# [molpro-user] multipole moments

cong.wang cong.wang at helsinki.fi
Tue Mar 9 12:44:09 GMT 2010

Dear Yulia,

Hello,

>Else: for the CH4 molecule the multipole moment of the 5th order is equal
to zero. here it is not zero. The hf values of >octupole and hexadecapole
moments are not correct at all (~5 times for hexa and ~3 times for oct mom).

I am not sure which data were compared for the octupole and  hexadecapole
moments. Seems experimental groups assigned quite different values of
octupole moments for methane. e.g.

http://zon8.physd.amu.edu.pl/historia/kielich-publ/048.pdf
5-6 * 10^{-34} esu, cm^3

2.12 * 10^{-34} esu, cm^3

This could produce a factor of 2-3 difference. (I didn't check the
experimental detail, temperature, whether adopted the Buckingham convention
for prefactor etc....)

If we stick to computational vaules, e.g. G. Maroulis Chem. Phys. Lett. 226,
420-426 (1994)
\Omega 2.4601
\Phi -7.984 (in. a.u.)

and your output is (Molpro seems print out the non-zero expectation values
of the Cartesian product, except "qm")
!RHF expec           <1.1|XYZ|1.1>      .989606160302

The agreement is quite good . Since the CPL article uses the Buckingham
convention, \Omega_{ijk} = \sum_a e_a 5/2 a_i a_j a_k - 1/2 a^2 (a_i
\delta_{jk} + a_j \delta_{ik} + a_k \delta_{ij} ). For \Omega_{xyz} there is
a prefactor of 5/2. By 5/2*   .989606160302 =2.474 it is quite close to the
CPL value.

For the hexadecapole moment of Td symmetry
\Phi_{xxxx} =  \Phi_{yyyy} =\Phi_{zzzz} = z^4 - 3(x^2 +y^2) z^2 + 3/8 (x^4 +
2 x^2y^2 + y^4)  from Stone's the theory of intermolecular forces, P228
=7/4 z^4 - 21/4 x^2 y^2

!RHF expec          <1.1|XXXX|1.1>   -44.391072212534
!RHF expec          <1.1|XXYY|1.1>   -13.290016794368
!RHF expec          <1.1|XXZZ|1.1>   -13.290016817165
!RHF expec          <1.1|YYYY|1.1>   -44.391072212538
!RHF expec          <1.1|YYZZ|1.1>   -13.290016817164
!RHF expec          <1.1|ZZZZ|1.1>   -44.391072185833

=7/4 *( -44.391072185833) - 21/4 * (-13.290016794368)
=-7.91179

also agrees with the CPL value.

The 32th- pole of Td point group symmetry is zero only for the spherical
harmonics components, as expanded by the irreducible representations, does
not contain A_1; but not for any Cartesian product. e.g., for Td, <xx> !=0,
<3xx - r^2>=0