Physical meaning of natural orbitals and natural occupation numbers Member of the Helmholtz-Association 13.04.2016 N. Helbig Forschungszentrum Jülich
Outline 1 Introduction 2 General properties Non-interacting electrons Interacting electrons Correlation entropy 3 A toy model Natural orbitals and occupation numbers Description of excitations Member of the Helmholtz-Association 4 Conclusions and Outlook 13.04.2016 N. Helbig 2 22
Reduced density-matrix functional theory One-body reduced density matrix � d 3 r 2 ... d 3 r N Ψ ∗ γ gs ( r , r ′ ) gs ( r ′ ... r N )Ψ gs ( r ... r N ) = N ∞ � n j ϕ ∗ j ( r ′ ) ϕ j ( r ) = j = 1 Ground-state energy Member of the Helmholtz-Association E [ γ ] = E kin [ γ ]+ E ext [ γ ]+ E H [ γ ]+ E xc [ γ ] = E [ { n j } , { ϕ j ( r ) } ] 13.04.2016 N. Helbig 3 22
Reduced density-matrix functional theory Minimize total energy with respect to occupation numbers and natural orbitals N -representability conditions ∞ � � d 3 r ϕ ∗ 0 ≤ n j ≤ 1 , n j = N , j ( r ) ϕ k ( r ) = δ jk j = 1 Ensemble N -representability � w j | Ψ j �� Ψ j | Member of the Helmholtz-Association j instead of a pure state | Ψ � (see talks tomorrow) 13.04.2016 N. Helbig 4 22
Reduced density-matrix functional theory Problem The exact E xc [ γ ] is unknown From energy minimization one obtains approximate natural orbitals and occupation numbers For exact natural orbitals and occupation numbers one needs to calculate Ψ( r 1 , · · · r N ) Member of the Helmholtz-Association → Introduce a one-dimensional model system 13.04.2016 N. Helbig 5 22
Non-interacting electrons Slater determinant � � ϕ 1 ( r 1 ) · · · ϕ 1 ( r N ) � � � � 1 . . � � Ψ( r 1 ... r N ) = √ . . . . � � N ! � � ϕ N ( r 1 ) · · · ϕ N ( r N ) � � Density matrix N � γ ( r , r ′ ) = ϕ ∗ j ( r ′ ) ϕ j ( r ) j = 1 Member of the Helmholtz-Association Natural orbitals are single-particle orbitals Occupation numbers are either zero or one 13.04.2016 N. Helbig 6 22
Non-interacting electrons Single-particle orbitals satisfy � � −∇ 2 2 + v ext ( r ) ϕ j ( r ) = ǫ j ϕ j ( r ) Lowest energy states are occupied ǫ 1 ≤ ǫ 2 ≤ ... ⇒ n j = 1 for 1 ≤ j ≤ N n j = 0 j > N for Member of the Helmholtz-Association The same holds for Hartree-Fock theory (except that v ext is replaced by the HF potential) 13.04.2016 N. Helbig 7 22
Interacting electrons Many-body wave function � Ψ( r 1 ... r N ) = c j Φ j ( r 1 ... r N ) j with Slater determinants Φ j ( r 1 ... r N ) and � j | c j | 2 = 1. Density matrix ∞ � γ ( r , r ′ ) = n j ϕ ∗ j ( r ′ ) ϕ j ( r ) Member of the Helmholtz-Association j = 1 No single-particle equation associated to the natural orbitals 13.04.2016 N. Helbig 8 22
Interacting electrons Use M natural orbitals to set up the Slater determinants Minimizes � d 3 r 1 · · · d 3 r N | Ψ( r 1 · · · r N ) − Ψ M ( r 1 · · · r N ) | 2 compared to any other set of M orbitals. Relation between coefficients and occupation numbers � | c k | 2 n j = Member of the Helmholtz-Association k ,ϕ j ∈ Φ k If n j = 1 ( 0 ) the corresponding natural orbital appears in all (none) of the Slater determinants. 13.04.2016 N. Helbig 9 22
Correlation entropy Measure for correlation s = − � ∞ j = 1 n j log n j 0.4 -x*log(x) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Member of the Helmholtz-Association 0 0 0.2 0.4 0.6 0.8 1 For non-interacting electrons s = 0. 13.04.2016 N. Helbig 10 22
Small toy model One-dimensional system with two electrons v v ext ( x ) = − cosh 2 ( x ) For non-interacting electrons � √ ǫ j = − 1 � 2 1 + 8 v − 1 − 2 ( j − 1 ) 8 � �� � > 0 For v = 0 . 9: only one bound state For v = 2 . 0: two bound states Member of the Helmholtz-Association Interaction b v int ( x 1 , x 2 ) = cosh 2 ( x 1 − x 2 ) 13.04.2016 N. Helbig 11 22
