MOSELEY’S LAW and

NEW DEVELOPMENTS in

QUANTUM MECHANICS

I. I. GUSEINOV

Department of Physics, Faculty of Art and Science Çanakkale Onsekiz Mart University

2)1( ZK1151048.2 snK

)(

)(

Zf

Af

A- Atomic Weight

Z-Atomic Number

Na

1s2 2s2 2p6 3s1

Mg

1s2 2s2 2p6 3s2

Al

1s2 2s2 2p6 3s2 3p1

Si ...

1s2 2s2 2p6 3s2 3p2

1s2 2s2 2p6

3s1

4s

5s

3p

4p

5p

3d

4d

5d

4f

5f

p

d fs

Optical Spectra of Na

Röntgen Spectra of Na

1s2 2s2 2p6 3s1 1s1 2s2 2p6 3s1

h

Röntgen Spectra of Mg

1s2 2s2 2p6 3s2 1s1 2s2 2p6 3s2

h

Röntgen Spectra of Al

1s2 2s2 2p6 3s2 3p1 1s1 2s2 2p6 3s2 3p1

h

Z=11:

Z=12:

Z=13:

Moseley’s Law:

A (Atomic Weight) Z (Atomic Number)1)

2) Discovery of Nuclei (Rutherford)

.10 8 smae

ea

na .10 13 sman

e-

Z

3) Proton-Neutron Model of Nuclei

Z

e-

ZN p

ZANn

4) Shell Model of Atomic-Molecular and Nuclear Systems (Hartree-Fock and Hartree-Fock-Roothaan Theory)

npNnp NNNxxxHxxxxxxH

,),...,,(),...,,;,...,,( 2102121 ( 1 )

)),...,,(;,...,,( 12121 nN RRRfxxxH

( 2 )

)(2

1,xyzx,1 electrons

)(,,2 nucleons2

1

2

1,xyzx

PHHP ( 3 )

Np and Nn N (Nucleon)

,

EH asP

11 , (4)

For electrons and nucleons:

aaP (5)

aa EH

(6)

?a

cvcvcv )3,)2,)1

Hßw

=ãm=1

N ikjjjjj- 1

2 m w- 1 Ñ m

2 - dw1 ãa

nZa

r am

y{zzzzz+âm=1

N - 1 ân=m+1

N

f w Hh, rmnL+ V Js®, p ,®

...N

f w Hh , rL= H- gLw - 1 e - hr

r (8 )

0,...),(2

2

c

vpsV

(9 )

N

1μ

1N

1μ

N

1μνμν

ω

a a μ

aω1

2μ1ω

ω

)r,(fr

Zδ

2m

1H

(1 0 )

ωωωω

ΨEΨH

(1 1 )

?

(7 )

12 NN

ω

0

ω

WHH

( 1 2 )

N

1μ

μω2

μ1ω

ω

0 )r(V2 m

1H ( 1 3 )

Wß w

= ãm = 1

N ikjjjjj - d w 1 ãa

Z a

r a m- V w Ir

zm My{zzzzz+ â

m = 1

N - 1 ân = m + 1

N

f w Ih , r m n M ( 1 4 )

INDEPENDENT PARTICLES MODEL

:0W ω ˆ

N

1μ

ω0

ω0

ω0μ

ω2μ1ω

ω0

ω

0 ΨEΨ)r(V2m

1ΨH ( 1 5 )

NE ...210 , ( 1 6 )

?)()...()( 22110 aNN xuxuxu ( 1 7 )

)()()()( st mmin uvxyzuu x ( 1 8 )

mnimimnxyz st ,,x ( 1 9 )

2

1

2

1m

2

1

2

1m st ,,, ( 2 0 )

Symmetry Properties of Orbitals in Independent Particles Model for 1N and 2N

Atoms, Nuclei

NLTSNLS

L

nl

l

EE

DPSL

ML

dpsl

:ml

,

...,,,

...,2,1,0

:

...,,,

...,2,1,0

Linear Mol.

SN

Mn

E

M

,...,,

,...2,1,0

...δ,π,σ,

...2,1,0,mλ

:mλ

:

λ

Nonlinear Mol.

SN

n

E

M

presIrreduc

M

,...2,1

.Re.

:

1,2,...m

repres. irreduc.-

:mγ

γ

O p e n s h e l l s :

,...)mγ(n)mγ(n 2

2

1

1

kγ22

kγ11 ,

20,)!2(!

)!2(

ik k

kkN ( 2 1 )

D i s t r i b u t i o n o f p a r t i c l e s i n s h e l l s ( = 1 f o r e l e c t r o n s , = 2 f o r n u c l e o n s )

k = 1 k = 2 k = 3 k = 4

sm st mm ss mm stst mmmm , ststst mmmmmm ,, stststst mmmmmmmm ,,,

e

e

:2

1

:2

1

n

n

p

p

:2

1

2

1

:2

1

2

1

:2

1

2

1

:2

1

2

1

ee:2

1

2

1

nn

np

np

np

np

pp

:2

1

2

1,

2

1

2

1

:2

1

2

1,

2

1

2

1

:2

1

2

1,

2

1

2

1

:2

1

2

1,

2

1

2

1

:2

1

2

1,

2

1

2

1

:2

1

2

1,

2

1

2

1

pnn

pnn

npp

npp

:2

1

2

1,

2

1

2

1,

2

1

2

1

:2

1

2

1,

2

1

2

1,

2

1

2

1

:2

1

2

1

2

1

2

1,

2

1

2

1

:2

1

2

1,

2

1

2

1,

2

1

2

1

nnpp:2

1

2

1,

2

1

2

1,

2

1

2

1,

2

1

2

1

Orthonormality Properties of One-Particle Orbitals Occurring in Independent Particles Model

(22)

sssstttt mm

2

1σ

mm

2

1κ

mmmmiiii δ()u(u,δ)()v(v,δdvuu

)

,δ(x)dτu(x)u nnnn

(23)

Orthonormal Determinantal Wave Functions Constructed From Orthonormalized Spin-Isospin Orbitals

U1N

Aun1x1un2x2...un

NxN

)()...()(

......

