Bohr model and dimensional scaling Bohr model and dimensional scaling analysis of atoms and molecules analysis of atoms and molecules Physics Department, Physics Department, Institute for Quantum Studies, Institute for Quantum Studies, Texas A&M University Texas A&M University Faculty: Faculty: Postdocs Postdocs : Students: : Students: Marlan Scully Dudley Herschbach Siu Chin Gordon Chen Anatoly Svidzinsky Robert Murawski Rui-Hua Xie Moochan Kim Kerim Urtekin Han Xiong Zhigang Zhang Atomic and molecular physics group Atomic and molecular physics group
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Bohr model and dimensional scaling Bohr model and dimensional scaling analysis of atoms and moleculesanalysis of atoms and molecules
Physics Department,Physics Department,
Institute for Quantum Studies, Institute for Quantum Studies,
Research HighlightsSubject Category: Quantum physics | Atomic and molecular physics'Bohr'n againAndreas TrabesingerAbstractA look back at Bohr's molecular model offers a fresh perspective on the formation of chemical bonds between atoms in hydrogen and other molecules. Although it is possible to model the electronic structure of molecules with great accuracy, such numerical methods provide little intuitive insight into electron−electron interactions. In two papers, in Physical Review Letters and Proceedings of the National Academy of Sciences, Anatoly Svidzinsky and colleagues 1,2 have taken a trip down memory lane to uncover an intriguing approach to understanding the chemical bonds within molecules, and at the same time take a fresh perspective on the "old quantum theory" developed by Niels Bohr in 1913.The famous Bohr model introduced the quantized nature of electron orbits in one-electron atoms, long before wave mechanics was developed. Much later, in the 1980s, the so-called D-scale approach provided a quantitative description of the two electrons surrounding a helium nucleus, by generalizing the Schrödinger equation to D dimensions; the situation relevant to the three-dimensional world is deduced by interpolating between the D = 1 and the D→∞ limits.However, neither approach — although each successful in its own realm — has so far yielded satisfactory results for two-centre problems, such as the hydrogen molecule. Svidzinsky et al. have re-examined the D-scale approach and show how a simple modification can fix its shortcomings 1. Whereas the original did not even predict a bound ground state for the hydrogen molecule, their new version provides quantitative values that are remarkably close to those obtained from extensive computer simulations. Furthermore, the authors show that, in the large- Dlimit, dimensional scaling can reproduce the Bohr model — notably by bringing in quantum mechanical concepts that were completely unknown to Bohr at the time. Svidzinsky et al. explore further 2 this link between 'new' and 'old', to demonstrate that Bohr's planetary model is indeed able to quantitatively describe the hydrogen molecule and some more complicated molecules such as diatomic lithium — and gives a clear physical picture of how a chemical bond forms.References1. Svidzinsky, A. A., Scully, M. O. & Herschbach, D. R. Simple and surprisingly accurate approach to the chemical bondobtained from dimensionality scaling. Phys. Rev. Lett. 95, 080401 (2005) 2. Svidzinsky, A. A., Scully, M. O. & Herschbach, D. R. Bohr's 1913 molecular model revisited. Proc. Natl Acad. Sci. 102, 11985–11988 (2005)
• Editors' Choice: Highlights of the recent literature• Chemistry
• Reviving Bohr Molecules
• Before the Heisenberg-Schrödinger formulation of quantum mechanics, the semiclassical Bohr-Sommerfeld theory successfully accounted for quantized properties such as the energy levels in the hydrogen atom. However, the forcing of closed orbits for particle motion ran afoul of the uncertainty principle. Recently, the use of D scaling, in which the motion of each particle is described by a vector in Ddimensions, was used to reintroduce the uncertainty principle to this earlier theory. When properly done, such equations reduce to the correct Schrödinger form for D = 3 but can still be solved in the more tractable D→∞ limit. This D scaling approach was applied successfully to atoms but did not yield bound states for molecules.
• Svidzinsky et al. have developed a D scaling description that fully quantizes one of the angles describing the interelectron coordinates and properly weights the contribution of electron-electron repulsion. After application of a leading correction term in 1/D, the potential energy curves for the lowest singlet, triplet, and excited states of H2 are in good agreement with accepted values after minimal numerical calculation. The procedure also yields reasonable agreement for the ground state of BeH. -- (P. D. Szuromi)
• Svidzinsky, A. A., Scully, M. O. & Herschbach, D. R. Phys. Rev. Lett. 95, 080401 (2005).
Bohr model of HBohr model of H22 moleculemolecule
,22
22
21 V
mp
mpE ++=
Kh ,3,2,1, ==⋅ nnpρ
Energy of the system:
where V is the Coulomb potential energy. Quantization condition:
radius)(Bohr length ofunit 2
2
0 −=me
a h (Hartree)energy ofunit 0
2
−ae
RZ
rrZ
rZ
rZ
rZVVnnE
baba
2
12221122
22
21
21 1,
21
++−−−−=+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
ρρ
0 1 2 3 4 5 6-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3 H2
1Σg+
3Σu+
_1
3
2
1
4
___H2 >2H (R>1.2)
___H2 >2H++2e
___H2 >H++H (R>1.68)
"exact" values
E, a
.u.
