Nonlocal potential

T. L. Gilbert, Hohenberg-Kohn theorem for nonlocal potentials. Phys. Rev. B 12, 2111 (1975). 

Let s return to the problem of finding the optimal nonlocal potential Dee where the one-electron functions, (x) resulting from solving the eigenproblem... 

According to Eq. (63), the 1-RDM and the generalized Fockian F commute at the extremum hence the NSOs are the solutions of the eigenproblem (60) with the nonlocal potential defined by the identity (64). One should note, however, that Eq. (64) does not completely define v e- In fact, the diagonal elements... 

In this manner, we have arrived at the Pernal nonlocal potential [81]. It can be shown, using the invariance of Vee with respect to an arbitrary unitary transformation and its extremal properties [13] or by means of the first-order perturbation theory applied to the eigenequation of the 1-RDM [81], that the off-diagonal elements of Uee may also be derived via the functional derivative... 

The explicit form derived by Pernal for the effective nonlocal potential allows one to establish one-electron equations that may be of great value for the development of efficient computational methods in NOF theory. Although recent progress has been made, NOF theory needs to continue its assessment. Some other essential conditions such as the reproduction of the homogeneous electron gas should be utilized in the evaluation of approximate implementations. 

HFEc[n] is defined slightly differently from Ec[n]. The functional derivative of HFEc[n] is meant to be added to the Hartree-Fock (HF) nonlocal potential, leading... 

Cho CH, Singh S, Robinson GW. Understanding all of water s anomalies with a nonlocal potential. J. Chem. Phys. 1997 107 7979-7988. 

In problems involving the interaction of an electron with a system containing electrons an operator that frequently appears is a nonlocal potential. 

This is an extension of the Schrodinger equation for a bound state to include the nonlocal potential... 

A frozen-core calculation involves choosing a particular state (for example the one lowest in energy), performing a Hartree—Fock calculation to find the best orbitals, then using the orbitals of the core to generate a nonlocal potential (5.27), which is taken to represent the core in calculations of further states. 

Table 5.1 illustrates the frozen-core approximation for the case of sodium using a simple Slater (4.38) basis in the analytic-orbital representation. The core (Is 2s 2p ) is first calculated by Hartree—Fock for the state characterised by the 3s one-electron orbital, which we call the 3s state. The frozen-core calculation for the 3p state uses the same core orbitals and solves the 3p one-electron problem in the nonlocal potential (5.27) of the core. Comparison with the core and 3p orbitals from a 3p Hartree—Fock calculation illustrates the approximation. The overwhelming component of the 3p orbital agrees to almost five significant figures. 

This is a matrix element of a nonlocal potential Fx(xo,xi), which we express by splitting H into two terms. 

Shadwick, 1976) and thus by extrapolation to the fully correlated problem via application to the multiconfiguration self-consistent field (MCSCF) problem. In spite of the fact that the mathematics of simultaneously averaging a nonlocal potential and proceeding to self-consistency is cumbersome at present, this approach seems a most promising avenue toward Slater s goal of using a local potential for orbital generation in Cl calculations. 

Because of the nonlocality of the exchange potential (the terms U(a,/3,r), a / ) Eq. (2) cannot be used to eliminate the small component from Eq. (1) exactly. In the QR approaches the local exchange approximation is used in Eq. (2) and only the second-order equation for the large component is retained. However, if the nonlocal potential is written as the sum of its local and nonlocal parts then the small component can be expressed as... 

Local and Nonlocal Potential Functions in Density Functional Theory... 

This argument leads back to the OEL equations of OFT, which contain nonlocal potentials in general. 

---