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New practicable forms for exchange and correlation energy functionals of the CDFT
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 1065–1066 www.elsevier.com/locate/jmmm
New practicable forms for exchange and correlation energy functionals of the CDFT K. Higuchia,, M. Miyasitab, M. Koderab, M. Higuchib a
Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima 739-8527, Japan b Department of Physics, Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan Available online 3 November 2006
Abstract We propose the vorticity expansion approximation (VEA) formulas of both the exchange and correlation energy functionals of the current density functional theory (CDFT). Similarly to the strategy of the generalized gradient approximation of the density functional theory, the coefficients of expansions are determined by requiring the VEA formulas to satisfy exact relations that are derived from uniform and nonuniform coordinate scaling of electrons. As compared with formulas of the local density approximation of the CDFT, the resultant VEA formulas satisfy a large number of exact relations. Due to the well-behaved forms, the VEA formulas can be in quite good agreement with the exchange and correlation energies of the homogeneous electron liquid under a uniform magnetic field. r 2006 Elsevier B.V. All rights reserved. PACS: 71.15.Mb; 31.15.Ew; 31.25.Eb Keywords: Exchange energy functional; Correlation energy functional; Current density functional theory
The density functional theory (DFT) has been extended so that various physical quantities can be chosen as basic variables [1–3]. The current-density functional theory (CDFT) [4,5] and its relativistic version, relativistic currentand spin-density functional theory (RCSDFT) [6,7], are ones of such extensions. In these theories the current density is chosen as a basic variable. Therefore, they are useful for describing the ground-state properties of systems such as f-electron magnetic materials where an orbital current is induced from both the strong spin–orbit interaction and the intra-atomic Coulomb interaction. In order to calculate electronic structures with taking the current density into account, the exchange and correlation energy functionals of the CDFT or RCSDFT must be devised in practicable forms. In this paper, we propose the vorticity expansion approximation (VEA) formulas for both exchange and correlation energy functionals of the CDFT. They are derived by utilizing exact relations as constraints which should be fulfilled by the VEA formulas. This strategy is Corresponding author. Tel.: +81 424 7016; fax: +81 424 7014.
E-mail address: [email protected] (K. Higuchi). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.259
analogous to the development of the generalized gradient approximation (GGA) functionals of the DFT. The exchange-correlation energy functional of the CDFT depends on the paramagnetic current density jp(r) only through the vorticity v(r) [4,5], where v(r) is defined by vðrÞ ¼ rfj p ðrÞ=rðrÞg, and where r(r) denotes the electron density. The same is true for both the exchange energy functional and the correlation energy functional [8], which thus can be denoted by E¯ x ½r; v and E¯ c ½r; v, respectively. With reference to the GGA, we assume that the approximate form of E¯ x ½r; v is given by Z E¯ x ½r; v ¼ E x ½r þ rðrÞF x ðr; vÞr¼rðrÞ dr, (1) n¼nðrÞ
where Ex[r] is the exchange energy functional of the conventional DFT, and where Fx(r, v) is an energy density. Since E¯ x ½r; 0 should coincide with the exchange energy functional of the DFT, we can obtain F x ðr; 0Þ ¼ 0. In addition, if F x ðr; vÞ is assumed to be spherical with respect to v, then the first order in the Taylor expansion of Fx(r, v) around of v ¼ 0 becomes zero, and the second order can be rewritten as DðrÞjvj2 . Here, D(r) is the expansion coefficient which should be determined. Truncating the
ARTICLE IN PRESS 1066
K. Higuchi et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1065–1066
third- and higher-order terms of the expansion, we get Z 2 ¯ (2) E x ½r; v ¼ E x ½r þ rðrÞDðrðrÞÞvðrÞ dr.
