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 .
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
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
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.
(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.
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
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
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)
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