Outline Background 1 Electronic states of Confined quantum - - PowerPoint PPT Presentation

SLIDE 1

Background Model Results Summary and outlook

Electronic states of confined 2-electron quantum systems

Tokuei Sako1 Geerd HF Diercksen2

1Nihon University, College of Science and Technology

Funabashi, Chiba, JAPAN

2Max-Planck-Institut für Astrophysik

Garching, GERMANY

October 17, 2007

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Confined quantum systems

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Confined quantum systems

Quantum systems and potentials

Confined systems: electrons (quantum dots, artificial atoms and molecues), atoms, molecules Confining potentials: exponential potentials, Gaussian potentials, magnetic fields, electric fields

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 2

Background Model Results Summary and outlook Confined quantum systems

Artificial atoms

Artificial atoms are small boxes ≈ 100nm along a side, contained in a semiconductor, and holding a number of electrons. In artificial atoms electrons are typically traped in a bowl like parabolic potential.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Confined quantum systems

Structure of artificial atoms

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

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Outline

Schrödinger equation Configuration interaction (CI) method Confining potential Gaussian basis set

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 3

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Schrödinger equation

[H(r)] Ψ(1, 2, . . . , N) = EΨ(1, 2, . . . , N) H(r) =

N

• i=1
• −1

2∇2 i

• +

N

• i=1

M

• α=1
• −

Zα |ri − Rα|

• +

N

• i=1

w(ri) +

N

• i>j
• 1
• ri − rj
• Tokuei Sako, Geerd HF Diercksen

Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

One-determinant wavefunction

|Ψ = Ψ(x1x2 · · · xN) = (N!)− 1

2

• χi(x1)

χj(x1) · · · χk(x1) χi(x2) χj(x2) · · · χk(x2) . . . . . . . . . χi(xN) χj(xN) · · · χk(xN)

• χ =

ψ · α ψ · β

χ: one-electron spin function ψ: one-electron space function α, β: spin functions

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Hartree-Fock method

f(i)ψ(xi) = εiψ(xi) f(i) = −1 2∇2

i − M

• α=1
• Zα

|ri − Rα|

• + w(ri) + v(i)

f(i): Hartree-Fock operator ψ: one-electron space function ε: orbital energy v(i): averaged field of (N ∋ i) electrons

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

LCAO/LCGO approximation

ψi =

• m

cimξm

ψi: one-eletron space function cim: linear combination coefficient ξm ∝ re−αmr: hydrogenic function ≡ Slater function ξm ∝ re−αmr 2: Gaussian function

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 4

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Aufbau principle

✻

E 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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Configuration interaction wavefunction

|Φ = C0|Ψ0 + X

ra

Cr

a|Ψr a +

X

a<b r<s

Crs

ab|Ψrs ab +

X

a<b<c r<s<t

Crst

abc|Ψrst abc + · · ·

Ψ0 c • b • a • . . . r . . . s . . . t Ψr

a

c • b • a . . . r

• .

. . s . . . t Ψrs

ab

c • b a . . . r

• .

. . s • . . . t Ψrst

abc

c b a . . . r

• .

. . s • . . . t

• Tokuei Sako, Geerd HF Diercksen

Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Anisotropic harmonic oscillator potential

Anisotropic harmonic oscillator potential: w(ri) = 1 2

• ω2

xx2 i + ω2 yy2 i + ω2 zz2 i

• Tokuei Sako, Geerd HF Diercksen

Confined 2-electron quantum systems

SLIDE 5

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Anisotropic harmonic oscillator eigenvalues

Eigenvalues of an anisotropic harmonic oscillator: Eω

0 = 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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Spherical harmonic oscillator eigenvalues

Eigenvalues for an electron confined in a spherical harmonic

• scillator potential (ωx = ωy = ωz = ω):

Eω

0 = ω(2ν + ℓ + 3/2)

.

