Linear expansion DeltaSCF DFT calculations - Constraining projected reference orbitals

Basics

For molecules adsorbed at surfaces, $\Delta$ SCF calculations become challenging, simply because of the large number of states. Not only is it difficult to identify the correct state, often hybridization spreads adsorbate molecular orbitals across a large number of substrate bands. A more realistic excitation constraint would be to project on a gasphase molecular orbital and subsequently enforce occupation of this state.

This idea is referred to as linear expansion $\Delta$ SCF and has first been proposed by Gavnholt et al¹.

The CASTEP implementation is described in Maurer and Reuer (2013)². Herein, we constrain the occupation of a so-called resonance state built from a linear combination of Kohn-Sham states instead of a single KS state. We expand an abitrary reference state $|\phi_c\rangle$ in the space of Kohn-Sham states as follows:

$|\tilde{\psi}_c^{\mathbf{k}}\rangle = \sum_i^{\mathrm{states}} |\psi_i^{\mathbf{k}}\rangle\langle\psi_i^{\mathbf{k}}|\phi_c^{\mathbf{k}} \rangle$

At the same time we orthogonalize the remaining KS states

$|\tilde{\psi_i^{\mathbf{k}}}\rangle = |\psi_i^{\mathbf{k}}\rangle - \sum_c^{\mathrm{constr.}}|\phi_c^{\mathbf{k}}\rangle\langle\phi_c^{\mathbf{k}}|\psi_i^{\mathbf{k}}\rangle$

After an additional orthonormalization of all states we have constructed a resonance state $\tilde{\psi_c^{\mathbf{k}}}$ and removed all its contributions from all other states. We can now constrain the occupation of this state with the effect of describing an excitation of a specific molecular orbital. This is done in every SCF step until the calculation is converged.

For a le $\Delta$ SCF calculation, we have to set deltascf_method = linear expansion in the <seed>.param file. The constraint must also be specifed, as discussed in the overview section.

The only additional relevant $\Delta$ SCF keywords is deltascf_overlap_cutoff

Example .param file:

reuse               : <base>.check
calculate_deltascf  : true
deltascf_checkpoint : <base>.check
deltascf_method     : linear expansion
deltascf_overlap_cutoff : 0.01

#band  occ  spin
%block deltascf_constraints
35    1.0000  1
%endblock deltascf_constraints

In this example, we take state 35 from wavefunction file <base>.check and enforce an occupation of 1.00 electrons. In this runmode we do not have to pick constraints that yield a net change in charge equal to 0. Charge neutrality will be satisfied by modifying the Fermi level accordingly. However, this is only strictly sensible for metallic systems.

Jeppe Gavnholt, Thomas Olsen, Mads Engelund, and Jakob Schiøtz. Δ self-consistent field method to obtain potential energy surfaces of excited molecules on surfaces. Phys. Rev. B, 78:075441, Aug 2008. doi:10.1103/PhysRevB.78.075441. ↩
Reinhard J. Maurer and Karsten Reuter. Excited-state potential-energy surfaces of metal-adsorbed organic molecules from linear expansion Δ-self-consistent field density-functional theory (ΔSCF-DFT). The Journal of Chemical Physics, 139(1):014708, 07 2013. doi:10.1063/1.4812398. ↩