Quasipinning and an extended Hartree-Fock method based on GPC - - PowerPoint PPT Presentation

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: H = T + Vext + Vee =

N

• i=1

−1 2∆ri +

N

• i=1

V (ri) +

N

• i<j

1 |ri − rj|. Pure states ρN := |ψNψN| have skewsymmetric wave functions ψN ∈ ∧NH H⊗N, where H is the one-particle Hilbert space. For ensemble states: ρN =

• s

ps|ψsψs|.

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 xn+1,...,xN gives the so-called n-particle Reduced Density Matrix (n-RDM):

ρn(x1,...,xn;x′

1,...,x′ n)

= N n

• ρN(x1,...,...,xN;x′

1,...,x′ n,xn+1,...,xN)dxn+1 ...dxN.

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 xn+1,...,xN gives the so-called n-particle Reduced Density Matrix (n-RDM):

ρn(x1,...,xn;x′

1,...,x′ n)

= N n

• ρN(x1,...,...,xN;x′

1,...,x′ n,xn+1,...,xN)dxn+1 ...dxN.

x := (r,ς), being ς ∈ {↑,↓} the spin coordinate. The helium-like energy functional is given by:

E(ρ2) = Tr              − 2 N − 1       ∆r1 2 − V (r1)       + 1 |r1 − r2|      ρ2       .

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 xn+1,...,xN gives the so-called n-particle Reduced Density Matrix (n-RDM):

ρn(x1,...,xn;x′

1,...,x′ n)

= N n

• ρN(x1,...,...,xN;x′

1,...,x′ n,xn+1,...,xN)dxn+1 ...dxN.

x := (r,ς), being ς ∈ {↑,↓} the spin coordinate. The helium-like energy functional is given by:

E(ρ2) = Tr              − 2 N − 1       ∆r1 2 − V (r1)       + 1 |r1 − r2|      ρ2       .

The ground-state energy minimizes E(ρ2):

Egs = min{E(ρ2)| ρ2 ∈ B2

N}.

Here, B2

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

ρ2 ∈ B2

N ⇔ ∃ρN ∈ DMN : ρ2 =

N 2

• ρN dx3 ...dxn,

where DMN is the set of the N-particle density matrices. The N-representability problem consists in finding necessary and sufficient conditions for B2

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:

ρ2(x1,x2;x′

1,x′ 2) =

                ρ2(↑1↑2↑′

1↑′ 2)

ρ2(↑1↑2↑′

1↓′ 2)

ρ2(↑1↑2↓′

1↑′ 2)

ρ2(↑1↑2↓′

1↓′ 2)

ρ2(↑1↓2↑′

1↑′ 2)

ρ2(↑1↓2↑′

1↓′ 2)

ρ2(↑1↓2↓′

1↑′ 2)

ρ2(↑1↓2↓′

1↓′ 2)

ρ2(↓1↑2↑′

1↑′ 2)

ρ2(↓1↑2↑′

1↓′ 2)

ρ2(↓1↑2↓′

1↑′ 2)

ρ2(↓1↑2↓′

1↓′ 2)

ρ2(↓1↓2↑′

1↑′ 2)

ρ2(↓1↓2↑′

1↓′ 2)

ρ2(↓1↓2↓′

1↑′ 2)

ρ2(↓1↓2↓′

1↓′ 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:

ρ2(x1,x2;x′

1,x′ 2) =

                ρ2(↑1↑2↑′

1↑′ 2)

ρ2(↑1↑2↑′

1↓′ 2)

ρ2(↑1↑2↓′

1↑′ 2)

ρ2(↑1↑2↓′

1↓′ 2)

ρ2(↑1↓2↑′

1↑′ 2)

ρ2(↑1↓2↑′

1↓′ 2)

ρ2(↑1↓2↓′

1↑′ 2)

ρ2(↑1↓2↓′

1↓′ 2)

ρ2(↓1↑2↑′

1↑′ 2)

ρ2(↓1↑2↑′

1↓′ 2)

ρ2(↓1↑2↓′

1↑′ 2)

ρ2(↓1↑2↓′

1↓′ 2)

ρ2(↓1↓2↑′

1↑′ 2)

ρ2(↓1↓2↑′

1↓′ 2)

ρ2(↓1↓2↓′

1↑′ 2)

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:

• [1] ⊕ [3]

⊗2 = 2[1] ⊕ 3[3] ⊕ [5].

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:

                      ρsc1 ρsc2 ρv1 ρv2 ρv3 ρq                       =                       ρsc1 ρsc2 ρv1 ρv2 ρv3 ρq                       .

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:

                      ρsc1 ρsc2 ρv1 ρv2 ρv3 ρq                       =                      +1 −1 −1 −1 +1 −1                                            ρsc1 ρsc2 ρv1 ρv2 ρv3 ρq                       .

