Relativistic Many Body Relativistic Many Body studies in molecules - An studies in molecules - An

applicationapplicationof PT odd effects (EDM) in of PT odd effects (EDM) in

YbF/BaF systemsYbF/BaF systemsGeetha G, M. Abe, B. P. DasGeetha G, M. Abe, B. P. Das*, D. *, D.

MukherjeeMukherjee and K. Hirao and K. Hirao

Department of Applied Chemistry, School of Department of Applied Chemistry, School of Engineering,Engineering,

University of Tokyo, Tokyo 113-8656, JAPAN.University of Tokyo, Tokyo 113-8656, JAPAN.*Non-Accelerator Particle Physics Group, *Non-Accelerator Particle Physics Group,

Indian Institute of Astrophysics, Bangalore Indian Institute of Astrophysics, Bangalore 560034, INDIA.560034, INDIA.

AbstractAbstractObjective: Determination of PT-odd effects in polar molecules Objective: Determination of PT-odd effects in polar molecules likeYbF and BaF using 4-component many-body theories like likeYbF and BaF using 4-component many-body theories like DF, CASCI, CASPT2 and RASCI approaches.DF, CASCI, CASPT2 and RASCI approaches.

Results: Relativistic ground state spectroscopic constant Results: Relativistic ground state spectroscopic constant calculations (rcalculations (r

e e ,, ω ωee, B, Bee) using DF, CASPT2 and RASCI methods.) using DF, CASPT2 and RASCI methods.

Future Work: Calculation of WFuture Work: Calculation of Wd d (PT-odd) and isotopic hyperfine (PT-odd) and isotopic hyperfine

constants (A||,Aconstants (A||,A⊥)⊥) using the above mentioned methods in using the above mentioned methods in molecules like YbF, BaF, HgF and some excited state molecules like YbF, BaF, HgF and some excited state calculations in PbO relevant for PT-odd effect studies.calculations in PbO relevant for PT-odd effect studies.

Conclusion: CASPT2 calculations for the ground state of YbF Conclusion: CASPT2 calculations for the ground state of YbF and BaF systems (and BaF systems (22ΣΣ1/21/2) are found to be dominated by dynamic ) are found to be dominated by dynamic correlations and hence numerically equivalent to MP2 correlations and hence numerically equivalent to MP2 correlations. Some of the excited state calculations need correlations. Some of the excited state calculations need further investigation with respect to basis accuracy in terms of further investigation with respect to basis accuracy in terms of addition of polarization functions in the atomic basisaddition of polarization functions in the atomic basis..

IntroductionIntroductionWhat are P and T symmetry transformations?What are P and T symmetry transformations?

P- Parity (position vector r becomes -rP- Parity (position vector r becomes -r T- Time reversal (time t becomes -t)T- Time reversal (time t becomes -t) C- Charge conjugation ( charge q becomes -q) C- Charge conjugation ( charge q becomes -q)

Charge-Parity-Time reversal (CPT) theorem states that a Charge-Parity-Time reversal (CPT) theorem states that a physical system or process can violate each of these symmetries physical system or process can violate each of these symmetries individually as long as the combined CPT is conservedindividually as long as the combined CPT is conserved

How is PT-odd related to EDM in systems?How is PT-odd related to EDM in systems?

Molecules like ammonia and water have permanent EDMs due to Molecules like ammonia and water have permanent EDMs due to degeneracy of states. EDMs of interest here are purely arising degeneracy of states. EDMs of interest here are purely arising from P and T violations in non-degenerate systems.from P and T violations in non-degenerate systems.

Hydrogen atomHydrogen atom

Interaction energy = - dInteraction energy = - de e σ. Eσ. Eintint, ,

σ=2S where S is the angular momentumσ=2S where S is the angular momentum

EEint int - Electric field between nucleus and electron- Electric field between nucleus and electron

de – magnitude of electron EDMde – magnitude of electron EDM

H = pH = p22/2m + V(r) - d/2m + V(r) - de e σ. Eσ. Eintint

S

edme

Parity P → H = (-Parity P → H = (-p)p)22/2m + V(r) +/2m + V(r) + dde e σ. Eσ. Eintint

σ σ →→ pseudo vector pseudo vector →→ No change under parityNo change under parity p and E p and E → → vector vector → → changes under paritychanges under parity H is not invariant under parity H is not invariant under parity H, P] ≠ 0H, P] ≠ 0 Parity is violatedParity is violated

