Atomic electric dipole moment calculation with perturbed coupled cluster method

Atomic electric dipole moment calculation with perturbed coupled cluster method D. Angom Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380 009 Symposium on 50 Years of Coupled Cluster Theory July 2, 2008

Plan of the talk Introduction Permanent electric dipole moment Theoretical calculations Coupled-cluster and atomic EDM Perturbed coupled-cluster Perturbed cluster amplitudes Cluster equations Some issues related to T (1) Diagrams Properties of T (1) 6j symbols Results

Introduction Introduction Permanent electric dipole moment Theoretical calculations Coupled-cluster and atomic EDM Perturbed coupled-cluster Perturbed cluster amplitudes Cluster equations Some issues related to T (1) Diagrams Properties of T (1) 6j symbols Results

Introduction Permanent electric dipole moment Permanent eletric dipole moment Permanent electric dipole moment is signature of parity and time reversal violations Within standard model of particle physics, electric dipole moment of electron d e 10 37 e cm. Time reversal violation is inferred from CP violation in neutral kaon decay.

Introduction Permanent electric dipole moment why choose atoms Permanent electric dipole moment d of a particle is measured from the response to external electromagnetic fields H int = d E µ B. EDM measurements of charged particles is nontrivial. Neutron and atoms, charge neutral, are ideal candidates. Origin EDM of subatomic particles: electron, proton and neutron. P and T violating e-e, e-n and n-n interactions. Atomic probes electron sector: paramagnetic atoms, nuclear sector: diamagnetic atoms.

Introduction Permanent electric dipole moment Implications to particle physics Atomic EDM probes P and T violation effects in different sectors, schematically particle level CP model Higgs SUSY LR Strong CP elementary particle level e e q q d e d q d q c GGG q q q q GG nucleon level _ e e n n scalar pseudo scalar or Tensor pseudo tensor NNNN d N nuclear level _ e e N N MQM S (Schiff ) atomic, molecular level d para d dia

Introduction Theoretical calculations Atomic theory For heavy atoms relativistic effects are important and Dirac-Coulomb Hamiltonian is appropriate NX H DC = [cα i p i + c 2 (β i 1) Z(ri) i=1 Closed-shell atomic EDM originate from N,N X ] + r i i<j 1 r i r j. Tensor pseudotensor e-n interactions H PTV = igf CT 2 X σ N γ iρ N (r). i Nuclear Schiff moment H PTV = X i 4π S δ( R i). Electronic wave-functions near the origin contribute, due to ρ N (r) and δ(r) in the interaction Hamiltonians,

Introduction Coupled-cluster and atomic EDM Coupled-cluster state Atomic states are solutions of the Schrödinger equation H DC Ψ i = E i Ψ i. Partition the Hamiltonian H DC = H o + V es and evaluate the eigenstates of H o. Consider V es as perturbation and evaluate Ψ i perturbatively. Another option, calculate Ψ i nonperturbatively with coupled-cluster theory [Latha et al, J. Phys.: Conf. Ser. 80, 012049 (2007)] Cluster amplitudes are solutions of Ψ i = e T Φ i. Φ r a e T HN DC e T Φ i = 0, Φ rs ab e T HN DC e T Φ i = 0. These are set of nonlinear algebraic equations.

Introduction Coupled-cluster and atomic EDM Atomic EDM Total atomic Hamiltonian, including H PTV, is H atom = H DC + λh PTV. Eigenstates of H atom are mixed parity states Ψ f i and perturbatively, f Ψ i = X I Ψ I ΨI HPTV Ψi E i E I, Ψ I opposite parity to Ψ i. Atomic EDM D a = f Ψ i D f Ψ i = 2 X I Ψ i D Ψ I Ψ I H PTV Ψ i E i E I. Calculations of all Ψ I impractical with coupled-cluster. Most dominant contributions considered.

