Delta Self-Consistent Field

Linear expansion Delta Self-Consistent Field

The method of linear expansion Delta Self-Consistent Field [1] adds the density of a specified orbital \(\varphi_a(r)\) to the total density in each step of the self-consistency cycle. The extra charge is usually taken from the fermi level to keep the system neutral:

\[n(r) = \sum_nf_{N-1}(T,\varepsilon_n)|\varphi_n(r)|^2 + |\varphi_a(r)|^2.\]

with \(N\) being the total number of electrons and \(f_{N-1}(T,\varepsilon_n)\) is the Fermi-Dirac distribution of the \(N-1\) electron system . To get the band energy right \(\varphi_a(r)\) needs to be expanded in Kohn-Sham orbitals:

\[|\varphi_a\rangle = \sum_nc_{na}|\varphi_n\rangle, \qquad c_{na} = \langle\varphi_n|\varphi_a\rangle\]

and the band energy of the orbital becomes

\[\varepsilon_a = \sum_n|c_{na}|^2\varepsilon_n.\]

The method is a generalization of traditional Delta Self-Consistent Field where only the occupation numbers are modified and it will reduce to that, if only one (normalized) term is included in the expansion of \(\varphi_a(r)\).

Simple molecules

The example below calculates the excitation energy of the \(5\sigma\rightarrow2\pi\) transition in CO. We only specify that the \(2\pi\) orbital should be occupied ([[1.0, lumo, 1]] means 1.0 electrons in lumo with spin 1) and the method will take the electron from highest occupied orbital which in this case is \(5\sigma\).

The lumo is an instance of the class AEOrbital which calculates the expansion of the saved \(2\pi\) state in each iteration step. In order to obtain the all-electron overlaps \(\langle\varphi_n|2\pi\rangle\) we need to supply the projector overlaps in addition to the pseudowavefunction.

Exciting the LUMO in CO (

from import molecule
from gpaw import GPAW
from gpaw import dscf

# Ground state calculation
calc = GPAW(mode='fd',
            convergence={'energy': 100,
                         'density': 100,
                         'eigenstates': 1.0e-9,
                         'bands': -1})

CO = molecule('CO')
CO.calc = calc

E_gs = CO.get_potential_energy()

# Obtain the pseudowavefunctions and projector overlaps of the
# state which is to be occupied. n=5,6 is the 2pix and 2piy orbitals
n = 5
molecule = [0, 1]
wf_u = [kpt.psit_nG[n] for kpt in calc.wfs.kpt_u]
p_uai = [dict([(molecule[a], P_ni[n]) for a, P_ni in kpt.P_ani.items()])
         for kpt in calc.wfs.kpt_u]

# Excited state calculation
calc_es = GPAW(mode='fd',
               convergence={'energy': 100,
                            'density': 100,
                            'eigenstates': 1.0e-9,
                            'bands': -1})

CO.calc = calc_es
lumo = dscf.AEOrbital(calc_es, wf_u, p_uai)
# lumo = dscf.MolecularOrbital(calc, weights={0: [0, 0, 0,  1],
#                                             1: [0, 0, 0, -1]})
dscf.dscf_calculation(calc_es, [[1.0, lumo, 1]], CO)

E_es = CO.get_potential_energy()

print('Excitation energy: ', E_es - E_gs)

The commented lines lumo = dscf.Molecular... uses another class to specify the \(2\pi\) orbital of CO which does not require a ground state calculation of the molecule. In the simple example above the two methods give identical results, but for more complicated systems the AEOrbital class should be used [2]. When using the AEOrbital class a new calculator object must be constructed for the dscf calculation.

In the example above we only specify a single state, but the function dscf.dscf_calculation takes a list of orbitals as input and we could for example have given the argument [[1.0, lumo, 1], [-1.0, pi, 0]] which would force the electron to be taken from the \(\pi\) orbital with spin 0. The pi should of course be another instance of the AEOrbital class.

Exciting an adsorbate

The method of linear expansion Delta Self-Consistent Field was designed for calculations with strongly hybridized orbitals. For example molecules chemisorbed on transition metals. In such cases the traditional Delta Self-Consistent Field breaks down since the orbital to be occupied is no longer well described by a single Kohn-Sham state.

The script calculates the HOMO energy of CO adsorbed on-top Pt(111). The script starts from scratch, but usually one would start from an optimized configuration saved in a file gs.gpw. The script only calculates the total energy of the excited state so the excitation energy is obtained as the difference between ground and excited state energies.

First a calculation of gas-phase CO is performed and the HOMO pseudo-wavefunctions and the projector overlaps are saved. The energy range [-100.0, 0.0] means we only include states below the Fermi level (default is states above).

The script calculates the LUMO energy of the same system, but is slightly more complicated due to the degeneracy of the \(2\pi\) orbital. We would like to occupy the \(2\pi_y\) orbital and we need to figure out which band (5 or 6) this orbital corresponds to in each k-point before we start the slab calculation.