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

[H. -J. Werner]

Dear Dr. Eisfeld,
I hope the following explains your questions about the quadrupole moments
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
 Quadrupole moment (field-independent basis, Debye-Ang):
   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