Bender & Orszag -Advanced Mathematical Methods for Scientists and Engineers
Quantum Chemistry in the Complex DomainExpert in Mathematical Physics Homepage: Book: \Advanced...
Transcript of Quantum Chemistry in the Complex DomainExpert in Mathematical Physics Homepage: Book: \Advanced...
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Quantum Chemistry in the Complex Domain
Hugh G. A. Burton,1 Alex J. W. Thom,1 and Pierre-Francois Loos2
1Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, UK
2Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universite de Toulouse, CNRS,UPS, France
5th Mar 2019
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Collaborators and Funding
• Selected CI and QMC
Anthony Yann Michel DenisScemama Garniron Caffarel Jacquemin
• Green function methods
Arjan Pina MikaBerger Romaniello Veril
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Selected CI methods + range-separated hybrids
“Quantum Package 2.0: An Open-Source Determinant-Driven Suite ofPrograms”,
Garniron et al. , JCTC (submitted) arXiv:1902.08154
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Green functions & self-consistency: an unhappy marriage?
G
Γ
P
W
Σ
εHF/KSεG
W
�Self-consistency?or
“Green functions and self-consistency: insights from the spherium model”,Loos, Romaniello & Berger, JCTC 14 (2018) 3071
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There’s a glitch in GW
“Unphysical discontinuities in GW methods”,Veril, Romaniello, Berger & Loos, JCTC 14 (2018) 5220
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How to morph ground state into excited state?
PhysicalTransition
Complex Adiabatic Connection
“Complex Adiabatic Connection: a Hidden Non-Hermitian Path fromGround to Excited States”,
Burton, Thom & Loos, JCP Comm. 150 (2019) 041103
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Section 2
PT -symmetric Quantum Mechanics
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Carl Bender
• Professor of Physics at Washington Universityin St. Louis:Expert in Mathematical Physics
• Homepage:https://web.physics.wustl.edu/cmb/
• Book:“Advanced Mathematical Methods for Scientistsand Engineers”
• Series of 15 lectures (can be found onYouTube) on Mathematical Physics:
• summation of divergent series• perturbation theory• asymptotic expansion• WKB approximation
• He will be lecturing at the 3rd Mini-school onMathematics (19th-21st Jun, Jussieu)https://wiki.lct.jussieu.fr/gdrnbody
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PT -Symmetric Quantum Mechanics
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PT -Symmetric Quantum Mechanics
The spectrum of the Hamiltonian
H = p2 + i x3
is real and positive.
Why? Because it is PT symmetric, i.e. invariant under the combination of
• parity P: p → −p and x → −x• time reversal T : p → −p, x → x and i → −i
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PT -Symmetric Quantum Mechanics
H = p2 + x2(ix)ε
• ε ≥ 0: unbroken PT -symmetry region
• ε = 0: PT boundary
• ε < 0: broken PT -symmetry region(eigenfunctions of H aren’t eigenfunctions of PT simultaneously)
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Hermitian vs PT -symmetric
PT -symmetric QM is an extension of QM into the complexplane
• Hermitian: H = H† where † means transpose +complex conjugate
• PT -symmetric: H = HPT , i.e. H = PT H(PT )−1
• Hermiticity is very powerful as it guarantees realenergies and conserves probability
• (unbroken) PT symmetry is a weaker condition whichstill ensure real energies and probability conservation
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Hermitian vs PT -symmetric vs Non-Hermitian
Hermitian H PT -symmetric H non-Hermitian H
H† = H HPT = H H† 6= H
Closed systems PT -symmetric systems Open systems
〈a|b〉 = a† · b 〈a|b〉 = aCPT · b (scattering, resonances, etc)
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PT -symmetric QM is a genuine quantum theory
Take-home message:PT -symmetric Hamiltonian can be seen as analytic continuation of
Hermitian Hamiltonian from real to complex space. 1
1The relativistic version of PT -symmetric QM does exist.
