[Pw_forum] Answers and Questions

Fernando A Reboredo reboredofa at ornl.gov
Thu Aug 31 18:35:10 CEST 2006

You convinced me.  I will try my best.

The energy of a set of atoms near the minimum can be frequently approximated 
as second order expansion in the atomic displacements with respect to the 
minimum. Within this approximation, the classical evolution in time of the 
atomic positions leads to a set of coupled second order equations in the 
displacements that can be decoupled by a unitary transformation (canonical) 
acting on the vector of displacements (see Ashcrof/Mermin Solid State 
Physics Chapter 22). After this transformation [that digitalizes the 
dynamical matrix in Eq (22.54)] one obtains a set of decoupled equations 
equivalent to independent harmonic oscillators, which are called normal 
modes, in general, and phonons in a solid state context.

The classical evolution in time of an arbitrary set of small displacement 
{u(r,t)} can be decomposed as a superposition of the evolution of the normal 
modes (as explained in the same text) where the coefficients in the 
expansion are chosen to satisfy the initial conditions of the ions.

The differential equation of the harmonic oscillator
x''(t) = -\omega^2 x(t) being a second order equation has two independent 
solutions of the form A \exp( I \omega t) + B (I - \omega t) where t is the 
time I is \sqrt(-1) and \omega is associated  2 \pi times the oscillator 
frequency. A and B are arbitrary constants set to satisfy the initial 
conditions (position and velocity at time zero).

Prove that a change in A  by and arbitrary complex number of modulus 1 is 
equivalent to change the origin of  t.

Prove that if the eigenvectors of the dynamical matrix  are multiplied by an 
arbitrary complex number of modulus one, a) the result is also a normalized 
eigenvector of the same dynamical matrix b) the coefficients of the 
expansion describing a given atomic motion will change only by  a 
multiplicative factor equal to the complex conjugate of the that number.

Prove that if two normal modes have the same frequency (which is often the 
case of a system with symmetries), then an arbitrary linear combination of 
these modes (with arbitrary complex coefficients) is also a normal mode.

Prove that the same is valid in the quantum mechanical case.

Fernando A. Reboredo

----- Original Message ----- 
From: "Axel Kohlmeyer" <akohlmey at cmm.upenn.edu>
To: <pw_forum at pwscf.org>
Sent: Thursday, August 31, 2006 10:33 AM
Subject: Re: [Pw_forum] Answers and Questions

> On 8/31/06, Fernando A Reboredo <reboredofa at ornl.gov> wrote:
>> Dear All,
> dear fernando,
> thanks for sharing your thoughts with us.
>> I feel compelled to write because I am always amazed by the degree of
>> patience that the developers of these ESPRESO package show towards the
>> questions in this mailing list. I also confess that I feel embarrassed to
>> ask my own questions since I know that I will be stealing research time 
>> of
>> colleagues.  I thank the developers for making this package available.
> you may be 'stealing' some research time, but you also have to realize
> that even trivial beginner's questions occasionally point out flaws in 
> either
> the implementation or the documentation. furthermore, even if you may
> not be the most qualified person to respond, you may still have 
> encountered
> a similar problem, so you would actually be giving back some of the 
> 'stolen'
> time by responding and reporting your own experiences. this is how 
> communities
> do work: if everybody contributes, everybody benefits.
>> However, there are two types of questions 1) the ones that refer to the
>> inner reason of the error messages (that I am not qualified to answer) 2)
> well, in my personal experience, trying to figure out what error messages
> in a code mean (by looking at the sources), has helped me a lot to 
> understand
> how program packages work and trying to figure out whether this is a
> legitimate problem, a bug in the code or just a flaw in the input file,
> is a good training to avoid problems for the next input you may need to
> write and may make you more qualified.
>> the ones that refer to a chapter of "Introduction to Solid State Theory" 
>> or
>> "Quantum Mechanics". Since I feel I am not better teacher that Aschoft or
>> Kohen I also remain quiet there.
> see above. even when you may not be the better teacher right now,
> responding to the best of your knowledge may actually _make_ you
> a better teacher. a lot of it is just a matter of practice. you can always
> start small and only provide answers to problems where you feel confident
> and refer to the literature for the rest.
> on top of that, a good way to contribute (and that applies to everybody)
> would be to collect frequently asked questions and their answers from the
> mailing list archives and integrate them into the quantum-espresso wiki
> pages. i've done something similar for a different project a couple of 
> years
> back and it was _extremely_ helpful in getting a better understanding,
> while at the same time the works was more that of an editor, i.e., you
> didn't have to be an expert, but just take the available answers and edit
> them into one more consistent text. this is somewhat time consuming,
> but given the large number of people in this forum, it should not be
> so much if this is shared amongst them. as i wrote before: everybody
> contributes, everybody benefits.
>> I interrupt my silence to suggest to askers to think whether a question 
>> is
>> type 1 or 2.
> part of the problem of a beginner in using tools like quantum espresso
> is, that you frequently cannot tell, where this problem originates from.
> many people answering here realize this fact (everybody has been through
> that in some way at some point in time) and are willing to give people
> some leeway at the beginning (there is no real gain from being rude
> over e-mail regardless) and only get increasingly irritated when people
> start taking advantage of that.
> ok. i guess this is enough 'preaching' for me for today. please everybody
> give this some thought and (hopefully) help us to make the QE project
> even better and even more fun than it is already right now.
> thanks for reading and ciao,
>   axel.
>> Thanks again for the hard work and patience.
>> I am Fernando A. Reboredo ORNL (and I approve this message)
> -- 
> =======================================================================
> Axel Kohlmeyer   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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