Background to o b t a i n V2$E = 'l_d_ ( d_ pi dpi \dpi (ra,- - mA2 d2 P\ + dz2 c x E i p ^ Z i y ^ - " 1 ^ (3.21) It is t h e n a s t ra igh t - fo rward ma t t e r u p o n c o m p a r i n g E q . (3.19) w i t h E q . (3.21) to i m m e d i a t e l y see tha t " 1 d ( ' d \ ( m i - m,-) 2 a2 " , — a~ \Pia~) 2 + ^ ~ 2 ®E{pi,Zi) Pidpi \ dpij pi dzf\ = -AKip]{pi,zl)ipi{Puzi) (3.22) T h e e l l i p t i c a l p a r t i a l differential equa t ion , E q . (3.22) is solved n u m e r i c a l l y a n d we thus o b t a i n an es t imate for the func t ion ctE{pi,Zi) for each of the pa i r -wise in te rac t ions a m o n g the N electrons. K n o w i n g cxE(pi,Zi) we can then o b t a i n acco rd ing to E q . (3.20). O n c e a n d $r; have been o b t a i n e d we c a n subs t i tu te t h e m in to E q . (3.13) for the poten t ia l s due to the di rec t a n d exchange in te rac t ions respec t ive ly to o b t a i n 2(3z(mi-l)+f32p2-2 P i + Z i . rpi { P i , Z i ) e i m ^ 25 Chapter 3. Theoretical Background = e ^ ( A , ^ ) e i m ^ (3.23) T a k i n g the inner p r o d u c t w i t h / d(j)ie'~imi't'1 on b o t h sides of E q . (3.23) we ob t a in ; w r i t i n g i n a compac t fo rm, "V?,<*,«) + ^  + W ™ * - !) + - -7= pi yJPi + ipi (Pi,Zi) + V E zi) ~ aElpj(pi, Zi)] = Cilpi (pi, Zi) (3.24) ZJ • , • where i,j = 1, 2, 3 , N a n d - V 2 ( p i A ) P i 9 p i V 1 dpi J dzf i jjti -(tpiipi, Zi)\(XE\ll>j{Pi, Zz))} (3.25) E q . (3.24) is the f ina l fo rm for the genera l ized H a r t r e e - F o c k E q u a t i o n s a n d for two-e lec t ron systems we have two equat ions , however for a r b i t r a r y nuclear charge Z. T h e energy of the Ha r t r ee -Fock state is g iven by E q . (3.25). T h i s completes our de r iva t ion of the Ha r t r ee -Fock equat ions for a toms i n magne t i c fields of a r b i t r a r y s t rength . T h e fo l lowing chapter delineates the 26 Chapter 3. Theoretical Background n u m e r i c a l procedures employed i n the c a l c u l a t i o n of the energy eigenvalues a n d eigenfunct ions. Thereaf ter , results are presented a n d a d i scuss ion fol lows. 27 Chapter 4 Numerical Details D e s c r i b e d here in are the n u m e r i c a l procedures emp loyed i n finding the eigen-values a n d eigenvectors for the p rob lems descr ibed above. F i r s t , we descr ibe the procedures for the hydrogen a t o m a n d fo l lowing t ha t for he l i um- l ike species. 4.1 Numerical procedures for the hydrogen atom T h e eigenvalue p r o b l e m for the hyd rogen a t o m i n E q . (3.3) is so lved by dis-c re t i s ing the equa t ion a n d s o l v i n g the resul tant a lgebraic eigenvalue p r o b l e m . T h e d i sc re t i sa t ion is done us ing F i n i t e - E l e m e n t M e t h o d ( F E M ) [54], whe re in a mesh of finite elements is cons t ruc ted over the ent i re d o m a i n of the p rob -l e m . T h i s discret izes the d o m a i n a n d thus the p a r t i a l different ia l equat ions take the f o r m of a lgebraic equat ions; one for each f ini te element. T h e gen-era l ized eigenvalue p r o b l e m is t h e n so lved us ing a sparse m a t r i x genera l ized e igensystem solver. T h i s m e t h o d was found to y i e l d accura te results for the energy eigenvalues for the first few eigenstates w i t h different mi. 28 Chapter 4. Numerical Details 4.2 Numerical procedures for helium-like species A n atomic structure software was deve loped as a par t of th is s t u d y for the purpose of c a l c u l a t i n g the energies of different states of mu l t i - e l ec t ron a toms. T h e p r o g r a m takes as i ts i n p u t , the number of electrons i n the a t o m n e , the nuclear charge Z, a n d the magne t i c field s t r eng th paramete r Q and then proceeds to c o m p u t e sy s t ema t i ca l l y the eigenvalues a n d eigenfunct ions of the coup led sys t em of equat ions i n E q (3.24) acco rd ing to the i t e ra t ive procedures desc r ibed be low i n brief. T h e E q s . (3.16), (3.22) a n d (3.24) are so lved i n a three step process. F i r s t an i n i t i a l es t imate is o b t a i n e d for the eigenvectors by s o l v i n g E q . (3.24) m i -nus the con t r ibu t ions due the in t e rac t ion between the electrons. T h e wave funct ions are t h e n used as s t a r t i ng po in t s for further improvement s . T h e second step involves o b t a i n i n g est imates for the po ten t ia l s due to the d i -rect and exchange in te rac t ions amongst the electrons, v i s a v i s the e l l i p t i c a l p a r t i a l different ial E q s . (3.22) and (3.16) are solved us ing the es t imates for the wave funct ions o b t a i n e d i n the previous step. These es t imates are t h e n used to solve for bet ter est imates of the eigenfunct ions a long w i t h the re l -evant eigenvalues i n E q . (3.24). T h e above steps are i t e ra ted i n the order desb r ibed above save the first step to o b t a i n progress ively bet ter est imates for the eigenvalues a n d eigenvectors. It was observed d u r i n g our runs tha t fast convergence was achieved; w i t h i n the first few i te ra t ions . A convergence 29 Chapter 4. Numerical Details c r i t e r i on was emp loyed where in the difference between the so lu t ions for two consecut ive i t e ra t ions was tested. If the difference was smal le r t h a n the de-s i red to lerance for convergence t h e n i t was cons idered to be a s o l u t i o n of the i t e ra t ive scheme. T y p i c a l l y , a tolerance on the order of 1 0 - 6 was employed . Thereaf ter , the t o t a l energy of the Ha r t r ee -Fock state under cons ide ra t ion is r epor ted acco rd ing to E q . (3.25). R u n s were ca r r i ed out for different values of the magne t i c field s t r eng th parameter 6 i n the range 1 0 - 3 < 6 < 1. F o r every r u n we employed 5 different levels of mesh refinement, r a n g i n g f rom coarse to suff iciently fine mesh. T h e fine mesh ca lcu la t ions took up to two days of c o m p u t i n g t i m e o n A M D O p t e r o n ® 844 1.8 G H z processors. A F i n i t e - E l e m e n t M e t h o d ( F E M ) was emp loyed for s o l v i n g each two d i -m e n s i o n a l p a r t i a l different ia l equa t ion . Br ie f ly , the d o m a i n is d i v i d e d i n t o a mesh of t r i angu l a r finite elements. T h i s d i s c r e t i z a t i on conver ts the p a r t i a l different ial equat ions in to a set of a lgebraic equa t ion on each element of the d o m a i n . T h e set of a lgebraic equat ions are solved s imu l t aneous ly to o b t a i n eigenvalues a n d eigenvectors. F o r detai ls o n the F i n i t e - E l e m e n t M e t h o d the reader is referred to [54]. I t was found tha t d o m a i n compac t i f i c a t i on was c o m p u t a t i o n a l l y expensive i n t e rms of m e m o r y a n d c o m p u t a t i o n /time even w i t h appropr i a t e sca l ing for o b t a i n i n g sufficiently accura te results . O n the o ther h a n d , l i m i t i n g the d o m a