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Projector-Augmented-Wave Method

The projector-augmented-wave (PAW) method represents smooth auxiliary spinors while retaining an explicit transformation to all-electron valence spinors. It is normally combined with the frozen-core approximation. This chapter defines the transformation, overlap matrix, density matrices, and augmentation quantities needed to interpret PAW Hamiltonians and densities. Hartree atomic units are used unless stated otherwise.

PAW is independent of the basis used for the auxiliary spinors. Plane waves are the standard choice in VASP, but the transformation and the one-center quantities are not plane-wave definitions.

Partial Waves, Projectors, and Transformation

Let \(i\) label an atom at \(\boldsymbol R_i\), and let \(\mu\) label a partial-wave channel in its PAW dataset. A channel includes its angular and radial reference labels; it is not a combined LCAO–spin index. The dataset provides three atom-centered functions for every retained channel:

  • an all-electron partial wave \(\lvert\phi_{i\mu}\rangle\);
  • a smooth partial wave \(\lvert\widetilde\phi_{i\mu}\rangle\);
  • a localized projector \(\lvert\widetilde p_{i\mu}\rangle\).

The all-electron and smooth partial waves agree outside the augmentation region of atom \(i\). Within that region, the projectors are dual to the smooth partial waves,

\[ \langle \widetilde p_{i\mu} \vert \widetilde\phi_{i\nu} \rangle = \delta_{\mu\nu}. \]

Spin is explicit. This chapter assumes a spin-independent PAW transformation,

\[ \lvert\phi_{i\mu};s\rangle = \lvert\phi_{i\mu}\rangle \otimes \lvert s\rangle, \]

and likewise for the smooth partial waves and projectors. Noncollinear magnetism and spin–orbit coupling may still mix spin through the transformed Hamiltonian and density matrices. A dataset with spinor partial waves or projectors would require a spin-dependent transformation and lies outside the scope fixed here.

For a one-electron state labeled by \(a\), the all-electron valence spinor \(\lvert\psi_a\rangle\) is obtained from its smooth auxiliary counterpart \(\lvert\widetilde\psi_a\rangle\) by

\[ \lvert\psi_a\rangle = \hat{\mathcal T} \lvert\widetilde\psi_a\rangle, \]

where the PAW transformation is

\[ \hat{\mathcal T} = \hat I + \sum_i \sum_\mu \sum_s \left( \lvert\phi_{i\mu};s\rangle - \lvert\widetilde\phi_{i\mu};s\rangle \right) \langle\widetilde p_{i\mu};s\rvert. \]

Thus the auxiliary spinor is retained outside the augmentation regions, while its one-center expansion is replaced by the corresponding all-electron partial-wave expansion near each nucleus. The transformation is exact only within the completeness of the finite partial-wave and projector sets contained in the dataset.

PAW Overlap Matrix and Transformed Operators

All-electron orthonormality gives the following transformation product in the auxiliary space:

\[ \hat S \equiv \hat{\mathcal T}^\dagger \hat{\mathcal T}. \]

For nonoverlapping augmentation regions and a spin-independent transformation, this operator has the projector form

\[ \hat S = \hat I + \sum_i \sum_{\mu\nu} \sum_s \lvert\widetilde p_{i\mu};s\rangle \Delta S^i_{\mu\nu} \langle\widetilde p_{i\nu};s\rvert, \]

with the fixed on-site overlap corrections

\[ \Delta S^i_{\mu\nu} \equiv \langle \phi_{i\mu} \vert \phi_{i\nu} \rangle - \langle \widetilde\phi_{i\mu} \vert \widetilde\phi_{i\nu} \rangle. \]

A physical one-electron operator \(\hat X\) acts in the all-electron space. Its auxiliary-space representative is

\[ \hat{\widetilde X} \equiv \hat{\mathcal T}^\dagger \hat X \hat{\mathcal T}. \]

