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TF screening TF ϭ ln( p TF / p FG ) derived from literature
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Explicit functions are widely used to interpolate, extrapolate, and differentiate theoretical or experimental data on the equation of state (EOS) of a solid. We present two EOS functions which are theoretically motivated. The simplest realistic model for a simple metal, the stabilized jellium (SJ) or structureless pseudopotential model, is the para...
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Context 1
... mE 1( x ) xe E 1( x ) represents a ‘‘modified exponential integral’’ function with E 1( x ) ϭ ͐ ρ x ( e Ϫ z / z ) dz . SL ϭ ͓ ͚ L 2 c k ( c 0 ϩ k ) ͔ / c 0 is just a fitted parameter and DL ( x ) ϭ ͚ L 2 ( c k / c k 0 Ϫ 1 ) p k ( x , c 0 ) includes the simple polynomials p k ( x , c 0 ) in x and c 0 with, for instance, p 2 ϭ 2 Ϫ x and p 3 ϭ x ϩ c 0 (3 Ϫ 3 x ϩ x 2 ). One should notice, thereby, that E 1( x ) and mE 1( x ) are very well-behaved functions and very fast algorithms for E 1( x ) are incorporated in modern mathematical software. In any case, also very accurate rapid approximations for E 1( x ) have been published. 6 ͑ 2 ͒ p APL gives for any value of K 0 Ј ϭ 3 ϩ ( c 0 ϩ c 2 ) 3 2 and for any order L the correct Fermi gas limit at x → 0 as well as the correct limiting behavior p → 0 for x → ρ without any higher-order discontinuity. ͑ 3 ͒ When APBAF compared the second order form H 12 with their third-order form SJEOS ͑ with the two free parameters K 0 and K Ј 0 and the direct adjustment of d to E 0 ), they find rather better agreement for the form H 12 with their LCGTO calculations than with their SJEOS form. The ‘‘aug- mented’’ ͑ more reasonable ͒ form ASJEOS of APBAF uses the additional parameter f for some constraint at x → 0 as our forms H 1 L and APL, but one additional parameter h is also used. This means, that a fair comparison should use our fourth-order form AP4 with c 4 constraint to E 0 , leaving three free parameters K 0 , K Ј 0 , and c 3 for the fitting. Due to the superior fitting of the second-order form H 12 in comparison with SJEOS, there is no doubt that our fourth-order form AP4 will fit the data of ‘‘regular’’ solids ͑ without electronic anomalies ͒ at least as good as their fourth-order form ASJEOS. ͑ 4 ͒ One should also notice that Fig. 1 of APBAF is mis- leading, since it uses the divergent form SJEOS down to x ϭ 0.1 as reference for the nondivergent H 02 form. The augmented jellium model ASJEOS removes this divergence by partial adaptation of our approach ͑ Ref. 1 of APBAF ͒ , and would therefore show a similar divergence as H 02 in this diagram. ͑ 5 ͒ The form ASJEOS has built in a higher-order discontinuity at x → 1, whereas our forms H 1 L and APL have only continuous derivatives up to the highest order. Some resemblance of ASJEOS for x Ͻ 1 to the our form APL is also obvious. 7 ͑ 6 ͒ APBAF did not realize that our form H 1 L and simi- larly APL do not only incorporate the Fermi gas limit ͑ for x → 0), but also the Thomas-Fermi ͑ TF ͒ screening represented by the parameter c 0 in the term e Ϫ c 0 x . Thereby c 0 represents an adjusted TF screening, which deviates only slightly from the original TF screening as discussed by us previously 2 ͑ Ref. 1 of APBAF ͒ . 7 The core overlap under strong compression is taken into account in H 1 L and APL by incorporation of the ͑ effective ͒ TF limit ͑ not only the Fermi gas limit ͒ . Therefore, H 1 L and APL are not only interesting for ‘‘standard treatments of white dwarf stars,’’ but represent to a good approximation the quantum statistical local density approximation of the TF model, when one takes into account, that the TF model gives only the repulsive pressure p rep ( x ), but the total pressure p ϭ (1 Ϫ x ) p rep ( x ) can be interpreted in this case as resulting from an additional attractive pressure Ϫ x p rep ( x ) in the forms H 1 L and APL like in