# [Wannier] hamiltonian matrix in wannier

Ivo Souza ivo_souza at ehu.es
Sat Jan 18 19:10:22 CET 2014

```Dear Liping,

> Is there anyone who can help with the following questions about
> hamiltonian matrix in wannier90? Since the dimension of position matrix
> is AA_R(num_wann,num_wann,num_kpts,3) and the dimension of hamiltonian
> matrix is ham_r(num_wann,num_wann,nrpts), I don't know what's the
> difference between num_kpts and nrpts. Thanks!

1) Actually AA_R has dimensions (num_wann,num_wann,nrpts,3), with nrpts as
the third dimension, *not* num_kpts. Indeed, all operators in the Wannier
basis (Op = H, r, etc) are defined as <0m|Op|Rn>, where R runs over the
same set of nrpts lattice vectors R, which are obtained as expalined
below.

2) Regarding the difference between nrpts (number of R-vectors) and
num_kpts (number of k-points in the uniform grid used in the nscf
ab-initio calculation):

If your ab-initio k-point grid contains num_kpts = N_1*N_2*N_3 k-points,
the Wannier functions are periodic over a supercell with dimensions
A_i=N_i*a_i, where {a_i} are the crystal-cell lattice vectors, and
i=1,2,3. The vectors {A_i} are the primitive translations of a
superlattice of vectors RR.

The optimal choice of R-vectors to be used in the Fourier sums are the
ones which lie within a Wigner-Seitz (W-S) supercell centered at the
origin: they are closer to RR=0 than to any other superlattice vectors RR.

You are right to expect that there should be exactly num_kpts such
R-vectors (i.e., nrpts = num_kpts). But there is a subtlety: sometimes you
will find R-vectors which are equidistant to RR=0 and to one or several
adjacent RR-vectors. When that happens, the R-vector in question is added
to your set, but the code also records, in the variable ndegen, the number
of W-S cells which "share" this R-vector. At the end you arrive at a set
of nrpts >= num_kpts R-vectors, such that the "sum rule"

sum_R 1/ndegen(R) = num_kpts

is satisfied, with the sum over the nrpts R-vectors. Of course, ndegen=1
for "interior" R-vectors (those which are not at W-S supercell boundary).
If all R-vectors are interior, then nrpts=num_kpts.

Suppose you have calculated the Hamiltonian matrix H_mn(R)=<0m|H|Rn> in
the Wannier basis. The Fourier transform H_mn(k) is obtained as

H_mn(k) = sum_R e^{+ik.R}*H_mn(R)/ndegen(R)

[note the weighting factor 1/ndegen(R)].

3) Your last question: "What is irvec(:,irpt) for?"

This is how the R-vectors that enter the sum_R above are stored in the
code. The index irpt runs from 1 to nrpts, and irvec(1:3,irpt) are the
integer expansion coefficients of R in the basis of the lattice vectors
{a_i}.

Best,

Ivo

>
> Best,
>
> Liping
>
> ----- Original Message -----
> From: "Liping Chen" <liping.chen at rochester.edu>
> To: wannier at quantum-espresso.org
> Sent: Thursday, January 16, 2014 2:44:29 PM
> Subject: hamiltonian matrix in wannier
>
> Dear wannier users,
>
> From the manual, I can get hamiltonian in the WF basis by setting
> HR_PLOT = true. Below are the codes to write the hamiltonian matrix in
> hamiltonian.F90
> ###################################################################
>     write(file_unit,*) header ! Date and time
>     write(file_unit,*) num_wann
>     write(file_unit,*) nrpts
>     write(file_unit,'(15I5)') (ndegen(i),i=1,nrpts)
>     do irpt=1,nrpts
>        do i=1,num_wann
>           do j=1,num_wann
>              write(file_unit,'(5I5,2F12.6)') irvec(:,irpt), j, i,&
>                   ham_r(j,i,irpt)
>           end do
>        end do
>     end do
> ###################################################################
> What's the relationship between num_kpts and nrpts? The former should be
> the total number of the kpoints, right? It seems that there are
> degeneracies for nrpts. Does it mean that the sum(ndegen(i),i=1,nrpts) =
> num_kpts? The hamiltonian matrix elements with the same degeneracy are
> the same? What is irvec(:,irpt) for?
>
> Thanks,
>
> Liping Chen
> Department of Chemistry
> University of Rochester
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>

```

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