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Pseudopotentials and Effective-Core Operators

Pseudopotential methods replace the explicit all-electron core problem by an effective valence-electron problem. This chapter fixes the operator and density conventions for norm-conserving pseudopotentials. The PAW reconstruction is treated separately in Projector-Augmented-Wave Method. Hartree atomic units are used unless stated otherwise.

The pseudopotential is part of the definition of the electronic problem, not only a numerical approximation to a fixed Hamiltonian. It specifies the core–valence partition, the effective ionic potential, the reference exchange–correlation functional, and the relativistic treatment inherited from the atomic calculation.

Valence-Electron Problem

Let \(i\) label an ion at \(\boldsymbol R_i\). Electrons assigned to the frozen core are removed from the variational one-electron space, while the remaining valence states are represented by smooth pseudo spinors \(\lvert\widetilde\psi_a\rangle\), where \(a\) labels a one-electron state. The corresponding Kohn–Sham Hamiltonian has the schematic form

\[ \hat H_{\mathrm{PS}} = -\frac{1}{2}\nabla^2\sigma_0 +\hat V_{\mathrm{ion}} +v_{\mathrm H}[n_{\mathrm v}](\boldsymbol r)\sigma_0 +\mathbf v_{\mathrm{xc}} \left[ n_{\mathrm v}+n_{\mathrm c}^{\mathrm{model}}, \boldsymbol m_{\mathrm v} \right](\boldsymbol r). \]

Here \(n_{\mathrm v}\) and \(\boldsymbol m_{\mathrm v}\) are the valence electron density and spin-polarization density. The fixed model core density \(n_{\mathrm c}^{\mathrm{model}}\) is present only when a nonlinear core correction is used. It enters the exchange–correlation functional but is not an additional variational density and is not included in the valence Hartree term.

The effective ionic operator contains a local part and a nonlocal projector part,

\[ \hat V_{\mathrm{ion}} = \sum_i v_i^{\mathrm{loc}} \left( \hat{\boldsymbol r}-\boldsymbol R_i \right) \sigma_0 +\hat V_{\mathrm{NL}}. \]

For a separable pseudopotential, the nonlocal operator is written as

\[ \hat V_{\mathrm{NL}} = \sum_i \sum_{\mu\nu} \sum_{ss'} \lvert\beta_{i\mu};s\rangle v^i_{\mu s,\nu s'} \langle\beta_{i\nu};s'\rvert. \]

The indices \(\mu\) and \(\nu\) label atom-centered projector channels, including their radial and angular labels, while \(s\) and \(s'\) are explicit spin indices. The functions \(\beta_{i\mu}\) and the finite coefficient matrices \(\mathbf v^i\) belong to the pseudopotential dataset. In a scalar-relativistic dataset the nonlocal operator is spin diagonal. A fully relativistic dataset may couple spin components and supplies the spin–orbit part of the ionic operator.

For periodic systems, the atom-centered local potentials and projectors are translated over the lattice. Their matrices at each \(\boldsymbol k\) point are obtained using the Fourier conventions of Periodic Systems. The operator itself is independent of whether it is represented in a localized atomic-orbital basis or a plane-wave basis.

Norm-Conserving Pseudopotentials

A norm-conserving dataset constructs each reference pseudo partial wave to match its all-electron counterpart outside a channel-dependent core radius and to have the same integrated norm inside that radius. The pseudo partial wave is therefore smooth near the nucleus while preserving the scattering properties selected during dataset generation.

Norm conservation gives the ordinary inner product,

\[ \hat S=\hat I. \]

The pseudo spinors consequently satisfy the standard eigenproblem and orthonormality conditions,

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

Their 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 Brillouin-zone average is understood for a periodic system. The valence spin-density matrix is the diagonal real-space kernel of \(\hat{\widetilde\rho}\),

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

This density is the self-consistent valence density of the norm-conserving pseudo problem. It is not, in general, the pointwise all-electron density inside the core region. Core-sensitive observables require additional reconstruction information even though the state normalization is ordinary.

Core Density and Relativistic Content

The core–valence partition is dataset dependent. Semicore shells may be treated either as frozen core or as explicit valence states; this choice changes the electron number, the projector set, and the range of environments over which the dataset is transferable.

A nonlinear core correction supplies a fixed smooth approximation to the core electron density in the exchange–correlation evaluation. In the usual spin-unpolarized frozen-core approximation, its contribution to the local spin-density matrix is

\[ \mathbf n_{\mathrm c}^{\mathrm{model}}(\boldsymbol r) = \frac{1}{2} n_{\mathrm c}^{\mathrm{model}}(\boldsymbol r) \sigma_0. \]

This fixed core contribution is not part of the variational valence density.

Scalar-relativistic datasets include mass–velocity and Darwin effects in the atomic reference problem but remain spin diagonal. Fully relativistic datasets retain the angular-momentum-dependent spinor structure needed for spin–orbit coupling. Enabling spin–orbit coupling with a dataset that does not contain the corresponding relativistic channels cannot recover the missing operator.

The unique spin-traceless part of the fully relativistic nonlocal operator and its direct evaluation in a numerical LCAO basis are defined in Fixed Spin–Orbit Operators in an LCAO Basis.

References