Table 1 Comparison of the VEA and the CDFT–LDA [10]
Next, let us consider the properties of D(r) by utilizing the exact relation for the exchange energy functional. The scaling behavior of the exchange energy functional was given by E¯ x rl ; vl ¼ lE¯ x ½r; v [9], where l is the scaling factor, and where rl(r) and vl(r) are scaled electron density and vorticity, respectively. Substituting Eq. (2) into this relation, we obtain l3 Dðl3 rÞ ¼ DðrÞ as the sufficient condition. If D(r) is assumed to be expessed as the power of r, then we obtain DðrÞ / r1 . Accordingly, we get the VEA formula for the exchange energy functional; 2 Z vðrÞ2 dr, ¯x _ (3) E¯ x ½r; v ¼ E x ½r þ D 3 aH H ¯ x stand for the Bohr radius, Rydberg where aH, eH and D constant and dimensionless constant, respectively. As for E¯ c ½r; v, we also consider the following approximate form of E¯ c ½r; v; Z (4) E¯ c ½r; v ¼ E c ½r þ rðrÞF c ðr; vÞr¼rðrÞ dr,
E x ½rl ; j lp ¼ lE x ½r; j p
n¼nðrÞ
where Ec[r] denotes the correlation energy functional of the conventional DFT, and where Fc(r,v) stands for an energy density. Similarly to Fx(r, v), if we assume that Fc(r, v) is spherical with respect to v, we get F c ðr; vÞ ¼ CðrÞjvj2 . Here C(r) is the expansion coefficient. Then, we obtain Z 2 (5) E¯ c ½r; v ¼ E c ½r þ rðrÞCðrðrÞÞvðrÞ dr. The expansion coefficient C(r) is determined by requiring E¯ c ½r; v to satisfy exact relations which have been derived by the uniform and nonuniform scaling techniques [9,10]. We impose on E¯ c ½r; v, eighteen exact relations that are listed in Table 1. The resultant VEA formula for the correlation energy functional is given by Z ¯aa3 rðrÞ C¯ 0 _2 e H rðrÞ3 2 vðrÞ dr, (6) E¯ c ½r; v ¼ E c ½r þ 3 ¯ 3 g3 aH H frðrÞ d=a H ¯ 0 , a¯ and d¯ are dimensionless constants which where C 3 ¯ should be satisfied with C¯ 0 o0, a¯ 40 and 0od5a H r, respectively. It is striking that no matter how we give values of their constants within these conditions, Eq. (6) satisfies all exact relations as shown in Table 1. The VEA formulas satisfy more ‘‘sum rules’’ than those of the local density approximation (LDA) of the CDFT as shown in Table 1. Although sum rules do not always guarantee to make an accurate functional, to say the least of it, the sum rule may fairly get rid of the difficulties which lead to nonphysical results. In this sense, it can be expected that the present VEA formulas may lead to more reasonable results than the CDFT–LDA. Finally, the VEA formulas are applied to the homogeneous electron liquid under the uniform magnetic field. ¯ x , C¯ 0 , a¯ and d¯ are determined by fitting the The values of D
LDA
VEA
Yes
Yes
E c ½rl ; j lp XlE c ½r; j p ;
lX1
Yes
Yes
E c ½rl ; j lp plE c ½r; j p ; lim E c ½rxl ; j xpl ¼ 0
lp1
Yes
Yes
—
Yes
—
Yes
Yes
Yes
No
Yes
l!1
lim lE c ½rxl ; j xpl ¼ const l!1 lim E c ½rxl ; j xpl ¼ 0 l!0 lim l1 E c ½rxl ; j xpl ¼ 0 l!0 lim l2 E c ½rxl ; j xpl ¼ const l!0 xy lim E c ½rxy ll ; j pll ¼ 0 l!1 xy xy lim lE c ½rll ; j pll ¼ const l!1 xy lim E c ½rxy ll ; j pll ¼ 0 l!0 xy lim l1 E c ½rxy ll ; j pll ¼ 0 l!0 xy lim l2 E c ½rll ; j xy pll ¼ const l!0 xyz ¼0 lim E c ½rlll1 ; j xyz plll1 l!1 xyz xyz lim lE c ½rlll1 ; j plll1 ¼ 0 l!1 lim l2 E c ½rxyz ; j xyz ¼ const lll1 plll1 l!1 xyz xyz lim E c ½rlll1 ; j plll1 ¼ 0 l!0 ; j xyz ¼ 0 lim l1 E c ½rxyz lll1 plll1 l!0 ; j xyz ¼ const lim l2 E c ½rxyz lll1 plll1 l!0
No
Yes
—
Yes
—
Yes
—
Yes
Yes
Yes
No
Yes
—
Yes
—
Yes
Yes
Yes
No
Yes
No
Yes
No
Yes
