Pauli Principle Borland-Dennis GPEP Pinning EHF
Beyond the Pauli exclusion prinicple
Carlos L. Benavides-Riveros
MLU Halle-Wittenberg, Germany
Universidad Nacional de Colombia, 12 Sep 2016
[joint work with: J. M. Gracia-Bondı́a (Zaragoza & Madrid), M. Springborg
(Saarbrücken), C. Schilling (Oxford), J. Várilly (San José),
J. Sánchez-Dehesa (Granada), M. Marques (Halle), N. Lathiotakis (Athens).]
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 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
Bogotá, 12 Sep 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
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The electronic Hamiltonian
On configuration space, in the Born-Oppenheimer regime the
electronic Hamiltonian reads:
H = T + Vext + Vee =
N
N
N
X
X
X
1
1
V (ri ) +
.
− ∆ri +
2
|ri − rj |
i=1
i=1
i<j
Pure states ρN := |ψN ihψN | have skewsymmetric wave
functions
ψN ∈ ∧N H ( H⊗N ,
where H is the one-particle Hilbert space. For ensemble states:
X
ρN =
ps |ψ s ihψ s |.
s
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Reduced Density Matrices
Integrating out the coordinates xn+1 , . . . , xN gives the so-called
n-particle Reduced Density Matrix (n-RDM):
ρn (x1 , . . . , xn ; x01 , . . . , x0n )
!Z
N
=
ρN (x1 , . . . , . . . , xN ; x01 , . . . , x0n , xn+1 , . . . , xN ) dxn+1 . . . dxN .
n
x := (r, ς), being ς ∈ {↑, ↓} the spin coordinate.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Reduced Density Matrices
Integrating out the coordinates xn+1 , . . . , xN gives the so-called
n-particle Reduced Density Matrix (n-RDM):
ρn (x1 , . . . , xn ; x01 , . . . , x0n )
!Z
N
=
ρN (x1 , . . . , . . . , xN ; x01 , . . . , x0n , xn+1 , . . . , xN ) dxn+1 . . . dxN .
n
x := (r, ς), being ς ∈ {↑, ↓} the spin coordinate. The helium-like
energy functional is given by:



 




 
2  ∆r1
1


ρ2 
E(ρ2 ) = Tr 
−
−
V
(r
)
+
.




1 





 N −1 2

|r1 − r2 |
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Reduced Density Matrices
Integrating out the coordinates xn+1 , . . . , xN gives the so-called
n-particle Reduced Density Matrix (n-RDM):
ρn (x1 , . . . , xn ; x01 , . . . , x0n )
!Z
N
=
ρN (x1 , . . . , . . . , xN ; x01 , . . . , x0n , xn+1 , . . . , xN ) dxn+1 . . . dxN .
n
x := (r, ς), being ς ∈ {↑, ↓} the spin coordinate. The helium-like
energy functional is given by:



 




 
2  ∆r1
1


ρ2 
E(ρ2 ) = Tr 
−
−
V
(r
)
+
.




1 





 N −1 2

|r1 − r2 |
The ground-state energy minimizes E(ρ2 ):
2
Egs = min{E(ρ2 ) | ρ2 ∈ BN
}.
2
Here, BN
is the set of the 2-RDM such that they come from
N -particle density matrices by integration.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The representability problem
2
ρ2 ∈ BN
N
⇔ ∃ ρN ∈ DMN : ρ2 =
2
!Z
ρN dx3 . . . dxn ,
where DMN is the set of the N-particle density matrices.
The N-representability problem consists in finding necessary
2
and sufficient conditions for BN
.
David Mazziotti, CR 112, 244 (2012).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Something is already known
ρ2 must be...
Hermitian,
normalized (fixed trace),
antisymmetric under the exchange of particles, and
positive semidefinite ⇔ its eigenvalues are non-negative.
The set Bn2 is convex.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
G, Q, T1 and T2 conditions
By 2012, the known N-representability necessary conditions
Γ 0
G 0 Q 0 T1 0
Carlos L. Benavides-Riveros
T2 0
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
G, Q, T1 and T2 conditions
By 2012, the known N-representability necessary conditions
Γ 0
G 0 Q 0 T1 0
T2 0
Garrod and Percus (1964): The G and Q conditions establish that the
following two matrices
ij
ij
Gkl = hψ|a†i aj a†l ak |ψi and Qkl = hψ|ai aj a†l a†k |ψi,
must be positive semidefinite if ρ2 is N-representable.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
G, Q, T1 and T2 conditions
By 2012, the known N-representability necessary conditions
Γ 0
G 0 Q 0 T1 0
T2 0
Garrod and Percus (1964): The G and Q conditions establish that the
following two matrices
ij
ij
Gkl = hψ|a†i aj a†l ak |ψi and Qkl = hψ|ai aj a†l a†k |ψi,
must be positive semidefinite if ρ2 is N-representable.
Erdhal (1978): The other known N-representability conditions are the
T1 and T2 conditions:
ijk
T1 lmn := hψ|a†i a†j a†k an am al + an am al a†i a†j a†k |ψi 0
ijk
T2 lmn = hψ|a†i a†j ak a†n am al + a†n am al a†i a†j ak |ψi 0
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
G, Q, T1 and T2 conditions
By 2012, the known N-representability necessary conditions
Γ 0
G 0 Q 0 T1 0
T2 0
Garrod and Percus (1964): The G and Q conditions establish that the
following two matrices
ij
ij
Gkl = hψ|a†i aj a†l ak |ψi and Qkl = hψ|ai aj a†l a†k |ψi,
must be positive semidefinite if ρ2 is N-representable.
Erdhal (1978): The other known N-representability conditions are the
T1 and T2 conditions:
ijk
T1 lmn := hψ|a†i a†j a†k an am al + an am al a†i a†j a†k |ψi 0
ijk
T2 lmn = hψ|a†i a†j ak a†n am al + a†n am al a†i a†j ak |ψi 0
Mazziotti (2012): For three-electron systems there is a set of 27
necessary and sufficient conditions for the representability of ρ2 .
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 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:

0 0
ρ2 (↑1 ↑2 ↑1 ↑2 )

ρ (↑1 ↓2 ↑01 ↑02 )

ρ2 (x1 , x2 ; x01 , x02 ) =  2 (↓ ↑ ↑0 ↑0 )
ρ2 1 2 1 2

 (↓1 ↓2 ↑01 ↑02 )
ρ2
0
0
ρ2 (↑1 ↑2 ↑1 ↓2 )
0 0
ρ2 (↑1 ↓2 ↑1 ↓2 )
0 0
ρ2 (↓1 ↑2 ↑1 ↓2 )
0 0
ρ2 (↓1 ↓2 ↑1 ↓2 )
Carlos L. Benavides-Riveros
0
0
ρ2 (↑1 ↑2 ↓1 ↑2 )
0 0
ρ2 (↑1 ↓2 ↓1 ↑2 )
0 0
ρ2 (↓1 ↑2 ↓1 ↑2 )
0 0
ρ2 (↓1 ↓2 ↓1 ↑2 )
Bogotá, 12 Sep 2016
0 0 
ρ2 (↑1 ↑2 ↓1 ↓2 ) 
0 0 
ρ2 (↑1 ↓2 ↓1 ↓2 ) 
0 0 
.
ρ2 (↓1 ↑2 ↓1 ↓2 ) 
0 0 

ρ2 (↓1 ↓2 ↓1 ↓2 )
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:

0 0
ρ2 (↑1 ↑2 ↑1 ↑2 )

ρ (↑1 ↓2 ↑01 ↑02 )

ρ2 (x1 , x2 ; x01 , x02 ) =  2 (↓ ↑ ↑0 ↑0 )
ρ2 1 2 1 2

 (↓1 ↓2 ↑01 ↑02 )
ρ2
0
0
ρ2 (↑1 ↑2 ↑1 ↓2 )
0 0
ρ2 (↑1 ↓2 ↑1 ↓2 )
0 0
ρ2 (↓1 ↑2 ↑1 ↓2 )
0 0
ρ2 (↓1 ↓2 ↑1 ↓2 )
0
0
ρ2 (↑1 ↑2 ↓1 ↑2 )
0 0
ρ2 (↑1 ↓2 ↓1 ↑2 )
0 0
ρ2 (↓1 ↑2 ↓1 ↑2 )
0 0
ρ2 (↓1 ↓2 ↓1 ↑2 )
0 0 
ρ2 (↑1 ↑2 ↓1 ↓2 ) 
0 0 
ρ2 (↑1 ↓2 ↓1 ↓2 ) 
0 0 
.
ρ2 (↓1 ↑2 ↓1 ↓2 ) 
0 0 

ρ2 (↓1 ↓2 ↓1 ↓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
[1] ⊕ [3]
= 2[1] ⊕ 3[3] ⊕ [5].
Two scalars, Three vectors, One quadrupole.
CLBR and JM Gracia-Bondı́a, PRA 87, 022118 (2013).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
2-RDM
The exchange transformation rule for this function multiplet
comes out:
 sc1 
ρ 
 sc2 
ρ 
 v1 
 ρ 
 v2  .
 ρ 
 v3 
 ρ 
ρq
 sc1 
ρ 
 sc2 
ρ 
 v1 
 ρ 
 v2  =
 ρ 
 v3 
 ρ 
ρq
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
2-RDM
The exchange transformation rule for this function multiplet
comes out:
 sc1 
ρ 
 sc2 
ρ 
 v1 
 ρ 
 v2  =
 ρ 
 v3 
 ρ 
ρq

+1









−1
−1
Carlos L. Benavides-Riveros
−1
+1
  sc1 
 ρ 
 ρsc2 
 
  v1 
  ρ 
  v2  .
  ρ 
 

  ρv3 
 q 
−1 ρ
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The intracule distribution
We are interested in the ground state energy of the system.
There is the helium-like energy functional:


 
 



 
Z  
1
 2  ∆r1
ρ2 
+
−
+
E(ρ2 ) = Tr 




.


 N − 1 
2
|r1 |   |r1 − r2 |  



Z
 ∆r

I (r)
Z 
= Tr  − 1 +
+
dr.
ρ1 (x1 ; x01 )
2
|r1 |
r
0
x1 =x1
where the intracular distribution is defined in the following way:
Z
I (r) = ρ2 (x1 , x2 ; x1 , x2 ) δ(|r1 − r2 | − r) dx1 dx2 .
Crittenden and Gill, JCP 127, 014101 (2007).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Density functional theory on phase space
Let us consider the Wigner function:
Z
0
−N
w( r, ς; p, ς ) := π
ρ1 (r − z, ς; r + z, ς0 )e2i p·z dz1 .
This is a real function and Tr w = N . However, w can take some
negative values.

Z
Z 
 p Z 
I (r)
dr.
E(w, I ) =  + w(r, ς; p, ς)drdpdς +
2 |r|
r
There is a “phase space” Gilbert theorem.
P. Blanchard, JM Gracia-Bondia and J Várilly, IJQC 112, 1134 (2012).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Quantum marginal problem
The bulk of these issues is connected to a more general effort
in quantum information theory, addressing the so-called
quantum marginal problem:
Consider a system of N identical fermions, whose
corresponding Hilbert space of antisymmetric states is
∧N Hm . For a given n ∈ {1, 2, . . . , N − 1}, the problem of
determining the set of admissible n-RDM that arises via
partial integrations from a corresponding pure/ensemble
N -density operator is the quantum marginal problem, or,
in the jargon of quantum chemistry, the N -representability
problem.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Pauli exclusion principle: a bound in the NON
In 1963 Coleman proved that pure
state or ensemble 1-RDM satisfy:
ρ1 ≥ 0,
Tr ρ1 = N .
For fermions, the eigenvalues of
ρ1 , called natural occupation
numbers (NON), obey
0 ≤ ni ≤ 1.
Admisible 1-RDM forms a convex
set whose extremal states are
those coming from Hartree-Fock
states.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
NON: the case of Moshinsky atom
The Moshinsky atom describes two fermions interacting with
an external harmonic potential and repelling each other by a
Hooke-type force. The Hamiltonian is
p12 p22 k 2 2 δ
+
+ (r + r ) − |r1 − r2 |2 .
2
2 2 1 2
4
We study the ground and first-excited states of this system (t ≡ δ/k)
H=
1.0
0.8
n0,fs
0.6
n0,gs
0.4
n1,gs
n1,fs
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
t
CLBR, JM Gracia-Bondı́a and JC Várilly, PRA 86, 022525 (2012).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Borland & Dennis findings
It was long suspected that for pure states there are constraints
not implied by the original Pauli principle on the spectrum of
the 1-RDM (n1 ≥ n2 ≥ ...).
Consider the system ∧3 H6 of three electrons and a
six-dimensional one-particle Hilbert space. Apart from the
Coleman’s inequalities:
n1 ≤ 1
n3 ≤ 1
and
Carlos L. Benavides-Riveros
n2 ≤ 1
n1 + n2 ≤ 2.
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Borland & Dennis findings
It was long suspected that for pure states there are constraints
not implied by the original Pauli principle on the spectrum of
the 1-RDM (n1 ≥ n2 ≥ ...).
Consider the system ∧3 H6 of three electrons and a
six-dimensional one-particle Hilbert space. Apart from the
Coleman’s inequalities:
n6 + n1 ≤ 1
n4 + n3 ≤ 1
and
n5 + n2 ≤ 1
n4 + n1 + n2 ≤ 2.
stronger constraints!!!
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Borland & Dennis findings
It was long suspected that for pure states there are constraints
not implied by the original Pauli principle on the spectrum of
the 1-RDM (n1 ≥ n2 ≥ ...).
Consider the system ∧3 H6 of three electrons and a
six-dimensional one-particle Hilbert space. Apart from the
Coleman’s inequalities:
n6 + n1 =1
n4 + n3 =1
n5 + n2 =1
and n4 + n1 + n2 ≤ 2.
stronger constraints!!!
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The Borland-Dennis polytope
It defines a polytope of admissible states
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The Borland-Dennis polytope
It defines a polytope of admissible states (recall n1 ≥ n2 ≥ n3 ):
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The Borland-Dennis polytope
It defines a polytope of admissible states (recall n1 ≥ n2 ≥ n3 ):
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Generalized Pauli exclusion principle (GPEP)
In 2006-2008 A. Klyachko & M.
Altunbulak exhibited an algorithm
to compute in principle all such
Pauli-like inequalities.
For the general situation ∧N Hm
(with m ≥ 2N ) there is a finite set
of constraints:
µ
µ
µ
µ
dN ,m = κ0 + κ1 n1 + · · · + κm nm ≥ 0,
µ
κi integer.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
GPEP: examples
For example, for ∧3 H7 , there are four linear inequalities:
1
d3,7
= 2 − (n1 + n2 + n4 + n7 ) ≥ 0,
2
d3,7
= 2 − (n1 + n2 + n5 + n6 ) ≥ 0,
3
d3,7
= 2 − (n2 + n3 + n4 + n5 ) ≥ 0,
4
d3,7
= 2 − (n1 + n3 + n4 + n6 ) ≥ 0.
Generalized Pauli constraints (GPC) are consistent, so lower
rank ones can be derived from higher ones: for instance,
