Density of States

The density of states (DOS) counts how many Kohn-Sham states occur at a given energy. In practice it is used together with band structures, occupations, and orbital projections to identify the electronic states responsible for a material or molecular property.

In electronic-structure analysis, DOS and projected DOS (PDOS) are most often used to answer questions such as:

  • Is the system metallic, semiconducting, or insulating? A finite DOS at the Fermi level usually indicates metallic behavior, while a gap around the band edges indicates an insulator or semiconductor.

  • Which atoms and orbitals form the valence-band maximum, conduction-band minimum, frontier molecular orbitals, or states near the Fermi level?

  • Are defect, dopant, adsorbate, or surface states inside the gap, and are they localized on particular atoms?

  • Which angular-momentum channels dominate a feature, for example transition-metal d states, oxygen p states, or adsorbate frontier orbitals?

  • How strong is orbital hybridization, such as metal–oxygen d–p mixing, substrate–adsorbate coupling, or ligand-field splitting?

  • How do strain, oxidation state, magnetic ordering, charging, adsorption, or structural relaxation shift band edges and redistribute orbital character?

DOS gives the total number of available states. PDOS decomposes the same spectrum into selected atomic kinds, angular-momentum channels, individual components, atom lists, or real-space regions. For this reason, total DOS is usually used to locate the energy windows of interest, while PDOS is used to assign the chemical character of the peaks or band edges.

Important

Most of the DOS/PDOS functionality described below, including the unified &DFT%PRINT%DOS interface and the broadened, directly plottable .dos and .pdos files, is only available in CP2K 2026.2 and later versions. In earlier CP2K versions, .dos and .pdos referred to state-resolved raw outputs rather than broadened DOS/PDOS curves. The raw output of PDOS is preserved and now written as .pdos_raw through &DFT%PRINT%DOS%PDOS%RAW keyword.

Basic CP2K input

For DFT calculations, DOS output is controlled by &DFT%PRINT%DOS. Requesting this print section writes the total DOS. Set &PDOS under the same section to additionally write projected DOS files.

&FORCE_EVAL
  &DFT
    &PRINT
      &DOS
        ENERGY_UNIT EV
        ENERGY_ZERO AUTO
        BROADEN_WIDTH [eV] 0.1
        DELTA_E [eV] 0.01
        &PDOS
          COMPONENTS F
        &END PDOS
      &END DOS
    &END PRINT
  &END DFT
&END FORCE_EVAL

Useful input keywords are documented in the input reference:

For PDOS, there are some additional controls:

  • COMPONENTS: split angular-momentum channels into individual components. This keyword also applies to &LDOS and &R_LDOS.

  • RAW: print raw state-resolved projected weights.

Note

For diagonalization-based SCF calculations, the unoccupied states used for DOS/PDOS are those available from the Kohn-Sham diagonalization, controlled by ADDED_MOS. In this case, NLUMO acts mainly as a lower bound: CP2K may increase the number of additional MOs if needed, but not truncate unoccupied MOs that are already available. In OT calculations, unoccupied states for DOS/PDOS are computed in a separate post-SCF step.

Choosing the energy reference

Use ENERGY_ZERO to make different calculations and plots comparable.

  • FERMI is normally the appropriate zero for metals and smeared calculations.

  • HOCO places the highest occupied crystal orbital at zero. It is often more convenient for molecules, semiconductors, and insulators, especially when comparing DOS with band plots where the valence-band maximum is set to 0 eV.

  • ABSOLUTE prints the absolute Kohn-Sham eigenvalues.

  • AUTO selects FERMI when smearing or fractional occupations are present, and HOCO otherwise.

The output header reports both reference energies and the selected zero, for example:

# E(Fermi) = 0.223578 a.u. = 6.08391 eV
# E(HOCO)  = 0.190000 a.u. = 5.17018 eV
# Energy zero: AUTO -> HOCO

When comparing several structures or charge states, choose the same energy-zero convention for all plots. If an absolute alignment is needed across different cells or surfaces, additional electrostatic-potential alignment may be required; changing the DOS zero alone is not a substitute for such an alignment.

Broadening, k-points, and gaps

Electronic-structure features observed in DOS/PDOS, such as peak positions, band gaps, and peak widths, can be affected by the calculation setup. Appropriate choices of the functional, basis set, k-point sampling, smearing, unoccupied states, and broadening parameters are important for obtaining a reliable interpretation of the electronic structure.

The broadened DOS replaces each discrete eigenvalue by a normalized line shape. BROADEN_WIDTH is the full width at half maximum (FWHM). The default value of 0.1 eV is a reasonable choice for most applications. Smaller values can be useful for resolving narrow features, small gaps, spin splittings, or defect states, whereas larger values mainly produce smoother curves and may hide such details.

DOS and PDOS usually require a denser k-point mesh than calculations aimed only at total energies, forces, or geometry optimization. A too sparse k-point mesh gives a DOS dominated by the discrete sampling of the Brillouin zone, especially near the band edges or the Fermi level. Broadening helps produce a smooth curve, but it should not be used as a substitute for insufficient k-point sampling.

For molecular or very large supercell calculations with only the Gamma point, the DOS is a broadened representation of discrete levels. It can still be useful for assigning orbital character, but peak widths are chosen mainly for visualization rather than for Brillouin-zone integration.

