Electronic states of confined 2-electron quantum systems · Confined quantum systems Structure of...
Transcript of Electronic states of confined 2-electron quantum systems · Confined quantum systems Structure of...
BackgroundModel
ResultsSummary and outlook
Electronic states ofconfined 2-electron quantum systems
Tokuei Sako1 Geerd HF Diercksen2
1Nihon University, College of Science and TechnologyFunabashi, Chiba, JAPAN
2Max-Planck-Institut für AstrophysikGarching, GERMANY
October 17, 2007
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Confined quantum systems
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Confined quantum systems
Quantum systems and potentials
Confined systems: electrons (quantum dots, artificialatoms and molecues), atoms, moleculesConfining potentials: exponential potentials, Gaussianpotentials, magnetic fields, electric fields
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Confined quantum systems
Artificial atoms
Artificial atoms are small boxes ≈ 100nm along a side,contained in a semiconductor, and holding a number ofelectrons.In artificial atoms electrons are typically traped in a bowllike parabolic potential.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Confined quantum systems
Structure of artificial atoms
Figure: Quantum dot. Areasshown in blue are metallic,shown in white are insulating(AlGaAs), and shown in red aresemiconducting (GaAs).
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Outline
Schrödinger equationConfiguration interaction (CI) methodConfining potentialGaussian basis set
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Schrödinger equation
[H(r)]Ψ(1,2, . . . ,N) = EΨ(1,2, . . . ,N)
H(r) =N∑
i=1
[−1
2∇2i
]+
N∑i=1
M∑α=1
[− Zα
|ri − Rα|
]+
N∑i=1
w(r i)
+N∑
i>j
[1∣∣ri − rj
∣∣]
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
One-determinant wavefunction
|Ψ〉 = Ψ(x1x2 · · ·xN) = (N!)−12
∣∣∣∣∣∣∣∣∣χi(x1) χj(x1) · · · χk (x1)χi(x2) χj(x2) · · · χk (x2)
......
...χi(xN) χj(xN) · · · χk (xN)
∣∣∣∣∣∣∣∣∣χ =
{ψ · αψ · β
χ: one-electron spin functionψ: one-electron space functionα, β: spin functions
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Hartree-Fock method
f (i)ψ(xi) = εiψ(xi)
f (i) = −12∇2
i −M∑
α=1
[Zα
|ri − Rα|
]+ w(r i) + v(i)
f (i): Hartree-Fock operatorψ: one-electron space functionε: orbital energyv(i): averaged field of (N 63 i) electrons
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
LCAO/LCGO approximation
ψi =∑
m
cimξm
ψi : one-eletron space functioncim: linear combination coefficientξm ∝ re−αmr : hydrogenic function ≡ Slater functionξm ∝ re−αmr2
: Gaussian function
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Aufbau principle
6E
Ψg
↑↓↑↓↑
↑↓↑↓↓
Ψe
↑↓↑↑↓
↑↓↓↑↓
Ψg = |ψ1(1)ψ1(2)ψ2(3)ψ2(4)ψ3(5)〉±|ψ1(1)ψ1(2)ψ2(3)ψ2(4)ψ3(5)〉
Ψe = |ψ1(1)ψ1(2)ψ2(3)ψ3(5)ψ3(4)〉±|ψ1(1)ψ1(2)ψ2(3)ψ3(5)ψ3(4)〉
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Configuration interaction wavefunction
|Φ〉 = C0|Ψ0〉 +Xra
Cra|Ψ
ra〉 +
Xa<br<s
Crsab|Ψ
rsab〉 +
Xa<b<cr<s<t
Crstabc |Ψ
rstabc〉 + · · ·
Ψ0
c •b •a •
...
r
...
s
...
t
Ψra
c •b •a
...
r •...
s
...
t
Ψrsab
c •ba
...
r •...
s •...
t
Ψrstabc
cba
...
r •...
s •...
t •
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Anisotropic harmonic oscillator potential
Anisotropic harmonic oscillator potential:
w(r i) =12
[ω2
x x2i + ω2
y y2i + ω2
z z2i
]
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Anisotropic harmonic oscillator eigenvalues
Eigenvalues of an anisotropic harmonic oscillator:
Eω0 = ωx(νx + 1/2) + ωy (νy + 1/2) + ωz(νz + 1/2).
(νx , νy , νz): harmonic oscillator quantum numbers
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Anisotropic harmonic oscillator eigenfunctions
Anisotropic harmonic oscillator eigenfunctions:
χ~ω~ν (~r) = N~ω
~ν Hνx (x)Hνy (y)Hνz (z) exp[−1
2(ωxx2 + ωyy2 + ωzz2)
].
N~ω~ν : normalization constant
Hνx (x), etc.: Hermite polynomial
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Spherical harmonic oscillator eigenvalues
Eigenvalues for an electron confined in a spherical harmonicoscillator potential (ωx = ωy = ωz = ω):
Eω0 = ω(2ν + `+ 3/2)
.
ν, ν = 0,1,2, ... : principal quantum number`, ` = 0,1,2, ... : one-electron angular momentum quantum number
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy sequence for 1 electron
Sequence of the energies Eω0 [ν1`1] for one electron confined in
a spherical harmonic oscillator potential:
Eω0 [0s] = (3/2)ω,
Eω0 [0p] = (5/2)ω,
Eω0 [0d ] = Eω
0 [1s] = (7/2)ω,Eω
0 [0f ] = Eω0 [1p] = (9/2)ω,
Eω0 [0g] = Eω
0 [1d ] = Eω0 [3s] = (11/2)ω, ...
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy sequence for 2 non-interacting electrons
Sequence of the energies Eω0 [ν1`1ν2`2] for two non-interacting
electrons confined in a spherical harmonic oscillator potential:
Eω0 [(0s)2] = 3ω,
Eω0 [0s0p] = 4ω,
Eω0 [0s1s] = Eω
0 [0s0d ] = Eω0 [(0p)2] = 5ω,
Eω0 [0s1p] = Eω
0 [1s0p] = Eω0 [0s0f ] = Eω
0 [0p0d ] = 6ω, ...
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy sequence for 2 interacting electrons
Sequence of the singlet energies Eω0 [ν1`1ν2`2] for two
interacting electrons confined in a spherical harmonic oscillatorpotential for small splittings of the degenerate levels:
Eω[S(0s)2] < Eω[P(0s0p)] < Eω[D(0s0d)] <
< Eω[S(0s1s)] < Eω[S(0p)2] < Eω[D(0p)2] < · · · .
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Transition dipol matrix
The transition dipole matrix element between a fundamentalstate and a state constructed by applying the A†
ξ operator tothis fundamental state is given by
⟨ΨE0+ωξ
∣∣ N∑i=1
ξi∣∣ΨE0
⟩=
√N
2ωξ, (ξ = x , y , z).
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Dipol polarizability
The dipole polarizability of the harmonic oscillator quantum dotis determined by the analytically expression
αxx = −[∂2E(Fx)
∂F 2x
]Fx=0
=Nω2
x,
αyy = −
[∂2E(Fy )
∂F 2y
]Fy=0
=Nω2
y,
αzz = −[∂2E(Fz)
∂F 2z
]Fz=0
=Nω2
z.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Potential energy curves
-4
-3
-2
-1
0
1
2
-12 -8 -4 0 4 8 12
W
r
T+V+W+GT+V+G
T+V
T+W
T+W+G
Broken line:Free helium atom potential V (r) = −2/r
Dotted line:Confinement potential (ω = 1) W (r) = r2
/2
Solid line:Total external potential V (r) + W (r)
Horizontal lines:Eigenvalues of the Hamiltonian
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Hamiltonian terms
-4
-3
-2
-1
0
1
2
-12 -8 -4 0 4 8 12
W
r
T+V+W+GT+V+G
T+V
T+W
T+W+G
T : Kinetic electron energy
V : Helium atom potential
W : Confining (exponential) potential
G: Electron interaction potential
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy levels
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
E [
a.u.
]
E(T+V) E(T+V+G) E(T+V+W+G) E(T+W+G) - 3 E(T+W) - 3
S
PS
D
PS
D
PS
[E(T + V )]:He with electron interaction neglected
[E(T + V + G)]:He with electron interaction included
[E(T + V + W + G)]:He confined in a Hooke’s-law potential
[E(T + W + G)] − 3:2e quantum dot with e interaction included
[E(T + W )] − 3:2e quantum dot with e interaction neglected
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Anisotropic Gaussian functions
ξc~a,~ζ
(~r) = xax yay zaz exp(−ζxx2 − ζyy2 − ζzz2)
a = ax + ay + az = 0, 1, 2, ... : s-, p-, d-, ... type orbitals(ζx , ζy , ζz) = (ωx/2, ωy/2, ωz/2)
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy levels of a 1 electron quantum dot
0.6
0.8
1.0
1.2
1.4
1.6
1.8
[2][2]
[3]
[1]
[1] [1] [1] [1]
[1] [1] [1] [1] [1]
(0,2,0)(1,1,0)(2,0,0)
(1,0,2)(0,1,2)(0,0,4)
(1,0,1)(0,1,1)(0,0,3)
(1,0,0)(0,1,0)(0,0,2)
(0,0,1)
(0,0,0)
E /a
.u.
analytical
[1]
[1]
[1]
[2]
[2]
[1]
[1]
[2]
s-GTO(0.5,0.5,0.25)Prolate1e
[6]
[6][2]
[1]
[1][1][2]
[3][2]
[2][2][2]
[2]
(147)11s8p6d5f4g1h
(95)10s7p5d3f2g
(77)10s7p5d3f
(56)10s7p5d
[2]
[2]
[1]
[1]
[1]
[2]
[1]
[1]
[1]
[1]
[2](1,1,1)
(2,0,1)(0,2,1)
(1,0,3)(0,1,3)(0,0,5)
Figure: Energy levels of oneelectron confined by a prolateharmonic oscillator potentialwith(ωx , ωy , ωz) = (0.5,0.5,0.25) fordifferent size Gaussian basissets. The analytical spectrumlabeled by the harmonicoscillator quantum numbers(νx , νy , νz) is shown at the righthand side.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy levels of a 1 electron quantum dot
0.6
0.8
1.0
1.2
1.4
1.6
1.8
[3] [3] [3]
[1] [1] [1] [1] [1]
[1] [1] [1] [1] [1]
(0,2,0)(1,1,0)(2,0,0)
(1,0,2)(0,1,2)(0,0,4)
(1,0,1)(0,1,1)(0,0,3)
(1,0,0)(0,1,0)(0,0,2)
(0,0,1)
(0,0,0)
E /a
.u.
analytical
[3]
c-aniGTO(0.5,0.5,0.25)Prolate1e
[6]
[6][6]
[6][6]
[6]
[5][3]
[6][5]
[3][3][3][3][3]
[3]
(84)1s1p1d1f1g1h1i
(56)1s1p1d1f1g1h
(36)1s1p1d1f1g
(20)1s1p1d1f
(1,1,1)
(2,0,1)(0,2,1)
(1,0,3)(0,1,3)(0,0,5)
Figure: Energy levels of oneelectron confined by a prolateharmonic oscillator potentialwith(ωx , ωy , ωz) = (0.5,0.5,0.25) fordifferent size anisotropicGaussian basis sets. Theanalytical spectrum labeled bythe harmonic oscillator quantumnumbers (νx , νy , νz) is shown atthe right hand side.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy levels of a 1 electron quantum dot
0.6
0.8
1.0
1.2
1.4
1.6
1.8
[2] [2] [3]
[1] [1] [1] [1] [1]
[1] [1] [1] [1] [1]
(0,2,0)(1,1,0)(2,0,0)
(1,0,2)(0,1,2)(0,0,4)
(1,0,1)(0,1,1)(0,0,3)
(1,0,0)(0,1,0)(0,0,2)
(0,0,1)
(0,0,0)
E /a
.u.
analytical
[2]
[1]
[1][1][1][1]
[2]
[1][1]
[2]
[1][1][1]
s-aniGTO(0.5,0.5,0.25)Prolate1e
[6]
[6][2]
[2][2]
[2]
[2][2]
[2][2]
[3][2][2][2][2]
[2]
(49)1s1p1d1f1g1h1i
(36)1s1p1d1f1g1h
(25)1s1p1d1f1g
(16)1s1p1d1f
[1]
[2][2]
(1,1,1)
(2,0,1)(0,2,1)
(1,0,3)(0,1,3)(0,0,5)
Figure: Energy levels of oneelectron confined by a prolateharmonic oscillator potentialwith(ωx , ωy , ωz) = (0.5,0.5,0.25) fordifferent size sphericalanisotropic Gaussian basissets. The analytical spectrumlabeled by the harmonicoscillator quantum numbers(νx , νy , νz) is shown at the righthand side.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Full CI energies of a 2 electron quantum dot
Table: Full CI energies of the lowest four singlet states of an oblateharmonic oscillator 2-electron quantum dot with (ωx , ωy , ωz) =(0.1,0.1,0.5) for different size anisotropic Gaussian basis sets.
