Application of the Coulomb spheroidal basis for diatomic molecular - PowerPoint PPT Presentation

Application of the Coulomb spheroidal basis for diatomic molecular calculations T . Kereselidze and G. Chkadua Department of Exact and Natural Sciences, Tbilisi State University, 0179 Tbilisi, Georgia

Content: 1. Introduction 2. The Coulomb spheroidal wave functions 3. Basic equations E i 4. Obtained results and comparison with the characteristics of the hydrogen molecular ion H 2 + 5. Conclusion

S ym m e t rical d iat om ic m ol e cu l e S฀ + + H , H , C , N , O H , H , C , N , O 2 2 2 2 2 2 2 2 2 2 ( ) Heitler and London LCAO Hund and Mulliken

+ Hydrogen molecular ion H 2  E  r r b a R Z A =1 Z B =1

Prolate spheroidal coordinate system + − r r r r ξ = η = ϕ = a b a b , , arctg y x ( / ) R R ≤ ξ < ∞ − ≤ η ≤ ≤ ϕ < π 1 , 1 1 0 2

Schrödinger equation for hydrogen molecular ion   1 1 1 − ∆ − − Ψ ± = ε ± Ψ ± ( ) ( ) ( )   ( R )   2 r r a b − ϕ im e ± ± Ψ = ξ η ( ) ( ) X ( , R Y ) ( , R ) n n m n m n m π ξ ξ η η 2  ±  ε ( ) ( ) 2 2 d dX R m ξ − + λ + ξ + ξ − ξ = 2 2   1 2 ( , ) 0 R X R ξ ξ ξ − 2 d d  2  1 ±  ±  ε ( ) ( ) ( ) 2 2 d dY R m ± − η + − − λ η − η = 2 2 ( )   1 Y ( , ) R 0 η η − η 2 d d  2  1

+ Electornic energies of H 2

H + HE GR OUND AND FIR M S OF STEXCIT ED T ER 2

฀ i st OF P UBL ICATION : 1 T ฀ K e re Se l id ze ฀ Z ฀ S ฀ IA NI A ND G ฀ c h Kad u a฀ ฀ P HYS ฀ J ฀ D 6 3 8 1 ฀ 2 0 1 1 ฀ M A CHA VA R E UR 2 ฀ J ฀ M ฀ P e e K฀ T ฀ K e re Se l id ze ฀ I ฀ N oSe l id ze A ND J ฀ K ie l KoP f฀ J ฀ P HYS ฀ B ฀ A T ฀ m ol ฀ o P t ฀ P HYS ฀ 4 0 ฀ 5 6 5 ฀ 2 0 0 7 ฀ 3฀ a ฀ d e vd ariaN i฀ t฀ m ฀ K e re Se l id ze ฀ i฀ l ฀ N oSe l id ze ฀ e ฀ d al im ie r฀ P ฀ S au vaN ฀ P ฀ a N ge l o฀ aN d r ฀ S cot t ฀ P h yS฀ r e v฀ v฀ a 71฀ P ฀ 022512 ฀ 2005฀ 4฀ t฀ m ฀ K e re Se l id ze ฀ i฀ l ฀ N oSe l id ze aN d m ฀ i฀ c h ib iSov฀ J ฀ P h yS฀ b ฀ a t ฀ m ol ฀ o P t ฀ P h yS฀ 36 853 ฀ 2003฀ 5฀ a ฀ z ฀ d e vd ariaN i฀ t฀ m ฀ K e re Se l id ze aN d i฀ l ฀ N oSe l id ze ฀ K h im ich e SKaia P h ySica v฀ 22 P ฀ 3 ฀ 2003฀ 6฀ t฀ m ฀ K e re Se l id ze ฀ z ฀ S ฀ m ach avariaN i aN d i฀ l ฀ N oSe l id ze ฀ J ฀ P h yS฀ b ฀ a t ฀ m ol ฀ o P t ฀ P h yS฀ 31฀ 15 ฀ 1998฀ 7฀ t฀ m ฀ K e re Se l id ze ฀ z ฀ S ฀ m ach avariaN i aN d i฀ l ฀ N oSe l id ze ฀ J ฀ P h yS฀ b ฀ a t ฀ m ol ฀ o P t ฀ P h yS฀ 29฀ 257 ฀ 1996฀ 8฀ t฀ m ฀ K e re Se l id ze ฀ h ฀ a ฀ m ou rad aN d m ฀ f ฀ t zu l u Kid ze ฀ J ฀ P h yS ฀ b ฀ a t ฀ m ol ฀ o P t ฀ P h yS฀ 25฀ 2957 ฀ 1992฀ 9฀ t฀ m ฀ K e re Se l id ze ฀ S ov฀ P h yS฀ J e t f 100฀ 95 ฀ 1991฀ 10฀ m ฀ i฀ c h ib iSov aN d t฀ m ฀ K e re Se l id ze ฀ P re P riN t ia e ฀ 5410฀ 6฀ m oScow ฀ P ฀ 1฀ 43 ฀ 1991฀ 11฀ t฀ m ฀ K e re Se l id ze P roce e d iN g of g e orgiaN a cad e m y of S cie N ce S ฀ v฀ 139฀ P ฀ 481 ฀ 1990฀ 12฀ a ฀ z ฀ d e vd ariaN i฀ t฀ m ฀ K e re Se l id ze aN d a ฀ l ฀ z agre b iN ฀ J ฀ P h yS฀ b ฀ a t ฀ m ol ฀ o P t ฀ P h yS฀ 23฀ 2457 ฀ 1990฀ 13T.฀. ฀ er esel i dze, J. ฀ hys. ฀ : ฀ t . ฀o l . ฀ hys. 20, 1891 (1987) 14฀ t฀ m ฀ K e re Se l id ze aN d b ฀ i฀ K iKiaN i฀ S ov฀ P h yS฀ J e t f 87฀ 741 ฀ 1984฀ 15฀ t฀ m ฀ K e re Se l id ze ฀ S ov฀ P h yS฀ J e t f 69฀ 67 ฀ 1975฀ 16฀ t฀ m ฀ K e re Se l id ze aN d o ฀ b ฀ f irSov฀ S ov฀ P h yS฀ J e t f 65฀ 98 ฀ 1973฀

