Axis systems and symmetries for asymmetric tops
In setting up asymmetric top
calculations, care must be taken in setting the axis systems used,
particularly for results to be consistent with other work. The key
point to realize is that there are three more or less independent
body fixed (Cartesian) axis systems to be chosen, and the choice
is often arbitrary, or subject to a convention that is not
necessarily followed. The choice is determined in
PGOPHER by the settings
PointGroup,
Representation,
C2zAxis
and
C2xAxis at the
molecule level. The
SReduction setting
is also similarly important, though it only affects the form of
the Hamiltonian rather than axis choice.
Principal Axes  A, B and C
The principal axes (here written
as a, b and c) are chosen by convention
so that the rotational constants are in the order A > B > C, though PGOPHER does not check
for, or even require, this ordering. If only three constants are
used, then any permutation of these numbers will give exactly the
same energy levels. However, most other constants will require
adjustment if the axis choice is changed, as discussed in the next
section.
The Representation  x,
y and z
When calculating the energy levels
PGOPHER will use
standard symmetric top functions, implying a choice must be made
as to which axis is chosen for
K,
the body fixed projection of
J.
This in turn determines the basis functions used. There are three
possible choices for this axis, along the
A, B or
C inertial axes, which are
labeled I, II or III respectively. It is conventional to label
this axis as
z, so that
the
Jz
operator has eigenvalues of
K.
x and
y axes must also be chosen,
labeled
r and
l, giving six possibilities
overall:

Ir 
IIr 
IIIr 
Il 
IIl 
IIIl 
z

a 
b 
c 
a 
b 
c 
x

b 
c 
a 
c 
a 
b 
y

c 
a 
b 
b 
c 
a 

Ir 
IIr 
IIIr 
Il 
IIl 
IIIl 
a 
z 
y 
x 
z 
x 
y 
b 
x 
z 
y 
y 
z 
x 
c 
y 
x 
z 
x 
y 
z 
The labels I
r ... III
l are known as the
representation, and must
always be specified when quoting rotational constants (other than
A, B or
C) for asymmetric tops. The
obvious choice would seem to be I for near prolate tops and III
for near oblate tops, as this will result in the choice of basis
for which
K is closest
to a good quantum number. However, unless
PGOPHER is set to
truncate the basis in some way (which it will not do by default)
the basis is sufficiently complete that the choice of
representation should make no difference to the final energy
levels. In practice it will make a difference because most of the
rotational constants are expressed with respect to the (
x,
y,
z)
axis system, rather than the (
a,
b,
c) principal axis system. As
an example, consider the
D_{k}
or Δ
_{k} centrifugal
distortion term, which is defined as:
D_{k}J_{z}^{4}≡
D_{k}K^{4}
Changing the representation will therefore mean that the operator
corresponding to D_{k}
will change and the energy levels will also change. It will normally
be possible to regain the same energy level pattern after changing
the representation, but only by adjusting all of the constants. An
exact match will only be achievable by adding higher order
constants.
As an example, consider the ground vibronic level
of HCHO, for which the Ir
representation is commonly used. Using J ≤ 10 levels calculated using the Ir
representation as input to a PGOPHER
energy level fit yields the following sets of constants:

Ir 
IIr 
IIIr 
Il 
A 
9.405524207646(36) 
9.4055520(69) 
9.4055261(66) 
9.405524207646(36) 
B 
1.29543120899(13) 
1.294704(15) 
1.296011(14) 
1.29543120899(13) 
C 
1.13419132182(13) 
1.134891(15) 
1.133609(14) 
1.13419132182(13) 
Δ_{K} 
6.476176877(13)e4 
6.4792(26)e4 
6.4791(25)e4 
6.476176877(13)e4 
Δ_{JK} 
4.30670962(16)e5 
9.9265(35)e4 
9.9444(32)e4 
4.30670962(16)e5 
Δ_{J} 
2.51257682(34)e6 
3.47693(92)e4 
3.48291(87)e4 
2.51257682(34)e6 
δ_{K} 
3.4294099(51)e5 
3.16927(88)e4 
3.29653(85)e4 
3.4294099(51)e5 
δ_{J} 
3.4872359(41)e7 
1.72970(48)e4 
1.72662(47)e4 
3.4872359(41)e7 
σ 
2.7e9 
5.2e4 
5.2e4 
2.7e9 
Note that changing between r
and l only changes the
sign of the off diagonal centrifugal distortion constants, δ_{J}
and δ_{K}.
The Point Group  x',
y' and z'
The third axis system is that
defined by the symmetry of the molecule, and is set by the C2zAxis and C2xAxis variables. This
is most straightforward to understand for C_{2v} where two things
must be specified:
First is the
location of the C_{2}
axis, which is conventionally the z axis in the character table, so we use z' for this to distinguish
it from the z axis used
to define K as described
in the previous section. The C2zAxis variable must be set to control this;
possible values are a, b or c.
Secondly we
must choose one of the two mirror panes to be perpendicular to the
x' axis; the
C2xAxis variable must be
set to control this. This must be different from the
C_{2} axis but the
choice is otherwise in a sense arbitrary. However, it does have
the effect of swapping over the
B_{1}
and
B_{2}
symmetries, so it is important to be consistent, particularly when
comparing to the literature. The IUPAC convention (
Schutter et al,
1997) for planar molecules is to take the
x' axis out of the plane,
which means it will always be the
c axis for planar asymmetric tops. However,
expect to find both possible choices in the literature, and any
given paper will need to be checked for the axis choice made. The
overall character table then looks like this:
C_{2v}

