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.
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

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 wher 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
AJa2
A/S
02000 01
BJb2 A/S 00200 01
CJc2 A/S 00020 01
BDelta1/4(J+2 + J-2) A/S 00002 01/4
DJ-J4-J2(J+1)2 A/S 40000 0-1
DJK-J2Jz2-J2(J+1)K2 A/S 20000 2-1
DK-Jz4-K4 A/S 00000 4-1
deltaJ-J2(J+2 + J-2) A
20002 0-1
deltaJJ2(J+2 + J-2) S
20002 01
deltaK-1/2[Jz2, J+2 + J-2]+ A
00002 2-1/2
deltaKJ+4 + J-4 S
00004 01
HJJ6 A/S 60000 01
HJKJ4Jz2 A/S 40000 21
HKJJ2Jz4 A/S 20000 41
HKJz6 A/S 00000 61
phiJJ4(J+2 + J-2) A/S
40002 01
phiJK1/2J2[Jz2, J+2 + J-2]+ A
20002 21/2
phiJKJ2(J+4 + J-4) S
20004 01
phiK1/2[Jz4, J+2 + J-2]+ A
00002 41/2
phiKJ+6 + J-6 S
00006 01
LJJ8 A/S 80000 01/4
LJJKJ6Jz2 A/S 60000 21
LJKJ4Jz4 A/S 40000 41
LKKJJ2Jz6 A/S 20000 61
LKJz8 A/S 00000 81
llJJ6(J+2 + J-2) A/S
60002 01
llJK1/2J4[Jz2, J+2 + J-2]+ A
40002 21/2
llJKJ4(J+4 + J-4) S
40004 01
llKJ1/2J2[Jz4, J+2 + J-2]+ A
20002 41/2
llKJJ2(J+6 + J-6) S
20006 01
llK1/2[Jz6, J+2 + J-2]+ A
00002 61/2
llK'J+8 + J-8 S
00008 01