Schrödinger equation for hydrogen-like ion   Z 1 − ∆ − Ψ = ε Ψ  a b ,  a b , a b , a b , ( R )   2 r   a b , − ϕ im e Ψ = ξ η a b , a b , X ( , R Y ) ( , R ) n n m n m n m π ξ η ξ η 2   ε ( ) 2 2 d dX R m ξ − + λ + ξ + ξ − ξ = 2 2   1 ( , ) 0 ZR X R ξ ξ ξ − 2 d d  2  1   ε ( ) a b , 2 2 d dY R m − η + − − λ η η − η = 2 2 a b ,    1 ZR Y ( , ) R 0 η η − η 2 d d  2  1 = + + + n n n m 1 ξ η

The Coulomb spheroidal wave functions = + + = n m 1, n n 0; 1 ฀ ξ η − ξ = ξ ZR / 2 n X e W ( ) 0 m = η η  a b , ZR / 2 n Y e W ( ) 0 m = + + = n m 2, 2 ฀ n n 1,2; ξ η   nh = − ξ ξ − ξ 1,2 ZR / 2 n   X e W ( ) 0 m m ,1   ZR   nh = η η η  , / 2 1,2 a b ZR n    Y e W ( ) 1 ,0 m m   ZR

= + + = n m 3, n n 1,2,3; 3 ฀ ξ η   2 nh n h − ξ = ξ − ξ + − − − ξ ZR / 2 n  2 1,2,3 1,2,3  X e ( h 2 m 4) 1 W ( )   0 m m ,1 ,2 m 1,2,3 2 2  ZR 2 Z R    2 nh n h η = η η + − − − η  a b , ZR / 2 n  2 1,2,3 1,2,3   Y e ( h 2 m 4) 1 W ( )   2 m m ,1 ,0 m 1,2,3 2 2  ZR 2 Z R  W ξ = ξ − W η = − η 2 1) m / 2 2 ) m / 2 ( ) ( ( ) (1 = + + + n n n m 1 ξ η

The basic equations { } { } ξ η − ϕ ξ η − ϕ a im a im X ( , ) R Y ( , ) R e X ( , ) R Y ( , ) R e nn m nn m nn m nn m ξ η ξ η ± = η ± η ( ) a b ( , ) ( , ) Y Y R Y R nn m nn m nn m η η η ± ± ϕ Ψ ξ η ϕ = Φ ξ η ( ) ( ) im ( , , , R ) ( , , R e ) ∞ − − = ∑ ∑ n m 1 ± ± ± Φ ξ η ( ) ( ) ( ) C ( ) R X ( , R Y ) ( , R ) nn m nn m nn m η ξ η = = n 1 n 0 η

