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Hubbard U Correction

DFT+\(U\) augments a Kohn–Sham functional by an orbital-dependent correction within selected localized subspaces. This chapter fixes the simplified, rotationally invariant form of Dudarev et al., which depends only on \(U_{\mathrm{eff}}=U-J\). Forms with separate anisotropic \(U\) and \(J\), intersite interactions, and procedures for determining \(U_{\mathrm{eff}}\) are outside the present scope. Hartree atomic units are used unless stated otherwise.

The correction is defined jointly by the correlated subspaces, their projectors, and \(U_{\mathrm{eff}}\). The value of \(U_{\mathrm{eff}}\) alone does not define the electronic problem. Changing any of these choices changes the energy functional and therefore the Hamiltonian, density, energy, and forces.

Correlated Subspaces and Occupation Matrices

Let \(\mathcal C_{i\ell}\) be a correlated angular-momentum shell on atom \(i\), spanned by localized Hubbard orbitals \(\lvert w_{i\ell m};s\rangle\). The index \(m\) labels the orbital components of the selected shell and \(s\) is an explicit spin index. The main formulas assume an orthonormal basis within each correlated subspace,

\[ \langle w_{i\ell m};s \vert w_{i\ell m'};s' \rangle = \delta_{mm'}\delta_{ss'}. \]

The correlated-subspace occupation matrix is

\[ N^{i\ell}_{ms,m's'} \equiv \langle w_{i\ell m};s \vert \hat\rho \vert w_{i\ell m'};s' \rangle. \]

The complete matrix \(\mathbf N^{i\ell}\) is Hermitian and acts on the orbital–spin product space of the selected shell. Its trace is the occupation of that correlated subspace, not the total electron number. It must not be identified with either the global density matrix \(\mathbf D\) or the PAW projector density matrix \(\mathbf P^i\).

For a periodic system, let \(\lvert w_{i\ell m\boldsymbol k};s\rangle\) denote the corresponding Bloch sum with the cell normalization of Periodic Systems. Then

\[ N^{i\ell}_{ms,m's'} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \sum_a f_{a\boldsymbol k} \langle w_{i\ell m\boldsymbol k};s \vert \psi_{a\boldsymbol k} \rangle_\Omega \langle \psi_{a\boldsymbol k} \vert w_{i\ell m'\boldsymbol k};s' \rangle_\Omega. \]

The occupations refer to explicit spinor states. When this integral is evaluated on an irreducible mesh, symmetry-related contributions must be restored with their atomic, orbital, and spin transformations.

Simplified Rotationally Invariant Functional

The DFT+\(U\) total energy is

\[ E_{\mathrm{DFT}+U} = E_{\mathrm{KS}} + \sum_{i\ell} E_U^{i\ell}, \]

with the net Hubbard correction

\[ E_U^{i\ell} = \frac{U_{\mathrm{eff}}^{i\ell}}{2} \operatorname{tr} \left[ \mathbf N^{i\ell} \left( \mathbf I-\mathbf N^{i\ell} \right) \right]. \]

The trace is over both orbital and explicit spin indices. In a collinear calculation, \(\mathbf N^{i\ell}\) is spin block diagonal and this trace reduces to the usual sum over the two spin channels. A noncollinear or spin–orbit-coupled calculation retains the off-diagonal spin and complex orbital components. No implicit factor of two is present.

The expression already contains the double-counting subtraction of the simplified Dudarev functional. Only \(U_{\mathrm{eff}}=U-J\) enters; separate values of \(U\) and \(J\) have no independent meaning within this form. For the orthonormal projector convention above, the eigenvalues of \(\mathbf N^{i\ell}\) lie between zero and one. Positive \(U_{\mathrm{eff}}\) penalizes fractional eigenvalues, and the correction vanishes when \(\mathbf N^{i\ell}\) is idempotent.

