Download Relativistic molecular structure calculations for the detection of CP

Symposium: New Generation Quantum Theory
-Particle Physics, Cosmology, and ChemistryKyoto University, Yoshida Campus
7-9 March 2016
TMU
Relativistic molecular electronic structure
calculations for the detection of CP violation
Be a chemist now!
Minori Abe 阿部穣里
Tokyo Metropolitan University
[email protected]
Outline
1. CP symmetry and
electron’s electric dipole moment (eEDM)
2. eEDM operator in molecule
and effective electric field (Eeff)
3. Eeff in YbF, HgX and BiO molecules:
Relativistic CCSD and CASPT2 results
Outline
1. CP symmetry and
electron’s electric dipole moment (eEDM)
2. eEDM operator in molecule
and effective electric field (Eeff)
3. Eeff in YbF, HgX and BiO molecules:
Relativistic CCSD and CASPT2 results
A mystery in Physics
Why anti-particles almost die out
in our universe,
even though the same number of
particles and anti-particles
are created in Big-Bang ?
Disappearance of anti-particles
particles and anti-particles
obey different laws
Sakharov’s three necessity conditions
１．Charge symmetry and
Charge-Parity (CP) symmetry violation
２．Baryon number violation
３．Interactions out of
thermal equilibrium
CP（Charge-Parity) symmetry violation
Charge conjugate
+q
Parity inversion
-q
The standard model predicts CP violation,
but it is very weak and cannot explain
the too small amount of anti-particles
in the present universe.
We need a new theory and experimental evidences
which explains much larger CP violation.
Electric dipole moment (EDM)
of an elementary particle
EDM d
EDM -d
Time reversal
spin
(t  -t)
(p-p) (rr)
L-L
Assume that
EDM is parallel to spin axis
spin
dd
EDM becomes antiparallel to spin axis
Non-zero value of EDM = T symmetry violation
= CP symmetry violation (under CPT theorem)
Upper limit of electron EDM
observed in atoms and molecules
Why upper limit
because the errors are larger than
the observables at present.
Molecules are
hot now!
If an absolute value of EDM is
determined,
it will be a great hint for the models
beyond the standard theory.
WC, HfF+, BiO, FrSr,…
YbF
ThO
Outline
1. CP symmetry and
electron’s electric dipole moment (eEDM)
2. eEDM operator in molecule
and effective electric field (Eeff)
3. Eeff in YbF, HgX and BiO molecules:
Relativistic CCSD and CASPT2 results
Observable
in atoms or molecules for EDM
Interaction energy between electric field and de
Electronic wave function
E=-de・E
E
spin matrices
Ne
E    d e σ j  Eint 
j
E=0
EDM de
spin
de
Yb
When spherically
symmetric orbital
Internal electric field Eint
Observable
in atoms or molecules for EDM
Interaction energy between electric field and de
Electronic wave function
E=-de・E
Ne
E    de σ j  Eint 
j
E≠0
EDM de
spin
E
de
Yb
when symmetry
is broken by Eext
Eext
Observable
in atoms or molecules for EDM
Interaction energy between electric field and de
Electronic wave function
E=-de・E
de
Ne
E    de σ j  Eint 
j
E≠0
EDM de
spin
E
Yb
F
In molecule, symmetry is strongly broken
without the external electric field.
Problem of EDM operator
Ne
Effective electric field
E   d e   σ j  Eint   d e Eeff
j
σ j  Eint  σ j , V (r )   σ j , Hˆ 0 
Unperturbed 2-component nonrelativistic Schrödinger eq.
1 2
ˆ
H 0    i  V
2
i
Ĥ 0   E0 
Ne
Eeff    σ , Hˆ 0    0
j
We cannot measure E. (Schiff’s theorem)
Relativistic EDM operator
Ne
E   d e    σ j  Eint   d e Eeff
j
2
ˆ


  σ j  Eint    σ j , H 0   2ic 5p j
Unperturbed 4-component relativistic Schrödinger eq.
Ne


Hˆ 0   c pi  βmc 2  V
Ĥ 0   E0 
i
Ne
Eeff    2ic 5p 2   0
j
Relativity is important to observe EDM.
Relativistic quantum chemistry is necessary
Ne
H eEDM   d e  σ j  Eint
E   Hˆ eEDM 
j
N nuc

ZA
2
ˆ
H 0   c pi  βmc  
i 
A ri  R A

N elec
 L 
 L

   S 
  
 S
 
 
4-cmp spinor
 Nelec 1

 i  j ri  r j
Dirac-Coulomb
Hamiltonian
Eeff  1 d e  Hˆ eEDM 
We cannot measure Eeff
experimentally and need
to calculate.
Collaborations of three fields
are very important!
Particle Physics
(Theory)
 d e    i  Eint 
i
Atomic, Molecular,
and Optical Physics
(Experiment)
Relativistic
Quantum Chemistry
to calculate Eeff
Effective one-body operator
Ne
H eEDM   d e  σ j  Eint
E   Hˆ eEDM 
j
2


  σ  Eint    σ , Ĥ 0   2ic 5p
Eeff
 1 d e  Hˆ eEDM 
Ne
   2ic 5p 2 
j
Effective one-body operator
The expectation value of
the effective operator is
exactly same to the
original one when the
wave function is exact
solution of Ĥ 0 .
One electron orbital form

2ic i4 cmp  5 p 2 i4 cmp  2ic iL p 2 iiS  iiS p 2 iL
 2ic i iL p 2 iS  i iL p 2 iS