the ‘‘universal EOS’’ ͑ Ref. 7 ͒ based on an effective Rydberg potential. With this rather arbitrary but useful approximation for the attractive part of the total pressure the TF model accounts typically for about 50% of the repulsive pressure near x ϭ 1 or for about 80% of the total pressure already in the zeroth order approximation of APL. ͑ 8 ͒ The ASJEOS approach uses an exponentially screened polynomial expansion for the total energy in powers of x starting with the zeroth-order term x Ϫ 2 for the screened Fermi gas whereas the APL approach is more directly related to the ͑ repulsive ͒ TF pressure , as illustrated in Fig. 1, where the TF ͑ universal ͒ screening TF ϭ In( p TF / p FG ) is plotted with respect to the TF radius r TF ϭ Z 1/3 r WS . Z is thereby the atomic number of the atom with the Wigner-Seitz radius r WS and p FG stands for the pressure of the corresponding Fermi gas. The first order form AP1 results here just in a straight line, which is fitted here to the initial slope of TF at x ϭ 0. The second order form AP2 gives in this case already an almost perfect fit of the TF pressure. If one considers instead of the total energy from the ASJEOS approach at first only a screened expansion of the repulsive potential ͑ SPL ͒ ...
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... mE 1( x ) xe E 1( x ) represents a ‘‘modified exponential integral’’ function with E 1( x ) ϭ ͐ ρ x ( e Ϫ z / z ) dz . SL ϭ ͓ ͚ L 2 c k ( c 0 ϩ k ) ͔ / c 0 is just a fitted parameter and DL ( x ) ϭ ͚ L 2 ( c k / c k 0 Ϫ 1 ) p k ( x , c 0 ) includes the simple polynomials p k ( x , c 0 ) in x and c 0 with, for instance, p 2 ϭ 2 Ϫ x and p 3 ϭ x ϩ c 0 (3 Ϫ 3 x ϩ x 2 ). One should notice, thereby, that E 1( x ) and mE 1( x ) are very well-behaved functions and very fast algorithms for E 1( x ) are incorporated in modern mathematical software. In any case, also very accurate rapid approximations for E 1( x ) have been published. 6 ͑ 2 ͒ p APL gives for any value of K 0 Ј ϭ 3 ϩ ( c 0 ϩ c 2 ) 3 2 and for any order L the correct Fermi gas limit at x → 0 as well as the correct limiting behavior p → 0 for x → ρ without any higher-order discontinuity. ͑ 3 ͒ When APBAF compared the second order form H 12 with their third-order form SJEOS ͑ with the two free parameters K 0 and K Ј 0 and the direct adjustment of d to E 0 ), they find rather better agreement for the form H 12 with their LCGTO calculations than with their SJEOS form. The ‘‘aug- mented’’ ͑ more reasonable ͒ form ASJEOS of APBAF uses the additional parameter f for some constraint at x → 0 as our forms H 1 L and APL, but one additional parameter h is also used. This means, that a fair comparison should use our fourth-order form AP4 with c 4 constraint to E 0 , leaving three free parameters K 0 , K Ј 0 , and c 3 for the fitting. Due to the superior fitting of the second-order form H 12 in comparison with SJEOS, there is no doubt that our fourth-order form AP4 will fit the data of ‘‘regular’’ solids ͑ without electronic anomalies ͒ at least as good as their fourth-order form ASJEOS. ͑ 4 ͒ One should also notice that Fig. 1 of APBAF is mis- leading, since it uses the divergent form SJEOS down to x ϭ 0.1 as reference for the nondivergent H 02 form. The augmented jellium model ASJEOS removes this divergence by partial adaptation of our approach ͑ Ref. 1 of APBAF ͒ , and would therefore show a similar divergence as H 02 in this diagram. ͑ 5 ͒ The form ASJEOS has built in a higher-order discontinuity at x → 1, whereas our forms H 1 L and APL have only continuous derivatives up to the highest order. Some resemblance of ASJEOS for x Ͻ 1 to the our form APL is also obvious. 7 ͑ 6 ͒ APBAF did not realize that our form H 1 L and simi- larly APL do not only incorporate the Fermi gas limit ͑ for x → 0), but also the Thomas-Fermi ͑ TF ͒ screening represented by the parameter c 0 in the term e Ϫ c 0 x . Thereby c 0 represents an adjusted TF screening, which deviates only slightly from the original TF screening as discussed by us previously 2 ͑ Ref. 1 of APBAF ͒ . 