Generalizations of Bohr Model and D-scaling Method Alexey Sergeev 1.Hydrogen atom 2.Helium 3.H 2...
47
Generalizations of Bohr Model and D- scaling Method Alexey Sergeev 1.Hydrogen atom 2.Helium 3.H 2 molecule 4.Many-electron atoms (with new results)
Transcript of Generalizations of Bohr Model and D-scaling Method Alexey Sergeev 1.Hydrogen atom 2.Helium 3.H 2...
Generalizations of Bohr Model and D-scaling Method
Alexey Sergeev
1. Hydrogen atom2. Helium3. H2 molecule4. Many-electron atoms (with new results)
WittenMlodinow
andPapanicolaou …
http://www.dimensionality.info/ refs/index.htm
LewisLangmuir
BohrGeneralization
of angular momentum
M. Scully et al.
Herschbachet al.
Trend?
2
2010
3
4
5
6
7
8
Hydrogen atom, Bohr model and large D limit
),()/2(121
/ ml
lln
nrlnlm YnrLer
Bohrmodel
3D circular Rydberg states
at l→∞
Ground state at D→∞
9
Ground state
re100
Bohr model
10
states
400
4n
410 411 420 421
422 430 431 432 433
11
Circular Rydberg stateslmnl ,1
)1,1,( nnn
5n
12
)10(9,9,10 n
13
)20(19,19,20 n
14
)50(49,49,50 n
15
)100(99,99,100 n
n
16
)( 0
22 rrr
17
Interdimensional degeneracy
m=l state: )()()( riyxr l
0)()(2
1 )1(2)1(2
rErVdr
dr
dr
d
rl
l
l=0 state in D dimensions: )()()( rRrD
0)()(2
1 11
rRErVdr
dr
dr
d
rD
D
m=l state (D=3) l=0 state (D=3+2l)18
Two-electron atomsl=0 state in D dimensions:
),,(),( 2121)( rrrrD
0),,(),,(sinsin
111
11
2
1
21212
222
21
2
12
21
21
11
11
1
rrErrVrr
dr
dr
dr
d
rdr
dr
dr
d
r
DD
D
D
D
D
Circular states of heliumin four dimensions
),,()()(),( 21221121 rriwziyxrr ll
000
000
000
000
l
l
l
l
L
ijjiijjiij pqpqpqpqL 22221111
20
Interdimensional degeneracy
l-state (D=4) l=0 state (D=4+2l)
24 whichin
function for the Equation
4)(D function the
for Equation
lD
R
21
Bohr orbits for helium in four dimensions (x, y, z, w)
0/1 12 r
0)(
0)(
sin)(
cos)(
1
1
1
1
tw
tz
tRty
tRtx
tRtw
tRtz
ty
tx
sin)(
cos)(
0)(
0)(
2
2
2
2
Rwwzzyyxxtr
Rwzyxtr
Rwzyxtr
2)()()()()(
)(
)(
2/1221
221
221
22112
2/122
22
22
222
2/121
21
21
211
90
22
J. A. West et al., Classical limit states of the helium atom. Phys. Rev. A, 58,
186, 1998
23
Probability distribution of an electron in a helium atom in D dimensions
24
L. G. Yaffe, Physics Today
(1982)
25
26
1/D-expansion
30.95
738.2min E
27
Atoms in the large D limit
28
Hydrogen molecule
30
Traditional D-scaling
Alternative D-scaling
31
ground state
Classical limit of large D
32
0 1 2 3 4-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
Alternative scaling
exact
Traditional scaling
E, a
.u.
R, a.u.
Results for H2
33
Many-electron atoms
35
36
37
38
39
40
41
Alternative D-scaling for many-electron atoms
Svidzinsky et al. considered minimization in 3D space(when r1, …, rN have 3 components).
Total number of minimization variables 3N-3
General theory requires minimization in arbitrary dimensional space. It lowers the energy.The total number of variables is only nmax
because of symmetry.
42
Comparison of 3D vs. arbitrary D minimization
43
44
45
46
Conclusions
In the generalized Bohr model for helium, electrons move on stable circular orbits in 4D space and form rigid triangle configuration.
Alternative D-scaling for atoms requires consideration of arbitrary dimensional configurations of electrons. It gives some improvement in accuracy in comparison with the earlier model in 3D space.
Combining Bohr model with Thomas-Fermi theory considerably improves accuracy