State [1s1p1d] [1s1p1d1f] [1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h1i](10) (20) (35) (56) (84)
11Σ+g 0.9317 0.9314 0.9311 0.9310 0.9309
11Πu 1.0590 1.0316 1.0314 1.0312 1.031011∆g 1.0423 1.0375 1.0362 1.0361 1.036121Σ+
g 1.1472 1.1099 1.1050 1.1044 1.1040
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Full CI energies of a 2 electron quantum dot
Table: Full CI energies of the lowest six singlet and triplet states,respectively, of an anisotropic harmonic oscillator 2-electron quantumdot with (ωx , ωy , ωz) = (0.1, 0.15, 0.2) for different size anisotropicGaussian basis sets.
[1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h1i] [3s3p1d1f1g1h1i]State (35) (56) (84) (92)
Singlet11Ag 0.6876 0.6874 0.6873 0.687211B3u 0.7879 0.7876 0.7875 0.787421Ag 0.8180 0.8176 0.8174 0.817311B2u 0.8379 0.8376 0.8375 0.837411B1g 0.8448 0.8447 0.8447 0.844711B1u 0.8879 0.8876 0.8874 0.8874Triplet13B3u 0.7186 0.7185 0.7185 0.718413B2u 0.7852 0.7851 0.7850 0.785013Ag 0.8190 0.8187 0.8185 0.818513B1u 0.8449 0.8447 0.8447 0.844713B1g 0.8688 0.8686 0.8685 0.868523B1g 0.8854 0.8852 0.8851 0.8851
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Transition probabilities of a 2 electron quantum dot
Table: Transition probabilities between the ground state and thelowest three excited states having non-zero probability of a harmonicoscillator 2-electron quantum dot with (ωx , ωy , ωz) = (0.1, 0.15, 0.2) fordifferent size anisotropic Gaussian basis sets. The number in theround bracket represents the total number of basis functions.
[1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h1i] [3s3p1d1f1g1h1i] analyticalTransition (35) (56) (84) (92)
Singlet11B3u−11Ag 10.012 10.011 10.006 9.998 10.011B2u−11Ag 6.671 6.671 6.669 6.664 6 2
311B1u−11Ag 5.002 5.002 5.001 4.998 5.0
Triplet13Ag−13B3u 10.101 10.010 10.007 10.003 10.013B1g−13B3u 6.676 6.667 6.667 6.666 6 2
313B2g−13B3u 5.004 5.000 5.000 4.999 5.0
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Energy levels of a 3 electron quantum dot
0.326
0.328
0.330
0.332
0.334
0.336
E /
a.u.
1 2Σg
-
1 2Σg
+2 2∆g
1 2Γg
1 2∆g
1 2Πu
(0.01, 0.01, 0.1)3e
reduced c-aniGTO
(81)(165)(120)(84)(56)
normal c-aniGTO
Figure: Energy levels of the lowlying doublet states of an oblateharmonic oscillator 3-electronquantum dot with (ωx , ωy , ωz) =(0.01, 0.01, 0.1) for differentsize anisotropic Gaussian basissets.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Full CI energy levels of the helium atom
CI(0.5,0.5,0.25)Prolate
c-GTOc-GTO + c-aniGTO
E /a
.u.
He
-2.5
-2.0
-1.5
-1.0
-0.5
11∆g
(117) [10s7p5d][1s1p1d1f1g1h]
(61)[10s7p5d]
(96) [10s7p5d][1s1p1d1f1g]
(93) [7s4p3d][1s1p1d1f1g1h]
(81) [10s7p5d][1s1p1d1f]
21Πu
41Σg
+
21Σu
+
11Πg 31Σg
+
11Πu
21Σg
+
11Σu
+
11Σg
+
Figure: Full CI energy levels ofthe helium atom confined by aprolate harmonic oscillatorpotential with(ωx , ωy , ωz) = (0.5,0.5,0.25) fordifferent size anisotropicGaussian basis sets. The totalnumber of basis functions isgiven in round brackets.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Full CI energies of the helium atom
Table: Full CI energies of the lowest singlet 11Σ+g state of the helium
atom confined by an oblate harmonic oscillator potential with(ωx , ωy , ωz) = (0.1,0.1,0.5) for different size anisotropic Gaussianbasis sets.
