[Pw_forum] Coordinate transorms: Crystal to Cartesian and back
Gabriele Sclauzero
sclauzer at sissa.it
Fri May 22 12:06:26 CEST 2009
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)
>
> 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)
Are you sure? If I understand correctly the meaning of "matrix in cartesian/crystal
coordinates" (which for me means that v_cart'*W_cart*v_cart = v_crys'*W_crys*v_crys),
then, following your notation
v_cart = A*v_cryst
so that
v_cart' = v_cryst' * A'
It follows
v_cart' * W_cart * v_cart = v_crys' * A' * W_cart * A* v_crys
Then
W_crys = A' * W_cart * A
I suppose that also (4b) is not correct...
GS
>
>
>
> 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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>
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--
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| Gabriele Sclauzero, PhD Student |
| c/o: SISSA & CNR-INFM Democritos, |
| via Beirut 2-4, 34014 Trieste (Italy) |
| email: sclauzer at sissa.it |
| phone: +39 040 3787 511 |
| skype: gurlonotturno |
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