Interpreting PDOS

With &PDOS, CP2K writes projected DOS, usually one file per atomic kind and spin channel. A .pdos file contains a projected total column followed by angular-momentum or component-resolved columns:

# Energy[eV]  total  s  p  d

With COMPONENTS T, the channels are split into individual components:

# Energy[eV]  total  s  py  pz  px  ...

The total column in one .pdos file is the projected total for that atomic kind, not the total DOS of the full system. Summing the projected totals over all kinds should reproduce the total DOS up to numerical and output precision, provided that the same states and broadening are used.

Warning

Component-resolved k-point PDOS with full k-point symmetry reduction should be interpreted with care, because symmetry-reduced k-points can rotate the orbital components. It is suggested to disable k-point symmetry when analyzing individual components.

Practical PDOS analysis usually proceeds by selecting an energy window and assigning the dominant contributions:

  • band-edge states: compare the PDOS around the valence-band maximum and conduction-band minimum;

  • defect or dopant levels: check whether in-gap peaks are localized on the defect atoms or nearby host atoms;

  • surface and adsorption states: compare slab atoms, adsorbate atoms, and bulk-like atoms;

  • transition-metal compounds: inspect the relative positions and spin splitting of metal d and ligand p channels;

  • molecular frontier levels: identify which fragments contribute to HOMO/HOCO and LUMO/LUCO features.

For quantitative orbital populations, integrated PDOS values should be used with care because they depend on the projection scheme, basis set, and chosen energy window.

RAW T prints state-resolved projected weights in addition to the broadened .pdos files:

&DOS
  &PDOS
    RAW T
  &END PDOS
&END DOS

The RAW option is currently not available with k-points calculations.

Projection used for PDOS

CP2K’s standard PDOS is a Löwdin-type projection in the atomic-orbital basis. For non-k-point calculations, the molecular-orbital coefficients C are first transformed with the square root of the overlap matrix S,

\[ \widetilde C = S^{\frac{1}{2}} C \]

and the projected weight of a state is accumulated from \(|\widetilde C_{\mu i}|^2\) over atomic orbitals belonging to a given atomic kind and angular-momentum channel. This gives a positive, orthogonalized AO decomposition of the DOS. It is not a Mulliken population analysis.

For k-point calculations, CP2K performs the same Löwdin-type projection separately at each k point, using the corresponding overlap matrix and Bloch MO coefficients. The projected contributions are then accumulated with the k-point weights to form the broadened PDOS.

Plotting DOS/PDOS

Using gnuplot

The DOS and PDOS files can be plotted directly with gnuplot, since comment lines starting with # are ignored. For example, if the first column is the energy and the second column is the total DOS, a simple plot is

plot "project-1.dos" using 1:2 with lines title "DOS"

The plotted energy range can be restricted directly in the plot command. For example, to show only states within 5 eV of the chosen energy zero,

plot [-5:5] "project-1.dos" using 1:2 with lines title "DOS"

Both the energy and DOS ranges can be limited in the same way:

plot [-5:5][0:*] "project-1.dos" using 1:2 with lines title "DOS"

For PDOS files, different columns can be plotted together. For example, if the second column is the kind-resolved total PDOS and the following columns are angular-momentum components,

plot [-5:5] "project-k1-1.pdos" using 1:2 with lines title "total", \
            "" using 1:3 with lines title "s", \
            "" using 1:4 with lines title "p"

When comparing DOS/PDOS with band structures, use the same energy zero, for example FERMI for metals or HOCO for many insulating and molecular systems.

Local projected DOS

&LDOS projects onto explicitly listed atoms. &R_LDOS projects onto a real-space volume defined relative to listed atoms. These outputs are useful for checking whether a DOS feature is localized in a defect region, adsorbate, surface layer, or selected fragment.They are currently not implemented for k-points.

Caution

The definition of LDOS in CP2K is different from the real-space local density of states \(\rho(\mathbf{r}, E)\) often used in, for example, STM-related analysis. In this DOS/PDOS print section, LDOS denotes a projected DOS for user-defined lists of atoms, i.e. a local PDOS over selected atoms. R_LDOS further restricts the analysis to a user-defined spatial region around selected atoms. These outputs are integrated projected quantities, not three-dimensional real-space LDOS maps.

XAS and XAS_TDP projected DOS

XAS and XAS_TDP have their own dedicated &PDOS print sections. They are intentionally separate from the DFT &DOS section because their output is used for spectral-feature analysis rather than ordinary ground-state density-of-states analysis.

For XAS calculations based on transition-potential or core-hole electronic structures, the projected spectrum is computed from the XAS electronic structure. It can be used to inspect which atoms and angular-momentum channels contribute to states relevant for the near-edge spectrum. The interpretation is therefore tied to the core-excited or transition-potential state, not to the ground-state DOS.

For XAS_TDP, the distinction is stronger. The .pdos files reuse the standard PDOS projection and file format, but the states being projected are linear-response orbitals and the energy column represents excitation energies. Occupation numbers and references to molecular orbitals in the generic PDOS format should not be interpreted as ordinary ground-state Kohn-Sham occupations.

A typical XAS_TDP use is to assign intense spectral features to atomic or orbital character, for example to distinguish metal-centered, ligand-centered, or mixed transitions. In this context, broadening and energy-grid settings remain useful plotting controls, while ordinary DFT-DOS concepts such as Fermi-level alignment, HOCO alignment, or total DOS are usually not the relevant analysis target. See &XAS%PRINT%PDOS and &XAS_TDP%PRINT%PDOS for the corresponding input sections.