Analytic Evaluation of two-electron Atomic Integrals involving Extended Hylleraas-CI functions with STO basis

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1 Analytic Evaluation of two-electon Atomic Integals involving Extended Hylleaas-CI functions with STO basis B PADHY (Retd.) Faculty Membe Depatment of Physics, Khalikote (Autonomous) College, Behampu , Odisha bholanath.padhy@gmail.com Abstact : Some typical ovelap/potential enegy integals which occu in the use of extended Hylleaas-configuation inteaction (E-Hy-CI) functions with Slate-type obital (STO) basis fo two-electon atomic stuctue calculations, have been evaluated analytically. The coesponding kinetic enegy integals have been simplified fist by using fomulas deived fom Gauss divegence theoem in vecto calculus, and then expessed in tems of ovelap / potential enegy matix elements. Also closed-fom expessions fo such integals which aise in the application of Hylleaas-CI functions, and CI functions have been obtained as special cases, and the calculated values ae found to agee well with coect esults published by othe investigatos. Keywods : Exponentially coelated integals, E-Hy-CI calculations, Two-electon atoms. 1. Intoduction It is well-accepted that electon-electon coelations in a multielecton atom ae equied to be included in the quantum mechanical calculations in ode to obtain vey accuate wave functions and enegies fo the atom. The methods that have been employed fo this pupose ae (i) Hylleaas (Hy) method [1,], (ii) configuation-inteaction (CI) method [3], (iii) Hylleaas-CI (Hy-CI) method [4] and (iv) extended Hy-CI (E-Hy-CI) method [5]. The E-Hy-CI method is an extension of Hy-CI method in the sense that exponential coelations ae included in the fome. Details of the pogessive development of Hy-CI method and its applications can be obtained by going though the ecent papes [6-10] and the efeences theein. Fo knowledge about vaious calculations with E-Hy-CI method, the eade is advised to go though the ecent papes [5,10-14]. 1

2 In an ealie aticle [14], heeinafte efeed as pape I, kinetic enegy matix elements fo a two-electon atom have been expessed in tems ovelap/potential enegy integals employing E-Hy-CI method. In this communication, which is athe temed as extension to pape I, a method of analytic evaluation of ovelap/potential enegy integals involving E-Hy-CI functions has been outlined. As a esult, it became possible to obtain values of seveal kinetic enegy matix elements easily.. Definition of two-electon ovelap / potential enegy integals Assuming the nucleus to be at est, let 1 and be the position vectos of the two electons with espect to the nucleus. The distance between two electons is denoted as 1 1. With spheical pola coodinates, the following set of unnomalised atomic Slate-type obitals (STOs) is taken as the basis : na 1 aj a j j e Yl a, m j a a ( ) ( ); 0, (1) whee Y, ( j) is an othonomal spheical hamonic with its agument being la m a the angula coodinates of j, and is defined in [4,8]. The subscipt a signifies a paticula STO. The most geneal type of ovelap/potential enegy integals which aise in calculations with E-Hy-CI functions fo a two-electon atomic system ae of the fom. I (1) () 1 a b 1 e d (1) e() () HCI with 1and 0. Integals ( I ) coesponding to Hy-CI functions can be obtained fom Eq. (), as a special case, by taking 1 and 0 simultaneously. Similaly if 0 and 0 simultaneously, one gets integals CI ( I ) coesponding to CI functions only..1 Analytic evaluation of I Fom Eq. () it is obseved that one may come acoss the poduct of two spheical hamonics of the same agument in the integand. Such a poduct can be expanded as a seies containing seveal tems with each tem being a poduct of Clebsch-Godan coefficients and an individual spheical hamonic of the same agument [15]. Hence, expessions fo all the I integals can be obtained by paametic diffeentiation of a basic integal given by

3 whee b n11 11 * 1 1 l1, m ˆ 1 1 I d e Y ( ) H, n,,, l, m, (3) 1 1 n 1 e H 1, n,,, l, m d e Yl, m ( ˆ ), (4) with the eplacement of 1 and by a, b, etc. wheeve it is equied. It is to be pointed out hee that a method was outlined by Calais and Lowdin [16] fo evaluating such integals as in Eq. (3), by employing the method of otation of coodinate system, and subsequent tansfomation of spheical hamonics. In what follows, howeve, an altenative method of evaluation is employed which does not involve otation of coodinate system. Accodingly, the above integal has been simplified in the Appendix-A to obtain whee H (, n,,, l, m ) Y (, ) G (, n,, ), (5) 1 l, m 1 1 l 1 Gl (, n,, ) d e 1 0 n e 1 1 d 1 Pl 1 1 1, (6) fo which a closed-fom expession can be obtained, as well as fo H though Eq. (5). Inseting Eq.(5) in Eq.(3) Ib becomes n b l1, l m1, m 1 1 l 1 0 I d e G (, n,, ), (7) which can be easily evaluated analytically tem by tem using the fomula n ax ( n 1) dx x e n1. (8) a 0 Altenatively, evaluation of Ib can be done by witing 1 3