[13s7p5d] [13s7p5d] [13s7p5d] [13s7p5d] [9s3p2d][1s1p1d] [1s1p1d1f] [1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h]
(74) (84) (99) (120) (86)-0.2111 -0.2113 -0.2114 -0.2116 -0.2116
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Polarizability of the helium atom
Table: Polarizability tensor components of the lowest singlet 11Σ+g
state of the helium atom confined by an oblate harmonic oscillatorpotential with (ωx , ωy , ωz) = (0.1,0.1,0.5) for different size anisotropicGaussian basis sets.
[13s7p5d] [13s7p5d] [13s7p5d] [13s7p5d] [9s3p2d][1s1p1d] [1s1p1d1f] [1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h]
(74) (84) (99) (120) (86)αxx 19.0 19.2 19.2 19.2 19.2αzz 3.92 3.96 3.96 3.96 3.96
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Multi-reference CI energies of the lithium atom
Table: Multi-reference CI energies of the lowest eight doublet statesof the lithium atom confined by a harmonic oscillator potential with(ωx , ωy , ωz) = (0.1, 0.15, 0.2) for different size anisotropic Gaussianbasis sets. The number in the round bracket represents the totalnumber of basis functions.
[10s5p3d] [10s5p3d] [10s5p3d] [10s5p3d] [10s5p3d][1s1p1d] [1s1p1d1f] [1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h1i]
State (53) (63) (78) (99) (127)12Ag -7.3255 -7.3255 -7.3256 -7.3256 -7.325612B3u -7.2645 -7.2651 -7.2651 -7.2652 -7.265212B2u -7.2386 -7.2388 -7.2388 -7.2389 -7.238912B1u -7.2077 -7.2080 -7.2080 -7.2082 -7.208222Ag -7.0476 -7.0476 -7.0481 -7.0481 -7.048312B1g -7.0224 -7.0225 -7.0234 -7.0234 -7.023722B3u -6.9785 -6.9840 -6.9840 -6.9843 -6.984312B2g -6.9798 -6.9799 -6.9819 -6.9820 -6.9825
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Transition probabilities of the lithium atom
Table: Transition probabilities between the ground state and thelowest three excited states with non-zero probability of the lithiumatom confined by a harmonic oscillator potential with (ωx , ωy , ωz) =(0.1, 0.15, 0.2) for different size anisotropic Gaussian basis sets. Thenumber in the round bracket represents the total number of basisfunctions.
[10s5p3d] [10s5p3d] [10s5p3d] [10s5p3d] [10s5p3d][1s1p1d] [1s1p1d1f] [1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h1i]
Transition (53) (63) (78) (99) (127)12B3u−12Ag 3.513 3.523 3.523 3.523 3.52312B2u−12Ag 2.664 2.666 2.666 2.666 2.66612B1u−12Ag 2.073 2.074 2.073 2.073 2.07222B3u−12Ag 0.430 0.391 0.391 0.389 0.389
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
Outline
Schrödinger equationConfiguration interaction (CI) methodConfining potentialGaussian basis set
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
HF orbital energies
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0 2p1f
2s1d
1f2p1d
2s
1f2p
1d
1p2s
2p1f2s1d1p
1s
1f
1d
2s
1f2p
1d
1p2s
1s
2eHe H-
2eH-He
ω = 0.5Spherical
Orb
ital e
nerg
y /a
.u.
1p
1s
1p
1s
-0.5
0.0
0.5
1.0
3s2p
1p
1s
Sphericalω = 0.1
Orb
ital e
nerg
y /a
.u.
1s HF
HF
Figure: Hartree-Fock orbitalenergies of the helium atom,the hydrogen negative ion andof two electrons confined by aspherical harmonic oscillatorpotential with (ωx , ωy , ωz) = (0.1,0.1,0.1) (upper fig.) and (0.5,0.5, 0.5) (lower fig.).
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
HF orbital densities
Spherical
1f
2s
1d
1p
1s
2p
ω = 0.5
He
HF
Figure: Hartree-Fock orbitaldensities of the helium atomconfined by a sphericalharmonic oscillator potentialwith (ωx , ωy , ωz) = (0.5, 0.5,0.5).
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
HF orbital densitiesO
rb
ita
l e
ne
rg
y /a
.u
.