ν, ν = 0,1,2, ... : principal quantum number ℓ, ℓ = 0,1,2, ... : one-electron angular momentum quantum number

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Energy sequence for 1 electron

Sequence of the energies Eω

0 [ν1ℓ1] for one electron confined in

a spherical harmonic oscillator potential: Eω

0 [0s] = (3/2)ω,

Eω

0 [0p] = (5/2)ω,

Eω

0 [0d] = Eω 0 [1s] = (7/2)ω,

Eω

0 [0f] = Eω 0 [1p] = (9/2)ω,

Eω

0 [0g] = Eω 0 [1d] = Eω 0 [3s] = (11/2)ω, ...

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 6

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Energy sequence for 2 non-interacting electrons

Sequence of the energies Eω

0 [ν1ℓ1ν2ℓ2] for two non-interacting

electrons confined in a spherical harmonic oscillator potential: Eω

0 [(0s)2] = 3ω,

Eω

0 [0s0p] = 4ω,

Eω

0 [0s1s] = Eω 0 [0s0d] = Eω 0 [(0p)2] = 5ω,

Eω

0 [0s1p] = Eω 0 [1s0p] = Eω 0 [0s0f] = Eω 0 [0p0d] = 6ω, ...

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Energy sequence for 2 interacting electrons

Sequence of the singlet energies Eω

0 [ν1ℓ1ν2ℓ2] for two

interacting electrons confined in a spherical harmonic oscillator potential 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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Transition dipol matrix

The transition dipole matrix element between a fundamental state and a state constructed by applying the A†

ξ operator to

this 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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Dipol polarizability

The dipole polarizability of the harmonic oscillator quantum dot is determined by the analytically expression αxx = − ∂2E(Fx) ∂F 2

x

• Fx=0

= N ω2

x

, αyy = −

• ∂2E(Fy)

∂F 2

y

• Fy=0

= N ω2

y

, αzz = − ∂2E(Fz) ∂F 2

z

• Fz=0

= N ω2

z

.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 7

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Potential energy curves

• 4
• 3
• 2
• 1

1 2

• 12
• 8
• 4

4 8 12

W r

T+V+W+G T+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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Hamiltonian terms

• 4
• 3
• 2
• 1

1 2

• 12
• 8
• 4

4 8 12

W r

T+V+W+G T+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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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 P S D P S D P S

[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

SLIDE 8

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Anisotropic Gaussian functions

ξc

• a,

ζ(

r) = xaxyayzaz 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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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)

Prolate

1e

[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 one electron confined by a prolate harmonic oscillator potential with (ωx, ωy, ωz) = (0.5, 0.5, 0.25) for different size Gaussian basis

• sets. The analytical spectrum

labeled by the harmonic

• scillator quantum numbers

(νx, νy, νz) is shown at the right hand side.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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)

Prolate

1e

[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 one electron confined by a prolate harmonic oscillator potential with (ωx, ωy, ωz) = (0.5, 0.5, 0.25) for different size anisotropic Gaussian basis sets. The analytical spectrum labeled by the harmonic oscillator quantum numbers (νx, νy, νz) is shown at the right hand side.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 9

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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)

Prolate

1e

[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 one electron confined by a prolate harmonic oscillator potential with (ωx, ωy, ωz) = (0.5, 0.5, 0.25) for different size spherical anisotropic Gaussian basis

• sets. The analytical spectrum

labeled by the harmonic

• scillator quantum numbers

(νx, νy, νz) is shown at the right hand side.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Full CI energies of a 2 electron quantum dot

Table: Full CI energies of the lowest four singlet states of an oblate harmonic 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 1.0310 11∆g 1.0423 1.0375 1.0362 1.0361 1.0361 21Σ+

g

1.1472 1.1099 1.1050 1.1044 1.1040 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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 quantum dot with (ωx, ωy, ωz) = (0.1, 0.15, 0.2) for different size anisotropic Gaussian basis sets.

[1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h1i] [3s3p1d1f1g1h1i] State (35) (56) (84) (92) Singlet 11Ag 0.6876 0.6874 0.6873 0.6872 11B3u 0.7879 0.7876 0.7875 0.7874 21Ag 0.8180 0.8176 0.8174 0.8173 11B2u 0.8379 0.8376 0.8375 0.8374 11B1g 0.8448 0.8447 0.8447 0.8447 11B1u 0.8879 0.8876 0.8874 0.8874 Triplet 13B3u 0.7186 0.7185 0.7185 0.7184 13B2u 0.7852 0.7851 0.7850 0.7850 13Ag 0.8190 0.8187 0.8185 0.8185 13B1u 0.8449 0.8447 0.8447 0.8447 13B1g 0.8688 0.8686 0.8685 0.8685 23B1g 0.8854 0.8852 0.8851 0.8851 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Transition probabilities of a 2 electron quantum dot

Table: Transition probabilities between the ground state and the lowest three excited states having non-zero probability of a harmonic

• scillator 2-electron quantum dot with (ωx, ωy, ωz) = (0.1, 0.15, 0.2) for

different size anisotropic Gaussian basis sets. The number in the round bracket represents the total number of basis functions.

[1s1p1d1f1g] [1s1p1d1f1g1h] [1s1p1d1f1g1h1i] [3s3p1d1f1g1h1i] analytical Transition (35) (56) (84) (92) Singlet 11B3u−11Ag 10.012 10.011 10.006 9.998 10.0 11B2u−11Ag 6.671 6.671 6.669 6.664 6 2

3

11B1u−11Ag 5.002 5.002 5.001 4.998 5.0 Triplet 13Ag−13B3u 10.101 10.010 10.007 10.003 10.0 13B1g−13B3u 6.676 6.667 6.667 6.666 6 2

3

13B2g−13B3u 5.004 5.000 5.000 4.999 5.0 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 10

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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 low lying doublet states of an oblate harmonic oscillator 3-electron quantum dot with (ωx, ωy, ωz) = (0.01, 0.01, 0.1) for different size anisotropic Gaussian basis sets.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Full CI energy levels of the helium atom

CI

(0.5,0.5,0.25)

Prolate

c-GTO c-GTO + c-aniGTO E /a.u.

He

• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

1

1∆g (117) [10s7p5d] [1s1p1d1f1g1h] (61) [10s7p5d] (96) [10s7p5d] [1s1p1d1f1g] (93) [7s4p3d] [1s1p1d1f1g1h] (81) [10s7p5d] [1s1p1d1f]

2

1Πu

4

1Σg +

2

1Σu +

1

1Πg 3 1Σg +

1

1Πu

2

1Σg +

1

1Σu +

1

1Σg +

Figure: Full CI energy levels of the helium atom confined by a prolate harmonic oscillator potential with (ωx, ωy, ωz) = (0.5, 0.5, 0.25) for different size anisotropic Gaussian basis sets. The total number of basis functions is given in round brackets.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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 Gaussian basis 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 Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis 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 oscillator potential with (ωx, ωy, ωz) = (0.1,0.1,0.5) for different size anisotropic Gaussian 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 19.2 αzz 3.92 3.96 3.96 3.96 3.96 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 11

Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Multi-reference CI energies of the lithium atom

Table: Multi-reference CI energies of the lowest eight doublet states

• f the lithium atom confined by a harmonic oscillator potential with

(ωx, ωy, ωz) = (0.1, 0.15, 0.2) for different size anisotropic Gaussian basis sets. The number in the round bracket represents the total number 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
• 7.3256

12B3u

• 7.2645
• 7.2651
• 7.2651
• 7.2652
• 7.2652

12B2u

• 7.2386
• 7.2388
• 7.2388
• 7.2389
• 7.2389

12B1u

• 7.2077
• 7.2080
• 7.2080
• 7.2082
• 7.2082

22Ag

• 7.0476
• 7.0476
• 7.0481
• 7.0481
• 7.0483

12B1g

• 7.0224
• 7.0225
• 7.0234
• 7.0234
• 7.0237

22B3u

• 6.9785
• 6.9840
• 6.9840
• 6.9843
• 6.9843

12B2g

• 6.9798
• 6.9799
• 6.9819
• 6.9820
• 6.9825

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Transition probabilities of the lithium atom

Table: Transition probabilities between the ground state and the lowest three excited states with non-zero probability of the lithium atom confined by a harmonic oscillator potential with (ωx, ωy, ωz) = (0.1, 0.15, 0.2) for different size anisotropic Gaussian basis sets. The number in the round bracket represents the total number of basis functions.