R, a.u.eV2.73a.u.100.0a.u.,10.1 === Be ER
eV4.74a.u.,4.1 == Be ERBohr model:
“Exact” value:
For n1=n2=1 there are four extremum configurations
Possible electron configurations correspond to extrema of E.
2,1,0,0 ==∂∂
=∂∂ iEzE
ii ρExtremum equations:
Charge distribution in HCharge distribution in H22 moleculemolecule
0 1 2 3 40.0
0.5
1.0
1.5
2.0Position of charge peak density z relative to the center of H2 molecule
R/2
Hund-Mulliken
Bohr model
Z, a
.u.
R, a.u.-1 0 1
1.0
1.5
2.0
e
e
-2 -1 0 1 20.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
BA
e
e
char
ge d
ensi
ty
H2 R=0.8 a.u.
z, a.u.
BA
R=1.4 a.u.
char
ge d
ensi
ty
z, a.u.
Bohr model of Bohr model of HeHHeH
),,,(21
21
12
2
RrrrVnE N
N
i i
i rK
rr∑=
+=ρ
0 1 2 3 4 5 6-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
8
"exact" values
HeH
E-E(
),
a.u.
R, a.u
For N electrons the model reduces to finding extrema of the energy
For HeH the three electrons cannot occupy the same lowest level of HeH++. For the configuration n1=n2=n3=1 the right energy corresponds to a saddle-pointrather then to a global minimum.
1)1)
2)2)
Bohr model
0 1 2 3 4 5 6 7 8 9 10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
"exact" values
r2r2 r1r1
e ee e HeHe
He2
8
E(R
)-E(
), a.
u.
R
Bohr model of HeBohr model of He22
Vrr
E ++= 22
21
11
R
Ground state
Bohr model of atomsBohr model of atoms
-2.2
-1.7
-1.2
-0.8
-0.08
0.6
1.2
2.8
5.5
∆, %
-128.919-126.043Ne10
-99.7190-98.0507F9
-75.0590-74.1716O8
-54.5840-54.1564N7
-37.8420-37.8128C6
-24.6520-24.7906B5
-14.6670-14.8381Be4
-7.4780-7.6890Li3
-2.9037-3.0625He2
Eexact a.u.EBohr a.u.SymbolZ
),,(21
21
12
2
N
N
i i
i rrrVrnE r
Krr∑
=
+=
Outer-shell electrons of Carbon form a regular tetrahedron in Bohr model. This is similar to bond structure of methane CH4.
DD--scaling analysis of H atom (ground sate)scaling analysis of H atom (ground sate)
Ψ=Ψ⎟⎠⎞
⎜⎝⎛ −∇− E
rZ
21 2
radius)(Bohr length ofunit 2
2
0 −=me
a h
(Hartree)energy ofunit 0
2
−ae
Schrödinger equation in 3 dimensions
D-scaling transformation
,12
211
12
rL
rr
rr= DD
D−−
− −⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
∇ ,2/1)-(D Φr=Ψ −
,4
)1( 2
rDr −→ ,
)1(4
2 EDE
−→
D=3 recovers 3-dimensional values
In the scaled variables Schrödinger equation reads
Φ=Φ⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
+∂∂
− ErrD
Dr
Z1)1()3(
21
1)-(D2
22
2
2
effective potential
Φ=Φ⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
+∂∂
− ErrD
Dr
Z1)1()3(
21
1)-(D2
22
2
2
Limit D → ∞:rZ
rE −= 22
1
dimensions 3 inanswer the withcoinsides2
2
←−=∞ZE
⎟⎟⎠
⎞⎜⎜⎝
⎛+++−= K2
212
2 21)-(D4
Dc
DcZED
The energy function is minium at
→−
− 2
2
)1(2DZ
1/D expansion
Zr 10 =
Exact hydrogen energy in D dimensions
( )2
2
D 232
nDZE+−
−=
Φ=Φ⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
+∂∂
− ErrD
Dr
Z1)1()3(
21
1)-(D2
22
2
2
1/D correction
Φ=Φ⎥⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛ −+
∂∂
− ErrDrZ11
21
D2
22
2
2
Keep terms contributing in 1/D
rZ
r ~1+=
Harmonic oscillator
Near the minimum
Φ=Φ⎥⎦
⎤⎢⎣
⎡−−+
∂∂
− ED
ZrZr
222
4
2
2
2Z
2~
2~D2
Shift in the effective potential
22
2222 ZDZ
DZZE −=−+−= 1/D correction = 0
Conventional DConventional D--scaling analysis of scaling analysis of HeHe atomatom
Goodson, Lopez-Cabrera, Herschbach & Morgan [J. Chem. Phys. 97, 8481 (1992)]calculated the expansion coefficients to order 30 by a recursive procedure. Analysis of the coefficients elucidates the singularity structure in the D→∞ limit which exhibits aspects of a square-root branch point. Using Pade-Borel summation they obtained 9significant figures for the He ground state energy.