Perturbed coupled-cluster Introduction Permanent electric dipole moment Theoretical calculations Coupled-cluster and atomic EDM Perturbed coupled-cluster Perturbed cluster amplitudes Cluster equations Some issues related to T (1) Diagrams Properties of T (1) 6j symbols Results

Perturbed coupled-cluster Perturbed cluster amplitudes Perturbed cluster amplitudes Mixed parity coupled-cluster ground state f Ψ o = e T (0) +λt (1) Φ o. Cluster amplitude T (1) incorporates H PTV to first order and residual Coulomb to all order. Diagrammatically represented as Perturbed states are solutions of Schrödinger equation (H DC + λh PTV)e T (0) +λt (1) Φ o = E ie T (0) +λt (1) Φ o. Perturbed state f Ψ 0 does not require sum over intermediate states.

Perturbed coupled-cluster Cluster equations Cluster equations Take exp(λt (1) ) = 1 + λt (1), the zeroth and first order equations are H DC e T (0) Φ o = E ie T (0) Φ o (H DC T (1) + H PTV)e T (0) Φ o = E ie T (0) T (1) Φ o. Multiply the zeroth order equation by T (1) T (1) H DC e T (0) Φ o = T (1) E ie T (0) Φ o (H DC T (1) + H PTV)e T (0) Φ o = E ie T (0) T (1) Φ o. Operating by e T (0) from left and writing H DC and H atom in normal form T (1) H DC N Φ o = T (1) E corr Φ o (H DC N T (1) + H PTV) Φ o = E corrt (1) Φ o.

Perturbed coupled-cluster Cluster equations Cluster equations Subtracting the two equations [H DC N, T (1) ] Φ o = H PTV Φ o. Projecting on Φ r a and Φ rs ab Φ r a [H DC N, T (1) ] Φ o = Φ r a H PTV Φ o Φ rs ab [H DC N, T (1) ] Φ o = Φ r a H PTV Φ o. These are set of linear algebraic equations and solved after calculating T (0).

Some issues related to T (1) Introduction Permanent electric dipole moment Theoretical calculations Coupled-cluster and atomic EDM Perturbed coupled-cluster Perturbed cluster amplitudes Cluster equations Some issues related to T (1) Diagrams Properties of T (1) 6j symbols Results

Some issues related to T (1) Diagrams Diagrams For close-shell atoms, coupled-cluster singles and doubles (CCSD) approximation consist of 79 T (0) diagrams. Whereas, there are 137 T (1) diagrams. Examples of Goldstone double substitution diagrams Identitcal diagrams when T (0) 2 replaces T (1) 2. Angular factors involve larger number of angular momentum couplings ( 1) j b j a+k p 2j b + 1δ(j a, j q)δ(j b, j p)

Some issues related to T (1) Properties of T (1) Properties of T (1) Only s-p matrix elements of H PTV are nonzero, igf CT X H PTV = σ N γ iρ N (r), H PTV = X 2 i i 4π S δ( R i). G F is parameter of calculations. Consequently, H PTV matrix elements are large in magnitude and slow the convergence of cluster amplitudes. As H PTV is one-body operator, among the perturbed cluster operators T (1) 1 are dominant. Number of cluster amplitudes is very large: different parity selection rules ( 1) la+lp ( 1) l b+l q and multipole coupling λ 1 1 λ 2 λ 1 + 1.

Some issues related to T (1) 6j symbols 6j symbols Angular factors have large number of 6j symbols and 9j symbols. Computationally expensive. Calculation of 6j symbols take 30% run time. To improve performance, tabulate it. Optimal implementation is imposing partial triangular condition and selected symmetry properties [Latha, DA and Das, arxiv:0805.2723]. Binary operations in retrieval from table is far less than numerical evaluations.

Some issues related to T (1) 6j symbols D a from Coupled-cluster Ground state D a is expectation of D in the perturbed ground atomic state D a = f Ψ o D f Ψ o f Ψ o f Ψ o From the definition f Ψ o = exp[t (0) + λt (1) ] Φ o, D a = Φo [DT (1) + T (1) D] Φ o, where D = e T (0) De T (0). Φ o Φ o In this expression D has infinite terms and calculation is truncated at some order of T (0). The dressed operator D can be partitioned as D = (1 + T (0) + 1 2! T (0) 2 + 1 3! T (0) 3 +...)De T (0), = De T (0) + X n=1 1 n! (T (0) ) n De T (0).

