Electronic states of confined 2-electron quantum systems · Conﬁned quantum systems Structure of artiﬁcial atoms Figure: Quantum dot. Areas shown in blue are metallic, shown in

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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

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Outline1 Background

Confined quantum systems2 Model

Computational methodsHarmonic oscillatorInterplay of potentialsBasis sets

3 ResultsEnergy and electron densityDipole polarizability

4 Summary and outlookOutlookDownloadsAcknowledgement


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Confined quantum systems

Outline1 Background






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Quantum systems and potentials

Confined systems: electrons (quantum dots, artificialatoms and molecues), atoms, moleculesConfining potentials: exponential potentials, Gaussianpotentials, magnetic fields, electric fields


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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.


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Structure of artificial atoms

Figure: Quantum dot. Areasshown in blue are metallic,shown in white are insulating(AlGaAs), and shown in red aresemiconducting (GaAs).


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Outline

Schrödinger equationConfiguration interaction (CI) methodConfining potentialGaussian basis set


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Outline1 Background






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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

∣∣]


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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


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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


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LCAO/LCGO approximation

ψi =∑

m

cimξm

ψi : one-eletron space functioncim: linear combination coefficientξm ∝ re−αmr : hydrogenic function ≡ Slater functionξm ∝ re−αmr2

: Gaussian function


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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)〉


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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 •


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Outline1 Background






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Anisotropic harmonic oscillator potential

Anisotropic harmonic oscillator potential:

w(r i) =12

[ω2

x x2i + ω2

y y2i + ω2

z z2i

]


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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


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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


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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


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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, ...


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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, ...


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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] < · · · .


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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).


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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.


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Outline1 Background






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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


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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


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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


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Outline1 Background






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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)


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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.


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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.


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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.


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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


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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


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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


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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.


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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.


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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


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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


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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


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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


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Outline

Schrödinger equationConfiguration interaction (CI) methodConfining potentialGaussian basis set


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Energy and electron densityDipole polarizability

Outline1 Background






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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.).


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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).


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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)


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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.).


BackgroundModel



Electron densities

��

��ω � ��

��

� �

��

�

��

��

��

��

��

��

��

��

��

��

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).


BackgroundModel



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


BackgroundModel




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


BackgroundModel




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


BackgroundModel



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.


BackgroundModel



Outline1 Background






BackgroundModel



Full CI electron density distribution

��

��

��

��

��

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.


BackgroundModel



Polarizability tensor components


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


BackgroundModel



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.


BackgroundModel




��

��

��

��

��


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.


BackgroundModel





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


BackgroundModel





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


BackgroundModel



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.


BackgroundModel



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.


BackgroundModel




��

��

��

��

��


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.


BackgroundModel



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.


BackgroundModel



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.


BackgroundModel


OutlookDownloadsAcknowledgement

Outline1 Background






BackgroundModel



Outlook

Anharmonic oscillator potentials, chaosGaussian potentials, double quantum dots, surfacesIntense laser fieldsStrong magnetic fields


BackgroundModel



Outline1 Background






BackgroundModel



Downloads

The lecture and relevant papers may be downloaded from:

URL: http://www.mpa-garching.mpg.de/mol_physics


BackgroundModel



Outline1 Background






BackgroundModel



Acknowledgement: Institutions


Documents

Electronic states of confined 2-electron quantum systems · Conﬁned quantum systems Structure of artiﬁcial atoms Figure: Quantum dot. Areas shown in blue are metallic, shown in