For asymmetric tops, the pure
rotational Hamiltonian only contains operators involving even
powers of the angular momentum, so is always symmetric with
respect to a C_{2}
rotation about any of the principal axes. The resulting group has
the same form as the D_{2}
point group, though it is often known as the V group in this context. For
molecules with point groups C_{2v} D_{2} and D_{2h} the rotational
operations acting on the rotational Hamiltonian are equivalent to
operations on the molecule as a whole and no special
considerations are required.
For lower symmetry groups it is
not possible to map all of the
C_{2}
rotations about each of the principal axes to operations of the
overall point group of the molecule, and indeed the full
Hamiltonian will not have these symmetries. However, in the
absence of perturbations acting between different vibronic states,
there will be no operators that break the
V symmetry and this can be
used to simplify calculations. This is controlled by the
PseudoC2v setting at the
molecule level  if this is set a higher effective symmetry is
used for calculations. In the most favourable cases, such as
C_{1}, this can lead
to a reduction in memory usage by 4 and in run time by 16, though
the savings are only going to be noticeable for complicated
systems. An additional advantage is in handling levels that are
degenerate to the machine precision (or close to it) where the
eigenvalue numbers (used by default in labelling states for
fitting) can change unexpectedly. The
PseudoC2v will
typically place these near degenerate states
in separate matrices, removing the problem.
The action on setting the
PseudoC2v
flag depends on the point group and is described individually
below. Note that is this flag is set, both the
C2zAxis and
C2xAxis must be set, and
the values must be different.
An alternative approach is to set the symmetry of the molecule to
a higher symmetry than it actually has, but then a symmetry check
on allowed transitions must be disabled. This is the purpose of
the
FakeSym flag,
which is also described below.
C_{1}
For this case the single symmetry becomes four, and the molecule
is treated exactly as
C_{2v}, with the
C2zAxis and
C2xAxis settings
controlling the symmetry labels used.
C_{s}
In this case turning
PseudoC2v
on splits each irreducible representation into two:
C_{s} 


C_{2v}


E

σ(x'y')




Sym 


Sym 

R^{π}(z')

R^{π}(y') 
R^{π}(x') 
A' 
0


A_{1}

0

1

1

1

1

A' 
0


A_{2} 
1

1

1

1

1

A" 
1


B_{1} 
2

1

1

1

1

A" 
1


B_{2} 
3

1

1

1

1

and
C2xAxis will
control the labels used for the split state
C_{2}
In this case turning
PseudoC2v
on splits each irreducible representation into two:
C_{2} 


C_{2v}


E

C_{2}(z')




Sym 


Sym 

R^{π}(z')

R^{π}(y') 
R^{π}(x') 
A 
0


A_{1}

0

1

1

1

1

A 
0


A_{2} 
1

1

1

1

1

B 
1


B_{1} 
2

1

1

1

1

B 
1


B_{2} 
3

1

1

1

1

and
C2xAxis will
control the labels used for the split state.
C_{i}
For this case
PseudoC2v
will actually force the use of
D_{2h} symmetry, with the
g states becoming one of
the four
g symmetries in
D_{2h} and the
u states becoming one of
the four
u
symmetries in
D_{2h}. The
C2zAxis and
C2xAxis settings control
the symmetry labels used.
C_{2}_{h}
For this case
PseudoC2v
will actually force the use of
D_{2h} symmetry, with each
irreducible representation split into two:
C_{2}_{h} 


D_{2h}


E

C_{2}(z')



i


Sym 


Sym 

R^{π}(z')

R^{π}(y') 
R^{π}(x') 
E*

A_{g} 
0


A_{g}

0

1

1

1

1

1

A_{g} 
0


B_{1}_{g} 
1

1

1

1

1

1

B_{g} 
1


B_{2}_{g} 
2

1

1

1

1

1

B_{g} 
1


B_{3}_{g} 
3

1

1

1

1

1

A_{u} 
2


A_{u}

4

1

1

1

1

1

A_{u} 
2


B_{1}_{u} 
5

1

1

1

1

1

B_{u} 
3


B_{2}_{u} 
6

1

1

1

1

1

B_{u} 
3


B_{3}_{u} 
7

1

1

1

1

1

and
C2xAxis will
control the labels used for the split state.
FakeSym
An alternative approach is to set the point group to the
higher symmetry; this will give equivalent results provided that
the normal check for allowed transitions from the overall
rovibronic symmetry is disabled by setting
FakeSym to true. As an
example consider the possible rotational transitions starting from
the 0
_{00} level of a totally symmetric vibronic state to
J = 0 and 1 levels of
another totally symmetric vibronic state. The allowed transitions
will depend on the point group and the alignment of the principal
axes with the symmetry elements. For two specific choices we have:
Upper State

Rovibronic
Symmetry in C_{2v}
(C2zAxis = a, C2xAxis = c) 
Rovibronic
Symmetry in C_{s
}(C2zAxis
= c)

0_{00}

A_{1}  forbidden_{
} 
A'  forbidden 
1_{01}

A_{2}  allowed

A"  allowed 
1_{11}

B_{1}  forbidden 
A"  allowed 
1_{10}

B_{2}  forbidden 
A'  forbidden 
In
C_{2v} symmetry only one
component of the transition dipole can give a transition, and with
the example axis choice above this is the
a component. For this
transitions with Δ
K_{a} = 0, Δ
K_{c}
= ±1 are allowed so only one of the four listed transitions
is possible. (This also follows as the the rovibronic symmetry of
the transition dipole moment is always A
_{2} in
C_{2v}.) In
C_{s }symmetry, given
that the symmetry axes are chosen such that the
a dipole is still symmetric,
the same transition is still allowed but another component, here
the
b component, now has
the same symmetry as the
a
component and thus also gives an allowed transition. To use
C_{2v} settings (with the
consequent reduction in matrix sizes) to calculate
b type transitions thus
requires a symmetry check on allowed transitions to be disabled,
which setting
FakeSym
to true will do. Similar logic allows
C_{2v}
settings to be applied to a
C_{1} molecule, where
c type transitions can also
be allowed.