[CPMD-list] Fd and LR in vibrational analysis

[Balasubramanian Sundaram]

Hi Yong,
    I am not sure what exactly is implemented in the CPMD code. However, the 
expression that Axel has written (V(d)+V(-d)-2V(0))/d^2 is for obtaining the 
diagonal terms of the Hessian matrix from the potential/function. The off-diagonal 
terms can also be found in a similar fashion. The expression is Equation 25.3.26 
of Abramovitz and Stegun (Handbook of Mathematical Functions). These expressions 
give the second derivatives, directly from the function/potential (without use of 
first derivative).

     If I understood Juerg's mail correctly, this is not how the second 
derivatives are calculated in any case, in CPMD. The first derivative is 
calculated analytically, and the second derivative is calculated numerically from 
the first derivative, using Equations 25.3.22 and others in Abramovitz+Stegun.

Cheers,
Bala


Yong Zhang wrote:
> 
> 
> On 11/28/06, *Axel Kohlmeyer* <akohlmey at cmm.chem.upenn.edu 
> <mailto:akohlmey at cmm.chem.upenn.edu>> wrote:
> 
>     On 11/28/06, Yong Zhang <zhangymall11 at gmail.com
>     <mailto:zhangymall11 at gmail.com>> wrote:
>      >
>      > I am a little confused.
> 
>     me too, just to make sure we are on the same page. FD stands for
>     finite differences.
>     if you compute the energy at point x and x+d and x-d you can
>     approximate the
>     _second_ derivative of the energy in the direction of d by putting a
>     parabola through
>     those points and thus you _can_ get the second derivative with
>     respect to
>     the displacement (in harmonic approximation, that is).
> 
> 
>  
> 
> Thanks Axel.
> In this case, the second derivative can be calculated by something like 
> (V(+d)+V(-d)-2V(0))/d**2. But that is only the one variable case, or say 
> the diagonal terms in the Hessian matrix. How about the off-diagonal terms?
>  
> 
>     in linear response you just compute the linear response to a
>     displacement
>     operator (without a real displacement and thus you avoid half
>     the calculations as the response is symmetric).
>     as a result you also get the second derivative in harmonic
>     approximation.
> 
> 
> For a molecule of 12 atoms, I actually found that FD is faster than LR.  
> Is this reasonable?
> 
>      > In vibrational analysis, FD is used for first derivative, then
>     how is the
>      > second derivative calculated for a job "VIBRATIONAL ANALYSIS FD"
>     ? FD or LR?
> 
>     for more details, please have a look at the code. secder.F and secdpt.F.
> 
>     note, that there is a second, more generic linear response kernel
>     implemented
>     into CPMD (job type LINEAR RESPONSE) that also allows to do a
>     vibrational
>     analysis via linear response via the PHONON keyword.
> 
>     axel.
> 
>      >
>      > Best wishes,
>      >
>      >
>      >
>      >
>      > On 11/28/06, Juerg Hutter < hutter at pci.unizh.ch
>     <mailto:hutter at pci.unizh.ch>> wrote:
>      > > Hi
>      > > > Thanks for the reply. I found the paper, which discussed the
>     LR method.
>      > > > For the FD method in CPMD, I guess the first order derivative is
>      > calculated
>      > > > from finite difference method as indicated in the output
>     file. Is the
>      > second
>      > > > order derivative calculated from those first order derivative
>     simply
>      > using
>      > > > finite difference method one more time? If not, how is the
>     second order
>      > > > derivative calculated?
>      > > > Thanks a lot
>      > > >
>      > >
>      > > You are one off. All first derivatives in CPMD are calculated
>     using
>      > > analytic formulas. Linear Response (LR) is needed to calculate
>      > > second order derivatives (e.g. dynamical matrix). The finite
>      > > difference (FD) method therefore is used on gradients (=first
>      > > derivatives) of the energy in the vibrational analysis.
>      > >
>      > > regards
>      > >
>      > > Juerg Hutter
>      > >
>      > >
>      > >
>      > > > Yong
>      > > >
>      > > >
>      > > > On 11/23/06, Juerg Hutter <hutter at pci.unizh.ch
>     <mailto:hutter at pci.unizh.ch> > wrote:
>      > > >>
>      > > >> Hi
>      > > >>
>      > > >> >
>      > > >> > I am just wondering where I can get detailed description
>     of the FD
>      > and
>      > > >> LR
>      > > >> > methods used for vibrational analysis in CPMD, for
>     example, the
>      > original
>      > > >> > references. Thanks a lot.
>      > > >> >
>      > > >>
>      > > >> See Reference 94 in the manual
>      > > >>
>      > > >> \bibitem{apdsmp} A. Putrino, D. Sebastiani and M. Parrinello,
>      > > >> J. Chem. Phys. {\bf 113}, 7102 (2000)
>      > > >>
>      > > >>
>      > > >> regards
>      > > >>
>      > > >> Juerg Hutter
>      > > >>
>      > > >
>      > >
>      >
>      >
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> 
>     --
>     =======================================================================
>     Axel Kohlmeyer   akohlmey at cmm.chem.upenn.edu
>     <mailto:akohlmey at cmm.chem.upenn.edu>   http://www.cmm.upenn.edu
>       Center for Molecular Modeling   --   University of Pennsylvania
>     Department of Chemistry, 231 S.34th Street, Philadelphia, PA 19104-6323
>     tel: 1-215-898-1582,  fax: 1-215-573-6233,  office-tel: 1-215-898-5425
>     =======================================================================
>     If you make something idiot-proof, the universe creates a better idiot.
> 
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