Non-local potential

Slater then proposed returning to one-electron Schrodinger equations (and therefore one-electron wave functions i and corresponding eigenvalues ,) but now using not the Hartree-Fock (non-local) potential in these equations but the simplified potential... 

In general, however, in the reduction of the many electron to single-electron equations there appears an additional, non-local potential. Then (10) is replaced by... 

Let us consider a few examples. If we replace the non-local potential by a local one, then we arrive at the homogeneous equation for each shell... 

Hohenberg-Kohn theorems, but use the Kohn-Sham construction and local approximations to such non-local potentials and often lump together the exchange and the correlation energies into an exchange-correlation energy Exc[n], This yields a local exchange-correlation potential vxc(t) in the Kohn-Sham equations that determine the Kohn-Sham spin orbitals j, i.e. 

The Use of Pseudopotentials in Molecular Calculations arbitrary non-local potential ... 

Since the Gaunt Coefficient (2 k CfJ- > 2 k) may be expressed in terms of Clebsch-Gordon coefficients it is easy to show that the parameters Ejm and 22jm are proportional so that the reduction of the matrix elements of the static potential to combinations of radial integrals over the r/-orbital basis (potential parameters) has the same structure for both local and non-local potentials which can therefore be treated on the same footing. 

Other alternatives to the FNA include the A-function method, [58-60]. The method builds nodal surfaces from a sum of Gaussian functions centered on the DMC walkers. There is also lattice-regularized DMC (LR-DMC) in which the kinetic energy is discretized by a finite difference Laplacian with two meshes. The use of a regularized Hamiltonian also permits a consistent variational treatment of non-local potentials [61]. 

In other words, the potential term in the equation determining the singleparticle function ij/, is to be found self-consistently from the mean field of the other electrons. The Hartree-Fock (HF) approximation incorporates Fermi statistics into this picture by replacing the product wavefunction (Eq. (1)) by a single determinantal function (Slater determinant) constructed from the tjf,. In this case, the single-particle functions satisfy an equation similar to (2) but with an additional non-local potential term. 

To this we follow closely Baym and Baym and Kadanoff in their discussion of perturbation theory for non-uniform systems. They consider non-equilibrium thermal Green s functions at an arbitrary temperature B in the presence of an external, in general non-local potential U that acts in the imaginary time interval O < it < 6. Writing the operators in the interaction representation for the external disturbance, the one-particle Green s function is... 

For the total energy, useful explicit reciprocal-space expressions were given by Ihm, Zunger and Cohen , who also covered non-local potentials. The formulas were elaborated by Yin and Cohenand Nielsen and Martin." The expression for the total energy (in atomic units where m = A = e=1) per unit cell volume Q is ... 

The non-local densetj/-dependence of the xc potential should not be confused with whether the Vxc (r) is a local or non-local potential in its dependence on its spatial argument r in Kohn-Sham theory the xc potential is always a local potential in the latter sense. 

We have seen that many-body-based methods provide an ab-initio way to treat the Coulomb correlation in an N electron system without the expensive cost of QMC calculations. However, they are computationally more demanding than routine LDA-KS calculations and, hence, the feasibility of their application to complex systems is unclear, especially in the context of ab-initio molecular dynamics calculations, where many total-energy evaluations are required. As described in Sect. 5.3, the main problem when constructing approximations to E c [n] is related to its inherent non-analytical character which is due to the specific way in which the KS mapping between the real and the fictitious systems is done. However, this is not the only possible realisation of DFT and recently, new DFT methods have been proposed [112,113]. In these generalised Kohn-Sham schemes (GKS) the actual electron system is mapped onto a fictitious one in which particles move in an effective non-local potential. As a result of this, it is possible to describe structmal properties at the same (or better) level than LDA/GGA but improving on its description of quasiparticle properties. 

There are some aspects of the pseudopotential, at least in the way that was formulated above, that make it somewhat suspicious. First, it is a non-local potential ... 

Pseudopotentials in DMC computations lead to special consideration owing to their non-locality. There is no difficulty with VMC calculations using a non-local potential. As long as one has a method to evaluate the operation of the pseudopotential operator on the trial function then it contributes to the local energy as any other term. In the DMC method, however, one does not have a method of evaluating the effect of the pseudopotential operator on the fixed-node wave function. This problem has led to the introduction of a localization approximation in which the non-local pseudopotential operator is replaced by a local potential. 

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