Natural orbitals 0,6 b=0.0 1st nat. orb. b=0.5 b=0.9 0,4 b=1.0 0,2 0 0,6 2nd nat. orb. 0,4 0,2 0 -0,2 b=0.01 -0,4 b=1.3 -0,6 0,4 3rd nat. orb. b=1.5 0,2 b=3.0 Member of the Helmholtz-Association 0 -0,2 -0,4 -10 0 10 x (a.u.) 13.04.2016 N. Helbig 12 22
Occupation numbers 2 n 1 n 2 n 3 Occ. number/entropy 1.5 s 1 0.5 Member of the Helmholtz-Association 0 0 0.5 1 1.5 2 2.5 3 Interaction strength (a.u.) 13.04.2016 N. Helbig 13 22
Excited state 0.6 0.6 b=0.01 b=1.0 b=0.5 b=1.3 2nd nat. orb. 0.4 b=0.9 b=1.5 0.4 1st nat. orb. 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 x (a.u.) x (a.u.) Member of the Helmholtz-Association One natural orbital always unbound. 13.04.2016 N. Helbig 14 22
Excitations For b < 1 . 0 the natural orbitals of the ground state are localized. Excited state always has one unbound natural orbital. First excited state of this system is ionized. Excitations cannot be described by just changing the occupation of the ground-state natural orbitals. This is however what we always do for non-interacting Member of the Helmholtz-Association electrons, even for ionization. 13.04.2016 N. Helbig 15 22
More bound states, v = 2 . 0 0.8 b=0.0 0.8 1st nat. orb. 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.6 0.6 b=0.01 b=1.0 b=0.5 b=1.3 2nd nat. orb. 0.3 b=0.9 b=1.5 0.3 0 0 Member of the Helmholtz-Association -0.3 -0.3 -0.6 -0.6 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 x (a.u.) x (a.u.) 13.04.2016 N. Helbig 16 22
Excitations For b < 1 . 0 the first two natural orbitals of the ground state and the excited state are localized. Excitation can be approximately described by just changing the occupations of the ground-state natural orbitals. Member of the Helmholtz-Association 13.04.2016 N. Helbig 17 22
Excitations For b < 1 . 0 the first two natural orbitals of the ground state and the excited state are localized. Excitation can be approximately described by just changing the occupations of the ground-state natural orbitals. Can excitations be described from ground-state natural orbitals? It depends, sometimes yes, sometimes no. Member of the Helmholtz-Association 13.04.2016 N. Helbig 17 22
Molecular dissociation Two potential wells at distance d v v v ext ( x ) == − cosh 2 ( x − d / 2 ) − cosh 2 ( x + d / 2 ) Interaction 1 v int ( x 1 , x 2 ) = cosh 2 ( x 1 − x 2 ) Interaction decays exponentially with distance. Member of the Helmholtz-Association 13.04.2016 N. Helbig 18 22
Molecular dissociation 0.8 1st nat. orb. d=1.0 0.6 d=3.0 0.4 d=5.0 0.2 0 0.6 2nd nat. orb. 0.4 0.2 0 -0.2 -0.4 -0.6 0.8 d=7.0 0.6 KS orb. d=11.0 Member of the Helmholtz-Association 0.4 d=13.0 0.2 d=15.0 0 -20 -15 -10 -5 0 5 10 15 20 x 13.04.2016 N. Helbig 19 22
Molecular dissociation 2 n 1 n 2 s 1.5 Correlation entropy 1 0.5 Member of the Helmholtz-Association 0 0 2 4 6 8 10 12 14 Distance (a.u.) 13.04.2016 N. Helbig 20 22
Conclusions Natural orbitals change dramatically from non-interacting to interacting particles. Excitations can be described by a change in the occupation numbers if the two states are similar in their localization. Occupation numbers provide a measure for correlation. n j = 0 and n j = 1 give more information on wave function. Member of the Helmholtz-Association 13.04.2016 N. Helbig 21 22
Work done together with... I.V. Tokatly UPV/EHU, San Sebastián (Spain) A. Rubio UPV/EHU, San Sebastián (Spain), MPI, Hamburg (Germany) References: Phys. Rev. A 81 , 022504 (2010) Member of the Helmholtz-Association 13.04.2016 N. Helbig 22 22
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