[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 3.523 12B2u−12Ag 2.664 2.666 2.666 2.666 2.666 12B1u−12Ag 2.073 2.074 2.073 2.073 2.072 22B3u−12Ag 0.430 0.391 0.391 0.389 0.389 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Computational methods Harmonic oscillator Interplay of potentials Basis sets

Outline

Schrödinger equation Configuration interaction (CI) method Confining potential Gaussian basis set

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 12

Background Model Results Summary and outlook Energy and electron density Dipole polarizability

HF orbital energies

• 0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2p 1f 2s 1d 1f 2p 1d 2s 1f 2p 1d 1p 2s 2p 1f 2s 1d 1p 1s 1f 1d 2s 1f 2p 1d 1p 2s 1s

2e He H

• 2e

H

• He

ω = 0.5 Spherical

Orbital energy /a.u. 1p 1s 1p 1s

• 0.5

0.0 0.5 1.0

3s 2p 1p 1s

Spherical ω = 0.1

Orbital energy /a.u. 1s

HF HF

Figure: Hartree-Fock orbital energies of the helium atom, the hydrogen negative ion and

• f two electrons confined by a

spherical harmonic oscillator potential 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 Background Model Results Summary and outlook Energy and electron density Dipole polarizability

HF orbital densities

• ω
• Figure: Hartree-Fock orbital

densities of the helium atom confined by a spherical harmonic oscillator potential with (ωx, ωy, ωz) = (0.5, 0.5, 0.5).

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

HF orbital densities

• σ
• π
• σ
• σ
• π
• σ
• σ
• σ
• σ
• σ
• σ
• π
• σ
• π

σ

• σ
• ω
• σ
• σ
• σ
• σ
• π
• π

σ

• σ
• π

δ

π

• δ

π

• δ

Figure: Hartree-Fock orbital densities and energies for the helium atom, the hydrogen negative ion and two electrons confined by a prolate harmonic

• scillator potential with

(ωx, ωy, ωz) = (0.5, 0.5, 0.25)

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Full CI energies

• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5 1.0

E /a.u.

2

1D

3

1S

1

1F

1

1D

1

1P

2

1S

1

1S

Spherical

2

1D

3

1S

2

1P

1

1F

1

1D

2

1S

1

1P

3

1S

1

1S

2

1D

3

1S

He

1

1P

1

1S

2

1S

Spherical ω = 0.1

1

1S

2e 2e H

• H
• 2

1P

3

1D

1

1F

2

1D

1

1D

1

1P

1

1P

2

1D

3

1S

2

1S

1

1D

1

1P

1

1S

2

1P

1

1F

3

1D

2

1P

1

1S

• 2
• 1

1 2 3 1

1F

2

1P

3

1S

2

1D

1

1D

2

1S

1

1F

2

1P

1

1D

2

1S

He ω = 0.5

Figure: Full CI energies of the helium atom, the hydrogen negative ion and of two electrons confined by a spherical harmonic oscillator potential 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

SLIDE 13

Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Electron densities

• ω
• Figure: Electron densities of the

helium atom, the hydrogen negativ ion and of two electrons confined by a spherical harmonic oscillator potential with (ωx, ωy, ωz) = (0.5, 0.5, 0.5).