i n size i n the two o r thogona l d i rec t ions to several B o h r r a d i i ( « 20) was observed to give reasonably accura te results . C o n c o r d a n t l y the c o m p u t a t i o n a l expense was m a n y t imes less i n c o m p a r i s o n to the V former case. N u m e r i c a l errors a r i s ing f rom such an a p p r o x i m a t i o n 30 Chapter 4. Numerical Details were not s ignif icant . O n c e aga in , the genera l ized sys t em of eigenvalue equat ions is solved us ing appropr i a t e F E M d i sc re t i sa t ion y i e l d i n g accura te resul ts for the eigenvalues a n d eigenvectors for the first few l o w - l y i n g states. T h e software deve loped for the purposes of th is s t u d y to de te rmine the e igensolut ions of b o t h the h y d r o g e n a n d the h e l i u m a toms c a n be found i n A p p e n d i x B. 31 Chapter 5 Results &; Discussion In the current chapter we sha l l present the results f rom the ca lcu la t ions ca r r i ed out for hyd rogen a n d h e l i u m i n s t rong magne t i c fields. A d i scuss ion of these results is presented concurren t ly . T h e key results are c o m p a r e d w i t h those of other researchers i n the past a n d inferences are d r a w n . T h e chapter is o rgan ized such t ha t first we sha l l present i n br ie f the results p e r t a i n i n g to the g r o u n d a n d first two exc i t ed states of the hyd rogen a t o m i n s t rong magne t i c fields. Thereaf te r we show the resul ts for h e l i u m . It is t o be m e n t i o n e d t ha t the present s t u d y focuses o n ca lcu la t ions for magne t i c field s t r eng th w i t h i n the so ca l l ed strong regime ( 1 0 - 3 < 6 < 1). C a l c u l a t i o n s outs ide th is regime for b o t h hyd rogen a n d h e l i u m a toms sha l l be pub l i shed elsewhere. 5.1 The Hydrogen Atom F o r magne t i c fields i n the range I O - 3 < 6 < 1 we so lved the t ime- independen t Schrod inger equa t ion g iven i n Eq . (3 .3 ) for different values of the magne t i c field s t r eng th parameter . W e sha l l however o n l y present the resul ts o b t a i n e d for the states co r resspond ing to ra = 0, —1 a n d —2 for the most t i g h t l y 32 Chapter 5. Results & Discussion b o u n d states co r re spond ing to each m. T h e b i n d i n g energies are r epor t ed i n R y d b e r g un i t s . T h e v a r i a t i o n i n the b i n d i n g energy for the state c o r r e s p o n d i n g to m = 0 , 7 T = +1 is s h o w n i n F i g u r e 5.1. T h e q u a n t i t y 7r repor ts p a r i t y w i t h respect to the z-axis . T h e d a t a po in t s are eigenvalues o b t a i n e d f rom the n u m e r i c a l s o l u t i o n o f E q . (3.3). T h e energy eigenvalues are r epo r t ed as the values co r re spond ing to in f in i t e ly fine mesh sizes, or i n o ther words w h e n the area of a finite element approaches zero. A short d i scuss ion of th is e s t ima t ion p rocedure is g iven later . A s c a n be seen i n the figure, the e lec t ron becomes more a n d more b o u n d as the magne t i c field s t r eng th increases. T h e l ine t h r o u g h the d a t a represents a fit to the da ta . A r a t i o n a l func t ion was used to m o d e l the d a t a i n th i s reg ime us ing a robus t L e v e n b e r g - M a r q u a r d t m e t h o d [55]. T h e values for the coefficients of the i n t e r p o l a t i n g func t ion are g iven i n T a b l e 5.1 w i t h 99% confidence bounds . T h e fit i n F i g u r e 5.1 was seen to be accura te t o w i t h i n 0.02 % d e v i a t i o n f rom the ca l cu la t ed values. T h i s was cons idered to be reasonably g o o d for the present s tudy. A l s o shown i n F i g u r e 5.1for i l l u s t r a t i ve purposes, are the resul ts f rom a pe r tu rba t i ve c a l c u l a t i o n for the s ta te i n cons ide ra t ion . A t y p i c a l Z e e m a n H a m i l t o n i a n was emp loyed w i t h b o t h the l inear and the quad ra t i c t e rms c o n t r i b u t i n g to the energy shift. T h e results p lo t t ed are for first order in p e r t u r b a t i o n theory. A s c a n be seen clearly, p e r t u r b a t i o n theory o n l y works for a l i m i t e d range of the magne t i c field s t r eng th paramete r 6, b e y o n d Q ~ 1 0 - 1 , the pe r tu rba t i ve results s tar t to diverge r ap id ly , y i e l d i n g progress ively erroneous 33 Chapter 5. Results Sz Discussion b i n d i n g energies w i t h increas ing magne t i c field s t rength . S i m i l a r l y , figure 5.2 shows a l l three of the states desc r ibed above p l o t t e d together for the sake of i l l u s t r a t i n g the differences be tween the b i n d i n g en-ergies w i t h v a r y i n g magne t i c field s t r eng th B. T h e states m — 0,7r = + 1 , m — — l , 7 r = +1 a n d m = —2,TT = +1 are p lo t t ed there in . A g a i n the l ines t h r o u g h the d a t a po in t s represent fits t o the da t a . E v e r y d a t a po in t i n these figures was o b t a i n e d as a n es t imate co r re spond ing to l i m i t of the finite ele-ment size go ing to zero; see la ter . T h e fits to the d a t a are r a t i o n a l funct ions ob t a ined as desc r ibed above. T h e coefficients are r epo r t ed to w i t h i n 9 9 % confidence bounds i n T a b l e 5.1. It is to to be no ted tha t the r a t i o n a l funct ions were so chosen as to reflect the fact tha t even tua l ly at large values of the magne t i c field s t rength pa ramete r (3 the energy w o u l d be p r o p o r t i o n a l to (52 [26]. Fo r the sake of c o m p a r i s o n w i t h p r e v i o u s l y de t e rmined accura te resul ts , the fit t o the ca l cu l a t ed d a t a for the g r o u n d state of hyd rogen cor resspond ing to m = 0 7 r = + l was p l o t t e d against the ca l cu la t ed d a t a of the researchers i n Ref [ l ] i n F i g u r e 5.5. T h e l ine t h r o u g h the i r d a t a is a sp l ine fit. A s can be seen i n the figure, the two l ines are nea r ly co inc iden ta l . T h e differences were i n the 5 t h or 6 t h d e c i m a l place be tween the results of the present s t udy a n d those of R u d e r et a l [1]; or i n other words the results were cons idered to be of comparab le accuracy. T h i s was cons idered to be suff icient ly accura te for the purposes of th is s tudy. F i g u r e 5.4 shows how the eigenvalues ob t a ined f rom the so lu t ion of E q . 