Asymmetric Top Perturbation
There will be one of these for each perturbation parameter in each
state.
Settings
| Nucleus |
Index, starting from 1, of the nuclear spin involved in
perturbation; 0 (default) for those not involving a nuclear spin. |
| SymSelect |
Symmetry select |
| ScalePrev |
Scale factor with respect to preceding perturbation |
| JPower |
Twice power of N(N+1) |
| aPower |
Power of Ja |
| bPower |
Power of Jb |
| cPower |
Power of Jc |
| pPower |
Power of J+
or J-. Currently
this can't be combined with the two of the three operators above that
correspond to the x and y axes in the current
representation.
|
KSelect
|
K
that the perturbation applies to; set to "all" (default) to apply to all K. Currently requires that K= K' = Kselect
|
| SaPower |
Power of Sa |
| SbPower |
Power of Sb |
| ScPower |
Power of Sc |
| SpPower |
Power of S+- |
| NSPower |
Power of N.S |
The effective operator for rotational perturbations (i.e. where Nucleus = 0) when pPower is zero is:
pNJPower [JaaPower, [JbbPower, JccPower]+]+
where:
[An,
Bm]+ = AnBm
+ AnBm
provided that the special cases with n
or m zero are taken as:
[An,
B0]+ = An
[A0, Bm]+ = Bm
p is a
phase factor to ensure that all the matrix elements are real; it is
equal to 1 except where the overall operator involves an odd power of Jy, when it
is equal to i.
When pPower is non zero
the effective operator is:
NJPower [JzzPower, (J+pPower + J-pPower)]+
with the same definitions as above. z
will correspond to a, b or c depending on the representation.
As an example, consider the operator bPower=1, cPower=1, which PGOPHER displays as bc, This
notionally corresponds to the operator JbJc+JcJb.
In the Ir representation this
is equivalent to JxJy+JyJx so the full operator is i(JxJy+JyJx) as it contains Jy.
Expressed in terms of raising and lowering operators this -i(J+2+J-2)/2 so the
matrix elements are:
<N,K-2|bc|N,K>
= sqrt((N*(N+1)+(-K+1)*(K-2))*(N*(N+1)-K*(K-1)))/2
<N,K+2|bc|N,K>
= -sqrt((N*(N+1)+(-K-1)*(K+2))*(N*(N+1)-K*(K+1)))/2
Parameters
| Value |
Size of perturbation. |
Rotational Hamiltonian expressed as perturbations
The standard rotational Hamiltonian can be expressed entirely in terms
of diagonal perturbations, as shown in the table below. This is useful
as it allows higher powers of the centrifugal distortion terms to
be added without altering the program. zPower is not an actual
setting; it corresponds to aPower,
bPower or cPower depending on the
representatiion used.
|
Operator
|
Reduction
|
JPower
|
aPower
|
bPower
|
cPower
|
pPower
|
zPower
|
Scale
Factor
|
| A |
Ja2
|
A/S
|
0 |
2 |
0 |
0 |
0 |
0 |
1
|
| B |
Jb2 |
A/S |
0 |
0 |
2 |
0 |
0 |
0 |
1
|
| C |
Jc2 |
A/S |
0 |
0 |
0 |
2 |
0 |
0 |
1
|
| BDelta |
1/4(J+2 + J-2) |
A/S |
0 |
0 |
0 |
0 |
2 |
0 |
1/4 |
| DJ |
-J4 ≡ -J2(J+1)2 |
A/S |
4 |
0 |
0 |
0 |
0 |
0 |
-1 |
| DJK |
-J2Jz2 ≡ -J2(J+1)K2 |
A/S |
2 |
0 |
0 |
0 |
0 |
2 |
-1 |
| DK |
-Jz4≡ -K4 |
A/S |
0 |
0 |
0 |
0 |
0 |
4 |
-1 |
| deltaJ |
-J2(J+2 + J-2) |
A
|
2 |
0 |
0 |
0 |
2 |
0 |
-1 |
| deltaJ |
J2(J+2 + J-2) |
S
|
2 |
0 |
0 |
0 |
2 |
0 |
1
|
| deltaK |
-1/2[Jz2, J+2 + J-2]+ |
A
|
0 |
0 |
0 |
0 |
2 |
2 |
-1/2 |
| deltaK |
J+4 + J-4 |
S
|
0 |
0 |
0 |
0 |
4 |
0 |
1
|
| HJ |
J6 |
A/S |
6 |
0 |
0 |
0 |
0 |
0 |
1
|
| HJK |
J4Jz2 |
A/S |
4 |
0 |
0 |
0 |
0 |
2 |
1
|
| HKJ |
J2Jz4 |
A/S |
2 |
0 |
0 |
0 |
0 |
4 |
1
|
| HK |
Jz6 |
A/S |
0 |
0 |
0 |
0 |
0 |
6 |
1
|
| phiJ |
J4(J+2 + J-2) |
A/S
|
4 |
0 |
0 |
0 |
2 |
0 |
1
|
| phiJK |
1/2J2[Jz2, J+2 + J-2]+ |
A
|
2 |
0 |
0 |
0 |
2 |
2 |
1/2
|
| phiJK |
J2(J+4 + J-4) |
S
|
2 |
0 |
0 |
0 |
4 |
0 |
1
|
| phiK |
1/2[Jz4, J+2 + J-2]+ |
A
|
0 |
0 |
0 |
0 |
2 |
4 |
1/2
|
| phiK |
J+6 + J-6 |
S
|
0 |
0 |
0 |
0 |
6 |
0 |
1
|
| LJ |
J8 |
A/S |
8 |
0 |
0 |
0 |
0 |
0 |
1/4
|
| LJJK |
J6Jz2 |
A/S |
6 |
0 |
0 |
0 |
0 |
2 |
1
|
| LJK |
J4Jz4 |
A/S |
4 |
0 |
0 |
0 |
0 |
4 |
1
|
| LKKJ |
J2Jz6 |
A/S |
2 |
0 |
0 |
0 |
0 |
6 |
1
|
| LK |
Jz8
|
A/S |
0 |
0 |
0 |
0 |
0 |
8 |
1
|
| llJ |
J6(J+2 + J-2) |
A/S
|
6 |
0 |
0 |
0 |
2 |
0 |
1
|
| llJK |
1/2J4[Jz2, J+2 + J-2]+ |
A
|
4 |
0 |
0 |
0 |
2 |
2 |
1/2
|
| llJK |
J4(J+4 + J-4) |
S
|
4 |
0 |
0 |
0 |
4 |
0 |
1
|
| llKJ |
1/2J2[Jz4, J+2 + J-2]+ |
A
|
2 |
0 |
0 |
0 |
2 |
4 |
1/2
|
| llKJ |
J2(J+6 + J-6) |
S
|
2 |
0 |
0 |
0 |
6 |
0 |
1
|
| llK |
1/2[Jz6, J+2 + J-2]+ |
A
|
0 |
0 |
0 |
0 |
2 |
6 |
1/2
|
| llK' |
J+8 + J-8 |
S
|
0 |
0 |
0 |
0 |
8 |
0 |
1
|
Molecule Types Asymmetric Tops