)(...)()()(...)()(

!

1

21

21

21

222

111

NnNnNn

nnn

nnn

xuxuxu

xuxuxuxuxuxu

NU

N

N

N

(24)

UUdUU ( 2 5 )

iiiM UD

( 2 6 )

MMd τΨΨ Γ

MΓM ΓΓ ( 2 7 )

Molec.NonlinearforΨ

Molec.LinearforΨ

AtomsforΨ

NucleiforΨ

Ψ

ΓSMM

ΛSMM

LSMM

LTSMMM

ΓM

SΓ

SΛ

SL

STL

Γ

( 2 8 )

P O S T U L A T E D T O T A L E N E R G Y

:0W

C l o s e d S h e l l s :

n

i

n

ki

i ik k

i ik ki KJhUdHUE

,

)]2(2[

( 2 9 )

O p e n S h e l l s :

C . C . J . R o o t h a a n , S e l f C o n s i s t e n t F i e l d T h e o r y f o r O p e n S h e l l s o f E l e c t r o n i c

S y s t e m s , R e v . M o d . P h y s . , 3 2 ( 1 9 6 0 ) 1 7 9 :

k km

kmkmmn

mnmnm

mlk

klklk KJb Ka JfHfKJHE )]2(2)2(2[)2(2, ( 3 0 )

I . I . G u s e i n o v , R e s t r i c t e d O p e n S h e l l H a r t r e e - F o c k T h e o r y , J . M o l . S t r u c t .

( T h e o c h e m ) , 4 2 2 ( 1 9 9 8 ) 6 9 .

n

i

n

lkji

ijkl

ijkl

ijkl

ijklii KBJAhfE

,,,

)]2(2[

( 3 1 )

w h e r e 0N

Nf i

i

6

2:2,6,1)221( 220

222 pp fNNpssC

21212121

21212121

11

211111

)()(),()()(

)()(),()()(

2

1,)()(

dvdvrururfruruK

dvdvrururfruruJ

r

Z

mhdvruhruh

jlkiij

kl

ljkiij

kl

n

a a

aiii

( 3 2 )

ijkl

ijkl B,A :

F o r c l o s e d - c l o s e d , c l o s e d - o p e n s h e l l s i n t e r a c t i o n s :

klijkikl

ijij

klkl

ijij

kl ffBBAA ( 3 3 )

F o r o p e n - o p e n s h e l l s i n t e r a c t i o n s :

d τΨHΨN

1E

Γ

ΓΓM

ΓM

ΓM

Γ

ωΓ

( 3 4 )

(35)

Molec.Nonlinearfor)12(

Molec.Linearfor)12)(2(

Atomsfor)12)(12(

Nucleifor)12)(12)(12(

N

d

0

S

S

SL

STL

H A R T R E E - F O C K E Q U A T I O N S

V a r i a t i o n a l P r i n c i p l e : 0 E

L a n g r a n g i a n ( U n d e t e r m i n e d ) M u l t i p l i e r s : i2

ssisi

i

εuuF ( 3 6 )

n

k lj ,

k l

i j

k lk l

i j

k l

ii

i

i

)KBJA(2 ωG,GhfF ( 3 7 )

jijk li

ij

k l uAuA

, jijk li

ij

k l uBuB

( 3 8 )

)r())dvr()ur,)f(r(u()r()r(J 122l2 12k11k l ( 3 9 )

)r()u)dvr()r,)f(r(u()r()r(K 1l22212k11kl ( 4 0 )

U n i t a r y T r a n s f o r m a t i o n o f O r b i t a l s

iiiii

iiiii )Qr(u)r(u,)Qr(u)r(u

( 4 1 )

εQQε,εuuFs

issi

i

( 4 2 )

O p e n S h e l l s H F E q u a t i o n s :

iii

i

uεuF

( 4 3 )

n

iiii

ω )εh(fωE ( 4 4 )

C l o s e d S h e l l s H F E q u a t i o n s :

iii uεuF

( 4 5 )

n

k

kkkk )KJ(2 ωG,GhF ( 4 6 )

i

iiω )ε(hωE

( 4 7 )

S c h r ö d i n g e r E q u a t i o n F o r O n e - p a r t i c l e :

)(2

1 21

rVm

FH

( 4 8 )

n

k

kkkk

n

aa

a

r

zrVuurV

mKJ

2)(,)(2

11

2

1 ( 4 9 )

??

Viu ( 5 0 )

H A R T R E E - F O C K - R O O T H A A N E Q U A T I O N S

q

qiqi Cχu ( 5 1 )

O p e n S h e l l s H F R E q u a t i o n s :

qqipqi

i

pq )CSεF( ( 5 2 )

dvχχS qppq ( 5 3 )

i

p qp qi

i

p q GhfF

( 5 4 )

1qpp q χˆχh dvh ( 5 5 )

r sj ,

pqrs

i j

r spqrs

i j

r s

i

pq )KbIa(2 ωG ( 5 6 )

CACai ji j

,

CBCbi ji j

( 5 7 )

qji j

qi

i j

CaCa

, q ji j

q i

i j

CbCb

( 5 8 )

212s1q212r1ppqrs dv)dvr() χr() χr,)f(r() χr(χI ( 5 9 )

psrq212q1s212r1p

pqrs Idv)dvr() χr() χr,)f(r() χr(χK ( 6 0 )