VEA formulas to the calculation results for the homogeneous electron liquid under a uniform magnetic field [11]. ¯ x ¼ 3:76 104 , C¯ 0 ¼ 4:669 104 , These fits lead to D ¯ a¯ ¼ 0:653 and d ¼ 1 1030 , respectively. Owing to the well-behaved forms, the VEA formulas can reproduce the calculations of Ref. [11] within the error of about 5%. This means the validity of the VEA formulas. Furthermore, the above values may be useful for practical calculations for various systems. This work was partially supported by a Grant-in-Aid for Scientific Research in Priority Areas ‘‘Development of New Quantum Simulators and Quantum Design’’ of The Ministry of Education, Culture, Sports, Science, and Technology, Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M. Higuchi, K. Higuchi, Phys. Rev. B 69 (2004) 035113. K. Higuchi, M. Higuchi, Phys. Rev. B 69 (2004) 165118. K. Higuchi, M. Higuchi, Phys. Rev. B 71 (2005) 035116. G. Vignale, M. Rasolt, Phys. Rev. Lett. 59 (1987) 2360. G. Vignale, M. Rasolt, Phys. Rev. B 37 (1988) 10685. M. Higuchi, A. Hasegawa, J. Phys. Soc. Jpn. 66 (1997) 149. M. Higuchi, A. Hasegawa, J. Phys. Soc. Jpn 67 (1998) 2037. K. Higuchi, M. Higuchi, Phys. Rev. B., in press. S. Erhard, E.K.U. Gross, Phys. Rev. A 53 (1996) R5. M. Higuchi, K. Higuchi, Phys. Rev. B 65 (2002) 195122. Y. Takada, H. Goto, J. Phys.: Condens. Matter 10 (1998) 11315.
Journal of Magnetism and Magnetic Materials 310 (2007) 1065–1066 www.elsevier.com/locate/jmmm
New practicable forms for exchange and correlation energy functionals of the CDFT K. Higuchia,, M. Miyasitab, M. Koderab, M. Higuchib a
Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima 739-8527, Japan b Department of Physics, Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan Available online 3 November 2006
Abstract We propose the vorticity expansion approximation (VEA) formulas of both the exchange and correlation energy functionals of the current density functional theory (CDFT). Similarly to the strategy of the generalized gradient approximation of the density functional theory, the coefficients of expansions are determined by requiring the VEA formulas to satisfy exact relations that are derived from uniform and nonuniform coordinate scaling of electrons. As compared with formulas of the local density approximation of the CDFT, the resultant VEA formulas satisfy a large number of exact relations. Due to the well-behaved forms, the VEA formulas can be in quite good agreement with the exchange and correlation energies of the homogeneous electron liquid under a uniform magnetic field. r 2006 Elsevier B.V. All rights reserved. PACS: 71.15.Mb; 31.15.Ew; 31.25.Eb Keywords: Exchange energy functional; Correlation energy functional; Current density functional theory
The density functional theory (DFT) has been extended so that various physical quantities can be chosen as basic variables [1–3]. The current-density functional theory (CDFT) [4,5] and its relativistic version, relativistic currentand spin-density functional theory (RCSDFT) [6,7], are ones of such extensions. In these theories the current density is chosen as a basic variable. Therefore, they are useful for describing the ground-state properties of systems such as f-electron magnetic materials where an orbital current is induced from both the strong spin–orbit interaction and the intra-atomic Coulomb interaction. In order to calculate electronic structures with taking the current density into account, the exchange and correlation energy functionals of the CDFT or RCSDFT must be devised in practicable forms. In this paper, we propose the vorticity expansion approximation (VEA) formulas for both exchange and correlation energy functionals of the CDFT. They are derived by utilizing exact relations as constraints which should be fulfilled by the VEA formulas. This strategy is Corresponding author. Tel.: +81 424 7016; fax: +81 424 7014.
