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

Background Background Model Model Results Results Summary and outlook Summary and outlook Outline Background 1 Electronic states of Confined quantum systems confined 2-electron quantum systems Model 2 Computational methods Harmonic oscillator Tokuei Sako 1 Geerd HF Diercksen 2 Interplay of potentials Basis sets 1 Nihon University, College of Science and Technology Results 3 Funabashi, Chiba, JAPAN Energy and electron density 2 Max-Planck-Institut für Astrophysik Dipole polarizability Garching, GERMANY Summary and outlook 4 Outlook October 17, 2007 Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Background Model Model Confined quantum systems Confined quantum systems Results Results Summary and outlook Summary and outlook Outline Quantum systems and potentials Background 1 Confined quantum systems Model 2 Computational methods Harmonic oscillator Confined systems: electrons (quantum dots, artificial Interplay of potentials atoms and molecues), atoms, molecules Basis sets Confining potentials: exponential potentials, Gaussian Results 3 potentials, magnetic fields, electric fields Energy and electron density Dipole polarizability Summary and outlook 4 Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Background Model Model Confined quantum systems Confined quantum systems Results Results Summary and outlook Summary and outlook Artificial atoms Structure of artificial atoms Artificial atoms are small boxes ≈ 100 nm along a side, Figure: Quantum dot. Areas contained in a semiconductor, and holding a number of shown in blue are metallic, electrons. shown in white are insulating In artificial atoms electrons are typically traped in a bowl (AlGaAs), and shown in red are semiconducting (GaAs). like parabolic potential. Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Computational methods Background Computational methods Model Harmonic oscillator Model Harmonic oscillator Results Interplay of potentials Results Interplay of potentials Summary and outlook Basis sets Summary and outlook Basis sets Outline Outline Background 1 Confined quantum systems Model 2 Computational methods Harmonic oscillator Schrödinger equation Interplay of potentials Configuration interaction (CI) method Basis sets Confining potential Results 3 Gaussian basis set Energy and electron density Dipole polarizability Summary and outlook 4 Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

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

Background Computational methods Background Computational methods Model Harmonic oscillator Model Harmonic oscillator Results Interplay of potentials Results Interplay of potentials Summary and outlook Basis sets Summary and outlook Basis sets Aufbau principle 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 E r < s r < s < t • t t t t ↑ ↓ ↑↓ ↑↓ . . . . . . . . . . . . ↑↓ ↑↓ ↑ ↓ s s s • s • ↑↓ ↑↓ ↑↓ ↑↓ . . . . . . . . . . . . Ψ g r r • r • r • Ψ e . . . . . . . . . . . . a • a a a Ψ g = | ψ 1 ( 1 ) ψ 1 ( 2 ) ψ 2 ( 3 ) ψ 2 ( 4 ) ψ 3 ( 5 ) � b • b • b b ±| ψ 1 ( 1 ) ψ 1 ( 2 ) ψ 2 ( 3 ) ψ 2 ( 4 ) ψ 3 ( 5 ) � c • c • c • c Ψ e = | ψ 1 ( 1 ) ψ 1 ( 2 ) ψ 2 ( 3 ) ψ 3 ( 5 ) ψ 3 ( 4 ) � Ψ r Ψ rs Ψ rst Ψ 0 a ab abc ±| 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 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Computational methods Background Computational methods Model Harmonic oscillator Model Harmonic oscillator Results Interplay of potentials Results Interplay of potentials Summary and outlook Basis sets Summary and outlook Basis sets Outline Anisotropic harmonic oscillator potential Background 1 Confined quantum systems Model 2 Computational methods Harmonic oscillator Anisotropic harmonic oscillator potential: Interplay of potentials w ( r i ) = 1 Basis sets � � ω 2 x x 2 i + ω 2 y y 2 i + ω 2 z z 2 i 2 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 Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems

Background Computational methods Background Computational methods Model Harmonic oscillator Model Harmonic oscillator Results Interplay of potentials Results Interplay of potentials Summary and outlook Basis sets Summary and outlook Basis sets Anisotropic harmonic oscillator eigenvalues Anisotropic harmonic oscillator eigenfunctions Anisotropic harmonic oscillator eigenfunctions: Eigenvalues of an anisotropic harmonic oscillator: � � − 1 2 ( ω x x 2 + ω y y 2 + ω z z 2 ) χ � ω r ) = N � ω E ω ν ( � ν H ν x ( x ) H ν y ( y ) H ν z ( z ) exp . 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 N � ω ν : normalization constant � H ν x ( x ) , etc.: Hermite polynomial Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Background Computational methods Background Computational methods Model Harmonic oscillator Model Harmonic oscillator Results Interplay of potentials Results Interplay of potentials Summary and outlook Basis sets Summary and outlook Basis sets Spherical harmonic oscillator eigenvalues Energy sequence for 1 electron Eigenvalues for an electron confined in a spherical harmonic Sequence of the energies E ω 0 [ ν 1 ℓ 1 ] for one electron confined in oscillator potential ( ω x = ω y = ω z = ω ): a spherical harmonic oscillator potential: E ω E ω 0 = ω ( 2 ν + ℓ + 3 / 2 ) 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, ν , ν = 0,1,2, ... : principal quantum number E ω 0 [ 0 g ] = E ω 0 [ 1 d ] = E ω 0 [ 3 s ] = ( 11 / 2 ) ω, ... ℓ , ℓ = 0,1,2, ... : one-electron angular momentum quantum number Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems Tokuei Sako, Geerd HF Diercksen Confined 2-electron quantum systems