Post on 21-Jan-2016
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