Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX
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A Parameter-Free Semilocal Exchange Energy Functional for Two-Dimensional Quantum Systems Abhilash Patra,* Subrata Jana,* and Prasanjit Samal* School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India ABSTRACT: The method of constructing semilocal density functional for exchange in two dimensions using one of the premier approaches, i.e., density matrix expansion, is revisited, and an accurate functional is constructed. The form of the functional is quite simple and includes no adjustable semiempirical parameters. In it, the kinetic energy dependent momentum is used to compensate nonlocal effects of the system. The functional is then examined by considering the very well-known semiconductor quantum dot systems. And despite its very simple form, the results obtained for quantum dots containing a higher number of electrons agrees pretty well with that of the standard exact exchange theory. Some of the desired properties relevant for the two-dimensional exchange functional and the lower bound associated with it are also discussed. It is observed that the above parameter-free semilocal exchange functional satisfies most of the discussed conditions.
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meta-GGA exchange energy functionals28 in 2D. The DMEbased functionals that are constructed from the exchange hole has the following important and unique features, i.e., in the limiting case, it correctly recovers the exchange hole with uniform density and substantiates overcoming the convergence issue without considering any cutoff procedure. To develop an exchange energy functional in 2D, we follow the work of Tsuneda et al.29 constructed for 3D. They derived an exchange energy functional to obey the fundamental conditions of the exact exchange functional, and that exchange energy functional contains kinetic energy density as a progressive part. Recently Jana et.al.28 proposed a DME-based exchange hole using the reverse engineering technique and shown that the full exchange hole obeys the exact expansion of the 2D exchange hole proposed by Pittalis et al.16 Whereas, in the present work, we have proposed a parameter-free exchange energy functional by restricting ourselves to the first part of the density matrix expansion, i.e., without considering the series resummation technique as is done by Jana et al.28 To enhance the efficiency and accuracy of the newly proposed functional, we incorporated the nonuniformity effects present in the system by modifying the expression for the Fermi momentum. The newly constructed exchange functional in 2D is a meta-GGA type functional, which contains reduced density gradient and KS kinetic energy density, as its main ingredients. The comprehensive testing of the functional is demonstrated for different quantum dots (QD), by varying the number of confined particles and confinement strength. We have benchmarked the above proposed parameter free exchange functional with regard to the 2D exact exchange (2D-EXX) within the Krieger-Li-Iafrate (KLI)
INTRODUCTION The Hohenberg−Kohn (HK) and Kohn−Sham (KS) density functional theory1,2 has become the de facto standard and widely used for studying the electronic structure of the quantum systems in varied dimensions. The efficiency, accuracy, and adaptability of the density functional theory depend on the performance of the approximated exchange-correlation (XC) functionals. Handling inhomogeneous systems in semilocal level is computationally affordable and the performance is satisfactory. Therefore, the development of accurate semilocal XC functionals is always been an active research field with promisingly new perspectives. In three dimensions (3D), generalized gradient approximations (GGAs)3−6 are preferred over the local density approximation (LDA). Beyond the GGAs, the Kohn−Sham kinetic energy dependent functionals are known as the meta-GGA.7−10 The meta-GGAs are known to be the most promising functionals for studying the properties of atoms, molecules, and solids in 3D with greater accuracy.9−11 However, during the last couple of decades, increasing attention has been paid toward understanding the electronic structure of low-dimensional quantum systems. It is noteworthy to mention that the applications involving low-dimensional quantum systems containing semiconductor quantum dots, quantum point contacts, quantum Hall devices have grown rapidly in recent years. However, the density functionals developed for 3D quantum systems cannot be applied directly to the 2D ones due to problems such as dimension crossover from 3D to quasi- and pure 2D systems.12−14 Hence, accurate approximations such as the semilocal XC functionals relevant for the 2D systems are always a thriving research topic. So far beyond the 2D-LDA,15 more attention has been paid to the development of semilocal exchange energy functionals16−27 with different formal exact properties. However, very recently density matrix expansion (DME) has been used to construct the © XXXX American Chemical Society
Received: January 14, 2018 Revised: March 21, 2018 Published: March 21, 2018 A
DOI: 10.1021/acs.jpca.8b00429 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A approximation, 2D-LDA exchange, and two well-known 2D-GGA functionals.16,26 The work is organized in the following manner. In the next section, we will discuss the DME in 2D. Then, the same will be used to construct the desired parameter-free exchange functional. Lastly, the functional will be applied to QDs, and we will discuss further the fundamental conditions desirable for the functionals in 2D.
