Pauli Principle Borland-Dennis GPEP Pinning EHF Quasipinning and an extended Hartree-Fock method based on GPC Carlos L. Benavides-Riveros MLU Halle-Wittenberg, Germany University of Oxford, 14th April 2016 [joint work with: J. M. Gracia-Bond´ ıa (Zaragoza & Madrid), M. Springborg (Saarbr¨ ucken), C. Schilling (Oxford), J. V´ arilly (San Jos´ e), J. S´ anchez-Dehesa (Granada)] Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF Pauli exclusion principle I In January 1925 Wolfgang Pauli announced his famous principle: in an atom there cannot be two electrons for which the value of all quantum numbers coincide. As Paul Dirac pointed out in 1926, this exclusion rule is the manifestation of a mathematical fact: the antisymmetric character of fermionic wavefunctions. Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF Pauli exclusion principle II Among other things, Pauli exclusion principle explains the electronic structure of atoms and molecules and in the end the stability of matter. The entire principle can be understood as a constitutively a priori element of quantum mechanics. Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF The electronic Hamiltonian On configuration space, in the Born-Oppenheimer regime the electronic Hamiltonian reads: N N N − 1 1 � � � V ( r i ) + | r i − r j | . H = T + V ext + V ee = 2 ∆ r i + i =1 i =1 i<j Pure states ρ N := | ψ N �� ψ N | have skewsymmetric wave functions ψ N ∈ ∧ N H � H ⊗ N , where H is the one-particle Hilbert space. For ensemble states: � ρ N = p s | ψ s �� ψ s | . s Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF Reduced Density Matrices Integrating out the coordinates x n +1 ,..., x N gives the so-called n -particle Reduced Density Matrix ( n -RDM): ρ n ( x 1 ,..., x n ; x ′ 1 ,..., x ′ n ) � N �� ρ N ( x 1 ,...,..., x N ; x ′ 1 ,..., x ′ n , x n +1 ,..., x N ) d x n +1 ...d x N . = n x := ( r ,ς ) , being ς ∈ {↑ , ↓} the spin coordinate. Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF Reduced Density Matrices Integrating out the coordinates x n +1 ,..., x N gives the so-called n -particle Reduced Density Matrix ( n -RDM): ρ n ( x 1 ,..., x n ; x ′ 1 ,..., x ′ n ) � N �� ρ N ( x 1 ,...,..., x N ; x ′ 1 ,..., x ′ n , x n +1 ,..., x N ) d x n +1 ...d x N . = n x := ( r ,ς ) , being ς ∈ {↑ , ↓} the spin coordinate. The helium-like energy functional is given by: ∆ r 1 2 1 E ( ρ 2 ) = Tr − − V ( r 1 ) + ρ 2 . N − 1 2 | r 1 − r 2 | Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF Reduced Density Matrices Integrating out the coordinates x n +1 ,..., x N gives the so-called n -particle Reduced Density Matrix ( n -RDM): ρ n ( x 1 ,..., x n ; x ′ 1 ,..., x ′ n ) � N �� ρ N ( x 1 ,...,..., x N ; x ′ 1 ,..., x ′ n , x n +1 ,..., x N ) d x n +1 ...d x N . = n x := ( r ,ς ) , being ς ∈ {↑ , ↓} the spin coordinate. The helium-like energy functional is given by: ∆ r 1 2 1 E ( ρ 2 ) = Tr − − V ( r 1 ) + ρ 2 . N − 1 2 | r 1 − r 2 | The ground-state energy minimizes E ( ρ 2 ) : E gs = min {E ( ρ 2 ) | ρ 2 ∈ B 2 N } . Here, B 2 N is the set of the 2-RDM such that they come from N -particle density matrices by integration. Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF The representability problem � N �� ρ 2 ∈ B 2 N ⇔ ∃ ρ N ∈ DM N : ρ 2 = ρ N d x 3 ...d x n , 2 where DM N is the set of the N-particle density matrices. The N-representability problem consists in finding necessary and su ffi cient conditions for B 2 N . Gross’, Mazziotti’s & Christandl’s talks. D. Mazziotti, CR 112 , 244 (2012). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF 2-RDM For fermions the reduced 2-RDM is a 4 × 4 matrix in the spin space. In matrix form: ρ 2( ↑ 1 ↑ 2 ↑ ′ 1 ↑ ′ ρ 2( ↑ 1 ↑ 2 ↑ ′ 1 ↓ ′ ρ 2( ↑ 1 ↑ 2 ↓ ′ 1 ↑ ′ ρ 2( ↑ 1 ↑ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) ρ 2( ↑ 1 ↓ 2 ↑ ′ 1 ↑ ′ ρ 2( ↑ 1 ↓ 2 ↑ ′ 1 ↓ ′ ρ 2( ↑ 1 ↓ 2 ↓ ′ 1 ↑ ′ ρ 2( ↑ 1 ↓ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) ρ 2 ( x 1 , x 2 ; x ′ 1 , x ′ . 2 ) = ρ 2( ↓ 1 ↑ 2 ↑ ′ 1 ↑ ′ ρ 2( ↓ 1 ↑ 2 ↑ ′ 1 ↓ ′ ρ 2( ↓ 1 ↑ 2 ↓ ′ 1 ↑ ′ ρ 2( ↓ 1 ↑ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) ρ 2( ↓ 1 ↓ 2 ↑ ′ 1 ↑ ′ ρ 2( ↓ 1 ↓ 2 ↑ ′ 1 ↓ ′ ρ 2( ↓ 1 ↓ 2 ↓ ′ 1 ↑ ′ ρ 2( ↓ 1 ↓ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF 2-RDM For fermions the reduced 2-RDM is a 4 × 4 matrix in the spin space. In matrix form: ρ 2( ↑ 1 ↑ 2 ↑ ′ 1 ↑ ′ ρ 2( ↑ 1 ↑ 2 ↑ ′ 1 ↓ ′ ρ 2( ↑ 1 ↑ 2 ↓ ′ 1 ↑ ′ ρ 2( ↑ 1 ↑ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) ρ 2( ↑ 1 ↓ 2 ↑ ′ 1 ↑ ′ ρ 2( ↑ 1 ↓ 2 ↑ ′ 1 ↓ ′ ρ 2( ↑ 1 ↓ 2 ↓ ′ 1 ↑ ′ ρ 2( ↑ 1 ↓ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) ρ 2 ( x 1 , x 2 ; x ′ 1 , x ′ . 2 ) = ρ 2( ↓ 1 ↑ 2 ↑ ′ 1 ↑ ′ ρ 2( ↓ 1 ↑ 2 ↑ ′ 1 ↓ ′ ρ 2( ↓ 1 ↑ 2 ↓ ′ 1 ↑ ′ ρ 2( ↓ 1 ↑ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) ρ 2( ↓ 1 ↓ 2 ↑ ′ 1 ↑ ′ ρ 2( ↓ 1 ↓ 2 ↑ ′ 1 ↓ ′ ρ 2( ↓ 1 ↓ 2 ↓ ′ 1 ↑ ′ ρ 2( ↓ 1 ↓ 2 ↓ ′ 1 ↓ ′ 2 ) 2 ) 2 ) 2 ) By employing Wigner quasidistributions, we sought to endow the spin representation with ostensible physical meaning, by grouping their entries into tensors under the rotation group: � ⊗ 2 = 2[ 1 ] ⊕ 3[ 3 ] ⊕ [ 5 ] . � [ 1 ] ⊕ [ 3 ] Two scalars, Three vectors, One quadrupole. CLBR and JM Gracia-Bond´ ıa, PRA 87 , 022118 (2013). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF 2-RDM The exchange transformation rule for this function multiplet comes out: ρ sc 1 ρ sc 1 ρ sc 2 ρ sc 2 ρ v 1 ρ v 1 . = ρ v 2 ρ v 2 ρ v 3 ρ v 3 ρ q ρ q Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF 2-RDM The exchange transformation rule for this function multiplet comes out: ρ sc 1 ρ sc 1 +1 ρ sc 2 ρ sc 2 − 1 ρ v 1 ρ v 1 − 1 . = ρ v 2 ρ v 2 − 1 ρ v 3 ρ v 3 +1 ρ q ρ q − 1 Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016
Recommend
More recommend