 022
 5  
  122




 4c Re iL p 2 iS
122 

022 
Coupling of large
and small components
N elec
 Eeff ROHF  4c  Re
 p 
L
i
2
S
i
i
At the Kramers
restricted ROHF level,
the p2 value of SOMO
is only remained.
 4c Re 
p 
singly occupied molecular orbital (SOMO)
L
SOMO
2
S
SOMO
Large and small component
of atomic spinor (basis sets)
Large component
Small component
Yanai et al.
JCP, 114,
6526, 2001.
Differentiation and angular moment

c
σ  p L
L
σ  p  
 

2
2mc  V  
2mc
c
L
σ  p  S
 
V  
S
 p 
L
2
S
 
L
σ  p σ  p  
S
Differentiation of s  p
Differentiation of p
…
s and d
Kinetic balance
condition
V   L

 σ  p   L
c
s
p
p
Mixture of s and p spinor in SOMO is important.
(s can be close to the nucleus having large Eint.)
Larger Eeff is better for experiment
Larger Eeff provides larger value of E in experiment.
When Eeff can be large?
• Paramagnetic (otherwise zero)
• Molecule with heavy element nucleus X
because of its large Eint
• SOMO is important!  SOMO  cs  s , X  c p  p , X
 SOMO distributing to the heavy nucleus X
 SOMO with parity mixing
Mixing of s and p-spinor may be the best.
Diatomic molecule with large Eeff
Q. Choose two atoms to create
diatomic molecule with large Eeff
Ask Mr. Sunaga
For example of XF molecules
SOMO mainly consists of s.
Virtual p will be mixed in SOMO.
X: (ns)2
F: (2p)5
X+: (ns)1
F-: (2p)6
Outline
1. CP symmetry and
electron’s electric dipole moment (eEDM)
2. eEDM operator in molecule
and effective electric field (Eeff)
3. Eeff in YbF, HgX and BiO molecules:
Relativistic CCSD and CASPT2 results
Previous theoretical works (Eeff)
Atomic EDM:
Dirac-Coulomb + CCSD, etc
・Accurate calculations are reported
for both relativity and electron correlation.
Molecular EDM (PbO,TlF,ThO,YbF…) :
（One or two-component ） GRECP＋RCCSD, GRECP+SOCI
（Four-component ） RASCI, GASCI
・Only a few works are reported,
enough accurate in relativity,
electron correlation, and basis set size.
Our motivation
Calculated Eeff  cannot be compared
with any experiments
We should calculate Eeff as precise as possible
with the techniques of quantum chemistry
and available computational resource!
Relativity (Dirac-Coulomb)
Basis sets (Dyall QZ)
Electron correlation
(CCSD, CASPT2)
YbF molecule
SOMO mainly consists of s.
Virtual p will be mixed in SOMO.
X: (ns)2
F: (2p)5
X+: (ns)1
F-: (2p)6
YbF molecule
 Experimentally reported
 Simple electronic structure
 Bench mark of calculations
Couple cluster singles and doubles (CCSD)
b
a
Virtual
orbitals
a
j
Occupied
orbitals
i
i
ΦHF
HF determinant
Φ𝑖𝑎 =𝐸𝑖𝑎 ΦHF
Singly excited det.
𝑇1 =
𝑡1,𝑖𝑎 𝐸𝑖𝑎
𝑖,𝑎
𝑎𝑏
𝑎𝑏
Φ𝑖𝑗
= 𝐸𝑖𝑗
ΦHF
Doubly excited det.
𝑇2 =
𝑡2,𝑖𝑗𝑎𝑏 𝐸𝑖𝑗𝑎𝑏
𝑖,𝑗,𝑎,𝑏
ΦCCSD = exp(𝑇1 + 𝑇2 ) ΦHF
An effective method when the HF determinant
is the only dominant determinant.
Computational details for YbF
Phys. Rev. A 90, 022501 (2014)
Wave function： Dirac-Fock, Dirac-CCSD
Program : UTChem + DIRAC08
Basis sets: Dyall’s DZ, TZ, and QZ for Yb
+ Watanabe’s basis for F
+ Sapporo (DK3) polarization
in uncontracted form
We approximate the expectation value
with CCSD wave function as follows.

 



 ˆ


ˆ
ˆ
ˆ
ˆ
O  HF 1  T1  T2 ON 1  T1  T2 HF
ˆ HF

HF
O
C
Measurable properties related with Eeff
• Hyperfine coupling constant (parallel) A//:
Interaction between
electron spin and Yb nuclear spin
• Molecular dipole moment PDM:
s-p hybridization
QZ
79e-CCSD(293)
Experiments
Error from
experiments
Eeff (GV/cm)
A// (MHz)
PDM (D)
23.1
7913
7424
7%
3.60
3.91
8%
-
Eeff in HgX (X=F, Cl, Br, I)
Basis set
Molecule
type
Mr. Prasannaa
HgF
HgCl
HgBr
HgI
DZ
DZ
DZ
DZ
Method
Eeff
(GV/cm)
CCSD
CCSD
CCSD
CCSD
115.42
113.56
109.29
109.30
HgX has much larger Eeff than YbF (23.1 GV/cm).
“Mercury Monohalides: Suitability for Electron Electric Dipole Moment
Searches” V. S. Prasannaa, A. C. Vutha, M. Abe, and B. P. Das
Phys. Rev. Lett. 114, 183001, (2015)