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Relativistic molecular physics: electronic properties insearch of fundamental EDMs
Harry QuineyTheoretical Condensed Matter Physics Group
School of PhysicsThe University of Melbourne
27 November 2019
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 1 / 36
Relativistic electronic structure: motivation
Relativistic atomic physics:
I Superheavy element research.I Tests of quantum electrodynamics and electroweak interactions.I Physics beyond the Standard Model.
Calculations simplified by separation into radial and spin-angular parts.
Relativistic molecular physics and quantum chemistry.
I Recovery and treatment of nuclear waste.I Heavy element materials science.I Electroweak interactions in chiral moleculesI New generation of Beyond Standard Model experiments.
Calculations require evaluation of multicentre integrals involving four-componentspinors and complicated two-body operators.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 2 / 36
Talk overview
1 Relativistic electronic structure
2 The electronic structure program BERTHA
3 Tests of fundamental physics
4 Reduced density matrix theory
5 Summary
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 3 / 36
The Dirac equation
Relativistic equation of motion for the wave function of an electron.[γµ(pµ + eAµ)−m0c
]Ψ(x) = 0.
I Proposed by Paul Dirac in 1928.
I Predicted the existence of positron – the Dirac sea.
I Relativistic self-consistent field proposed by Bertha Swirles in 1935.
I Douglas Hartree suggested the project on the platform of Euston Station,1934.
I Atomic structure codes in development since 1970’s.
I Program GRASP developed by Ian Grant et al.
Figure: Paul Dirac, Bertha Swirles (Lady Jeffreys), Ian Grant.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 4 / 36
Dirac energy spectrum
Einstein’s special relativity admits two roots:
E2 = (mc2)2 + (pc)2 → E = ±√
(mc2)2 + (pc)2
Finite basis method produces discrete representation of continuous positive/negativeenergy spectra.
I Half of the solution space is in the negative-energy spectrum!
I Anything that relies on a complete space requires this ‘Dirac sea’.
I∑
bound +∫+ve
+∫-ve
becomes∑
all.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 5 / 36
Familiar things from Szabo and Ostlund that stillwork here
I Variational principle (with some qualifications)I Hamiltonian formalismI Slater determinantsI Mean-field theories (Dirac-Hartree-Fock, Dirac-Kohn-Sham)I Gaussian basis setsI Brillouin’s TheoremI Koopmans’ TheoremI Slater-Condon rules for matrix element of two-body operatorsI Many-body theories: MCSCF, MBPT, CC, CI)
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 6 / 36
Relativistic N-electron problem
I Second quantisation → electron (ai, a†i ) and positron (bi, b
†i ) states.
I Fermion spin-statistics: Ψ(r1, . . . , rN ) is anti-symmetric with 4-spinorstructure
I Independent particle model: electrons move in an average potential u(r):
H =
[∑i
hD(ri) + u(ri)
]+
[−∑i
u(ri) +∑i>j
vCij
]= H0 + V .
I Dirac-Hartree-Fock model treats H0; V requires many-body theory (MBPT,CC, CI).
I Negative energy states include O((Z/c)4
)energy correction:
E2 =∑ab
[++∑rs
〈ab||rs〉〈rs||ab〉εa + εb − εr − εs
−−−∑cd
〈ab||cd〉〈cd||ab〉εa + εb − εc − εd
]
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 7 / 36
The electron-electron interaction
I In the Feynman gauge, the relativistically-covariant interaction is
vF12 = (1−α1 ·α2)eiωr12
r12, ω = photon frequency.
I Electronic structure calculations are made under the Coulomb gauge (∇ ·A = 0):
limω→0
vC12 =1
r12− 1
2
(α1 ·α2
r12+
(α1 · r12)(α2 · r12)
r312
)+O
(1/c4
)= Coulomb + Breit +O
(1/c4
)I Field-consistent interaction between charge-current pairs:
(ij|vC12|kl) =
∫∫%ij(r1)%kl(r2)
r12dr1 dr2 +
1
2c2
∫∫jij(r1) · jkl(r2)
r12dr1 dr2
+1
2c2
∫∫[jij(r1) · r12][jkl(r2) · r12]
r12dr1 dr2 +O
(1/c4
).