34 Chapter 5. Results & Discussion (3.3) depend u p o n the mesh size. A s was men t ioned earl ier , the ca lcu la t ions were ca r r i ed out o n a finite d o m a i n of several B o h r r a d i i i n each of the two d i rec t ions b o t h pa ra l l e l a n d pe rpend icu la r to the field. W h i l e keeping the d o m a i n fixed, the n u m b e r of finite elements c o n s t i t u t i n g the mesh was var ied . R u n s were pe r fo rmed o n each mesh size for every value of magne t i c field s t r eng th cons idered i n the s t u d y for each of the three states of hydrogen , m = 0 , -1 a n d -2. T h e eigenvalues thus o b t a i n e d were p l o t t e d against the average area per finite element i n the mesh co r re spond ing to each value of the magne t i c field s t r eng th paramete r 0. E x t r a p o l a t i o n of the d a t a to the l i m i t of zero mesh size i n each case y i e lded the values t ha t w o u l d co r respond to an in f in i t e ly fine mesh. These values were r epor t ed as the ca l cu l a t ed d a t a po in t s i n the p reced ing figures. T h e average accu racy o f the es t imate of the a s y m p t o t i c value was de t e rmined to be on the order of 3 x I O - 6 [55]. T h e l ine t h r o u g h the d a t a poin ts is mere ly a guide to the eye. T h e wave funct ions for the most t i g h t l y b o u n d state de t e rmined f rom the ca lcu la t ions are p lo t t ed i n F i g u r e 5.5 for four different values of the magne t i c field s t r eng th 3. T h e p lo ts represent a sl ice t h r o u g h the three d i m e n s i o n a l a t o m i n the pos i t ive z- p lane . In order to o b t a i n the comple te representa t ion i n three d imens ions one c o u l d imag ine p l a c i n g such slices at every angle t h r o u g h 27r a n d t h e n ref lect ing i t abou t the xy -p l ane . T h e l eng th un i t s are B o h r r a d i i of the hydrogen a t o m and the x -ax i s represents the d i r e c t i o n pe rpend icu l a r to the magne t i c field d i r e c t i o n w h i l e the axis l abe led " y " represents the d i r ec t i on pa ra l l e l to the magne t i c field, i.e. the z- d i r e c t i o n 35 Chapter 5. Results Sz Discussion State F o r m Coefficients m = 0 , 7 r=+l ax3+bx2+cx+d x+e a=-0.01905 b=0 .4236 c=2.326 d=0 .6567 e=0.6567 m = - l , 7 r = + l u=0 ax4 +bx3+cx2+dx+e x2+fx+g a=0.005058 b=0 .0513 c=2 .347 d=1.274 e=0.04658 f=1.913 g=0.1929 m = - 2 , 7 r = + l u=0 ax5+bx4+cx3+dx2+ex+f x3+gx2+hx+i a=-0.01436 b=0 .2984 c=0.9101 d=0.05033 e=0.01379 f=0.00007748 g=0.3168 h=0.01621 i=0 .00224 T a b l e 5.1: T a b l e l i s t i n g the coefficients of the different r a t i o n a l funct ions emp loyed i n the fits for the three states of hyd rogen discussed above. T h e coefficient values have confidence bounds of 99%. i n three d i m e n s i o n a l c y l i n d r i c a l po la r coordinates . T h e purpose here is to i l lus t ra te the d r a m a t i c change tha t occurs near ^ « 1 w h e n the e lec t ron becomes t i g h t l y b o u n d a n d the b i n d i n g energy increases d r a m a t i c a l l y w i t h increas ing 8. It is i m m e d i a t e l y evident u p o n i n s p e c t i o n t ha t the spher ica l s y m m e t r y of the a t o m is c lea r ly b roken as we app roach higher a n d higher magne t i c fields. 36 Chapter 5. Results & Discussion 5.2 The Helium Atom F o r magne t i c fields i n the range 1 0 ~ 3 < (3 < 1 we so lved the set of cou-p l e d H a r t r e e - F o c k equat ions g iven i n Eq . (3 .24) for different values of the magne t i c f ield s t r eng th parameter . W e sha l l however be showing the results o b t a i n e d for the states cor ressponding to M = 0 , - 1 a n d —2 for the mos t t i g h t l y b o u n d states co r re spond ing to each M. T h e two electrons are t a k e n to posess a z i m u t h a l angular m o m e n t u m q u a n t u m number g iven b y m\ and ra2 respect ively. W e consider here i n the s t udy on ly the ful ly sp in -po la r i sed states, thus each of the electrons has i ts sp in an t i - a l igned w i t h the magne t i c f ield as assumed i n the de r iva t i on of E q . (3.24). These states have been seen to be the mos t t i g h t l y b o u n d states of h e l i u m i n s t rong magne t i c fields [1]. It is to be no t ed tha t the b i n d i n g energies are r epor t ed i n un i t s of R y d b e r g energy i n the C o u l o m b p o t e n t i a l of nuclear charge Ze. T h e v a r i a t i o n i n the b i n d i n g energy for the state co r re spond ing to M = — l,Sz — —l7r2 — +1 is s h o w n i n F i g u r e 5.6. T h e d a t a po in t s are eigenvalues o b t a i n e d f rom the n u m e r i c a l s o l u t i o n of E q . (3.24) acco rd ing to the n u m e r i c a l procedures descr ibed i n C h a p t e r 4. T h e energy eigenvalues are once aga in r epor t ed as the values co r respond ing to in f in i t e ly fine mesh sizes, or i n other words w h e n the area of a finite element approaches zero. A shor t d i scuss ion of th i s e s t i m a t i o n procedure can be found i n the previous sec t ion . A s can be .seen i n the figure, the b i n d i n g energy of the state increases w i t h increas ing magne t i c field s t r eng th (0). T h e l ine t h r o u g h the d a t a represents a fit to the 37 Chapter 5. Results & Discussion da ta . A s i n the case of the hyd rogen a t o m , a r a t i o n a l func t ion was used to m o d e l the d a t a i n th is regime us ing a robus t L e v e n b e r g - M a r q u a r d t m e t h o d [55]. T h e values for the coefficients of the i n t e r p o l a t i n g func t ion are g iven i n T a b l e 5.2 w i t h 9 9 % confidence bounds . T h e fit i n F i g u r e 5.8 was seen to be accura te to w i t h i n 0.18 % d e v i a t i o n f rom the ca l cu la t ed values. A l s o shown i n F i g u r e 5.6for i l l u s t r a t i ve purposes, are the results f rom a pe r tu rba t i ve c a l c u l a t i o n for the state i n cons ide ra t ion . A t y p i c a l Z e e m a n H a m i l t o n i a n was emp loyed w i t h b o t h the l inear a n d the q u a d r a t i c t e rms c o n t r i b u t i n g to the energy shift . T h e results p l o t t e d are for first order i n p e r t u r b a t i o n theory. A s c a n be seen clearly, p e r t u r b a t i o n theory o n l y works for a l i m i t e d range of the magne t i c field s t r eng th paramete r 0, b e y o n d 0 ~ 1 0 ~ 2 , the p e r t u r b a t i v e results s tar t to diverge r ap id ly , y i e l d i n g progress ively erroneous b i n d i n g energies w i t h inc reas ing magne t i c field s t rength . S i m i l a r l y , figure 5.7 shows the v a r i a t i o n i n the b i n d i n g energy w i t h 0Z for a l l the three states co r re spond ing to the q u a n t u m numbers M = —l,Sz = — 1,7T = +1 M = -2,SZ = —l,7r = +1 a n d M = 0, Sz = - 1 , vr = +1 re-spect ively . A g a i n the l ines t h r o u g h the d a t a po in t s represent fits to the da ta . E v e r y d a t a po in t i n these figures was ob t a ined as an es t imate co r re spond ing to l i m i t of the mesh size go ing to zero. A g o o d measure of the d i s c r e t i z a t i on error was cons idered to be the difference between the c o m p u t e d eigenvalues for the two mos t f inely refined mesh sizes employed . T h e average error re l -a t ive to the eigenvalue c o m p u t e d for the finest mesh size, for a l l the three states of h e l i u m a n d a l l values of the magne t i c field s t rength , was seen to 38 Chapter 5. Results & Discussion be o n the order of 0.008%. T h i s was cons idered to be suff