putting n7 = 0 we obtain the former restrictions for ∧3 H6 .
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
(Quasi)pinning I
For the Borland-Dennis setting ∧3 H6 , the GPEP states that:
n4 + n1 + n2 ≤ 2.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
(Quasi)pinning I
For the Borland-Dennis setting ∧3 H6 , the GPEP states that:
n4 + n1 + n2 ≤ 2.
The tantalizing suggestion is that the inequality is nearly
saturated for ground states. This is the “quasipinning”
phenomenon.
C Schilling, D Gross and M Christandl, PRL 110, 040404 (2013).
Actually, for not highly-correlated systems and
spin-compensated configurations one can show that the NON
are pinned to the boundary of the polytope:
n4 + n1 + n2 = 2.
CLBR and M Springborg, PRA 92, 012512 (2015).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Borland-Dennis polytope (reloaded)
The Borland-Dennis polytope
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Borland-Dennis polytope (reloaded)
The Borland-Dennis polytope
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Selection rules for pinned systems
µ
Let us consider one of the GPC for which pinning dN ,m = 0
holds. For pinned systems, the corresponding wave function
belongs to the 0-eigenspace of the operator
µ
µ
µ
µ
DN ,m = κ0 1 + κ1 a†1 a1 + · · · + κm a†m am ,
where a†i and ai are the creation and annihilation fermionic
operators of the state i.
Therefore, for the wave function, expanded in Slater
determinants,
X
µ
cK |Ki,
if DN ,m |Ki , 0, then cK = 0.
ψN =
K
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
energy
Hartree-Fock, correlation and pinned energy
(+)
Eex
(-)
Eex
EHF
Ecorr
ED
E0
Carlos L. Benavides-Riveros
E
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Towards a SL functional for more than 2 electrons
The pinned wave function ∧3 H6 reads:
√
√
√
|ψ3 i = n3 |α1 α2 α3 i + n5 |α1 α4 α5 i + n6 |α2 α4 α6 i,
containing only double excitations |αi αx αy i of the initial state
|α1 α2 α3 i.
Far from the Hartree-Fock state:
√
√
√
|ψ3 i = n4 |α1 α2 α4 i + n5 |α1 α3 α5 i + n6 |α2 α3 α6 i,
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The vanishing role of single excitations
Figure: Coefficients of the single excitations in the valence full CI
wave function of BH in HF MO (red) and in the NO basis (black).
ŁM Mentel, R van Meer, OV Gritsenko and EJ Baerends, JCP 140 214105 (2014).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The best’s still to come
Aimed to examine the nature of quasipinning in real systems,
we started to explore radial configurations of lithium based on
off-the-shelf basis sets.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Lithium
We may consider two different basis types for six-rank
approximations for lithium-like ions:
For the constraints within our calculation 6b , we find
b
0 ≤ d 6 = 2 − n1 − n2 − n4 = 2.146 × 10−5
and
b
d 6 /n6 ≈ 0.97.
For the restricted spin orbital case 6a , which actually delivers
a
the best energy, one finds d 6 = 0. Pinning!
CLBR, JM Gracia-Bondı́a and M Springborg, PRA 88, 022508 (2013).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Lithium series at ranks seven and eight
For a rank seven calculation the Klyachko constraints read
1
= 0,
0 ≤ d3,7
2
= 1.304 × 10−5 ,
0 ≤ d3,7
3
= 7.741 × 10−5 ,
0 ≤ d3,7
4
= 8.002 × 10−5 .
0 ≤ d3,7
We see three scales of (quasi)pinning!
For ∧3 H8 there are 31 inequalities. Some of them are given by
µ
µ
0 ≤ d3,8 = d3,7
5
0 ≤ d3,8
6
0 ≤ d3,8
7
0 ≤ d3,8
8
0 ≤ d3,8
µ = 1, ..., 4
= 1 − (n1 + n2 − n3 )
= 1 − (n2 + n5 − n7 )
= 1 − (n1 + n6 − n7 )
= 1 − (n2 + n4 − n6 ) and so on...
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The molecule He+2
Energy (a.u.)
-4.70
HF
Rank six
-4.75
Rank seven
Rank eight
-4.80
-4.85
-4.90
-4.95
0
2
4
R (a.u.)
6
8
CLBR and M Springborg, PRA 92 012512 (2015).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The molecule He+2
0.00025
D13,8
GPC
0.00020
D23,8
0.00015
D53,8
0.00010
0.00005
0.00000
0
2
4
6
8
R (a.u.)
CLBR and M Springborg, PRA 92 012512 (2015).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
The molecule He+2
GPC
0.0020
0.0015
0.0010
D33,8
D43,8
0.0005
0.0000
0
2
4
6
8
R (a.u.)
CLBR and M Springborg, PRA 92 012512 (2015).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Quasipinning and double excitations
2