I Think of these as scalar Coulomb, current-current and magnetic dipole interactions.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 8 / 36
Talk overview
1 Relativistic electronic structure
2 The electronic structure program BERTHA
3 Tests of fundamental physics
4 Reduced density matrix theory
5 Summary
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 9 / 36
The electronic structure program BERTHA
I Dirac-Hartree-Fock self-consistent fields.I Relativistic density functional theory.I Relativistic spinor and charge-current structures naturally built in at code
level.I Breit interaction in self-consistent field.I Detailed treatment near the nucleus (Fermi nuclear models and leading-order
QED).I Many-body perturbation theory corrections (MBPT2) – Coulomb and Breit.I Electric/magnetic properties, electroweak and PT -odd matrix elements.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 10 / 36
G-spinors and many-centre spinor integrals
I G-spinors: Gaussian spinor basis functions
I Atom-centred radial and spin-angular parts: M [T, µ; r] = 1rfT (r)χT (θ, ϕ).
I T labels ”large” and ”small” components related by σ.p operator
I Gaussian product theorem:
M†[T, µ; r]σqM [T ′, ν; r] =Λ∑αβγ
ETT′
q [µ, ν;α, β, γ]H(p, rp;α, β, γ)
I The expansion is finite: 0 ≤ α+ β + γ ≤ Λ with Λ = `µ + `ν + ηTT′
I One-centre terms handled with atomic Racah algebra methods.
I Coulomb integrals involve σ0 operator.
I Breit integrals involve σq (q = 1, 2, 3) operators.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 11 / 36
Dirac-Hartree-Fock-Breit equations
Finite basis set method relies on matrix rep. of the Hamiltonian (the Fock matrix)
Fc = ESc.
Self-consistent procedure (' 30 iterations) until solutions converge.
I E is a diagonal matrix of eigenvalue energies.
I Finite representation of the + and − energy continua.I Occupied bound states.
I c lists expansion coefficients or linear combinations that expand spinors.
I Separated into L and S components (equal numbers).I Orthonormal spinor wave functions.
I F = H +G+B is the Fock matrix taken in this SCF approach.
I H → One-electron terms (overlap, kinetic, nuclear attraction, QED).I G→ Coulomb two-electron interactions.I B → Breit two-electron interactions.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 12 / 36
Many-centre matrix elements
Full suite of matrix elements available:(µ, T | σq | ν, T ′
) (µ, T | σqxi | ν, T ′
)(µ, T | σqxixj | ν, T ′
) (µ, T
∣∣∣∣ σq xi|x−C|3
∣∣∣∣ ν, T ′)(µ, T
∣∣∣∣ σq xixj|x−C|3
∣∣∣∣ ν, T ′) (µ, T
∣∣∣∣ σq xixjxk|x−C|3
∣∣∣∣ ν, T ′)(µ, T | σqVnuc(|x−C|) | ν, T ′
) (µ, T
∣∣∣∣ σq∇i( xi|x−C|
) ∣∣∣∣ ν, T ′)(µ, T | σq%nuc(|x−C|) | ν, T ′
) (µ, T | σ · p | ν, T ′
)(µ, T
∣∣ σq∇2∣∣ ν, T ′) (
µ, T∣∣∣ σqe−ik·x ∣∣∣ ν, T ′)(
µ, T ; ν, T
∣∣∣∣ σq,1σq,2 1
r12
∣∣∣∣σ, T ′, τ, T ′) (µ, T ; ν, T
∣∣∣∣ (xi,1σi,1)(xj,2σj,2)
(r12)3
∣∣∣∣σ, T ′; τ, T ′)
I Physical properties involve linear combinations of these matrix elements.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 13 / 36
Realistic nuclear charge models
High-Z nuclei are poorly-described by Gaussian charge models.
I Fermi charge model more appropriate.
I Many-centre integrals for Fermi not known.
I Use linear combination of 20 Gaussian-type fitting functions.
I Point-wise matching of nuclear potential near the origin.I Demand radial moments match experimental data.
I Critical details for electroweak current interactions.