Results Validation: Coupled perturbed Hartree-Fock Dirac-Fock and coupled perturbed Hatree-Fock (CPHF) are the dominant contributions. It induces opposite parity admixture (h 0 + g 0 ɛ 0 a) ψ 1 a = ( h PTV g 1 ) ψ 0 a. arises from ab 1/r 12 pq class of two-electron interactions. has D a contribution equivalent to part of perturbed cluster terms (D a) CPHF = DT 1 (1) eff (1) (0) +T 1 DT 2.

Results Validation: Coupled perturbed Hartree-Fock Dirac-Fock and coupled perturbed Hatree-Fock (CPHF) are the dominant contributions. It induces opposite parity admixture (h 0 + g 0 ɛ 0 a) ψ 1 a = ( h PTV g 1 ) ψ 0 a. Hg EDM in units of 10 21 e-m Virtuals Normal (Normal+Pseudo) s 1/2 -p 3/2-0.630-0.548 s 1/2 -d 5/2-0.631-0.553 s 1/2 -f 7/2-0.616-0.581 s 1/2 -g 9/2-0.616-0.581 Basis consists of 109 orbitals: (1-14)s 1/2, (2-14)p 1/2,3/2,(3-12)d 3/2,5/2, (4-8)f 5/2,7/2 and (5-9)g 7/2,9/2 arises from ab 1/r 12 pq class of two-electron interactions. has D a contribution equivalent to part of perturbed cluster terms (D a) CPHF = DT 1 (1) eff (1) (0) +T 1 DT 2. Atomic Hg EDM (DF + CPHF) [Latha et al, JPB 41, 035005 (2008)] 0.581 10 21 σ N C T e m is in good agreement with earlier work [Martensson, PRL 54, 1153 (1985)].

Results Results: linear in perturbed coupled-cluster Atomic EDM linear in the cluster operators is D a = 2 Φ 0 [T (1) 1 (1) D+T 1 (0) DT 1 T (1) 1 DT (0) 2 +T (1) 2 DT (0) 1 +T (1) 2 DT (0) 2 Φ 0 Total value is 4.865 10 22 σ N C T e m, in which the contribution from the singles T (1) 1 is 4.886 10 22 σ N C T. The contribution from the doubles T (1) 2 is 0.021 10 22 σ N C T. The most dominant contribution is from the Dirac-Fock term. Contributions from various terms ( in units of 10 22 σ N C T e m ) Term Contribution D -4.635 T (1) 1 T (1) (0) 1 DT T (1) (0) 1 DT T (1) (0) 2 DT T (1) (0) 2 DT 1 0.022 2-0.273 1 0.002 2 0.019

Results Results: linear in perturbed coupled-cluster Atomic EDM linear in the cluster operators is D a = 2 Φ 0 [T (1) 1 (1) D+T 1 (0) DT 1 T (1) 1 DT (0) 2 +T (1) 2 DT (0) 1 +T (1) 2 DT (0) 2 Φ 0 Total value is 4.865 10 22 σ N C T e m, in which the contribution from the singles T (1) 1 is 4.886 10 22 σ N C T. The contribution from the doubles T (1) 2 is 0.021 10 22 σ N C T. The most dominant contribution is from the Dirac-Fock term. C T (0.94 ± 0.44 ± 0.36) 10 8 σ N η np (8.4 ± 3.9 ± 3.2) 10 3 ḡ πnn (0.92 ± 0.43 ± 0.35) 10 10 θ QCD (3.4 ± 1.6 ± 1.3) 10 9 ( d u d d ) (0.46 ± 0.22 ± 0.18) 10 24 ecm These have important implications for physics beyond the standard model.

Results I thank my group members Sandeep Gautam Brajesh K. Mani Salman A. Silotri K. V. P. Latha S. Ravichandran and collaborators B. P. Das R. K. Chaudhury D. Mukherjee M. S. Santhanam H. Mishra A. Mishra

Results