C l o s e d S h e l l s H F R E q u a t i o n s :

0Sεq

pqipq qiCF ( 6 1 )

pqpqpq GhF ( 6 2 )

)2(* pqrs

pqrs

rsrspq KIG

( 6 3 )

BASIS FUNCTIONS and INTERACTION POTENTIALS

 STO (Slater, 1929), GTO (Boys, 1951), Coulomb Sturmians (Shull-Löwdin, 1959), -ETO (Guseinov, 2002). I. I. Guseinov, New Complete Orthonormal Sets of Exponential Type Orbitals and Their Application to Translation of Slater Orbitals, Int. J. Quantum Chem., 90 (2002) 114-118. I. I. Guseinov, Addition and Expansion Theorems for Complete Orthonormal Sets of Exponential-Type Orbitals in Coordinate and Momentum Representations, J. Mol. Model., 9 (2003) 135-141. I. I. Guseinov, New Complete orthonormal sets of Hyperspherical Harmonics and Their One-Range Addition and Expansion Theorems (submitted).

B A S I S F U N C T I O N S

C o o r d i n a t e S p a c e :

),(r)(2Le)(q!(2n)

)!(q)(21)()r,(Ψ l

pq

rζ2

1

3α

3α

m

αnlm S

p

( 6 4 )

w h e r e 1,22 lnqlp a n d

2,...1,1,0,α ( 6 5 )

),()!2(

)2(),( 1

21

lm

rnn

nlm Sern

r

( 6 6 )

n

lnlmn

lnnnlm rr

1

),(),(

( 6 7 )

n

lnlmn

lnnnlm rr

1

),(),(

( 6 8 )

Illll ( 6 9 )

M o m e n t u m S p a c e :

),(),( kr nlmnlm ( 7 0 )

),(),( kUr nlmnlm ( 7 1 ) F o u r D i m e n s i o n a l S p a c e :

)(),( nlmnlm Zk ( 7 2 )

)(),( nlmnlm VkU ( 7 3 )

H y d r o g e n - l i k e a t o m s :

:0E

) ,((r)YR,n

RE lmn ln lm2n ( 7 4 )

?0E ( 7 5 )

n

ZandforZZ n lmn lmn lmn lmn lmn lm 1,,

. ( 7 6 )

I N T E R A C T I O N P O T E N T I A L S

f u v s Ih , r® M= r u - 1 e - h r J 4 p

2 v + 1N1 2

S v s Hq , j L, ( 7 7 )

f Hr L= f 0 0 0 H0 , r L=1

r ( 7 8 )

f Hh , r L= f 0 0 0 Hh , r L=e - h r

r ( 7 9 )

)(4)()(2

2

2

2

2

2

rrfzyx

( 8 0 )

)(4),()( 22

2

2

2

2

2

rrfzyx

( 8 1 )

Coulomb potential  

Yukawa potential

LCAOETOofSetslOrthonormaComplete

?? LCNOofSetslOrthonormaComplete

O N E - R A N G E A D D I T I O N a n d E X P A N S I O N T H E O R E M S f o r O R B I T A L S , P O T E N T I A L S a n d T H E I R D E R I V A T I V E S

1

r 2 1

= ãl = 0

¥ ãm = - l

l4 p

2 l + 1 S l m

* Hq 2 , j 2 L S l m Hq 1 , j 1 L : r2l ‘ r 1

l + 1 r 2 < r 1

r1l ‘ r 2

l + 1 r 2 > r 1 ( 8 2 )

U S E o f i n - E T O H F R T H E O R Y f o r A T O M I C - M O L E C U L A R

S Y S T E M S ( = 1 )

q q

q iqq iqi CCu ,...2,1,0,1,

( 8 3 )

CCl

( 8 4 )

CC l ( 8 5 )

,...2,1,0,1, ICSCSCC ( 8 6 )

1ar

Matrix elements of arbitrary multi electron operators over

ijiijiiji fff ,,,,,,,, ,, overlap integrals with STO:

11'

1*'

,),(),();,( '''''' dvrrRS bmlnanlmabmlnnlm

(87)

z 1 a

b

o y x

abR

1br

APPLICATIONS: ATOMIC-MOLECULAR SYSTEMS (=1)

1. )221( 222 pssC

])()()()()()([A!6

16215214

2

1200

3

2

1200

2

2

1100

1

2

1100

xuxuxuxuxuxuUslsl mmmm

(88)

S1 (f=1)

)2p2sC(1s 222

D1 (f=5)

f=15

P3(f=9)

W=0

W0

T a b l e 1 . T h e i n d e p e n d e n t d e t e r m i n a n t a l w a v e f u n c t i o n s f o r t h e e l e c t r o n i c c o n f i g u r a t i o n )221( 222 pssC

55:5 sl mmn

66

:6 sl mmn

LM S

M )2121(6655 slsl mmmmU

211:2

2 0 )212112

1211(1 U

210:3

1 1 )212102

1211(2U

210:4

1 0 )212102

1211(3 U

211:5

0 1 )211212

1211(5 U

211:1

211:6 0 0 )2

112121211(6 U

210:3 1 0 )2

121021211(4 U

210:4 1 1 )2

121021211(9 U

211:5 0 0 )2

112121211(7 U

211:2

211:6 0 1 )2

112121211(10 U

210:4 0 0 )2

121021210(8 U

211:5 1 1 )2

112121210(11 U

210:3

211:6 1 0 )2

112121210(12 U

211:5 1 0 )2

112121210(1 3 U 2

10:4 2

11:6 1 1 )211212

1210(14 U

211:5 2

11:6 2 0 )211212

1121(15 U

T a b l e 2 . T h e t e r m s o f e l e c t r o n i c c o n f i g u r a t i o n )221( 222 pssC a n d t h e i r m u l t i d e t e r m i n a n t a l w a v e f u n c t i o n s T e r m s LS

MM SL

S1

)(3

1876

0000 UUU

D1

120

20 U )(2

143

2010 UU

)2(6

1876

2000 UUU

)(2

11312

2010 UU 15

2020 U

P3

211

11 U )(2

143

1110 UU

911

11 U

511

01 U )(2

176

1100 UU

1011

10 U

1111

11 U )(2

11312

1110 UU

1411

11 U

T a b l e 3 . T h e v a l u e s o f c o u p l i n g - p r o j e c t i o n c o e f f i c i e n t s ij

klA a n d

ij

klB f o r e l e c t r o n i c

c o n f i g u r a t i o n )221( 222 pssC .

C l o s e d - c l o s e d a n d c l o s e d - o p e n s h e l l s O p e n - o p e n s h e l l s 111

11A 111

11B

122

11

11

22 AA 122

11

11

22 BB

3

133

11

11

33 AA 3

133

11

11

33 BB

3

144

11

11

44 AA

3

144

11

11

44 BB

3

155

11

11

55 AA 3

155

11

11

55 BB

122

22A 122

22B

3

133

22

22

33 AA

3

133

22

22

33 BB

3

144

22

22

44 AA 3

144

22

22

44 BB

3

155

22

22

55 AA 3

155

22

22

55 BB

12

144

33

33

44 AA

6

144

33

33

44 BB

12

155

33

33

55 AA

6

155

33

33

55 BB

12

155

44

44

55 AA

6

155

44

44

55 BB

T a b l e 4 . N u m e r i c a l l i n e a r c o m b i n a t i o n c o e f f i c i e n t s o f S l a t e r a t o m i c o r b i t a l s )(

5

1

qqiqi

Cu f o r t h e g r o u n d s t a t e o f ),221( 3222 PpssC a n d o r b i t a l e n e r g i e s ( i n

a . u . ) .

lnlmi uu 1001 uu 2002 uu 2113 uu 1214 uu 2105 uu

i 301550.11

11

s

774946.6

22

s

338743.1

23

xp

338743.1

24

zp

338743.1

25

yp

q q qiC

)1(1 sC

)2(2 sC

)2(3 xpC

)2(4 zpC

)2(5 ypC

5 . 6 7 2 7

1 . 6 0 8 3

1 . 5 6 7 9

1 . 5 6 7 9 1 . 5 6 7 9

0 . 9 9 7 4 3 8

0 . 0 1 1 4 3 8

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

- 0 . 2 3 5 0 7 8

1 . 0 2 4 7 0 2

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

1 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

1 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

0 . 0 0 0 0 0 0

1 . 0 0 0 0 0 0

T o t a l e n e r g y K i n e t i c e n e r g y V i r i a l r a t i o

- 3 7 . 6 2 2 3 8 9 3 7 . 6 2 2 6 9 1 - 1 . 9 9 9 9 9 2

- 3 7 . 6 2 2 3 8 9 ( E . C l e m e n t i , D . L . R a i m o n d i , J . C h e m . P h y s . , 3 8 ( 1 9 6 3 ) 2 6 8 6 . )

- 3 7 . 5 7 9 0 1 8 ( I . E m a , J . V e g a , B . M i g u e l , J . D o t t e r w e i c h , H . M e i t n e r , E . O . S t e i n b o r n , A t o m i c D a t a a n d N u c l e a r D a t a T a b l e s , 7 2 ( 1 9 9 9 ) 5 7 . )

x

y

z

2222222 )1()1(54321( CO2.

vC )(2 zC zC2)( v

1A

2A

1E

2cos22E

cos2

E

1 1 1 1

1 1 1 -1

2 -2 0

2 2 0

... ... ... ... ...

a1

a2

e1

e2

...

Table 5. Numerical linear combination cofficients of molecular orbitals )( q

qiqi Cu for the ground

electronic state of molecule CO( 2222222 )1()1(54321 , 3 ) and orbital energies (in.a.u.)

E=-112.3243, Virial=-2.0014

mni uu 1001 uu 2002 uu 3003 uu 4004 uu 5005 uu 6006 uu 1117 uu 2118 uu 1119 uu 12110 uu

2 3 4 5 6 1 2 1 2

i

-20.81314

-11.44460

-1.53738

-0.76039

-0.50739

0.87014

-0.61518

0.22317

-0.61518

0.22317

q qiC

)1(1 sC 0.0001 -0.9970 -0.1147 0.1449 -0.1360 -0.0885 0. 0. 0. 0.

)2(2 sC -0.0059 -0.0141 0.2227 -0.6306 0.7656 1.0198 0. 0. 0. 0.

)2(3 zpC

0.0052 0.0052 -0.1549 0.0593 0.5605 -1.2925 0. 0. 0. 0.

)1(4 sO 0.9968 0.0003 -0.2116 -0.1234 0.0024 0.1171 0. 0. 0. 0.

)2(5 sO 0.0167 0.0004 0.7682 0.6525 0.0375 -1.1904 0. 0. 0. 0.

)2(6 zpO

0.0051 0.0002 0.2416 -0.6103 -0.4484 -0.9477 0. 0. 0. 0.

)2(7 xpC

0. 0. 0. 0. 0. 0. 0.4607 -0.9320 0. 0.

)2(8 xpO

0. 0. 0. 0. 0. 0. 0.7703 0.6981 0. 0.

