Molecular-Orbital Projected DOS

• Molecular-Orbital Projected Density of States is the energy-resolved decomposition of a material's total electronic states onto selected molecular orbitals.
• It is employed in both finite molecules and extended systems to analyze orbital hybridization, level alignment, and bonding characteristics.
• The method uses projector sets such as atomic, Wannier, or molecular orbitals, forming the foundation for applications from interface studies to machine learning descriptors.

Molecular-orbital projected density of states (MO-PDOS) is the energy-resolved decomposition of a density of states onto a chosen set of molecular orbitals. In single-particle electronic-structure theory, it assigns to each energy the weight with which a Hartree–Fock, Kohn–Sham, or related eigenstate overlaps a selected orbital; in finite molecules, the discrete spectrum is commonly broadened to form an effective total density of states (TDOS) before projection; and in adsorbates, crystals, and cluster solids, the same construction is applied by projecting extended states onto freestanding-layer orbitals, Wannier-based cluster orbitals, or other localized subspaces (Contreras et al., 2021, Haags et al., 2022, Yanagi et al., 28 Apr 2026). Conceptually, it is a special case of projected density of states in which the projection subspace is a molecular orbital rather than an atomic orbital (Pires et al., 8 Jul 2026).

1. Formal definition and scope

The basic density of states is written, in terms of discrete eigenvalues $E_i$, as

$D(E) = \sum_i \delta(E - E_i),$

or, in extended systems, as a sum over band and crystal-momentum indices. A projected density of states resolves this spectral measure onto a chosen orbital, fragment, or localized basis function. In the general single-particle form used across the literature,

$$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$

where $\phi_{\alpha}$ may be atomic-like, Wannier-like, or molecular. For a molecular orbital $i$, the same structure becomes

$$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$

and this is the defining expression of MO-PDOS in the interface and slab literature (Haags et al., 2022, Haags et al., 9 Jan 2025).

The term is not entirely uniform across subfields. In molecular spectroscopy and wavefunction analysis, one encounters TDOS, PDOS, and OPDOS; in adsorbate and interface studies, the same object is often called molecular-orbital projected DOS or MOPDOS; in crystalline materials, site- and orbital-projected DOS onto atom-centered projectors is frequently denoted pDOS, with the explicit observation that the construction is conceptually identical to molecular-orbital projection when the projector subspace is changed from atomic orbitals to molecular orbitals (Contreras et al., 2021, Pires et al., 8 Jul 2026).

Quantity	Definition or weight	Role
TDOS	$\sum_i g_{\text{type}}(E-E_i)$	Continuous representation of discrete molecular levels
PDOS / MO-PDOS	$$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$	Orbital-resolved spectral weight
OPDOS	$\sum_i X_{A,B}^i\, g_{\text{type}}(E-E_i)$	Bonding vs antibonding analysis
PDoT	$$\sum_\alpha \langle \Psi_\alpha^\dagger \Psi_\alpha \rangle \delta(\varepsilon-\varepsilon_\alpha)$$	Many-body generalization of PDOS

This formalism is neutral with respect to basis choice, but its interpretation is not. The central methodological issue is always the definition of the projector set $D(E) = \sum_i \delta(E - E_i),$0: isolated-molecule orbitals, freestanding-layer orbitals, symmetry-adapted cluster orbitals, Wannier functions, or many-body transition operators lead to related but not identical projected spectra (Rangel et al., 2015, Haydock, 2014).

2. Finite-molecule constructions: TDOS, PDOS, and OPDOS

For isolated molecules, the electronic spectrum from Hartree–Fock or comparable methods is discrete, so the conventional macroscopic picture of DOS as $D(E) = \sum_i \delta(E - E_i),$1 is not directly applicable. The standard workaround is artificial broadening of the molecular eigenvalues to define an effective TDOS,

$D(E) = \sum_i \delta(E - E_i),$2

with $D(E) = \sum_i \delta(E - E_i),$3 chosen as Gaussian, Lorentzian, or pseudo-Voigt. The width parameter $D(E) = \sum_i \delta(E - E_i),$4 controls the trade-off between visual continuity and retention of fine structure. In this setting, PDOS is obtained by weighting each broadened level by a fragment population, and OPDOS is obtained by weighting it by an overlap population between two fragments (Contreras et al., 2021).