Time T → H = (-Time T → H = (-p)p)22/2m + V(r) + d/2m + V(r) + de e σ. Eσ. Eintint

All time dependent quantities changes sign under TAll time dependent quantities changes sign under T

σ,p σ,p →→ angular momentum angular momentum → → changes under Tchanges under T E E → no change under T→ no change under T H is not invariant under Time reversal H is not invariant under Time reversal H, T] ≠ 0H, T] ≠ 0 Time reversal is violatedTime reversal is violated

Hence net EDM could exist if electron were to have an EDM.Hence net EDM could exist if electron were to have an EDM. HH0 0 is even but is even but de σ. Ede σ. Eintis intis odd odd under both P and T under both P and T Using perturbation theory, we can express the wave function as Using perturbation theory, we can express the wave function as

||ΨΨαα>= | Ψ>= | Ψαα00> +> +λλ | Ψ| Ψαα11> +> +λλ22 | Ψ| Ψαα22> + …….> + ……. EDM = <EDM = < Ψ|D| Ψ|D| Ψ> = <ΨΨ> = <Ψαα00|D| |D| ΨΨαα11> + > + <Ψ<Ψαα11|D| |D| ΨΨαα00>≠0>≠0

+

+

P

+ +T

What are the implications of EDM is systems?What are the implications of EDM is systems?Measurement of electron EDM directly searches for physics beyond the Measurement of electron EDM directly searches for physics beyond the Standard ModelStandard Model

What is the present limit for EDM and its implications for particle What is the present limit for EDM and its implications for particle physics? physics? Experiment limit Experiment limit (2002)(2002) (d(dee < 1.6 X 10 < 1.6 X 10-27 -27 e.cm)e.cm) Berkely Thallium Berkely Thallium experimentexperiment (B C Regan et al, PRL, (B C Regan et al, PRL, Feb 2002)Feb 2002)

e.cme.cm

Why molecules ?Why molecules ?

How to measure electron EDM ?How to measure electron EDM ?

Why YbF?Why YbF?

LeftRight

SUSYΦ̴1

SUSYΦ̴α/∏

Standard Model

MultiHiggs

10-22

10-24

10-26

10-28

10-30

10-32

10-34

10-36

system

E

Interaction energy -deη E.σ

η de σ

can be <1 or >1

Why YbFWhy YbF

1.1. The interaction energy WThe interaction energy Wedm edm = = ηηddeeEE [[ηη = enhancement factor= d = enhancement factor= datomatom/d/dee] ]

In TlIn Tl expt., expt., ηη = 585 and E = 123 kV/cm = 585 and E = 123 kV/cm

For de ̴10For de ̴10-27-27 e.cm, W e.cm, Wedmedm ̴ 0.1eV ̴ (20µHz) ̴ 0.1eV ̴ (20µHz)

But in heavy polar molecules like YbF, the value of But in heavy polar molecules like YbF, the value of ηη can be 10 can be 1066. Hence the . Hence the interaction energy interaction energy WWedm edm to be measured is around 300 times larger than for to be measured is around 300 times larger than for Tl.Tl.

2.2. Require excellent control over magnetic fieldsRequire excellent control over magnetic fields

In Tl expt., the error in dIn Tl expt., the error in dee ̴ 10 ̴ 10 -27-27 B B corrcorr (fT) (fT)

As electric field is much larger the relative sensitivity to correlated As electric field is much larger the relative sensitivity to correlated magnetic fileds is correspondingly smaller. In YbF expt., the error in dmagnetic fileds is correspondingly smaller. In YbF expt., the error in dee ̴ 5 ̴ 5 X10 X10 -30-30 B B corrcorr (fT) (fT)

3.3. Motional magnetic field B = v X EMotional magnetic field B = v X E

BBmotmoti = 4 x 10i = 4 x 1077 fT fT

As molecules are strongly polarized in the direction of the applied electric As molecules are strongly polarized in the direction of the applied electric fields its interaction with small magnetic field like Bfields its interaction with small magnetic field like Bmotimoti is anisotropic. – the is anisotropic. – the effect is perpendicular and highly suppressed. In YbF expt., Beffect is perpendicular and highly suppressed. In YbF expt., Bmotmoti is i is suppressed by 9 orders of magnitude.suppressed by 9 orders of magnitude.

