Canonical Orbitals, Unoccupied Orbitals

To calculate the canonical Kohn-Sham orbitals and their energies/eigenvalues (as in Eq.27), we have to performan a new calculation based on an already converged wavefunction from a RESTART file. This calculation is initiated by the keyword KOHN-SHAM ENERGIES which has to be followed by a number indicating how many unoccupied orbitals (i.e. ``empty'' states) should be added to the calculation. The calculation will then be starting a diagonalization (Lanczos by default) to make certain that all states are diagonal to each other. The individual orbitals (or ``bands'' in terms of solid state electronic structure calculations) can be written to disk with the RHOOUT command. Please note that the results have to be analyzed with caution. Since we are doing a pseudopotential calculation, we cannot enforce diagonalization across the ``missing'' states due to the removed core electrons. For most pseudopotentials, there is little overlap between the core states and the valence states of the individual atoms (you can do a population analysis of an all-electron calculation to verify this), and as a result the orbitals calculated by CPMD are very close to the corresponding ones from equivalent all-electron calculations. One has to be more careful about how to interpret the unoccupied Kohn-Sham orbitals (in general and in particular from pseudopotential calculations), but the frontier orbitals (HOMO, LUMO) and neighboring states are usually quite well represented.

The following shows the &CPMD section to calculate and write out the two lowest states as densities ( $\left\vert\phi_i\cdot\phi_i^*\right\vert$ ) and the HOMO and LUMO as wavefunctions ( $\phi_i\cdot\phi_i^*$ ) (see RHOOUT for details). We use the DAVIDSON DIAGONALIZATION instead of the default (Lanczos), since it is much faster in this case as the default.

&CPMD
  KOHN-SHAM ENERGIES
   2
  RESTART WAVEFUNCTION COORDINATES
  DAVIDSON DIAGONALIZATION
  RHOOUT BANDS
    4
   1 2 -4 -5
&END

In the resulting output you can see that the job type has been recognized ...

 SINGLE POINT DENSITY OPTIMIZATION

 EXACT DIAGONALIZATION OF KOHN-SHAM MATRIX

[...]

 NUMBER OF STATES:                                              6
 NUMBER OF ELECTRONS:                                     8.00000
 CHARGE:                                                  0.00000
 ELECTRON TEMPERATURE(KELVIN):                            0.00000
 OCCUPATION
  2.0  2.0  2.0  2.0  0.0  0.0

[...]

 RV30! NUMBER OF STATES   HAS CHANGED                      4    6

 RESTART INFORMATION READ ON FILE                       ./RESTART

and that two unoccupied states have been added to the system. The warning RV30! indicates that only 4 states were read from the RESTART file.

Later in the output you can see the progress of the diagonalization. The already converged states from the restart become diagonal in the first step and so does one of the added empty states.

 ITER  STATES  SUBSPACE    STATE     ENERGY     RESIDUAL     TCPU
    0      4       6          5     0.136953    0.50E-01     0.40
    1      4       4          5     0.130553    0.47E-01     0.14
    2      4       6          5     0.051382    0.30E-01     0.15
    3      4       8          5     0.005339    0.25E-01     0.14
    4      4       4          5    -0.027323    0.85E-02     0.14
    5      4       6          5    -0.038423    0.10E-02     0.15
    6      4       8          5    -0.038810    0.28E-03     0.15
    7      4       4          5    -0.038834    0.16E-03     0.14
    8      4       6          5    -0.038838    0.27E-04     0.15
    9      6       8          6     0.012297    0.51E-05     0.14

 EIGENVALUES(EV) AND OCCUPATION:
       1    -25.2030827       2.00000000        2    -13.0331379       2.00000000
       3     -9.2770150       2.00000000        4     -7.2112201       2.00000000
       5     -1.0568488       0.00000000        6      0.3346207       0.00000000
 CHEMICAL POTENTIAL =                            -7.2112214091 EV

At the end of the diagonalization the eigenvalues of the Kohn-Sham states and their occupation are listed (in electron volt). After the energy summary output, the requested orbital files (and the total density) are written out.

 DENSITY WRITTEN TO FILE DENSITY
 DENSITY WRITTEN TO FILE DENSITY.1
 DENSITY WRITTEN TO FILE DENSITY.2
 DENSITY WRITTEN TO FILE WAVEFUNCTION.4
 DENSITY WRITTEN TO FILE WAVEFUNCTION.5

The resulting files have to be converted to cube files for visualization using the cpmd2cube.x tool. For the orbitals that are quite local and with an isovalue of 0.1 for visualization, using the -trim 0.01 will reduce the size of some of the individual cube files enormously without losing any important information.