In the Mulliken-based formulation used with Multiwfn, the projected DOS for fragment $D(E) = \sum_i \delta(E - E_i),$5 is

$D(E) = \sum_i \delta(E - E_i),$6

where $D(E) = \sum_i \delta(E - E_i),$7 is the Mulliken-type composition of fragment $D(E) = \sum_i \delta(E - E_i),$8 in molecular orbital $D(E) = \sum_i \delta(E - E_i),$9. The overlap population DOS between fragments $$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$0 and $$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$1 is

$$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$2

with

$$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$3

Large positive $$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$4 indicates bonding character, large negative $$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$5 indicates antibonding character, and values near zero indicate little contribution to $$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$6–$$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$7 bonding (Contreras et al., 2021).

This framework was illustrated pedagogically for H$$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$8O and N$$D_{\alpha}(E) = \sum_{n\mathbf{k}} \big|\langle \psi_{n\mathbf{k}} \mid \phi_{\alpha} \rangle\big|^2 \delta(E - E_{n\mathbf{k}}),$$9 at the HF/6-311G* level using Gaussian 09 and Multiwfn. In that setting, TDOS peaks correspond to individual molecular levels or groups of near-degenerate levels, PDOS separates atomic or fragment contributions, and OPDOS identifies bonding, nonbonding, and antibonding states. The explicit objective was to relate DOS-based visualizations to molecular orbital diagrams rather than replace them. A plausible implication is that MO-PDOS is most informative when interpreted as a complement to orbital topology, symmetry labels, and occupancy, not as a stand-alone descriptor (Contreras et al., 2021).

3. Adsorbates, interfaces, and cluster orbitals in solids

At organic/metal interfaces, MO-PDOS is used to track how gas-phase-like orbitals survive, shift, broaden, and hybridize upon adsorption. For bisanthene on Cu(110), the interface Kohn–Sham states of the slab are projected onto a chosen orthonormal set of molecular orbitals $\phi_{\alpha}$0 of a freestanding bisanthene layer, yielding

$\phi_{\alpha}$1

This produces an orbital-by-orbital DOS for specific $\phi_{\alpha}$2 and $\phi_{\alpha}$3 states, such as $\phi_{\alpha}$4 and $\phi_{\alpha}$5, and enables direct comparison with experimental orbital-resolved pDOS extracted from photoemission orbital tomography. In one study, 13 $\phi_{\alpha}$6 and 12 $\phi_{\alpha}$7 orbitals were identified and used for benchmarking functionals; in a later extension, 15 $\phi_{\alpha}$8 and 23 $\phi_{\alpha}$9 orbitals were extracted over a binding-energy range larger than 10 eV (Haags et al., 2022, Haags et al., 9 Jan 2025).

The same logic extends beyond isolated molecules adsorbed on metals. In PbRe$i$0O$i$1, the relevant local states are not best described as isolated atomic orbitals, but as molecular-like orbitals on Re hexagons. Symmetry-adapted combinations of six $i$2 orbitals on a hexagon are classified by irreducible representations of $i$3,

$i$4

and the DOS is projected onto these cluster orbitals. In this MO-resolved description, $i$5 states dominate flat bands just below $i$6, $i$7 states dominate nearly dispersionless bands around $i$8 eV, and $i$9/$$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$0 states form quasi-one-dimensional Fermi surfaces. MO-PDOS in this sense resolves spectral weight by cluster symmetry rather than by isolated-molecule identity (Yanagi et al., 28 Apr 2026).

