[Wien] Augmented Plane Wave

Fri Mar 22 17:20:39 CET 2019

Dear Sam,

Thank you for your contribution; once I saw a figure in which the plane wave continues into the MT region but enlarged, which confused me.

________________________________
De: Wien <wien-bounces at zeus.theochem.tuwien.ac.at> en nombre de Sam Trickey <trickey at qtp.ufl.edu>
Enviado: viernes, 22 de marzo de 2019 08:13 a. m.
Para: wien at zeus.theochem.tuwien.ac.at
Asunto: Re: [Wien] Augmented Plane Wave

Rarely do I contribute to this list, though I benefit from reading it.

I must respectfully contradict Martin.  Professor Slater clearly intended
that the augmentation be of the plane wave.  The history, at least as I
know it, is this.

Professor Slater did not use the word "augment" or "augmented" in his
1937 paper that usually is cited as the original APW paper [Rev. 51, 846 (1937)].
So far as I am aware, the first time he used the phrase "augmented plane
wave" is in Phys. Rev. 92, 603 (1953).  The phrase appears in the title but,
more important for discerning his intent is the discussion beginning
on the bottom right of p. 603 and continuing on 604.  He introduces Herring's
orthogonalized plane waves and summarizes their applications and then says
(lines 7-9, LH column,  p. 604)  "Relatively few such orthogonalized or
augmented plane waves suffice to give a rather good approximate wave
function."  Noting the problem of non-existent orthogonalization of a 2p-like
state to a deeper core state found by Frank Herman, Prof. Slater goes on to
say "Herman has suggested that in such a case we could augment the plane
wave ...".  He opens the next paragraph with "The present method may be regarded
as a straight-forward procedure for augmenting a plane wave by adding to it
a contribution near each nucleus ...".

It may be useful to add that I was part of Prof. Slater's group within QTP for the
last 7-1/2 years of his life and one year was assigned (even though I was an
Asst. Prof. not a post-doc!) to be his teaching assistant in his "Quantum Theory
of Matter" course.  Those experiences were consistent with the sentences just quoted.

Peace, Sam

On 3/22/19 7:04 AM, pieper wrote:
Dear Pablo,

I suspect your problem occurs because you left out the word which "Augmented" refers too: It is an "Augmented Plane Wave Method",
that is, the Method is augmented (including additional basis functions), not the plane waves (in amplitude or intensity).

Best regards,

Martin Pieper

---
Dr. Martin Pieper
Karl-Franzens University
Institute of Physics
Universitätsplatz 5
A-8010 Graz
Austria
Tel.: +43-(0)316-380-8564

Am 2019-03-22 02:49, schrieb delamora:
Dear Wien users,
 I have a question about the name of

 "Augmented Plane Wave"
 I had the idea that when the wave enters the Muffin Tin sphere the
amplitude of the wave increased.
 Trying to see this I found that when a wave crosses a step function,
 https://urldefense.proofpoint.com/v2/url?u=https-3A__quantummechanics.ucsd.edu_ph130a_130-5Fnotes_node149.html&d=DwIGaQ&c=sJ6xIWYx-zLMB3EPkvcnVg&r=3u4OCHnAtdypMfYN_K57COVk2XByDoKwnLuc_zRENzE&m=x11-msafgQ72gWZKP5gGcbCizsHiCDlxxznut2Zo-ug&s=CV5sXDCtZrDlO2GFM8sW4hec5wD0wT0O34PcltW3ZfI&e=

 When the incoming wave

 exp(ikx)

 reaches an upwards step function there is a reflected wave
 R exp(-ikx)

 and a transmitted wave
 T exp(ik'x)

 what this article shows is;
 1 + R = T

 That is, the amplitudes of the incoming wave and the reflected wave
add to the amplitude of the transmitted wave

 If I take this into a square well then I would understand that the
waves inside the well have the total amplitude equal to the incoming
and transmitted wave. That is, when the wave enters the Muffin Tin the
amplitude of wave is not AUGMENTED. So why is this method called
"Augmented Plane Wave"?

 Saludos

 Pablo
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Samuel B. Trickey
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