Semilocal Exchange Energy Functional for Two-Dimensional

Article pubs.acs.org/JPCA

Semilocal Exchange Energy Functional for Two-Dimensional Quantum Systems: A Step Beyond Generalized Gradient Approximations Subrata Jana* and Prasanjit Samal* School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India ABSTRACT: Semilocal density functionals for the exchangecorrelation energy of electrons are extensively used as they produce realistic and accurate results for finite and extended systems. The choice of techniques plays a crucial role in constructing such functionals of improved accuracy and efficiency. An accurate and efficient semilocal exchange energy functional in two dimensions is constructed by making use of the corresponding hole which is derived based on the density matrix expansion. The exchange hole involved is localized under the generalized coordinate transformation and satisfies all the relevant constraints. Comprehensive testing and excellent performance of the functional is demonstrated versus exact exchange results. The accuracy of results obtained by using the newly constructed functional is quite remarkable as it substantially reduces the errors present in the local and nonempirical exchange functionals proposed so far for two-dimensional quantum systems. The underlying principles involved in the functional construction are physically appealing and hold promise for developing range separated and nonlocal exchange functionals in two dimensions.

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INTRODUCTION Density-functional theory (DFT)1,2 is most successful in addressing the complex effects due to electron−electron interactions. Tremendous advances beyond the local density approximation (LDA) have been achieved through the development of accurate nonlocal, semilocal and hybrid exchangecorrelation (XC) functionals.3−22 However, construction of XC functionals in low dimension become indispensable as cutting edge research in such dimensions have also gained momentum as far as the theoretical and experimental findings25,26 are concerned. In spite of the promising applications in three dimensions (3D), the dimensional crossover of the XC energy functional from 3D to twodimensional (2D) regime, still remains one of the most difficult problem.27,28 Thus, wide use of DFT in 2D principally demands potentiality in this direction. So the systematic DFT calculations and proper explanations of numerous properties of low-dimensional systems ranging from atomistic to artificial structures, e.g., quantum dots, modulated semiconductor layers and surfaces, quantum Hall systems, spintronic devices, quantum rings, and artificial graphene pose a great challenge. As a matter of which, the construction of accurate nonlocal and semilocal XC functionals to appropriately describe the effects and phenomena occurring in 2D quantum systems has become an enthralling and growing research field. In this regard, the first step among the available theoretical methods is the wellknown 2D-LDA.29 The 2D-LDA, combined with the 2D correlation,30,31 leads to intriguing results and establishes its superiority over quantum Monte Carlo calculations.32 In recent © XXXX American Chemical Society

years, several advances have been made beyond the 2D-LDA, e.g., generalized gradient approximations (2D-GGAs),33−43 which perform in a more excellent manner. Not only that, various types correlation functionals compatible with the 2DGGAs are also constructed.38−41,44−46 In principle, the exchange functionals can be constructed from the exchange hole. In 3D, it is done by making use of Taylor series expansion,3,8 real space cutoff procedure,4 modeling the exchange hole,7 and the density matrix expansion (DME) based on general coordinate transformation.10−13,15,22 It is to note that the Taylor series expansion method has been applied to construct 2D-GGA.34 However, unlike Taylor expansion, DME10,11,15,22,47 based approaches are not only correct for small separation limit, but do converge in case of large separation22 and recover the correct uniform gas behavior. Prompted by these, in this work, we have formulated the 2D counterpart of the above DME-based exchange energy functionals. Advanced DME techniques are also proposed for constructing the exchange hole and the corresponding energy functional. Then, the functional is bench-marked against the optimized effective potential (OEP) based exact exchange (EXX)48 and local and gradient approximations for exchange in 2D systems.33,34,43 The OEP-based EXX functional is used as standard reference because it is the most accurate approach that is routinely applied for studying quantum dots.49 Furthermore, Received: April 19, 2017 Revised: May 26, 2017 Published: June 5, 2017 A

DOI: 10.1021/acs.jpca.7b03686 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A the newly constructed functional is applied to study the few electrons trapped inside parabolic and Gaussian quantum dots.

⟨ρxt2D ⟩ = −

■

THEORETICAL BACKGROUND The quantum mechanical exchange energy is nothing but the electrostatic interaction between the electron located at r ⃗ and the exchange hole at r ⃗ + u⃗ surrounding it. Thus, the spinunpolarized exchange functional in 2D is defined as Ex2D[ρ] =

1 2

ρ( r ⃗)ρx ( r ⃗ , r ⃗ + u ⃗)

∫ d2r ∫

d2u

u

+

|Γ( r ⃗ , r ⃗ + u ⃗)|2 2ρ( r ⃗)

(1)

1 2

EXCHANGE HOLE AND THE SEMILOCAL FUNCTIONAL Now, for modeling the exchange hole and then constructing the exchange functional, we seek an expansion based on the DME. The DME, that we intend should quite rightly obtain (i) the correct uniform density limit, (ii) the cylindrically averaged exchange hole similar to that given in eq 7, when terms up to u2 will be taken into consideration, and (iii) the large u-limit (i.e., 0 to ∞ integral limit of u) that converges even without considering any cutoff procedure. So to construct the desired semilocal functional, we begin by considering the DME given in eq 5 along with the following plane wave expansion in terms of the Bessel and Hypergeometric functions i.e.

(2)

ρxt2D ( r ⃗ λ , u)

∫ d2r λ ρ( r ⃗ λ) ∫

u

d2u

ekucosϕy / k = ( + )

(3)

ρxt2D = −

λ

(=

2

− (1 − λ)u ⃗ , r ⃗ + λu ⃗)| 2ρ( r ⃗)

with the generalized Hypergeometric functions, 2F1, the Bessel functions, J2n+1 and y = −(1 − λ)∇⃗ 1 + λ∇⃗ 2. The series resummation technique along with the Gegenbauer addition theorem51 are used to arrive at the above expansion (i.e., eq 8 and eq 9). In the appendix, we have derived a scheme of generalized Gegenbauer addition theorem which has been used to obtain eq 8. Actually, the Gegenbauer addition theorem51 is the method of expressing plane wave in terms of Bessel and Hypergeometric functions. However, using only the above theorem, one can obtain part of our expansion, and the corresponding exchange hole that will not preserve the conditions i−iii mentioned in the beginning of this section. Therefore, we have added a series (i.e., known as series resummation) with the expansion obtained using the Gegenbauer addition theorem51 and then determined the

i

(5)

where ∇⃗ 1 and ∇⃗ 2 operate on Ψi* and Ψi respectively. The exchange energy, E2D involves cylindrical average of the x exchange hole ⟨ρx(r,⃗ r ⃗ + u⃗)⟩cyl over the direction of u⃗ i.e. dΩ u 2π

are

(10)

⃗

∫ ρx ( r ⃗ , r ⃗ + u⃗)

Cm2n

⎛ ν + m − 1⎞ ⎛ 1 2⎞⎟ ⎟ F ⎜−ν , ν + m; C2mν(x) = ( −1)ν ⎜ ;x ⎠ ⎝ ⎠2 1⎝ ν 2

= eu ⃗[−(1 − λ)∇1 + λ∇2 ] ∑ Ψ*i ( r ⃗ λ − (1 − λ)u ⃗)Ψi

⟨ρx ( r ⃗ , r ⃗ + u ⃗)⟩cyl =

2 ∑ (−1)n (2n + 1)J2n+ 1(ku) 1 2 2 cos ϕ ku n = 0 ⎡ ⎤ ⎛ ⎞ ϕ y cos ∂ 1 ⎟⎥ × ⎢ C 2n ⎜ − i ∂y ⎣ ⎝ k ⎠⎦

and ϕ be the azimuthal angle. The polynomials, expressed as

occ

( r ⃗ + λu ⃗)|u ⃗ = 0

n=0

y cos ϕ ⎞ ⎟ k ⎠

(9)

⃗

λ

⎛

∞

∑ (−1)n (2n + 1)J2n+ 1(ku)C21n⎜⎝−i ∞

)=

Γ12tD( r ⃗ , u ⃗) = eu ⃗[−(1 − λ)∇1 + λ∇2 ]Γ12tD( r ⃗ , u ⃗)|u ⃗ = 0 ⃗

2 ku

(4)

with Γ2D 1t be the KS single particle density matrix. The real parameter, λ involved in the generalized coordinate transformation can take values 1/2 → 1 (or, 0 → 1/2). In particular, the conventional and on top exchange holes (which is 1 maximally localized in 2D50) correspond to λ = 1 and λ = 2 respectively. Also for λ = 1, eq 4 reduces to eq 2 i.e. the conventional exchange hole. Now the transformed single particle KS density matrix around u = 0 becomes ⃗

(8)

where

where ρtx2D is the transformed exchange hole defined by |Γ12tD( r ⃗ λ

(7)

■

with Γ(r,⃗ r ⃗ + u)⃗ = 2∑occ ⃗ . The exchange hole i ψ* i (r)⃗ ψi(r ⃗ + u) obeys two important properties: (i) the normalization sum rule, ∫ ρx(r,⃗ r ⃗ + u⃗) d2u = −1, and (ii) the negativity constraint, ρx(r,⃗ r ⃗ + u⃗) ≤ 0. Now, under general coordinate transformation (i.e., (r1⃗ ,r2⃗ ) → (rλ⃗ ,u)), where rλ⃗ = λr1⃗ + (1 − λ)r2⃗ , the exchange energy functional given by eq 1 reduces to Ex2D =

|∇⃗ρ( r ⃗)|2 ⎤ 2 1 ⎥u (2λ − 1)2 4 ρ( r ⃗) ⎦

The expression in eq 7 was originally proposed for the conventional exchange hole in 3D3 and then extended to 2D.34 However, the same method has failed to recover the uniform density limit. In order to recover this limit, the whole term was multiplied by the exchange hole of the uniform electron gas,34 whereas, in this work, all the above deficiencies are overcome through a proposed novel approach based on the DME, which will be discussed in the following section.

where ρx(r,⃗ r ⃗ + u⃗) is the exchange hole associated with the electron at r.⃗ In terms of the 1st order reduced density matrix Γ(r,⃗ r ⃗ + u⃗) = 2∑occ i ψ* i (r)⃗ ψi(r ⃗ + u⃗) and the Kohn−Sham (KS) orbitals ψi, the exchange hole is defined as ρx ( r ⃗ , r ⃗ + u ⃗) = −

ρ( r ⃗) 1 ⎡⎛ 1⎞ − ⎢⎜λ 2 − λ + ⎟∇2 ρ( r ⃗) − 2τ ⎝ 2 4⎣ 2⎠

(6)

On taking the cylindrical average of the density matrix given in eq 5 after Taylor series expansion yields the correct small u behavior and the same is given by B


Article

The Journal of Physical Chemistry A

2

Figure 1. eq 15 is plotted (with jp = 0) for two noninteracting electrons with density ρ(r ) = π exp(− 2r 2) parabolically confined in 2D. Shown are the exchange holes at the reference point r = 0.5 au for several λ values, where r is the radial distance from the origin. For λ = 1.0, some portion of the exchange hole is +ve indicating that the exchange hole violates the negativity property and will underestimate the magnitude of exchange energy. This implies the requirement for normalization of the exchange hole.

exchange hole that satisfies all the desired conditions. Finally, eq 8 together with eq 5 have produced the generalized coordinate transformed density matrix J1(ku)

Γ12tD = 2ρ

ku

+

6J3(ku) 3

ku

.+

24J3(ku) k3u 2

/

where jp2 ⎛ 1⎞ 1 3 = ⎜λ 2 − λ + ⎟∇2 ρ − 2τ + 4 + k 2ρ ⎝ 2⎠ 2 ρ

(11)

4 = (2λ − 1)2

where ⎧⎛ ⎫ 1⎞ . = 4 cos2 ϕ⎨⎜λ 2 − λ + ⎟∇2 ρ − 2τ ⎬ + k 2ρ ⎠ ⎩⎝ ⎭ 2 / = cos ϕ(2λ − 1)|∇ρ|

(12)

jp 2 ρ

(13)

where jp =

1 2i

occ

∑ {ψi*( r ⃗)[∇⃗ψi( r ⃗)] − [∇⃗ψi*( r ⃗)]ψi( r ⃗)} i

(14)

is the paramagnetic current density. By doing this, the functional also becomes gauge invariant and fulfills all the above-mentioned criteria. Inclusion of current density is particularly important whenever there happens to be the radiation matter interactions. It is relevant to use the following cylindrical average of the exchange hole (e.g., shown in FIG.1) corresponding to the above density matrix which will be used for the construction of the desired 2D semilocal exchange functional, i.e. ⟨ρxt2D ⟩ = −

2J12 (ku) 2 2

ku 2 144J3 (ku) − 4 k6u 4

ρ( r ⃗) −

24J1(ku)J3(ku) k 4u 2

(16)

By virtue of the above exchange hole, the functional retains the most unique features like uniform density limit for k = kF = (2πρ)1/2 and correct u2 behavior. So the above DME based exchange hole is more general in nature than its predecessors.34 Actually, in the earlier attempt,34 the small u expansion of cylindrical average exchange hole was multiplied by the corresponding averaged exchange hole of uniform gas and the parameters were determined using the sum rule, whereas, in the present attempt, all these are automatically taken care. Thus, the uniform density limit is trivially recovered when k = kF. But for inhomogeneous systems, the extent of inhomogeneity is included through use of a parameter f (need to be determined analytically) so that k = f kF. Then, f is being obtained from the normalization of the cylindrically averaged exchange hole i.e.

⃗ 2 with τ = ∑occ i |∇ψi| , the KS kinetic energy density. Now, in order to make τ gauge-invariant, we have modified it such that τ → τ̃ = τ − 2

|∇ρ|2 ρ

1 6 + 4y = 1 f f2

where y = (2λ − 1)2p and p = s 2 =

(17) 2

|∇⃗ρ| (2kFρ)2

is the square of the

reduced density gradient in 2D. For slowly varying density limit, eq 17 demands that f ≈ 1 + 6y and in the limit of large density gradient, f → y1/4 , similar to that proposed earlier.34 However, it is quite desirable to obtain a more intuitive form of f by examining the solution of eq 17 which can retain all the above low and high density features. Thus, analytic solution of eq 17 is obtained by applying successive root finding method and we propose that for any arbitrary density the dimensionless parameter f should satisfy the following relation

3

(15) C


Article

The Journal of Physical Chemistry A f = [1 + 90(2λ − 1)2 p + β(2λ − 1)4 p2 ]1/15

⎡ jp 2 ⎤ jp 2 ⎛ 2 1 ⎞⎟⎢ unif ⎥ ⎜ 3=6 λ −λ+ τ − τ2D − 2 − 2τ + 4 ⎝ ρ ⎥⎦ ρ 2 ⎠⎢⎣ 1 2 + kF ρ 2

(18)

The above expression for f is arrived at by rigorously analyzing the solution of eq 17 in the following manner. As eq 17 is a nonlinear equation. So we have used the binomial expansion technique and have dropped the higher powers of the reduced density gradient (s) to find the approximate solution of it. In particular, we have kept only the terms up to ’s4’ as other higher powers are insignificant for calculating the exchange energy. In this way, we have obtained eq 18. It is noteworthy to mention that in case of low density, the binomial expansion of eq 18 leads to f ≈ 1 + 6y. Thus, the proposition, eq 18 is in right spirit. Here, the parameter λ is determined along with β by fitting with the exact results known for physical systems of importance. In the present case, we have found these parameters by comparing the results obtained with that of the exact exchange for the few electron quantum dots. Nonetheless, the Laplacian term present in the exchange hole expression is difficult to handle numerically in various cases. For example, in 3D, it has been observed that, near the nucleus, the exchange hole diverges due to the presence of the Laplacian term in it.14,22−24 Besides the above divergence problem, another important motivation behind the replacement of Laplacian term is that the exchange hole should be finite everywhere in space.22 In this way, the higher rung of the Jacob’s ladder is constructed. Though, in the present study of harmonically confined quantum dots in 2D at the origin (i.e., r → 0), the Laplacian term does not have any divergence character. But, to obtain the higher rung of the Jacob’s ladder in 2D, we tried to replace the Laplacian term analogous to it is 3D counterpart. In fact, a Laplacian free exchange energy functional in 2D is already proposed in reference 33, where they have used the integration by parts technique. Therefore, either one can adopt the previous approach33 or try to replace the Laplacian term by the semiclassical approximation of kinetic energy density52 as this method has been successfully employed in designing the meta-GGA type functional in 3D. Here, we have preferred the second procedure in order to keep the functional construction similar to that of 3D meta-GGAs.14,22 Thus, ∇2ρ is replaced by ⎡ jp 2 ⎤ unif ⎢ ∇ ρ ≈ 6 τ − τ2D − 2 ⎥ ⎢ ρ ⎥⎦ ⎣

4 = (2λ − 1)2

(21)

Now, the semilocal exchange energy functional in 2D can be obtained by substituting eq 21 back in eq 1. Thus, the exchange energy expression becomes Ex2D − mGGA = −

where ϵ2xD − LDA = in FIG.2),

−

2

2 2

f kF u

ρ( r ⃗) −

(19)

144J32 (fkFu) 6

6 4

f kF u

24J1(fkFu)J3(fkFu) f 4 kF 4u 2

and the enhancement factor (e.g., shown 1 2R + 3 f 5f

(23)

Figure 2. Enhancement factor F2D−mGGA eq 23 (with jp = 0) is plotted x as a functional of s in the iso-orbital (α = 0.0) and orbital overlap regions (α = 0.5 and α = 1.0). The enhancement factors of 2D-GGA and 2D-B88 are shown in the subfigure for comparison.

with

πρ2

=−

4kF 3π

Fx2D − mGGA[p , τ , jp ] =

where τ2unif D = 2 . The right side of the above expression trivially reduces to ∇2ρ for slowly varying density limit. This justifies why it is a good approximation to remove the Laplacian and keep the exchange hole as well as energy Laplacian free. So this completes the motivation of replacing ∇2ρ by the semiclassical approximation of KE. Now, with the help of the above replacement, the modified exchange hole takes the mathematical form 2J12 (fkFu)

∫ ρ( r ⃗)ϵ2xD−LDAFx2D−mGGA[p , τ , jp ] d2r (22)

2

⟨ρxt2D ⟩

|∇ρ|2 ρ

128 (2λ − 1)2 p 21 jp 2 ⎞ jp 2 1 ⎛ 3 λ 2 − λ + 2 ⎜τ − τ2unif D − 2ρ⎟ − τ + 2ρ ⎝ ⎠

R=1+

(

+

)

τ2unif D (24)

Now, in order to obtain the behavior of the newly proposed enhancement factor, F2D−mGGA , as a functional of reduced x density gradient, it is desirable to consider the iso-orbital indicator as is done in 3D.22 This is because the new enhancement factor is of meta-GGA type and also depends on the KS kinetic energy density. Thus, we have considered the

3

4

iso-orbital indicator α =

(20)

τ − τW τ2unif D

, where τ W =

|∇ ρ|2 8ρ

be the Von

Weizsäcker kinetic energy density and have replaced the KS kinetic energy density by the parameter α. Upon substituting τ

where D


Article

The Journal of Physical Chemistry A Table 1. Exchange Energies (in au) for Parabolically Confined Few Electron Quantum Dotsa N 2 2 2 2 2 2 2 6 6 6 6 6 6 6 6 12 12 12 12 12 20 20 20 20 20 Δ(MPE)

ω

−E2D−EXX x

−E2D−LDA x

−E2D−GGA x

−E2D−B88 x

−E2D−BR x

−EmGGA x

1/6 0.25 0.50 1.00 1.50 2.50 3.50 1/1.892 0.25 0.421 68 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

0.380 0.485 0.729 1.083 1.358 1.797 2.157 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

0.337 0.431 0.649 0.967 1.214 1.610 1.934 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

0.368 0.470 0.707 1.051 1.319 1.748 2.097 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

0.364 0.464 0.699 1.039 1.304 1.728 2.074 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

0.375 0.480 0.722 1.080 1.354 1.794 2.020 1.775 1.655 2.281 2.529 3.824 4.845 6.492 7.846 5.728 8.572 10.915 14.716 17.858 10.167 15.573 19.892 26.935 32.777

0.386 0.492 0.735 1.085 1.354 1.776 2.113 1.736 1.620 2.226 2.466 3.716 4.699 6.279 7.573 5.415 8.231 10.461 14.063 17.019 9.805 14.894 19.007 25.698 31.230

5.7

1.7

3.9

2.8

0.7

The 1st and 2nd columns contain the number of particles and confinement strengths used for finding the parameters of the proposed functional. Results for EXX, 2D-LDA, 2D-GGA, 2D-B88 and 2D-BR are also shown for comparison with that obtained using the constructed 2D-mGGA functional. The last row contains the mean percentage error, Δ. a

W 2D−mGGA in terms of α i.e., τ = ατunif as a 2D + τ , an expression for Fx functional of only reduced density gradient is obtained. Now, using this the behavior of the enhancement factor can be studied for different values of α and is shown in Figure 2). In particular, α = 0 (i.e., τ = τW) is known as iso-orbital region. Whereas, α > 0 defines the orbital overlapping region. So the above procedure is similar in spirit with the techniques used earlier for studying the meta-GGA enhancement factor in 3D.14,22

exact exchange of hydrogen atom along with the smooth behavior of the enhancement factor at the iso-orbital region in order to remove the spurious divergence of the exchange potential.54 Finally, the newly constructed exchange functional is tested and the performance of it is given in Table 1. Trivially, the results are quite superior as it yields error that are smaller by at least a factor of 8.1, 2.4, 5.6, and 4.0 with respect to 2D-LDA, 2D-GGA,34 2D-B88,43 and 2D-BR33 respectively. Finally, the comprehensive assessment of the functional is being performed for Gaussian quantum dots by simultaneously varying the number of electrons trapped (N), depth of the potential and confinement strength (ω). For this case, the performance is presented in Table 2 and Figure 3, respectively. Here too, the results are found to be in excellent agreement with KLI-EXX. Actually, the new semilocal exchange energy functional reduces the error by a factor of 2.2 compared to 2D-GGA for the whole set of test cases.

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NUMERICAL DEMONSTRATION To study the performance of the above newly constructed meta-GGA type semilocal functional and obtain the parameters β and λ present in it, a test set of parabolic quantum dots having varying confinement strengths with few electrons embedded into it is considered. This type of system is also reported earlier for testing the 2D-GGA functional34 and to fix the corresponding parameters involved therein. A selfconsistent calculation with KLI-OEP exact-exchange method using OCTOPUS code53 has been performed and the density is used as the reference input. As our system is nonmagnetic, so the paramagnetic current density is irrelevant because it vanishes (jp = 0). Then, the value of λ is obtained by fitting with different confinement strengths such that the mean percentage error gets reduced, whereas β is fixed so as to confirm the smooth behavior of the enhancement factor in the s ≈ 0 region.54 The value of the parameters λ and β are obtained to be 0.74 and 30.0 respectively. In 3D,22 same set of parameters are also used. However, those are fixed by taking the

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CONCLUSIONS

To summarize, a meta-GGA type semilocal functional in two dimensions is constructed based on DME. The beauty of this functional is that the exchange hole involved in it has correct short-range behavior and recovers the uniform density limit quite accurately. The convergence of the exchange hole in large separation limit leads to an analytic expression for the corresponding energy functional even without applying any cutoff procedure which are essentially lacked by 2D functionals proposed in recent years. The most appealing feature of the E


Article


plane wave expansion in accordance with the above theorem is as follows:

Table 2. Comparison of Exchange Energies (au) for 2 2 Gaussian Quantum Dots (vext = −V0e−ω r ) for Low Densitya V0

N

ω2

−E2D−EXX x

−E2D−LDA x

−E2D−GGA x

−E2D−mGGA x

10 10 10 10 10 40 40 40 40

2 2 2 2 2 6 6 6 6

0.05 0.10 0.25 1/6 0.50 0.05 0.10 0.25 1/6

1.047 1.255 1.573 1.427 1.839 5.416 6.525 8.255 7.454

0.934 1.120 1.405 1.274 1.643 5.139 6.194 7.840 7.076

1.017 1.219 1.529 1.386 1.788 5.354 6.450 8.160 7.367

1.048 1.250 1.555 1.416 1.804 5.372 6.460 8.142 7.364

8.3

2.0

0.9

Δ a

∞

e

iz cosϕ

ν

= 2 Γ(ν)

∑ (−1)m (2m + ν) m=0

J2m + ν (z) zν

C2νm(i

cos ϕ)

(25)

Here J2m+ν(z) is the Bessel function and Cν2m(i cos ϕ) is associated with the generalized Hypergeometric function through ⎛ ⎛ m + ν − 1⎞ 1 2⎞⎟ ⎟ F ⎜−m , m + ν ; C2νm(x) = ( −1)m ⎜ ;x ⎝ ⎠2 1⎝ ⎠ m 2 (26)

In the present study, the left side of eq 25 is recognized as

Mean percentage error given in the last row.

∞

eiku(−icosϕy)/ u = 2νΓ(ν)

∑ (−1)m (2m + ν) m=0

J2m + ν (ku) (ku)ν

C2νm( −i cos ϕy/u)

(27)

For ν = 1, the right side of our eq 27 is recognized as ( of eq 8. The only remaining part of eq 8 is ) . Part ( of eq 8 only gives the first two terms of exchange hole expression, i.e., eq 15. However, in order to obtain the remaining part of exchange hole, which is necessary to produce the exact small u behavior of the exchange hole, we have added a series of Bessel and hypergeometric functions using reverse engineering technique (i.e., knowing small u expansion of exchange hole before hand). We found that derivative of Hypergeometric functions is useful to get the full behavior of exchange hole. This complete the resultant expression of eq 8.

Figure 3. Shown in the figure, the exchange energy per electron (in au) plotted versus ω2 for a series of Gaussian quantum dots with N electrons and confinement strength ω.

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AUTHOR INFORMATION

Corresponding Authors

*(S.J.) E-mail: [email protected]. *(P.S.) E-mail: [email protected].

present semilocal functional is that it is derived from the full exchange hole and thus having a strong physical basis. The functional remains one step ahead of the 2D-GGA because of the significant reduction of error compare to its counterparts. Thus, the functional in principle can enable us for making precise many-electron calculations involving larger structures such as arrays of quantum-dots, quantum-Hall devices, semiconductor quantum dots, and quantum-Hall bars on a regular basis. Also, the proposed exchange hole can be used to construct meta-GGA level exchange only pair-distribution function, static structure factor, and nonlocal and rangeseparated functionals in 2D. The present construction can be further extended to the recently developed density functional formalism for strictly correlated electrons. The obvious next step is to construct a functional for correlation energy that will be compatible with the exchange proposed here. The functional is not only physically appealing but also practically useful as it opens the path for constructing exchange correlation functionals in two dimensions analogue to Jacob’s Ladder in three dimensions.

ORCID

Prasanjit Samal: 0000-0002-0234-8831 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Esa Räsänen, Stefano Pittalis, and Jianmin Tao for helpful discussions and critically examining the work. The authors also thank Manoj K. Harbola and S. D. Mahanti for their valuable comments and suggestions on the manuscript. The financial support from the Department of Atomic Energy, Government of India is highly acknowledged.

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REFERENCES

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APPENDIX: GEGENBAUER ADDITION THEOREM The generalized Gegenbauer addition theorem of Bessel functions is obtained by expressing the plane wave in terms of Bessel and Hypergeometric functions. The expression of the F


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Semilocal Exchange Energy Functional for Two-Dimensional

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