7 The core overlap under strong compression is taken into account in H 1 L and APL by incorporation of the ͑ effective ͒ TF limit ͑ not only the Fermi gas limit ͒ . Therefore, H 1 L and APL are not only interesting for ‘‘standard treatments of white dwarf stars,’’ but represent to a good approximation the quantum statistical local density approximation of the TF model, when one takes into account, that the TF model gives only the repulsive pressure p rep ( x ), but the total pressure p ϭ (1 Ϫ x ) p rep ( x ) can be interpreted in this case as resulting from an additional attractive pressure Ϫ x p rep ( x ) in the forms H 1 L and APL like in the ‘‘universal EOS’’ ͑ Ref. 7 ͒ based on an effective Rydberg potential. With this rather arbitrary but useful approximation for the attractive part of the total pressure the TF model accounts typically for about 50% of the repulsive pressure near x ϭ 1 or for about 80% of the total pressure already in the zeroth order approximation of APL. ͑ 8 ͒ The ASJEOS approach uses an exponentially screened polynomial expansion for the total energy in powers of x starting with the zeroth-order term x Ϫ 2 for the screened Fermi gas whereas the APL approach is more directly related to the ͑ repulsive ͒ TF pressure , as illustrated in Fig. 1, where the TF ͑ universal ͒ screening TF ϭ In( p TF / p FG ) is plotted with respect to the TF radius r TF ϭ Z 1/3 r WS . Z is thereby the atomic number of the atom with the Wigner-Seitz radius r WS and p FG stands for the pressure of the corresponding Fermi gas. The first order form AP1 results here just in a straight line, which is fitted here to the initial slope of TF at x ϭ 0. The second order form AP2 gives in this case already an almost perfect fit of the TF pressure. If one considers instead of the total energy from the ASJEOS approach at first only a screened expansion of the repulsive potential ͑ SPL ͒ ...
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... the correct Fermi gas limit at x 0), the corresponding repulsive pressure p repSPL gives with a best fitting of d 0 to the TF data near x → 0 the first order form represented by SP1 in Fig. 1 with its unfavorable ͑ convex ͒ curvature. A second order form SP2 gives no essential improvement and even a third-order form SP3 does not fit as well as AP2 especially for r TF Ͼ 4 nm. The ASJEOS approach would correspond in this range just to a fourth-order form SP4. This comparison indicates, that the APL expansion is specially suited to fit the TF pressure . Due to its expansion around x ϭ 1 it can take into account also more flexibly higher order corrections for exchange, correlation and atomic shell structure. SPL on the other hand represents a series expansion in the energy around x ϭ 0. p repSPL therefore starts with the wrong curvature and appears to be less flexible with respect to EOS data near x ϭ 1. All these disadvantages of SPL are encountered also in the ASJEOS approach, which includes in addition only a higher-order discontinuity at x ϭ 1. ͑ 9 ͒ If one adds to the repulsive potential E repSPL one more term x L to include the attractive pressure components, one obtains a modified jellium model, which gives for the pressure with the constraint p MJL ( x ϭ 1) ϭ 0 the second-order ...
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Citations
... A 12-point series of values was fitted to the stabilized jellium equation of state (SJEOS). 31 Static-crystal lattice constants and cohesive energies were calculated for 55 solids 12 and bulk moduli for 44 solids, 13 as were Kohn-Sham band gaps of 21 insulators and semiconductors. 14 All the values were compared with the experimental values. ...
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... where are parameters obtained from a polynomial fit; and the stabilized jellium EOS [135,136]: ...
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