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Poster Icqc

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  • 1. Relativistic Many Body studies in molecules - An application of PT odd effects (EDM) in YbF/BaF systems
      • Geetha G, M. Abe, B. P. Das *, D. Mukherjee and K. Hirao
      • Department of Applied Chemistry, School of Engineering,
      • University of Tokyo, Tokyo 113-8656, JAPAN.
      • *Non-Accelerator Particle Physics Group, Indian Institute of Astrophysics, Bangalore 560034, INDIA.
  • 2. Abstract
      • Objective: Determination of PT-odd effects in polar molecules likeYbF and BaF using 4-component many-body theories like DF, CASCI, CASPT2 and RASCI approaches.
      • Results: Relativistic ground state spectroscopic constant calculations (r e , ω e , B e ) using DF, CASPT2 and RASCI methods.
      • Future Work: Calculation of W d (PT-odd) and isotopic hyperfine constants (A||,A ⊥) using the above mentioned methods in molecules like YbF, BaF, HgF and some excited state calculations in PbO relevant for PT-odd effect studies.
      • Conclusion: CASPT2 calculations for the ground state of YbF and BaF systems ( 2 Σ 1/2 ) are found to be dominated by dynamic correlations and hence numerically equivalent to MP2 correlations. Some of the excited state calculations need further investigation with respect to basis accuracy in terms of addition of polarization functions in the atomic basis .
  • 3. Introduction
    • What are P and T symmetry transformations?
      • P- Parity (position vector r becomes -r
      • T- Time reversal (time t becomes -t)
      • C- Charge conjugation ( charge q becomes -q)
      • Charge-Parity-Time reversal (CPT) theorem states that a physical system or process can violate each of these symmetries individually as long as the combined CPT is conserved
    • How is PT-odd related to EDM in systems?
      • Molecules like ammonia and water have permanent EDMs due to degeneracy of states. EDMs of interest here are purely arising from P and T violations in non-degenerate systems.
      • Hydrogen atom
      • Interaction energy = - d e σ. E int ,
    • σ=2S where S is the angular momentum
    • E int - Electric field between nucleus and electron
    • de – magnitude of electron EDM
    • H = p 2 /2m + V(r) - d e σ. E int
    S edm e
  • 4.
    • Parity P -> H = (- p) 2 /2m + V(r) + d e σ. E int
    • σ -> pseudo vector -> No change under parity
    • p and E -> vector -> changes under parity
    • H is not invariant under parity
    •  H, P] ≠ 0
    • Parity is violated
    • Time T -> H = (- p) 2 /2m + V(r) + d e σ. E int
    • All time dependent quantities changes sign under T
    • σ,p -> angular momentum -> changes under T
    • E -> no change under T
    • H is not invariant under Time reversal
    •  H, T] ≠ 0
    • Time reversal is violated
    • Hence net EDM could exist if electron were to have an EDM.
    • H 0 is even but de σ. E intis odd under both P and T
    • Using perturbation theory, we can express the wave function as
    • | Ψ α >= | Ψ α 0 > + λ | Ψ α 1 > + λ 2 | Ψ α 2 > + …….
    • EDM = < Ψ|D| Ψ> = <Ψ α 0 |D| Ψ α 1 > + <Ψ α 1 |D| Ψ α 0 >≠0
    + + P + + T
  • 5.
      • What are the implications of EDM is systems?
      • Measurement of electron EDM directly searches for physics beyond the Standard Model
      • What is the present limit for EDM and its implications for particle physics?
      • Experiment limit (2002)
      • (d e < 1.6 X 10 -27 e.cm)
      • Berkely Thallium experiment
      • (B C Regan et al, PRL, Feb 2002)
      • e.cm
      • Why molecules ?
      • How to measure electron EDM ?
      • Why YbF?
