Dear Dr. Drummond,

Thank you for your message.

Let me make some general comments first and then try to answer your
specific questions below.
Feel free to contact me directly with follow-up questions on this topic.

Hyperpolarizability calculations are technically difficult. This is
even more true for
gamma (2nd hyperpolarizability) than for beta (1st hyperpolarizability).

* if you use basis sets with many diffuse functions, as needed to get
reliable hyperpolarizabilities
   for small molecules, you should also check the effect of using the
DEPENDENCY keyword to
   remove linear dependencies in the basis set.

* because the hyperpolarizability results are so sensitive to many
numerical issues, the
   symmetry relations are not always fulfilled perfectly.

* ADF will automatically rotate a system (molecule + external field) if
it is not oriented in the way ADF wants.
   ADF wants the main symmetry axis (defined by the external field in
the case of gamma calculations)
   to be the z-axis. So, the meaning of x, y, and z might change
compared to your expectations.
   This would not happen if you would specify SYMMETRY NOSYM, but that
has its own obvious
   disadvantages. ADF prints which atomic coordinates it used.

* You should focus on the results for the individual tensor components
that ADF prints.
    The rest can be deduced from those components, by taking suitable
linear combinations,
    as explained in the literature and implemented in
$ADFHOME/adf/hyperpol/PrintBeta.d90

    subroutine PrintBeta (betatt, beta33, thresh)

!
========================================================================
=====
! purpose: to print the hyperpolarizability tensor components.
!
! input : betatt - contains all components of static beta, SHG, EOPE
! and OR tensors (the latter three in a DYNAHYP run
only)

    Furthermore, some of the printed quantities are useful only for
special cases (e.g. linear molecules
    or other high-symmetry cases). So, the important basic output is
given by the individual beta tensor
    components, possibly as a function of external electric field in the
case of gamma calculations.
    To compare to experiment, it is typically needed to combine this
with the result for the dipole moment vector.

In addition to those general comments, I will try to answer your
specific questions below.

On Apr 11, 2005, at 8:35 PM, Mike Drummond wrote:

> I am attempting to use ADF to calculate frequency-dependent
> hyperpolarizabilities. Upon the advice of the ADF manual, I have read
> some "introductory" papers, and quickly realized that, amidst the
> complicated landscape of nomenclature and notation, perhaps replicating
> someone else's results would be a nice initial step.
>
> Therefore, I have performed a Response calculation of CO2, following
> the
> same computational details of van Gisbergen et al., J. Chem. Phys.,
> 109,
> 10657 (1998). The output is, to me, somewhat cryptic, and I cannot
> figure
> out the manipulations required to transform the ADF output into the
> numbers in Table IV of the aforementioned paper.
>
> In my input file, the key lines are:
>
> EFIELD 0 0 0.001
>
> Response
> Allcomponents
> Hyperpol 0.0656
> Dynahyp
> End
>
> To simulate an electric field along the z-axis of D(inf)h CO2, and to
> calculate hyperpolarizabilities at a laser frequency of 694.3 nm
> (0.0656
> au). In the output, there are eight sections of hyperpolarizability
> data:
>
> (1) The STATIC hyperpolarizability tensor beta, followed by Non-zero
> components
>
> (2) The SHG hyperpolarizability tensor beta, followed by Non-zero
> components
>
> (3) The EOPE hyperpolarizability tensor beta, followed by Non-zero
> components
>
> (4) The OR hyperpolarizability tensor beta, followed by Non-zero
> components
>
> (5) - (8) are (1) - (4) repeated, except with output consisting of
> single
> values for quantities labelled beta_z, mu_z, beta_bar, and beta-vec
> (and beta_Kerr for the EOPE section).
>
> My questions are:
>
> [A]: What manipulations of Sections (1) through (8) above give the
> correct
> gamma (z z z z) values of Table IV? It seems that the value beta_z of
> Section 5 above divided by the electric field strength of 0.001 au
> might
> be right, in that it gives me (to two significant figures) Stan's
> values
> for BLYP and LB94, but I get 2000 for LDA instead of 1900.