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Leading configurations

Table: Leading configurations and their squared norms for the lowest four singlet states of the helium atom, the hydrogen negative ion and

• f two electrons confined by a spherical harmonic oscillator potential

with ω = 0.5. He H− 2e State config. norm config. norm config. norm 11S (1s)2 0.994 (1s)2 0.987 (1s)2 0.973 11P (1s)(1p) 0.970 (1s)(1p) 0.970 (1s)(1p) 0.966 21S (1s)(2s) 0.973 (1s)(2s) 0.969 (1p)2 0.487 11D (1s)(1d) 0.965 (1s)(1d) 0.902 (1s)(1d) 0.480

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Leading configurations

Table: Leading configurations and their squared norms for the lowest four singlet states of the helium atom, the hydrogen negative ion and

• f two electrons confined by a prolate harmonic oscillator potential

with (ωx,ωy,ωz) = (0.5,0.5,0.25).

He H− 2e State config. norm config. norm config. norm 11Σ+

g

(1σg)2 0.993 (1σg)2 0.985 (1σg)2 0.942 11Σ+

u

(1σg)(1σu) 0.951 (1σg)(1σu) 0.950 (1σg)(1σu) 0.910 21Σ+

g

(1σg)(2σg) 0.960 (1σg)(2σg) 0.936 (1σg)(2σg) 0.489 11Πu (1σg)(1πu) 0.963 (1σg)(1πu) 0.964 (1σg)(1πu) 0.937 31Σ+

g

(1σg)(3σg) 0.915 (1σu)2 0.648 (1σg)(2σg) 0.446 11Πg (1σg)(1πg) 0.952 (1σg)(1πg) 0.905 (1σg)(1πg) 0.482 21Σ+

u

(1σg)(2σu) 0.927 (1σg)(2σu) 0.947 (1σg)(2σu) 0.729 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Leading configurations

Table: Leading configurations and their squared norms for the lowest four singlet states of the helium atom, the hydrogen negative ion and

• f two electrons confined by a oblate harmonic oscillator potential

with (ωx,ωy,ωz) = (0.5,0.5,0.25).

He H− 2e State config. norm config. norm config. norm 11Σ+

g

(1σg)2 0.993 (1σg)2 0.983 (1σg)2 0.946 11Πu (1σg)(1πu) 0.946 (1σg)(1πu) 0.946 (1σg)(1πu) 0.925 21Σ+

g

(1σg)(2σg) 0.959 (1σg)(2σg) 0.944 (1σg)(2σg) 0.487 11Σ+

u

(1σg)(1σu) 0.956 (1σg)(1σu) 0.959 (1σg)(1σu) 0.941 11∆g (1σg)(1δg) 0.949 (1σg)(1δg) 0.913 (1σg)(1δg) 0.483 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 14

Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Electron correlation energy

0.0 0.1 0.2 0.3 0.4 0.5 0.02 0.03 0.04 0.05 0.06

2e

Spherical Oblate Prolate |ECI-ESCF| /a.u.

ω

He

・

0.0 0.1 0.2 0.3 0.4 0.5 2 4 6 8 10 12

He

・

2e

Spherical Oblate Prolate x100 [ECI-ESCF]/ECI

ω

Figure: Electron correlation energy of a spherical, a prolate, and an oblate 2 electron harmonic oscillator quantum dot as function of ω.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Full CI electron density distribution

• Figure: Electron density

distribution of the lowest singlet 11Σ+

g state of He, H−, and of

two electrons confined by a spherical harmonic oscillator potential 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 Background Model Results Summary and outlook Energy and electron density Dipole 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 spherical harmonic oscillator potential for different values of ω. (ωx, ωy, ωz) He H− 2e αzz (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 5

9

(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

SLIDE 15

Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Dipole polarizability and correlation contribution

0.0 0.2 0.4 0.6 0.8 1.0 1.2 5 10 15 20 25

ω

αzz(CI)

2e H

• He

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.01 0.1 1

Spherical Spherical

αzz(CI) - αzz(HF)

H

• He

ω

Figure: Dipole polarizability (upper fig.) and electron correlation contribution (lower fig.) of the lowest singlet 11Σ+

g

state of He and H− confined by a spherical harmonic oscillator potential with (ωx, ωy, ωz) = (ω, ω, ω), ω = 0.1 - 1.2.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Full CI electron density distribution

• Figure: Electron density

distribution of the lowest singlet 11Σ+

g state of He, H−, and of

two electrons confined by a prolate-type harmonic oscillator potential 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 Background Model Results Summary and outlook Energy and electron density Dipole 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-type harmonic oscillator potential for different values of confinement parameters (ωx, ωy, ωz) = (ω, ω, 0.1).