Sym

E

C_{2}(z')

σ(x'z') 
σ(y'z') 



R^{π}(z')

R^{π}(y') 
R^{π}(x') 
A_{1}

0

1

1

1

1

A_{2} 
1

1

1

1

1

B_{1} 
2

1

1

1

1

B_{2} 
3

1

1

1

1

The Sym column gives
the numbers PGOPHER
uses internally to represent the symmetry.
Symmetry of Rotational Wavefunctions
The C_{2v}
group is the point group of the molecule, and as such is only
appropriate for determining vibronic symmetry. However, when
considering the symmetry of the overall wavefunction, the symmetry
of the rotational wavefunction is also required, and this is best
done using the permutation inversion group of the molecule. For
molecules with C_{2v }symmetry
this is straightforward, as each overall operation in the
permutation inversion group can be taken as an operator acting on
the rotational wavefunction combined with an operation of the
point group. The operations on the rotational wavefunctions
are written as R^{π}
here, and rotate the rotational wavefunction by 180° about the
given body fixed axis. The paring with the operations of the
conventional point group are indicated in the tables above. The
effect on a symmetric top basis function defined such that K is along the z' axis is:
R^{π}(z') J, K>
= (1)^{K}J, K>
R^{π}(y') J, K> = (1)^{JK}J,
K>
R^{π}(x') J, K> = (1)^{J}J,
K>
As these symmetry operations
changes the sign of K,
symmetry adapted functions are obtained by taking linear
combinations of the +K
and K functions:
J,
K, ±>
= 2^{1/2}[J, K>±J, K>]
This is the
Wang transformation, and
gives the basis functions
PGOPHER
uses. Given the alternation of symmetries with
K the odd and even
K levels will be different
giving four different sorts of basis functions, designated
E+ (
K even, positive combination),
E,
O+ and
O.
K = 0 is a special case;
there is no
E function
and the
E+ function is
just 
J, 0>. In
the
C_{2v} point group these
correspond to four different symmetries; the actual symmetry
depends on the axis choice, but can be worked out from the tables
above.

R^{π}(z)

R^{π}(y) 
R^{π}(x) 
E+

1

(1)^{J}

(1)^{J} 
E 
1

(1)^{J} 
(1)^{J} 
O+ 
1

(1)^{J} 
(1)^{J} 
O

1

(1)^{J} 
(1)^{J} 
ee eo
oe oo Notation
In discussing the symmetry of rotational wavefunctions, the ee/eo/oe/oo notation is often used,
which specifies whether the K_{a}
and K_{c} quantum
numbers respectively are even or odd. This notation is used
internally by PGOPHER
to handle rotational symmetries, and is used to specify statistical
weights. The corresponding symmetries are:
RoSym 
K_{a}K_{c} 
R^{π}(a)

R^{π}(b) 
R^{π}(c) 
0

ee

1

1

1

1 
eo

1

1

1

2 
oe

1

1

1

3 
oo

1

1

1

The statistical weights are set in the constants window for a
totally symmetric vibronic state for the molecule under
consideration; PGOPHER
will convert the symmetries as required.
Overall symmetry
PGOPHER requires the overall rovibronic
symmetry (rotational × vibrational × electronic) to
calculate any given state as, in general, Hamiltonian matrices
containing all states of the same total angular momentum and
rovibronic symmetry are set up and diagonalized. (The point group
symmetry number given above will be used internally.) In the
simple case that the vibronic wavefunction is totally symmetric (
A_{1} in
C_{2v}) this means that
the rotational symmetry will be the same as the overall symmetry.
The
ee rotational
functions are always totally symmetric, and thus correspond to
A_{1} symmetry, but
for the other rotational functions the matching function will
depend on the mapping between the
a,
b
and
c axes and the
x',
y' and
z'
axes, which is in turn determined by the
C2zAxis and
C2xAxis settings. For the
HCHO case the conventional choice is
C2zAxis = a and C2xAxis = c, giving the following
character table:
C_{2v}