The constraint d3,8
≥ 0 appears to be saturated exactly for the
diatomic ion; and nearly so the constraints
1
d3,8
≥ 0,
5
d3,8
:= 1 − n1 − n2 + n3 ≥ 0.
There is evidence showing that, when using natural orbitals (as
distinct from Hartree–Fock molecular orbitals, say) to study
bond weakening and breaking, doubly excited determinants
are enhanced dramatically with respect to singly and triply
ones.
For few-electron systems this outstanding phenomenon could
be explained from quasipinning, which first and foremost
eliminates approximately the oddly-excited configurations.
CLBR, JM Gracia-Bondı́a and M Springborg, arXiv:1409.6435 (2014).
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
MCSCF: a summary
Electronic wave functions often call for more than one
configuration to correctly describe quantum systems for which
the single-determinantal Hartree-Fock description is not
suitable.
The standard solution to this problem is to conduct a full
optimization calculation where both the orbitals and the
coefficients of the electronic configurations are optimized
simultaneously.
This latter approach is known as the multiconfigurational
self-consistent field (MCSCF) method. For several reasons, the
optimization of a MCSCF wave function is a very demanding
computational problem.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
An extended Hartree-Fock method
Pinned wave functions undergo remarkable structural
simplifications, which suggest a natural extension of the
Hartree-Fock ansatz of the form:
X
cK |Ki.
|Ψ i =
K
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
An extended Hartree-Fock method
Pinned wave functions undergo remarkable structural
simplifications, which suggest a natural extension of the
Hartree-Fock ansatz of the form:
X
cK |Ki.
|Ψ i =
K∈Id
Here Id stands for the family of configurations that may
contribute to the wave function in case of pinning to some
µ
facet dN ,m = 0 of the polytope of pure-realizable states.
The minimization of the following energy functional:
E[{cK }K∈ID , {|αi i}] = hΨ |H|Ψ i,
gives the ground-state energy of the system.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
energy
Hartree-Fock, correlation and pinned energy
(+)
Eex
(-)
Eex
EHF
Ecorr
ED
E0
Carlos L. Benavides-Riveros
E
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Numerical test
We apply this new method to a concrete system: the
one-dimensional N -Harmonium whose Hamiltonian is given by:
H=
N
N
N
i=1
i=1
i <j
1 X 2 ω2 X 2 δ X 2
pi +
xi +
xij ,
2
2
2
where xij := xi − xj . For fermions, the ground-state energy of
this system and the wave function can be found analytically.
In order to apply the pinning Ansatz, the configuration space is
divided in (N − 3) core, 3 active and 3 virtual orbitals (say,
∧N HN +3 ), in such a way that the Borland-Dennis state reads:
|Ψ i3,6 = c123 |α1 · · · αN −3 αN −2 αN −1 αN i
+ c145 |α1 · · · αN −3 αN −2 αN +1 αN +2 i
+ c246 |α1 · · · αN −3 αN −1 αN +1 αN +3 i.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Percentage of the correlation energy
CLBR and CS, Zeitschrift für Physikalische Chemie 230, 703 (2016).
CLBR and CS, To appear.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016
Pauli Principle Borland-Dennis GPEP Pinning EHF
Quasipinning and correlation energy
Theorem 1.
Let H be an N -fermion Hamiltonian on ∧N Hm with a unique
ground state with NON n
~ = (n1 , n2 , . . . , nm ). The error ∆E in the
energy of the ansatz based on pinning to a given facet Fd of
the polytope is bounded from above:
dN ,m
∆E
,
≤K
Ecorr
S(~
n)
(1)
(+)
where Ecorr is the correlation energy and K =
S(~
n) :=
N
X
(1 − ni ) +
i=1
Carlos L. Benavides-Riveros
m
X
ni .
i=N +1
Bogotá, 12 Sep 2016
C Eex −E0
N E (−) −E
ex
0
and
Pauli Principle Borland-Dennis GPEP Pinning EHF
A growing field...
International workshop on ”Reduced Density Matrices in
Quantum Physics and Role of Fermionic Exchange
Symmetry”, 12-15 April, 2016, Oxford, UK.
Workshop on ”Generalized Pauli Constraints and Fermion
Correlation”, 11-13 August, 2016, Vienna, Austria.
International Workshop on ”New challenges in Reduced
Density Matrix Functional Theory: Symmetries,
time-evolution and entanglement”, 26-29 September,
2017, Lausanne, Switzerland.
Carlos L. Benavides-Riveros
Bogotá, 12 Sep 2016