Herschbach, J. Chem. Phys. 84, 838 (1986)
WittenPhys. Today 33, 38 (1980)
Witten
Herschbach
-3.2%-4.2%-12.1%-27.2%-58.1%
-1.0%1.8%-1.0%0.96%-5.7%
1/D41/D31/D21/DD→∞
HH22 moleculemolecule
2
2
22
22
22 11
φρ+
z+
ρρ
ρρ= D
D ∂∂
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
∇ −−
( ) Φρρ=Ψ D 2)/2(21
−−
( )( ) ( ) ( )Rz,ρ,DRz,ρ,E,
DE
41
14 2
2−
→−
→
( )Rφ,zzρρV+= ,,,,21
21H 2121
22
21 ∇−∇−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
∇ −−
−− φ
φφφρ
+z
+ρ
ρρρ
= DD
DD
3322
22
22 sin
sin111
V+ρ
+ρ
=Eφ22
221 sin
11121
⎟⎟⎠
⎞⎜⎜⎝
⎛
Conventional DConventional D--ScalingScaling
V+ρ
+ρ
=E ⎟⎟⎠
⎞⎜⎜⎝
⎛22
21
1121
Bohr model DBohr model D--scalingscaling
( ) Φρρ=Ψ DD )(sin 2/)3(2)/2(21 ϕ−−−−
D → ∞ limit:
ρ1
ρ2
z1 z2
R
Comparison of different D-scalings for H2
VE +⎟⎟⎠
⎞⎜⎜⎝
⎛+=
φρρ 222
21 sin
11121)1(
VE +⎟⎟⎠
⎞⎜⎜⎝
⎛+= 2
221
1121)2(
ρρ
Ground state E(R) of H2 molecule in the limit D=∞ calculated in two D-scaling schemes (solid lines) and the ``exact” energy in three dimensions (dots).
1/D correction for H1/D correction for H22 moleculemolecule
ρ1
ρ2
z1 z2
KK +−=⇒Φ=Φ⎥⎦
⎤⎢⎣
⎡+−+
∂∂
− ||2~||~D2 2
2
2
2 ss RRDAEEzRRA
z
R
0 1 2 3 4-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
including modified first 1/D correction8
D-scaling analysis
Bohr model (D= )
H2
E, a
.u.
R, a.u.
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
E, a
.u.
R, a.u.
Ground state E(R) of H2 molecule in the limit D=∞ and including modified 1/D correction
Interpolated D-scaling E(R) of H2
Modified 1/D correctionModified 1/D correction
2
1
0 1 2 3 4-3.0
-2.5
-2.0
-1.5
-1.0
Bohr model
Including 1/D correction
interpolation by 3rd order polynomial
E-1/
R, a
.u.
R, a.u.
Constrained Bohr modelConstrained Bohr model
0 1 2 3 4 5 6-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
21
H2
E, a
.u.
R, a.u.
VE +⎟⎟⎠
⎞⎜⎜⎝
⎛+= 2
221
1121
ρρ
Rρ1
ρ2
Molecular axis quantization (curve 1)
Atomic quantization (curve 2) Vrr
Eba
+⎟⎟⎠
⎞⎜⎜⎝
⎛+= 2
221
1121
Solution exists at R>2.77 a.u.
A B
r1b
R
r
r
1
2
e
e
HH22
In quantum mechanics the electron 1 is a cloud with characteristic size r. Interaction potential between the cloud and the nucleus B is Φ(r,R).In the Bohr model we treat electron as a point particle located distance rfrom A. Position of the point electron on the sphere gives right quantum mechanical answer for the particle interaction with the nucleus B if
),(1
1
Rrr b
Φ=−
HH22 moleculemoleculeΨ−Ψ=Φ
br1
1
)2()1()2()1( abba ±=Ψ
area 13
)1( α
πα −=
R
r1a
r2b
r1b
r2a
r
EffectiveEffective interaction potential between electron and the interaction potential between electron and the opposite nucleusopposite nucleus
Heitler-London trial function
breb 13
)1( α
πα −= r
1=α
,1)1()1(21)1(1
11
11
1
22
⎭⎬⎫
⎩⎨⎧
±±
−=Φ ∫∫ rdr
baSrdr
aS bb
rr∫= 1)1()1( rdbaS r
After integration
,11111
1),( //22
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +±⎟
⎠⎞
⎜⎝⎛ +−
±−=Φ −−
rRe
rS
Rre
RSRr rRrR
Coulomb integral
Exchange integral
⎟⎟⎠
⎞⎜⎜⎝
⎛++= −
2
2/
31
rR
rReS rR
Constrained Bohr model of HConstrained Bohr model of H22
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