)2(9 ypC

0. 0. 0. 0. 0. 0. 0. 0. 0.4607 -0.9320

)2(10 ypO

0. 0. 0. 0. 0. 0. 0. 0. 0.7703 0.6981

1

2222222222 )1()1()1()1(32211

gguugugugF 3.

z

y

xhD

)(2 zC zC2)( v i )(2 zS h

2C

g

u

g

u

g

u

cos2 cos2

g2cos2

u

E

1 1 1 1 1 1 1 1

1 1 1 1 -1 -1 -1 -1

1 1 1 -1 1 1 1 -1

1 1 1 -1 -1 -1 -1 1

2 -2 0 2 -2 0

2 -2 0 -2 2 0

2 2 0 2 2 0

2 2 0 -2 -2 0

… … … … … … … … …

cos2

2cos2

cos2

2cos2

2cos2

g

u

g

u

g

u

g

u

...

Table 6. Numerical linear combination cofficients of molecular orbitals for the ground electronic

state of molecule and orbital energies (in.a.u.)

)( q

qiqi Cu

2222222222 )1()1()1()1(32211

gguugugugF  

                   

 -26.38175

 -26.38159

 -1.63256

 -1.36781

 -0.65672

 -0.328112

 -0.47923

 -0.61417

 -0.47923

 -0.61417

0.7330 -0.6763 -0.1689 0.1823 -0.0378 0.0426 0. 0. 0. 0.

0.0089 -0.0068 0.66888 -0.7606 0.1803 -0.2130 0. 0. 0. 0.

-0.0013 -0.0002 -0.0845 -0.0710 0.6503 0.7929 0. 0. 0. 0.

0.7332 0.6760 -0.1689 -0.1823 -0.0378 -0.00425 0. 0. 0. 0.

0.0083 0.0075 0.6687 0.7607 0.1803 0.2218 0. 0. 0. 0.

0.0013 -0.0002 0.0845 -0.0710 -0.6503 0.7929 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0.7250 0.6807 0. 0.

0. 0. 0. 0. 0. 0. -0.7261 0.6894 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0.7259 0.6807

0. 0. 0. 0. 0. 0. 0. 0. -0.7262 0.6894

Imni uu 10011 uu 11002 uu 20013 uu 12004 uu 30015 uu 13006 uu 11117 uu 11118 uu 11119 uu 111110 uu

iq qiC

g1 u

1 g 2 u

2 g 3 u

2 g

1

u

1 g

1

u

1

)1(11 sF

)2(12 sF

)2(13 zpF

)1(24 sF

)2(25 sF

)2(26 zpF

)2(17 xpF

)2(28 xpF

)2(19 ypF

)2(210 ypF

E= -199.5695, Virial=-2.0033

)1121( 2221

213 yx eeaaBH4.

B(0,0,0)

0,

3

3,01 aH

0,

6

3,

23 aa

H

0,

6

3,

22 aa

H

22U 3

2U

a

x

y

E 32C23U

D3

a1 A11 1 1

a2 A21 1 -1

e E 2 -1 0

Table 7. Numerical linear combination cofficients of molecular orbitals for the

ground electronic state of nonlinear molecule and orbital energies (in a.u.)

)( q

qiqi Du

)1121( 2221

213 yx eeaaBH

mmi uu 111 auu 122 auu

133 auu 214 auu

xeuu 15 xeuu 26

yeuu 17 yeuu 28

111 a 122 a

133 a 214 a

xe15 xe26

ye17

ye28 i

- 7 . 8 2 8 6 1 1 - 0 . 8 2 9 6 1 0

0 . 8 2 9 5 1 7 0 . 0 2 8 4 8 - 0 . 5 9 9 1 1 0 8 0 . 7 7 2 8 6 1 - 0 . 5 9 9 1 1 0 8 0 . 7 7 2 8 6 1

q qiD

111 a

0 . 0 0 7 8 6 7 1

- 0 . 5 6 4 7 4 4 5

2 . 3 2 9 3 9 7 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

122 a - 0 . 9 9 6 7 7 2 3 2 0 . 2 0 5 9 1 7 7 0 . 1 1 2 3 3 8 7 3 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

- 0 . 0 2 1 2 9 0 2 6 - 0 . 4 7 7 7 9 2 2 - 2 . 3 4 8 5 5 4 7 3 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 1 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 4 6 7 9 6 - 1 . 6 3 1 8 8 5 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 . 5 8 2 5 6 2 1 . 6 0 2 6 1 2 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 4 6 8 2 0 3 4 - 1 . 6 3 1 8 8 5

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 , 5 8 2 5 6 2 1 . 6 0 2 6 1 2

133 a

214 a

xe15

xe26

ye17

ye28

E=-29.119467

62 HB5.

62HB molekülünün geometrisi (a=1,89547500, b= 1,94740340,

c=1,67715313 d=2,82426137).

  H1 H2 H3 H4 H5 H6 B1 B2

X 0 0 0 0 a -a 0 0

Y -b b -b b 0 0 0 0

Z d d -d -d 0 0 c -c

Tablo 8. Numerical linear combination cofficients of molecular orbitals , for the ground electronic

state of and orbital energies (in a.u.) 62

HBqiC

i

iu

i

-7.7872 -7.7890 -0.9296 -0.6890 -0.5961 -0.5816 -05622 -05105 0.0843 0.3528 0.4165 0.5063 0.5755 0.6828 0.7333 0.8393

q

0.0011 -0.0027 -0.0910 0.2110 0.0001 -0.2244 -0.2414 0.2912 0.0002 0.2104 0.2791 0.7071 -0.0007 -0.9633 -0.2282 -0.7747

0.0012 -0.0027 -0.0908 0.2110 0.0001 0.2196 -0.2451 -0.2915 0.0002 0.2088 0.2835 -0.7166 -0.0007 -0.9650 -0.2162 0.7662

0.0079 -0.0010 -0.0794 -0.2328 0.0001 -0.2405 -0.2120 -0.2755 -0.0002 -0.1609 0.2092 0.7166 -0.0008 0.2505 -0.9850 0.7702

0.0079 -0.0010 -0.0794 -0.2328 0.0001 0.2363 -0.2165 0.2755 -0.0002 -0.1625 0.2128 - 0.7210 -0.0007 0.2649 -0.9852 -0.7596

0.0029 -0.0054 -0.2154 0.02516 0.4414 0.0020 0.1862 0.0000 0.0280 -0.4202 -0.8342 -0.0052 1.0332 -0.4133 -0.3916 -0.0004

0.0030 -0.0053 -0.2155 0.0252 -0.4415 0.0020 0.1857 0.0000 -0.0282 -0.4194 -0.8327 -0.0053 -1.0362 -0.4118 -0.3892 -0.0004

0.2402 0970 0.1355 -0.1159 -0.0000 0.0004 0.0435 0.0000 0.0001 0.0852 -0.0750 -0.0012 -0.0002 -0.1447 -0.0656 -0.0005

-0.0030 0.0219 -0.3268 0.4061 0.0001 -0.0022 -0.2284 -0.0001 -0.0006 -0.9886 0.5607 0.0117 0.0019 1.3618 0.7350 0.0040

0.0000 0.0000 -0.0000 0.0000 0.3207 0.0000 0.0000 -0.0000 0.7974 0.0000 0.0007 -0.0000 -0.8997 0.0008 0.0009 -0.0000

-0.0000 0.0000 -0.0001 -0.0002 -0.0000 0.3889 -0.0038 -0.4627 -0.0000 0.0010 -0.0024 0.7888 -0.0000 0.0027 -0.0080 -1.1785

0.0010 -0.0033 0.1465 0.1239 0.0001 -0.0038 -0.3868 -0.0002 0.0002 0.6773 -1.1125 -0.0018 -0.0002 0.6494 -0.1853 0.0044

0.9681 -0.2405 0.1324 0.1303 -0.0000 0.0003 0.0266 -0.0000 -0.0001 -0.1437 0.0059 -0.0003 -0.0002 0.0027 -0.1164 0.0005

0.0050 -0.0031 -0.3016 -0.4734 0.0001 -0.0016 -0.1505 -0.0001 0.0007 1.4182 -0.2531 0.0029 0.0017 0.0250 1.2639 -0.0049

0.0000 0.0000 -0.0000 -0.0001 0.2952 0.0000 -0.0000 0.0000 -0.8656 0.0006 0.0005 -0.0000 -0.8437 0.0005 0.0010 -0.0000

0.0000 -0.0000 -0.0000 0.0001 -0.0000 0.4147 -0.0045 0.4426 0.0000 0.0010 -0.0019 0.7961 -0.0000 -0.0097 0.0015 1.1725

0.0064 -0.0003 -0.1322 0.1606 -0.0000 0.0037 0.3935 0.0001 0.0003 1.1214 0.1262 -0.0021 -0.0001 0.4648 -0.8071 0.0057

1u

2u 3

u4

u5

u 6u

7u 8

u9

u10

u11

u12

u 13u

14u 15

u16

u

1qC

2qC

3qC

4qC

5qC

6qC

7qC

8qC

9qC 10q

C11q

C12q

C13q

C14q

C15q

C16q

C)1(

11sH

)1(22

sH

)1(33

sH

)1(24

sH

)1(55

sH

)1(66

sH

)1(17

sB

)2(18 sB

)2(19 xpB

)2(110 ypB

)2(111 zpB)1(212 sB

)2(213 sB

)2(214 xpB

)2(215 ypB

)2(216 zpB

E= -52.6325, Virial= 2.0188

1

2 3

4

5

6

9

7 8

10

11

12 13

14 15

16

62

21

214 121 taaCH5.

4CH

C(0,0,0)

Y

Z

aaaH3

1,

3

1,

3

11

H1

aaaH

3

1,

3

1,

3

13

aaaH

3

1,

3

1,

3

12

zS4

yS4

xS 4

13C

43C

23C

33C

14

13

X

a

aaaH3

1,

3

1,

3

14

(a=1.19309433 a.u.)

Td E 8C3 3C2 6S4 6d

a1 A1 1 1 1 1 1

a2 A2 1 1 1 -1 -1

e E 2 -1 2 0 0

t1 T1 3 0 -1 1 -1

t2 T2 3 0 -1 -1 1

Table 9. Linear combination coefficients of molecular orbitals ( for the ground electronic state of

molecule CH4 ( ) and orbital energies. (in a.u.)

)( q

qiqi Du

622

12

1 121 taa

mmi uu 111 auu

1212 auu 133 auu

xtuu214

xtuu225

ytuu216

ytuu227

ztuu218

111 a 122 a

1323 a xt214

xt225 yt216

yt227 zt218

zt228

i

-11.32619 -0.947618 0.6419439 -0.5524043 0.62064 -0.5524043 0.62064. -0.5524043. 0.62064

q qiD

-0.0122119 0.4259840 -2.0935808 0. 0. 0. 0. 0. 0.

0.99613990 -0.19903203 -0.19984831 0. 0. 0. 0. 0. 0.

0.02263477 0.66968797 1.75394974 0. 0. 0. 0. 0. 0.

0. 0. 0. 0.18389415 0.43912572 0. .0 0. 0.

0. 0. 0. 0.60937449 1.15256433 0. 0. 0. 0.

0. 0. 0. 0. 0 0.18389415 0.43912572 0. 0.

0. 0. 0. 0. 0. 0.60937449 1.15256433 0. 0.

0. 0. 0. 0. 0. 0. 0. 0.18389415 0.43912572

0. 0. 0. 0. 0. 0. 0. 0.60937449 1.15256433

111 a

122 a

133 a

xt214

xt225

yt216

yt227

zt218

zt229

E=-40.10133, V=2.01774

ztuu

229

APPLICATIONS: NUCLEAR SYSTEMS (=2)

)221( 1146 pssLi1.

