PHYSICAL REVIEW E 90, 012141 (2014)
Structure and thermodynamic properties of relativistic electron gases Yu Liu and Jianzhong Wu* Department of Chemical and Environmental Engineering and Department of Mathematics, University of California, Riverside, California 92521, USA (Received 25 April 2014; published 30 July 2014) Relativistic effect is important in many quantum systems but theoretically complicated from both fundamental and practical perspectives. Herein we introduce an efficient computational procedure to predict the structure and energetic properties of relativistic quantum systems by mapping the Pauli principle into an effective pairwiseadditive potential such that the properties of relativistic nonquantum systems can be readily predicted from conventional liquid-state methods. We applied our theoretical procedure to relativistic uniform electron gases and compared the pair correlation functions with those for systems of nonrelativistic electrons. A simple analytical expression has been developed to correlate the exchange-correlation free energy of relativistic uniform electron systems. DOI: 10.1103/PhysRevE.90.012141
PACS number(s): 05.30.Fk, 03.65.Pm, 05.20.Jj, 71.10.Ca
I. INTRODUCTION
Relativistic effect is of great importance in modern physics at both microscopic and astronomical scales [1–13]. It is manifested not only in high-energy physics but also in many conventional chemical systems [6,14]. In particular, relativistic effect makes significant contributions to the electronic properties of heavy atoms such as Cs, Rb, Xe, and Au, and is non-negligible even for light atoms such as O, S, and F in order to predict their electronic properties with high accuracy [14,15]. Because in microscopic systems the relativistic effect is usually coupled together with quantum interactions, a faithful description of such coupling demands extremely complicated mathematical procedures, viz., the Dirac equation and the Pauli matrices. In this work, we introduce a relatively simple procedure to account for the relativistic effect in uniform electron systems. While the model system does not represent specific quantum systems of practical concern, it provides a good starting point toward understanding the correlation functions and the energetic properties of realistic inhomogeneous systems [15]. Consider a homogeneous system containing N electrons at finite temperature T . At each microstate, the wave function of the system, ψ(x1 ,x2 , . . . ,x3N ,t), where xi=1,2,...,3N and t are spatial and temporal coordinates, respectively, is described by the Dirac equation ∂ψ = Hψ, (1) ∂t where is the reduced Planck constant or the Dirac constant. In general, the Hamiltonian depends on the electron rest mass (m), the speed of light (c), momentum operator p = (i ∂x∂ 1 ,i ∂x∂ 2 , . . . ,i ∂x∂3N ), scalar potential , and vector potential A:
and I is the unit matrix with the rank of 2n (n 2). For calculating time-dependent properties such as high-frequency electromagnetic waves, we must solve the Dirac equation together with the Maxwell equations wherein relativity is described by the Lorentz covariance. These complicated equations are known as the Einstein-Maxwell-Dirac equations [16,17]. Whereas the Dirac equation is formally exact, its numerical implementation is extremely complicated even for the simplest case, viz., N = 1 and n = 2. Indeed, the numerical complexity represents one of the main obstacles to the broad utilization of the Dirac equation for conventional quantum-mechanical calculations including quantum Monte Carlo (QMC) simulation or electronic density functional theory (DFT) [18,19]. The purpose of this work is to introduce an efficient computational procedure to predict the structure and energetic properties of relativistic homogeneous quantum systems without explicitly solving the Dirac equation. We will demonstrate that our theoretical approach is able to predict both the pair correlation functions and the exchangecorrelation free energies of relativistic uniform electron gases with little computational cost. A similar approach was proposed before for nonrelativistic quantum systems [20–23].
i
H = βmc2 + c α · p + e( − α · A),
(2)
where e represents the unit charge, and α = {α1 ,α2 , · · · α3N } and β are a set of Pauli matrices that satisfy αi2 = β 2 = I αi αj + αj αi = 0 i = j, αi β + βαi = 0 *
Corresponding author: [email protected]
1539-3755/2014/90(1)/012141(6)
(3)
II. MODELS AND METHODS
As in our previous work for describing the properties of nonrelativistic-quantum (NRQ) systems [22,23], we intend to predict the structure and energetic properties of the relativistic quantum (RQ) system by constructing a relativistic nonquantum (RNQ) reference system that reproduces the pair correlation functions. In the so-called classical mapping, we first calculate the static properties of the electronic system without electrostatic interactions, i.e., a system of relativistic noninteracting fermions. In that case, the Hamiltonian is time independent and thus the Dirac equation becomes H0 ψ = Eψ,
(4)
where H0 = βmc2 + c α · p. Applying the operator H0 on both sides of Eq. (4) leads to the free-particle Klein-Gordon (KG) 012141-1
©2014 American Physical Society
YU LIU AND JIANZHONG WU
PHYSICAL REVIEW E 90, 012141 (2014)
equation or Schr¨odinger’s relativistic wave equation, (−2 c2 ∇ 2 + m2 c4 )ψ = E 2 ψ.