A direct bridge to crystalline materials appears in work on graph neural networks augmented by orbital-projected density of states. There, site- and orbital-projected DOS functions $$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$1 are discretized into fingerprints over orbital channels $$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$2 and energy windows aligned to the Fermi level or to band edges. The authors explicitly note that this site- and orbital-projected DOS is conceptually identical to projecting onto localized molecular orbitals or basis functions in a molecule, except that the projectors are atomic-like orbitals in a periodic crystal (Pires et al., 8 Jul 2026).

4. Computational and experimental realizations

Several computational realizations of MO-PDOS are in active use. For isolated molecules, Hartree–Fock or DFT eigenvalues are broadened and analyzed with packages such as Multiwfn, which implement TDOS, fragment PDOS, and OPDOS from AO coefficients and overlap matrices (Contreras et al., 2021). For surfaces and interfaces, slab calculations in VASP provide the adsorbed-system states $$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$3, while freestanding-layer orbitals $$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$4 furnish the projector basis; the overlaps $$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$5 define the MOPDOS curves (Haags et al., 2022). For cluster solids, Wien2k together with Wannier90 and wien2wannier supplies localized $$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$6-derived Wannier functions and symmetry-adapted hexagon orbitals used as molecular projectors (Yanagi et al., 28 Apr 2026). For transition-metal complexes, Quantum ESPRESSO and pmw.x are used to build localized molecular orbital projectors from a selected frontier-band subspace via AO projection, Löwdin orthonormalization, and Fourier transformation to real-space Wannier functions (Bajaj et al., 2021).

Photoemission orbital tomography provides an experimental reconstruction of orbital-resolved pDOS. Within the plane-wave approximation for the final state, the momentum map of a given orbital is proportional to the squared modulus of its Fourier transform,

$$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$7

Measured intensity maps $$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$8 are then decomposed as

$$D_i(E) = \sum_n |\langle \phi_i \mid \psi_n \rangle|^2\, \delta(E - E_n),$$9

with the weights $\sum_i g_{\text{type}}(E-E_i)$0 obtained by least-squares minimization. Those weights are the experimental orbital-resolved pDOS. This procedure enabled a one-to-one comparison between experimental pDOS and theoretical MO-PDOS for 38 bisanthene orbitals on Cu(110) (Haags et al., 2022, Haags et al., 9 Jan 2025).

The same projector machinery also underlies corrective electronic-structure frameworks. In jmDFT, a molecular orbital projector is

$\sum_i g_{\text{type}}(E-E_i)$1

with occupations

$\sum_i g_{\text{type}}(E-E_i)$2

These are the same overlaps that enter an MO-PDOS,

$\sum_i g_{\text{type}}(E-E_i)$3

so the distinction between spectral analysis and projector-based correction is one of use, not of basic formalism (Bajaj et al., 2021).

5. Interpretive roles and applications

MO-PDOS is used to localize orbital character in energy, to analyze hybridization, and to relate spectral features to bonding or transport observables. At organic/metal interfaces, it resolves level alignment and resonance broadening orbital by orbital. For bisanthene/Cu(110), a comparison across PBE, HSE, PBE0, and B3LYP found that the range-separated hybrid HSE performs best for the investigated interface. More fundamentally, the remarkable agreement between the experimental and the Kohn–Sham orbital energies over a binding energy range larger than 10 eV suggests that Kohn–Sham orbitals approximate Dyson orbitals in a much better way than previously thought (Haags et al., 9 Jan 2025).

In finite molecules, OPDOS adds an explicitly bonding-sensitive complement to MO-PDOS. Positive OPDOS peaks identify energies where fragment overlap is constructive; negative peaks identify antibonding contributions. For H$\sum_i g_{\text{type}}(E-E_i)$4O, orbitals above about $\sum_i g_{\text{type}}(E-E_i)$5 a.u. were described as not leading to bond formation because the green OPDOS curve is negative, while the occupied lone-pair-like HOMO shows nearly zero OPDOS and strong oxygen localization. For N$\sum_i g_{\text{type}}(E-E_i)$6, the HOMO $\sum_i g_{\text{type}}(E-E_i)$7 has positive OPDOS and the LUMO $\sum_i g_{\text{type}}(E-E_i)$8 has negative OPDOS, reproducing the familiar bonding–antibonding distinction in an energy-resolved form (Contreras et al., 2021).

