Hartree-Fock equations. Spontaneous symmetry breaking

Ammonia molecule NH 3
Total energy (a.u.)
Symmetry-breaking
configurations
Symmetry-conserving
configuration
Distance of N from the H 3 plane
(a.u.)
The height of the barrier: 0.254 eV at 0 K
Minima separation of about 0.75 A
Parity splitting (inversion doubling): 0.0001 eV
Application: ammonia maser based on the inversion doubling 1954
(Nobel Prize for Charles Townes in 1964)

Laboratory frame and intrinsic frame
(ammonia molecule)
H
H
H
|R› = |L› =
" #
r
D " #
= 0
N
)
P L = R " #
= 1 2 1+ # P
)
but
( ) L
L r
D L " 0
!
Spontaneous symmetry breaking gives a description of the system in
terms of wave packets instead of eigenstates. The wave packets
corresponds to given configurations of constituents. If the
configuration interaction energy is very small, the wave packets live
!
a very long time and behave like classical objects. If configurations
are orthogonal and degenerate, the wave packets are also eigenstates
(infinite systems).
Symmetry restoration amounts to projecting states with good
quantum numbers from the symmetry-breaking states. If the
configuration interaction energy is very small, the energies of
projected states are very close to those of the symmetry-breaking
states. Symmetry restoration can be essential for calculating average
values of symmetry-conserving observables other than the
Hamiltonian. After the symmetry restoration, the symmetry-breaking
solutions do not break symmetry!

Don’t let yourself confuse by
the confusing traditional
terminology
When you hear about:
Think about:
State in the intrinsic
reference frame
State before the
symmetry restoration
State in the laboratory
reference frame
State after the
symmetry restoration
Mexican-hat potential

Intrinsic symmetries
5 - ν=1
3 -
1 -
(parallel band)
D •h (C 2 , P)
O
C
O
6 +
4 +
2 +
0 +
ν=0
(ground band)
CO 2
- an axially symmetric molecule with no static dipole moment.
O
C
O
C •ν (S=C 2 P)
3 -
2 +
O N
4 +
1 -
0 +
N 2
O - an axially symmetric molecule with static dipole moment.

Symmetry breaking
q = !
= 0 but
0 0 0
Q ˆ
! crit
E
ω RPA
=0
ω RPA
>0 ω RPA
-
imag.
Vibrator
(weak
coupling)
Soft
(interm.
coupling)
Rotor
(strong
coupling)
No phase
Band structures
transition
labelled by
in the finite
different q.n.
system!
[ H ˆ S ˆ ] h ˆ , S ˆ
deformed
intrinsic
system!
Q 2 !
“exact”
mean field
of the internally
broken symmetries
[ ] ! 0

Laboratory frame and intrinsic frame
(molecular or nuclear quadrupole moment)
Q =
2K 2 -I(I+1)
Q
(I+1)(2I+3) 0
MEASURED
INTRINSIC
RELATIVE
MOTION
Q=0 for I=K=0, independently of Q 0 !

Excitation spectrum of N 2 molecule
excited 1 +
Σ u
and 1 Π u
states
Rotational Transitions ~ 10 meV
Vibrational Transitions ~ 100 meV
Electronic Transitions ~ 1 eV
r
N
N
Diabatic potential energy
surfaces for excited electronic
configurations of N 2

Nuclear collective motion
Rotational Transitions ~ 0.2-2 MeV
Vibrational Transitions ~ 0.5-12 MeV
Nucleonic Transitions ~ 7 MeV