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

SLIDE 1

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

SLIDE 2

i

E

Content:

1. Introduction
2. The Coulomb spheroidal wave functions
3. Basic equations
4. Obtained results and comparison with the characteristics
f the hydrogen molecular ion H2

+

5. Conclusion

SLIDE 3

S ym m e t rical d iat

m ic

m ol e cu l e S฀

2 2 2 2 2

, , , , H H C N O

+ 2 2 2 2 2

, , , , H H C N O

+ ( )

Heitler and London LCAO

Hund and Mulliken

SLIDE 4

a

r 

b

r 

Hydrogen molecular ion H 2

+

ZA=1 ZB=1

E

R

SLIDE 5

Prolate spheroidal coordinate system

, , ( / )

a b a b

r r r r arctg y x R R ξ η ϕ + − = = =

1 , 1 1 2 ξ η ϕ π ≤ < ∞ − ≤ ≤ ≤ <

SLIDE 6

Schrödinger equation for hydrogen molecular ion

( ) ( ) ( )

1 1 1 ( ) 2

a b

R r r ε

± ± ±

  − ∆ − − Ψ = Ψ    

( ) ( )

( , ) ( , ) 2

im n m n n m n m

e X R Y R

ξ ξ η η

ϕ

ξ η π

− ± ±

Ψ =

( ) ( )

( ) 2 2 2 2 2 ( ) ( ) 2 2 2 2 ( ) 2

1 2 ( , ) 2 1 1 ( , ) 2 1 d dX R m R X R d d d dY R m Y R d d ε ξ λ ξ ξ ξ ξ ξ ξ ε η λ η η η η η

± ± ± ±

  − + + + − =   −     − + − − − =   −  

SLIDE 7

Electornic energies of H 2

+

SLIDE 8

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

2

H +

SLIDE 9

฀ i st OF P UBL

ICATION: 1 T฀ K e re Se l

id ze ฀

Z ฀ S ฀ M A CHA VA R IA NI A ND G ฀ c h Kad u a฀ E UR

฀

P HYS฀ J ฀ D 6 3 8 1 ฀ 2 0 1 1 ฀ 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

฀

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

฀

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 ฀

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 ฀

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 ฀

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 ฀

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

f

g e orgiaN a cad e m y

f

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 ฀

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

฀

b ฀ f irSov฀ S ov฀ P h yS฀ J e t f 65฀ 98 ฀ 1973฀

SLIDE 10

Schrödinger equation for hydrogen-like ion

, , , , ,

1 ( ) 2

a b a b a b a b a b

Z R r ε   − ∆ − Ψ = Ψ      

, ,

( , ) ( , ) 2

im a b a b n n m n m n m

e X R Y R

ξ η ξ η

ϕ

ξ η π

−

Ψ =

( ) ( )

2 2 2 2 2 , 2 2 2 2 , 2

1 ( , ) 2 1 1 ( , ) 2 1

a b a b

d dX R m ZR X R d d d dY R m ZR Y R d d ε ξ λ ξ ξ ξ ξ ξ ξ ε η λ η η η η η η   − + + + − =   −     − + − − − =   −   

1 n n n m

ξ η

= + + +

SLIDE 11

/ 2 , / 2

( ) ( )

ZR n m a b ZR n m

X e W Y e W

ξ η

−

= =



1,2 / 2 ,1 1,2 , / 2 1 ,0

( ) ( )

ZR n m m a b ZR n m m

nh X e W ZR nh Y e W ZR

ξ η

ξ ξ η η

−

  = −       =    



The Coulomb spheroidal wave functions

1, n m = +

0; n n

ξ η

+ =

1 ฀

2, n m = +

1,2; n n

ξ η

+ =

2 ฀

SLIDE 12

2 1,2,3 1,2,3 / 2 2 ,1 ,2 1,2,3 2 2 2 1,2,3 1,2,3 , / 2 2 2 ,1 ,0 1,2,3 2 2

( 2 4) 1 ( ) 2 ( 2 4) 1 ( ) 2

ZR n m m m a b ZR n m m m

nh n h X e h m W ZR Z R nh n h Y e h m W ZR Z R

ξ η

ξ ξ ξ η η η

−

  = − + − − −         = + − − −      



3, n m = +

1,2,3; n n

ξ η

+ =

3 ฀

2 / 2

( ) ( 1)m W ξ ξ = −

2 / 2

( ) (1 )m W η η = −

1 n n n m

ξ η

= + + +

SLIDE 13

The basic equations

{ }

( , ) ( , )

a im nn m nn m

X R Y R e

ξ η

ϕ

ξ η

−

( )

( , ) ( , )

a b nn m nn m nn m

Y Y R Y R

η η η

η η

±

= ±

{ }

( , ) ( , )

a im nn m nn m

X R Y R e

ξ η

ϕ

ξ η

−

1 ( ) ( ) ( ) 1

( ) ( , ) ( , )

n m nn m nn m nn m n n

C R X R Y R

η ξ η η

ξ η

∞ − − ± ± ± = =

Φ = ∑ ∑

( ) ( )

( , , , ) ( , , )

im

R R e

ϕ

ξ η ϕ ξ η

± ±

Ψ = Φ

SLIDE 14

The basic equations

( )

