Connection to Hartree Fock RDMFT Functionals and Minimization Applications Introduction and Overview of the Reduced Density Matrix Functional Theory N. N. Lathiotakis Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens April 13, 2016 Oxford, 13 April 2016
Connection to Hartree Fock RDMFT Functionals and Minimization Applications Outline Connection to Hartree Fock 1 Density matrices and N -representability 2 Reduced density matrix functional theory (RDMFT) 3 Functionals minimization/performance 4 Comparison with DFT 5 Application to prototype systems: Correlation energies, 6 Molecular dissociation, Homogeneous electron gas, Spectra, IPs, Electronic Gap Conclusion 7 Oxford, 13 April 2016
Connection to Hartree Fock RDMFT Functionals and Minimization Applications Hartree Fock Wave function is one Slater Determinant: ϕ 1 ( x 1 ) ϕ 1 ( x 2 ) · · · ϕ 1 ( x N ) ϕ 2 ( x 1 ) ϕ 2 ( x 2 ) · · · ϕ 2 ( x N ) Φ( x 1 , x 2 , · · · x N ) = . . . ... . . . . . . ϕ N ( x 1 ) ϕ N ( x 2 ) · · · ϕ N ( x N ) We need to minimize: E tot = � Φ | ˆ H | Φ � � Φ | Φ � Minimization chooses N orbitals out an infinite dimension space (or of dimension M > N for practical applications). Oxford, 13 April 2016
Connection to Hartree Fock RDMFT Functionals and Minimization Applications Energy in Hartree-Fock Spin-orbitals ϕ ( x ) = ϕ ( r ) α ( ω ) . For spin compensated systems: N/ 2 N/ 2 N/ 2 h (1) � � � E tot = 2 jj + 2 J jk − K jk j =1 j,k =1 j,k =1 � � � − 1 h (1) d 3 r ϕ ∗ 2 ∇ 2 jj = j ( r ) r + V ( r ) ϕ j ( r ) d 3 r ′ | ϕ j ( r ) | 2 | ϕ k ( r ′ ) | 2 � � d 3 r J jk = | r − r ′ | d 3 r ′ ϕ j ( r ) ϕ ∗ j ( r ′ ) ϕ k ( r ′ ) ϕ ∗ k ( r ) � � d 3 r K jk = | r − r ′ | Oxford, 13 April 2016
Connection to Hartree Fock RDMFT Functionals and Minimization Applications Energy in Hartree-Fock Spin-orbitals ϕ ( x ) = ϕ ( r ) α ( ω ) . For spin compensated systems: ∞ ∞ ∞ n j h (1) � � � E tot = 2 jj + 2 n j n k J jk − n j n k K jk j =1 j,k =1 j,k =1 � � � − 1 h (1) d 3 r ϕ ∗ 2 ∇ 2 jj = j ( r ) r + V ( r ) ϕ j ( r ) d 3 r ′ | ϕ j ( r ) | 2 | ϕ k ( r ′ ) | 2 � � d 3 r J jk = | r − r ′ | d 3 r ′ ϕ j ( r ) ϕ ∗ j ( r ′ ) ϕ k ( r ′ ) ϕ ∗ k ( r ) � � d 3 r K jk = | r − r ′ | Where n j and n k occupation numbers Oxford, 13 April 2016
Connection to Hartree Fock RDMFT Functionals and Minimization Applications Hartree Fock Functional in RDMFT ∞ ∞ ∞ n j h (1) � � � E tot = 2 jj + 2 n j n k J jk − n j n k K jk j =1 j,k =1 j,k =1 Assume that this functional is minimized w.r.t. n j , ϕ j . It is not bound! n j should satisfy extra conditions. Ensemble N -representability conditions of Coleman: ∞ � 0 ≤ n j ≤ 1 , and 2 n j = N j =1 The first reflects the Pauli principle and the second fixes the number of particles. No extrema between 0 and 1: collapses to HF Theory Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications Density matrices N -body density matrix ( N RDM) Γ ( N ) ( r 1 , r 2 .. r N ; r ′ 1 , r ′ 2 .. r ′ N ) = Ψ ∗ ( r ′ 1 , r ′ 2 .. r ′ N ) Ψ( r 1 , r 2 .. r N ) Reduce the order of the density matrix ( p RDM) Γ ( p ) ( r 1 , .. r p ; r ′ 1 , .. r ′ p ) = � N � � d 3 r p +1 ..d 3 r N Ψ ∗ ( r ′ 1 , .. r ′ p , r p +1 .. r N ) Ψ( r 1 , .. r N ) p Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications Density matrices N -body density matrix ( N RDM) Γ ( N ) ( r 1 , r 2 .. r N ; r ′ 1 , r ′ 2 .. r ′ N ) = Ψ ∗ ( r ′ 1 , r ′ 2 .. r ′ N ) Ψ( r 1 , r 2 .. r N ) Reduce the order of the density matrix ( p RDM) Γ ( p ) ( r 1 , .. r p ; r ′ 1 , .. r ′ p ) = � N � � d 3 r p +1 ..d 3 r N Ψ ∗ ( r ′ 1 , .. r ′ p , r p +1 .. r N ) Ψ( r 1 , .. r N ) p Recurrence relation Γ ( p − 1) ( r 1 , .. r p − 1 ; r ′ 1 , .. r ′ p − 1 ) = p � d 3 r p Γ ( p ) ( r 1 , .. r p ; r ′ 1 , .. r ′ p − 1 , r p ) N − p + 1 Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications Density matrices One-body reduced density matrix (1RDM) 2 � Γ (1) ( r , r ′ ) = d 3 r 2 Γ (2) ( r , r 2 ; r ′ , r 2 ) =: γ ( r ; r ′ ) N − 1 Expectation value of p -body operator: � Γ (p) ˆ � � ˆ O � = Tr O Total energy: expectation value of the Hamiltonian (2-body) The e-e interaction energy simple functional of 2RDM: ρ (2) ( r 1 , r 2 ) � � d 3 r 1 d 3 r 2 E ee = | r 1 − r 2 ) ρ (2) ( r 1 , r 2 ) = Γ (2) ( r 1 , r 2 ; r 1 , r 2 ) (second reduced density) Why don’t we minimize the total energy with respect to Γ (2) ? Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications N -representability of the 2RDM Remember Γ (2) ( r 1 , r 2 ; r ′ 1 , r ′ 2 ) = N ( N − 1) � d 3 r 3 ..d 3 r N Ψ ∗ ( r ′ 1 , r ′ 2 , r 3 .. r N )Ψ( r 1 .. r N ) 2 with Ψ : antisymmetric, normalized wave function For Γ (2) several necessary N -representability conditions are known 1 . These conditions are not sufficient. If they were sufficient they would provide the solution of the many electron problem. 1work and talk of D. A. Mazziotti. Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications N -representability of the 1RDM For γ ensemble N -representability, necessary and sufficient conditions were proven by Coleman 2 : � 0 ≤ n j ≤ 1 , n j = N j occupation numbers n j and the natural orbitals ϕ j : ∞ � γ ( r , r ′ ) = n j ϕ ∗ j ( r ′ ) ϕ j ( r ) j =1 Plus orthonormality of ϕ j 2Rev. Mod. Phys. 35 , 668 (1963) Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications N -representability of the 1RDM For γ ensemble N -representability, necessary and sufficient conditions were proven by Coleman 2 : � 0 ≤ n j ≤ 1 , n j = N j occupation numbers n j and the natural orbitals ϕ j : ∞ � γ ( r , r ′ ) = n j ϕ ∗ j ( r ′ ) ϕ j ( r ) j =1 Plus orthonormality of ϕ j 2RDM vs 1RDM functional theory: (simple functional but complicated N-rep) vs (totally unknown functional but simple N-rep.) 2Rev. Mod. Phys. 35 , 668 (1963) Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications N -representability of the 1RDM Coleman’s conditions are also sufficient for pure state N-rep for even number of electrons and spin compensated systems. Given the exact functional of the 1RDM the ensemble N-rep conditions are enough. Are there pure state N-rep conditions? Generalized Pauli constrains 3 for ( N, M ) ; N : number of electrons, M : size of the Hilbert space. Recently: a method to generate them for all pairs ( M, N ) . Unfortunately, their number explodes as N , M increase. Their incorporation in RDMFT calculations for 3-electron systems improves the results for approximate 1RDM functionals. 3Talks on Thursday Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications RDMFT Foundations Gilbert’s Theorem (T. Gilbert Phys. Rev. B 12 , 2111 (1975)): γ gs ( r ; r ′ ) 1 − 1 ← → Ψ gs ( r 1 , r 2 ... r N ) Every ground-state observable is a functional of the ground-state 1RDM. The exact total energy functional E tot = E kin + E ext + E ee E ee [ γ ] = min Ψ → γ � Ψ | V ee | Ψ � (Domain of γ : Pure state N-representable (Lieb 1979)) E ee [ γ ] = Γ ( N ) → γ � Ψ | V ee | Ψ � min (Domain of γ : Ensemble state N-representable (Valone 1980)) Oxford, 13 April 2016
Connection to Hartree Fock Density Matrices RDMFT N -representability Functionals and Minimization Foundations Applications RDMFT Foundations Total energy E tot = E kin + E ext + E ee −∇ 2 � � � � d 3 r d 3 r ′ δ ( r − r ′ ) γ ( r ; r ′ ) E kin = 2 � d 3 r v ext ( r ) γ ( r ; r ) E ext = E ee = E H + E xc d 3 r d 3 r ′ γ ( r ; r ) γ ( r ′ ; r ′ ) � � E H = | r − r ′ | Exchange-correlation energy does not contain any kinetic energy contributions Oxford, 13 April 2016
Connection to Hartree Fock Functionals RDMFT Minimization Functionals and Minimization Comparison with DFT Applications M¨ uller type functionals ∞ d 3 rd 3 r ′ ϕ j ( r ) ϕ ∗ j ( r ′ ) ϕ k ( r ′ ) ϕ ∗ k ( r ) E xc = − 1 � � f ( n j , n k ) | r − r ′ | 2 j,k =1 Hartree-Fock: f ( n j , n k ) = n j n k uller functional 4 : f ( n j , n k ) = √ n j n k M¨ Goedecker-Umrigar 5 : f ( n j , n k ) = √ n j n k (1 − δ jk ) + n 2 j δ jk Power functional 6 f ( n j , n k ) = ( n j n k ) α , α ∼ 0 . 6 . ML: Pade approximation for f , fit for a set of molecules 7 4A. M¨ uller, Phys. Lett. 105A , 446 (1984); M. A. Buijse, E. J. Baerends, Mol. Phys. 100 , 401 (2002) 5S. Goedecker, C. J. Umrigar, Phys. Rev. Lett. 81 , 866 (1998). 6Sharma et al, PRB 78 , 201103R (2008) 7Marques, et al, PRA 77 , 032509 (2008). Oxford, 13 April 2016
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