])()()()()()([A!6

16215204

2

1

2

1100

3

2

1

2

1100

2

2

1

2

1100

1

2

1

2

1100 222111

xuxuxuxuxuxuUstlstl mmmmmm

(89)

P13 ( f = 9 )

P33 ( f = 2 7 )

6 L i ( 1 s 4 2 s 1 2 p 1

)

P11 ( f = 3 )

P3

1 ( f = 9 )

0W 0W

f = 4 8 LS

T

12

12

Table 9. The independent determinantal wave functions for the nucleonic configuration

)221( 1146 pssLi

1lm1tm

1sm 2lm

2tm2sm LM TM SM U [

1lm ,1tm ,

1sm ,2lm ,

2tm ,2sm ]

1 0 12

12

1 12

12

1 , 1 , 1 U 1 A0 ,12

,12

, 1 ,12

,12E

2 0 12

12 1 1

2- 1

2 1 , 1 , 0 U 2 A0 ,

12

,12

, 1 ,12

, -12E

3 0 12

12 1 - 1

212

1 , 0 , 1 U 4 A0 ,12

,12

, 1 , -12

,12E

4 0 12

12 1 - 1

2- 1

2 1 , 0 , 0 U 6 A0 ,

12

,12

, 1 , -12

, -12E

5 0 12

12 0 1

212

0 , 1 , 1 U 1 0 A0 ,12

,12

, 0 ,12

,12E

6 0 12

12 0 1

2- 1

2 0 , 1 , 0 U 1 1 A0 ,

12

,12

, 0 ,12

, -12E

7 0 12

12 0 - 1

212

0 , 0 , 1 U 1 3 A0 ,12

,12

, 0 , -12

,12E

8 0 12

12 0 - 1

2- 1

2 0 , 0 , 0 U 1 5 A0 ,

12

,12

, 0 , -12

, -12E

9 0 12

12 - 1 1

212

- 1 , 1 , 1 U 1 9 A0 ,12

,12

, - 1 ,12

,12E

1 0 0 12

12 - 1 1

2- 1

2 - 1 , 1 , 0 U 2 0 A0 ,

12

,12

, - 1 ,12

, -12E

1 1 0 12

12

- 1 - 12

12

- 1 , 0 , 1 U 2 2 A0 ,12

,12

, - 1 , -12

,12E

1 2 0 12

12

- 1 - 12

- 12

- 1 , 0 , 0 U 2 4 A0 ,12

,12

, - 1 , -12

, -12

E

1 3 0 12

- 12

1 12

12

1 , 1 , 0 U 3 A0 ,12

, -12

, 1 ,12

,12

E

1 4 0 12

- 12

1 12

- 12

1 , 1 , - 1 U 2 8 A0 ,12

, -12

, 1 ,12

, -12E

1 5 0 12

- 12

1 - 12

12

1 , 0 , 0 U 7 A0 ,12

, -12

, 1 , -12

,12E

1 6 0 12

- 12

1 - 12

- 12

1 , 0 , - 1 U 1 6 A0 , -12

, -12

, 0 , -12

, -12

E

1 7 0 12

- 12

0 12

12

0 , 1 , 0 U 2 9 A0 ,12

, -12

, 1 , -12

, -12

E

1 8 0 12

- 12

0 12

- 12

0 , 1 , - 1 U 1 2 A0 ,12

, -12

, 0 ,12

,12

E

1 9 0 12

- 12

0 - 12

12

0 , 0 , 0 U 3 1 A0 ,12

, -12

, 0 ,12

, -12E

2 0 0 12

- 12

0 - 12

- 12

0 , 0 , - 1 U 1 6 A0 ,12

, -12

, 0 , -12

,12E

2 1 0 12

- 12

- 1 12

12

- 1 , 1 , 0 U 3 2 A0 ,12

, -12

, 0 , -12

, -12

E

2 2 0 12

- 12

- 1 12

- 12

- 1 , 1 , - 1 U 2 1 A0 ,12

, -12

, - 1 ,12

,12E

2 3 0 12

- 12

- 1 - 1

212

- 1 , 0 , 0 U 3 4 A0 ,

12

, -12

, - 1 ,12

, -12

E

2 4 0 12

- 12

- 1 - 1

2- 1

2 - 1 , 0 , - 1 U 2 5 A0 ,

12

, -12

, - 1 , -12

,12

E

2 5 0 - 12

12

1 12

12

1 , 0 , 1 U 3 5 A0 ,12

, -12

, - 1 , -12

, -12

E

2 6 0 - 12

12

1 12

- 12

1 , 0 , 0 U 5 A0 , -12

,12

, 1 ,12

,12

E

2 7 0 - 12

12

1 - 12

12

1 , - 1 , 1 U 8 A0 , -12

,12

, 1 ,12

, -12E

2 8 0 - 12

12

1 - 12

- 12

1 , - 1 , 0 U 3 7 A0 , -12

,12

, 1 , -12

,12E

2 9 0 - 12

12

0 12

12

0 , 0 , 1 U 3 8 A0 , -1

2,

1

2, 1 , -

1

2, -

1

2E

3 0 0 - 12

12

0 12

- 12

0 , 0 , 0 U 1 4 A0 , -12

,12

, 0 ,12

,12

E

3 1 0 - 12

12

0 - 12

12

0 , - 1 , 1 U 1 7 A0 , -1

2,

1

2, 0 ,

1

2, -

1

2E

3 2 0 - 12

12

0 - 12

- 12

0 , - 1 , 0 U 4 0 A0 , -12

,12

, 0 , -12

,12E

3 3 0 - 12

12

- 1 12

12

- 1 , 0 , 1 U 4 1 A0 , -12

,12

, 0 , -12

, -12

E

3 4 0 - 12

12

- 1 12

- 12

- 1 , 0 , 0 U 2 3 A0 , -12

,12

, - 1 ,12

,12E

3 5 0 - 12

12

- 1 - 12

