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Ghost bands

Approximate linear dependence of the basis set or the linearization of the energy dependence of the radial wave functions (see section 2.2) can lead to spurious eigenvalues, termed ``ghost bands''.

The first case may occur in a system which has atoms with very different atomic sphere radii. Suppose you calculate a hydroxide with very short O-H bonds so that you select small RMT radii for O and H such as e.g. 1.0 and 0.6 a.u., respectively. The cation may be large and thus you could choose a large RMT of e.g. 2.4 a.u. However, this gives a four time larger effective RKmax for the cation than for H, (e.g. 16.0 when you select RKmax=4.0 in case.in1). This enormous difference in the convergence may lead to unphysical eigenvalues. In such cases choose lmax=12 in case.in1 (in order to get a very good re-expansion of the plane waves) and reduce RMT for the cation to e.g. 1.8 a.u.

The second case can occur when you don't use a proper set of local orbitals. In this situation the energy region of interest (valence bands) falls about midway between two states with different principle quantum numbers, but with the same l-value (for one atom).

Take for example Ti with its 3p states being occupied as (semi-core) states, while the 4p states remain mostly unoccupied. In the valence band region neither of those two states (Ti 3p, 4p) should appear. If one uses 0.2 Ry for the expansion energy E(1) for the p states of Ti, then Ti-p states do appear as ghost bands. Such a run is shown below for TiO2 (rutile).

The lowest six eigenvalues at GAMMA fall between about -1.30 and -1.28 Ry. They are ghost bands derived from fictitious Ti-p states. The next four eigenvalues between -0.94 and -0.78 Ry correspond to states derived from O 2s states, which are ok, since there are four O's per unit cell, four states are found.

The occurrence of such unphysical (indeed, unchemical!) ghostbands is the first warning that something went wrong. A more definite warning comes upon running LAPW2, where the corresponding charge densities are calculated. If the contribution to the charge density from the energy derivative of the basis function [the Blm coefficient in equ. 2.5,2.8] is significant (i.e. much more than 5 per cent) then a warning is issued in LAPW2.

In the present example it reads:

QTL-B VALUE .EQ. 40.35396 !!!!!!

This message is found in both the case.scf file and in case.output2.

When such a message appears, one can also look at the partial charges (QTL), which are printed under these conditions to OUTPUT2, and always appear in the files case.helpXX, etc., where the last digit refers to the atomic index.

In the file below, note the E(1) energy parameter as well as the 6 ghost band energies around -1.29.

--------------- top of file:tio2.scf -----------------------------
          ATOMIC SPHERE DEPENDENT PARAMETERS FOR ATOM  Titanium 
          OVERALL ENERGY PARAMETER IS     .2000
          E( 0)=     .2000
--->      E( 1)=     .2000
          E( 2)=     .2000   E(BOTTOM)=    -.140   E(TOP)= -200.000


          ATOMIC SPHERE DEPENDENT PARAMETERS FOR ATOM  Oxygen 
          OVERALL ENERGY PARAMETER IS     .2000
          E( 0)=    -.7100   E(BOTTOM)=   -2.090   E(TOP)=     .670

       K=    .00000    .00000    .00000            1
:RKM  : MATRIX SIZE= 599  RKM= 6.99  WEIGHT= 8.00  PGR:    
       EIGENVALUES ARE:
        -1.2970782   -1.2970782   -1.2948747   -1.2897193  -1.2897193
        -1.2882306    -.9389111    -.8484857    -.7880729   -.7880729
         -.0484830    -.0162982     .0121181     .0976534    .0976534
          .1914068     .1914068     .2341991     .3286919    .3477629
          .3477629     .3809219     .5143729     .5356211    .5550735
          .5617155     .5617155     .7087550     .7197110    .8736991
          .8736991     .9428865     .9533619    1.2224570   1.2224570
         1.4285169
       ********************************************************
       NUMBER OF K-POINTS:          1 

:NOE  : NUMBER OF ELECTRONS          =  48.000
:FER  : F E R M I - ENERGY           =    .53562

:POS01: AT.NR. -1  POSITION =  .00000  .00000  .00000  MULTIPLICITY=  2
       LMMAX=10
       LM=  0 0 2 0 2 2 4 0 4 2 4 4 6 0 6 2 6 4 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0
:CHA01: TOTAL CHARGE INSIDE SPHERE   1 =     8.802166
:PCS01: PARTIAL CHARGES SPHERE =  1 S,P,D,F,PX,PY,PZ,D-Z2,D-X2Y2,D-XY,D-XZ,D-YZ 
:QTL01:  .127 6.080 2.518  .067 2.011 2.047 2.022 1.090  .760 .155  .480  .034
                      VXX         VYY         VZZ       UP TO R
:VZZ01:            -4.96856     8.48379    -3.51524       2.000
:POS02: AT.NR. -2  POSITION =  .30500  .30500  .00000  MULTIPLICITY=  4
       LMMAX=16
       LM=  0 0 1 0 2 0 2 2 3 0 3 2 4 0 4 2 4 4 5 0 5 2 5 4 6 0 6 2 6 4 6 6 0 0
:CHA02: TOTAL CHARGE INSIDE SPHERE   2 =     5.486185
:PCS02: PARTIAL CHARGES SPHERE =  2 S,P,D,F,PX,PY,PZ,D-Z2,D-X2Y2,D-XY,D-XZ,D-YZ 
:QTL02: 1.559 3.902  .022  .002 1.296 1.306 1.300  .014  .004 .000  .003  .001
                      VXX         VYY         VZZ       UP TO R
:VZZ02:              .25199     -.55091      .29892       1.600