iciently accura te for the present s tudy. It is to be men t ioned at th i s j u n c t u r e t ha t the error bars were too s m a l l to be shown on the p lo ts . T h e fits to the d a t a are ra t io -n a l funct ions ob t a ined as desc r ibed above. T h e coefficients are r epor t ed to w i t h i n 9 9 % confidence bounds i n T a b l e 5.2. F i g u r e 5.8 shows how the eigenvalues o b t a i n e d f rom the so lu t i on of E q . (3.24) depend u p o n the mesh size. A s i n the case of hyd rogen runs were per formed o n each mesh size for every value of magne t i c field s t rength con-s idered i n the s t udy for each of the three states of h e l i u m , M = 0 , -1 and -2. T h e q u a n t u m number M is the s u m of the i n d i v i d u a l a z i m u t h a l q u a n t u m number s of the two electrons w i t h the s ign preserved. T h e eigenvalues thus o b t a i n e d were p l o t t e d agains t the average a rea per f ini te e lement i n the mesh co r re spond ing to each value of the magne t i c f ield s t r eng th paramete r Bz- E x -t r a p o l a t i o n of the d a t a to the l i m i t of zero mesh size i n each case y i e lded the values tha t w o u l d co r respond to an in f in i t e ly fine mesh. These values were r epor t ed as the ca l cu la t ed d a t a po in t s i n the p reced ing figures. T h e average accuracy of the es t imate of the a s y m p t o t i c value was de t e rmined to be on the order of 2 x 1 0 - 5 [55]. T h e l ine t h r o u g h the d a t a po in t s is mere ly a guide to the eye. F o r the sake of c o m p a r i s o n w i t h p rev ious ly de t e rmined accura te results , the fit to the ca l cu la t ed d a t a for the g r o u n d state of h e l i u m cor respond-i n g to M = - l 7 r = + l , the most t i g h t l y b o u n d state, was p l o t t e d against the ca l cu la t ed d a t a of the researchers i n R e f [1] a n d R e f [2, 3] i n F i g u r e 5.9. 39 Chapter 5. Results & Discussion T h e l ines t h r o u g h the i r d a t a are spl ine fits. It can be seen tha t the ca lcu -l a t ed eigenvalues for the b i n d i n g energy of the g r o u n d state of the h e l i u m a t o m are i n agreement w i t h p rev ious work . I n p a r t i c u l a r t h e y are seen here to be improvement s u p o n the est imates of R u d e r et a l [1] over the entire range of Bz cons idered i n th i s s tudy. Jones et a l o n the other h a n d h a d employed a M o n t e - C a r l o approach [2] to s o l v i n g the H F equat ions . T h i s was accompl i shed by a s suming a large number of basis funct ions w i t h var iab le parameters tha t c o u l d be fine t u n e d w i t h i n the f ramework of M o n t e - C a r l o s imu la t ions to a r r ive at upper bounds for the energies. T h o u g h th is m e t h o d is effective, as can be seen f rom the i r es t imates s h o w n i n F i g u r e 5.9 i t is c o m p u t a t i o n a l l y d e m a n d i n g and i t restr icts the wave funct ions of the elec-t rons to be expressed us ing a finite number of basis funct ions. T h e m e t h o d desc r ibed i n the current s t u d y does not impose such a c o n d i t i o n and thus the wave funct ions tha t are de t e rmined are i n effect super pos i t ions of an inf in i te n u m b e r of such basis funct ions a n d arise n a t u r a l l y f rom the so l u t i on itself. However , i t is to be men t ioned t ha t for h i g h magne t i c field s t r eng th paramete r Bz ~ 20 the eigenvalues were seen to d rop be low those of R u d e r et al 's es t imates . It is to be no ted tha t th is effect was due to the fact tha t the m a x i m u m mesh refinement for the g iven d o m a i n size (see above) was s t i l l inadequate for a n accurate es t imate of the eigenvalues. See C h a p t e r 6 for a short d i scuss ion o n the mat te r . F i n a l l y , the wave funct ions for the most t i g h t l y b o u n d state de t e rmined f rom the ca lcu la t ions are p l o t t e d i n F i g u r e 5.10 for four different values of 40 Chapter 5. Results & Discussion the magne t i c field s t r eng th f3z. However th is t i m e we choose to show the e lec t ron i n the state w i t h q u a n t u m numbers m = - l , s ^ = - l / 2 , 7 r = + l , w h i c h is equivalent to 2 p _ i o r b i t a l i n the low field l i m i t . T h e l eng th un i t s are B o h r r a d i i of the h e l i u m a t o m , i.e. ci0/Z and the x -ax i s represents the d i r ec t i on pe rpend icu l a r to the magne t i c field d i r e c t i o n w h i l e the axis l abe led " y " rep-resents the d i r ec t i on pa ra l l e l to the magne t i c field, i.e. the z- d i r e c t i o n i n three d i m e n s i o n a l c y l i n d r i c a l po l a r coordina tes . T h e purpose once aga in is to i l l u s t r a t e the sharp change i n the fo rm of the wave func t ion tha t occurs near Bz « 1 w h e n the e lec t ron becomes t i g h t l y b o u n d a n d the b i n d i n g energy increases d r a m a t i c a l l y w i t h inc reas ing /?_. T h e ca l cu la t ed eigenvalues for the three states of hyd rogen a n d the three states of h e l i u m descr ibed above are l i s t ed i n the tables i n the A p p e n d i x . T h e reader s h o u l d refer the re in for accura te values of the energy eigenvalue for the states show i n the figures. 41 Chapter 5. Results Sz Discussion State F o r m Coefficients M = - l l s 0 ; 2 p _ i ax3+bx2+cx+d x+e a=-0.2438 b=1 .18 c=1 .987 d=0.2411 e=0.2077 M = - l l s 0 ; 3 d _ 2 ax4+bx3+cx2+dx+e a=33.54 b=-107.2 c=193.6 d=154 .7 e=4.611 f=117.1 g=3.967 x2+fx+g M = 0 l s 0 ; 2 p 0 ax4+bx3+cx2+dx+e a=1.121 b=-2 .583 c=7.404 d=3.194 e=0.0419 f=2.568 g=0.03593 x2+fx+g T a b l e 5.2: T a b l e l i s t i n g the coefficients of the different r a t i ona l funct ions emp loyed i n the fits for the three states of hydrogen discussed above. T h e coefficient values have confidence bounds of 99%. 42 Chapter 5. Results & Discussion T 1—I 1 II 1 0 F i g u r e 5.1: F i g u r e showing the v a r i a t i o n i n the b i n d i n g energy of the g r o u n d state of hyd rogen w i t h the magne t i c field s t r eng th pa ramete r i n the range 1 0 ~ 3 < 8 < I. T h e energy is r epor ted i n un i t s of R y d b e r g energy. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u l a t i o n w h i l e the l ine represents a fit t o the da ta . T h e dashed l ine represents the results f rom first order p e r t u r b a t i o n theory. T h e purpose b e i n g to i l lus t ra te the fact tha t p e r t u r b a t i o n theory breaks d o w n w i t h inc reas ing the magne t i f field s t rength . 43 Chapter 5. Results Sz Discussion F i g u r e 5.2: F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of three lowest m - states of hydrogen co r re spond ing to m = 0 , - l , - 2 , w i t h 7 r = + l w i t h the magne t i c f ield s t r eng th paramete r i n the range 1 0 - 3 < (3 < 1. T h e energy is r epo r t ed i n un i t s of R y d b e r g energy. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u a t i o n w h i l e the l ines represent fits to the da ta . 44 Chapter 5. Results & Discussion 2.5 1.5 C d I 0.5 - Calculated -Ruder et al i o - io - io - 10° F i g u r e 5.3: F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of the state of hydrogen co r re spond ing to m=0 w i t h 7r=+l w i t h the magne t i c field s t reng th paramete r i n the range 10~ 3 < 3 < 1. Resu l t s f rom b o t h the present s t udy as w e l l as those f rom R e f [1] are p l o t t e d here in . T h e energy is r epor t ed i n un i t s of R y d b e r g energy. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u a t i o n w h i l e the l ines represent fits to the da ta . 45 Chapter 5. Results &: Discussion i—i—i 111111 1—i—i 111111 - Q B . — • M I I I I I I I I I I I I I 1 I 10"3 10-2 10"1 10° Area Per Finite-Element [au] F i g u r e 5.4: F i g u r e showing a p lo t of the eigenvalues ca l cu l a t ed as a func t ion of the are per f ini te element i n the mesh for a c a l c u l a t i o n for the g r o u n d state of hydrogen at /3=10~\ W i t h inc reas ing ly finer meshes for a sufficient d o m a i n size the eigenvalue ob t a ined approaches an a sympto t e co r re spond ing to the l i m i t of in f in i t e ly fine mesh. T h e parameter desc r ib ing the mesh size is the average area per f ini te element i n the mesh 46 Chapter 5. Results & Discussion Figure 5.5: Figure showing a plot of the real part of the wave functions, i.e. tp(p,z), of the ground state of the hydrogen atom with changing magnetic field strength. The wave functions are for (a) /?=1(T 3 , (b) /3=1CT 2 , (c) / ?=10~ 1 and (d) j3=l. The effect of the increasing binding energy is clearly visible in the shrinking of the wave function with increasing (3. These plots are not to be confused with electron density. 47 Chapter 5. Results & Discussion F i g u r e 5.6: F i g u r e showing the v a r i a t i o n i n the b i n d i n g energy of the g r o u n d state of h e l i u m w i t h the magne t i c field s t r eng th paramete r i n the range 10~ 3 < Pz < 1- T h e energy is r epor ted i n un i t s of R y d b e r g energy for nuclear charge Ze. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u l a t i o n wh i l e the l ine represents a fit to the da ta . 48 Chapter 5. Results Sz Discussion F i g u r e 5.7: F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of three lowest M - states of h e l i u m co r re spond ing to M=0,- l , -2 , w i t h 7T=+1 w i t h the magne t i c f ield s t r eng th paramete r i n the range 1 0 - 3 < /?z < 1. T h e energy is r epor t ed i n un i t s of R y d b e r g energy for nuclear charge Ze. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u a t i o n wh i l e the l ines represent fits to the da ta . 49 Chapter 5. Results & Discussion F i g u r e 5.8: F i g u r e showing the v a r i a t i o n i n the b i n d i n g energies of the state of h e l i u m co r r e spond ing to M = - l w i t h 7T=+1 w i t h the magne t i c field s t r eng th paramete r i n the range 1 0 ~ 3 < 0Z < 1. Resu l t s f rom b o t h the present s t u d y as we l l as those f rom R e f [1] a n d Jones et a l [2, 3] are p l o t t e d here in . T h e energy is r epor t ed i n un i t s of R y d b e r g energy for nuclear charge Ze. T h e d a t a po in t s are results of the n u m e r i c a l c a l c u a t i o n w h i l e the l ines represent fits to the da ta . 50 Chapter 5. Results & Discussion i—i—i—i i 111 I O " 3 I O " 2 10"1 Area Per F in i t e -E lement [au] F i g u r e 5.9: F i g u r e showing a p lo t of the eigenvalues ca l cu la t ed as a func t ion of the are per finite element i n the mesh for a c a l c u l a t i o n for the g r o u n d state of hyd rogen at Bz=l- W i t h inc reas ing ly finer meshes for a sufficient d o m a i n size the eigenvalue o b t a i n e d approaches an a sympto t e co r re spond ing to the l i m i t of in f in i t e ly fine mesh. T h e parameter desc r ib ing the mesh size is the average area per finite element i n the mesh 51 Chapter 5. Results Sz Discussion Figure 5.10: Figure showing a plot of the real part of the wave functions, i.e. ip(p,z), of one of the electrons comprising the ground state of the helium atom with changing magnetic field strength. The quantum numbers corre-spond to m 2 =- l , s_=-l/2, 7r=+l.The wave functions are for (a) 3Z—10~3, (b) /3Z=10~2, (c) / 5 2 = 1 0 _ 1 and (d) Bz=l. The effect of the increasing binding energy is clearly visible in the shrinking of the wave function with increasing /3Z. These plots are not to be confused with electron density. 52 Chapter 6 Summary & Conclusions W e present here i n th i s chapter a s u m m a r y of our f indings a n d the conc lus ions tha t c a n be d r a w n f rom the w o r k descr ibed i n th i s thesis . T h e w o r k desc r ibed here was m o t i v a t e d by the need to have accura te ly d e t e r m i n e d values for the upper bounds for the energy levels of a toms i n s t rong magne t i c fields. A s was discussed earl ier , th i s need has ar isen due to the presence of s t rong magne t i c fields i n n e u t r o n stars a n d w h i t e d w a r f stars. M u c h of the prev ious w o r k discussed i n the l i t e ra ture rev iew has focused o n the a toms hydrogen a n d h e l i u m . T h u s these a toms were s tud ied i n th is thesis w i t h the i n t en t ion of o b t a i n i n g accura te a n d perhaps more b o u n d est i-mates of the energy levels of the first few low l y i n g states i n s t rong magne t i c fields. W e desc r ibed a' m e t h o d a d o p t i n g a p h y s i c a l l y m o t i v a t e d approach governed by the inherent symmet r i e s of the p r o b l e m . W e s imul t aneous ly c i r -cumvent the need for a d o p t i n g a definite basis of funct ions to descr ibe the wave funct ions of the e lec t ron i n ei ther of the d i rec t ions , pa ra l l e l a n d perpen-d i c u l a r to the magne t i c field. T h e app roach is unres t r i c t ive w i t h regard to the wave func t ion ; i t has the d i s t inc t advantage over me thods t ha t require a basis of funct ions to descr ibe the- wave funct ions because i n n u m e r i c a l so lu-53 Chapter 6. Summary & Conclusions t ions one c a n o n l y have a finite number of such funct ions . T h e wave funct ions de t e rmined i n the present s t u d y came abou t n a t u r a l l y f rom the symmet r i e s of the p r o b l e m a n d are thus i n effect super pos i t ions of an inf in i te number of basis funct ions . S u c h an approach however resul ted i n e l l i p t i c a l p a r t i a l different ia l equa-t ions for the electrons tha t were subsequent ly so lved us ing finite element techniques. It is to be no ted tha t the m e t h o d adop ted for d e t e r m i n i n g the d i rec t and exchange in te rac t ions between the electrons in the h e l i u m a t o m is also exact i n the sense tha t i t does not re ly u p o n any ab i n i t i o assumpt ions to a p p r o x i m a t e the integrals . These i n t e r ac t i on po ten t ia l s are solved i n a n a t u r a l manne r by s o l v i n g the i r poisson- l ike equat ions , E q s . (3.16 a n d 3.22). T h e eigenvalues found i n the range of the magne t i c f ield s t r eng th pa ramete r I O " 3 < Qz < 1 cons idered i n th is s t u d y were seen to be i n agreement w i t h p rev ious findings [1] a n d [2, 3]. R a t i o n a l funct ions were also used to find suf-ficiently accura te i n t e r p o l a t i n g funct ions for the b i n d i n g energies of var ious states of b o t h the hyd rogen a n d h e l i u m a toms i n the range of magne t i c fields cons idered here in . These were seen to be accuara te to (an average for a l l s ix fits) w i t h i n 0.2%. P o t e n t i a l l y such i n t e r p o l a t i n g funct ions c o u l d be used i n a tmosphere mode ls of neu t ron stars a n d w h i t e d w a r f stars thus o b v i a t i n g the need for i nvo lved and l abor ious ca lcu la t ions of the same. T h u s , the current work describes an unres t r i c ted a n d c o m p u t a t i o n a l l y less in tensive m e t h o d for c a l c u l a t i n g the energy levels of a toms i n s t rong magne t i c fields. 54 Chapter 6. Summary & Conclusions, 6.1 Future Work T h e r e were i n essence three d i rec t ions i n w h i c h the current w o r k is i n the process of be ing ex tended . F i r s t , an inves t iga t ion of the effects of the d o m a i n size is underway. In the current s t udy a d o m a i n size of several B o h r r a d i i i n each of the two d i rec t ions , pa ra l l e l a n d pe rpend icu l a r to the f ield was em-ployed . T h u s the i m p a c t of different d o m a i n sizes a n d mesh sizes the re in is be ing inves t iga ted a n d the results are to be p u b l i s h e d elsewhere. S i m u l t a n e -ously, improvemen t s w i t h regard to ex t end ing the resul ts to h igher magne t i c field s t rengths is i m m i n e n t . T h e ca lcu la t ions a n d the software deve loped as a pa r t of th i s s t u d y are r ead i ly ex tendable to systems w i t h more t h a n two electrons, i.e., L i , B e , B , C , O etc. F i n a l l y , the procedures i m p l e m e n t e d here in can also be ex tended towards a m u l t i - c o n f i g u r a t i o n f ramework [36]. I n essence, the ca lcu la t ions emp loyed here in are for a s ingle conf igura t ion of the electrons. T h u s h a v i n g found the energies of the different conf igura t ions , i t is poss ible to ex t end the theory to incorpora te a m u l t i - c o n f i g u r a t i o n approach w h i c h is l i ke ly to improve the results o b t a i n e d here. 55 Bibliography [1] H . R u d e r et a l . Atoms in Strong Magnetic Fields: Quantum Mechan-ical Treatment and Applications in Astrophysics and Quantum Chaos. A s t r o n o m y a n d A s t r o p h y s i c s L i b r a r y . Sp r inge r -Ver l ag , N e w Y o r k , 1994. [2] M . D . Jones, G . 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The Finite Element Method for Elliptic Problems. N o r t h - H o l l a n d P u b . C o , N e w Y o r k , 1978. [55] S a u l A . et a l Teuko l sky . Numerical Recipes in C. C a m b r i d g e U n i v e r s i t y Press , N e w Y o r k , 1992. 63 Appendix A Energy Values W e present here the energy values for the mos t t i g h t l y b o u n d states of hyd ro -gen a n d h e l i u m w i t h v a r y i n g magne t i c f ield s t r eng th parameter . T h e eigen-values quo ted are as de t e rmined by the e x t r a p o l a t i o n procedure ou t l i ned in § 5 . 1 a n d § 5.2 to w h i c h the reader is referred. T h e r a t i o n a l funct ions g iven i n Tab les 5.1 a n d 5.2 give accura te p red ic t ions for values of the energies i n the range of magne t i c f ield s t r eng th 1 0 ~ 3 < /3, /?z < 1 for each of the states of hyd rogen a n d h e l i u m l i s t ed be low. T a b l e A . l l is ts the b i n d i n g energies of the states of hyd rogen w h i l e T a b l e A . 2 conta ins d a t a for the energies of the three mos t t i g h t l y b o u n d states h e l i u m . T h e magne t i c f ield s t r eng th paramete r is defined i n un i t s of the c r i t i c a l f ield s t r eng th as desc r ibed earl ier , wh i l e the b i n d i n g energies are r epor t ed i n un i t s of R y d b e r g energies for hyd rogen and i n un i t s of Ezoo w i t h Z = 2 for h e l i u m . 64 Appendix A. Energy Values p l s o / 0 0 + 1 2 p _ i / - 1 0 + l 3 d _ 2 / - 2 0 + l 1 x 1 0 - -3 1.0020431140 0.246199255 0.111700270 2 x 1 0 " -3 1.0040371140 0.250143677 0.122673450 5 x 1 0 " -3 1.0099951247 0.261754987 0.135754880 7 x 1 0 ' -3 1.0139471450 0.269311532 0.146756935 1 x 1 0 " -2 1.0198452260 0.280371872 0.153586459 2 x 1 0 " -2 1.0392466270 0.314916706 0.180915515 5 x 1 0 " -2 1.0950987030 0.399746961 0.254178450 7 x 1 0 " -2 1.1304424140 0.444548841 0.305533689 1 x 1 0 " -1 1.1808101480 0.500974961 0.360939849 2 x 1 0 " -1 1.3292614470 0.642705806 0.481963967 5 x 1 0 " -1 1.6624036380 0.913181157 0.706096051 7 x 1 0 - -1 1.8324084570 1.041992602 0.812476599 1 2.0445217070 1.199185965 0.942343873 T a b l e A . l : T a b l e l i s t i n g the b i n d i n g energies of the three states of the hy-drogen a t o m cons idered i n th is s tudy. T h e q u a n t u m number s co r re spond to m, vz a n d irz. T h e energies are i n R y d b e r g un i t s . p 2 p _ 1 / - 1 0 3 d _ 2 / - 2 0 2 P o / 0 1 1 x 1 0 " -3 1.164695861 1.172671163 1.167909206 2 x 1 0 " -3 1.168743777 1.176762708 1.171831677 5 x 1 0 " -3 1.180585498 1.188758114 1.183289656 7 x 1 0 " -3 1.188240029 1.196509113 1.190671633 1 x 1 0 " -2 1.199348874 1.207790181 1.201363034 2 x 1 0 " -2 1.233502737 1.242415362 1.233796322 5 x 1 0 " -2 1.330316305 1.322544944 1.306799557 7 x 1 0 " -2 1.391217761 1.361558006 1.343610424 1 x 1 0 " -1 1.468427732 1.424874414 1.394367666 2 x 1 0 " -1 1.675435070 1.629324420 1.549721855 5 x 1 0 " -1 2.119084148 2.060660277 1.896753578 7 x 1 0 " -1 2.342544449 2.277403651 2.071315489 1 2.620407779 2.546854461 2.287514063 T a b l e A . 