4 n11 11 n1 b l 1, l m1, m I d e d e 1 1 e 1 1 d1 Pl 1 1 1, (9) and following a method employed by Ruiz [17]. 3. Reduction of kinetic enegy integals to integable fom CI HCI HCI HCI Vaious kinetic enegy integals, such as K, K1, K, K 3,, 1,, 3 K K K K have been defined and evaluated analytically in pape I. Hence, it is felt that thee is no necessity of epeating those details hee. Howeve, simplification of two basic integals will be done hee by making use of Gauss divegence theoem in vecto calculus, and two elated fomulae, which have been deived in the Appendix-B. The following two integals HCI 1 1 a b 1 d e 1 K (, ) (1) () (1) () ;, 0 (10) w1 1 w1 a b 1 d e K ( w, w) (1) () e (1) () e ; w, w 0 (11) have been educed to integable fom as given in the Appendix-C and Appendix-D, espectively. The simplified expessions ae HCI 1 * * K (, ) d b() e () d1 a(1) d (1) 1 * * 1 d (1) 1a (1) 1 a (1) 1d (1), [1] 1 * K ( w, w) d b() e() d 1 e ( ww) 1 * w * w * a (1) d (1) ww d (1) 1a (1) a (1) 1d (1). w w w w The integal in Eq. (10) has also been simplified by Ruiz [17]. [13] 4

5 The method followed hee fo simplification of the kinetic enegy integals has been known as Kolos-Roothan tansfomation method [18] in the liteatue. Stating with Eqs. (10) and (1), definition of integals and espective CI HCI HCI integable expessions fo K, K1, K and K3 HCI can be obtained by choosing suitable values of and. By paametic diffeentiation of Eqs. (11) and (13) with espect to w and/o w, the definition of integals K1, K and K3 epoduced. as well as thei integable expessions as given in pape I can be 3.1 Analytic evaluation of kinetic enegy integals Expessing in spheical pola coodinates and employing some elated fomulae as given in the Appendix-B, it is staightfowad to deive the following equations [8]: l n l n 1 n 1 * a a a a a a a * 1 a(1) (1), a 1 1 l n l n 1 n 1 d d d d d d d 1 d(1) (1). d 1 1 Inseting Eqs. (14) and (15) in vaious equations containing integable expessions, each of the kinetic enegy integals can be expessed as a combination of seveal ovelap/potential enegy integals evaluated analytically in Appendix-A. Thus all the kinetic enegy integals ae evaluated analytically. The closed-fom expessions have been given in pape-i. Hence, it is not necessay to epeat the expessions hee. (14) (15) The values of seveal kinetic enegy integals have been calculated and obseved to be in close ageement with those epoted by Ruiz [17] and by Hais [9]. 4. Conclusion The matix elements fo the kinetic enegy integals fo a two-electon atom coesponding to CI, Hy-CI and E-Hy-CI functions could be expessed in tems of the matix elements fo the ovelap/potential enegy integals fo which closedfom expessions have been obtained hee. Calculated values of vaious such 5

6 integals will be displayed in the fom of tables and compaed with those published by othe investigatos [9,17] in a futue publication. Acknowledgements The autho is highly indebted to D. J.S. Sims and D. M.B. Ruiz fo peiodic discussions elating to this wok though exchange of s. Also, I am vey much gateful to Pof. N. Baik fo seveal useful suggestions, and to D. P.K. Panda fo help duing computation and fo citically going though the manuscipt. Appendix A: Equation (5) established In this Appendix, a closed-fom expession fo the following integal H will be deived: n1 ( ˆ ) H 1 n l m d e Yl, m 1 e,,,,,. (A1) To evaluate this integal spheical pola coodinates ae employed, in which the volume element d is eplaced by witten as Hee 1 d sin d d, and hence Eq. (A1) is 1 n 1 e d e d d Yl, m H= sin (, ). (A) ˆ1 ˆ ( cos ) ; cos ( ). (A3) 1 It is clea that e / 1 is a function of 1,, and cos 1. It is also known that the Legende polynomials PL(cos 1) of degee L = 0,1,,, etc. constitute a complete set of othogonal polynomials satisfying the following othogonality condition : P L (cos 1 ) P L (cos 1 )sin 1 d1,. L 1 L L (A4) 0 6