Form of PT-odd Interaction operatorForm of PT-odd Interaction operatorMajor contributions to EDM in YbF/BaF (open shell systems) areMajor contributions to EDM in YbF/BaF (open shell systems) are

Electron EDM (dElectron EDM (dee)) Scalar- pseudo scalar (CScalar- pseudo scalar (Css) electron-nucleus interaction) electron-nucleus interaction

Electron EDM (dElectron EDM (dee))

General form for the interaction of intrinsic electron electric dipole moment (EDM) General form for the interaction of intrinsic electron electric dipole moment (EDM) with molecular electric field (E) is defined by the operatorwith molecular electric field (E) is defined by the operator

HHdd = -d = -dee (β-1) Σ. E (β-1) Σ. E

where dwhere dee – EDM of electron – EDM of electron Σ, β – Dirac matrices and E – Electric fieldΣ, β – Dirac matrices and E – Electric field

Using Σ, β matrices, we get Using Σ, β matrices, we get HHd d = 2d= 2dee 0 0 . E 0 0 . E 0 0 σσ

where E = Ewhere E = Eiimolmol = = ΣΣ

mm E Eii

mm+ + ΣΣj≠ij≠i

EEijij

EEiim m = field due to m= field due to mthth nucleus at the site of i nucleus at the site of ithth electron (individual nucleus) electron (individual nucleus)

ΣΣj≠ij≠i

EEij ij = field due to j= field due to jthth electron at the site of i electron at the site of ithth electron electron (shielding effect)(shielding effect)

The value of the matrix element of the operator depends mainly on electric field and The value of the matrix element of the operator depends mainly on electric field and the small component of the wave function.the small component of the wave function.

ApproximationApproximation : :(1) Screening term is neglected ( this accelerates the decline of the E wrt distance (1) Screening term is neglected ( this accelerates the decline of the E wrt distance

from the center)from the center) (2) The electrons of each atom have completely screened their nuclei at the (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 location of any other nucleus and hence the problem is uncoupled for various nuclear regionsnuclear regions

||ψψ(YbF) > = |(YbF) > = |ψψ(Yb)> + |(Yb)> + |ψψ(F)> (F)>

The PT-odd constant for YbF (ground state = The PT-odd constant for YbF (ground state = 22ΣΣ1/21/2))

WWdd = 2/d = 2/dee < < 22ΣΣ1/21/2| H| Hdd | | 22ΣΣ1/21/2> [ expectation value in the ground state].> [ expectation value in the ground state].

Exact method – which involves transition moments and hence requireExact method – which involves transition moments and hence requireexcited states.excited states.

Starting from the Hamiltonian given byStarting from the Hamiltonian given by

HH00 = c = c αα.p.pii + + ββmmiicc22 + V + Vii(r) + (r) + ΣΣi≠ji≠j

e/r e/rij ij – d– dee ββ σσ. E. Eintint

||ψψ˜̃αα> = |> = |ψψαα

00> + |> + |ψψαα11> >

wherewhere

||ψψαα11> = > = ΣΣ I≠I≠αα | | ψψII

00><>< ψψII00|-|- ddee ββ σσ. E. Eintint | | ψψαα

00> > ----------------------------------------------------------------------------

(E(Eαα00 - - EEII

00) ) Hence we can look for an electric dipole transition between states of same parity.Hence we can look for an electric dipole transition between states of same parity.

Measurable D = < Measurable D = < ψψ˜̃αα|D||D| ψψ˜̃

αα> = > = = < = < ψψ00

α α |D||D| ψψ00αα> +< > +< ψψ00

α α |D||D| ψψ11αα> + < > + < ψψ11

α α |D||D| ψψ00αα> ------A> ------A

For atoms, because of spherical symmetry, < For atoms, because of spherical symmetry, < ψψ00α α |D||D| ψψ00

αα> = 0. So all the > = 0. So all the contributions are mixed (1,1) perturbed terms.contributions are mixed (1,1) perturbed terms.