Your procedure looks correct. The gamma-results are very sensitive.
The 5% difference might be explained by the use of the exact density
[a non-standard option] instead of fitted density for LDA, or perhaps a
different setting for the DEPENDENCY keyword, or another small
difference in settings.

> [B]: How do I manipulate the data of Sections (1) through (8) to get
> gamma
> (x x x x) and gamma (x x z z)? For the former, my first inclination
> would
> be to change the EFIELD line in my input file to: EFIELD 0.001 0 0, and
> then manipulate the beta (x x x) values, but no such values are
> reported.
> Do I use the beta (z z z) values with the EFIELD line which sets up a
> field polarized along the x-axis?

Your first calculation has a field along the z-axis.
You can get gamma_xxzz from beta_xxz as a function of E_z.

For gamma_xxxx you will indeed need to know beta_xxx as function of E_x.
However, applying a field in the x-direction changes the symmetry of
the molecule,
if any, which means that ADF will reorient it ...

One way around this is to impose SYMMETRY NOSYM.
Another way is not to change the direction of the external electric
field (keep it in
the z-direction), but to rotate the molecule instead.

Or just remember that everything has been rotated and realize that "x"
now means "z" ...

> [C]: What about getting gamma (||), which is also known as <gamma> (z
> z z
> z)? I know Equation 3 explains its derivation, but I must confess
> bafflement about the meaning of the indices i and j.

All the averaging procedures are indeed subtle and involve various
summations
over indices which can all be x, y, z. For this I should refer to the
literature
(e.g. reviews by Bishop and refs. therein). As I said, it's best to use
the individual
tensor components from ADF and then do the linear combinations yourself.

> [D]: What do I have to do to obtain the value EOKE? I believe it has
> something to do with Equation 9.20 from Stan van Gisbergen's thesis
> (available from the SCM website), but said equation is utterly
> inscrutable
> to me (no offense, Stan -- it's my ignorance, not your writing :). It
> seems I have to manipulate the EOPE values, but I'm not sure how.
> Also,
> does the reported value beta_Kerr get used at all?

The basics of what ADF prints are beta tensor components for
the static first hyperpolarizability and three different
frequency-dependent
1st-order hyperpolarizabilities.

One gets EOKE gamma components from EOPE beta components
in the same way as one gets the static gamma components from static
beta components.

> [E]: Finally, is there a difference between the Electro-Optical Kerr
> Effect (EOKE) terminology used in ADF and the Optical Kerr Effect (OKE)
> designation I've seen elsewhere? How do these (or does this, if
> they're
> one and the same) correspond to the phenomena termed "dc Kerr effect"
> and
> "ac Kerr effect" by Shelton and Rice in their review (Chem. Rev., 94, 3
> (1994))?

For the many different terms in use (and many different conventions,
prefactors
etc.) the review papers by David Bishop should shed some light.
This is very confusing to me as well.
Some of these references should be mentioned in thesis, e.g. Ref [251].
As other ADF and BAND related Ph.D. thesis, it can be downloaded from
the SCM website: http://www.scm.com/Doc/theses.html

I hope I have not increased the confusion with these comments ..

Best regards,
Stan van Gisbergen

>
> I apologize in advance for the neophyte (and time-consuming) questions.
>
> Thank you,
> Michael Drummond
>
> Postdoctoral Research Associate
> Computational Chemical Sciences Group
> Computer Science and Mathematics Division
> Oak Ridge National Laboratory
>
Dr. S.J.A. van Gisbergen      
Scientific Computing & Modelling NV
Theoretical Chemistry, Vrije Universiteit
De Boelelaan 1083
1081 HV Amsterdam
The Netherlands                                
vangisbergen_at_scm.com  
http://www.scm.com

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Received on 2005-04-12 12:37:54