This shoud be true for any, so the equations are equivaent to the Schrodingerequation (H E) : () Another important property of the (2) functiona is, t

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1 I. Variationa method Introduction Usefu for the determination of energies and wavefunctions of different atomic states. H the time-independent Hamitonian of the system E n eigenenergies Ψ n - eigenfunctions Schrödinger-equation HΨ n E n Ψ n () -wavefunction with finite norm E[] functiona E[] hjhji hji : (2) If Ψ n than E[] E n eigenstates. We prove, that if Ψ n + ; (3) than We make the variation on (2) E : (4) E[] Λ Hdfi + Λ Hdfi Λ dfi Λ dfi + Λ dfi Λ Hdfi Λ dfi 2 ; (5) where we have used (H)H. Using (2) and making E one obtains Λ (H E)dfi + Λ (H E)dfi : (6) Making! i we get i Λ (H E)dfi + i Λ (H E)dfi : (7) Using the above two equations, we can write Λ (H E)dfi (8) Λ (H E)dfi : (9)

2 This shoud be true for any, so the equations are equivaent to the Schrodingerequation (H E) : () Another important property of the (2) functiona is, that gives a superior imit for the ground-state energy E of the system. We expand in terms of the Ψ n eigenfunctions of the Hamitonian H. X n a n Ψ n : () Introducing this expansion to (2) E[] n aλ n a nhψ n jhjψ n i n aλ n a nhψ n jψ n i n ja nj 2 E n n ja nj 2 ; (2) where we have usedhψ n E n Ψ n. Subtracting the ground-state energy E, we get n E[] E ja nj 2 (E n E ) n ja nj 2 : (3) Using E n E, the right side of the equation is nonnegative, so E» E[]: (4) In practice we perform the variation ony for a cass of functions. E.g.: ayeigh itz method In this case the tria function (ff i ;fi) depends on some parameters, so E E(ff i ); i ;n; (5) If we make the variation on the cass of of the tria functions, the E condition is i ; i ;n (6) We can use this method aso for the excited states by imposing the orthogonaity of to a states with ower energy hjψ n i ; n ;i ; (7) 2 The ayeigh itz method for the ground state of the heium 2. Simpe variationa method We appy the independent eectron approximation product wavefunction (r ;r 2 )ψ s (r )ψ s (r 2 ); (8) 2

3 Eachs wavefunction is a hydrogenike function with parameter the effective charge ff ff 3 2 ψ s (r) e ffr : (9) ß The energy functiona: The Hamitonian: where 2 is the atomic number 2.. Cacuating the matrix eement E[] hjhji: (2) H r2 2 r2 2 2 r r 2 + ; (2) hψ i (r )ψ i(r 2 )jhjψ j (r )ψ j(r 2 )i Let ψ i and ψ j being one-eectron normaized wavefunction cacuated in a spherica potentia. In this case ψ i (r) i (r)y imi( ; ') (22) ψ j (r) j (r)y j mj ( ; '); where i (r) and j (r) are radia wavefunctions. The matrix eement can be expressed as hψ i (r )ψ i(r 2 )jhjψ j (r )ψ j(r 2 )i hψ i (r )ψ i(r 2 )j r2 2 jψ j(r )ψ j(r 2 )i + hψ i (r )ψ i(r 2 )j r2 2 2 jψ j(r )ψ j(r 2 )i hψ i (r )ψ i(r 2 )j r jψ j (r )ψ j(r 2 )i hψ i (r )ψ i(r 2 )j r 2 jψ j (r )ψ j(r 2 )i +hψ i (r )ψ i(r 2 )j jψ j (r )ψ j(r 2 )i: (23) In the first 4 terms the operator acts ony on the wavefunction of one eectron, so we can separate the integras for the coordinates of the 2 eectrons hψ i (r )ψ i(r 2 )jhjψ j (r )ψ j(r 2 )i hψ i (r )j r2 2 jψ j(r )i i j + hψ i(r 2 )j r2 2 2 jψ j(r 2 )i ij hψ i (r )j jψ j (r )i i j r hψ i(r 2 )j jψ r j(r 2 )i ij 2 +hψ i (r )ψ i(r 2 )j jψ j (r )ψ j(r 2 )i: (24) 3

4 The integras containing, k ; 2 can be easiy cacuated hψ i ( )j jψ j ( )i r 2 k d Λ i ( ) j ( ) d ^ im i ( ^ )Y jm j ( ^ ) r 2 k d Λ i ( ) j ( ) i j mim j : (25) In genera numerica integration is needed for the cacuation of the radia matrix eements. If the wavefunction is cacuated in a ff Couomb-potentia and i j, than we can cacuate anayticay r 2 k d Λ i ( ) i ( ) ff n 2 (26) The matrix eements of the kinetic energy operator r 2 k 2 can be cacuated directy, orifψ i is eigenfunction of r 2 k 2+V () with eigenvaues E i then hψ i ( )j r2 k 2 jψ j( )i hψ i ( )j r2 k 2 + V ()jψ j ( )i hψ i ( )jv ( )jψ j ( )i hψ i ( )je j jψ j ( )i hψ i ( )jv ( )jψ j ( )i E j ij hψ i ( )jv ( )jψ j ( )i: (27) If the potentia is Couombian ff ff 2 2n 2, and then the eigenvaue of the energy is hψ i ( )j r2 k 2 jψ j( )i ff2 2n 2 ij + ffhψ i ( )j jψ j ( )i: (28) j The matrix eement of we expand the potentia in terms of Legendre poynomias X jr r 2 j r< + (cos ); (29) mutipoe expansion We express (cos ) in terms of spherica harmonics (cos ) X 4ß 2 + X m X 4ß r< 2 +r + > m m(^r )Y m (^r 2 ): (3) m(^r )Y m (^r 2 ): (3) 4

5 Further hψ i (r )ψi(r 2 )j jψ j (r )ψ r j(r 2 )i 2 X 4ß 2 + dr r 2 Λ i (r ) j (r ) dr 2 r 2 2 Λ i (r 2 ) r < + j(r 2 ) X m im i (^r ) m(^r )Y j m j (^r )d^r i m i(^r 2 )Y m (^r 2 )Y The integra of the product of 3 spherica harmonics is am a (^r)y b m b (^r)y cm c (^r)d^r s j m j (^r 2 )d^r 2 : (32) (2 b +)(2 c +) C a 4ß(2 a +) b c Cama b m b cmc; (33) C Cebsch Gordan coecient In order to have nonzero terms a shoud be the vectoria sum of b and c, meaning that and C a b c is nonzero ony for even a + b + c The matrix eement of wi be nonzero if In these cases we obtain hψ i (r )ψ i(r 2 )j jψ j (r )ψ j(r 2 )i X minfi+j; i jg + maxfji jj;j jg i j s (2 i + )(2 j +) (2 j +)(2 i +)C j i C i m a m b + m c ; (34) j a c j» b» a + c (35) m j m i m i m j (36) j i jj» i + j (37) j i j j» i + j (38) i + j + i + j even (39) dr r 2 Λ i (r ) j (r ) dr 2 r 2 2 Λ i (r 2 ) r < + j(r 2 ) X j m im jm i m j C jm j ;m j m i; im i C i m i m j m i j m j : (4) 5