Under a unitary change of basis within a fixed correlated subspace,

\[ \mathbf N^{i\ell} \longmapsto \mathbf U^\dagger \mathbf N^{i\ell} \mathbf U, \]

the energy is unchanged. This rotational invariance does not make the choice of correlated subspace or projector convention irrelevant.

Differentiating the correction in the matrix trace pairing gives the on-site potential matrix

\[ \mathbf V_U^{i\ell} = U_{\mathrm{eff}}^{i\ell} \left( \frac{1}{2}\mathbf I - \mathbf N^{i\ell} \right). \]

Its one-electron operator is

\[ \hat V_U = \sum_{i\ell} \sum_{mm'} \sum_{ss'} \lvert w_{i\ell m};s\rangle \left( V_U^{i\ell} \right)_{ms,m's'} \langle w_{i\ell m'};s'\rvert, \]

and the self-consistent Hamiltonian is

\[ \hat H_{\mathrm{DFT}+U} = \hat H_{\mathrm{KS}} + \hat V_U. \]

Because \(\mathbf N^{i\ell}\) depends on \(\hat\rho\), the Hubbard potential must be updated within the self-consistent cycle.

Relation to Global Bases

The plane-wave or LCAO basis used to represent the Kohn–Sham states is conceptually distinct from the correlated subspace. The Hubbard orbitals may be expanded in that global basis, but they define an additional set of local projectors.

For a finite ordinary nonorthogonal LCAO problem, let the columns of \(\mathbf W^{i\ell}\) contain the coefficients of the orthonormal Hubbard orbitals in the LCAO basis. Using the conventions of Localized Atomic-Orbital Basis,

\[ \mathbf N^{i\ell} = \left( \mathbf W^{i\ell} \right)^\dagger \mathbf S \mathbf D \mathbf S \mathbf W^{i\ell}. \]

For a periodic LCAO problem, the corresponding expression is

\[ \mathbf N^{i\ell} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \left( \mathbf W^{i\ell}(\boldsymbol k) \right)^\dagger \mathbf S(\boldsymbol k) \mathbf D(\boldsymbol k) \mathbf S(\boldsymbol k) \mathbf W^{i\ell}(\boldsymbol k). \]

For an orthonormal plane-wave basis, the ordinary basis overlap is the identity. These relations describe how the same occupation matrix is evaluated in different global bases; they do not redefine \(\mathbf N^{i\ell}\).

In PAW, the physical correlated-subspace projector is represented by its auxiliary-space transformed operator. The PAW matrix \(\mathbf P^i\) contains projector-channel information for the complete PAW augmentation space, whereas \(\mathbf N^{i\ell}\) is the occupation matrix of a selected Hubbard shell. A method- and dataset-specific contraction may relate them, but they are not universally identical. The PAW object hierarchy is defined in Projector-Augmented-Wave Method.

If the raw localized functions defining a correlated subspace are nonorthogonal, a dual-projector or orthonormalization convention is required. That convention changes the occupation matrix and is part of the definition of the DFT+\(U\) calculation.

Forces and Dataset Identity

Forces and stresses are derivatives of the complete self-consistent DFT+\(U\) energy. When the Hubbard projectors move or change with the nuclei, their derivatives contribute in addition to the ordinary Kohn–Sham force terms. Derivatives of any orthonormalization or dual-projector construction must be treated consistently.

The energy landscape may contain multiple self-consistent orbital and magnetic solutions with different occupation matrices. A calculation or dataset must therefore identify at least:

  • the underlying exchange–correlation functional;
  • the corrected atoms and angular-momentum shells;
  • \(U_{\mathrm{eff}}^{i\ell}\) and its units;
  • the correlated-orbital and projector construction, including any orthonormalization or dual convention;
  • the simplified Dudarev functional and the spin treatment;
  • the converged orbital and magnetic state when multiple solutions are relevant.

Calculations that differ in any of these defining choices must not be treated as labels of the same electronic problem merely because the atomic structure is unchanged.

References