฀฀ ฀฀

1 ( ) ( ) ( ) ( ) , , 1 n m n nn n n nn n n nn n n

E U V C

η η η η η η

ε

∞ − − ± ± ± ± = =

  − + =    

∑ ∑

฀฀ ฀฀ ฀฀ ฀฀ ฀฀ ฀฀ ฀฀ ฀฀ ฀฀ ฀฀

( ) 2 ( ) ( ) ( ) 2 ( ) , ( ) ( ) ( ) , ( ) ( ) ( )

2 (2 )

nn nn nn nn n n n n n n n n n n n n nn nn n n n n n n n n nn nn nn n n n n

U X X X X V Z X X R Z X X d

ξ η ξ η η η ξ η ξ η ξ η η η ξ η ξ η η ξ η

ξ η ξ η

± ± ± ± ± ± ± ± ± ±

= 〈 〉〈ϒ ϒ 〉 − 〈 〉 〈ϒ ϒ 〉  = − 〈 〉〈ϒ ϒ 〉    + 〈 〉 〈ϒ ϒ 〉  

 2 2

/ 2

n

E Z n = −

( )

฀฀ ฀฀

( ) ( ) ( ) , , n n n nn n n nn

E U V

η η η η

ε ±

± ±

− + =

SLIDE 15

฀ xac t VAL

UES

0.25

1.4754
1.8980
1.8981
1.8986

0.5

1.4213
1.7318
1.7319
1.7350

0.75

1.3560
1.5753
1.5757
1.5824

1.0

1.2884
1.4410
1.4418
1.4518

1.25

1.2230
1.3283
1.3295
1.3418

1.50

1.1617
1.2338
1.2353
1.2490

1.75

1.1053
1.1541
1.1559
1.1701

2.0

1.0538
1.0865
1.0885
1.1026

2.25

1.0071
1.0286
1.0307
1.0444

2.5

0.9648
0.9788
0.9808
0.9938

2.75

0.9267
0.9355
0.9376
0.9497

3.0

0.8924
0.8978
0.8998
0.9109

3.5

0.8339
0.8357
0.8375
0.8466

4.0

0.7869
0.7873
0.7898
0.7961

4.5

0.7493
0.7493
0.7506
0.7562

5.0

0.7192
0.7192
0.7202
0.7244

6.0

0.6757
0.6757
0.6763
0.6786

7.0

0.6469
0.6469
0.6472
0.6485

8.0

0.6267
0.6267
0.6269
0.6276

9.0

0.6118
0.6118
0.6119
0.6123

10.0

0.6003
0.6003
0.6004
0.6006

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

R au 1 Z = 1 ≠ Z 1 ≠ Z

e l e ct roN ic e N e rgie S for St at e

1sσ

( ) 1

( )

s

R au

σ

ε +

SLIDE 16

e l e ct roN ic e N e rgie S for St at e

2pπ

฀ xac t VAL

UES

0.25

0.3746
0.4880
0.4980
0.4980

0.5

0.3735
0.4923
0.4923
0.4923

0.75

0.3716
0.4839
0.4839
0.4841

1.0

0.3692
0.4736
0.4737
0.4741

1.25

0.3662
0.4622
0.4623
0.4631

1.50

0.3627
0.4503
0.4504
0.4517

1.75

0.3589
0.4380
0.4382
0.4402

2.0

0.3548
0.4259
0.4261
0.4288

2.25

0.3504
0.4140
0.4143
0.4176

2.5

0.3458
0.4025
0.4029
0.4068

2.75

0.3411
0.3914
0.3919
0.3964

3.0

0.3363
0.3808
0.3814
0.3864

3.5

0.3266
0.3610
0.3619
0.3678

4.0

0.3169
0.3432
0.3443
0.3508

4.5

0.3073
0.3272
0.3285
0.3354

5.0

0.2979
0.3129
0.3143
0.3214

6.0

0.2803
0.2884
0.2899
0.2970

7.0

0.2642
0.2684
0.2699
0.2766

8.0

0.2499
0.2519
0.2534
0.2595

9.0

0.2374
0.2383
0.2397
0.2450

10.0

0.2265
0.2269
0.2282
0.2327

12.0

0.2092
0.2092
0.2102
0.2133

16.0

0.1871
0.1871
0.1876
0.1888

20.0

0.1745
0.1745
0.1747
0.1751

( ) 2

( )

p

R au

π

ε +

R au 1 Z = 1 ≠ Z 1 ≠ Z

SLIDE 17

v ariat ioN al P riN ciP l e

( )(

, )

i

d Z R dZ ε ± =

( ) ( ) ( ) ( ) ( )

฀ ( , , ) ( , , ) ( , ) ( , , ) ( , , )

i i i i i

R H R Z R R R ξ η ξ η ε ξ η ξ η

± ± ± ± ±

〈Φ Φ 〉 = 〈Φ Φ 〉

SLIDE 18

t h e e ffe ct ive ch arge aS a fu N ct ioN

f

r

SLIDE 19

SLIDE 20

m ol e cu l ar

rb it

al corre l at ioN

R UL E 2

u u

n n n l n m

η η

= + = +

1 2 1

u u

n n n l n m

η η

= + + = + + 2 2 1

u u

l m n l m n

η η

− = − = +

1

r u u

n n l nξ ≡ − − =

฀

r

sym m et r i c st at es ฀

r

ant i sym m et r i c st at es

SLIDE 21