12

- 1 , - 1 , 1 U 2 6 A0 , -12

,12

, - 1 ,12

, -12

E

3 6 0 - 12

12

- 1 - 1

2- 1

2

- 1 , - 1 , 0 U 4 3 A0 , -12

,12

, - 1 , -12

,12

E

3 7 0 - 12

- 12

1 12

12

1 , 0 , 0 U 4 4 A0 , -12

,12

, - 1 , -12

, -12

E

3 8 0 - 12

- 12

1 12

- 12

1 , 0 , - 1 U 9 A0 , -12

, -12

, 1 ,12

,12E

3 9 0 - 12

- 12

1 - 12

12

1 , - 1 , 0 U 3 0 A0 , -12

, -12

, 1 ,12

, -12

E

4 0 0 - 12

- 12

1 - 12

- 12

1 , - 1 , - 1 U 3 9 A0 , -12

, -12

, 1 , -12

,12

E

4 1 0 - 12

- 12

0 12

12

0 , 0 , 0 U 4 6 A0 , -12

, -12

, 1 , -12

, -12

E

4 2 0 - 12

- 12

0 12

- 12

0 , 0 , - 1 U 1 8 A0 , -12

, -12

, 0 ,12

,12E

4 3 0 - 12

- 12

0 - 12

12

0 , - 1 , 0 U 3 3 A0 , -12

, -12

, 0 ,12

, -12

E

4 4 0 - 12

- 12

0 - 12

- 12

0 , - 1 , - 1 U 4 2 A0 , -12

, -12

, 0 , -12

,12

E

4 5 0 - 12

- 12

- 1 12

12

- 1 , 0 , 0 U 4 7 A0 , -12

, -12

, 0 , -12

, -12

E

4 6 0 - 12

- 12

- 1 12

- 12

- 1 , 0 , - 1 U 2 7 A0 , -12

, -12

, - 1 ,12

,12

E

4 7 0 - 12

- 12

- 1 - 12

12

- 1 , - 1 , 0 U 3 6 A0 , -12

, -12

, - 1 ,12

, -12

E

4 8 0 - 12

- 12

- 1 - 12

- 12

- 1 , - 1 , - 1 U 4 8 A0 , -12

, -12

, - 1 , -12

, -12E

Table 10. The terms of nucleonic configuration and their multideterminantal wave functions

),221( 31

1146 PpssLi

Y 1 0 11 0 1 = 1"####2

HU 4 - U 5 L Y 1 0 01 0 1 = 1

2 HU 6 + U 7 - U 8 - U 9 L Y - 1 0 1

1 0 1 = 1"####2 HU 2 2 - U 2 3L

Y 1 0 - 11 0 1 = 1"####2

HU 2 9 - U 3 0LY 0 0 01 0 1 = 1

2 HU 1 5 + U 1 6 - U 1 7 - U 1 8 LY 0 0 - 1

1 0 1 = 1"####2 HU 3 2 - U 3 3L

Y 0 0 11 0 1 = 1"####2

HU 1 3 - U 1 4 LY - 1 0 01 0 1 = 1

2 HU 2 4 + U 2 5 - U 2 6 - U 2 7LY - 1 0 - 1

1 0 1 = 1"####2 HU 3 5 - U 3 6L

Table 11. The terms of nucleonic configuration and their

multideterminantal wave functions

),221( 33

1146 PpssLi

Y111111 =U1 Y101

111 = 1"####2 HU2 +U3LY1- 11111 =U37

Y011111 =U10 Y001

111 = 1"####2 HU13 +U14LY0- 11111 =U40

Y- 111111 =U19 Y- 101

111 = 1"####2 HU22 +U23LY- 1- 11111 =U43

Y101111 = 1"####2 HU4 +U5LY100

111 = 1

2 HU6 +U7 +U8 +U9LY1- 10

111 = 1"####2 HU38 +U39LY010

111 = 1"####2 HU11 +U12LY000111 = 1

2 HU15 +U16 +U17 +U18LY0- 10

111 = 1"####2 HU41 +U42LY- 110

111 = 1"####2 HU20 +U21LY- 100111 = 1

2 HU24 +U25 +U26 +U27LY- 1- 10

111 = 1"####2 HU44 +U45LY11- 1

111 =U28 Y10- 1111 = 1"####2 HU29 +U30LY1- 1- 1

111 =U46

Y01- 1111 =U31 Y00- 1

111 = 1"####2 HU32 +U33LY0- 1- 1111 =U47

Y- 11- 1111 =U34 Y- 10- 1

111 = 1"####2 HU35 +U36LY- 1- 1- 1111 =U48

Table 12. The values of coupling-projection coefficients and for nucleonic

Configuration

ij

klA ij

klB

),221( 33

1146 PpssLi 1s4 - 1s4

( Closed- Closed)

11111 A

11111 B

1s4 - 2s1

(Closed –Open) 4

11122 A

4

12211 A

4

11122 B

4

12211 B

1s4 -2p1

(Closed –Open)

12

11133 A

12

13311 A

12

11144 A

12

14411 A

12

11155 A

12

15511 A

12

11133 B

12

13311 B

12

11144 B

12

14411 B

12

11155 B

12

15511 B

2s1 -2p1

(Open-Open)

48

12233 A

48

13322 A

48

12244 A

48

14422 A

48

12255 A 48

15522 A

12

12233 B

12

13322 B

12

12244 B

12

14422 B

12

12255 B

12

15522 B