The basic equations ∞ − − ( ) n m 1   ∑ ∑ ± ± ± ± ε − + = ( ) ( ) ( ) ( ) E U V C 0     ฀฀ ฀฀ n nn η n n , nn n n , nn η η η η = = n 1 n 0 η ( ) ε ± ± ± − + = ( ) ( ) ( ) E U V 0 ฀฀ ฀฀ n n n , nn n n , nn η η η η ± = 〈 ξ 〉〈ϒ ± ϒ ± 〉 − 〈 〉 〈ϒ ± η ϒ ± 〉 ( ) 2 ( ) ( ) ( ) 2 ( ) U X X X X ฀฀ ฀฀ ฀฀ ฀฀ ฀฀ nn nn nn nn n n , n n n n ξ n n η n n ξ n n η η η ξ η ξ η 2 (2  ± = − 〈 ξ 〉〈ϒ ± ϒ ± 〉 ( ) ( ) ( ) V Z ) X X   ฀฀ ฀฀ ฀฀ nn nn n n , n n n n ξ n n η R η η ξ η  ± ± + 〈 〉 〈ϒ η ϒ 〉  ( ) ( ) ( ) Z X X d   ฀฀ ฀฀ nn nn nn ξ η η n n n n ξ η = − 2 2 E Z / 2 n n

1 s σ e l e ct roN ic e N e rgie S for St at e ε + ( ) ( ) R au σ 1 s ฀ xac t VAL UES Z = ≠ ≠ R au 1 Z 1 Z 1 0.25 -1.4754 -1.8980 -1.8986 -1.8981 0.5 -1.4213 -1.7318 -1.7319 -1.7350 0.75 -1.3560 -1.5753 -1.5824 -1.5757 1.0 -1.2884 -1.4410 -1.4418 -1.4518 1.25 -1.2230 -1.3283 -1.3418 -1.3295 1.50 -1.1617 -1.2338 -1.2490 -1.2353 1.75 -1.1053 -1.1541 -1.1559 -1.1701 2.0 -1.0538 -1.0865 -1.1026 -1.0885 2.25 -1.0071 -1.0286 -1.0444 -1.0307 2.5 -0.9648 -0.9788 -0.9808 -0.9938 2.75 -0.9267 -0.9355 -0.9497 -0.9376 3.0 -0.8924 -0.8978 -0.9109 -0.8998 3.5 -0.8339 -0.8357 -0.8375 -0.8466 4.0 -0.7869 -0.7873 -0.7961 -0.7898 4.5 -0.7493 -0.7493 -0.7562 -0.7506 5.0 -0.7192 -0.7192 -0.7202 -0.7244 6.0 -0.6757 -0.6757 -0.6786 -0.6763 7.0 -0.6469 -0.6469 -0.6485 -0.6472 8.0 -0.6267 -0.6267 -0.6269 -0.6276 9.0 -0.6118 -0.6118 -0.6123 -0.6119 10.0 -0.6003 -0.6003 -0.6006 -0.6004 12.0 -0.5834 -0.5834 -0.5834 -0.5835 16.0 -0.5625 -0.5625 -0.5625 -0.5625 20.0 -0.5500 -0.5500 -0.5500 -0.5500

2 p π e l e ct roN ic e N e rgie S for St at e ε + ( ) ( ) R au π 2 p ฀ xac t VAL UES Z = ≠ ≠ 1 1 1 Z Z R au 0.25 -0.3746 -0.4980 -0.4880 -0.4980 0.5 -0.3735 -0.4923 -0.4923 -0.4923 0.75 -0.3716 -0.4841 -0.4839 -0.4839 1.0 -0.3692 -0.4736 -0.4737 -0.4741 1.25 -0.3662 -0.4631 -0.4622 -0.4623 1.50 -0.3627 -0.4517 -0.4503 -0.4504 1.75 -0.3589 -0.4380 -0.4382 -0.4402 2.0 -0.3548 -0.4259 -0.4288 -0.4261 2.25 -0.3504 -0.4176 -0.4140 -0.4143 2.5 -0.3458 -0.4025 -0.4029 -0.4068 2.75 -0.3411 -0.3964 -0.3914 -0.3919 3.0 -0.3363 -0.3808 -0.3864 -0.3814 3.5 -0.3266 -0.3610 -0.3619 -0.3678 4.0 -0.3169 -0.3508 -0.3432 -0.3443 4.5 -0.3073 -0.3354 -0.3272 -0.3285 5.0 -0.2979 -0.3129 -0.3143 -0.3214 6.0 -0.2803 -0.2970 -0.2884 -0.2899 7.0 -0.2642 -0.2766 -0.2684 -0.2699 8.0 -0.2499 -0.2519 -0.2534 -0.2595 9.0 -0.2374 -0.2450 -0.2383 -0.2397 10.0 -0.2265 -0.2327 -0.2269 -0.2282 12.0 -0.2092 -0.2092 -0.2102 -0.2133 16.0 -0.1871 -0.1888 -0.1871 -0.1876 20.0 -0.1745 -0.1745 -0.1747 -0.1751