In particular,

\[ \hat{\widetilde H} = \hat{\mathcal T}^\dagger \hat H_{\mathrm{KS}} \hat{\mathcal T}, \]

and the PAW Kohn–Sham equations are

\[ \hat{\widetilde H} \lvert\widetilde\psi_a\rangle = \varepsilon_a \hat S \lvert\widetilde\psi_a\rangle, \qquad \langle \widetilde\psi_a \vert \hat S \vert \widetilde\psi_b \rangle = \delta_{ab}. \]

After either an LCAO or plane-wave basis is chosen, the corresponding matrices obey the same generalized eigenproblem

\[ \mathbf H\mathbf C = \mathbf S\mathbf C\mathbf E, \qquad \mathbf C^\dagger \mathbf S \mathbf C = \mathbf I. \]

The columns of \(\mathbf C\) are the auxiliary-spinor coefficient vectors, and \(\mathbf E\) is the diagonal matrix of Kohn–Sham eigenvalues. The matrix \(\mathbf S\) is the complete PAW overlap matrix in the chosen basis. For plane waves, the identity part of \(\hat S\) gives the identity matrix. For an LCAO basis, it gives the ordinary basis-overlap matrix, to which the projector corrections in \(\hat S\) are added. There is only one overlap matrix on the right-hand side of the generalized eigenproblem.

Density Operators and Expectation Values

The auxiliary one-particle density operator is

\[ \hat{\widetilde\rho} = \sum_a f_a \lvert\widetilde\psi_a\rangle \langle\widetilde\psi_a\rvert. \]

Here \(f_a\) is the occupation of an explicit spinor state, with no implicit factor of two. The normalized Brillouin-zone average is included for a periodic system. The all-electron valence density operator is

\[ \hat\rho_{\mathrm v} = \hat{\mathcal T} \hat{\widetilde\rho} \hat{\mathcal T}^\dagger. \]

This relation distinguishes the auxiliary density operator from the physical all-electron object. For any one-electron operator,

\[ \langle\hat X\rangle_{\mathrm v} = \operatorname{tr} \left( \hat\rho_{\mathrm v}\hat X \right) = \operatorname{tr} \left( \hat{\widetilde\rho}\hat{\widetilde X} \right). \]

The valence electron number is therefore

\[ N_{\mathrm v} = \operatorname{tr} \left( \hat{\widetilde\rho}\hat S \right). \]

After a finite basis is chosen, the density matrix is

\[ \mathbf D = \mathbf C\mathbf f\mathbf C^\dagger. \]

Here \(\mathbf f\) is the diagonal occupation matrix. The matrix \(\mathbf D\) represents \(\hat{\widetilde\rho}\) through the chosen basis expansion; in a nonorthogonal LCAO basis it is not the matrix of covariant operator elements. It does not represent \(\hat\rho_{\mathrm v}\) directly. Consequently, \(\operatorname{tr}(\mathbf D\mathbf S)\) gives the valence electron number, while a physical observable uses the matrix of its transformed PAW operator.

On-Site Projector Density Matrices

The state-dependent information entering the one-center PAW corrections is obtained by projecting \(\hat{\widetilde\rho}\) onto the localized projectors. For spinful data, define

\[ P^i_{\mu s,\nu s'} \equiv \langle \widetilde p_{i\mu};s \vert \hat{\widetilde\rho} \vert \widetilde p_{i\nu};s' \rangle. \]

The complete matrix \(\mathbf P^i\) is Hermitian and is the on-site projector density matrix of atom \(i\). It is a PAW augmentation quantity and must not be confused with the global density matrix \(\mathbf D\). In a periodic calculation, let \(\lvert\widetilde p_{i\mu\boldsymbol k};s\rangle\) denote the \(\boldsymbol k\)-compatible Bloch sum of the translated projectors, with the cell normalization of Periodic Systems. Then

\[ P^i_{\mu s,\nu s'} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \sum_a f_{a\boldsymbol k} \langle \widetilde p_{i\mu\boldsymbol k};s \vert \widetilde\psi_{a\boldsymbol k} \rangle_\Omega \langle \widetilde\psi_{a\boldsymbol k} \vert \widetilde p_{i\nu\boldsymbol k};s' \rangle_\Omega. \]

When this integral is evaluated on an irreducible mesh, symmetry-related contributions must be restored with the transformations of atomic sites, projector channels, and spin; multiplicity weights alone are not generally sufficient.