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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: E(ρ2) = Tr              2 N − 1       − ∆r1 2 + Z |r1|       +       1 |r1 − r2|            ρ2       . = Tr             − ∆r1 2 + Z |r1|      ρ1(x1;x′

1)

     

x1=x′

1

+ I(r) r dr. where the intracular distribution is defined in the following way: I(r) =

• ρ2(x1,x2;x1,x2)δ(|r1 − r2| − r)dx1 dx2.

Crittenden and Gill, JCP 127, 014101 (2007). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Density functional theory on phase space

Let us consider the Wigner function: w(r,ς;p,ς′) := π−N

• ρ1(r − z,ς;r + z,ς′)e2ip·z dz1.

This is a real function and Trw = N. However, w can take some negative values. E(w,I) =       p 2 + Z |r|      w(r,ς;p,ς)drdpdς + I(r) r dr. There is a “phase space” Gilbert theorem.

P . Blanchard, JM Gracia-Bondia and J V´ arilly, IJQC 112, 1134 (2012). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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 ∧NHm. 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 University of Oxford, 12-15 April 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, Trρ1 = 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 University of Oxford, 12-15 April 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

H = p2

1

2 + p2

2

2 + k 2(r2

1 + r2 2) − δ

2|r1 − r2|2. We study the ground and first-excited states of this system

n0,fs n0,gs n1,gs n1,fs 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t CLBR, JM Gracia-Bond´ ıa and JC V´ arilly, PRA 86, 022525 (2012). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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 ∧3H6 of three electrons and a six-dimensional one-particle Hilbert space. Apart from the Coleman’s inequalities:

n1 ≤ 1 n2 ≤ 1 n3 ≤ 1 and n1 + n2 ≤ 2.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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 ∧3H6 of three electrons and a six-dimensional one-particle Hilbert space. Apart from the Coleman’s inequalities:

n6 + n1 ≤ 1 n5 + n2 ≤ 1 n4 + n3 ≤ 1 and n4 + n1 + n2 ≤ 2.

stronger constraints!!!

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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 ∧3H6 of three electrons and a six-dimensional one-particle Hilbert space. Apart from the Coleman’s inequalities:

n6 + n1=1 n5 + n2=1 n4 + n3=1 and n4 + n1 + n2 ≤ 2.

stronger constraints!!!

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

The Borland-Dennis polytope

It defines a polytope of admissible states

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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 University of Oxford, 12-15 April 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 University of Oxford, 12-15 April 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 ∧NHm (with m ≥ 2N) there is a finite set

• f constraints:

dµ

N,m = κµ 0 + κµ 1n1 + ··· + κµ nnn ≥ 0,

κµ

i integer.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

GPEP: examples

For example, for ∧3H7, there are four linear inequalities: d1

3,7 = 2 − (n1 + n2 + n4 + n7) ≥ 0,

d2

3,7 = 2 − (n1 + n2 + n5 + n6) ≥ 0,

d3

3,7 = 2 − (n2 + n3 + n4 + n5) ≥ 0,

d4

3,7 = 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 ∧3H6.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

(Quasi)pinning I

For the Borland-Dennis setting ∧3H6, the GPEP states that: n4 + n1 + n2 ≤ 2.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

(Quasi)pinning I

For the Borland-Dennis setting ∧3H6, 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 University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Borland-Dennis polytope (reloaded)

The Borland-Dennis polytope

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Borland-Dennis polytope (reloaded)

The Borland-Dennis polytope

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Pinning in Hubbard model (3 fermions, 3 sites)

C Schilling, PRB 92, 155149 (2015). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Pinning in Hubbard model (3 fermions, 4 sites)

(s,m,k) = (1

2, 1 2,1)

C Schilling, PRB 92, 155149 (2015). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Selection rules for pinned systems

Let us consider one of the GPC for which pinning dµ

N,m = 0

• holds. For pinned systems, the corresponding wave function

belongs to the 0-eigenspace of the operator Dµ

N,m = κµ 01 + κµ 1a† 1a1 + ··· + κµ ma† mam,

where a†

i and ai are the creation and annihilation fermionic

• perators of the state i.

Therefore, for the wave function, expanded in Slater determinants, ψN =

• K

cK|K, if Dµ

N,m|K 0, then cK = 0.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Hartree-Fock, correlation and pinned energy

E E E E E

HF ex ex (-) (+)

E E

energy

D corr

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Towards a SL functional for more than 2 electrons

The pinned wave function ∧3H6 reads: |ψ3 = √n3|α1α2α3 + √n5|α1α4α5 + √n6|α2α4α6, containing only double excitations |αiαxαy of the initial state |α1α2α3. Far from the Hartree-Fock state: |ψ3 = √n4|α1α2α4 + √n5|α1α3α5 + √n6|α2α3α6,

Christandl’s talk. Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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 University of Oxford, 12-15 April 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

• ff-the-shelf basis sets.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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

0 ≤ d6b = 2 − n1 − n2 − n4 = 2.146 × 10−5 and d6b/n6 ≈ 0.97.

