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This method presumes that the pseudopotential W € ” W…r† ˆ j w…r À Rj †, where w…r† is the pseudopotential of a separate ion. Then, ” is reduced to finding an electron’s pseudopotential in a the task of defining W free ion’s field. Abarenkov and Heine (1964) proposed defining a model pseudopotential for an ion in the following way. A sphere of radius r0 is drawn around the ion. Outside the sphere the potential affecting an electron is just ÀZe2 ar. The form of electronic wave functions beyond the sphere depends on the azimuthal and magnetic quantum numbers, the energy E (or the main quantum number), and on the values of the logarithmic derivative of the wave function at the distance r0 .

G. of the liquid metal with given number of particles. For amorphous metals, however, as seen below, for a given ionic configuration i, only the trace over the electronic states is performed, and then the result Xi is averaged over the essentially fixed typical configuration of the ions. e. ‘self-averaging’, sample, is called a ‘quenched average’, in contrast to the 37 38 2 The Pseudopotential Method ‘annealed averaging’ performed by taking the trace over all electronic plus ionic states. It is also necessary to point out here that all the consequences following from Eq.

The matrix element for this transition is À4pe2 a…Vjk ‡ k ‡ qj †. The minus sign reflects the fact that now the creation and annihilation operators in Eq. 7) stand in a different order. We should point out that the exchange process can only take place if the electrons k and ÀkH have parallel spins, while they can have both parallel and antiparallel spins in the ‘direct process’. A typical exchange diagram is shown in Fig. 3, and its contribution to e2cor is …b† e2cor ˆ 2 1 ˆ 4pe2 4pe2 m Á 2 ZNV 2 4 H q2 …k ‡ kH ‡ q†2 h " q…k ‡ kH ‡ q† kYk Yq …qTˆ0†  nkr …1 À nk‡qYr † nkH r …1 À nk‡qYr † …2X4X14† …a† The contribution e2cor of direct processes diverges logarithmically at small transferred momenta; in order to be convinced of this, we may observe that, due to the requirement k ` kF and jk ‡ qj b kF , the values of k can only be near the Fermi surface in a narrow layer of the order q.

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Analysis of the Stability of Astrophysical Objects and a New Mechanism for Freeing Their Energy


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