σg
2
πu
2
σu
1
σu
2
πu
1
σg
3
σg
1
σg
4
σg
1
σu
1
σg
2
πu
1
σu
2
1πg
σg
3
σg
4
Prolate
ω = 0.5
HF
He H-
2e
σg
1
σu
1
σg
2
σu
2
πu
1
1πg
σg
3
σg
4
1πg
1δg
πu
2
1δg
πu
2
1δg
Figure: Hartree-Fock orbitaldensities and energies for thehelium atom, the hydrogennegative ion and two electronsconfined by a prolate harmonicoscillator potential with(ωx , ωy , ωz) = (0.5, 0.5, 0.25)
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Full CI energies
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
E /a
.u.
21D31S
11F 11D
11P21S
11S
Spherical
21D31S21P
11F
11D
21S11P
31S
11S21D
31S
He
11P
11S
21S
Sphericalω = 0.1
11S
2e 2eH-H-
21P31D11F
21D
11D
11P
11P
21D31S21S
11D 11P11S
21P
11F31D21P
11S
-2
-1
0
1
2
3
11F21P 31S
21D
11D21S
11F21P
11D
21S
He
ω = 0.5
Figure: Full CI energies of thehelium atom, the hydrogennegative ion and of twoelectrons confined by aspherical harmonic oscillatorpotential with (ωx , ωy , ωz) = (0.1,0.1, 0.1) (left fig.) and (0.5, 0.5,0.5) (right fig.).
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Electron densities
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Figure: Electron densities of thehelium atom, the hydrogennegativ ion and of two electronsconfined by a sphericalharmonic oscillator potentialwith(ωx , ωy , ωz) = (0.5,0.5,0.5).
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Leading configurations
Table: Leading configurations and their squared norms for the lowestfour singlet states of the helium atom, the hydrogen negative ion andof two electrons confined by a spherical harmonic oscillator potentialwith ω = 0.5.
He H− 2eState config. norm config. norm config. norm11S (1s)2 0.994 (1s)2 0.987 (1s)2 0.97311P (1s)(1p) 0.970 (1s)(1p) 0.970 (1s)(1p) 0.96621S (1s)(2s) 0.973 (1s)(2s) 0.969 (1p)2 0.48711D (1s)(1d) 0.965 (1s)(1d) 0.902 (1s)(1d) 0.480
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Leading configurations
Table: Leading configurations and their squared norms for the lowestfour singlet states of the helium atom, the hydrogen negative ion andof two electrons confined by a prolate harmonic oscillator potentialwith (ωx ,ωy ,ωz) = (0.5,0.5,0.25).
He H− 2eState config. norm config. norm config. norm11Σ+
g (1σg )2 0.993 (1σg )2 0.985 (1σg )2 0.94211Σ+
u (1σg )(1σu) 0.951 (1σg )(1σu) 0.950 (1σg )(1σu) 0.91021Σ+
g (1σg )(2σg ) 0.960 (1σg )(2σg ) 0.936 (1σg )(2σg ) 0.48911Πu (1σg )(1πu) 0.963 (1σg )(1πu) 0.964 (1σg )(1πu) 0.93731Σ+
g (1σg )(3σg ) 0.915 (1σu)2 0.648 (1σg )(2σg ) 0.44611Πg (1σg )(1πg ) 0.952 (1σg )(1πg ) 0.905 (1σg )(1πg ) 0.48221Σ+
u (1σg )(2σu) 0.927 (1σg )(2σu) 0.947 (1σg )(2σu) 0.729
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Leading configurations
Table: Leading configurations and their squared norms for the lowestfour singlet states of the helium atom, the hydrogen negative ion andof two electrons confined by a oblate harmonic oscillator potentialwith (ωx ,ωy ,ωz) = (0.5,0.5,0.25).
He H− 2eState config. norm config. norm config. norm11Σ+
g (1σg )2 0.993 (1σg )2 0.983 (1σg )2 0.94611Πu (1σg )(1πu) 0.946 (1σg )(1πu) 0.946 (1σg )(1πu) 0.92521Σ+
g (1σg )(2σg ) 0.959 (1σg )(2σg ) 0.944 (1σg )(2σg ) 0.48711Σ+
u (1σg )(1σu) 0.956 (1σg )(1σu) 0.959 (1σg )(1σu) 0.94111∆g (1σg )(1δg ) 0.949 (1σg )(1δg ) 0.913 (1σg )(1δg ) 0.483
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Electron correlation energy
0.0 0.1 0.2 0.3 0.4 0.50.02
0.03
0.04
0.05
0.06
2e
Spherical
Oblate
Prolate
|EC
I-ES
CF|
/a.u
.