III. RESULTS AND DISCUSSION
(5)
While the KG equation results in a problematic probability density if ψ is interpreted as the one-particle wave function, the interpretation problems may be avoided in the context of the multiparticle quantum field theory as opposed to the single-particle nature of the quantum-mechanical wave function. From a phenomenological perspective, the loss of positive definiteness in the KG probability density can be attributed to particle creation or annihilation, i.e., that the number of particles is not constant in the relativistic theory. For simplicity, the negative energy states and positrons are not considered in this work. As for nonrelativistic quantum systems [24,25], Eq. (5) can be solved analytically for a system of spin unpolarized uniform electrons, leading to the following expressions for the total energy and the first-order density matrix, respectively: E 2 = 2 k 2 c 2 + m 2 c 4 = ρ1 (r1 ,r2 ) =
1 4π 3
4π 2 2 c2 2 n + m2 c 4 , V 2/3
exp[ik · (r1 − r2 )] √ 2 2 2 2 c4 −μ dk, 1 + exp k ckB+m T
(6)
In our previous work [22,23], we demonstrated that the pair correlation function of noninteracting fermions can be reproduced by that of a classical system with an effective Pauli potential. Upon the addition of a diffraction-corrected Coulomb energy to the effective Pauli potential, we found that the pair correlation functions of the classical system are in excellent agreement with those of uniform electron gases. The good agreement in the microscopic structure can be explained by the similarity in the Coulomb interactions between quantum and classical particles. Because the “classical mapping” entails no information on electron momentum, we expect that the procedure is similarly applicable to relativistic quantum systems. The theoretical methods for calculating the pair correlation functions of a classical system have been well established [26]. In this work, we follow the integration equation theory (IET). For a binary system of spherical particles with a pairwiseadditive potential uij (r), the pair correlation functions, hij (r), and the direct correlation functions, cij (r), are related through the Ornstein-Zernike (OZ) equation, ρk cik (r )hkj (|r − r |)dr . (12) hij (r) = cij (r) + k
(7)
where kB is the Boltzmann constant, V is the system volume, and μ is the electron chemical potential. In Eq. (7), the chemical potential is determined from the normalization condition, 1 1 (8) √2 k2 c2 +m2 c4 −μ dk = ρ, 4π 3 1 + exp kB T where ρ stands for the average number density. From the firstorder density matrix, ρ1 (r1 ,r2 ), we obtain the pair correlation functions, 2 2 ∞ rsinhx) − πzξ2 rρ 0 sinhxcoshxsin(ξ dx i = j, NI z+exp(ηcoshx) hij (r) = (9) 0 i = j
As discussed in our previous work [22,23], the bridge functions make negligible contributions to the properties of uniform electron gases. For all calculations reported in this work, we thus solve the OZ equation numerically together with the hyper-netted-chain (HNC) closure: uij (r) hij (r) = exp − + hij (r) − cij (r) − 1, (13) k B TC where TC stands for the system temperature without the relativistic effect. For a relativistic nonquantum system, TC is proportional to the absolute temperature TRC ,
mc2 , (14) TC = TRC κ kB TRC where
2/3 2 3/2 x ∞ −xcosht κ(x) = x e e sinh2 tcoshte−xcosht dt . π 0
2
, η = kmc , and k = ξ sinhx. In the where z = exp( kBμT ), ξ = mc BT noninteracting quantum system (NI), correlation exists only between electrons of the same spin. The adiabatic connection provides a quantitative relationship between the properties of the relativistic noninteracting fermions and real electrons. As in our previous work for nonrelativistic electrons [22,23], the exchange-correlation free energy of the relativistic electronic system is given by Fxc =
ρ2 4
1
dλ 0
h↑↑ (r,λ) + h↑↓ (r,λ) dr, r
(10)
where hij (r,λ) is the pair correlation function of a reference quantum system that has a reduced Hamiltonian, Hλ = βmc2 + c α · p + λ.
(15) As shown in the Appendix, the relativistic effect is insignificant for classical systems when TRC < 103 a.u. At higher temperature, however, the relativistic effect quickly becomes dominant. Equations (12) and (13) allow us to calculate the pair correlation functions from the reduced pairwise-additive potential, uij (r)/kB TC . Conversely, we can determine the reduced pairwise-additive potential from the pair correlation functions. The latter approach allows us to define an effective Pauli potential, P (r), for a system of noninteracting fermions using the pair correlation functions, given by Eq. (9), as an input [20,23]. Because the Pauli exclusion principle applies only to electrons of the same spin, the Pauli potential can be written as
(11)
Equation (10) is formally exact and easy to implement numerically if hij (r,λ) is available for 0 λ 1.
uNI ij (r) = P (r)δij ,
(16)
where δij stands for the Kronecker delta function. Figure 1 shows the effective Pauli potential for relativistic and
012141-2
STRUCTURE AND THERMODYNAMIC PROPERTIES OF . . .
PHYSICAL REVIEW E 90, 012141 (2014)
and Tq =
εF . √ 1.594 − 0.316 rs + 0.024rs
(18)
In Eq. (18), εF denotes the Fermi energy, rs = (3/4πρ)1/3 is the Wigner-Seitz radius, and the numerical coefficients were obtained by reproducing the correlation energy of uniform electrons by that of the classical system with a diffractioncorrected Coulomb potential [20,23,27,28],
r 1 , (19) 1 − exp − √ ∗ uij (r) = P (r)δij + r π m TC
FIG. 1. (Color online) Effective Pauli potential for a uniform electron gas at different temperatures (ρ = 0.033 77 a.u.).
nonrelativistic electronic systems. The trends of these curves are similar to each other, but the effective Pauli potential in relativistic systems is less repulsive in comparison with that in nonrelativistic systems. The differences are most significant at high temperature. To fix the classical temperature, we adopt a relationship established for nonrelativistic quantum systems (in atomic units) [20,23,27]: TRC =
T 2 + Tq2
(17)
where m∗ = 1 (atom unit) represents the effective mass of electrons. Because TRC was introduced in terms of the electron correlation energy, we expect that Eqs. (17)–(19) are equally applicable to relativistic quantum systems. From the effective potential given by Eq. (19), we can then predict the correlation functions of the relativistic uniform electron gases using the liquid-state theory. Figure 2 presents the radial distribution functions of spin unpolarized uniform electron gases at different temperatures. Here the average radial distribution function is defined as g(r) = 12 [h↑↑ (r) + h↑↓ (r)] + 1.
(20)
At all temperatures, g(r) approaches zero at small r due to the electron-hole correlations. The relativity effect reduces the absolute value of the average radial distribution function. Intuitively, the relativity makes electrons become slightly less repulsive to each other. Figure 2 shows that the relativistic
FIG. 2. (Color online) The radial distribution functions of uniform electron gas at different temperatures (ρ = 0.033 77 a.u.). 012141-3
YU LIU AND JIANZHONG WU
PHYSICAL REVIEW E 90, 012141 (2014)
effect is most significant at high temperature, which agrees with the fact that main relativistic contributions to electronic properties stem from the relativistic kinetic energy [15]. The temperature dependence of relativistic effect is also consistent with that predicted from classical systems. As shown in Fig. 4 in Appendix, parameter κ, defined by Eq. (15), is always greater than 1, implying TC TRC [Eq. (14)]. For both quantum and classical systems, inclusion of relativistic effect increases the magnitude of the kinetic energy. In contrast, contributions from the Coulomb and Pauli interactions become less important at high temperature. Figure 2 suggests that relativistic effect makes significant contributions to the properties of uniform electron gases only at very high temperature, i.e., T > 1000 a.u. This conclusion is probably not applicable to nonuniform quantum systems because the momentum of electrons is strongly influenced by nuclear-electron attractions. As shown in Eq. (17), the relativistic effect is dependent not only on the thermodynamic temperature T but also on quantum temperate Tq . For nonuniform quantum systems, the definition of Tq is totally different from Eq. (18). Because of the quantum temperature, the effective classical temperature can be on the order of 103 a.u. even when T is low. The high value of the effective classical temperature explains why the relativistic effect could affect the electronic properties of atoms even at low temperature. One important property of uniform electron gases is the exchange-correlation free energy, which is often used in DFT calculations. Figure 3 presents the theoretical predictions of the exchange-correlation free energies for relativistic uniform electron gases at different temperature and density. In these calculations, the pair correlation functions appeared in the adiabatic connection, i.e., hij (r,λ) in Eq. (10) are calculated from a classical reference system with a reduced Coulomb potential:
r λ . (21) 1 − exp − √ ∗ uij (r,λ) = P (r)δij + r π m Tc Figure 3 shows that |Fxc | increases with both temperature and density. In the high-density region, we can identify a linear relationship between log(−Fxc ) and log(ρ). At low density,
FIG. 3. (Color online) Exchange-correlation free energy as a function of the electron density at several temperatures. The lines are calculated from Eq. (23).
however, the relationship between Fxc and density is more complicated. The different trends at the low- and high-density regions arise from the distinctive behaviors of the exchange free energy and the correlation free energy, the two components of Fxc . At high density, Fxc is dominated by the exchange free energy, which has an exact expression for nonrelativistic system at 0 K: Fx(0)
=
Ex(0)
3 3 1/3 4/3 =− ρ . 4 π
(22)
Equation (22) suggests that we may fit the exchangecorrelation free energy of a relativistic uniform electron gas at high density (ρ > 0.1 a.u.) into a simple function:
3 3 1/3 exp(k1 T k2 )ρ k3 . Fxc = − 4 π Correspondingly, the exchange-correlation energy is
∂Fxc Exc = Fxc − T ∂T N,V
1/3 3 3 =− (1 − k1 k2 T k2 )exp(k1 T k2 )ρ k3 , 4 π
(23)
(24)
where k1 = −2.333 83, k2 = 0.158 05, and k3 = 1.313 27. Parameter k3 is close to 4/3, implying that the relativistic effect does not change the density dependence of the exchange free energy very much. At 0 K, the exchange-correlation energy of a relativistic system is higher than that of the correR NR sponding nonrelativistic system, i.e., Exc = Exc − Exc > 0. Equation (24) predicts that Exc increases with the overall density, which is consistent with the MacDonald-Vosko approximation [29] and quantum Monte Carlo calculations by Kenny et al. [30]. At low density, the correlation free energy is comparable with the exchange-correlation energy, leading to a more complicated relationship between the exchangecorrelation energy and the electron density. IV. CONCLUSIONS
In summary, we have introduced a computational procedure for efficient prediction of the pair correlation functions and the exchange-correlation free energies of uniform relativistic quantum systems. For uniform electron gases, the relativistic effect is most significant at high temperature (T > 103 a.u.), and it leads to an effective attraction between electrons. We find that the exchange-correlation free energy exhibits different trends at different density regions and that such difference can be explained in terms of the competition of exchange and correlation effects. By using an effective pairwise-additive Pauli potential, our theoretical method provides a computationally convenient procedure to account for the relativistic effects in uniform quantum systems. Although the model system does not represent a specific quantum system of practical concern, we expect that the structure and energetic properties obtained from classical mapping will be useful for future DFT calculations (e.g., via either local or weighted density approximations).
012141-4
STRUCTURE AND THERMODYNAMIC PROPERTIES OF . . .
PHYSICAL REVIEW E 90, 012141 (2014)
ACKNOWLEDGMENTS
J.W. is grateful to the US Department of Energy (Grant No. DE-FG02-06ER46296) for the financial support of this research. The authors are also grateful to the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC03-76SF0009.
APPENDIX: STRUCTURE AND THERMODYNAMIC PROPERTIES OF RELATIVISTIC HOMOGENEOUS QUANTUM SYSTEMS
The main difference between a relativistic nonquantum (RNQ) system and a conventional classical system lies in the kinetic energy, id RNQ , which can be found from the idea part of the grand canonical partition function. For a RNQ system, the noninteracting (NI) grand canonical partition function is given by
NI RNQ [μRNQ ,V ,TRNQ ] −
μRNQ
e kB TRNQ = N !h3N
2 2 − c p + m2 c 4 dpN drN exp kB TRNQ
μ −mc2 − RNQ kB T
V Ne = N !h3N
N
mc2 3 3 4π m c R , kB TRNQ
FIG. 4. Parameter κ versus temperature. It deviates from unity significantly only at high temperature.
we find that these two systems are equivalent if we define
mc2 , (A3) μC = (μRNQ − mc2 )κ kB TRNQ
mc2 TC = TRNQ κ , (A4) kB TRNQ with κ(x) =
(A1)
2 3/2 x x e π
∞
e
−xcosht
2/3 2
sinh tcoshtdt
. (A5)
0
(A2)
With the above definitions, one can easily verify that Eqs. (A1) and (A2) become identical. Parameter κ is a key factor indicating the importance of relativistic effect. If κ = 1, there is no relativistic effect; the more κ deviates from 1, the more important the relativistic effect is. Figure 4 shows κ as a function of temperature. Clearly, κ does not change so much when TRNQ < 103 a.u. and increases dramatically when TRNQ > 104 a.u. In other words, the relativistic effect is important when TRNQ > 103 a.u., and quickly becomes the essential contribution as the temperature increases. Because κ 1, the relativistic effect enhances the kinetic energy.
[1] O. D. Haberlen, S. C. Chung, M. Stener, and N. Rosch, J. Chem. Phys. 106, 5189 (1997). [2] H. Hakkinen, M. Moseler, and U. Landman, Phys. Rev. Lett. 89, 033401 (2002). [3] P. Pyykko, J. Li, and N. Runeberg, J. Phys. Chem. 98, 4809 (1994). [4] M. Barysz and P. Pyykko, Chem. Phys. Lett. 285, 398 (1998). [5] J. Vaara and P. Pyykko, J. Chem. Phys. 118, 2973 (2003). [6] P. Pyykko, Annu. Rev. Phys. Chem. 63, 45 (2012). [7] P. Pyykko, J. Comput. Chem. 34, 2667 (2013). [8] V. Roy, A. K. Chaudhuri, and B. Mohanty, Phys. Rev. C 86, 014902 (2012). [9] L. Xue, Y. G. Ma, J. H. Chen, and S. Zhang, Phys. Rev. C 85, 064912 (2012). [10] P. Pyykko, Chem. Rev. 88, 563 (1988).
[11] P. Pyykko and Y. F. Zhao, Angew. Chem., Int. Ed. Engl. 30, 604 (1991). [12] P. Pyykko and Y. F. Zhao, Chem. Phys. Lett. 177, 103 (1991). [13] J. Li and P. Pyykko, Chem. Phys. Lett. 197, 586 (1992). [14] C. Michauk and J. Gauss, J. Chem. Phys. 127, 044106 (2007). [15] E. Engel and R. M. Dreizler, Density Functional Theory: An Advanced Course (Springer, Heidelberg, Germany, New York, 2011). [16] F. Finster, J. Smoller, and S. T. Yau, Mod. Phys. Lett. A 14, 1053 (1999). [17] C. Saclioglu, Classical Quantum Gravity 17, 485 (2000). [18] W. Liu and I. Lindgren, J. Chem. Phys. 139, 014108 (2013). [19] W. Liu, Mol. Phys. 108, 1679 (2010). [20] M. W. C. Dharma-Wardana and F. Perrot, Phys. Rev. Lett. 84, 959 (2000).
∞ where R(x) = ex 0 sinh2 tcoshte−xcosht dt, and kB is the Boltzmann constant. Comparing equation (A1) with the noninteracting grand canonical partition function for a system of classical particles, −
NI C [μC ,V ,TC ]
μC
V N e k B TC = (2π mkB TC )3N/2 , N !h3N
012141-5
YU LIU AND JIANZHONG WU
PHYSICAL REVIEW E 90, 012141 (2014)
[21] F. Perrot and M. W. C. Dharma-Wardana, Phys. Rev. B 62, 16536 (2000). [22] S. Zhao, P. Feng, and J. Wu, Chem. Phys. Lett. 556, 336 (2013). [23] Y. Liu and J. Wu, J. Chem. Phys. 140, 084103 (2014). [24] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). [25] M. Klawunn, A. Recati, L. P. Pitaevskii, and S. Stringari, Phys. Rev. A 84, 033612 (2011).
[26] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). [27] G. Ortiz and P. Ballone, Phys. Rev. B 50, 1391 (1994). [28] H. Minoo, M. M. Gombert, and C. Deutsch, Phys. Rev. A 23, 924 (1981). [29] A. H. MacDonald and S. H. Vosko, J. Phys. C: Solid State Phys. 12, 2977 (1979). [30] S. D. Kenny, G. Rajagopal, R. J. Needs, W. K. Leung, M. J. Godfrey, A. J. Williamson, and W. M. C. Foulkes, Phys. Rev. Lett. 77, 1099 (1996).
012141-6
Structure and thermodynamic properties of relativistic electron gases Yu Liu and Jianzhong Wu* Department of Chemical and Environmental Engineering and Department of Mathematics, University of California, Riverside, California 92521, USA (Received 25 April 2014; published 30 July 2014) Relativistic effect is important in many quantum systems but theoretically complicated from both fundamental and practical perspectives. Herein we introduce an efficient computational procedure to predict the structure and energetic properties of relativistic quantum systems by mapping the Pauli principle into an effective pairwiseadditive potential such that the properties of relativistic nonquantum systems can be readily predicted from conventional liquid-state methods. We applied our theoretical procedure to relativistic uniform electron gases and compared the pair correlation functions with those for systems of nonrelativistic electrons. A simple analytical expression has been developed to correlate the exchange-correlation free energy of relativistic uniform electron systems. DOI: 10.1103/PhysRevE.90.012141
PACS number(s): 05.30.Fk, 03.65.Pm, 05.20.Jj, 71.10.Ca
I. INTRODUCTION
Relativistic effect is of great importance in modern physics at both microscopic and astronomical scales [1–13]. It is manifested not only in high-energy physics but also in many conventional chemical systems [6,14]. In particular, relativistic effect makes significant contributions to the electronic properties of heavy atoms such as Cs, Rb, Xe, and Au, and is non-negligible even for light atoms such as O, S, and F in order to predict their electronic properties with high accuracy [14,15]. Because in microscopic systems the relativistic effect is usually coupled together with quantum interactions, a faithful description of such coupling demands extremely complicated mathematical procedures, viz., the Dirac equation and the Pauli matrices. In this work, we introduce a relatively simple procedure to account for the relativistic effect in uniform electron systems. While the model system does not represent specific quantum systems of practical concern, it provides a good starting point toward understanding the correlation functions and the energetic properties of realistic inhomogeneous systems [15]. Consider a homogeneous system containing N electrons at finite temperature T . At each microstate, the wave function of the system, ψ(x1 ,x2 , . . . ,x3N ,t), where xi=1,2,...,3N and t are spatial and temporal coordinates, respectively, is described by the Dirac equation ∂ψ = Hψ, (1) ∂t where is the reduced Planck constant or the Dirac constant. In general, the Hamiltonian depends on the electron rest mass (m), the speed of light (c), momentum operator p = (i ∂x∂ 1 ,i ∂x∂ 2 , . . . ,i ∂x∂3N ), scalar potential , and vector potential A:
and I is the unit matrix with the rank of 2n (n 2). For calculating time-dependent properties such as high-frequency electromagnetic waves, we must solve the Dirac equation together with the Maxwell equations wherein relativity is described by the Lorentz covariance. These complicated equations are known as the Einstein-Maxwell-Dirac equations [16,17]. Whereas the Dirac equation is formally exact, its numerical implementation is extremely complicated even for the simplest case, viz., N = 1 and n = 2. Indeed, the numerical complexity represents one of the main obstacles to the broad utilization of the Dirac equation for conventional quantum-mechanical calculations including quantum Monte Carlo (QMC) simulation or electronic density functional theory (DFT) [18,19]. The purpose of this work is to introduce an efficient computational procedure to predict the structure and energetic properties of relativistic homogeneous quantum systems without explicitly solving the Dirac equation. We will demonstrate that our theoretical approach is able to predict both the pair correlation functions and the exchangecorrelation free energies of relativistic uniform electron gases with little computational cost. A similar approach was proposed before for nonrelativistic quantum systems [20–23].
i
H = βmc2 + c α · p + e( − α · A),
(2)
where e represents the unit charge, and α = {α1 ,α2 , · · · α3N } and β are a set of Pauli matrices that satisfy αi2 = β 2 = I αi αj + αj αi = 0 i = j, αi β + βαi = 0 *
Corresponding author: [email protected]
1539-3755/2014/90(1)/012141(6)
(3)
II. MODELS AND METHODS
As in our previous work for describing the properties of nonrelativistic-quantum (NRQ) systems [22,23], we intend to predict the structure and energetic properties of the relativistic quantum (RQ) system by constructing a relativistic nonquantum (RNQ) reference system that reproduces the pair correlation functions. In the so-called classical mapping, we first calculate the static properties of the electronic system without electrostatic interactions, i.e., a system of relativistic noninteracting fermions. In that case, the Hamiltonian is time independent and thus the Dirac equation becomes H0 ψ = Eψ,
(4)
where H0 = βmc2 + c α · p. Applying the operator H0 on both sides of Eq. (4) leads to the free-particle Klein-Gordon (KG) 012141-1
©2014 American Physical Society
YU LIU AND JIANZHONG WU
PHYSICAL REVIEW E 90, 012141 (2014)
equation or Schr¨odinger’s relativistic wave equation, (−2 c2 ∇ 2 + m2 c4 )ψ = E 2 ψ.
III. RESULTS AND DISCUSSION
(5)
While the KG equation results in a problematic probability density if ψ is interpreted as the one-particle wave function, the interpretation problems may be avoided in the context of the multiparticle quantum field theory as opposed to the single-particle nature of the quantum-mechanical wave function. From a phenomenological perspective, the loss of positive definiteness in the KG probability density can be attributed to particle creation or annihilation, i.e., that the number of particles is not constant in the relativistic theory. For simplicity, the negative energy states and positrons are not considered in this work. As for nonrelativistic quantum systems [24,25], Eq. (5) can be solved analytically for a system of spin unpolarized uniform electrons, leading to the following expressions for the total energy and the first-order density matrix, respectively: E 2 = 2 k 2 c 2 + m 2 c 4 = ρ1 (r1 ,r2 ) =
1 4π 3
4π 2 2 c2 2 n + m2 c 4 , V 2/3
exp[ik · (r1 − r2 )] √ 2 2 2 2 c4 −μ dk, 1 + exp k ckB+m T
(6)
In our previous work [22,23], we demonstrated that the pair correlation function of noninteracting fermions can be reproduced by that of a classical system with an effective Pauli potential. Upon the addition of a diffraction-corrected Coulomb energy to the effective Pauli potential, we found that the pair correlation functions of the classical system are in excellent agreement with those of uniform electron gases. The good agreement in the microscopic structure can be explained by the similarity in the Coulomb interactions between quantum and classical particles. Because the “classical mapping” entails no information on electron momentum, we expect that the procedure is similarly applicable to relativistic quantum systems. The theoretical methods for calculating the pair correlation functions of a classical system have been well established [26]. In this work, we follow the integration equation theory (IET). For a binary system of spherical particles with a pairwiseadditive potential uij (r), the pair correlation functions, hij (r), and the direct correlation functions, cij (r), are related through the Ornstein-Zernike (OZ) equation, ρk cik (r )hkj (|r − r |)dr . (12) hij (r) = cij (r) + k
(7)
where kB is the Boltzmann constant, V is the system volume, and μ is the electron chemical potential. In Eq. (7), the chemical potential is determined from the normalization condition, 1 1 (8) √2 k2 c2 +m2 c4 −μ dk = ρ, 4π 3 1 + exp kB T where ρ stands for the average number density. From the firstorder density matrix, ρ1 (r1 ,r2 ), we obtain the pair correlation functions, 2 2 ∞ rsinhx) − πzξ2 rρ 0 sinhxcoshxsin(ξ dx i = j, NI z+exp(ηcoshx) hij (r) = (9) 0 i = j
As discussed in our previous work [22,23], the bridge functions make negligible contributions to the properties of uniform electron gases. For all calculations reported in this work, we thus solve the OZ equation numerically together with the hyper-netted-chain (HNC) closure: uij (r) hij (r) = exp − + hij (r) − cij (r) − 1, (13) k B TC where TC stands for the system temperature without the relativistic effect. For a relativistic nonquantum system, TC is proportional to the absolute temperature TRC ,
mc2 , (14) TC = TRC κ kB TRC where
2/3 2 3/2 x ∞ −xcosht κ(x) = x e e sinh2 tcoshte−xcosht dt . π 0
2
, η = kmc , and k = ξ sinhx. In the where z = exp( kBμT ), ξ = mc BT noninteracting quantum system (NI), correlation exists only between electrons of the same spin. The adiabatic connection provides a quantitative relationship between the properties of the relativistic noninteracting fermions and real electrons. As in our previous work for nonrelativistic electrons [22,23], the exchange-correlation free energy of the relativistic electronic system is given by Fxc =
ρ2 4
1
dλ 0
h↑↑ (r,λ) + h↑↓ (r,λ) dr, r
(10)
where hij (r,λ) is the pair correlation function of a reference quantum system that has a reduced Hamiltonian, Hλ = βmc2 + c α · p + λ.
(15) As shown in the Appendix, the relativistic effect is insignificant for classical systems when TRC < 103 a.u. At higher temperature, however, the relativistic effect quickly becomes dominant. Equations (12) and (13) allow us to calculate the pair correlation functions from the reduced pairwise-additive potential, uij (r)/kB TC . Conversely, we can determine the reduced pairwise-additive potential from the pair correlation functions. The latter approach allows us to define an effective Pauli potential, P (r), for a system of noninteracting fermions using the pair correlation functions, given by Eq. (9), as an input [20,23]. Because the Pauli exclusion principle applies only to electrons of the same spin, the Pauli potential can be written as
(11)
Equation (10) is formally exact and easy to implement numerically if hij (r,λ) is available for 0 λ 1.
uNI ij (r) = P (r)δij ,
(16)
where δij stands for the Kronecker delta function. Figure 1 shows the effective Pauli potential for relativistic and
012141-2
STRUCTURE AND THERMODYNAMIC PROPERTIES OF . . .
PHYSICAL REVIEW E 90, 012141 (2014)
and Tq =
εF . √ 1.594 − 0.316 rs + 0.024rs
(18)
In Eq. (18), εF denotes the Fermi energy, rs = (3/4πρ)1/3 is the Wigner-Seitz radius, and the numerical coefficients were obtained by reproducing the correlation energy of uniform electrons by that of the classical system with a diffractioncorrected Coulomb potential [20,23,27,28],
r 1 , (19) 1 − exp − √ ∗ uij (r) = P (r)δij + r π m TC
FIG. 1. (Color online) Effective Pauli potential for a uniform electron gas at different temperatures (ρ = 0.033 77 a.u.).
nonrelativistic electronic systems. The trends of these curves are similar to each other, but the effective Pauli potential in relativistic systems is less repulsive in comparison with that in nonrelativistic systems. The differences are most significant at high temperature. To fix the classical temperature, we adopt a relationship established for nonrelativistic quantum systems (in atomic units) [20,23,27]: TRC =
T 2 + Tq2
(17)
where m∗ = 1 (atom unit) represents the effective mass of electrons. Because TRC was introduced in terms of the electron correlation energy, we expect that Eqs. (17)–(19) are equally applicable to relativistic quantum systems. From the effective potential given by Eq. (19), we can then predict the correlation functions of the relativistic uniform electron gases using the liquid-state theory. Figure 2 presents the radial distribution functions of spin unpolarized uniform electron gases at different temperatures. Here the average radial distribution function is defined as g(r) = 12 [h↑↑ (r) + h↑↓ (r)] + 1.
(20)
At all temperatures, g(r) approaches zero at small r due to the electron-hole correlations. The relativity effect reduces the absolute value of the average radial distribution function. Intuitively, the relativity makes electrons become slightly less repulsive to each other. Figure 2 shows that the relativistic
FIG. 2. (Color online) The radial distribution functions of uniform electron gas at different temperatures (ρ = 0.033 77 a.u.). 012141-3
YU LIU AND JIANZHONG WU
PHYSICAL REVIEW E 90, 012141 (2014)
effect is most significant at high temperature, which agrees with the fact that main relativistic contributions to electronic properties stem from the relativistic kinetic energy [15]. The temperature dependence of relativistic effect is also consistent with that predicted from classical systems. As shown in Fig. 4 in Appendix, parameter κ, defined by Eq. (15), is always greater than 1, implying TC TRC [Eq. (14)]. For both quantum and classical systems, inclusion of relativistic effect increases the magnitude of the kinetic energy. In contrast, contributions from the Coulomb and Pauli interactions become less important at high temperature. Figure 2 suggests that relativistic effect makes significant contributions to the properties of uniform electron gases only at very high temperature, i.e., T > 1000 a.u. This conclusion is probably not applicable to nonuniform quantum systems because the momentum of electrons is strongly influenced by nuclear-electron attractions. As shown in Eq. (17), the relativistic effect is dependent not only on the thermodynamic temperature T but also on quantum temperate Tq . For nonuniform quantum systems, the definition of Tq is totally different from Eq. (18). Because of the quantum temperature, the effective classical temperature can be on the order of 103 a.u. even when T is low. The high value of the effective classical temperature explains why the relativistic effect could affect the electronic properties of atoms even at low temperature. One important property of uniform electron gases is the exchange-correlation free energy, which is often used in DFT calculations. Figure 3 presents the theoretical predictions of the exchange-correlation free energies for relativistic uniform electron gases at different temperature and density. In these calculations, the pair correlation functions appeared in the adiabatic connection, i.e., hij (r,λ) in Eq. (10) are calculated from a classical reference system with a reduced Coulomb potential:
r λ . (21) 1 − exp − √ ∗ uij (r,λ) = P (r)δij + r π m Tc Figure 3 shows that |Fxc | increases with both temperature and density. In the high-density region, we can identify a linear relationship between log(−Fxc ) and log(ρ). At low density,
FIG. 3. (Color online) Exchange-correlation free energy as a function of the electron density at several temperatures. The lines are calculated from Eq. (23).
however, the relationship between Fxc and density is more complicated. The different trends at the low- and high-density regions arise from the distinctive behaviors of the exchange free energy and the correlation free energy, the two components of Fxc . At high density, Fxc is dominated by the exchange free energy, which has an exact expression for nonrelativistic system at 0 K: Fx(0)
=
Ex(0)
3 3 1/3 4/3 =− ρ . 4 π
(22)
Equation (22) suggests that we may fit the exchangecorrelation free energy of a relativistic uniform electron gas at high density (ρ > 0.1 a.u.) into a simple function:
3 3 1/3 exp(k1 T k2 )ρ k3 . Fxc = − 4 π Correspondingly, the exchange-correlation energy is
∂Fxc Exc = Fxc − T ∂T N,V
1/3 3 3 =− (1 − k1 k2 T k2 )exp(k1 T k2 )ρ k3 , 4 π
(23)
(24)
where k1 = −2.333 83, k2 = 0.158 05, and k3 = 1.313 27. Parameter k3 is close to 4/3, implying that the relativistic effect does not change the density dependence of the exchange free energy very much. At 0 K, the exchange-correlation energy of a relativistic system is higher than that of the correR NR sponding nonrelativistic system, i.e., Exc = Exc − Exc > 0. Equation (24) predicts that Exc increases with the overall density, which is consistent with the MacDonald-Vosko approximation [29] and quantum Monte Carlo calculations by Kenny et al. [30]. At low density, the correlation free energy is comparable with the exchange-correlation energy, leading to a more complicated relationship between the exchangecorrelation energy and the electron density. IV. CONCLUSIONS
In summary, we have introduced a computational procedure for efficient prediction of the pair correlation functions and the exchange-correlation free energies of uniform relativistic quantum systems. For uniform electron gases, the relativistic effect is most significant at high temperature (T > 103 a.u.), and it leads to an effective attraction between electrons. We find that the exchange-correlation free energy exhibits different trends at different density regions and that such difference can be explained in terms of the competition of exchange and correlation effects. By using an effective pairwise-additive Pauli potential, our theoretical method provides a computationally convenient procedure to account for the relativistic effects in uniform quantum systems. Although the model system does not represent a specific quantum system of practical concern, we expect that the structure and energetic properties obtained from classical mapping will be useful for future DFT calculations (e.g., via either local or weighted density approximations).
012141-4
STRUCTURE AND THERMODYNAMIC PROPERTIES OF . . .
PHYSICAL REVIEW E 90, 012141 (2014)
ACKNOWLEDGMENTS
J.W. is grateful to the US Department of Energy (Grant No. DE-FG02-06ER46296) for the financial support of this research. The authors are also grateful to the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC03-76SF0009.
APPENDIX: STRUCTURE AND THERMODYNAMIC PROPERTIES OF RELATIVISTIC HOMOGENEOUS QUANTUM SYSTEMS
The main difference between a relativistic nonquantum (RNQ) system and a conventional classical system lies in the kinetic energy, id RNQ , which can be found from the idea part of the grand canonical partition function. For a RNQ system, the noninteracting (NI) grand canonical partition function is given by
NI RNQ [μRNQ ,V ,TRNQ ] −
μRNQ
e kB TRNQ = N !h3N
2 2 − c p + m2 c 4 dpN drN exp kB TRNQ
μ −mc2 − RNQ kB T
V Ne = N !h3N
N
mc2 3 3 4π m c R , kB TRNQ
FIG. 4. Parameter κ versus temperature. It deviates from unity significantly only at high temperature.
we find that these two systems are equivalent if we define
mc2 , (A3) μC = (μRNQ − mc2 )κ kB TRNQ
mc2 TC = TRNQ κ , (A4) kB TRNQ with κ(x) =
(A1)
2 3/2 x x e π
∞
e
−xcosht
2/3 2
sinh tcoshtdt
. (A5)
0
(A2)
With the above definitions, one can easily verify that Eqs. (A1) and (A2) become identical. Parameter κ is a key factor indicating the importance of relativistic effect. If κ = 1, there is no relativistic effect; the more κ deviates from 1, the more important the relativistic effect is. Figure 4 shows κ as a function of temperature. Clearly, κ does not change so much when TRNQ < 103 a.u. and increases dramatically when TRNQ > 104 a.u. In other words, the relativistic effect is important when TRNQ > 103 a.u., and quickly becomes the essential contribution as the temperature increases. Because κ 1, the relativistic effect enhances the kinetic energy.
[1] O. D. Haberlen, S. C. Chung, M. Stener, and N. Rosch, J. Chem. Phys. 106, 5189 (1997). [2] H. Hakkinen, M. Moseler, and U. Landman, Phys. Rev. Lett. 89, 033401 (2002). [3] P. Pyykko, J. Li, and N. Runeberg, J. Phys. Chem. 98, 4809 (1994). [4] M. Barysz and P. Pyykko, Chem. Phys. Lett. 285, 398 (1998). [5] J. Vaara and P. Pyykko, J. Chem. Phys. 118, 2973 (2003). [6] P. Pyykko, Annu. Rev. Phys. Chem. 63, 45 (2012). [7] P. Pyykko, J. Comput. Chem. 34, 2667 (2013). [8] V. Roy, A. K. Chaudhuri, and B. Mohanty, Phys. Rev. C 86, 014902 (2012). [9] L. Xue, Y. G. Ma, J. H. Chen, and S. Zhang, Phys. Rev. C 85, 064912 (2012). [10] P. Pyykko, Chem. Rev. 88, 563 (1988).
[11] P. Pyykko and Y. F. Zhao, Angew. Chem., Int. Ed. Engl. 30, 604 (1991). [12] P. Pyykko and Y. F. Zhao, Chem. Phys. Lett. 177, 103 (1991). [13] J. Li and P. Pyykko, Chem. Phys. Lett. 197, 586 (1992). [14] C. Michauk and J. Gauss, J. Chem. Phys. 127, 044106 (2007). [15] E. Engel and R. M. Dreizler, Density Functional Theory: An Advanced Course (Springer, Heidelberg, Germany, New York, 2011). [16] F. Finster, J. Smoller, and S. T. Yau, Mod. Phys. Lett. A 14, 1053 (1999). [17] C. Saclioglu, Classical Quantum Gravity 17, 485 (2000). [18] W. Liu and I. Lindgren, J. Chem. Phys. 139, 014108 (2013). [19] W. Liu, Mol. Phys. 108, 1679 (2010). [20] M. W. C. Dharma-Wardana and F. Perrot, Phys. Rev. Lett. 84, 959 (2000).
∞ where R(x) = ex 0 sinh2 tcoshte−xcosht dt, and kB is the Boltzmann constant. Comparing equation (A1) with the noninteracting grand canonical partition function for a system of classical particles, −
NI C [μC ,V ,TC ]
μC
V N e k B TC = (2π mkB TC )3N/2 , N !h3N
012141-5
YU LIU AND JIANZHONG WU
PHYSICAL REVIEW E 90, 012141 (2014)
[21] F. Perrot and M. W. C. Dharma-Wardana, Phys. Rev. B 62, 16536 (2000). [22] S. Zhao, P. Feng, and J. Wu, Chem. Phys. Lett. 556, 336 (2013). [23] Y. Liu and J. Wu, J. Chem. Phys. 140, 084103 (2014). [24] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). [25] M. Klawunn, A. Recati, L. P. Pitaevskii, and S. Stringari, Phys. Rev. A 84, 033612 (2011).
[26] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). [27] G. Ortiz and P. Ballone, Phys. Rev. B 50, 1391 (1994). [28] H. Minoo, M. M. Gombert, and C. Deutsch, Phys. Rev. A 23, 924 (1981). [29] A. H. MacDonald and S. H. Vosko, J. Phys. C: Solid State Phys. 12, 2977 (1979). [30] S. D. Kenny, G. Rajagopal, R. J. Needs, W. K. Leung, M. J. Godfrey, A. J. Williamson, and W. M. C. Foulkes, Phys. Rev. Lett. 77, 1099 (1996).
012141-6