PAW literature and code outputs often denote its elements by \(D^i_{\mu\nu}\) or \(\rho^i_{\mu\nu}\). The symbol \(\mathbf P^i\) is used here so that \(\mathbf D\) remains reserved for the global density matrix. A common alternative orders the amplitudes as

\[ \langle \widetilde\psi \vert \widetilde p_{i\mu};s \rangle \langle \widetilde p_{i\nu};s' \vert \widetilde\psi \rangle. \]

After the same occupation and \(\boldsymbol k\)-point sums, the matrix element defined with this order is \(P^i_{\nu s',\mu s}\). This index mapping matters when the matrices are complex.

The off-diagonal spin elements of \(\mathbf P^i\) carry the local noncollinear spin coherence. A collinear calculation retains only its spin-diagonal elements in the chosen global frame.

Smooth and One-Center Densities

The smooth valence spin-density matrix is the diagonal real-space kernel of the auxiliary density operator,

\[ \widetilde n_{ss'}(\boldsymbol r) = \langle \boldsymbol r,s \vert \hat{\widetilde\rho} \vert \boldsymbol r,s' \rangle. \]

The all-electron and smooth one-center densities associated with atom \(i\) are reconstructed from the same projector density matrix,

\[ n_{ss'}^{1,i}(\boldsymbol r) = \sum_{\mu\nu} \phi_{i\mu}(\boldsymbol r) P^i_{\mu s,\nu s'} \phi_{i\nu}^*(\boldsymbol r), \]
\[ \widetilde n_{ss'}^{1,i}(\boldsymbol r) = \sum_{\mu\nu} \widetilde\phi_{i\mu}(\boldsymbol r) P^i_{\mu s,\nu s'} \widetilde\phi_{i\nu}^*(\boldsymbol r). \]

The complete one-center spin-density matrices are

\[ \mathbf n^{1,i} = \left( n_{ss'}^{1,i} \right)_{ss'}, \qquad \widetilde{\mathbf n}^{1,i} = \left( \widetilde n_{ss'}^{1,i} \right)_{ss'}. \]

Under the same nonoverlapping-augmentation-region assumption used for the projector form of \(\hat S\), and within the partial-wave completeness stated above, the corrections from different atoms have disjoint support. The partial waves are centered at \(\boldsymbol R_i\), and the all-electron and smooth partial waves agree outside the augmentation region. Their difference, and hence the one-center density correction, vanishes there. The physical valence spin-density matrix is

\[ n_{ss'}^{\mathrm v}(\boldsymbol r) = \widetilde n_{ss'}(\boldsymbol r) + \sum_i \left[ n_{ss'}^{1,i}(\boldsymbol r) - \widetilde n_{ss'}^{1,i}(\boldsymbol r) \right]. \]

The scalar valence density and spin-polarization density follow from the same Pauli decomposition as in Kohn–Sham Density Functional Theory. The fixed frozen-core density must be added when the total all-electron density is required. In the usual spin-unpolarized frozen-core approximation,

\[ \mathbf n_{\mathrm c}(\boldsymbol r) = \frac{1}{2} \sum_i n_{\mathrm c}^i \left( \boldsymbol r-\boldsymbol R_i \right) \sigma_0. \]

The smooth density, the one-center densities, and the frozen-core density are distinct objects even when a code combines some of them in a stored output.

Compensation Charge and Electrostatics

Define the scalar one-center densities by tracing over spin,

\[ n^{1,i}(\boldsymbol r) \equiv \sum_s n_{ss}^{1,i}(\boldsymbol r), \qquad \widetilde n^{1,i}(\boldsymbol r) \equiv \sum_s \widetilde n_{ss}^{1,i}(\boldsymbol r). \]

The difference \(n^{1,i}-\widetilde n^{1,i}\) is localized but generally has nonzero electrostatic multipole moments. Evaluating its long-range Coulomb interaction directly would couple smooth three-dimensional grids to atom-centered radial grids.