For the restricted spin orbital case 6a, which actually delivers the best energy, one finds d6a = 0. Pinning!

CLBR, JM Gracia-Bond´ ıa and M Springborg, PRA 88, 022508 (2013). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Lithium series at ranks seven and eight

For a rank seven calculation the Klyachko constraints read

0 ≤ d1

3,7 = 0,

0 ≤ d2

3,7 = 1.304 × 10−5,

0 ≤ d3

3,7 = 7.741 × 10−5,

0 ≤ d4

3,7 = 8.002 × 10−5.

We see three scales of (quasi)pinning! For ∧3H8 there are 31 inequalities. Some of them are given by

0 ≤ dµ

3,8 = dµ 3,7

µ = 1,...,4 0 ≤ d5

3,8 = 1 − (n1 + n2 − n3)

0 ≤ d6

3,8 = 1 − (n2 + n5 − n7)

0 ≤ d7

3,8 = 1 − (n1 + n6 − n7)

0 ≤ d8

3,8 = 1 − (n2 + n4 − n6)

and so on...

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

The molecule He+

2

HF Rank six Rank seven Rank eight

2 4 6 8

• 4.95
• 4.90
• 4.85
• 4.80
• 4.75
• 4.70

R (a.u.) Energy (a.u.)

CLBR and M Springborg, PRA 92 012512 (2015). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

The molecule He+

2

D3,8

1

D3,8

2

D3,8

5

2 4 6 8 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025

R (a.u.) GPC

CLBR and M Springborg, PRA 92 012512 (2015). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

The molecule He+

2

D3,8

3

D3,8

4

2 4 6 8 0.0000 0.0005 0.0010 0.0015 0.0020

R (a.u.) GPC

CLBR and M Springborg, PRA 92 012512 (2015). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Quasipinning and double excitations

The constraint d2

3,8 ≥ 0 appears to be saturated exactly for the

diatomic ion; and nearly so the constraints d1

3,8 ≥ 0,

d5

3,8 := 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

• nes.

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-Bond´ ıa and M Springborg, arXiv:1409.6435 (2014). Carlos L. Benavides-Riveros University of Oxford, 12-15 April 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

• ptimization 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

• ptimization of a MCSCF wave function is a very demanding

computational problem.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

An extended Hartree-Fock method I

Pinned wave functions undergo remarkable structural simplifications, which suggest a natural extension of the Hartree-Fock ansatz of the form: |Ψ =

• K

cK|K.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

An extended Hartree-Fock method I

Pinned wave functions undergo remarkable structural simplifications, which suggest a natural extension of the Hartree-Fock ansatz of the form: |Ψ =

• K∈Id

cK|K. Here Id stands for the family of configurations that may contribute to the wave function in case of pinning to some facet dµ

N,m = 0 of the polytope of pure-realizable states.

The minimization of the following energy functional: E[{cK}K∈ID,{|αi}] = Ψ|H|Ψ, gives the ground-state energy of the system.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

An extended Hartree-Fock method II

The NO |αi are rotated according to |αi = e− ˆ

κ| ˜

αi, ∀i, with an antihermitian operator ˆ κ. The orbital optimization is accomplished in form of an

• ptimization of ˆ

κ and the unitarity of e− ˆ

κ.

The energy functional becomes: E[{cK}K∈ID,κ] = ˜ Ψ|e ˆ

κHe− ˆ κ| ˜

Ψ.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Numerical test

We apply this new method to a concrete system: the

• ne-dimensional N-Harmonium whose Hamiltonian is given by:

H = 1 2

N

• i=1

p2

i + ω2

2

N

• i=1

x2

i + δ

2

N

• i <j

x2

ij,

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, ∧NHN+3), in such a way that the Borland-Dennis state reads: |Ψ3,6 =c123|α1 ···αN−3αN−2αN−1αN +c145|α1 ···αN−3αN−2αN+1αN+2 +c246|α1 ···αN−3αN−1αN+1αN+3.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Correlation energy?

CLBR and C Schilling, To appear. Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Quasipinning and correlation energy

E E E E E

HF ex ex (-) (+)

E E

energy

D corr

Figure: Energy spectrum of the Hamiltonian H.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Quasipinning and correlation energy

Theorem 1. Let H be an N-fermion Hamiltonian on ∧NHm 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: ∆E Ecorr ≤ K dN,m S( n) , (1) where Ecorr is the correlation energy and K = C

N E(+)

ex −E0

E(−)

ex −E0

and S( n) :=

N

• i=1

(1 − ni) +

m

• i=N+1

ni.

CLBR and C Schilling, Z. Phys. Chem. (in press). CLBR and C Schilling, To appear soon. Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016

Pauli Principle Borland-Dennis GPEP Pinning EHF

Thanks.

Carlos L. Benavides-Riveros University of Oxford, 12-15 April 2016