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.

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.

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 Ir ... IIIl 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:

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 C₂ 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 C₂ axis but the choice is otherwise in a sense arbitrary. However, it does have the effect of swapping over the B₁ and B₂ symmetries, so it is important to be consistent, particularly when comparing to the literature. The convention 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. The overall character table then looks like this:

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)^J-K|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.

	Rπ(z)	Rπ(y)	Rπ(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

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 (A₁ 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 A₁ 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:

C2v		Sym	E	C2(a)	σ(ac)	σ(ab)
				Rπ(a)	Rπ(b)	Rπ(c)	Even J	Odd J
A1	ee	0	1	1	1	1	E+	E-
A2	eo	1	1	1	-1	-1	E-	E+
B1	oo	2	1	-1	1	-1	O+	O-
B2	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 A₁ symmetry for even J but A₂ for odd J. In considering the energy level and transition patterns it is helpful to consider the E+ levels together, rather than the A₁ 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 A₁ and A₂ and the complete table is:

			Even J				Odd J
E+	Ka+Kc=J		A1	0	ee		A2	1	eo
E-	Ka+Kc=J+1		A2	1	eo		A1	0	ee
O+	Ka+Kc=J		B1	2	oo		B2	3	oe
O-	Ka+Kc=J+1		B2	3	oe		B1	2	oo

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

D₂

This group is the same size as C_2v and therefore many of the ideas used above carry over directly:

D2	Sym	E	C2(z')	C2(y')	C2(x')
			Rπ(z')	Rπ(y')	Rπ(x')
A	0	1	1	1	1
B1	1	1	1	-1	-1
B2	2	1	-1	1	-1
B3	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 - C₁, C₂ 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

C₁

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.

C₂

This and C_s have a single symmetry operation, and thus only two possible symmetries:

C2	Sym	E	C2(z')
			Rπ(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 C₂ 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 E-O+, 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 C₂ in only having  a single symmetry operation, and thus only two possible symmetries:

Cs	Sym	E	σ(x'y')
			Rπ(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 E-O+, 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 C₁:

Ci	Sym	E	i
			E*
Ag	0	1	1
Au	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_2h

This is generated by adding a centre of symmetry to C₂:

C2h	Sym	E	C2(z')	i
			Rπ(z')	E*
Ag	0	1	1	1
Bg	1	1	-1	1
Au	2	1	1	-1
Bu	3	1	-1	-1

Only the C2zAxis variable is used, and specifies which of the principal axes corresponds to the C₂ 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 E-O+_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_2h

This is generated by adding a centre of symmetry to D₂:

D2h	Sym	E	C2(z')	C2(y')	C2(x')	i
			Rπ(z')	Rπ(y')	Rπ(x')	E*
Ag	0	1	1	1	1	1
B1g	1	1	1	-1	-1	1
B2g	2	1	-1	1	-1	1
B3g	3	1	-1	-1	1	1
Au	4	1	1	1	1	-1
B1u	5	1	1	-1	-1	-1
B2u	6	1	-1	1	-1	-1
B3u	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, E-g, E-u, O+g, O+u, O-g and O-u.

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, D₂ or D_2h symmetry. To use line list files generated for the old version, add a

pgopherversion 5.1