The Hubbard U Parameter

An additional benefit of using DFT over Hartree-Fock, besides the decrease in com- putational cost, is that dynamic electron correlation - the probability density of one electron reduces the probability of finding another electron in the same space - is

built into DFT explicitly, whereas this is not the case for Hartree-Fock, see sec- tion 2.3.2. However, Hartree-Fock includes the exact electron repulsion integrals whereas this is missing from DFT when using either LDA or GGA-based function- als.

There is, therefore, a need for a DFT-based method that includes a correct descrip- tion for the strong Coulomb repulsions between localised electrons e.g. those occu- pying d or f orbitals. The Hubbard model was first designed after it was established that many electrons retain well localised character in some solid state materials, as opposed to to being a delocalised cloud in the weak electrostatic field of the nuclei - i.e. modelled via the homogeneous electron gas as in LDA - which has a tendency to over-delocalise the electrons.[251]

The key error this over-delocalisation introduces into the system,[252] particu- larly for the work presented here, is the band gap problem - with DFT predicting metallic ground states for a number of transition metal oxides instead of insulating ones.[253] Hybrid functionals, described earlier in the chapter, can be implemented to correct for this over-delocalisation, however they are computationally expensive and the amount of HF exchange to include can be difficult to assess, depending on the property of interest. An alternative method is to introduce the Hubbard U parameter on specific ions that require a correct description of localised electrons. This method is referred to as DFT+U.

The Hubbard Model[121] is able to explain the transition between the limits of the band model in a conductor and the localised limit in an insulator, using only two interaction terms. Competing interactions - the kinetic energy operator and the weak electron-nuclear attraction, which favour delocalisation, and a Coulomb repulsion term resulting from the pairing of two electrons on an atomic site - are tuned, and the full Hamiltonian is represented as follows:

H=

_∑

i j

∑

µ ν ˆ T_{i j}µ νc†_iµcjν+

_∑

i jklµ ν σ τ

∑

hµiνj|σkτlic†iµcjνc † lτckσ (2.59)

This expression can be split into two parts: the first is the transfer integral where the electrons are described as being able to move only between orbitals µ and ν

on nearest neighbour sites: i and j. The second part corresponds to the electron- electron repulsion integrals of four orbitals; µ, ν, σ and τ on four atomic sites: i, j, k and l. The original model can be greatly simplified by only considering electron- electron repulsions on the same atom, which are an order of magnitude greater than other electron-electron repulsions:[121]

H= −t

_∑

hi, ji,σ (c†_i,σc_j,σ+ c†_j,σc_i,σ) +U N

∑

i=1 NI↑NI↓ (2.60)

Here the transfer integral is treated as a single parameter; t, and is the kinetic energy associated with delocalising the electrons, U is the Coulomb energy required to pair two electrons on one atom, N is the number of atoms, and NI↑ and NI↓ are the number of spin up or spin down electrons on each atom, respectively.[121] The original work only considered a single pair of electrons;[121] however it is usually employed across all d or f electrons in the system.[251]

Anderson Model The Anderson impurity model[122] includes all the elements of Hubbard’s model described above, but additionally considers the hybridisation be- tween the localised d electrons with the delocalised s electrons. His model is built on the basis that it is more important to describe the repulsion between opposite spin electrons, rather than the attraction between electrons with the same spins; known as the Coulomb (J) and Exchange (K) integrals respectively, in order to describe localised magnetic moments correctly. In implementing the Anderson Model the U term is represented as Ueff:

U_{e f f} = U − J (2.61) Where U is the energy penalty of pairing two electrons on one atom, as is the case with Hubbard’s model.

Implementing DFT+U An energy calculation in DFT+U generally involves com- bining 3 terms; the DFT calculation (EDFT), the Hubbard term (EHub) and a double counting term (EDC):[251]

EDFT+U= EDFT+ EHub− EDC (2.62) The Hubbard term is the contribution to the energy from the Coulomb interaction between localised orbitals e.g. the d orbitals, whereas the double counting term removes the mean field description of these orbitals obtained from the homoge- neous electron gas model. A common form of the double counting terms is as follows:[254] EDC[N_nlI ] = 1 2U I nlNnlI (NnlI − 1) − 1 2J[N I↑ nl(N I↑ nl − 1) + N I↓ nl(N I↓ nl − 1)] (2.63)

where NnlI is the total number of electrons in the localised subshell, with NnlI↑

and NnlI↓ being the number of spin up and spin down electrons in each subshell

respectively.

There are a number of ways to implement DFT+U, which differ by a number of factors, for example the simplifications used, e.g. Ueff, or by the choice of lo-

cal projection used.[255] In this work, DFT+U is implemented as LDSA+U in- troduced by Dudarev et al.[256] using a projector-augmented wave (PAW)-based approach,[257] as used in many electronic calculations in VASP.[258] This form of DFT+U combines the rotationally invariant functional from Liechtenstein[259] and the orbital-dependent formulation by Anisimov et al.[260], while retaining the simplified Ueff parameter employed in the Anisimov formulation. The LDSA+U

functional defined as:[256]

ELSDA+U= ELSDA+( ¯U− ¯J) 2

∑

_σ "

∑

j ρσ_{j j} ! −

_∑

j,l ρσ_jlρ_{l j}σ !# (2.64)

Where ¯U and ¯J are spherically average matrix elements of the screened Coulomb electron-electron interaction and ρσ

jl is the density matrix of d electrons. At the

limit of integer occupation numbers - where terms representing interger number of delectrons and non-integer numbers of d electrons cancel - the second term on the right hand side of equation 2.64 vanishes and the energy can be expressed in terms of Kohn-Sham eigenvalues: εi

ELSDA+U= ELSDA[εi] +

( ¯U− ¯J) 2 _{l, j,σ}

∑

ρ

σ

l jρσjl (2.65)

with the final term in equation 2.65 representing the double counting term as defined above.

2.3.4

Defining the Wave function: Plane waves and Pseudopo-

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