ω
He・
0.0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
12
He・
2e
Spherical
Oblate
Prolatex100
[EC
I-ES
CF]/
EC
I
ω
Figure: Electron correlationenergy of a spherical, a prolate,and an oblate 2 electronharmonic oscillator quantum dotas function of ω.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Full CI electron density distribution
���� ��
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����������
Figure: Electron densitydistribution of the lowest singlet11Σ+
g state of He, H−, and oftwo electrons confined by aspherical harmonic oscillatorpotential with(ωx , ωy , ωz) = (ω, ω, ω), ω = 0.1,0.2, 0.4, and 0.8.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Polarizability tensor components
Table: Polarizability tensor components of the lowest singlet 11Σ+g
state of He, H−, and of two electrons confined by a sphericalharmonic oscillator potential for different values of ω.
(ωx , ωy , ωz) He H− 2eαzz (0.1, 0.1, 0.1) 1.31 2.74×101 2×102
(0.2, 0.2, 0.2) 1.16 1.14×101 5×101
(0.4, 0.4, 0.4) 0.850 4.24 1.25×101
(0.6, 0.6, 0.6) 0.631 2.27 5 59
(0.8, 0.8, 0.8) 0.483 1.43 3.125(1.0, 1.0, 1.0) 0.382 0.988 2(1.2, 1.2, 1.2) 0.309 0.728 1 7
18
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Dipole polarizability and correlation contribution
0.0 0.2 0.4 0.6 0.8 1.0 1.20
5
10
15
20
25
ω
α zz(C
I)
2e
H-
He
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.01
0.1
1Spherical
Spherical
α zz(C
I) - α
zz(H
F)
H-
He
ω
Figure: Dipole polarizability(upper fig.) and electroncorrelation contribution (lowerfig.) of the lowest singlet 11Σ+
gstate of He and H− confined bya spherical harmonic oscillatorpotential with(ωx , ωy , ωz) = (ω, ω, ω), ω = 0.1 -1.2.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Full CI electron density distribution
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����������
����������
����������
Figure: Electron densitydistribution of the lowest singlet11Σ+
g state of He, H−, and oftwo electrons confined by aprolate-type harmonic oscillatorpotential with(ωx , ωy , ωz) = (ω, ω,0.1), ω =0.1, 0.2, 0.4, and 0.8.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Polarizability tensor components
Table: Polarizability tensor components of the lowest singlet 11Σ+g
state of He, H−, and of two electrons confined by a prolate-typeharmonic oscillator potential for different values of confinementparameters (ωx , ωy , ωz) = (ω, ω,0.1).
(ωx , ωy , ωz ) He H− 2eαzz (0.1, 0.1, 0.1) 1.31 2.74×101 2×102
(0.2, 0.2, 0.1) 1.26 2.12×101 2×102
(0.4, 0.4, 0.1) 1.11 1.52×101 2×102
(0.6, 0.6, 0.1) 0.969 1.22×101 2×102
(0.8, 0.8, 0.1) 0.855 1.03×101 2×102
(1.0, 1.0, 0.1) 0.762 9.08 2×102
(1.2, 1.2, 0.1) 0.685 8.15 2×102
αxx (0.1, 0.1, 0.1) 1.31 2.74×101 2×102
(0.2, 0.2, 0.1) 1.18 1.23×101 5×101
(0.4, 0.4, 0.1) 0.898 4.81 1.25×101
(0.6, 0.6, 0.1) 0.682 2.61 5 59
(0.8, 0.8, 0.1) 0.530 1.66 3.125(1.0, 1.0, 0.1) 0.423 1.15 2(1.2, 1.2, 0.1) 0.345 0.844 1 7
18
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Polarizability tensor components
Table: Polarizability tensor components of the lowest singlet 11Σ+g
state of He, H−, and of two electrons confined by a prolate-typeharmonic oscillator potential for different values of (ωx , ωy , ωz).