Thus, using eq 6 and eq 7, the density matrix expression, i.e., eq 5 reduces to
DENSITY MATRIX EXPANSION IN TWO DIMENSIONS In principle, the DME technique is proved to be one of the best strategies for constructing an analytic exchange energy functional in 3D.29−32 Here, we have explored a similar expansion in 2D. The density functional exchange energy is recognized as the electrostatic interaction between the density of an electron at reference point r with its exchange hole surrounding itself.33 So, the exchange energy is given by
⃗ 2 where τ = ∑occ i |∇ϕi| be the KS kinetic energy density. Now, the cylindrically averaged exchange hole can be obtained by considering the density matrix expansion and realizing that
⎛ u u⎞ Γ12D⎜R + , R − ⎟ ⎝ 2 2⎠ ⎤ ⎧ ∇2 ρ ⎫ 6J (ku) ⎡ 2J (ku) = 1 ρ(R ) + 3 3 ⎢4 cos2 ϕ⎨ − 2τ ⎬ + k 2ρ⎥ ku k u ⎢⎣ ⎩ 4 ⎭ ⎦⎥
■
Ex =
1 2
∫ ρ(r)d2r ∫ d2r′
(8)
⎛ u u⎞ Γ12D⎜R + , R − ⎟|2 ⎝ 2 2⎠
(1)
⎛ u u⎞ ⎜ ,R− ⎟ Γ12D t ⎝R + 2 2⎠
occu
(3)
where ϕi are the occupied KS orbitals. Hence, knowing the density matrix, the exchange hole can be modeled and the same can be further used in the development of the exchange energy functional. Here, we have taken advantage of the exchange hole construction procedure proposed by Negele and Vautherin (NV)34 based on the DME. It follows from the expansion in the relative and center-of-mass coordinates u and R as, ⎛ u u⎞ Γ1⎜R + , R − ⎟ = ⎝ 2 2⎠
⎛
∑ ϕi*⎝R + ⎜
i
u ⎟⎞ ⎛⎜ u⎞ ϕ R− ⎟ 2 ⎠ i⎝ 2⎠
⟨ρx (R , u)⟩cyl = −
(4)
2 z
∞
∑ (2n + 1)(−1)n J2n+ 1(z)C21n(y) n=0
with z = ku and y = expressed as
i(∇⃗ − ∇⃗ ) cos ϕ − 1 22k ,
the polynomial
⎛ ν + m − 1⎞ ⎛ 1 2⎟⎞ ⎟ F ⎜−ν , ν + m; C2mν(x) = (− 1)ν ⎜ ;x ⎠ ⎝ ⎠2 1⎝ ν 2
k 4u 2
Ex =
(6)
Cm2n
(11)
PARAMETER-FREE EXCHANGE ENERGY FUNCTIONAL Now, based on the methodology presented in the previous section, we will construct a parameter-free exchange energy functional based on the density matrix expansion and the modeled exchange hole expression discussed above. In the center-of-mass coordinate system, the exchange energy expression eq 1 becomes
(5)
where ∇⃗ 1 acts on R1 and ∇⃗ 2 acts on R2. Now, to obtain the NV type exchange hole34 in 2D, we have expanded the plane wave, i.e., expression eq 5 in terms of the Bessel and hypergeometric functions using the generalized Gegenbouer additional theorem28 and is given by eizy =
12J1(ku)J3(ku)
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i
i
ku
−
For homogeneous systems, the terms present inside the square bracket in eq 11 vanish, and one automatically ends up with the uniform exchange hole. Given the above exchange hole, the corresponding parameter-free exchange energy functional will be constructed in the following section.
⃗ ⃗ Γ12D = e u·(∇1 −∇2 )/2 ∑ ϕi*(R1)ϕi(R 2)|u = 0
∑ ϕi*(R1)ϕi(R 2)|u= 0
2 2
⎡∇ ρ ⎤ − 4τ + k 2ρ⎥ ⎢ ⎣ 2 ⎦
r+ r′
=e
2J12 (ku)ρ(R ) 2
where R = 2 and u = r − r′. By virtue of the Taylor series expansion, eq 4 can be written as
u(∇1⃗ −∇2⃗ )/2cosϕ
(10)
The above expansion recovers correctly the small u expansion of eq 9 upon taking the expansion of the Bessel and hypergeometric functions. The present expansion is also similar to that of the Negele-Vautherin34 and later used by Scuseria and co-workers30−32 and Tsuneda et al.29 The main idea is to keep the expansion of the exchange hole up to u2 without applying any series resummation technique that has been proposed recently in 3D10 as well as in 2D.28 As a matter of which, the cylindrical averaged exchange hole becomes
with
i=1
(9)
cyl
⎤ 6J (ku) ⎡ ∇2 ρ 2J (ku) = 1 ρ(R ) + 3 3 ⎢ − 4τ + k 2ρ⎥ ku ⎦ ku ⎣ 2
(2)
Γ1(r, r′) = 2 ∑ ϕi*(r)ϕi(r′)
cyl
manifests to
2
⟨|Γ1(r, r′)| ⟩ 2ρ(r)
+ O(u 4)
⎡1 ⎤ u2 = ρ(R) + ρ(R)⎢ ∇2 ρ(R) − 2τ ⎥ + O(u 4) ⎣4 ⎦ 2
where ⟨ρx(r, r′)⟩ is the cylindrically averaged exchange hole in 2D. In terms of the first-order density matrix Γ1(r, r′), the cylindrically averaged exchange hole is expressed as ⟨ρx (r, r′)⟩ = −
2
⎛ u u⎞ Γ12D⎜R + , R − ⎟ ⎝ 2 2⎠
=
⟨ρx (r, r′)⟩ |r − r′|
cyl
1 2
∫ d2R ∫ d2u
ρ(R )⟨ρx (R , u)⟩cyl (12)
u
Upon using the cylindrically averaged exchange hole, the above exchange energy expression reduces to
is
Ex = −
(7) B
1 2
⎡
∫ d2R ⎢⎢⎣ 163kρ
2
+
⎞⎤ 32ρ ⎛ ∇2 ρ 2 − τ + ρ 4 k ⎜ ⎟⎥ 15k3 ⎝ 2 ⎠⎥⎦
(13)
DOI: 10.1021/acs.jpca.8b00429 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A
which upon using the functional form of kσ[τσ] becomes
The Laplacian term present in the above exchange energy expression is the most difficult component to handle numerically. In 3D, near the nucleus, the Laplacian term diverges. Some previous studies in 2D17,28 also suggest the necessity for removing the Laplacian of density from the exchange energy functional. Thus, we have eliminated it using integration by parts along with the identity proposed by Voorhis and Scuseria31 in 3D and used by Pittalis et al.16 for 2D. This leads to Ex = −
1 2
⎡
∫ d2R ⎢⎢ 163kρ
2
+
⎣
ExPF[ρσ , ∇ρσ , τσ ] = −
⎪
⎪
⎪
|∇ ρ| ρ3/2
is the reduced den-
sity gradient in 2D. Now, using the spin density scaling relation, i.e., Ex[ρ↑ , ρ↓ ] =
1 1 Ex[2ρ↑ ] + Ex[2ρ↓ ] 2 2
Table 1. Tabulated Exchange Energies in Atomic Units (a.u.) for Parabolic Quantum Dots Using EPF−KS , with τσ = τKS x σ and MPFa Ex
(15)
the spin polarized exchange energy becomes, Ex[ρσ , kσ , τσ ] = −
1 2
⎡ 32ρ2
∑ ∫ d2R ⎢
σ
⎢⎣ 3kσ
σ
⎞⎤ ⎛ x2 ⎜ σ − zσ ⎟⎥ ⎠⎥⎦ ⎝8
+
128ρσ3 15kσ3
(16)
Here, we have defined a dimensionless quantity zσ = (4τσ − 2k2σρσ)/4ρ2σ. The first choice for kσ is quite apparent, i.e., the Fermi momentum kσ = kFσ = 4πρσ . The momentum kσ has been used as an arbitrary quantity with the only constraint of dimension, i.e., it should follow the dimension of length inverse.31 Now, we have defined kσ as a functional of the kinetic energy density τσ(R), as was done by Tsuneda et al.29 in 3D through the following relation, i.e., τσ(R) = 2
(18)
If τσ is equal to the Thomas−Fermi (TF) kinetic-energy density, τσTF = 2πρσ2 , then the above-defined momentum vector kσ becomes identical to the Fermi momentum kFσ = 4πρσ for the noninteracting system. Using kσ from eq 18 in the exchange energy expression eq 16, the dimensionless parameter zσ vanishes. Hence, a simple parameter-free τσ dependent exchange functional (EPF) for 2D is obtained as
∑∫ σ
− E2D−EXX x
− E2D−LSDA x
− E2D−B86 x
− E2D−B88 x
− EPF−KS x
−EMPF x
2 2 2 2 2 2 2 Δ
1/6 0.25 0.50 1.00 1.50 2.50 3.50
0.380 0.485 0.729 1.083 1.358 1.797 2.157
0.337 0.431 0.649 0.967 1.214 1.610 1.934 16.55
0.368 0.470 0.707 1.051 1.319 1.748 2.097 2.94
0.364 0.464 0.699 1.039 1.344 1.728 2.074 3.59
0.373 0.479 0.726 1.085 1.365 1.812 2.175 0.83
0.413 0.527 0.789 1.167 1.457 1.908 2.258 7.35
and Table 2. These are obtained by applying it to the parabolic quantum dot with varying number of electrons and confinement strength. However, some unsatisfactory outcome of eq 20 prompted us to propose more appropriate and allowed modifications to the functional form of eq 19. The reasons for the unsatisfactory results are also addressed in next section in more details. As pointed above, the modifications of the functional are done from the physical point of view. Thus, for any inhomogeneous systems, we seek an expansion whose first term will give the LDA and subsequent terms as the leading order corrections to it. To do this, we have used kσ = kFσ in the first term and have used kσ from eq 18 in the second term of eq 19. So, finally the modified parameter free (MPF) exchange functional (EMPF x ) becomes
In eq 17, fσ(R, k) is the distribution function for the first order spin-polarized density matrix and is approximated with the help of step function in the momentum space.29 Also, the kσ present in the above eq 17 can be used in the DME based exchange energy functional. This is because the measurement of kσ in both the cases is identical. On using eq 17, one can express kσ as
1 ExPF[ρσ , ∇ρσ , kσ[τσ ]] = − 2
ω
The first column contains the number of particles. The second column contains different confinement strengths. The results obtained and EMPF functionals are presented in the last two with EPF−KS x x columns. Results obtained with the exact exchange 2D-EXX, 2D-LSDA, 2D-B88, and 2D-B86 are shown for comparison. The last row contains the MAPE, Δ.
(17)
2τσ ρσ
N
a
∫ kσ2fσ (R , kσ )d2kσ kσ2 k2 ρσ (R ) = ρσ (R) = ρ 2 σ 2 ∫ fσ (R , kσ )d2kσ
kσ = kσ[τσ ] =
σ
32ρσ5/2 ⎡ x 2ρ2 ⎤ ⎢1 + σ σ ⎥ 3 2τσ ⎢⎣ 20τσ ⎥⎦ (20)
(14)
where the dimensionless quantity x =
∑ ∫ d2R
The assessment of the above proposed exchange energy functional, eq 20, which contains τσ as a progressive part, fully depends on the choice of τσ. The first choice for it is τσ = τTF σ , i.e., Thomas-Fermi kinetic energy density. This homogeneous density approximated value τTF σ is way off from the exact kinetic energy density. So, the use of τTF σ in eq 20 leads to unsatisfactory results. Hence, we have not discussed those results for this choice in our calculation section. On the other hand, we have used the Kohn−Sham kinetic energy density τKS σ in place of the τσ as our next choice and denote the corresponding functional obtained with eq 20 as EPF−KS . The results of this choice are given in Table 1 x
⎛ 4τ − k 2ρ ⎞⎫⎤ 32ρ3 ⎧ x 2 ⎨ − ⎟⎬⎥ ⎜ ρ2 ⎠⎭⎥⎦ 15k3 ⎩ 4 ⎝ ⎪
1 2
ExMPF[ρσ , ∇ρσ , τσ ] = −
1 2
∑∫ σ
⎡ ⎤ ⎢ ⎥ 2 xσ ρσ ⎥ 2 8kFσ ρσ ⎢ 1+ dR 3π ⎢ ⎛ 2τσ ⎞2 ⎥ ⎢ ⎟ ⎥ 10⎜ ⎢⎣ ⎝ ρσ ⎠ ⎥⎦ (21)
EMPF x ,
The exchange energy functional, has a very simple analytic form. The first term corresponds to the LDA exchange energy. The presence of the LDA exchange term makes the proposed functional more stable, and all the nonuniform effects are taken care of by the second term.
32ρσ2 ⎡ x 2ρ ⎤ ⎢1 + σ σ2 ⎥ d 2R 3kσ ⎢⎣ 10kσ ⎥⎦ (19) C
DOI: 10.1021/acs.jpca.8b00429 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A Table 2. Table Caption Is Identical to Table 1 but with N = 6 to 56 N
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6 6 6 6 6 6 6 6 12 12 12 12 12 20 20 20 20 20 30 30 30 30 42 42 42 42 56 56 56 56 Δ
ω 2
1/1.89 0.25 0.42168 0.50 1.00 1.50 2.50 3.50 0.50 1.00 1.50 2.50 3.50 0.50 1.00 1.50 2.50 3.50 1.00 1.50 2.50 3.50 1.00 1.50 2.50 3.50 1.00 1.50 2.50 3.50
−E2D−EXX x
−E2D−LSDA x
−E2D−B86 x
−E2D−B88 x
−EPF−KS x
−EMPF x
1.735 1.618 2.229 2.470 3.732 4.726 6.331 7.651 5.431 8.275 10.535 14.204 17.237 9.765 14.957 19.108 25.875 31.491 23.979 30.707 41.718 50.882 35.513 45.560 62.051 75.814 49.710 63.869 87.164 106.639
1.642 1.531 2.110 2.339 3.537 4.482 6.008 7.264 5.257 8.013 10.206 13.765 16.709 9.553 14.638 18.704 25.334 30.837 23.610 30.237 41.085 50.115 35.099 45.032 61.339 74.946 49.256 63.289 86.378 105.684 2.75
1.719 1.603 2.206 2.444 3.690 4.672 6.258 7.562 5.406 8.230 10.476 14.122 17.136 9.746 14.919 19.053 25.796 31.392 23.953 30.665 41.651 50.794 35.503 45.538 62.007 75.748 49.722 63.871 87.148 106.609 0.45
1.749 1.594 2.241 2.431 3.742 4.648 6.226 7.525 5.387 8.311 10.444 14.080 17.086 9.722 15.029 19.188 25.973 31.603 24.091 30.836 41.878 51.068 35.671 45.747 62.286 76.085 49.919 64.117 87.479 107.010 0.58
1.775 1.655 2.283 2.532 3.830 4.851 6.491 7.829 5.559 8.474 10.789 14.535 17.620 9.968 15.266 19.496 26.376 32.063 24.410 31.249 42.420 51.694 36.071 46.259 62.946 76.837 50.407 64.739 88.281 107.918 1.97
1.743 1.626 2.238 2.481 3.745 4.740 6.337 7.642 5.442 8.282 10.534 14.169 17.153 9.783 14.977 19.123 25.867 31.445 24.005 30.722 41.686 50.778 35.561 45.606 62.062 75.771 49.771 63.921 87.160 106.542 0.17
delocalization of τKS σ relative to the electron density ρσ. In 3D, a similar outcome is also encountered Tsuneda et al.29 So, for betterment, we have modified eq 19 into eq 21. Next, we have tested the performance of the new exchange i.e. eq 21. Table 1 is only for N = 2 and the energy functional EMPF x MAPE for a set of values ω is better than the 2D-LSDA and inferior to 2D-B88 and 2D-B86. From Table 2, it is clear that the functional eq 21 gives the best result for more than two electrons and can be applied to practical systems. Here, we have emphasized on the calculated MAPE from the Table 2, i.e., 0.17 which is the best MAPE calculated thus far. The Figure 1 is the plot of the mean error (ME) for a set of electron numbers up to 56. These electrons are confined by the external parabolic potential with varied confinement strengths. The various color bars represent different exchange energy functionals. The deviations of MPF functional from the 2D-KLI (origin) are very less in comparison to all other considered functionals. In addition to exchange energies, the exchange potentials for aforementioned functionals are plotted in Figure 2. A parabolic QD having 6-electrons and confinement strength ω = 1 is considered for plotting the potentials. The potential of the new functional agrees well with the OEPKLI potential. But, all the discussed semilocal potentials fail to produce exact potential in the asymptotic region. Next, we have examined the performance of the newly proposed exchange energy functional by considering the Gaussian quantum dot system with confinement potential V(r) = −V0 exp(−ω2r2), where −V0 is the depth of the potential. Here, the shape of the potential changes with different values of ω and V0.
NUMERICAL DEMONSTRATION AND DISCUSSIONS The numerical demonstration reported below is based on multielectron quantum dots (QD). The external confining potential for parabolic QD is V0 = ω2r2/2, where ω is the confinement strength. For comparison of the results obtained with the newly constructed functionals, we have also calculated the 2D-KLI,35 2D-LSDA,36 2D-B88,26 and 2D-B8616 within the spin DFT implemented in the OCTOPUS code.37 For all the new functionals, we have used the self-consistent density and kinetic energy density of the KLI-OEP as the reference input. The performances of the new 2D exchange energy functionals, eq 20 with τσ = τKS σ and eq 21 are given in Table 1 for two electrons and in Table 2 for N = 6 to N = 56. In these tables, we have varied ω from 1/6 to 3.5. Presentation of the results of the same functional in different tables are intentional and this is to show the effectiveness of our functional eq 21 for N > 2. It can be clearly observed from Table 1 that replacing τσ by τKS σ in eq 20, the results for EPF−KS improves over existing exchange x energy functionals for two electrons and the mean absolute percentage error (MAPE) is the best among all the existing given in eq 20 is considered functionals. So, the functional EPF−KS x for studying the properties of the two-electron quantum dots. The presence of the KS-KE density in the first term of eq 20 makes the functional unstable. Hence, it is observed that with increasing number of electrons and ω value, the difference between exact and calculated values of the exchange energies increases, which makes the resulting MAPE greater than that obtained with 2D-GGA. The cause for this inefficient performance is the D
DOI: 10.1021/acs.jpca.8b00429 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A
chosen 2D-B86 because of its close result with our functional in case of the parabolic quantum dot. The reduction of error with the present functional in both types of quantum dot systems adds to the merit of the construction.
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FUNDAMENTAL CONDITIONS In two dimensions, the exchange energy functional can be expressed in terms of the enhancement factor as, Ex2D[ρ↑ , ρ↓ ] = −
− LDA 2D Fx , σ [xσ , τσ ] ∫ d2rρ(r)ϵ2D x ,σ
(22)
where − LDA ϵ2D = x, σ
Figure 1. Shown are the mean error associated with the exchange energies for few electron parabolic quantum dots with the variation of the electron number. The mean error is calculated for all the systems given in Table 2 with corresponding ω values. For systems having N = 72, 90, and 110, ω is varied from 1 to 3.5.
4kF, σ (23)
3π
From eq 21, the enhancement factor is given by Fx,2Dσ − DME[xσ , τσ ] = 1 +
xσ2ρσ ⎛ 10⎜ ⎝
2τσ ρσ
⎞2 ⎟ ⎠
(24)
Here, some of the fundamental conditions, which should be taken care of while constructing the exchange energy functionals in 2D, are discussed. (a) The exchange energy is negative for any nonzero densities, i.e., Ex[ρ] < 0
Figure 2. Shown are the exchange potentials for a six-electron parabolic quantum dot with ω = 1 using relevant functionals.
In the present case, we have fixed V0 at 40 au to calculate exchange energies by varying the number of confined electrons (N) and the value of confinement strength ω. Actually, in this case, the error reduction for the new functional is quite encouraging. The results obtained are shown in Table 3. The improvement
Ex[ρ↑ , ρ↓ ] ≥ −1.84
Table 3. Tabulated Exchange Energies (in Atomic Units) for Gaussian Quantum Dots Using the New Functional EMPF x , i.e., eq 21a N
ω2
−E2D−EXX x
−E2D−LSDA x
−E2D−B86 x
−E2D−MPF x
6 6 6 6 12 12 12 12 20 20 20 Δ
0.10 1/6 0.25 0.50 0.10 1/6 0.25 0.50 0.1 1/6 0.25
6.525 7.454 8.255 9.744 14.324 16.304 17.986 20.908 25.386 28.692 31.349
6.193 7.076 7.840 9.260 13.887 15.812 17.437 20.295 24.871 28.122 30.736 3.47
6.450 7.367 8.160 9.635 14.241 16.211 17.873 20.795 25.311 28.611 31.263 0.70
6.521 7.444 8.232 9.688 14.255 16.187 17.824 20.634 25.354 28.606 31.197 0.48
(25)
All the GGA and meta-GGA functionals obey this property. So also the newly constructed exchange energy functional eq 21 follows the condition as the enhancement factor from eq 24 is positive for all positive densities. (b) For homogeneous systems, the exchange energy functionals should produce the LSDA. Here, eq 21 perfectly reduces to LSDA functional in the uniform density limit as the change in the density vanishes for this case. (c) The local bound for the exchange-correlation functional in two dimensions,38,39
∫ d2rρ3/2
(26)
needs to be satisfied as well. With this, the enhancement factor in two dimensions follows FX(s) ≤ 1.301. The Figure 3 is an enhancement factor plot of the discussed functional
a
For comparison, the results obtained with other relevant functionals are also provided in various columns. The mean absolute percentage error Δ is listed in the last row.
Figure 3. Shown in the plot, the enhancement factor corresponding to the exchange functional EMPF x , i.e., Fx(s, α) versus s for different values of α. Here, α is a dimensionless ingredient of the enhancement factor and varies from 0 to 1.
over 2D-LSDA is clear from the above table. In this comparison, we also have included results obtained with 2D-B86. We have E
DOI: 10.1021/acs.jpca.8b00429 J. Phys. Chem. A XXXX, XXX, XXX−XXX
The Journal of Physical Chemistry A
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eq 24. We have used the parameter α = τ − τw/τunif for plotting, similar to 3D.10 Here, τw = |∇ρσ|2/4ρσ and τunif = 2πρ2σ. The entity α varies from the iso-orbital region (α = 0) to orbital overlapping region (α > 0). It shows that the enhancement factor for the newly constructed functional obeys the lower bound, and it increases from 1 to a value nearly equal to 1.2, which is less than 1.3. (d) Under uniform scaling of coordinates, the exchange energy functional scales as Ex[ρλ ] = λEX [ρ]
*E-mail:
[email protected]. *E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
Prasanjit Samal: 0000-0002-0234-8831 Notes
The authors declare no competing financial interest.
(27)
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ACKNOWLEDGMENTS Our research was supported by the Department of Atomic Energy (DAE), Government of India.
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(28)
Thus, with these scaling, the enhancement factor in eq 24 and the enhancement factor for eq 20 scales as a constant. (e) The exchange functional should obey the following scaling relation in order to be self-interaction free,40 i.e., for ρq = qρ1, Ex[qρ1] = q2Ex[ρ1]
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and the enhancement factor Fx should be dimensionless under the above scaling. In two dimension, the density scales as ρλ(r) = λ2ρ(λr), and the kinetic energy density scales as τσ[ϕi , λ] = λ 4τσ[ϕi]
Article
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where ρ1 is the one particle density and 0 < q ≤ 1. In 2D, to satisfy this scaling, the enhancement factor should scale as O (q1/2). The enhancement factor for eq 20 satisfies the above property (i.e., e), but eq 24 does not satisfy this scaling. The presence of the LSDA energy expression as the first term of EMPF makes the functional not follow this scaling. x
CONCLUSIONS An exchange energy functional for two-dimensional quantum dot systems is constructed and can be used within the framework of DFT. The functional depends on density as well as the kinetic energy density and has a simple mathematical form without any adjustable parameters. One of the best approaches in the DFT, i.e., DME is followed and statistically averaged kinetic energy density is used. The functional contains the LDA as the first term and corrections to LSDA in the successive terms. The choice of the kinetic energy density played an important role in this work. The exchange energy functional can be updated with the help of appropriately approximated kinetic energy density. Assessment of exchange energies for parabolically confined many-electron quantum dots and Gaussian quantum dots are demonstrated by varying the number of electrons from 2 to 56 along with the confinement strength. Even though the functional has very simple form without any adjusted semiempirical parameters, the exchange energies for these systems are satisfactory and gives minimum mean absolute percentage error when compared with other popularly used existing exchange energy functionals. The result is optimistic for electron numbers more than two. Some important properties such as the lower bound and the scaling relations in 2D are also discussed. The tightest upper bound and the uniform coordinate scaling are satisfied by the enhancement factor of the newly constructed exchange energy functional. The exchange energy functionals having LDA as a first-term similar to the newly constructed functional fail to satisfy the selfinteraction free scaling relation. We believe the functional will be very effective in modeling and testing the 2D artificial atom within density functional theory. The many-electron interactions of these artificial atoms can be studied efficiently using this functional. F
DOI: 10.1021/acs.jpca.8b00429 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
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DOI: 10.1021/acs.jpca.8b00429 J. Phys. Chem. A XXXX, XXX, XXX−XXX
