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Atomic Reference Data for Electronic Structure Calculations, Relativistic corrections to the energy-density functional

The energy-density functional described above can be modified to take account of relativistic effects, such as the retardation of the Coulomb interaction and the magnetic interaction between moving electrons. We employ the modification to the exchange portion of the local energy-density functional that has been proposed by MacDonald and Vosko [7].

In this scheme, the exchange energy is partitioned as

$$E_{\rm xc}[n] = E_{\rm x}^{\rm DF}[n] + E_{\rm x}^{\rm T}[n] + E_{\rm c}[n]$$

(Eq. 12)

where n is the number density of electrons. Here, "DF" refers to the "Dirac-Fock" model; "T" is for "transverse" and represents the terms which are first order in the fine structure constant α.

Their corrections are multiplicative, i.e.,

$$\varepsilon_x^{\rm DF}(n) = \varepsilon(n) \phi_{\rm C }(n)$$

(Eq. 13)

and

$$\varepsilon_{\rm x}^{\rm T}(n) = \varepsilon(n) \phi_{\rm T} (n) .$$

(Eq. 14)

where ε(n) is the non-relativistic exchange energy density. The corrections are given by

$$\phi_C (n) = \left[ {{5} \over{6}} + {{1} \over{3\beta^2}} + {{2\eta \ln \xi} \over{3\beta}} - {{2\eta^4 \ln \eta}\over{3\beta^4}} - {1\over2} \left( {{\beta \eta - \ln \xi}\over{\beta^2}} \right)^2 \right]$$

(Eq. 15)

$$\phi_T (n) = \left[ {1\over6} - {{1}\over{3\beta^2}} - {{2\eta \ln \xi}\over{3\beta}} + {{2\eta^4\ln \eta}\over{3\beta^4}} - \left( {{\beta \eta - \ln \xi}\over{\beta^2}} \right)^2 \right] ,$$

(Eq. 16)

where:

$$\beta = v_F/c = (\hbar/(mc))(3\pi^2 n)^{1/3},$$

(Eq. 17)

with υF being the Fermi velocity;

$$\eta = (1+\beta^2)^{(1/2)} ;$$

(Eq. 18)

and

$$\xi = \beta + \eta .$$

(Eq. 19)

Only the sum,

$$ \phi_{\rm C}(n) + \phi_{\rm T}(n) = \left[ 1 - {3\over2} \left( {{\beta \eta - \ln \xi}\over{\beta^2}} \right)^2 \right].$$

(Eq. 20)

enters into the final formula

$$E_{xc}[n] = \varepsilon(n) [\phi_C (n) +\phi_T (n)] + E_c[n].$$

(Eq. 21)

 

Created October 15, 2015, Updated November 4, 2019