Z = 100 E1s 〈1/r2〉1spoint –5245.5257 65135.6725Gaussian –5232.4427 60296.7617uniform –5232.2469 60165.0237Fermi –5232.2803 60187.7620
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 14 / 36
QED effects: vacuum polarisation
Vacuum polarisation from high-Z nuclei affects electronic structure.
I Leading-order effect is the Uehling potential VUeh(r).
I Depends on nuclear charge distribution %(r).I No closed-form expression for most realistic %(r).
I Difficult to apply molecular integrals to this.
I Could instead regard as a polarised charge density %(r).I Know that Qtot = 0 and 〈r2〉 = 8
5c3.
I Use Gaussians to match VUeh(r) pointwise and meet moment conditions.I Need 26 Gaussians to generate reliable VUeh(r) for all Z.I Many-centre self-consistent vacuum polarisation treatment in BERTHA.
Paper in preparation – “Vacuum polarisation potential from polarized charge-density”.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 15 / 36
Talk overview
1 Relativistic electronic structure
2 The electronic structure program BERTHA
3 Tests of fundamental physics
4 Reduced density matrix theory
5 Summary
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 16 / 36
Applications to tests of fundamental physics
I Properties of superheavy elementsI Electric dipole momentsI Interactions with dark matterI Electroweak interactions and biomolecular chirality
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 17 / 36
Proton EDM in Thallium Fluoride
I Sandars and Hinds (1980) formulated the coupling of a proton EDM to theelectronic structure in terms of a volume effect and a magnetic effect. Therelevant single-particle matrix elements are (respectively)
Xj =2π
3
∂
∂zψ†j (r)ψj(r)
∣∣∣r=0
Mj =
⟨ψj
∣∣∣∣α× `r3
∣∣∣∣ψj
⟩I The results of Quiney et al., (1998) were a direct reappraisal of the earlier
work of Coveney and Sandars (1983).I The study underlined the importance of basis set quality and, in particular,
determining the correct ratio of the large- and small-component spinoramplitudes (p0/q0) at the nuclear positions.
I Very large basis sets are required (uncontracted 34s34p16d9f for Tl).
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 18 / 36
Electron EDM experiments
I How to choose a molecule (ref Timo Fleig’s talk):
I A heavy centre.I Highly-polarised.I Large enhancement factor.I Simple hyperfine spectrum.I Stable in lab conditions.I Within capabilities of electronic structure programs.
I Current experimental favourites are YbF, ThO, HfF+.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 19 / 36
Electron EDM experiments
I How to choose a molecule (ref Timo Fleig’s talk):
I A heavy centre.I Highly-polarised.I Large enhancement factor.I Simple hyperfine spectrum.I Stable in lab conditions.I Within capabilities of electronic structure programs.
I Current experimental favourites are YbF, ThO, HfF+.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 19 / 36
Electron EDM calculations in YbF
I Sandars (1965) transformed eEDM operator into an effective single articleoperator
Hd = −de(γ0 − I)σ.E
= −de2icγ0γ5p2
I Both forms were calculated using BERTHA long ago (1998), withsecond-order MBPT corrections; more sophisticated calculations essentially inagreement.
I These calculations place an upper bound on the electron EDM of1.1× 10−29 e.cm
I Reliability of calculations assessed by magnetic dipole interactions
A‖(MHz) A⊥(MHz)DHF 5987 5883DHF + MBPT (2) 7985 7805Exp. 7822 7513
Table: Parallel (A‖) and perpendicular (A⊥) hyperfine constants for YbF.
I A goal of the proposed CTP program was to develop the tools to treat alleffects on the same footing.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 20 / 36
Talk overview
1 Relativistic electronic structure
2 The electronic structure program BERTHA
3 Tests of fundamental physics
4 Reduced density matrix theory
5 Summary
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 21 / 36
A brief history of a famous problem
N -particle Schrodinger equation
HΨ(r!, r2, . . . , rN ) = EΨ(r!, r2, . . . , rN )
N -particle density
D(N) = Ψ(r!, r2, . . . , rN )Ψ∗(r!, r2, . . . , rN )
I The wavefunction or the N -particle density provide a complee description of thesystem. If the system involves only pairwise interactions, the descriptions areovercomplete. The complexity of Ψ and D(N) grow exponentially with the numberof particles, so they are out of reach for real systems containing more than a fewparticles.