7 Theefoe, 1 / 1 can be expanded in the fom of the following seies : 1 e e a (,, ) P (cos ), (A5) 1 L0 L 1 L 1 whee alsae the coefficients in the expansion to be detemined. Multiplying both sides of Eq. (A5) by PL (cos 1 )sin 1 d1, then integating (0 ), and employing Eq. (A4), it is easy to obtain ove 1 1 L 1 e a (,, ) P (cos )sin d. (A6) L 1 1 L Inseting Eq. (A5) in Eq. (A), then employing the spheical hamonic addition theoem L 4 * L 1 L, M 1 1 L, M L 1ML P (cos ) Y (, ) Y (, ), (A7) and making use of the othonomality popety of the spheical hamonics, the angula integation in Eq. (A) is done to get whee H (, n,,, l, m ) Y (, ) G (, n,, ), (A8) 1 l, m 1 1 l 1 4 n 1 l 1 l 1 l 1 0 G (, n,, ) d e a (,, ). (A9) Inseting Eq. (A6) in Eq. (A9), one obtains 1 n 1 e l l G d e P (cos )sin d, (A10) which indicates that Gl depends on l but not on m, though the pesence of the Legende polynomial in the integand. To obtain a closed-fom expession fo Gl, and hence fo H though Eq. (A8), change of vaiable fom 1 to 1 is made using Eq. (A3). Thus cos (A11) 1 7

8 and 8 d sin d1 (A1) 1 Incopoating these changes in Eq. (A10), the following integable expession fo is obtained: Gl Gl (, n,, ) d e 1 0 n e 1 1 d 1 Pl (A13) Fo a paticula value of l, the Legende polynomial P (cos ) is eplaced by a l 1 function of 1, and 1 using Eq. (A11). Then integation in Eq. (A13) is pefomed easily to obtain a closed-fom expession fo G l. Thus the integal H as defined in Eq. (A1) is analytically evaluated using Eq. (A8). Appendix B: Some peliminaies elating to The opeato nabla, also called del, and atled (invese of delta), is a vecto diffeential opeato defined in Catesian coodinates by the elation ˆ ˆ i j kˆ, x y z (B1) and is widely used in vecto calculus. Only some peliminaies, which ae useful fo the pesent investigation, ae outlined below. Let and be two scala point functions, and F be a vecto point function, which ae, in geneal, complex well-behaved functions vanishing at infinity. The Gauss divegence theoem in vecto calculus is mathematically stated as ( F) dv F d s (B) V S with the condition that V is the volume enclosed by the closed suface S. If the volume integal on the left hand side of Eq. (B) is evaluated ove the entie space, then the suface integal on the ight hand side of Eq.(B) vanishes

9 because F vanishes at each point on the suface enclosing the infinite volume as pe supposition. This conclusion will be used in the following paagaph. Using the vecto identity ( F) ( ) F ( F), ( B3) and then integating both sides of Eq.(B3) ove the whole space, the integal on the left hand side is educed to a suface integal as pe Eq. (B), and vanishes at infinity since the poduct F 0 at each point on the infinite suface as pe supposition. Hence the following elation is established: ( ) F dv ( F) dv, (B4) with the integation to be done ove the entie space. Similaly, integating both sides of the following identity * * * ( ) ( ) ( ) ( ), (B5) ove the whole space, then employing Eq. (B) and noting that point on the infinite suface, the following equation is obtained: * 0 at each * * ( ) dv ( ) ( ) dv. (B6) In paticula, if, then Eq.(B6) becomes * dv ( ) dv. (B7) Anothe popety of the nabla opeato is that it is antihemitian, which follows fom the definition of adjoint of an opeato, and a Hemitian opeato. Since the linea momentum opeato ˆp i, that is, the nabla opeato is antihemitian. In spheical pola coodinates (,, ) is Hemitian, it follows that ˆ 1 1 ˆ ˆ, sin (B8) and 9

10 Lˆ, (B9) whee ˆL is the opeato coesponding to the squae of the angula momentum and is given by ˆ 1 1 sin. L sin sin The following ae few established equations: (B10) (B14) ˆ m m l (, ) ( 1) l (, ), 0,1,, etc. ; L Y l l Y l (B11) ˆ m m L Y (, ) m Y (, ), m l, l 1,...,0, 1,,..., l ; (B1) z l l m m Yl l l Yl (, ) ( 1) (, ) / ; (B13) f ( ) f ( ) f ( ) ˆ ; ˆ ; (B15) ˆ. (B16) Appendix-C : Equation (1) established The geneal kinetic enegy integal to be evaluated in Hy-CI calculations, as defined by Eq. (10) in the text, is 1 1 a b 1 d e 1 HCI K (, ) (1) () (1) (), (C1) whee, 0(integes). The above equation can be ewitten as HCI * a b e K (, ) d () () K (,, ), (C) 10

