# [molpro-user] spacial extent <R**2>

H. -J. Werner werner at theochem.uni-stuttgart.de
Sat Jul 2 10:55:19 BST 2005

Dear Dr. Eisfeld,
in Molpro.
Best regards
Joachim Werner

1.) Definition of second moments: The electronic contributions are negative,
i.e., the operators are

\hat xx = -\sum_i x_i^2 + \sum_N Z_N X_N^2
\hat yy = -\sum_i y_i^2 + \sum_N Z_N Y_N^2
\hat zz = -\sum_i z_i^2 + \sum_N Z_N Z_N^2

where i runs over electrons, N over nuclei. This explains why the printed

rr=xx+yy+zz

is negative.

2.) Definition of traceless quadrupole moments in Molpro (according to Buckingham):

qmxx=0.5*(3*xx-rr)
qmyy=0.5*(3*yy-rr)
qmzz=0.5*(3*zz-rr)
qmxy=1.5*xy etc.

Aparently, in Gaussian they are smaller by a factor of 3/2, i.e.

qmxx=xx-rr/3
qmyy=yy-rr/3
qmzz=zz-rr/3
qmxy=xy etc.

3.) The second moments and quadrupole moments are origin dependent.
For instance, moving the molecule along the z-axis changes the
second moments as:

szz(2)=szz(1)+2*(z(2)-z(1))*dipz

The Gaussian geometry in your test is relative to the center of nuclear charge, while
the default Molpro one is relative to the center of nuclear mass. The CHARGE keyword
can be used in the Molpro geometry input to use the center of charge (but for spectroscopy
and dynamics one usually needs the moments relative to the center of mass). It also
matters for the quadrupole moments which masses are used for computing the center
of mass. By default, Molpro uses average isotope masses from the from CRC Handbook
of Chemistry and Physics.

4.) Comparison with Gaussian for H2O, (R=1.0 ANG, THETA=104.5 Degree)

Gaussian results:
Dipole moment (field-independent basis, Debye):
X=     0.0000    Y=     0.0000    Z=    -2.0955  Tot=     2.0955
XX=    -7.1302   YY=    -4.0304   ZZ=    -5.8997

Molpro results in atomic units:

ORIG                           Z(O)         DIPZ          SXX           SYY           SZZ
center of mass              -0.12945776   0.82442133   -5.30114009   -2.99652392   -4.21824164
center of charge            -0.23138458   0.82442133   -5.30114010   -2.99652392   -4.38630293

Molpro results in Ang, Debye, Ang*Debye:

ORIG                           Z(O)         DIPZ          SXX           SYY           SZZ
center of mass              -0.06850610   2.09533278   -7.12974882   -4.03016380   -5.67330853
center of charge            -0.12244346   2.09533277   -7.12974882   -4.03016380   -5.89934196

Conversion factors in Molpro:
TOANG   = 0.52917725
TODEBYE = 2.54158000

Conversion factor for second moments or quadrupole moments: toang*todebye=1.34494631

Note that in this example the (arbitrary) orientation of the Molecule with reflection
of the Z-coordinates is opposite in Gaussian and Molpro, which changes the sign of the
dipole moment.

The small deviations between Gaussian and Molpro results may be due to different conversion
factors or less accurate HF convergence in Gaussian. For accurate comparisons between different
programs, only atomic units should be used for the geometry and results.

--
Prof. Hans-Joachim Werner
Institute for Theoretical Chemistry
University of Stuttgart
Pfaffenwaldring 55
D-70569 Stuttgart, Germany
Tel.: (0049) 711 / 685 4400
Fax.: (0049) 711 / 685 4442
e-mail: werner at theochem.uni-stuttgart.de