[molpro-user] Calculation of C6 and C9 dispersion coefficients in propag_util.F and new_propag_util.F

[Tatiana Korona]

Dear Paul,

The "strange" factors you cite come from numerical quadrature I used.

Best wishes,

Tatiana

On Fri, 23 Dec 2016, Paul Jerabek wrote:

>
> Dear Molpro community,
>
> I'm interested in calculation non-additive dispersion coefficients for
> homo-nuclear, closed-shell three-body systems. They can be calculated
> from the multipole polarizabilities:
>
> Z_{l_1, l_2, l_3} = (1 / \pi) * int_0^{\infty} (alpha^{l_1}(iw)
> alpha^{l_1}(iw) alpha^{l_1}(iw))
>
> [ Equation (13) in https://arxiv.org/abs/physics/0505131 ]
>
> To my understanding, one requires the dynamic multipole polarizabilities
> for that and needs to integrate over them.
>
> Molpro provides such an functionality via
>
> {CCSD;  CPROP,PROPAGATOR=1,EOMPROP=1, PROP_ORDER=3, HIGHW=0, DISPCOEF=12}
>
> which I basically understand: Calculation of polarizabilities for
> different imaginary frequencies and then Laguerre-Gauss summation with
> certain weights.
>
> Molpro even calculates the C_6 and C_9 coefficients (which are the
> two-body and three-body dipole-dipole and dipole-dipole-dipole
> interactions, respectively) and I thought it would be a good idea to try
> to reproduce those numbers and then go to higher interactions (like e.g.
> dipole-dipole-quadrupole).
>
> For my attached example of Argon, the Molpro results
>
> C_6 = 66.0
> C_9 = 535.9
>
> are in the right ball-park compared to literature:
>
> C_6 =  61.833-64.543  [Ref: Table IV in
> http://aip.scitation.org/doi/abs/10.1063]/1.463012]
> C_9 =  484.2-521.7  [Ref: Table V (Z_{111} * 3 because of the
> definition) http://aip.scitation.org/doi/abs/10.1063/1.463012]
>
>
> But here the problems start: I don't really understand why the C_6 and
> C_9 coefficients are calculated the way they are!
>
> By looking into the source of the modules
>
> src/eom/propag_util.F
> src/eom/new_propag_util.F
>
> I could deduce that
>
> C_6 = 2/3 * w0 * 1/PI * Sum[ (alpha_xx+alpha_yy+alpha_zz)^2 *
> weighting_factor * 1/((1+x)^2) ]
>
> C_9 = 2/9 * w0 * 1/PI * Sum[ (alpha_xx+alpha_yy+alpha_zz)^3 *
> weighting_factor * 1/((1+x)^2) ]
>
> (with w0=0.3 <- took my a while to find this!)
>
> and then I could reproduce the numbers.
>
>
> But how does this relate to the usual definitions? E.g., for C_6 it
> should be something like
>
> C_6 = 3/PI * int_0^{\infty} (alpha^{a}(iw) alpha^{b}(iw))
>
> [Eq. (26) in http://pubs.acs.org/doi/abs/10.1021/acs.chemrev.5b00533]
>
>
> I understand that there is some averaging over the components of the
> polarizabilities involved, but I cannot figure out what's going on in
> these equation used in the Molpro module.
>
> Help would be really appreciated!
>
> Thanks in advance and best wishes,
>
> 	Paul
>
> -- 
> Dr. Paul Jerabek
> Schwerdtfeger Group, Centre for Theoretical Chemistry and Physics
> Bob Tindall Bldg., NZ Institute for Advanced Study
> Massey University (Albany Campus)
> Private Bag 102904
> North Shore MSC, Auckland
> New Zealand
> Phone +64 9 414 0800 ext. 41698
>
>
> -- 
> Dr. Paul Jerabek
> Fachbereich Chemie, AK Frenking, Theoretische Chemie
> Philipps-Universität Marburg
> Hans-Meerwein-Strasse
> D-35032 Marburg
> Germany
>
> Phone: +49-(0)6421-28-27001

Dr. Tatiana Korona http://tiger.chem.uw.edu.pl/staff/tania/index.html
Quantum Chemistry Laboratory, University of Warsaw, Pasteura 1, PL-02-093 Warsaw, POLAND