hydrogen energy levels

[sand7000]

I attempted to calculate the hydrogen energy levels, by plotting the bands of a lone hydrogen atom in a large box. I expected the lowest band to lie at -13.6eV but it was actually at around -6eV. The band is flat, as it should be, so I believe it is well converged. Anyone have any suggestions for what I am doing wrong?

I am using the Troullier-Martins LDA psuedopotential and my input file is as follows:

Code: Select all

#########
#hydrogen
#########
chkprim 0     # the unit cell is not primitive so do not check

ndtset 2

##########################################################################
#Spin Info taken from .../tutorial/Input/t15.in.
##########################################################################
nsppol1 2          # Allow a spin-polarized calculation
occopt1 2          # Allow occupation numbers to be set by hand
nband1  1 1        # Number of bands for spin up and spin down
occ1    1.0  0.0   # Occupation numbers for spin up state and spin down state.
                  # The spin up state is occupied, not the spin down state.
spinat1 0.0 0.0 1.0  # Gives an initial estimation of the spin on the atom (easy)

############################
#Definition of the unit cell
############################
acell    3*30

rprim    1.0  0.0  0.0
            0.0  1.0  0.0
            0.0  0.0  1.0

#############################
#Definition of the atom types
#############################
ntypat 1          # There is only one type of atom
znucl 1           # The keyword "znucl" refers to the atomic number of the 
                  # possible type(s) of atom.
                         
########################
#Definition of the atoms
########################
natom 1           # There is one atom
typat 1           # It is hydrogen.

xred
1/2 1/2 1/2

######################################
#Definition of the planewave basis set
######################################
ecut 40.0

###########################################
#Definition of the k-point grid # First Run
###########################################
kptopt 1          # Option for the automatic generation of k points, taking
                  # into account the symmetry
ngkpt  1 1 1

################################
#Definition of the SCF procedure
################################
nstep 30          # Maximal number of SCF cycles
toldfe 1.0d-8     # Will stop when, twice in a row, the difference 
                  # between two consecutive evaluations of total energy 
                  # differ by less than toldfe (in Hartree) 
diemac 1.0        # Hydrogen requires a number between 1 and 3


########################
#Plot the band structure
########################
iscf2   -2
getden2   -1
nband2   10
kptopt2   -1

ndivk2   10
kptbounds2   0.0  0.0  0.0                   # Gamma point
             1.0  1.0  0.0                   # Gamma point in another cell.
tolwfr2  1.0d-12
enunit2  1                                   

[gonze]

The major problem is with the exchange-correlation functional ...
None of the LDA or GGAs is self-interaction free and exact for the Hydrogen case.
By contrast, for the hydrogen atom, using -1/r, that is
the so called Fermi-Amaldi correction (ixc=20), is the exact XC functional.

Actually, this example is treated in test/v1 #25 (see tests/v1/Input/t25.in,
as well as the documentation in tests/v1/README ). There, one uses the bare
hydrogenic potential (instead of a pseudopotential), and not sufficiently
large values of acell and ecut (for testing purposes), and you will see
that the 1s-2s splitting is obtained as 0.368 Ha, instead of the exact 0.375 Ha.
Simply increasing the ecut and acell will give a better (closer to the exact value) splitting.
One is able to obtain a good value for the ionisation potential, but the acell
has to be quite large.

So, using ixc=20, wth the base hydrogen nuclei potential, with a high ecut and large acell
will give you the value of 0.5 Ha.

X.

[sand7000]

Thanks gonze, that makes sense. I have one further question. If the band structure for a lone hydrogen is so far off, am I able to have any faith in the band structure for a hydrogen compound, such as hydrogen on a silicon substrate?

[mverstra]

Self interaction tends to be a smaller effect in larger systems. H is the extreme case : 50% correction for self interaction energy.

There are, of course, leftovers, and errors which can be treated with other methods (GW, hybrids, SIC...) in particular the band gap discontinuity.

matthieu