Non-uniqueness of the Pseudopotential

Non-uniqueness of the Pseudopotential.—The Phillips-Kleinman pseudopotential contains operators 2( v- k) j k) .p which shifts the eigenvalues of the core func- 

This arbitrariness was noticed by Cohen and Heine and by Schlosser28 who suggested that one way of resolving the difficulty would be to place extra conditions on the pseudopotential operator. In particular they proposed that the eigenfunctions xm should be required to have the minimum possible radial kinetic energy  

This is intuitively appealing since the resulting eigenfunctions would also tend to be smoother since the kinetic energy is proportional to the curvature of the wave-function. 

The non-uniqueness problem is more apparent in the case where the pseudopotential is to be defined by localizing the non-local Hartree-Fock (HF) potential due to 

In order to ensure that the orbitals are smooth and that there is a prescription for choosing them Kahn and Goddard2 have minimized, with respect to the coefficients bni, the functional 

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We can see that the non-uniqueness of the pseudopotential and of the open-shell hamiltonian have similar origins. Following Roothaan36 the total open-shell hamiltonian may be written in terms of the basic operator Pa by using projection operators to define the particular form of the operator for each sub-space ... 

If we do this then both the form of and the local form of the pseudopotential are fixed since the latter contains y- In this case it is now worth looking briefly at the physical interpretation of the pseudopotential and the orbital x which are inextricably linked together in the local form. The orbital calculated by the solution of an effective Schrodinger equation is often called a pseudo-orbital to emphasise its origins and to remind us of its non-unique nature. 

This is a inhomogeneous set of linear equations in the expansion coefficients < X n >) und the solutions are thus non-zero and unique, except for the special pseudopotential (3.7). 

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