Sym

E

C_{2}(a)

σ(ac) 
σ(ab) 






R^{π}(a)

R^{π}(b) 
R^{π}(c) 
Even J

Odd J

A_{1}

ee

0

1

1

1

1

E+

E

A_{2} 
eo

1

1

1

1

1

E

E+

B_{1} 
oo

2

1

1

1

1

O+

O

B_{2} 
oe

3

1

1

1

1

O

O+

J adjusted symmetry
From the table above it can be seen that the form of the basis
functions alternates with
J,
in that E+ functions have
A_{1}
symmetry for even
J but
A_{2} for odd
J. In considering the energy
level and transition patterns it is helpful to consider the E+
levels together, rather than the
A_{1} functions. (Similar considerations
arise in linear molecules, where
e and
f
parity labels (which alternate with
J) are more useful than the absolute parity.) It
is therefore helpful to introduce the idea of a
J adjusted symmetry. There is
no standard notation for this, so
PGOPHER uses the Wang labels. For the case
illustrated here E+ alternates between
A_{1} and
A_{2} and the complete table is:



Even
J 

Odd J

E+

K_{a}+K_{c}=J 

A_{1}

0

ee


A_{2} 
1

eo

E 
K_{a}+K_{c}=J+1 

A_{2} 
1

eo


A_{1} 
0

ee

O+ 
K_{a}+K_{c}=J 

B_{1} 
2

oo


B_{2}

3

oe

O

K_{a}+K_{c}=J+1 

B_{2} 
3

oe


B_{1}

2

oo

Note that for cases where the statistical weights are all 1 (i.e. no
equivalent nuclei) the C2zAxis
and C2xAxis settings
will not affect the form of the spectra, but will affect the
symmetry labels given.
_{}
PGOPHER input and
display of symmetry
When displaying the symmetry
PGOPHER will normally use
the
J adjusted symmetry
labels described above or, if
JAdjustSym for the Mixture is set to false,
the standard irreducible representation labels for the point
group. On input (typically in a
line list
file) either notation is accepted or the symmetry number (the
Sym column in the tables
above) can also be given. To display the various labels used above
for any given state, right click on the state name in the
constants window and select "Symmetry Table". The result will be
found in the log window.
Other Point Groups
D_{2}
This group is the same size as
C_{2v} and therefore many
of the ideas used above carry over directly:
D_{2}

Sym

E

C_{2}(z')

C_{2}(y') 
C_{2}(x') 



R^{π}(z')

R^{π}(y') 
R^{π}(x') 
A

0

1

1

1

1

B_{1} 
1

1

1

1

1

B_{2} 
2

1

1

1

1

B_{3} 
3

1

1

1

1

The
C2zAxis and
C2xAxis variables
indicate the principal axes that correspond to the
z' and
x' axes of the point group.
The
J adjusted symmetry
notation is essentially the same as for
C_{2v}.
Lower Symmetry Groups  C_{1},
C_{2} and C_{s}
The character tables for these point groups can be derived from
C_{2v} by removing one or
more of the operations of the group
C_{1}
This has no symmetry operations, so all states have the same
symmetry,
A or symmetry
number 0. The
C2zAxis
and
C2xAxis
variables are therefore both ignored, and the discussion about
J adjusted symmetry does not
apply to this point group.
C_{2}
This and
C_{s}
have a single symmetry operation, and thus only two possible
symmetries:
C_{2}

Sym

E

C_{2}(z')




R^{π}(z')

A

0

1

1

B 
1

1

1

Only the
C2zAxis
variable is used, and specifies which of the principal axes
corresponds to the
C_{2}
axis of the molecule; the
C2xAxis
variable is ignored. For the
J
adjusted symmetry labels there are three possible set of labels,
depending on which of the Wang symmetries pair up:
 E and O, used where E+ has the same symmetry as E and O+
has the same symmetry as O
 + and , used where E+ has the same symmetry as O+ and E
has the same symmetry as O
 E+O and EO+, used where E+ has the same symmetry as O and
E has the same symmetry as O+
The appropriate set depends on the representation and
C2zAxis.
C_{s}
This is similar to
C_{2}
in only having a single symmetry operation, and thus only
two possible symmetries:
C_{s} 
Sym

