Pseudo C_2v (or D_2h) Symmetry

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 C₂ rotation about any of the principal axes. The resulting group has the same form as the D₂ point group, though it is often known as the V group in this context. For molecules with point groups C_2v D₂ 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 C₂ 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 C₁, 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.

C₁

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:

Cs			C2v		E	σ(x'y')
	Sym			Sym		Rπ(z')	Rπ(y')	Rπ(x')
A'	0		A1	0	1	1	1	1
A'	0		A2	1	1	1	-1	-1
A"	1		B1	2	1	-1	1	-1
A"	1		B2	3	1	-1	-1	1

and C2xAxis will control the labels used for the split state

C₂

In this case turning PseudoC2v on splits each irreducible representation into two:

C2			C2v		E	C2(z')
	Sym			Sym		Rπ(z')	Rπ(y')	Rπ(x')
A	0		A1	0	1	1	1	1
A	0		A2	1	1	1	-1	-1
B	1		B1	2	1	-1	1	-1
B	1		B2	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_2h

For this case PseudoC2v will actually force the use of D_2h symmetry, with each irreducible representation split into two:

C2h			D2h		E	C2(z')			i
	Sym			Sym		Rπ(z')	Rπ(y')	Rπ(x')	E*
Ag	0		Ag	0	1	1	1	1	1
Ag	0		B1g	1	1	1	-1	-1	1
Bg	1		B2g	2	1	-1	1	-1	1
Bg	1		B3g	3	1	-1	-1	1	1
Au	2		Au	4	1	1	1	1	-1
Au	2		B1u	5	1	1	-1	-1	-1
Bu	3		B2u	6	1	-1	1	-1	-1
Bu	3		B3u	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 0₀₀ 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 C2v (C2zAxis = a, C2xAxis = c)	Rovibronic Symmetry in Cs (C2zAxis = c)
000	A1 - forbidden	A' - forbidden
101	A2 - allowed	A" - allowed
111	B1 - forbidden	A" - allowed
110	B2 - 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 ΔK_a = 0, ΔK_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 A₂ 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 C₁ molecule, where c type transitions can also be allowed.