E-mail address: [email protected] (K. Higuchi). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.259
analogous to the development of the generalized gradient approximation (GGA) functionals of the DFT. The exchange-correlation energy functional of the CDFT depends on the paramagnetic current density jp(r) only through the vorticity v(r) [4,5], where v(r) is defined by vðrÞ ¼ rfj p ðrÞ=rðrÞg, and where r(r) denotes the electron density. The same is true for both the exchange energy functional and the correlation energy functional [8], which thus can be denoted by E¯ x ½r; v and E¯ c ½r; v, respectively. With reference to the GGA, we assume that the approximate form of E¯ x ½r; v is given by Z E¯ x ½r; v ¼ E x ½r þ rðrÞF x ðr; vÞr¼rðrÞ dr, (1) n¼nðrÞ
where Ex[r] is the exchange energy functional of the conventional DFT, and where Fx(r, v) is an energy density. Since E¯ x ½r; 0 should coincide with the exchange energy functional of the DFT, we can obtain F x ðr; 0Þ ¼ 0. In addition, if F x ðr; vÞ is assumed to be spherical with respect to v, then the first order in the Taylor expansion of Fx(r, v) around of v ¼ 0 becomes zero, and the second order can be rewritten as DðrÞjvj2 . Here, D(r) is the expansion coefficient which should be determined. Truncating the
ARTICLE IN PRESS 1066
K. Higuchi et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1065–1066
third- and higher-order terms of the expansion, we get Z 2 ¯ (2) E x ½r; v ¼ E x ½r þ rðrÞDðrðrÞÞvðrÞ dr.
Table 1 Comparison of the VEA and the CDFT–LDA [10]
Next, let us consider the properties of D(r) by utilizing the exact relation for the exchange energy functional. The scaling behavior of the exchange energy functional was given by E¯ x rl ; vl ¼ lE¯ x ½r; v [9], where l is the scaling factor, and where rl(r) and vl(r) are scaled electron density and vorticity, respectively. Substituting Eq. (2) into this relation, we obtain l3 Dðl3 rÞ ¼ DðrÞ as the sufficient condition. If D(r) is assumed to be expessed as the power of r, then we obtain DðrÞ / r1 . Accordingly, we get the VEA formula for the exchange energy functional; 2 Z vðrÞ2 dr, ¯x _ (3) E¯ x ½r; v ¼ E x ½r þ D 3 aH H ¯ x stand for the Bohr radius, Rydberg where aH, eH and D constant and dimensionless constant, respectively. As for E¯ c ½r; v, we also consider the following approximate form of E¯ c ½r; v; Z (4) E¯ c ½r; v ¼ E c ½r þ rðrÞF c ðr; vÞr¼rðrÞ dr,
E x ½rl ; j lp ¼ lE x ½r; j p
n¼nðrÞ
where Ec[r] denotes the correlation energy functional of the conventional DFT, and where Fc(r,v) stands for an energy density. Similarly to Fx(r, v), if we assume that Fc(r, v) is spherical with respect to v, we get F c ðr; vÞ ¼ CðrÞjvj2 . Here C(r) is the expansion coefficient. Then, we obtain Z 2 (5) E¯ c ½r; v ¼ E c ½r þ rðrÞCðrðrÞÞvðrÞ dr. The expansion coefficient C(r) is determined by requiring E¯ c ½r; v to satisfy exact relations which have been derived by the uniform and nonuniform scaling techniques [9,10]. We impose on E¯ c ½r; v, eighteen exact relations that are listed in Table 1. The resultant VEA formula for the correlation energy functional is given by Z ¯aa3 rðrÞ C¯ 0 _2 e H