Molecule Types Asymmetric Tops  <Prev Next> 
Currently the N_{±} and S_{±} operators can only be combined with the a/b/c style operators for the axis corresponding to the z axis so, for example, in the Ir representation N_{±} can be combined with N_{a }or S_{a} but not N_{b }or S_{b}.
The effective operator for rotational perturbations (i.e. where Nucleus = 0) when pPower is zero is:N^{JPower} [N_{z}^{zPower}, (N_{+}^{pPower} + N_{}^{pPower})]_{+}_{ } with the same definitions as above. z will correspond to a, b or c depending on the representation.
[S_{p}, N.S^{NSPower}S^{SPower/2}]_{+}or just S_{p} or N.S^{NSPower}S^{SPower/2} if only one is present. Currently the N.S^{NSPower}S^{SPower/2} operator is the only spin operator that can be combined with another spin operator.
If both rotation and spin operators are present, the most general form is:
N^{JPower/2}N_{z}^{zPower}(N_{+}^{pPower} + N_{}^{pPower}) S_{p} N.S^{NS}^{Power}S^{SPower/2} + S^{SPower/2}N.S^{NS}^{Power} S_{p}(N_{+}^{pPower} + N_{}^{pPower})N_{z}^{zPower}N^{JPower/2}
or
N^{JPower/2}N_{z}^{zPower}[N_{x}^{xPower}, N_{y}^{yPower}]_{+} S_{p} N.S^{NS}^{Power}S^{SPower/2} + S^{SPower/2}N.S^{NS}^{Power} S_{p} [N_{x}^{xPower}, N_{y}^{yPower}]_{+}_{ }N_{z}^{zPower}N^{JPower/2}
As an example, consider the operator bPower = 1, cPower = 1, which PGOPHER displays as bc. This notionally corresponds to the operator N_{b}N_{c} + N_{c}N_{b}. In the Ir representation this is equivalent to N_{x}N_{y} + N_{y}N_{x}. Expressed in terms of raising and lowering operators this is i(N_{+}^{2} N_{}^{2})/2 so the matrix elements are:
<N,K2bcN,K> = i sqrt((N*(N+1)+(K+1)*(K2))*(N*(N+1)K*(K1)))/2Note that, for this matrix element, or any other matrix element that is imaginary, allowing the matrix elements to be complex is required to ensure that the overall Hamiltonian matrix is Hermitian. If AllowComplex is false then the matrix elements used are simply the imaginary matrix elements multiplied by i, which can result in a Hamiltonian matrix that is not symmetric. In this case <N,K2bcN,K> = sqrt(x) and <N,K+2bcN,K> = sqrt(x), but these two matrix elements should have the same sign.
<N,K+2bcN,K> = i sqrt((N*(N+1)+(K1)*(K+2))*(N*(N+1)K*(K+1)))/2
Value  Size of perturbation; the expressions and operators given here should be understood to be multiplied by this. 
Operator 
Reduction 
JPower 
aPower 
bPower 
cPower 
pPower 
zPower 
Scale Factor 

A  N_{a}^{2} 
A/S 
0  2  0  0  0  0  1 
B  N_{b}^{2}  A/S  0  0  2  0  0  0  1 
C  N_{c}^{2}  A/S  0  0  0  2  0  0  1 
BDelta  N_{+}^{2} + N_{}^{2}  A/S  0  0  0  0  2  0  ¼ 
DJ  N^{4} ≡ N^{2}(N+1)^{2}  A/S  4  0  0  0  0  0  1 
DJK  N^{2}N_{z}^{2} ≡ N(N+1)K^{2}  A/S  2  0  0  0  0  2  1 
DK  N_{z}^{4}≡ K^{4}  A/S  0  0  0  0  0  4  1 
deltaJ  N^{2}(N_{+}^{2} + N_{}^{2})  A 
2  0  0  0  2  0  1 
N^{2}(N_{+}^{2} + N_{}^{2})  S 
2  0  0  0  2  0  1 

deltaK  [N_{z}^{2}, N_{+}^{2} + N_{}^{2}]_{+}  A 
0  0  0  0  2  2  ½ 
N_{+}^{4} + N_{}^{4}  S 
0  0  0  0  4  0  1 

HJ  N^{6}  A/S  6  0  0  0  0  0  1 
HJK  N^{4}N_{z}^{2}  A/S  4  0  0  0  0  2  1 
HKJ  N^{2}N_{z}^{4}  A/S  2  0  0  0  0  4  1 
HK  N_{z}^{6}  A/S  0  0  0  0  0  6  1 
phiJ  N^{4}(N_{+}^{2} + N_{}^{2})  A/S 
4  0  0  0  2  0  1 
phiJK  N^{2}[N_{z}^{2}, N_{+}^{2} + N_{}^{2}]_{+}  A 
2  0  0  0  2  2  ½ 
N^{2}(N_{+}^{4} + N_{}^{4})  S 
2  0  0  0  4  0  1 

phiK  [N_{z}^{4}, N_{+}^{2} + N_{}^{2}]_{+}  A 
0  0  0  0  2  4  ½ 
N_{+}^{6} + N_{}^{6}  S 
0  0  0  0  6  0  1 

LJ  N^{8}  A/S  8  0  0  0  0  0  ¼ 
LJJK  N^{6}N_{z}^{2}  A/S  6  0  0  0  0  2  1 
LJK  N^{4}N_{z}^{4}  A/S  4  0  0  0  0  4  1 
LKKJ  N^{2}N_{z}^{6}  A/S  2  0  0  0  0  6  1 
LK  N_{z}^{8}  A/S  0  0  0  0  0  8  1 
llJ  N^{6}(N_{+}^{2} + N_{}^{2})  A/S 
6  0  0  0  2  0  1 
llJK  N^{4}[N_{z}^{2}, N_{+}^{2} + N_{}^{2}]_{+}  A 
4  0  0  0  2  2  ½ 
N^{4}(N_{+}^{4} + N_{}^{4})  S 
4  0  0  0  4  0  1 

llKJ  N^{2}[N_{z}^{4}, N_{+}^{2} + N_{}^{2}]_{+}  A 
2  0  0  0  2  4  ½ 
N^{2}(N_{+}^{6} + N_{}^{6})  S 
2  0  0  0  6  0  1 

llK  [N_{z}^{6}, N_{+}^{2} + N_{}^{2}]_{+}  A 
0  0  0  0  2  6  ½ 
N_{+}^{8} + N_{}^{8}  S 
0  0  0  0  8  0  1 
Operator 
aPower 
bPower 
cPower 
SaPower 
SbPower 
ScPower 
Scale Factor 
Notes 

eaa  N_{a}S_{a} + S_{a}N_{a} =2N_{a}S_{a} 
1 
0 
0 
1 
0 
0 
½  = 2N_{a}S_{a} = 2S_{a}N_{a} as N_{a}
and S_{a} commute 
ebb  N_{b}S_{b} + S_{b}N_{b
}
=2N_{b}S_{b}

0 
1 
0 
0 
1 
0 
½  = 2N_{b}S_{b} = 2S_{b}N_{b} as N_{b} and S_{b} commute 
ecc  N_{b}S_{b} + S_{b}N_{b
}
=2N_{b}S_{b}

0 
0 
1 
0 
0 
1 
½  = 2N_{c}S_{c} = 2S_{c}N_{c} as N_{c} and S_{c} commute 
eabbar 
N_{a}S_{b} + S_{b}N_{a}  1 
0 
0 
0 
1 
0 
½  Two
perturbations required 
N_{b}S_{a} + S_{a}N_{b}  0 
1 
0 
1 
0 
0 
½  
eacbar 
N_{a}S_{c} + S_{c}N_{a}  1 
0 
0 
0 
0 
1 
½  Two
perturbations required 
N_{c}S_{a} + S_{a}N_{c}  0 
0 
1 
1 
0 
0 
½  
ebcbar 
N_{b}S_{c} + S_{c}N_{b}  0 
1 
0 
0 
0 
1 
½  Two
perturbations required 
N_{c}S_{b} + S_{b}N_{c}  0 
0 
1 
0 
1 
0 
½ 
Operator 
Reduction 
JPower 
pPower 
zPower 
SpPower 
SzPower 
NSPower 
Scale Factor 

DsN 
N^{2}N.S 
A/S 
2 
0 
0 
0 
0 
1 
1 
DsNK 
[N^{2}, N_{z}S_{z}]_{+}  A/S  2 
0 
1 
0 
1 
0 
½ 
DsKN 
N.S N_{z}^{2}  A/S  0 
0 
2 
0 
0 
1 
1 
DsK 
[N_{z}^{3}, S_{z}] = 2N_{z}^{3}S_{z}  A/S  0 
0 
3 
0 
1 
0 
½ 
ds1 
N.S(N_{+}^{2} + N_{}^{2})  A/S  0 
2 
0 
0 
0 
1 
1 
ds2 
[N_{+}^{2} + N_{}^{2}, N_{z}S_{z}]_{+}  A 
0 
2 
1 
0 
1 
0 
½ 
[N_{+}^{3}, S_{+}]_{+}^{} + [N_{}^{3}, S_{}]_{+} = 2(N_{+}^{3}S_{+}+N_{}^{3}S_{}) 
S 
0 
3 
0 
1 
0 
0 
½ 
Operator 
SxPower 
SyPower 
SzPower 
SpPower 
Scale Factor 

alpha 
S_{x}^{2} 
2 
0 
0 
0 
1  3S_{z}^{2}S^{2} = 2S_{z}^{2} S_{x}^{2}S_{y}^{2} 
S_{y}^{2} 
0 
2 
0 
0 
1  
S_{z}^{2} 
0 
0 
2 
0 
2 

beta 
S_{+}^{2} + S_{}^{2} = 2(S_{x}^{2}S_{y}^{2}) 
0 
0 
0 
2 
½ 