I On the other hand, the procedure for determining Ψ is simple, direct and is basedon a simple variation principle (full configuration interaction). It is reassuring sucha scheme exists in principle even though it is not really practical.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 22 / 36
Reduced density matrices: p-RDM
Dirac (1929) formulated Hartree-Fock theory in terms of 1-RDM and 2-RDM:
D(1)(r, r′) =∑i
φi(r)φ∗i (r′)
D(2)(r1, r2; r′1, r′2) = D(1)(r1, r
′1)D(1)(r2, r
′2)−D(1)(r1, r
′2)D(1)(r2, r
′1)
Husimi (1940) generalized this to the p-RDM
D(p) =
∫D(N)drp+1drp+2 . . . drN
The energy of a system involving only p-particle interactions can be writtenexactly using only the p-RDM.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 23 / 36
Reduced density matrices: fermions and two-bodyoperators
For fermions and two-body operators we have, in general:
E = Tr[HD(2)]
=∑pqst
K2,pqst D2,pq
st
where
D2,pqst =
⟨Ψ∣∣∣Γ2,pq
st
∣∣∣Ψ⟩Γ2,pqst = a†pa
†qatas
K2,pqst =
1
N − 1〈p|h|s〉δqt + 〈pq|g|st〉.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 24 / 36
Reduced density matrices: properties
1 Hermiticity
D1,ji =
(D1,i
j
)∗; D2,kl
ij =(D2,ij
kl
)∗2 Antisymmetry
D2,klij = −D2,kl
ji = −D2,lkij = D2,lk
ji
3 Contraction
D1,ij =
1
N − 1
∑k
D2,ikjk
4 TraceTr[D(1)] = N ; Tr[D(2)] = N(N − 1).
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 25 / 36
A brief history of 2-RDM theory
Figure: Charles Coulson, Roy McWeeny, Per-Olaf Lowdin, John Coleman, DavidMazziotti
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 26 / 36
Variational methods and N-representability
I Starting in 1955, several people had the bright idea of minimizing the energy,E, with respect to the elements of a trial form for D(2).
I It was apparent that that this approach had the potential to ’solve’ themany-electron problem in atomic and molecular physics, because thecomplexity of D(2) is very much less than the complexity ofΨ(r1, r2, . . . , rN ).
I It rapidly became apparent that there is a very serious catch: not all trialmatrices D(2) of the appropriate dimensions are related to a valid D(2) thatis derivable from an N -fermion wavefunction.
I Of course, this variational approach does work if one constructs trial matricesD(2) from a trial Ψ(r1, r2, . . . , rN ), but that is a rather pointless way ofproceeding.
I John Coleman (Queen’s University, Canada, 1958) was the first to describethis is the ”N -representability problem”.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 27 / 36
Density Functional Theory
I In 1964, the Hohenberg-Kohn theorems established that E = E[ρ], where theelectron density ρ is just the 1-RDM.
I Parametrization of ρ in an orbital basis facilitiates the valid N -representationof the kinetic energy functional, T [ρ].
I The orbitals are generated using an ”exchange-correlation” potential, VXC [ρ]I In practice, VXC [ρ] is approximated by parametrized models, based on the
Dirac free-electron gas functional.I DFT hs revolutionized quantum chemistry; relativistic extensiions are
available.I In practice its semi-empirical treatment of the many-body problem is not
really consistent with the CTP philosophy of achieving high precision throughefficient implementation of ab initio theories.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 28 / 36
N-representability of the 2-RDM
I Unconstrained variation of D(2) generally leads to unphysical eigenvalues: theN -density and all p-RDM are all positive semi-definite matrices.
I Coulson (1960) declared the solution to the N -representability problem to bethe most challenging in electronic structure theory, in order to escape theexponential scaling of wavefunction-based methods, the limitations imposedby approximation schemes and to understand the nature of electroncorrelation in atoms molecules and solids,
I It took more than 50 years to determine a complete set of N -representabilityconditions on D(2).