rðrÞ3 2 vðrÞ dr, (6) E¯ c ½r; v ¼ E c ½r þ 3 ¯ 3 g3 aH H frðrÞ d=a H ¯ 0 , a¯ and d¯ are dimensionless constants which where C 3 ¯ should be satisfied with C¯ 0 o0, a¯ 40 and 0od5a H r, respectively. It is striking that no matter how we give values of their constants within these conditions, Eq. (6) satisfies all exact relations as shown in Table 1. The VEA formulas satisfy more ‘‘sum rules’’ than those of the local density approximation (LDA) of the CDFT as shown in Table 1. Although sum rules do not always guarantee to make an accurate functional, to say the least of it, the sum rule may fairly get rid of the difficulties which lead to nonphysical results. In this sense, it can be expected that the present VEA formulas may lead to more reasonable results than the CDFT–LDA. Finally, the VEA formulas are applied to the homogeneous electron liquid under the uniform magnetic field. ¯ x , C¯ 0 , a¯ and d¯ are determined by fitting the The values of D
LDA
VEA
Yes
Yes
E c ½rl ; j lp XlE c ½r; j p ;
lX1
Yes
Yes
E c ½rl ; j lp plE c ½r; j p ; lim E c ½rxl ; j xpl ¼ 0
lp1
Yes
Yes
—
Yes
—
Yes
Yes
Yes
No
Yes
l!1
lim lE c ½rxl ; j xpl ¼ const l!1 lim E c ½rxl ; j xpl ¼ 0 l!0 lim l1 E c ½rxl ; j xpl ¼ 0 l!0 lim l2 E c ½rxl ; j xpl ¼ const l!0 xy lim E c ½rxy ll ; j pll ¼ 0 l!1 xy xy lim lE c ½rll ; j pll ¼ const l!1 xy lim E c ½rxy ll ; j pll ¼ 0 l!0 xy lim l1 E c ½rxy ll ; j pll ¼ 0 l!0 xy lim l2 E c ½rll ; j xy pll ¼ const l!0 xyz ¼0 lim E c ½rlll1 ; j xyz plll1 l!1 xyz xyz lim lE c ½rlll1 ; j plll1 ¼ 0 l!1 lim l2 E c ½rxyz ; j xyz ¼ const lll1 plll1 l!1 xyz xyz lim E c ½rlll1 ; j plll1 ¼ 0 l!0 ; j xyz ¼ 0 lim l1 E c ½rxyz lll1 plll1 l!0 ; j xyz ¼ const lim l2 E c ½rxyz lll1 plll1 l!0
No
Yes
—
Yes
—
Yes
—
Yes
Yes
Yes
No
Yes
—
Yes
—
Yes
Yes
Yes
No
Yes
No
Yes
No
Yes
VEA formulas to the calculation results for the homogeneous electron liquid under a uniform magnetic field [11]. ¯ x ¼ 3:76 104 , C¯ 0 ¼ 4:669 104 , These fits lead to D ¯ a¯ ¼ 0:653 and d ¼ 1 1030 , respectively. Owing to the well-behaved forms, the VEA formulas can reproduce the calculations of Ref. [11] within the error of about 5%. This means the validity of the VEA formulas. Furthermore, the above values may be useful for practical calculations for various systems. This work was partially supported by a Grant-in-Aid for Scientific Research in Priority Areas ‘‘Development of New Quantum Simulators and Quantum Design’’ of The Ministry of Education, Culture, Sports, Science, and Technology, Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M. Higuchi, K. Higuchi, Phys. Rev. B 69 (2004) 035113. K. Higuchi, M. Higuchi, Phys. Rev. B 69 (2004) 165118. K. Higuchi, M. Higuchi, Phys. Rev. B 71 (2005) 035116. G. Vignale, M. Rasolt, Phys. Rev. Lett. 59 (1987) 2360. G. Vignale, M. Rasolt, Phys. Rev. B 37 (1988) 10685. M. Higuchi, A. Hasegawa, J. Phys. Soc. Jpn. 66 (1997) 149. M. Higuchi, A. Hasegawa, J. Phys. Soc. Jpn 67 (1998) 2037. K. Higuchi, M. Higuchi, Phys. Rev. B., in press. S. Erhard, E.K.U. Gross, Phys. Rev. A 53 (1996) R5. M. Higuchi, K. Higuchi, Phys. Rev. B 65 (2002) 195122. Y. Takada, H. Goto, J. Phys.: Condens. Matter 10 (1998) 11315.