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PT -symmetric experiments
“Parity-time-symmetric whispering-gallery microcavities”Peng et al. Nature Physics 10 (2014) 394
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Highlight in Nature Physics (2015)
NATURE PHYSICS | VOL 11 | OCTOBER 2015 | www.nature.com/naturephysics 799
feature
© JO
RGE
CH
AM
© 2015 Macmillan Publishers Limited. All rights reserved
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PT -symmetry in Quantum and Classical Physics
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Section 3
Non-Hermitian quantum chemistry
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Hermitian Hamiltonian going complex
Let’s consider the Hamiltonian for two electrons on a unit sphere
H = −∇21 +∇2
2
2+
λ
r12
The CID/CCD Hamiltonian for 2 states reads
H = H(0)+λH(1) =
(λ λ/
√3
λ/√
3 2 + 7λ/5
)=
(0 00 2
)+λ
(1 1/
√3
1/√
3 7/5
)The eigenvalues are
E± = 1 +18λ
15±√
1 +2λ
5+
28λ2
75
For complex λ, the Hamiltonian becomes non Hermitian.There is a (square-root) singularity in the complex-λ plane at
λEP = −15
28
(1± i
5√3
)(Exceptional points)
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Hermitian Hamiltonian going complex
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• There is an avoided crossing at Re(λEP)
• The smaller Im(λEP), the sharper the avoided crossing is
• Square-root branch cuts from λEP running parallel to the Im axistowards ±i∞
• (non-Hermitian) exceptional points ≡ (Hermitian) conical intersection
• Im(λEP) is linked to the radius of convergence of PT
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Conical intersection (CI) vs exceptional point (EP)
Benda & Jagau, JPCL 9 (2018) 6978
• At CI, the eigenvectors stayorthogonal
• At EP, both eigenvalues andeigenvectors coalesce(self-orthogonal state)
• Encircling a CI, states do notinterchange but wave functionpicks up geometric phase
• Encircling a EP, states caninterchange and wave functionpicks up geometric phase
• encircling a EP clockwise oranticlockwise yields differentstates
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Hermitian Hamiltonian going PT -symmetric
How to PT -symmetrize a CI matrix?
H =
(λ iλ/
√3
iλ/√
3 2 + 7λ/5
)It is definitely not Hermitian but thePT -symmetry is not obvious...
H =
(ε1 iλiλ ε2
)=
((ε1 + ε2)/2 0
0 (ε1 + ε2)/2
)+ i
(i(ε2 − ε1)/2 λ
λ −i(ε2 − ε1)/2
)PT -symmetry projects exceptionalpoints on the real axis
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Section 4
non-Hermitian Quantum Chemistry
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The basic idea
• Quantum mechanics isquantized because we’re lookingat it in the real plane (Reimannsheets or parking garage)
• If you extend real numbers tocomplex numbers you lose theordering property of realnumbers
• So, can we interchange groundand excited states away fromthe real axis?
• How do we do it (in practice)?
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The basic idea
• Quantum mechanics isquantized because we’re lookingat it in the real plane (Reimannsheets or parking garage)
• If you extend real numbers tocomplex numbers you lose theordering property of realnumbers
• So, can we interchange groundand excited states away fromthe real axis?
• How do we do it (in practice)?
![Page 26: Quantum Chemistry in the Complex DomainExpert in Mathematical Physics Homepage: Book: \Advanced Mathematical Methods for Scientists and Engineers" Series of 15 lectures (can be found](https://reader033.fdocuments.in/reader033/viewer/2022060506/5f1f84dccdef73458378c1b9/html5/thumbnails/26.jpg)
Holomorphic HF = analytical continuation of HF
Let’s consider (again) the Hamiltonian for two electrons on a unit sphere
H = −∇21 +∇2
2
2+
λ
r12
We are looking for a UHF solution of the form
ΨUHF(θ1, θ2) = ϕ(θ1)ϕ(π − θ2)
where the spatial orbital is ϕ = s cosχ+ pz sinχ.
Ensuring the stationarity of the UHF energy, i.e., ∂EUHF/∂χ = 0
sin 2χ (75 + 6λ− 56λ cos 2χ) = 0
or
χ = 0 or π/2 χ = ± arccos
(3
28+
75
56λ
)
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HF energy landscape
-4 -2 0 2 4-6
-4
-2
0
2
4
6
E s2
RHF(λ) = λ Ep2z
RHF(λ) = 2 +29λ
25EUHF(λ) = − 75
112λ+
25
28+
59λ
84
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Analytical continuation and state interconversion
arccos(z) = π/2 + i log(i z +
√1− z2
)z = 3/28 + 75/(56λ)
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Complex adiabatic connection path
-3 -2 -1 0 1 2 3-4
-2
0
2
4
Coulson-Fisher points ≈ exceptional points ⇒ quasi-exceptional points
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That’s (almost) the end...
Acknowledgements
• Hugh Burton and Alex Thom (Cambridge)
• Emmanuel Giner and Julien Toulouse (Paris)
• Denis Jacquemin (Nantes)
• Emmanuel Fromager (Strasbourg)
• Pina Romaniello and Arjan Berger (Toulouse)
• Anthony Scemama and Michel Caffarel (Toulouse)