I The technical challenge of implementing efficiently this positive-definitevariational optimization problem is now a subject of active research.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 29 / 36
N-representability of the 2-RDM
I By 1964, Coleman conjectured (by analysing conventional many-bodytheories) that at least three of the variational constraints required to achievevariational behaviour should be based on the positve semi-definite characterof the 2-RDMs:
D2,ijkl =
⟨Ψ∣∣∣a†ia†jalak∣∣∣Ψ⟩
Q2,ijkl =
⟨Ψ∣∣∣aiaja†l a†k∣∣∣Ψ⟩
G2,ijkl =
⟨Ψ∣∣∣a†iaja†l ak∣∣∣Ψ⟩
I Together with the (simple) 1-RDM constraints, imposition of positivesemi-definite constraints on the 2-particle (D(2)), 2-hole (Q(2)) and1-particle-1-hole (G(2)) matrices restored strict variational behaviour totwo-fermion systems
I The problem rapidly becomes too complex to analyse in detail for more thana few particles.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 30 / 36
N-representability of the 2-RDM
I By 1978, Erdahl extended Coleman’s work to find one constraint for the3-RDM
D3,ijkpqr =
⟨Ψ∣∣∣a†ia†ja†karaqap∣∣∣Ψ⟩
which further constrains the 2-RDM by index contraction
D2,ijpq =
1
N − 2
∑k
D3,ijkpqk
I This gives some hint about the structure of the solution: the 2-RDM isrelated to the p-RDM for p > 2, but the energy does not depend explicitly onthe p-RDM. The constraints must reduce to the 2-RDM expression for Ethrough contraction.
I Despite another 25 years of effort, variational behaviour was not establishedfor N > 2.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 31 / 36
Constructive N-representability
I Necessary and sufficient conditions forN -representability of the 2-RDM werefinally obtained by Mazziotti (2012), based on Kummer’s Theorem (1975):
There exists a set of two-body operators, O(2), with the property
Tr[O(2)D(2)] ≥ 0
if, and only if, D(2) is N -representable. The operators represent the ”polar” ofthe set of N -representable 2-RDM, denoted P 2∗
N
I These operators may be constructed systemmatically in the form
O(2) =∑i
wiCiC†i
where the operators Ci are polynomials in ap and a†p and wi are non-negativeintegers.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 32 / 36
N-representability: the (2,q) conditions
The Ci-operators of order q are generated in two ways to form a complete set:
1 All permutations of products of order q > 2 involving ap and a†p2 All operators generated by ”lifting” the q = 2 operators by q − 2.
I The weights are determined by the condition that all 3-body and higher-orderterms cancel identically at each order, generating the polar set of two-bodyoperators, O(2).
I The order-by-order specification of the operators O(2) form the generators ofan N -representable 2-RDM.
I Extension to relativistic 2-RDM theory seems straightforward. The relativisticinteractions (including frequency-dependent Breit interactions) are two-bodyand the creation of particle-hole pairs adds two new operators to each Ci.
I These new terms correspond to many-body vacuum correlations.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 33 / 36
Talk overview
1 Relativistic electronic structure
2 The electronic structure program BERTHA
3 Tests of fundamental physics
4 Reduced density matrix theory
5 Summary
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 34 / 36
Summary
I Relativistic electronic structure theory gives access to physics across (and beyond)the Standard Model, i.e., proton radius, nuclear island of stability, electroweaktheory, EDM searches.
I N -representability problem in 2-RDM theory solved for all practical purposes
I Polynomial scaling with N suggest a new, complete and feasible approach to thefermion many-body problem, including a complete and consistent treatment ofpair-creation problems
I 2-RDM theory equally applicable to electnonic structure as well as strongcorrelations in Fermi gases
I Hybrid algorithms and quantum measurement approaches to 2-RDM theoryreported recently.
I A lesson in the power of perseverance.
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 35 / 36
Acknowledgements
I Daniel Flynn
I Andy Martin, Mitch Knight
I Alex Kozlov
I Ian Grant, Haakon Skaane
I Lady Bertha Jeffreys (Swirles)
Harry Quiney EDM Workshop, ANU, 25-27 November 2019 36 / 36