2 : T a b l e l i s t i n g the b i n d i n g energies of the three states of the h e l i u m a tom,cons ide red i n th is s tudy. T h e q u a n t u m numbers co r respond to m a n d uz. T h e energies are i n un i t s of Ezoo 65 Appendix B Source Code Presen ted here is the software w r i t t e n i n the M a t l a b ® p r o g r a m m i n g l an -guage t ha t was deve loped for th i s s tudy. T h e source code represents the m a i n d r ive r rou t ine a n d provides the a l g o r i t h m for the me thods descr ibed i n C h a p t e r 4 for f ind ing the eigenvalues and eigenfunct ions of the Ha r t r ee -Fock equat ions for the h e l i u m a t o m . T h e p o r t i o n of the code l abe led Hydrogenic Part relates to s o l v i n g the hydrogen ic eigenvalue p r o b l e m . % % P D E - H F CODE FOR HELIUM V. A u g 2 0 0 7 % . . . % '/, The c o d e o u t l i n e d h e r e s o l v e s t h e H a r t r e e - F o c k E q u a t i o n s f o r a 2 '/, e l e c t r o n s y s t e m , i . e . , H e l i u m . The E q u a t i o n s a r e s o l v e d u s i n g t h e s e l f '/, c o n s i s t e n t f i e l d m e t h o d due t o H a r t r e e . '/, The m e t h o d o f s o l u t i o n c o n s i s t s o f t h e f o l l o w i n g s t e p s : •/. '/, (a ) S o l v e t h e H y d r o g e n i c p r o b l e m w i t h m a g n e t i c f i e l d s t r e n g t h b e t a _ Z '/, o b t a i n i n g t h e w a v e f u n c t i o n s U l a n d U 2 ; s u b s e q u e n t l y n o r m a l i z e t h e m . '/, ( b ) U s e t h e n o r m a l i z e d wave f u n c t i o n s t h u s o b t a i n e d t o g e t t h e '/. e l e c t r o n - e l e c t r o n i n t e r a c t i o n p o t e n t i a l s , i . e . , t h e d i r e c t a n d e x c h a n g e '/, i n t e r a c t i o n s b e t w e e n t h e e l e c t r o n s . '/. ( c ) P l u g i n t h e p o t e n t i a l s o b t a i n e d i n (b) i n t o t h e H a r t r e e - F o c k 66 Appendix B. Source Code E q u a t i o n s w r i t t e n a b o v e . T h i s t h e n y i e l d s a s e t o f two c o u p l e d e q u a t i o n s t h a t a r e p a r t o f a n e i g e n s y s t e m . '/. ( d ) S o l v e t h e e i g e n s y s t e m P D E ' s t o o b t a i n new e s t i m a t e s f o r U l a n d U2. '/, ( e ) R e p e a t s t e p s (b ) t h r o u g h (d ) u n t i l t h e i t e r a t i v e p r o c e d u r e h a s c o n v e r g e d t o t h e d e s i r e d t o l e r a n c e . START c l e a r a l l ; c l e a r g l o b a l ; g l o b a l n o r m _ i n t p e t ; b e t a _ f i d = f o p e n ( ' b e t a f i l e ' , ' r ' ) ; [ b e t a _ n o w , b e t a _ c o u n t ] =f s c a n f ( b e t a . f i d , 'V.f' ,inf); f o r b e t a _ i = l : b e t a _ c o u n t , b e t a = b e t a _ n o w ( b e t a _ i ) ; '/, O b t a i n U s e r i n p u t f o r m a g n e t i c f i e l d s t r e n g t h 7 , b e t a = i n p u t ( ' P l e a s e e n t e r t h e v a l u e f o r b e t a : ' ) ; Z=2; '/, A t o m i c Number ne=2; '/, Number o f E l e c t r o n s b e t a _ Z = b e t a / ( Z * Z ) ; '/. D e f i n i n g B e t a . Z = B e t a / Z " 2 b e t a _ Z 2 = b e t a _ Z * b e t a _ Z ; c o u n t e r = l ; '/, i n t i a l i z e c o u n t e r ; n e e d e d f o r h y d r o g e n i c p a r t o f t h e c a l c u l a t i o n s '/, D e f i n i n g t h e r a n g e w i t h i n w h i c h we s h a l l be s e a r c h i n g f o r e i g e n v a l u e s . r m i n = - 5 ; rmax=0; '/, I n i t i a l i z i n g t h e d o m a i n a n d t h e g r i d f o r t h e p r o b l e m f a c t o r = l + l o g ( l + b e t a _ Z ) ; 67 Appendix B. Source Code 0 ; x m a x ; x m a x ; 0 ; 0 ; 0 ; y m a x ; y m a x ] ; xmax=10 ymax=20 gd= [ 3 ; 4 s f = ' R l ' n s = [ 8 2 ; 4 9 ] ; d l = d e c s g ( g d , s f , n s ) ; '/, Maximum X - c o o r d i n a t e o f D o m a i n '/, Maximum Y - c o o r d i n a t e o f D o m a i n '/, D e f i n i n g t h e G e o m e t r y o f d o m a i n '/. S e t F o r m u l a ; s e e PDE t o o l b o x H e l p '/, Named p r o p e r t y ; s e e P D E t o o l H e l p % d e c o m p o s e d G e o m e t r y m a t r i x '/, T o t a l number o f i t e r a t i o n s o f t h e HF p r o c e d u r e . m a x i t e r = 1 0 ; % % */, D e f n i i n g t e h b o u n d a r y c o n d i t i o n m a t r i c e s '/. '/, b d e f i n e s t h e b o u n d a r y c o n d i t i o n m a t r i x f o r t h e h y d r o g e n i c p r o b l e m a s '/. w e l l a s f o r t h e e l l i p t i c P D E ' s t o be s o l v e d f o r g e t t i n g t h e d i r e c t a n d % e x c h a n g e i n t e r a c t i o n p o t e n t i a l f u n c t i o n s . Z '/. b2 d e f i n e s t h e b o u n d a r y c o n d i t i o n m a t r i x f o r t h e c o u p l e d PDE s y s t e m . b =[ 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 48 1 1 48 48 1 1 48 48 48 48 48 48 48 48 48 48 49 49 48 48 48 48 48 68 Appendix B. Source Code b2 =[ 2 2 2 2 0 2 2 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 • 1 1 1 1 1 48 1 1 48 48 1 1 48 48 1 1 48 48 1 1 48 48 1 1 48 48 1 1 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 49 49 49 49 48 48 48 48 48 48 48 48 49 49 49 49 48 48 48 48 48 48 48 48 ] ; % ' / . I n i t i a l i z i n g a n d r e f i n i n g t h e mesh 69 Appendix B. Source Code jmax=3; '/, number o f t i m e s t h e mesh i s r e f i n e d [ p , e , t ] = i n i t m e s h ( d l ) ; f o r j = l : j m a x , [ p , e , t ] = r e f i n e m e s h ( d l , p , e , t ) ; '/. Performs I r e f i n e m e s h l j m a x t i m e s . e n d y> •/. HYDROGENIC PART BEGINS y> y> '/. Now s o l v i n g t h e h y d r o g e n i c p r o b l e m '/, The p r o b l e m g e t s s e t up a s '/. - d i v ( c * g r a d ( u ) ) + a u = l a m b d a * d * u '/. t h e c o e f f i c i e n t s ' c ' , ' a ' a n d ' d ' a r e s e t up a c c o r d i n g l y b e l o w a n d t h e '/, e i g e n v a l u e s o l v e r ' p d e e i g ' i s c a l l e d a n d s u b s e q u e n t l y t h e p r o b l e m i s '/. s o l v e d f o r a g i v e n b e t a a n d a z i m u t h a l q u a n t u m number m. f o r m = - l : 0 , m2=m*m; '/, d e f i n i n g c o e f f i e c i e n t s ' c ' ' a ' a n d ' d ' c = ' x ' ; a = s t r c a t ( n u m 2 s t r ( m 2 ) , ' . / x + ' , n u m 2 s t r ( m ) , ' . * 2 . * ' , n u m 2 s t r ( b e t a _ Z ) , . . . • . * x - 2 * x . / s q r t ( x . ~ 2 + y . ~ 2 ) + ' , n u m 2 s t r ( b e t a _ Z 2 ) , ' * x . * ( x . " 2 ) - 2 . * ' , . . . n u m 2 s t r ( b e t a _ Z ) , ' . * x ' ) ; d = ' x ' ; ' / . s o l v i n g t h e e i g e n v a l u e p r o b l e m [ vv ,1 ] = p d e e i g ( b , p , e , t , c , a , d , [ r m i n r m a x ] ) ; i f ( c o u n t e r == 1) v 2 = v v ( : , 1 ) ; o l d v 2 = v 2 ; e i g e n v a l 2 = l ; e l s e v l = v v ( : , 1 ) ; 70 Appendix B. Source Code o l d v l = v l ; e i g e n v a l l = l ; e n d c o u n t e r = c o u n t e r + l ; / d i s p O f i r s t p a r t d o n e ! H y d r o g e n i c p r o b l e m s o l v e d ' ) e n d c o u n t e r = l ; % '/, H y d r o g e n i c p a r t f i n i s h e d - . y> '/, N o r m a l i z i n g t h e e i g e n v e c t o r s n o r m _ i n t = 4 * p i * p ( l , : ) ' . * v l . " 2 ; n o r m = d b l q u a d ( S i n t e g r a n d , 0 . 0 , x m a x , 0 . 0 , y m a x ) ; v l = v l / s q r t ( n o r m ) ; n o r m _ i n t = 4 * p i * p ( l , : ) ' . * v 2 . " 2 ; n o r m = d b l q u a d ( © i n t e g r a n d , 0 . 0 , x m a x , 0 . 0 , y m a x ) ; v 2 = v 2 / s q r t ( n o r m ) ; v l _ c = p d e i n t r p ( p , t , v l ) ; D e f i n i n g t h e e i g e n v e c t o r s a t t h e t r i a n l g e c e n t r e s v 2 _ c = p d e i n t r p ( p , t , v 2 ) ; '/, D e f i n i n g t h e e i g e n v e c t o r s a t t h e t r i a n l g e c e n t r e s v l _ s q = v l _ c . " 2 ; v 2 _ s q = v 2 _ c . " 2 ; v l v 2 = v l _ c . * v 2 _ c ; d i s p C n o w d e f i n i n g v l , v 2 , a n d v l v 2 ' ) y > y> . '/, C r e a t i n g a p p r o p r i a t e o u p u t f i l e s d i s p C n o w s t a r t i n g i t e r a t i o n s ' ) '/, c r e a t i n g t h e o u t p u t f i l e s f n a m e = s t r c a t ( ' e i g v e c _ M ' , n u m 2 s t r ( m ) , ' _ j m a x ' , n u m 2 s t r ( j m a x ) , . . . 