Relativistic molecular physics: electronic properties in ... · The Dirac equation Relativistic equation of motion for the wave function of an electron. (p + eA ) m 0c (x) = 0: I

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.


Talk overview

1 Relativistic electronic structure

2 The electronic structure program BERTHA

3 Tests of fundamental physics

4 Reduced density matrix theory

5 Summary


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.


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.


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)


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

]


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.


Talk overview





5 Summary


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.


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.


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.


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.


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


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”.


Talk overview





5 Summary


Applications to tests of fundamental physics

I Properties of superheavy elementsI Electric dipole momentsI Interactions with dark matterI Electroweak interactions and biomolecular chirality


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).


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+.


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+.


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.


Talk overview





5 Summary


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.


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.


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〉.


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).


A brief history of 2-RDM theory

Figure: Charles Coulson, Roy McWeeny, Per-Olaf Lowdin, John Coleman, DavidMazziotti


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”.


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.


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.



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.



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.


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.


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.


Talk overview





5 Summary


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.


Acknowledgements

I Daniel Flynn

I Andy Martin, Mitch Knight

I Alex Kozlov

I Ian Grant, Haakon Skaane

I Lady Bertha Jeffreys (Swirles)


Documents

Relativistic molecular physics: electronic properties in ... · The Dirac equation Relativistic equation of motion for the wave function of an electron. (p + eA ) m 0c (x) = 0: I