:CHA  : TOTAL CHARGE INSIDE CELL =      48.000000
:SUM  : SUM OF EIGENVALUES =            -15.810906

   QTL-B VALUE .EQ.   40.35396   !!!!!!
      NBAND in QTL-file:         24 
----------------end of truncated file tio2.scf----------------------

Next we show tio2.output2 for the first of the ghost bands at -1.297 Ry. One sees that it corresponds mainly to a p-like charge, which originates from the energy derivative part Q(UE) of the Kohn-Sham orbital. Q(UE) contributes 40.1% compared with 8.5% from the main component Q(U). Q(UE) greater than Q(U) is a good indication for a ghost band.

----------------part of file tio2.output2 --------------------------
   QTL-B VALUE .EQ.   40.35396   !!!!!!
  K-POINT:   .0000   .0000   .0000   599  36           1
  BAND #  1  E= -1.29708  WEIGHT= 2.0000000
           L= 0     L= 1       PX:      PY:      PZ:    L= 2    DZ2:   DX2Y2:     DXY:     DXZ:     DYZ:    L= 3  
 QINSID:    .0000  48.6035  35.0996  13.5039    .0000    .0000    .0000    .0000    .0000    .0000    .0000    .0030
 Q(U)  :    .0000   8.4902   6.0125   2.4777    .0000    .0000    .0000    .0000    .0000    .0000    .0000    .0026
 Q(UE) :    .0000  40.1132  29.0871  11.0261    .0000    .0000    .0000    .0000    .0000    .0000    .0000    .0005
           L= 0     L= 1       PX:      PY:      PZ:    L= 2      DZ2:   DX2Y2:     DXY:     DXZ:     DYZ:    L= 3  
 QINSID:    .1294    .0707    .0000    .0055    .0653    .0088    .0038    .0049    .0000    .0000    .0000    .0022
 Q(U)  :    .1279    .0627    .0000    .0052    .0575    .0087    .0038    .0049    .0000    .0000    .0000    .0020
 Q(UE) :    .0016    .0081    .0000    .0003    .0077    .0001    .0000    .0000    .0000    .0000    .0000    .0002
 QOUT  : 1.9265
----------------------bottom of truncated file ----------------------

Another file in which the same information can be found is tio2.help31, since the ghost band is caused by a bad choice for the Ti-p energy parameter:

----------------------Top of file tio2.help31 ---------------------
  K-POINT:   .0000   .0000   .0000   599  36           1
  BAND #  1  E= -1.29708  WEIGHT= 2.0000000
  L= 0     .00000      .00000    .00000    .00000    .00000    .00000
  L= 1   48.60346     8.49022  40.11324    .00000    .00000    .00000
    PX:  35.09960     6.01247  29.08712    .00000    .00000    .00000
    PY:  13.50386     2.47774  11.02612    .00000    .00000    .00000
    PZ:    .00000      .00000    .00000    .00000    .00000    .00000
  L= 2     .00000      .00000    .00000    .00000    .00000    .00000
   DZ2:    .00000      .00000    .00000    .00000    .00000    .00000
 DX2Y2:    .00000      .00000    .00000    .00000    .00000    .00000
   DXY:    .00000      .00000    .00000    .00000    .00000    .00000
   DXZ:    .00000      .00000    .00000    .00000    .00000    .00000
   DYZ:    .00000      .00000    .00000    .00000    .00000    .00000
  L= 3     .00304      .00255    .00050    .00000    .00000    .00000
  L= 4     .00000      .00000    .00000    .00000    .00000    .00000
  L= 5     .00096      .00082    .00014    .00000    .00000    .00000
  L= 6     .00000      .00000    .00000    .00000    .00000    .00000
-------------------bottom of truncated file--------------------------

Note again for L=1 the percentage of charge associated with the primary (APW) basis functions ul (8.5%) versus that coming from the energy derivative component (40.1%).

If a ghost band appears, one should first analyze its origin as indicated above, then use appropriate local orbitals to improve the calculation and get rid of these unphysical states.

Do not perform calculations with ``ghost-bands'', even when the calculation converges.


Good luck !

next up previous contents
Next: References Up: Trouble shooting Previous: Trouble shooting

2000-04-11