11 whee 1 1 a 1 d 1 a K (,, ) (1) (1). (C3) Since and 1 1, Eq. (B6) is employed to get * a d a 1 K (,, ) d 1 1 ( 1 (1)) 1( (1) 1 ). (C4) Expanding the gadient of the scala point functions within each pai of culy backets and then taking the scala poduct, one obtains a 1 K (,, ) d ( ) ( ) * * a 1 d a d 1 v1 * * 1 ˆ 1 d 1 a a 1 d ( ) ( ) ( ), (C5) whee ˆ 1 1 / 1. Also a * (1) and d (1) ae eplaced by * a and d, espectively, fo convenience. Making use of the elation ˆ 1 1, (C6) the integal coesponding to the thid tem of the integand in Eq. (C5) can be simplified as pe Eq. (B4). Consequently, expanding the divegence of the vecto point function in the esulting integand and adding with the fist two tems of the integand in Eq. (C5), the following expession fo a 1 K (,, ) d (1) (1) * 1 a d 1 a K is obtained. * * 1 d ( 1 a ) 1 a ( 1 d ). (C7) Inseting Eq. (C7) in Eq. (C), the elation given in Eq.(1) in the text is established. Appendix D : Equation (13) established As pe Eq. (11) in the text, the integal K ( w, w ) is given by K ( w, w) (1) () e (1) () e, (D1) w1 1 w1 a b 1 d e 11

12 whee w and w ae the exponential paametes with ww, 0. The integal can be ecast as * K d () () K ( w, w, ), (D) whee b e a K (1) e (1) e. (D3) w1 w1 a a 1 d 1 Since and, Eq. (D3) becomes * w1 w1 K 1 a d 1 1( a(1) e ) 1( d (1) e ). (D4) Expanding the gadient of the scala point function within each pai of the culy backets, the scala poduct of the esulting vecto point functions is obtained. Taking note that 1 ˆ 1 1, Eq. (D4) is simplified to get 1 ( ww ) 1 K * * a d 1 e ( 1 a(1)) ( 1 d (1)) a(1) d (1) ww Since ( ww) 1 ( ww) e 1 e ˆ 1 1, ( ww) * * ˆ 1 wd (1)( 1 a (1)) w a (1)( 1 d (1)). (D5) (D6) the thid tem of the integal, denoted as T 3, in Eq. (D5) becomes ( ww) 1 1 e T3 d 1 1 ( ww) * * w d 1 a w a 1 d which is futhe simplified by using Eq. (B4). Thus T (1) (1) (1) (1), (D7) 1 ( ww) 1 * * 3 d 1e ( w w) w 1 d 1 a w a 1 d (1) (1) (1) (1). (D8) Expanding the divegence of the sum of the two vecto point functions within the culy backets in the integand in Eq. (D8), and then inseting the esulting expession in Eq. (D5), some cancellations ae made to obtain 1

13 1 ( ww ) 1 K * a d 1 e a(1) d (1) ww w * w * d (1) 1a (1) a (1) 1d (1). w w w w (D9) Inseting Eq. (D9) in Eq. (D), one gets 1 * K ( w, w) d b() e() d 1e ( ww) 1 * w * w * a (1) d (1) ww d (1) 1a (1) a(1) 1d (1), w w w w which is the equied Eq. (13) in the text. (D10) Refeences [1] EA Hylleaas, Adv. Quantum Chem. 1, 1(1964) [] FW King, Recent Advances in Computational Chemisty, Molecula Integals ove Slate Obitals, edited by T Ozdogan and MB Ruiz (Tanswold, Keala, India), pp (008) [3] JN Silveman and GH Bigman, Rev. Mod. Phys. 39, 8 (1967) [4] JS Sims and SA Hagstom, J. Chem. Phys. 55, 4699 (1971) [5] C Wang, P Mei, Y Kuokawa, H Nakashima and H Nakatsuji, Phys. Rev. A 85, 0451 (01) [6] JS Sims and SA Hagstom, J. Phys. B: At. Mol. Opt. Phys. 48, (015) [7] MB Ruiz, J. Math. Chem. 54, 1083 (016) [8] FE Hais, J. Chem. Phys. 144, (016); eatum 145, (016) [9] FE Hais, Mol. Phys. 115 (17-18), 048 (017) [10] FE Hais, Adv. Quantum Chem. 76, 187 (017) [11] B Padhy, Asian J. Spectosc. (special Issue) pp (01); axiv:

14 [1] FW King, J. Phys. B: At. Mol. Opt. Phys. 49, (016) [13] B Padhy, axiv: [14] B Padhy, axiv: ; Oissa J. Phys. 5 (1), 9 (018) [15] ME Rose, Elementay Theoy of Angula Momentum, Dove Publications, INC. New Yok (1957), pp.61. [16] JL Calais and PO Lowdin, J. Mol. Spectosc. 8, 03 (196) [17] MB Ruiz, J. Math. Chem. 49, 457 (011) [18] W Kolos and CCJ Roothan, Rev. Mod. Phys. 3, 19 (1960) 14