 Molecule Types Asymmetric Tops

# 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, RepresentationC2zAxis 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 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 Dk or Δk centrifugal distortion term, which is defined as:
DkJz4DkK4
Changing the representation will therefore mean that the operator corresponding to Dk 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) ΔK 6.476176877(13)e-4 6.4792(26)e-4 6.4791(25)e-4 6.476176877(13)e-4 ΔJK 4.30670962(16)e-5 -9.9265(35)e-4 -9.9444(32)e-4 4.30670962(16)e-5 ΔJ 2.51257682(34)e-6 3.47693(92)e-4 3.48291(87)e-4 2.51257682(34)e-6 δK 3.4294099(51)e-5 3.16927(88)e-4 -3.29653(85)e-4 -3.4294099(51)e-5 δJ 3.4872359(41)e-7 -1.72970(48)e-4 1.72662(47)e-4 -3.4872359(41)e-7 σ 2.7e-9 5.2e-4 5.2e-4 2.7e-9
Note that changing between r and l only changes the sign of the off diagonal centrifugal distortion constants, δJ and δK.

## 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 C2v 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 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:

 C2v Sym E C2(z') σ(x'z') σ(y'z') Rπ(z') Rπ(y') Rπ(x') A1 0 1 1 1 1 A2 1 1 1 -1 -1 B1 2 1 -1 1 -1 B2 3 1 -1 -1 1
The Sym column gives the numbers PGOPHER uses internally to represent the symmetry.

## Symmetry of Rotational Wavefunctions

The C2v 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 C2symmetry 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 Rπ 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:
Rπ(z') |J, K> = (-1)K|J, K>
Rπ(y') |J, K> = (-1)J-K|J, -K>
Rπ(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 C2v 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

## 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 Ka and Kc 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 KaKc Rπ(a) Rπ(b) Rπ(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 (A1 in C2v) 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+

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

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

### D2

This group is the same size as C2v 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 C2v.

## Lower Symmetry Groups - C1, C2 and Cs

The character tables for these point groups can be derived from C2v 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 Cs 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.

### Cs

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 - Ci,  C2h and D2h

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.

### Ci

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.

### C2h

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:
• Eg and Og, 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.

### D2h

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 C2v, though there are now 8 possibilities, E+g, E+u, E-g, E-u, O+g, O+u, O-g and O-u. 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 C2v, D2 or D2h 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.