    Left Right SUSY Φ ̴1 SUSY Φ ̴ α /∏ Standard Model Multi Higgs 10 -22 10 -24 10 -26 10 -28 10 -30 10 -32 10 -34 10 -36 system E Interaction energy -d e η E. σ η d e σ can be <1 or >1
  • 6.
    • Why YbF
    • The interaction energy W edm = η d e E [ η = enhancement factor= d atom /d e ]
    • In Tl expt., η = 585 and E = 123 kV/cm
    • For de ̴10 -27 e.cm, W edm ̴ 0.1eV ̴ (20µHz)
    • But in heavy polar molecules like YbF, the value of η can be 10 6 . Hence the interaction energy W edm to be measured is around 300 times larger than for Tl.
    • Require excellent control over magnetic fields
    • In Tl expt., the error in d e ̴ 10 -27 B corr (fT)
    • As electric field is much larger the relative sensitivity to correlated magnetic fileds is correspondingly smaller. In YbF expt., the error in d e ̴ 5 X10 -30 B corr (fT)
    • Motional magnetic field B = v X E
    • B mot i = 4 x 10 7 fT
    • As molecules are strongly polarized in the direction of the applied electric fields its interaction with small magnetic field like B moti is anisotropic. – the effect is perpendicular and highly suppressed. In YbF expt., B mot i is suppressed by 9 orders of magnitude.
  • 7. Form of PT-odd Interaction operator
    • Major contributions to EDM in YbF/BaF (open shell systems) are
    • Electron EDM (d e )
    • Scalar- pseudo scalar (C s ) electron-nucleus interaction
    • Electron EDM (d e )
    • General form for the interaction of intrinsic electron electric dipole moment (EDM) with molecular electric field (E) is defined by the operator
    • H d = -d e (β-1) Σ. E
    • where d e – EDM of electron
    • Σ, β – Dirac matrices and E – Electric field
    • Using Σ, β matrices, we get
    • H d = 2d e 0 0 . E
    • 0 σ
    • where E = E i mol = Σ m E i m + Σ j≠i E ij
    • E i m = field due to m th nucleus at the site of i th electron (individual nucleus)
    • Σ j≠i E ij = field due to j th electron at the site of i th electron (shielding effect)
    • The value of the matrix element of the operator depends mainly on electric field and the small component of the wave function.
  • 8.
    • Approximation :
    • (1) Screening term is neglected ( this accelerates the decline of the E wrt distance from the center)
    • (2) The electrons of each atom have completely screened their nuclei at the location of any other nucleus and hence the problem is uncoupled for various nuclear regions
    • | ψ (YbF) > = | ψ (Yb)> + | ψ (F)>
    • The PT-odd constant for YbF (ground state = 2 Σ 1/2 )
    • W d = 2/d e < 2 Σ 1/2 | H d | 2 Σ 1/2 > [ expectation value in the ground state].
    • Exact method – which involves transition moments and hence require
    • excited states.
    • Starting from the Hamiltonian given by
    • H 0 = c α .p i + β m i c 2 + V i (r) + Σ i≠j e/r ij – d e β σ . E int
    • | ψ ˜ α > = | ψ α 0 > + | ψ α 1 >
    • where
    • | ψ α 1 > = Σ I≠ α | ψ I 0 >< ψ I 0 |- d e β σ . E int | ψ α 0 >
    • --------------------------------------
    • (E α 0 - E I 0 )
    • Hence we can look for an electric dipole transition between states of same parity.
  • 9.
    • Measurable D = < ψ ˜ α |D| ψ ˜ α > =
    • = < ψ 0 α |D| ψ 0 α > +< ψ 0 α |D| ψ 1 α > + < ψ 1 α |D| ψ 0 α > ------A
    • For atoms, because of spherical symmetry, < ψ 0 α |D| ψ 0 α > = 0. So all the contributions are mixed (1,1) perturbed terms.
    • To get a matrix element without Electric field
    • --------------------------------------------------------
    • [ βσ . Δ ,H 0 ] = [ βσ . Δ , c α .p i + β m i c 2 + V i (r) ]
    • [ βσ . Δ , α .p i ] = [ β , α ] α { α = γ 5 β , σ = γ 5 α , [AB,C] =A[B,C] + [A,C]B
    • [ βσ . Δ , β ] = 0
    • [ βσ . Δ , V i (r) ] = -e βσ .E
    • -e βσ .E = [ βσ . Δ ,H 0 ] - [ βσ . Δ ,c α .p i ] = H`
    • Substituting them in equation A, we get
    • D = de/e [ Σ I≠0 < ψ 0 α |er| ψ 0 I > < ψ 0 I | H` | ψ 0 α > + c.c.]/ (E α 0 - E I 0 )
    • using H 0 | ψ 0 ( I, α ) > = E (I, α ) | ψ 0 ( I, α ) >
    • Σ I≠0 | ψ 0 I > < ψ 0 I | = 1- | ψ 0 α > < ψ 0 α |
    • 4cde < ψ 0 α |z| ψ 0 I > < ψ 0 I |i γ 5 β p 2 | ψ 0 α >
    • D= Σ I≠0 -------- --------------------------------------------
    • h (E α 0 - E I 0 )
  • 10.
    • Whereas for non centro symmetric molecules, < ψ 0 α |D| ψ 0 α > is ≠ 0.
    • Although the perturbing term is short range, for molecules any charge-transfer excitations (like from Yb to F) has large cross terms of < ψ 0 (yb/f) |z| ψ 0 (f/yb) > and that enhances the contribution to EDM of electron.
    • Scalar- pseudo scalar (C s ) electron-nucleus interaction
    • H S-PS = G F /2 C s A Σ e i β e γ e 5 ρ N (r e ) where G F is the Fermi constant
    • Cs is the s-ps constant
    • A – atomic mass number
    • ρ N (r e ) – nuclear density
    • γ 5 - i γ 0 γ 1 γ 2 γ 3
    • This can again be computed in two ways…
    • Compute the effect as the odd moment of the molecule by taking expectation value
    • Σ i nuclei < Ψ α |H S-PS | Ψ α > = E S-PS
    • OR
    • By starting from perturbation theory and measuring the transition moments as
    • D = Σ I≠α < Ψ α |z | Ψ I > < Ψ I | H S-PS | Ψ α >
    • -------------------------------------- + c.c.
    • (E α 0 - E I 0 )
  • 11. Step by step coding procedure
    • Understanding how the primitives GTOs for each symmetry (s (1/2), p (1/2,3/2), d(3/2,5/2), f(5/2,7/2) …….for both large (P) and small (Q) in the four components are written in UTChem code.
    • Checking the accuracy of the basis close to the nucleus by calculating the isotropic hyperfine constants and compare with available expt numbers– need to write the hyperfine integrals
    • Checking the accuracy of the basis in the large radial points by computing the dipole polarizability or dipole moment of the molecule (these integrals are also incorporated in UTChem code but not used), ground to excited transition energies and ionization potentials.
    • Taking out the already available p 2 and z integrals and form the necessary integrals required for the above calculation
    • Do the above calculation at the DF level followed by CASPT2 using the excited states to get the necessary correlation.
    • Do the same steps but by writing the integrals for the C S matrix using the primitive GTOs and compute the EDM at DF level followed by CASPT2 level
  • 12.
    • In the mean time some of the basic spectroscopic constant calculations like equilibrium bond length and vibrational frequencies are tried.
    • Previous calculations in YbF molecule
    • a) Approximate relativistic spinors (E A Hinds et al, PRA 1980)
    • b) Relativistic effective core potential method (RECP)
    • M G Kozlov and V F Ezhov semi-empirical approach ,PRA 1994
    • A V Titov, PRL (1996) using RASSCF orbitals
    • N.S.Mosyagin et al, J.Phys.B (1998)
    • c) All electron UDF calculation (F A Parpia, J.Phys.B 1998, H M Quiney,J. Phys.B 1998) (neglecting valence-valence correlations)
    • D) Restricted Active Space CI (RASCI) (Malaya et al, JCP 2006)
    • Experiment [ Hudson, Sauer, E A Hinds, PRL, 89,(2002).
    • d e = (-0.2 ±3.2) X 10- 26 e.cm
  • 13. Ground state calculations in YbF molecule
    • F : [15s10p]/(5s3p)
    • Yb: [26s26p15d8f]/(9s7p3d3f)
    • Total number of spinors =160
    • Ncore =48
    • |2s(2)2p(5)> + |5s(2)5p(6)4f(14)6s(2)>
    • RAS1(inactive) = 30, RAS2 =2 (2 singly occupied), RAS3 (secondary)=60
    • Nvir = 22, no. of active electron =1
    B(cm-1) Ω (cm-1) R (A) Method - 529 2.051 Malaya - 502 2.016 Expt 0.2476 596 1.994(.02) RASCI 0.2498 582.9 1.985(.03) CASPT2 0.2377 552.8 2.034(.018) CASCI 0.2369 545 2.038(.022) DF F Yb F - Yb +
  • 14. Ground state calculations in BaF molecule
    • F : [15s10p]/(5s3p)
    • Ba : [26s26p15d]/(9s7p3d)
    • Total number of spinors =176
    • Ncore =48
    • |2s(2)2p(5)> + |5s(2)5p(6)6s(2)>
    • RAS1 (inactive)= 14
    • RAS2 (active)=10 (2 doubly occupied, 2 singly occupied, 6 unoccupied)
    • RAS3 (nsec)=104
    • Nvir = 22, no. of active electrons =3
    B(cm-1) Ω (cm-1) R (A) Method - 2.16 Expt 0.2498 534.4 2.246 CASPT2 0.2377 429 2.276 CASCI 0.2369 427 2.278 DF
  • 15. BaF using less active space and more diffused orbitals
    • F : [15s10p]/(5s3p)
    • Ba : [26s26p15d]/(9s7p3d)
    • Total number of spinors =208
    • Ncore =48
    • |2s(2)2p(5)> + |5s(2)5p(6)6s(2)>
    • RAS1 = 16, RAS2 =2 (2 singly occupied) , RAS3=92
    • Nvir = 50, no. of active electrons =1
    B(cm-1) ω (cm-1) R (A) Method - 2.16 Expt 0.203 897 2.23 CASPT2 0.194 369.7 2.28 CASCI 0.194 369.7 2.28 DF
  • 16. Conclusions
    • YbF : the energy difference between first excited state ( ∏ 1/2 ) and ground state ( Σ 1/2 ) is 18000cm -1. As spin orbit interaction can mix states of same total angular momentum, may be the CAS space need to incorporate more of virtual spinors.
    • Calculations by Kozlov et al (J. Phys.B (1995)) points that spin orbital splitting of substates (∏ 1/2 , and ∏ 3/2 ) with the 6p state of Yb+ shows that the 6p-orbital contributes about 60% to the excited (∏) state whereas
    • to the ground state ( Σ ) is 23%. – different CAS space for ground and excited states?
    • BaF : The ground ( Σ 1/2 ) and the first excited state ( ∏ 1/2 ) state can also mix strongly because of the small energy gap between them.
    • From previous calculations on heavy atoms for properties like excitation energy and ionization potential we find that the accuracy depends very much on the kind of basis used – like addition of a minimum of two polarization functions.
    • Improved virtual orbitals as suggested by Nakajima sensei
    • Relativistic second order MBPT to the CF stretching mode in CHFClBr – Peter S etal (PRA, 2005) computed as expectation value of the PNC matrix element seems to be prospective with two/four component approaches. Problems handling bigger basis may be easier.
    • New calculations in other kinds of systems as demonstrated by Stauber (A185 poster of ICQC 2006)