In correlated and transition-metal systems, MO projectors support both analysis and correction. For a set of nine representative Ti(III) and V(IV) $\sum_i g_{\text{type}}(E-E_i)$9 complexes, jmDFT with a molecular orbital projector basis nearly eliminates energetic delocalization error and static correlation error, and in all cases MOP jmDFT outperforms AOP jmDFT. This suggests that when the frontier electron density is strongly metal–ligand hybridized, orbital-resolved analysis based on molecular projectors is more faithful than analysis based on purely atomic $$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$0-projectors (Bajaj et al., 2021).

Projected DOS has also been converted into machine-learning descriptors. In a graph neural network framework, atomic node representations are augmented by site-projected orbital density of states fingerprints computed directly from density functional theory. For superconducting critical temperature $$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$1 and optical dielectric constant $$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$2, this pDOS augmentation reduces prediction errors by 22.9% and 27.9%, respectively, relative to the elemental-descriptor baseline, and an attention-gating mechanism reveals which orbital channels and energy windows are most relevant. The paper explicitly states that this projection-and-discretization strategy transfers directly to molecular systems if molecular orbitals are chosen as the projector functions (Pires et al., 8 Jul 2026).

6. Ambiguities, limitations, and many-body extensions

A persistent difficulty is that MO-PDOS depends on the definition of the molecular orbital basis. In electronic-transport analysis, Rangel, Rignanese, and Olevano compared two standard constructions: diagonalization of the isolated-molecule Hamiltonian and diagonalization of a junction submatrix built from basis elements localized on the molecule. They found that these methods can lead to substantially different molecular orbitals and hence PDOS. Within the isolated-molecule approach, the PDOS can differ depending on whether the reference molecule is, for example, benzene-dithiol or benzene-dithiolate; within the junction-submatrix approach, the PDOS depends on the chosen basis set. Their conclusion was that these differences can be critical when PDOS is used to provide a physical interpretation of conductance, especially when it has small values at zero bias, and that local density of states is often a more robust interpretive quantity (Rangel et al., 2015).

This basis dependence is not merely technical. In strongly hybridized systems, a state that is chemically intuitive in a gas-phase picture may no longer correspond to an eigenstate-like object in the junction or on the surface. A plausible implication is that MO-PDOS is most stable when the projector set tracks the actual physical subspace of interest—freestanding-layer resonances for adsorbates, symmetry-adapted cluster orbitals for cluster solids, or Wannierized frontier manifolds for transition-metal complexes—rather than a nominal gas-phase reference.

For interacting electrons, the single-particle MO-PDOS is generalized by Haydock’s projected density of transitions (PDoT). Starting from a creation operator $$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$3 for a chosen localized spin orbital or molecular orbital, the Heisenberg evolution is decomposed into stationary transition operators $$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$4 with transition energies $$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$5, and the spectral measure is

$$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$6

In spectral form, this is close in spirit to

$$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$7

Haydock explicitly presents PDoT as the many-body generalization of projected density of states for independent electrons, and the formalism was applied both to a Hubbard model of H$$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$8 and to $$\sum_n |\langle \phi_i \mid \psi_n \rangle|^2 \delta(E-E_n)$$9-bands in transition metals (Haydock, 2014).

The main conceptual consequence is that MO-PDOS in the strict single-particle sense is not the final spectral object when electron correlation reorganizes addition and removal processes into multiple many-body transitions. In weakly correlated regimes, the projected spectrum may remain close to a one-orbital-one-peak picture. In strongly correlated regimes, the many-body analog contains lower and upper Hubbard bands, gap formation, satellite features, and redistribution of spectral weight. This suggests that MO-PDOS is best understood as a hierarchy of related constructions: broadened orbital analysis for isolated molecules, projector-based spectral decomposition for interfaces and solids, and transition-resolved spectral measures for interacting systems.