Polarizability of molecular chains: does one need exact exchange?
📝 Abstract
Standard density functional approximations greatly over-estimate the static polarizability of longchain polymers, but Hartree-Fock or exact exchange calculations do not. Simple self-interaction corrected (SIC) approximations can be even better than exact exchange, while their computational cost can scale only linearly with the number of occupied orbitals.
💡 Analysis
Standard density functional approximations greatly over-estimate the static polarizability of longchain polymers, but Hartree-Fock or exact exchange calculations do not. Simple self-interaction corrected (SIC) approximations can be even better than exact exchange, while their computational cost can scale only linearly with the number of occupied orbitals.
📄 Content
Ground-state Kohn-Sham (KS) density functional theory (DFT) has become extraordinarily popular for solving electronic structure problems in solid-state physics, quantum chemistry and materials science [1]. The accuracy of modern generalized gradient approximations (GGAs) and hybrid functionals has proven sufficient for many applications, often with surprisingly small errors. Bond dissociation energies, geometries, phonons, etc., are now routinely calculated with errors of 10-20%.
But local and gradient-corrected functionals overestimate massively the static polarizability and hyperpolarizability of molecular chains, especially conjugated polymers. This failure has been the subject of many studies over the last decade [2,3,4,5,6,7,8,9,10], studies which highlight the important role played by the response field originating from the exchange-correlation (XC) potential. The exact induced XC field counteracts the applied external field, keeping the polarization low. In the local (or gradient-corrected) density approximation (LDA), this field erroneously points in the same direction as the applied field [3,4,5]. Such failures of standard functionals appear in other contexts, such as transport through single molecules [11], or the polarizability of large molecules.
In contrast, these effects are easily captured within standard wavefunction theory. In particular, Hartree-Fock (HF) theory does not greatly overestimate the polarizabilities and provides a good starting point for more accurate wavefunction treatments, such as Möller-Plesset (MP) perturbation theory. Thus exact exchange (EXX) DFT, the KS-DFT method for minimizing the HF energy while retaining a single multiplicative potential, provides a promising alternative and indeed has been found to give results very similar to HF [4,8,10]. This improvement can be attributed to the orbital-dependence of EXX, and the lack of self-interaction error [8], i.e. EXX is exact for one electron, unlike LDA or GGA.
However, EXX is only one among many possible selfinteraction free functionals that one may construct. In fact any GGA can be corrected to become self-interaction free (self-interaction corrected -SIC) by direct subtraction of the XC functional evaluated on each of the individual orbitals [12]. While this can be performed for either LDA or GGA, only LDA has significantly improved energetics from this procedure, but many investigators are searching for useful methods to correct GGA’s for self-interaction [13]. More importantly, EXX includes a sum over all unoccupied orbitals, while SIC functionals use only occupied ones, i.e., EXX can often be substantially more expensive computationally. So the question then becomes: does one really need EXX, or will any self-interaction free functional perform equally well ?
We perform SIC calculations for the polarizabilities of hydrogenic chains using LDA and GGA. Our SIC potential is constructed using the optimized effective potential (OEP) framework within the Krieger-Li-Iafrate (KLI) approximation [14]. Using results from accurate wave-function methods as a benchmark, we find that the polarizabilities calculated with KLI-SIC are in better agreement than those obtained with KLI exact exchange (X-KLI), with the remaining error attributed to the KLI approximation [10,15]. This is an important result since KLI-SIC scales only linearly with the number of occupied Kohn-Sham (KS) orbitals, as compared with the quadratic scaling of X-KLI. Thus SIC becomes a valuable scheme for evaluating the static polarizability of polymers with large unit cells, and in other applications where these effects may be important [11].
We start with a brief description of the SIC method used in this work. In DFT [16], the total energy functional E[ρ ↑ , ρ ↓ ] (ρ σ is the spin σ =↑, ↓ density, ρ = σ ρ σ ) can be written as
(1) with T S the kinetic energy of the non-interacting KS orbitals, v(r) the external potential, U the Hartree energy, and E xc the XC energy. For any GGA
where ρ σ n = |ψ σ n | 2 is the density of the n-th KS orbital. Levy’s minimization [17] leads to a set of single particle KS-like equations for ψ σ n with corresponding eigenvalues ǫ σ,SIC n and occupation numbers
The effective potential v σ eff,n (r) is now KS-orbital dependent and cannot be classified as a standard multiplicative KS-potential. For instance it is ambiguously defined for unoccupied KS-orbitals since the SIC is only defined for the occupied ones. The solution of equation ( 3) has followed several approaches. The simplest one is to solve it directly under a normalization constraint with the resulting non-orthogonal orbitals undergoing an orthogonalization procedure [12]. This scheme however is not free of complications, since E SIC xc is not invariant under a unitary transformations of the occupied {ψ σ n }. The solution [18] is then to work with an auxiliary set of localized orbitals {φ σ n } used for constructing v σ eff,n (r) and related to {ψ σ n } by a uni
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