E

σ_{}(x'y')




R^{π}(z')

A'

0

1

1

A"

1

1

1

Only the
C2zAxis
variable is used, and specifies which of the principal axes is
perpendicular to the symmetry plane. For a planar molecule this
will always be the
c
axis
. The
C2xAxis variable is
ignored. For the
J
adjusted symmetry labels there are three possible set of labels,
depending on which of the Wang symmetries pair up:
 E and O, used where E+ has the same symmetry as E and O+
has the same symmetry as O
 + and , used where E+ has the same symmetry as O+ and E
has the same symmetry as O
 E+O and EO+, used where E+ has the same symmetry as O and
E has the same symmetry as O+
The appropriate set depends on the representation and
C2zAxis.
Groups with a centre of symmetry  C_{i}, C_{2h}
and D_{2h}
These can all be generated by adding a centre of symmetry, and
thus
g and
u labels to one of the above.
Rotational wavefunctions always have
g symmetry, so any given vibronic state will
only have half of the possible rovibronic symmetries.
C_{i}
This is generated by adding a centre of symmetry to
C_{1}:
C_{i} 
Sym

E

i




E*

A_{g}

0

1

1

A_{u} 
1

1

1

The
C2zAxis and
C2xAxis variables are
therefore both ignored, and the discussion about
J adjusted symmetry does not
apply to this point group as all the rotational states for a given
vibronic state have the same symmetry.
C_{2}_{h}
This is generated by adding a centre of symmetry to
C_{2}:
C_{2}_{h} 
Sym

E

C_{2}(z')

i




R^{π}(z')

E*

A_{g}

0

1

1

1

B_{g} 
1

1

1

1

A_{u} 
2

1

1

1

B_{u} 
3

1

1

1

Only the
C2zAxis
variable is used, and specifies which of the principal axes
corresponds to the
C_{2}
axis of the molecule; the
C2xAxis
variable is ignored. For the
J
adjusted symmetry labels there are 6 possible set of labels,
depending on which of the Wang symmetries pair up:
 E_{g} and O_{g}, used where E+ has the same
symmetry as E and O+ has the same symmetry as O
 +_{g} and _{g}, used where E+ has the same
symmetry as O+ and E has the same symmetry as O
 E+O_{g} and EO+_{g}, used where E+ has the
same symmetry as O and E has the same symmetry as O+
and another 3 sets with
u
symmetry. The appropriate set depends on the representation and
C2zAxis.
D_{2}_{h}
This is generated by adding a centre of symmetry to
D_{2}:
D_{2h}

Sym

E

C_{2}(z')

C_{2}(y') 
C_{2}(x') 
i




R^{π}(z')

R^{π}(y') 
R^{π}(x') 
E*

A_{g}

0

1

1

1

1

1

B_{1}_{g} 
1

1

1

1

1

1

B_{2}_{g} 
2

1

1

1

1

1

B_{3}_{g} 
3

1

1

1

1

1

A_{u}

4

1

1

1

1

1

B_{1}_{u} 
5

1

1

1

1

1

B_{2}_{u} 
6

1

1

1

1

1

B_{3}_{u} 
7

1

1

1

1

1

The
C2zAxis and
C2xAxis variables
indicate the principal axes that correspond to the
z' and
x' axes of the point group.
The
J adjusted symmetry
notation is similar to that for C
_{2v}, though there are now 8 possibilities,
E+g, E+u, Eg, Eu, O+g, O+u, Og and Ou. Note also that the
choice of
x' and
z' axis is not unique  as for
C2v there is a convention (
Schutter et al,
1997) but expect for the convention not to be followed.
PGOPHER 5.1 input
and display of symmetry
In version 5.1 of PGOPHER a different
numbering scheme was used to label the rovibrational symmetry.
This had a less obvious relationship with the various ways of
denoting symmetry as described above, hence the change. This
change only affects molecules with C_{2v},
D_{2 }or D_{2h} symmetry. To use
line list files generated for the old version, add a
pgopherversion 5.1
directive at the top of the file, as described here. The symmetry tables
displayed by PGOPHER
will
show this number (Labeled "Old sym" in brackets) where it differs
from
the current scheme.
References
 C. J. H. Schutte, J. E. Bertie, P. R. Bunker, J. T. Hougen, I. M. Mills, J. K. G. Watson and B. P. Winnewisser, "Notations and conventions in molecular spectroscopy: Part 2. Symmetry notation (IUPAC Recommendations 1997)", Pure Appl. Chem., 69, 1641 (1997), doi:10.1351/pac199769081641.