PAW therefore introduces a smooth localized compensation charge \(n_{\mathrm{comp}}^i(\boldsymbol r)\). Its multipole moments are chosen to match those of the relevant all-electron minus smooth one-center charge. In an electron-number-density convention, the defining condition has the form

\[ \int \mathrm d\boldsymbol r\, \left[ n^{1,i}(\boldsymbol r) - \widetilde n^{1,i}(\boldsymbol r) - n_{\mathrm{comp}}^i(\boldsymbol r) \right] \lvert\boldsymbol r-\boldsymbol R_i\rvert^L \mathcal Y_L^M \left( \widehat{\boldsymbol r-\boldsymbol R_i} \right) = 0 \]

Here \(\mathcal Y_L^M\) is a real spherical harmonic. The condition is imposed for every retained multipole \((L,M)\). Ionic and frozen-core terms are included in the corresponding code-specific charge convention.

The compensation charge is expanded in dataset-supplied localized functions, with state-dependent coefficients determined by \(\mathbf P^i\) and fixed core and ionic moments.

Some implementations call this quantity an augmentation charge. The term compensation charge is used here to distinguish it from the physical one-center density correction \(\mathbf n^{1,i}-\widetilde{\mathbf n}^{1,i}\).

The compensation charge allows the long-range electrostatics to be evaluated from smooth quantities. It is an auxiliary charge used by the PAW energy functional, not an additional contribution to the physical all-electron density defined above.

Hamiltonian and Total Energy

The transformed PAW Hamiltonian can be organized as a smooth local operator plus atom-centered projector terms,

\[ \hat{\widetilde H} = \hat H_{\mathrm{smooth}} + \sum_i \sum_{\mu\nu} \sum_{ss'} \lvert\widetilde p_{i\mu};s\rangle h^i_{\mu s,\nu s'} \langle\widetilde p_{i\nu};s'\rvert. \]

The on-site coefficient matrices \(\mathbf h^i\) are obtained by differentiating the PAW energy with respect to \(\mathbf P^i\). They depend on the self-consistent smooth fields and one-center densities; spin-dependent and spin–orbit terms may make them nondiagonal in spin.

The total energy has the corresponding smooth-plus-one-center structure,

\[ E_{\mathrm{PAW}} = \widetilde E + \sum_i \left( E^{1,i} - \widetilde E^{1,i} \right). \]

The smooth term contains the auxiliary-orbital kinetic energy and the smooth-grid electrostatic and exchange–correlation contributions, including the compensation charges. The one-center difference restores the all-electron partial-wave, frozen-core, and ionic contributions without double counting the smooth contribution. This equation specifies the decomposition rather than a separate approximation to the Kohn–Sham functional.

Forces and stresses are derivatives of this complete PAW energy. They include derivatives of projectors, compensation charges, one-center matrices, and the PAW overlap matrix. If the auxiliary spinors are expanded in atom-centered orbitals, ordinary basis-response terms are present as well.

Object Summary

The practical PAW objects should be kept distinct:

  • \(\mathbf H\) and \(\mathbf S\) are global matrices in the chosen auxiliary basis;
  • \(\mathbf D=\mathbf C\mathbf f\mathbf C^\dagger\) is the global auxiliary density matrix;
  • \(\mathbf P^i\) is an atom-centered projector density matrix derived from \(\mathbf D\);
  • \(\widetilde{\mathbf n}(\boldsymbol r)\) is the smooth grid density, whereas \(\mathbf n^{1,i}\) and \(\widetilde{\mathbf n}^{1,i}\) are one-center densities;
  • \(n_{\mathrm{comp}}^i\) is an auxiliary electrostatic quantity and is not part of the physical all-electron density.

PAW accuracy is controlled jointly by the auxiliary basis or grid, the partial-wave completeness, the augmentation radii, and the frozen-core partition. The reconstruction does not remove dataset error, basis incompleteness, or errors caused by strongly overlapping augmentation regions.

References