71 Appendix B. Source Code ' _ b e t a _ Z 1 , n u m 2 s t r ( b e t a _ Z ) , ' _ . t x t ' ) ; f n a m e 2 = s t r c a t ( ' e i g e n e r g y _ M ' , n u m 2 s t r ( m ) , ' _ j m a x ' , n u m 2 s t r ( j m a x ) , . . . ' _ b e t a _ Z ' , n u m 2 s t r ( b e t a _ Z ) , ' _ . t x t ' ) ; f i d = f o p e n ( f n a m e , ' w + ' ) ; f i d 2 = f o p e n ( f n a m e 2 , ' « + ' ) ; • / f •/, '/. d e f i n i n g t h e m i d p o i n t s o f t r i a n g l e s p _ m i d = p d e i n t r p ( p , t , p ' ) ; t •/, '/. Now s t a r t i n g H a r t r e e - F o c k I t e r a t i o n s f o r i t e r = l : m a x i t e r , f l = 4 * p i * v l _ s q ; '/. RHS f o r e q u a t i o n d e t e r m i n i n g t h e d i r e c t i n t e r a c t i o n f 2 = 4 * p i * v 2 _ s q ; '/. RHS f o r e q u a t i o n d e t e r m i n i n g t h e d i r e c t i n t e r a c t i o n f 1 2 = - 4 * p i * v l v 2 ; 7. RHS f o r e q u a t i o n d e t e r m i n i n g t h e e x c h a n g e i n t e r a c t i o n . '/. Make n o t e o f t h e s i g n . d i s p C f l , f 2 a n d f l 2 d e f i n e d ' ) x t i m e s f l = f 1 . * p _ m i d ( l , : ) ; x t i m e s f 2 = f 2 . * p _ m i d ( l , : ) ; x t i m e s f l 2 = f 1 2 . * p _ m i d ( l , : ) ; c l e a r f l ; c l e a r f 2 ; c l e a r f l 2 ; ft d i s p ( ' x t i m e s f f u n c t i o n s d e f i n e d ' ) a _ c o e f f = s t r c a t ( n u m 2 s t r ( - l ) , ' . / x ' ) ; '/. N u m 2 s t r o f m d i s p C N o w f i n d i n g d i r e c t a n d e x c h a n g e i n t e r a c t i o n s . . . ' ) '/, F i n d i n g d i r e c t a n d e x c h a n g e i n t e r a c t i o n s 72 Appendix B. Source Code u l = a s s e m p d e ( b , p , e , t , ' x ' , 0 , x t i m e s f 2 ) ; u 2 = a s s e m p d e ( b , p , e , t , ' x ' , 0 , x t i m e s f 1 ) ; a l p h a _ e x c h = a s s e m p d e ( b , p , e ) t , ' - x ' , a _ c o e f f , x t i m e s f l 2 ) ; 7. Make n o t e o f t h e s i g n c l e a r x t i m e s f l ; c l e a r x t i m e s f 2 ; c l e a r x t i m e s f l 2 ; d i s p ( ' d i r e c t a n d e x c h a n g e i n t e r a c t i o n f o u n d ' ) 7. I n t e r p o l a t i n g t o f i n d v a l u e o f p o t e n t i a l s a t t r i a n g l e c e n t r e s u l _ c = p d e i n t r p ( p , t , u l ) 1 ; u 2 _ c = p d e i n t r p ( p , t , u 2 ) ' ; a l p h a _ e x c h _ c = p d e i n t r p ( p , t , a l p h a _ e x c h ) ' ; 7 . D e f i n i n g t h e c o e f f i c i e n t s f o r t h e 2 - D PDE p r o b l e m d = ' x ' ; c = ' x ' ; c l e a r a ; d i s p C n o w d e f i n i n g a ' ) a ( l , : ) = - 2 . 0 * p _ m i d ( l , : ) . / s q r t ( p _ m i d ( l , : ) . " 2 + p _ m i d ( 2 , : ) . " 2 ) + b e t a _ Z 2 * p . m i d . . . ( i , : ) . " 3 - 2 . 0 * b e t a _ Z * p _ m i d ( l , : ) + 2 * u l _ c ' . * p _ m i d ( l , : ) / Z ; a ( 4 , : ) = l . / p _ m i d ( l , : ) - 2 . 0 * b e t a . Z * p . m i d ( l , : ) - 2 . 0 * p _ m i d ( l , : ) . / s q r t ( p _ m i d ( l , : ) . " . . . 2 + p _ m i d ( 2 , : ) . " 2 ) + b e t a _ Z 2 * p _ m i d ( l , : ) . ~ 3 - 2 . 0 * b e t a _ Z * p _ m i d ( l , : ) . . . + 2 * u 2 _ c ' . * p _ m i d ( l , : ) / Z ; a ( 2 , : ) = - 2 . 0 * a l p h a _ e x c h _ c ' . * p _ m i d ( l , : ) / Z ; a ( 3 , : ) = a ( 2 , : ) ; d i s p C . . . a d e f i n e d . . . Now s o l v i n g t h e PDE s y s t e m . . . ' ) d i s p C S o l v i n g t h e e i g e n v a l u e p r o b l e m . . . ' ) [ v , l ] = p d e e i g ( b 2 , p , e , t , c , a , d , [ r m i n r m a x ] ) ; l l = l e n g t h ( v ( : , l ) ) / 2 ; v l ( l : l l ) = v ( l : l l , l ) ; v 2 ( l : l l ) = v ( l l + l : e n d , 2 ) ; 73 Appendix B. Source Code '/, N o r m a l i z i n g t h e w a v e f u n c t i o n s n o r m _ i n t = 4 * p i * p ( l , : ) ' . * v l ( : ) . " 2 ; n o r m = d b l q u a d ( @ i n t e g r a n d , 0 . 0 , x m a x , 0 . 0 , y m a x ) ; v l = v l / s q r t ( n o r m ) ; n o r m _ i n t = 4 * p i * p ( l , : ) * . * v 2 ( : ) . " 2 ; n o r m = d b l q u a d ( ® i n t e g r a n d , 0 . O . x m a x , 0 . 0 , y m a x ) ; v 2 = v 2 / s q r t ( n o r m ) ; i f ( i t e r < m a x i t e r ) '/. D e f i n i n g t h e RHS o f t h e P o i s s o n - l i k e e q u a t i o n s f o r f i n d i n g t h e '/, e l e c t r o n - e l e c t r o n i n t e r a c t i o n p o t e n t i a l s v l _ c = p d e i n t r p ( p , t , v l ) ; v 2 _ c = p d e i n t r p ( p , t , v 2 ) ; v l _ s q = v l _ c . " 2 ; v 2 _ s q = v 2 _ c . " 2 ; v l v 2 = v l _ c . * v 2 _ c ; d i s p ( ' L o o p i n g t h e l o o p . . . t h e l a s t i t e r a t i o n w a s : ' ) i t e r a t i o n = i t e r f o r i = l : l e n g t h ( l ) e i g v s i t e r ( i t e r , l ) = i t e r ; e i g v s i t e r ( i t e r , i + l ) = l ( i ) ; e n d e l s e f p r i n t f ( f i d , 7 . - 1 8 s ' / , - 1 8 s \ n ' , ' v l ' , ' v 2 ' ) ; f o r j = l : l e n g t h ( v l ) , f p r i n t f ( f i d , " / . - 1 8 . 1 2 f ' / . - 1 8 . 1 2 f \ n ' , v l ( j ) , v 2 ( j ) ) ; e n d d i s p ( ' C a l c u l a t i o n f i n i s h e d ' ) 1 e n d e n d 74 Appendix B. Source Code y '/,'/, Now f i n d i n g t h e d i r e c t a n d e x c h a n g e i n t e r a c t i o n e n e r g i e s . d i r e c p r d p s i l = 0 . 0 ; d i r e c p r d p s i 2 = 0 . 0 ; e x c h a p r d p s i = 0 . 0 ; n o r m _ i n t = 4 * p i * p ( l , : ) ' . * v l ( : ) . " 2 . * u l ( : ) ; d i r e c p r d p s i l = d b l q u a d O i n t e g r a n d , 0 . 0 , x m a x , 0 . 0 , y m a x ) d i s p C ' d i r e c p r d p s i l f o u n d ' ) n o r m _ i n t = 4 * p i * p ( l , : ) ' . * v 2 ( : ) , " 2 . * u 2 ( : ) ; d i r e c p r d p s i 2 = d b l q u a d ( 3 i n t e g r a n d , 0 . 0 , x m a x , 0 . 0 , y m a x ) d i s p ( ' d i r e c p r d p s i 2 f o u n d ' ) n o r m _ i n t = 4 * p i * p ( l , : ) ' . * v l ( : ) . * v 2 ( : ) . * a l p h a _ e x c h ( : ) ; e x c h a p r d p s i = d b l q u a d ( 8 i n t e g r a n d , 0 . 0 , x m a x , 0 . 0 , y m a x ) d i s p ( ' e x c h a p r d p s i f o u n d ' ) y '/, T o t a l H a r t r e e - F o c k E n e r g y y E _ H F = l ( l ) + l ( 2 ) - ( d i r e c p r d p s i l + d i r e c p r d p s i 2 - 2 * e x c h a p r d p s i ) / 2 y f p r i n t f ( f i d 2 , , , / . - 1 8 s , / . - 1 8 s , / . - 1 8 s ' / . - 1 8 s ' / . - 1 8 s , / . - 1 8 s \ n ' , ' e i g e n v a l l ' , ' e i g e n v a l 2 ' , ' E _ H F ' , ' d i r e c t l ' , . ' d i r e c t 2 ' , ' e x c h a n g e ' ) ; f p r i n t f ( f i d 2 , ' " / . - 1 8 . 1 2 f V . - 1 8 . 1 2 f V . - 1 8 . 1 2 f * / . - 1 8 . 1 2 f ' / . - 1 8 . 1 2 f ' / . - 1 8 . 1 2 f \ n ' , 1 ( 1 ) , 1 ( 2 ) , E . H F , . . . d i r e c p r d p s i l , d i r e c p r d p s i 2 , e x c h a p r d p s i ) ; y s t a t u s = f c l o s e ( f i d 2 ) ; s t a t u s = f c l o s e ( f i d ) ; y e n d y 75 Appendix B. Source Code E N D '/.pdeplot(p,e,t, 'xydata' ,v, 'mesh', ' o f f ' ) ; 76 

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