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<DIV class=Section1>
<P class=MsoNormal><FONT face=Arial size=2><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US">Hi
everyone,</SPAN></FONT></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>I'm having problems with the math behind some
subroutines. I am talking about PH/trntnsc.f90 and PW/cryst_to_car.f90.
</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>These subroutines are supposed to convert vectors
(cryst_to_car.f90) or matrices (trntnsc.f90) from crystal coordinates to
Cartesian (flag=1) or vice versa (flag=-1). And it seems like a very
straightforward task - just a little bit of vector algebra. But when I
derive these transformations myself they don't check out. Here is what I've
got.</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>By definition of reciprocal vectors we have the
following property for dot-product:</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>(a(i),b(j))=
2*pi*delta(i,j),
(1a)</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>where a(i) - lattice vectors, b(j) - reciprocal lattice
vectors. In QE these vectors are stored in matrices at(i,j) and bg(i,j)
respectively. Matrix elements of at are in units a_0 and matrix elements of bg
are in units (2*pi/a_0). So the condition (1a) can be rewritten as
follows:</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p> SUM(k){at(k,i)*bg(k,j)}
= delta(i,j)<SPAN
class=201260921-21052009>.
(1b)</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>In matrix form it can be represented
as</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>B' * A = I, or B' =
inv(A),
(2)</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>where B' is a transposed matrix B and inv(A) is an
inversed matrix A, and I is the identity matrix. I checked these matrices
for the case of graphene and property (2) indeed holds
true.</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>Next, it's easy to show that transformation of an
arbitrary vector v from crystal to Cartesian coordinates is described by
the matrix equation:</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>v_cart =
A*v_cryst,
(3a)</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>where A is a matrix at(i,j) mentioned above, which
contains lattice vector components. If you multiply equation (3a) from the left
by inv(A) and use the property inv(A)=B' (see eq. (2)) you will get the
reverse transformation:</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>v_cryst = B' *
v_cart
(4a)</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>Obviously, for an arbitrary matrix W the transformation
equations will be</SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>W_cart = A' *W_cryst *
A
(3b)</SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>W_cryst = B * W_cart *
B'
(4b)</SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>But that's not what I see in the code. In case of
subroutine trntnsc.f90 matrices A and B switch places in (3b)
and (4b). And in cryst_to_car.f90 everything seems fine but only if you use
at(i,j) as a transformation matrix when flag=1 and bg(i,j) if flag=-1. However,
when it's called in subroutine lint.f90 it looks like
this:</SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>call cryst_to_cart
(nkh,xp,at,-1)</SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>Like I said, I think the correct result can be obtained
only in two cases:</SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>call lint(nkh,xp,at,+1) - Crystal ->
Cartesian</SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>call lint(nkh,xp,bg,-1) - Cartesian
-> Crystal
</SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009>Maybe I missed something but this seems like a pretty
simple problem from vector algebra. Any
ideas?</SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009><SPAN lang=EN-US
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p><SPAN
class=201260921-21052009></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN></SPAN></o:p></SPAN> </P>
<P class=MsoNormal><FONT face=Arial size=2><SPAN
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial"><o:p> </o:p></SPAN></FONT></P>
<P class=MsoNormal><A name=_MailAutoSig><FONT color=navy><SPAN lang=EN-US
style="FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: Arial; mso-ansi-language: EN-US; mso-no-proof: yes">Best
regards,<o:p></o:p></SPAN></FONT></A></P>
<P class=MsoNormal><SPAN style="mso-bookmark: _MailAutoSig"><FONT
color=navy><SPAN lang=EN-US
style="FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: Arial; mso-ansi-language: EN-US; mso-no-proof: yes"><o:p> </o:p></SPAN></FONT></SPAN></P>
<P class=MsoNormal><SPAN style="mso-bookmark: _MailAutoSig"><FONT
color=navy><SPAN lang=EN-US
style="FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: Arial; mso-ansi-language: EN-US; mso-no-proof: yes">Kostyantyn
Borysenko</SPAN></FONT></SPAN><SPAN
style="mso-bookmark: _MailAutoSig"></SPAN><FONT color=navy><SPAN lang=EN-US
style="FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: Arial; mso-ansi-language: EN-US"><o:p></o:p></SPAN></FONT></P><FONT
face="Times New Roman" size=3><SPAN style="FONT-SIZE: 12pt"><o:p><PRE><FONT face=Arial size=2>Department<SPAN class=201260921-21052009> of </SPAN>Electrical and Computer Engineering
North Carolina State University</FONT>
</PRE></o:p></SPAN></FONT></DIV></BODY></HTML>