(ωx , ωy , ωz) He H− 2e αzz (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 5

9

(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 Background Model Results Summary and outlook Energy and electron density Dipole 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-type harmonic oscillator potential for different values of (ωx, ωy, ωz).

(ωx , ωy , ωz) He H− 2e αzz (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 5

9

(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

18

αxx (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

SLIDE 16

Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Dipole polarizability

0.0 0.2 0.4 0.6 0.8 1.0 1.2 4 8 12 16 20 24 28

H

• He

αzz(CI)

ω

0.0 0.2 0.4 0.6 0.8 1.0 1.2 4 8 12 16 20 24 28

αxx(CI) (ω, ω, 0.1) (ω, ω, 0.1)

Prolate Prolate ω

H

• He

Figure: Dipole polarizability of the lowest singlet 11Σ+

g state of

He and H− confined by a prolate-type harmonic oscillator potential with (ωx, ωy, ωz) = (ω, ω, 0.1), ω = 0.1 - 1.2.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole 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

αzz(CI) - αzz(HF)

ω

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.01 0.1 1

αxx(CI) - αxx(HF) (ω, ω, 0.1) (ω, ω, 0.1)

Prolate Prolate ω

H

• He

Figure: Electron correlation contribution to the dipole polarizability of the lowest singlet 11Σ+

g state of He and H−

confined by a prolate-type harmonic oscillator potential with (ωx, ωy, ωz) = (ω, ω, 0.1), ω = 0.1 - 1.2.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Full CI electron density distribution

• Figure: Electron density

distribution of the lowest singlet 11Σ+

g state of He, H−, and of

two electrons confined by an

• blate-type harmonic oscillator

potential 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 Background Model Results Summary and outlook Energy and electron density Dipole polarizability

Dipole polarizability

0.0 0.2 0.4 0.6 0.8 1.0 1.2 4 8 12 16 20 24 28

αzz(CI)

ω ω Oblate H

• He

0.0 0.2 0.4 0.6 0.8 1.0 1.2 4 8 12 16 20 24 28

(0.1,0.1,ω) (0.1,0.1,ω) αxx(CI)

Oblate

H

• He

Figure: Dipole polarizability of the lowest singlet 11Σ+

g state of

He and H− confined by an

• blate-type harmonic oscillator

potential with (ωx, ωy, ωz) = (0.1, 0.1, ω), ω = 0.1 - 1.2.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 17

Background Model Results Summary and outlook Energy and electron density Dipole 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

αzz(CI) - αzz(HF)

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) αxx(CI) - αxx(HF)

Oblate

H

• He

Figure: Electron correlation contribution to the dipole polarizability of the lowest singlet 11Σ+

g state of He and H−

confined by an oblate-elliptical harmonic oscillator potential with (ωx, ωy, ωz) = (0.1, 0.1, ω), ω = 0.1 - 1.2.

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Outlook Downloads Acknowledgement

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Outlook Downloads Acknowledgement

Outlook

Anharmonic oscillator potentials, chaos Gaussian potentials, double quantum dots, surfaces Intense laser fields Strong magnetic fields

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Outlook Downloads Acknowledgement

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

SLIDE 18

Background Model Results Summary and outlook Outlook Downloads Acknowledgement

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 Background Model Results Summary and outlook Outlook Downloads Acknowledgement

Outline

1

Background Confined quantum systems

2

Model Computational methods Harmonic oscillator Interplay of potentials Basis sets

3

Results Energy and electron density Dipole polarizability

4

Summary and outlook Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Model Results Summary and outlook Outlook Downloads Acknowledgement

Acknowledgement: Institutions

Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems