Electronic states of confined 2-electron quantum systems Tokuei Sako - PowerPoint PPT Presentation

Background Model Results Summary and outlook Electronic states of confined 2-electron quantum systems Tokuei Sako 1 Geerd HF Diercksen 2 1 Nihon University, College of Science and Technology Funabashi, Chiba, JAPAN 2 Max-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 Background 1 Confined quantum systems Model 2 Computational methods Harmonic oscillator Interplay of potentials Basis sets Results 3 Energy and electron density Dipole polarizability Summary and outlook 4 Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Model Confined quantum systems Results Summary and outlook Outline Background 1 Confined quantum systems Model 2 Computational methods Harmonic oscillator Interplay of potentials Basis sets Results 3 Energy and electron density Dipole polarizability Summary and outlook 4 Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Model Confined quantum systems Results Summary and outlook 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

Background Model Confined quantum systems Results Summary and outlook Artificial atoms Artificial atoms are small boxes ≈ 100 nm 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 Confined quantum systems Results Summary and outlook 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 Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook 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 Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Outline Background 1 Confined quantum systems Model 2 Computational methods Harmonic oscillator Interplay of potentials Basis sets Results 3 Energy and electron density Dipole polarizability Summary and outlook 4 Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Schrödinger equation [ H ( r )] Ψ( 1 , 2 , . . . , N ) = E Ψ( 1 , 2 , . . . , N ) N N M N � Z α � � � � − 1 2 ∇ 2 � � � H ( r ) = + − + w ( r i ) i | r i − R α | i = 1 i = 1 α = 1 i = 1 N � � 1 � + � � � r i − r j � i > j Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets One-determinant wavefunction � � χ i ( x 1 ) χ j ( x 1 ) · · · χ k ( x 1 ) � � � � χ i ( x 2 ) χ j ( x 2 ) · · · χ k ( x 2 ) � � | Ψ � = Ψ( x 1 x 2 · · · x N ) = ( N !) − 1 2 � . . . � . . . � � . . . � � � � χ i ( x N ) χ j ( x N ) · · · χ k ( x N ) � � � ψ · α χ = 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 Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Hartree-Fock method f ( i ) ψ ( x i ) = ε i ψ ( x i ) M � � f ( i ) = − 1 Z α � 2 ∇ 2 i − + w ( r i ) + v ( i ) | r i − R α | α = 1 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 Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets LCAO/LCGO approximation � ψ i = c im ξ m m ψ i : one-eletron space function c im : linear combination coefficient ξ m ∝ re − α m r : hydrogenic function ≡ Slater function ξ m ∝ re − α m r 2 : Gaussian function Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Aufbau principle ✻ E ↑ ↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↓ ↑↓ ↑↓ ↑↓ ↑↓ Ψ g Ψ e Ψ g = | ψ 1 ( 1 ) ψ 1 ( 2 ) ψ 2 ( 3 ) ψ 2 ( 4 ) ψ 3 ( 5 ) � ±| ψ 1 ( 1 ) ψ 1 ( 2 ) ψ 2 ( 3 ) ψ 2 ( 4 ) ψ 3 ( 5 ) � Ψ e = | ψ 1 ( 1 ) ψ 1 ( 2 ) ψ 2 ( 3 ) ψ 3 ( 5 ) ψ 3 ( 4 ) � ±| ψ 1 ( 1 ) ψ 1 ( 2 ) ψ 2 ( 3 ) ψ 3 ( 5 ) ψ 3 ( 4 ) � Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Configuration interaction wavefunction C r a | Ψ r C rs ab | Ψ rs C rst abc | Ψ rst X X X | Φ � = C 0 | Ψ 0 � + a � + ab � + abc � + · · · ra a < b a < b < c r < s r < s < t • t t t t . . . . . . . . . . . . s s s • s • . . . . . . . . . . . . r r • r • r • . . . . . . . . . . . . a • a a a b • b • b b c • c • c • c Ψ r Ψ rs Ψ rst Ψ 0 a ab abc Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Outline Background 1 Confined quantum systems Model 2 Computational methods Harmonic oscillator Interplay of potentials Basis sets Results 3 Energy and electron density Dipole polarizability Summary and outlook 4 Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Anisotropic harmonic oscillator potential Anisotropic harmonic oscillator potential: w ( r i ) = 1 � � ω 2 x x 2 i + ω 2 y y 2 i + ω 2 z z 2 i 2 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Anisotropic harmonic oscillator eigenvalues Eigenvalues of an anisotropic harmonic oscillator: E ω 0 = ω x ( ν x + 1 / 2 ) + ω y ( ν y + 1 / 2 ) + ω z ( ν z + 1 / 2 ) . ( ν x , ν y , ν z ) : harmonic oscillator quantum numbers Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Anisotropic harmonic oscillator eigenfunctions Anisotropic harmonic oscillator eigenfunctions: � − 1 � 2 ( ω x x 2 + ω y y 2 + ω z z 2 ) χ � ω r ) = N � ω ν ( � ν H ν x ( x ) H ν y ( y ) H ν z ( z ) exp . � � N � ω ν : normalization constant � H ν x ( x ) , etc.: Hermite polynomial Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Spherical harmonic oscillator eigenvalues Eigenvalues for an electron confined in a spherical harmonic oscillator potential ( ω x = ω y = ω z = ω ): E ω 0 = ω ( 2 ν + 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 Computational methods Model Harmonic oscillator Results Interplay of potentials Summary and outlook Basis sets Energy sequence for 1 electron Sequence of the energies E ω 0 [ ν 1 ℓ 1 ] for one electron confined in a spherical harmonic oscillator potential: E ω 0 [ 0 s ] = ( 3 / 2 ) ω, E ω 0 [ 0 p ] = ( 5 / 2 ) ω, E ω 0 [ 0 d ] = E ω 0 [ 1 s ] = ( 7 / 2 ) ω, E ω 0 [ 0 f ] = E ω 0 [ 1 p ] = ( 9 / 2 ) ω, E ω 0 [ 0 g ] = E ω 0 [ 1 d ] = E ω 0 [ 3 s ] = ( 11 / 2 ) ω, ... Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems