[Pw_forum] Coordinate transorms: Crystal to Cartesian and back

[Dal Corso Andrea]

On Fri, 2009-05-22 at 01:26 -0400, Kostyantyn Borysenko wrote: 
> Hi everyone,
> 
> I'm having problems with the math behind some subroutines. I am
> talking about PH/trntnsc.f90 and PW/cryst_to_car.f90. 
> 
> 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.
> 
>  
> 
> By definition of reciprocal vectors we have the following property for
> dot-product:
> 
> (a(i),b(j))= 2*pi*delta(i,j),                (1a)
> 
> 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:
> 
>  SUM(k){at(k,i)*bg(k,j)} = delta(i,j).                (1b)
> 
> In matrix form it can be represented as
> 
> B' * A = I, or B' = inv(A),                    (2)
> 
> 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.
> 
>  
> 
> Next, it's easy to show that transformation of an arbitrary vector
> v from crystal to Cartesian coordinates is described by the matrix
> equation:
> 
> v_cart = A*v_cryst,                (3a)

I am not following all your notation, but it seems to me that you are
missing the difference between vectors that in crystal coordinates are
espressed in the basis of at, such as the position r, and vectors that
in crystal coordinates are expressed in the basis of bg such as the k
points for which Eq. 3a is not correct.

HTH

Andrea



> 
> 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:
> 
> v_cryst = B' * v_cart                (4a)
> 
>  
> 
> Obviously, for an arbitrary matrix W the transformation equations will
> be
> 
> W_cart = A' *W_cryst * A                (3b)
> 
> W_cryst = B * W_cart * B'                (4b)
> 
>  
> 
> 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:
> 
>  
> 
> call cryst_to_cart (nkh,xp,at,-1)
> 
>  
> 
> Like I said, I think the correct result can be obtained only in two
> cases:
> 
> call lint(nkh,xp,at,+1)   - Crystal -> Cartesian
> 
> call lint(nkh,xp,bg,-1)   - Cartesian -> Crystal 
> 
>  
> 
> Maybe I missed something but this seems like a pretty simple problem
> from vector algebra. Any ideas?
> 
>  
> 
>  
> 
> Best regards,
> 
>  
> 
> Kostyantyn Borysenko
> 
> Department of Electrical and Computer Engineering 
> 
> North Carolina State University
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-- 
Andrea Dal Corso                    Tel. 0039-040-3787428
SISSA, Via Beirut 2/4               Fax. 0039-040-3787528
34014 Trieste (Italy)               e-mail: dalcorso at sissa.it