To get a matrix element without Electric fieldTo get a matrix element without Electric field----------------------------------------------------------------------------------------------------------------[[βσβσ..ΔΔ,H,H00] = [] = [βσβσ..ΔΔ, c , c αα.p.pii + + ββmmiicc22 + V + Vii(r) (r) ]]

[[βσβσ..ΔΔ, , αα.p.pii] = [] = [ββ, , αα] ] αα { {αα= = γγ55 ββ , , σσ = = γγ55 αα , [AB,C] =A[B,C] + [A,C]B , [AB,C] =A[B,C] + [A,C]B [[βσβσ..ΔΔ, , ββ] = 0] = 0 [[βσβσ..ΔΔ, V, Vii(r) ] = -e (r) ] = -e βσβσ.E .E

-e-eβσβσ.E = [.E = [βσβσ..ΔΔ,H,H00] - [] - [βσβσ..ΔΔ,c ,c αα.p.pii] = H` ] = H`

Substituting them in equation A, we getSubstituting them in equation A, we get D = de/e [D = de/e [Σ Σ I≠0 I≠0 < < ψψ00

α α |er||er| ψψ00II> < > < ψψ00

II || H` | H` | ψψ00α α > + c.c.]/ (E> + c.c.]/ (Eαα

00 - - EEII00))

using Husing H0 0 || ψψ00((I, I, αα))> = > = EE

(I, (I, αα)) | | ψψ00((I, I, αα))> >

- Σ Σ I≠0 I≠0 || ψψ00II> < > < ψψ00

II | = 1- || = 1- | ψψ0 0 αα

> < > < ψψ00 αα ||- 4cde < 4cde < ψψ00

α α |z||z| ψψ00II> < > < ψψ00

II |i|iγγ55 ββ p p22||ψψ00αα> >

D= D= Σ Σ I≠0 I≠0 ---------------- ---------------------------------------------------------------------------------------- h (Eh (Eαα

00 - - EEII00))

Whereas for non centro symmetric molecules, < Whereas for non centro symmetric molecules, < ψψ00α α |D||D| ψψ00

αα> is ≠ 0. > is ≠ 0.

Although the perturbing term is short range, for molecules any charge-transfer Although the perturbing term is short range, for molecules any charge-transfer excitations (like from Yb to F) has large cross terms of < excitations (like from Yb to F) has large cross terms of < ψψ00

(yb/f)(yb/f) |z||z| ψψ00(f/yb)(f/yb)> and > and

that enhances the contribution to EDM of electron.that enhances the contribution to EDM of electron.

Scalar- pseudo scalar (CScalar- pseudo scalar (Css) electron-nucleus interaction) electron-nucleus interaction

HHS-PS =S-PS = G GFF/2 C/2 Css A A ΣΣee i iββeeγγee55ρρNN (r (ree) where G) where GFF is the Fermi constant is the Fermi constant

Cs is the s-ps constant Cs is the s-ps constant A – atomic mass numberA – atomic mass number

ρρNN (r (ree) – nuclear density) – nuclear density

γγ55 - i - i γγ00

γγ11 γγ22

γγ3 3

This can again be computed in two ways…This can again be computed in two ways…

Compute the effect as the odd moment of the molecule by taking expectation Compute the effect as the odd moment of the molecule by taking expectation valuevalue

ΣΣiinucleinuclei < <ΨΨαα |H |HS-PS | S-PS | ΨΨαα > = E > = ES-PSS-PS

ORORBy starting from perturbation theory and measuring the transition moments asBy starting from perturbation theory and measuring the transition moments as

D = D = ΣΣ I≠αI≠α < <ΨΨαα |z | |z |ΨΨII > < > <ΨΨII | H | HS-PSS-PS | |ΨΨαα > > -------------------------------------- + c.c.-------------------------------------- + c.c.

(E(Eαα00 - - EEII

00))

Step by step coding procedureStep by step coding procedure Understanding how the primitives GTOs for each symmetry (s (1/2), p 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 (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.the four components are written in UTChem code.

Checking the accuracy of the basis close to the nucleus by calculating Checking the accuracy of the basis close to the nucleus by calculating the isotropic hyperfine constants and compare with available expt the isotropic hyperfine constants and compare with available expt numbers– need to write the hyperfine integralsnumbers– need to write the hyperfine integrals

Checking the accuracy of the basis in the large radial points by Checking the accuracy of the basis in the large radial points by computing the dipole polarizability or dipole moment of the molecule computing the dipole polarizability or dipole moment of the molecule (these integrals are also incorporated in UTChem code but not used), (these integrals are also incorporated in UTChem code but not used), ground to excited transition energies and ionization potentials.ground to excited transition energies and ionization potentials.

Taking out the already available pTaking out the already available p22 and z integrals and form the and z integrals and form the necessary integrals required for the above calculationnecessary integrals required for the above calculation

Do the above calculation at the DF level followed by CASPT2 using the Do the above calculation at the DF level followed by CASPT2 using the excited states to get the necessary correlation.excited states to get the necessary correlation.

Do the same steps but by writing the integrals for the CDo the same steps but by writing the integrals for the CSS matrix using matrix using the primitive GTOs and compute the EDM at DF level followed by the primitive GTOs and compute the EDM at DF level followed by CASPT2 levelCASPT2 level

In the mean time some of the basic spectroscopic constant calculations In the mean time some of the basic spectroscopic constant calculations like equilibrium bond length and vibrational frequencies are tried. like equilibrium bond length and vibrational frequencies are tried.

Previous calculations in YbF moleculePrevious calculations in YbF molecule

a) Approximate relativistic spinors (E A Hinds et al, PRA 1980)a) Approximate relativistic spinors (E A Hinds et al, PRA 1980)

b) Relativistic effective core potential method (RECP) b) Relativistic effective core potential method (RECP) M G Kozlov and V F Ezhov semi-empirical approach ,PRA 1994M G Kozlov and V F Ezhov semi-empirical approach ,PRA 1994 A V Titov, PRL (1996) using RASSCF orbitalsA V Titov, PRL (1996) using RASSCF orbitals N.S.Mosyagin et al, J.Phys.B (1998)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. c) All electron UDF calculation (F A Parpia, J.Phys.B 1998, H M Quiney,J.

Phys.B 1998) (neglecting valence-valence correlations)Phys.B 1998) (neglecting valence-valence correlations)

D) Restricted Active Space CI (RASCI) (Malaya et al, JCP 2006)D) Restricted Active Space CI (RASCI) (Malaya et al, JCP 2006)

Experiment [ Hudson, Sauer, E A Hinds, PRL, 89,(2002). Experiment [ Hudson, Sauer, E A Hinds, PRL, 89,(2002). ddee = (-0.2 = (-0.2 ±3.2) X 10-±3.2) X 10-26 26 e.cme.cm

Ground state calculations in YbF Ground state calculations in YbF moleculemolecule

F : [15s10p]/(5s3p)F : [15s10p]/(5s3p)

Yb: [26s26p15d8f]/(9s7p3d3f)Yb: [26s26p15d8f]/(9s7p3d3f)

Total number of spinors =160Total number of spinors =160

Ncore =48Ncore =48

|2s(2)2p(5)> + ||2s(2)2p(5)> + |5s(2)5p(6)4f(14)6s(2)>5s(2)5p(6)4f(14)6s(2)>

RAS1(inactive) = 30, RAS2 =2 (2 singly RAS1(inactive) = 30, RAS2 =2 (2 singly occupied), RAS3 (secondary)=60occupied), RAS3 (secondary)=60

Nvir = 22, no. of active electron =1Nvir = 22, no. of active electron =1

MethodMethod R (A)R (A) ΩΩ(cm-(cm-1)1)

B(cm-1)B(cm-1)

DFDF 2.038(.0222.038(.022))

545545 0.23690.2369

CASCICASCI 2.034(.0182.034(.018))

552.8552.8 0.23770.2377

CASPT2CASPT2 1.985(.03)1.985(.03) 582.9582.9 0.24980.2498

RASCIRASCI 1.994(.02)1.994(.02) 596596 0.24760.2476

ExptExpt 2.0162.016 502502 --

MalayaMalaya 2.0512.051 529529 --

F Yb

F-Yb+

Ground state calculations in BaF Ground state calculations in BaF moleculemolecule

F : [15s10p]/(5s3p)F : [15s10p]/(5s3p)

Ba : [26s26p15d]/(9s7p3d)Ba : [26s26p15d]/(9s7p3d)

Total number of spinors =176Total number of spinors =176

Ncore =48Ncore =48

|2s(2)2p(5)> + |5s(2)5p(6)6s(2)>|2s(2)2p(5)> + |5s(2)5p(6)6s(2)>

RAS1 (inactive)= 14RAS1 (inactive)= 14

RAS2 (active)=10 (2 doubly RAS2 (active)=10 (2 doubly occupied, 2 singly occupied, 6 occupied, 2 singly occupied, 6 unoccupied)unoccupied)

RAS3 (nsec)=104RAS3 (nsec)=104

Nvir = 22, no. of active electrons Nvir = 22, no. of active electrons =3=3MethodMethod R (A)R (A) ΩΩ(cm-1)(cm-1) B(cm-1)B(cm-1)

DFDF 2.2782.278 427427 0.23690.2369

CASCICASCI 2.2762.276 429429 0.23770.2377

CASPT2CASPT2 2.2462.246 534.4534.4 0.24980.2498

ExptExpt 2.162.16 --

1.5 2 2.5 3 3.5 4 4.5 5 5.5

-8235.6000

-8235.5000

-8235.4000

-8235.3000

-8235.2000

-8235.1000

-8235.0000

-8234.9000

PES of ground state

DF

CASCI

CASPT2

R (A)E

nerg

y (a

u)

BaF using less active space and more diffused orbitalsBaF using less active space and more diffused orbitals

F : [15s10p]/(5s3p)F : [15s10p]/(5s3p)

Ba : [26s26p15d]/(9s7p3d)Ba : [26s26p15d]/(9s7p3d)

Total number of spinors =208Total number of spinors =208

Ncore =48Ncore =48

|2s(2)2p(5)> + ||2s(2)2p(5)> + |5s(2)5p(6)6s(2)>5s(2)5p(6)6s(2)>

RAS1 = 16, RAS2 =2 (2 singly RAS1 = 16, RAS2 =2 (2 singly occupied) , RAS3=92occupied) , RAS3=92

Nvir = 50, no. of active electrons Nvir = 50, no. of active electrons =1=1

MethodMethod R (A)R (A) ωω(cm-(cm-1)1)

B(cm-1)B(cm-1)

DFDF 2.282.28 369.7369.7 0.1940.194

CASCICASCI 2.282.28 369.7369.7 0.1940.194

CASPT2CASPT2 2.232.23 897897 0.2030.203

ExptExpt 2.162.16 --

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

-8236

-8235

-8234

-8233

-8232

-8231

-8230

-8229

PES of ground state

DF

CASCI

CASPT2

R (A)E

ne

rgy

(au

)

ConclusionsConclusions YbF : the energy difference between first excited state (YbF : the energy difference between first excited state (∏∏1/2 ) 1/2 ) and ground and ground

state (state (ΣΣ1/21/2) is 18000cm -1. As spin orbit interaction can mix states of ) is 18000cm -1. As spin orbit interaction can mix states of same total angular momentum, may be the CAS space need to same total angular momentum, may be the CAS space need to incorporate more of virtual spinors.incorporate more of virtual spinors.

Calculations by Kozlov et al (J. Phys.B (1995)) points that spin orbital Calculations by Kozlov et al (J. Phys.B (1995)) points that spin orbital splitting of substates (∏splitting of substates (∏1/2 1/2 , and, and ∏∏3/23/2) with the 6p state of Yb+ shows that ) with the 6p state of Yb+ shows that the 6p-orbital contributes about 60% to the excited (∏) state whereas the 6p-orbital contributes about 60% to the excited (∏) state whereas

to the ground state (to the ground state (ΣΣ) is 23%. – different CAS space for ground and ) is 23%. – different CAS space for ground and excited states? excited states?

BaF : The ground (BaF : The ground (ΣΣ1/21/2) and the first excited state ) and the first excited state ((∏∏1/2 ) 1/2 ) state can also state can also mix strongly because of the small energy gap between them. mix strongly because of the small energy gap between them.

From previous calculations on heavy atoms for properties like excitation From previous calculations on heavy atoms for properties like excitation energy and ionization potential we find that the accuracy depends very 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 much on the kind of basis used – like addition of a minimum of two polarization functions.polarization functions.

Improved virtual orbitals as suggested by Nakajima senseiImproved virtual orbitals as suggested by Nakajima sensei

Relativistic second order MBPT to the CF stretching mode in CHFClBr – Relativistic second order MBPT to the CF stretching mode in CHFClBr – Peter S etal (PRA, 2005) computed as expectation value of the PNC Peter S etal (PRA, 2005) computed as expectation value of the PNC matrix element seems to be prospective with two/four component matrix element seems to be prospective with two/four component approaches. Problems handling bigger basis may be easier.approaches. Problems handling bigger basis may be easier.

New calculations in other kinds of systems as demonstrated by Stauber (A185 poster of ICQC 2006)New calculations in other kinds of systems as demonstrated by Stauber (A185 poster of ICQC 2006)