(ωx , ωy , ωz ) He H− 2eαzz (0.1, 0.1, 0.1) 1.31 2.74×101 2×102
(0.1, 0.1, 0.2) 1.21 1.34×101 5×101
(0.1, 0.1, 0.4) 0.953 5.46 1.25×101
(0.1, 0.1, 0.6) 0.744 3.01 5 59
(0.1, 0.1, 0.8) 0.590 1.92 3.125(0.1, 0.1, 1.0) 0.476 1.33 2(0.1, 0.1, 1.2) 0.391 0.976 1 7
18αxx (0.1, 0.1, 0.1) 1.31 2.74×101 2×102
(0.1, 0.1, 0.2) 1.28 2.40×101 2×102
(0.1, 0.1, 0.4) 1.20 2.04×101 2×102
(0.1, 0.1, 0.6) 1.11 1.83×101 2×102
(0.1, 0.1, 0.8) 1.03 1.70×101 2×102
(0.1, 0.1, 1.0) 0.958 1.61×101 2×102
(0.1, 0.1, 1.2) 0.897 1.54×101 2×102
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Dipole polarizability
0.0 0.2 0.4 0.6 0.8 1.0 1.20
4
8
12
16
20
24
28
H-
He
α zz(C
I)
ω
0.0 0.2 0.4 0.6 0.8 1.0 1.20
4
8
12
16
20
24
28
α xx(C
I)
(ω, ω, 0.1)
(ω, ω, 0.1)
Prolate
Prolate
ω
H-
He
Figure: Dipole polarizability ofthe lowest singlet 11Σ+
g state ofHe and H− confined by aprolate-type harmonic oscillatorpotential with(ωx , ωy , ωz) = (ω, ω,0.1), ω =0.1 - 1.2.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Correlation contribution to the dipole polarizability
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.01
0.1
1
H-
He
α zz(C
I) - α
zz(H
F)
ω
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.01
0.1
1
α xx(C
I) - α
xx(H
F)
(ω, ω, 0.1)
(ω, ω, 0.1)
Prolate
Prolate
ω
H-
He
Figure: Electron correlationcontribution to the dipolepolarizability of the lowestsinglet 11Σ+
g state of He and H−
confined by a prolate-typeharmonic oscillator potentialwith (ωx , ωy , ωz) = (ω, ω,0.1), ω= 0.1 - 1.2.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Full CI electron density distribution
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����������
����������
����������
Figure: Electron densitydistribution of the lowest singlet11Σ+
g state of He, H−, and oftwo electrons confined by anoblate-type harmonic oscillatorpotential with(ωx , ωy , ωz) = (0.1,0.1, ωz), ωz= 0.1, 0.2, 0.4, and 0.8.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Dipole polarizability
0.0 0.2 0.4 0.6 0.8 1.0 1.20
4
8
12
16
20
24
28
α zz(C
I)
ω
ω
Oblate
H-
He
0.0 0.2 0.4 0.6 0.8 1.0 1.20
4
8
12
16
20
24
28
(0.1,0.1,ω)
(0.1,0.1,ω)
α xx(C
I)
Oblate
H-
He
Figure: Dipole polarizability ofthe lowest singlet 11Σ+
g state ofHe and H− confined by anoblate-type harmonic oscillatorpotential with(ωx , ωy , ωz) = (0.1,0.1, ω), ω =0.1 - 1.2.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
Energy and electron densityDipole polarizability
Correlation contribution to the dipole polarizability
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.01
0.1
1
α zz(C
I) - α
zz(H
F)
ω
ω
Oblate
H-
He
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.01
0.1
1
(0.1,0.1,ω)
(0.1,0.1,ω)
α xx(C
I) - α
xx(H
F)
Oblate
H-
He
Figure: Electron correlationcontribution to the dipolepolarizability of the lowestsinglet 11Σ+
g state of He and H−
confined by an oblate-ellipticalharmonic oscillator potentialwith (ωx , ωy , ωz) = (0.1,0.1, ω),ω = 0.1 - 1.2.
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
OutlookDownloadsAcknowledgement
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
OutlookDownloadsAcknowledgement
Outlook
Anharmonic oscillator potentials, chaosGaussian potentials, double quantum dots, surfacesIntense laser fieldsStrong magnetic fields
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
OutlookDownloadsAcknowledgement
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
OutlookDownloadsAcknowledgement
Downloads
The lecture and relevant papers may be downloaded from:
URL: http://www.mpa-garching.mpg.de/mol_physics
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
OutlookDownloadsAcknowledgement
Outline1 Background
Confined quantum systems2 Model
Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets
3 ResultsEnergy and electron densityDipole polarizability
4 Summary and outlookOutlookDownloadsAcknowledgement
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems
BackgroundModel
ResultsSummary and outlook
OutlookDownloadsAcknowledgement
Acknowledgement: Institutions
Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems