Test of the validity of the gradient expansion for slowly-varying extended systems and coefficients of the sixth order for the KE in 1d

Transcription

1 Test of the valiity of the graient expansion for slowly-varying extene systems an coefficients of the sixth orer for the KE in 1 Joseph B. Dizon Department of Chemistry, University of California, Irvine, CA Lucas O. Wagner, John C. Snyer, an Kieron Burke Department of Physics an Astronomy, University of California, Irvine, CA (Date: June 14, 2012 The calculation of the non-interacting KE is an increible challenge. One coul use the graient expansion (GE to calculate the non-interacting kinetic energy (KE by the knowlege of the ensity alone. The system of interest was the cosine potential, which has a perioic potential that can approximately moel solis. The ensity of cosine potential was are obtaine by summing singleparticle eigenfunctions that satisfying bounary conitions. The ensity epens on the potential V 0, the length of the unit cell, the number of electrons per unit η, an the number of unit cells N c, which was manipulate to explore the behavior of the GE. An extrapolation technique was introuce to calculate the KE within the thermoynamic limit where N c. An error analysis for the performance of the GE was implement by plotting the logarithm of percent errors against the orer of the GE. The aition of the GE terms systematically improve KE calculations, an the sixth orer GE performs the best for small V 0 an large η an since the ensity becomes more slowly-varing. The sixth orer term performs the best for large η an an small V 0. However, the GE orers, particularly, the sixth orer, breaks own completely for large V 0. CONTENTS I. Introuction 1 II. The graient expansion of the kinetic energy 2 III. The ensity of the cosine potential 3 IV. The KE of the cosine potential an tools for analyzing the KE 5 V. Performance of the GE for varying V 0 7 VI. Performance of the GE for varying η 8 VII. Discussion 8 VIII. Conclusion 9 IX. Acknowlegements 10 References 10 I. INTRODUCTION Electronic structure calculations in chemistry an physics may solving the interacting Schröinger equation for the energy of the system which inclues the kinetic energy (KE. However, calculations of the interacting KE Also at Department of Chemistry, University of California, Irvine, CA are too slow an ifficult to carry out, making them impractical to perform. Instea, the non-interacting KE is consiere because it can be calculate with reasonable accuracy. Density-functional theory has become a popular work horse for solving the electronic structure of atoms, molecules, an solis. The calculation of the noninteracting system, unlike the interacting system, involves the introuction of the Kohn-Sham (KS scheme that calculates the components of the energy incluing non-interacting KE is obtaine by using KS equations [1]. For the KS scheme to work, the energy must epen on the electronic ensity an from the HK theorem, the components of the energy are expresse as a functional of the ensity [2]. A functional, as efine in many DFT textbooks, assigns a number to a function [3]. In other wors, one can calculate the energy by simply knowing the ensity. The non-interacting KE is of important interest because this makes up the large portion of the energy that seemingly cannot be calculate from the electronic ensity alone (while the potential energy can be base solely on the ensity. Although the KS scheme has mae many tremenous contributions for fining electronic structure properties or molecules, atoms, an solis, the KS scheme becomes too slow an computationally costly for large systems. Even though the KS scheme was evelope in the 1960 s, there were attempts to write explicit formulas for the non-interacting KE functional without going through the KS scheme with reasonable efficiency an minimal computational cost since the birth of DFT in the late 1920 s. One of the first attempts to carry out KE calculations was one by Thomas an Fermi: they expresse the KE functional that is exact for the uniform electron gas of constant ensity an approximate for the slowly-

2 2 varying ensity, which later became famously known as the Thomas-Fermi moel [4, 5]. However, the KE from the TF moel suffers for rapily-varying ensity, as the latter fails to capture the changes of the ensity. Because of the ifficulties of the TF KE for rapilyvarying ensities, the graient expansion (GE has become a weapon of choice to correct the TF KE [6]. The GE is an asymptotic expansion that takes account of changes in the ensity by incluing erivative terms of the ensity. The GE, by construction, only inclues even erivative orers for the ensity, the o erivative orers canceling through translation invariance [6]. Various tests with the GE from previous works show that the GE corrects the non-interacting KE for secon an fourth orers an iverges beyon the fourth orer ue to asymptotic behavior at higher orers. As the result, the GE fails to correct the KE at higher orers [6]. Despite of the shortcomings of the GE ue to the ivergence at higher orers, there is still interest in unerstaning the behavior of the GE for simple 1-imensional (1 systems which will serve as motivation to generate new KE functionals. The GE from L. Samaj an J. K. Percus [7] was selecte for this stuy because it is believe that this GE can be use to calculate the KE for large 1 systems slowly-varying ensities. This GE is forme by using a series of recursion formulas to generate the KE functional by incluing correction terms for the TF KE. Presently, the GE is expane to the 6th orer by using a computer algebra system like Mathematica. In aition, the GE is expecte to perform better for slowly-varying ensity. Extene perioic systems are wiely stuie because these systems escribe an approximate the behavior of solis [8, 9]. One can figure out properties of solis such as ban structures, ban energies, an an ban gaps [9, 10]. Solis of interest are comprise of many repeating unit cells [6]. One can moel the soli by introucing a perioic potential that escribes the repeating cell [9]. By oing so, one can obtain eigenfunctions that have the same perioicity as a perioic potential an satisfy perioic bounary conitions, which is useful to generate electronic ensities that escribe the location of electrons in a perioic lattice. One of the common perioic systems that is investigate to stuy solis is the Kronig Penney, which obtains ban energies by solving the Schröinger equation to get eigenfunctions that satisfies bounary conitions [8]. The cosine moel is investigate in this stuy because this resembles soli that contains a cosine potential an one can etermine eigenfunctions that satisfy bounary conitions [11]. By knowing the ensity, one can etermine the ensity that retains bounary conitions from the eigenfunctions. In aition, there are a very few number of stuies that use the GE to stuy extene systems like the cosine potential. The motivation for this project is to assess the performance of the GE for slowly-varying ensities from the cosine potential to calculate the KE. The investigation of the GE ivie into multiple steps: Sec. II escribes how to obtain terms of the GE. Sec. III shows how to get ensities of the cosine potential from eigenfunctions with the requirement of satisfying perioic bounary conitions. Sec. IV inclues the formulation of calculating of the exact KE from eigenfunctions, the TF moel, an the GE using an extrapolation approach for an infinitely large system, an an error analysis of plotting the logarithm of the percent error against the orer of the GE. Analyses of the performance of the GE is ivie into two main areas: Sec. V assess the behavior of TF moel an the GE by changing the potential V 0 for varying the size of the unit cell an taking a high V 0 to examine where the GE fails. Sec. VI investigates the behavior of the TF moel an the orers of the GE for changing the length of the unit cell to ientify the regime where GE performs the best. These analyses aim to explore the effects of increasing the orer of the GE whether the KE becomes more accurate than the TF moel. In aition, these analyses serve as an assessment to etermine if the coefficients of the graient at the sixth orer is correct. Atomic units are carrie for all calculations presente in this paper an the KE will always be referre to as the non-interacting KE. II. THE GRADIENT EXPANSION OF THE KINETIC ENERGY The calculation of the non-interacting KE T s is one of the earliest challenges, an perhaps, one of the central problems in DFT. First, the formalism for expressing T s as a functional of the ensity involves on integrating over all space [7] T s [n] = x t s (x (1 where t s (x is the non-interacting KE ensity epenent on the electronic ensity n(x. Historically, there were attempts to express the KE ensitiy irectly to calculate the KE without going through the use of KS equation. The KE ensitiy from the TF moel in its alternative form [7, 12] for non-interacting same-spin fermions is t T F s = π2 6 n3 (x (2 where the factor π 2 /6 a correction factor for treating with large 1 systems, an the KE calculate from the TF moel by inserting into (1 becomes [12] Ts T F [n] = π2 x n 3 (x. (3 6 The TF moel oes not always get an accurate KE since this is a local approximation without taking account changes of the ensity. The GE serves as an approach

3 3 to correct the TF KE by taking account graient terms of the ensity. However, the GE is only use to treat slowly-varying ensities an there is hope that truncting a certain number of correction terms can systematically improve the accuracy of the KE, which is an approach to treat asymptotic expansions. Samaj an Percus generate their form of the GE by using a series of recursion formulas that obtains the KE ensity for various orers [7]. The recursion formulation requires a fair amount of numerical work to expan at the sixth orer. The GE for the KE incorporating (1 an (3 is formally written as Ts GE [n] = Ts T F [n] + x ( t (2 s (x + t (4 s (x +... (4 an the secon, fourth, an sixth orers of the KE ensities obtaine from the recursion formulas are [7] t (2 s (x = 1 n (x 2 24 n(x 1 2 n (x (5 t (4 s (x = 1 30 n (x 4 π 2 n(x n (x 2 n (x π 2 n(x n (x 2 π 2 n(x n (xn (x π 2 n(x n (4 (x π 2 n(x 2. (6 s (x = 25 n (x 6 81 π 4 n(x t ( n (xn (xn (x π 4 n(x n (xn (5 (x 1890 π 4 n(x n (x 4 n (x π 4 n(x n (x 2 n (x π 4 n(x n (x 2 π 4 n(x n (x 3 π 4 n(x n (x 3 n (x π 4 n(x 7 n (x 2 n (4 (x π 4 n(x 6 67 n (xn (4 (x 6048 π 4 n(x 5 n (6 (x π 4 n(x 4 (7 The recursion formula still obtains the TF portion of the kinetic energy an obtains the secon, fourth, an sixthorer KE ensity corrections for the KE ensity an these KE ensities are integrate over all space to obtain corrections to the KE. The orers of the GE are etermine by the combinations of the erivatives of the ensity, i.e. the secon-orer term inclues n (x an (n (x 2, the fourth-orer term inclues n (x 4, n (x 2, n (xn (x, an n (4 (x, an so forth. The sixth-orer has the most terms, which relies on having many graient terms for the ensity. In aition, the etermination of the sixth orer GE is still a great interest because it is ifficult to tell if this is the correct GE, but there is a chance to expan the GE to the eight orer to verify if the sixth orer is consistent. The secon an fourth orers were confirme to be consistent [7]. The GE from (4 an the correction terms (5, (6, an (7 were use for calculating the KE for the cosine potential. The knowlege of the ensity is an essential requirement to obtain T s. III. THE DENSITY OF THE COSINE POTENTIAL The cosine potential is use to moel extene perioic systems [11]. The cosine potential is expresse as ( V (x = V 0 sin 2 πx (8 which has a cosine term when one can rewrite (8 using trigonometric ientities. From (8, V 0 is the epth of the potential an is the length of the unit cell. The length of the crystal is L = N c where N c is the number of unit cells. The presence of the perioic potential like this cosine potential serves as a pertubation for the freeelectron moel, where the potential is constant an for convenience, is zero [9 11]. Perioic systems are treate as infinite systems to escribe a soli [6]. The number of unit cells shall become extremely large, perhaps N c. This is the thermoynamic limit for the perioic system. The first step of getting the ensity of the cosine potential is to obtain eigenfunctions using a single-particle Schröinger equation inserting (8 as the potential, which is 1 ( 2 ψ j (x V 0 sin 2 πx ψ j (x = ɛ j ψ j (x. (9 From (9, ɛ j is the energy eigenvalue, ψ j (x is the eigenfunction, an j inicates an iniviual orbital. In orer to solve (9, an expression for ɛ j must be known. Because of the cosine term, one can rewrite the Schröinger in the form of a Mathieu ifferential equation [11]

4 4 Vx x V 0 FIG. 1. Cosine potential with = 1 an N c = 2, an V 0 = 0.5. y (v + (a 2qcos(2vy(v = 0 (10 where a is the general characteristic value, q is the parameter, an v a variable to a function y(v. By rearranging (9 in the form of (10, one can etermine a, q, an ɛ j, which are an a = 2 π 2 (V 0 + 2ɛ, (11 q = V 0 2 2π 2, (12 ɛ = 1 ( π a V 0. (13 Inserting (11, (12, an (13 into (9 obtains the general solution for the cosine potential. With the help of Mathematica, solving the Schröinger equation gets an eigenfunction compose of Mathieu functions [11]. ( ψ j (x = A Ce a, q, πx ( + B Se a, q, πx (14 From (14, Ce is the even Mathieu function an Se is the o Mathieu function. A an B are coefficients for the even an o solution, respectively. The perioic bounary conitions are necessary requirements to obtain perioic eigenfunctions. The eigenfunction must be continuous an ifferentiable at the entire length of the unit cell. ψ j (0 = ψ j (L ψ j(0 = ψ j(l (15a (15b (15a requires the eigenfunctions to be continuous an (15b hols the conition of ifferentiability at the ens of the soli. (14 is broken into separate solutions that can satisfy conitions (15a an (15b. To o so, one can select an eigenfunction epening on j. Writing the eigenfunction can be tricky since one nees to properly express the characteristic values for the even an o Mathieu functions respectively. The characteristic values for the even Mathieu function is a j Mathieu function is b j+1 (q an for the o (q. The avantage of expressing these characteristic values is the proper selection eigenfunctions base on the value of j. Both even an o Mathieu functions behave as cosine an sine functions, respectively. Since normalization of sine an cosine functions are simple to carry, the normalization constants can be inserte into even an o Mathieu functions, which are an ψ e,j (x = ψ o,j (x = 2 L Ce ( a j 2 L Se ( b j+1 (q, q, πx (16 (q, q, πx. (17 Once proper characteristic values for the even an o Mathieu are obtaine, one can select proper eigenfunctions by choosing integer values of j. The even solution from (17 is acceptable for j = 0 an even j. On the other han, for o j, (17 is appropriate. By following the pattern of selecting respectable eigenfunctions that meet conitions (15a an (15b, The eigenfunction for the cosine potential selecting proper j values becomes ψ j (x = { ψ e,j (x j = 0, 2, 4,... ψ o,j (x j = 1, 3, 5,... (18 In aition, one can express the eigenvalues for each respective solutions from (16 an (17, which are essential for etermining correct eigenfunctions. ɛ j = ( π 2 ( π 2 2 a j 2 b j+1 (q V 0 (q V 0 j = 0, 2, 4,... j = 1, 3, 5,... (19 The eigenfunction from (18 are correct as long as this satisfies both sies of the Schröinger equation while inserting the eigenvalue expression from (9. The eigenfunctions from (18 are real solutions for the cosine potential. One can plot appropriate eigenfunctions for each j. From Fig. 2, the eigenfunction is even an has no noes at j = 0. When j = 1, the eigenfunction is o an has one noe. When j = 2, the eigenfunction has two

5 5 Ψx j 0 j 1 j 2 j nx N 10 N 20 N 40 N FIG. 2. Eigenfunctions for j = 0 (blue, 1 (re, 2 (green, an 3 (cyan obtaine from (18 an = 1 an N c = 4. Even eigenfunctions are soli an o eigenfunctions are ashe lines. noes, an so forth. The values of j inicate the number of noes for each eigenfunction an satisfying conitions (15a an (15b. The ensity of the cosine potential is compose of many single-particle eigenfunctions, which escribes orbital that a single electron can occupy. One can express the ensity by summing the square mouli for each eigenfunction. In orer to write the ensity, one munst know the number of electrons for the entire soli, which is N e. Because the soli is comprise of many unit cells, one nees to know the number of electrons per unit η, which is etermine from iviing the number of unit cells from N e by the number of unit cells in the soli N c. Therefore, the number of electrons is N e = ηn c. Using this information, the ensity now becomes n η (x = ηn c 1 j=0 ψ j (x 2. (20 The construction of the ensity requires that η = to set the average ensity as n = η/ = 1. This allows to a multiple number of electrons per unit cell in the soli for any given. The nature of the ensity epens on, η, an N c. N c is necessary to escribe the size of the soli for a given η. By having a certain number of unit cells N c, one can etermine the converge limit of the ensity for large N c with a given V 0. FIG. 3. Density for η = 1 with varying N c an V 0 = 1. nx N 10 N 20 N 40 N FIG. 4. Density for η = 2 with varying N c an V 0 = 1. ensities are continious an ifferentiable at the eges of the unit cell, which is a convenient feature for stuying perioic systems. This also reveals that one can construct the ensity of the entire soli by simply using a single unit cell alone. Asie from N c, V 0 ajusts the shape of the ensity. The ensity becomes more slowly-varying for small V 0, an when V 0 = 0, the ensity becomes constant, which is a istinct feature of a uniform electron gas [3] an present in the free-electron moel [8]. The challenge is to etermine which conition of the ensity maximizes the performance of the GE, either by having small V 0 or large η. There are ifferent qualitative features for the following ensites from Fig. 5 to 7. When η = = 1, a single bump is present, two bumps are seen for η = = 2, an for η = = 4, four bumps are observe. The presence of these bumps inicate the number of electrons in a unit cell. Increasing η an causes the ensities to become more slowly-varying where fluctuations of the ensity become small. The beauty of the perioic system is that one can raw the ensity of a single unit cell without picturing the ensity of the entire system. In aition, the IV. THE KE OF THE COSINE POTENTIAL AND TOOLS FOR ANALYZING THE KE With the knowlege of the eigenfunction for the cosine potential from (18, one can calculate the KE [9]. t j = { x ψ e,j (xψ 1 2 e,j (x j = 0, 2, 4,... 0 x ψ o,j (xψ o,j (x j = 1, 3, 5,... (21

6 6 nx N 10 N 20 N 40 N KE is plotte against 1/N c. To etermine the KE at the thermoynamic limit, one can obtain the y-intercept of the plot by taking constructing an approximate parabolic fit. By oing so, one can etermine the KE by setting 1/N c = 0, which follows that N c. T s as the KE for each N c is evaluate for the exact, TF, an the GE. The KE is plotte against 1/N c to generate an approximate quaratic plot. The KE at the thermoynamic limit is calculate by generating the fit an set 1/ = 0, which is N c. The quaratic extrapolation for getting the KE at the thermoynamic limit is FIG. 5. Density for η = 4 with varying N c an V 0 = 1. nx V 0 18 V 0 14 V 0 12 V FIG. 6. Density for η = 1 an N c = 80 for varying V 0. (21 calculates the KE per unit length for a single unit cell rather than the KE for the entire soli. The proceure to sum the exact non-interacting KE for the entire soli is just like the approach from(20, which is f(z = c 0 z 2 + T s (23 where c 0 is a coefficient etermine from the quaratic fit, z = 1/N c is the variable for (23, an Ts is the KE at the thermoynamic unit for the exact, TF, an the GE. Ts c Exact 0th 2n 4th 6th FIG. 7. Extrapolation plot for exact KE, TF, an the GE for η = = 2 an V 0 = 1/2 using extrapolation points N c = 80, 60, 320, an 640. ηn c 1 Ts,η EX = t j. (22 j=0 The TF moel an the GE calculates the KE by using (4 an iviing by to get the KE per unit length, which the principle applies from (21. Due to the perioicity of the ensity for a unit cell, one can take avantage of calculating the KE without consiering the entire soli. This proceure can minimize the cost of performing calculations for the KE for the remaining sections for this paper. So far, calculations of the KE involve with a finite system, which has a finite number of unit cells N c. The main goal is to calculate the KE at the thermoynamic limit, which is a key feature for a perioic system to escribe a soli. One approach to carry calculations at the thermoynamic limit by using an extrapolation approach. In the extrapolation approach, one calculates the KE for the exact, TF, an the GE by choosing finite N c values. After carrying calculations with ifferent N c, the From Fig. 7, the extrapolation fits only perform well when 1/N c is small. The ifference between the TF an GE an the exact KE is evient as inicate by having ifferent values of Ts. To compare the performance of of the TF an the GE at the thermoynamic limit, one must compare these KEs with respect of the exact KE by computing the percent error % T,p s = T s,p T,EX s T,EX s 100% (24 where Ts,p is the KE for TF an the GE Ts,EX is the exact KE, an p is the orer of the GE expansion (p = 0 for TF, p = 2, 4, an 6 for the orers of the GE. For convenience, one can take logarithm of the percent error to escribe orer of magnitue of the error. A plot of the logarithm of the percent error against the orer of the GE p serves as the benchmark to assess the performance of the GE.

7 7 V. PERFORMANCE OF THE GE FOR VARYING V 0 In this section, one can test the performance of the TF an the GE by ajusting V 0. A soli with unit cell length = 1 an η = 1 an the soli with = η = 2 are selecte for this investigation. The plot of the logarithm of the percent error against the GE assess the performance of the GE here. Extrapolation points are use for N c = 10, 20, 40, 80, an 160 to compute the KE at the thermoynamic limit. as the secon orer. From Fig. 9, the same behavior hols for the shape of the plots where the secon orer contributes larges to the KE correction while incluing the fourth an sixth orers slightly improve the accuracy of the KE. On the sie note, the performance of the GE performs better for η = = 2 an η = = 1 where the error improves by orer of magnitue btween 1 an 2. An in-epth focus for changing η an is available on Sec. VI. When V 0 becomes large, TF an the GE starts to fall apart for both η = 1 an η = 2. log Ts V 0 18 V 0 14 V 0 12 Ts Exact 0th 2n 4th 6th c FIG. 8. Plotting the percent error of T s vs. p for η = = 1 for various V 0. FIG. 10. Plotting T s vs. 1/N c for η = = 1 for V 0 = 5. log Ts V 0 18 V 0 14 V 0 12 Ts c Exact 0th 2n 4th 6th FIG. 9. Plotting the percent error of T s vs. p for η = = 2 for ifferent V 0. From Fig. 8, the errors for the KE calculations for ifferent V 0 are not esirably accurate with an orer of magnitue of 1 or 2. Each plot has a similar shape for ifferent V 0 an this inicating the varying performance of the GE terms. The secon orer makes up the majority of the KE corrections an aing the fourth an sixth orer improves the KE calculation although the contribution for the fourth an sixth orer are not as much FIG. 11. Plotting T s vs. 1/N c for η = = 2 for V 0 = 10. The logarithm of the percent error is not applicable because GE unerestimates the exact KE, which obtains a negative number. From Fig. 10 an 11, the plots of the TF an the GE o not line up with the exact KE, which reveals the breakown of TF an the GE at large V 0 even at the thermoynamic limit. The sixth-orer performs the worst, which severely unerestimates the KE for both η = 1 an 2. On the other han, the TF KE completely overestimates the KE for large V 0. Overall, the GE unerperforms at large V 0

8 8 As a sie note, the TF moel an the GE are exact within the thermoynamic unit when V 0 = 0 because the ensity is constant. VI. PERFORMANCE OF THE GE FOR VARYING η This section explores the behavior of the GE with ifferent η an, holing an important conition that η = as escribe in Sec. IV. Large N c values for the extrapolation approach are chosen to examine the behavior of the KE within the thermoynamic limit. In aition, calculations also inclue iffererent values of V 0 to etermine if the shapes of the log(% T s vs. p will have similar behavior an the shifts of the plots epen on V 0 log Ts Η 1 Η 2 Η 4 Η The extrapolation points are N c = 80, 160, 320, an 640, which are use to obtain more accurate KEs at the thermoynamic limit than choosing smaller extrapolation points. In Fig. 12, the KEs for each η are more accurate for V 0 = 1/10 as inicate by a larger negative log(% T s than for V 0 = 1/2 from Fig. 13. The shapes are similar for each η = 1 an 2 with p = 0, 2, an 4, which the plots have a convex shape ue to the slope between 0 an 2 is greater than the slope between 2 an 4. But incluing the sixth orer GE reveal subtle ifferent qualitative features for η = 4 an η = 6 with ifferent V 0. For V 0 = 1/2 on Fig. 13, the plot for η = 4 is convex, where the slope of the line between p = 4 an 6 is smaller than the slopes at p = 0 an 2 an p = 2 an 4. For η = 8, the line is approximately linear. When V 0 = 1/10, the plot is linear for η = 4 an the plot is concave for η = 8 as inicate by a bigger slope between p values between 4 an 6 than p values between 0 an 2 an between 2 an 4. Asie from ifferent V 0 values, one can explore the general features from Fig. 12. For η = 1, the error improves by magnitue of 1 an 2, but performs least accurately among other solis. For η = 2, the error improves by magnitue of 3 an 4, which is a slight improvement over η = 1. The secon orer term makes up the majority of the KE correction inicate by the steepest slope as escribe earlier. For η = 4, the error to the KE improve to about the orer of magnitue of 4 an 8, which is a better improvement over η = 1 an 2. The GE performs best at η = 8 since the orer of magnitue of errors is extremely low, with the error improves by a magnitue of 4 for aing each GE orer while the sixth-orer contributes the most for correcting the TF KE. FIG. 12. Plotting the percent error of Ts an an V 0 = 1/10. vs. p for various η VII. DISCUSSION log Ts Η 1 Η 2 Η 4 Η FIG. 13. Plotting the percent error of Ts an an V 0 = 1/2. vs. p for various η Calculations from Sec. V an Sec. VI present varying performances of the TF an the GE for ifferent V 0, η,, an extrapolation points for N c. One can assess that the GE performs better for smaller V 0 as inicate by ownwar shifts of the log(% T s vs. p from Fig. 8, 9, 12, an 13. From these results, one can imply that ecreasing V 0 makes the ensities become more slowly-varying, causing the graient of the ensity to become smaller. However from Fig. 8, the aition of terms of the GE at η = 1 improves the error slighly even for small V 0. One can say that the correction terms for higher orers of the GE become weaker even though aing those terms may improve the accuracy of obtaining the KE. In contrast, the errors for η = 2 improves by roughly an orer of magnitue of 1 an 2 compare to η = 1 even though the shape of the curves is similar an higher orer terms of the GE contributes less than at lower orers. Changing η has a significant effect on the performance of the TF an the GE, which shows that changing the length of the unit cell an putting more electrons in the unit cell

9 9 can make the ensity become more slowly-varying than changing the potential with a small V 0. Moreover, this is evient from Fig. 12 an 13 that the errors of the GE becomes better for η = 8 than η = 1, 2, an 4, implying that the graients of the ensity becomes weak. Therefore, the GE at the sixth orer can significantly perform better than at lower orers. The performance of each correction term of the GE varies with ifferent η. Looking back at Fig. 8 an 9, the GE incluing the secon-orer term corrects the KE the most while incluing the fourth an sixth orer slightly improves the KE, inicates that the epenence of the orer is weak even though aing more terms makes the total KE calculation more accurate. The sixth orer term only corrects the TF KE by a small fraction, which is smaller than incluing the secon orer term. But for high η, the aition of successive correction terms of the GE significantly improves the KE. Looking at η = 4 an V 0 = 1/2 from Fig. 13, aing orers of the GE improves the error by a magnitue of 2, which can tell that the epenence of the orer is greater. Aing the sixth orer term improves the TF KE, but oes not improve the accuracy signifcantly. However, when V 0 = 1/10, the sixth orer term significantly improves the TF KE by an orer of magnitue of 4 in aition to big contributions from the secon an fourth correction terms. This may show changing η has a tremenous effect on the overall performance of the GE. Perhaps one can assert that changing η has a substantial effect on changing behavior of the ensity, which influences the behavior of the GE. When V 0 is large, the terms of the GE becomes erratic particularly at the sixth orer. Before looking at the sixth orer an from Fig. 10 an 11, TF baly overestimates the KE. The TF moel is esigne for the uniform electron glass an expects to behave poorly since changes in the ensity in terms of the graients of the ensity are not inclue. The secon orer term for η = 1 an 2 significantly improves the KE but is still not close to the KE. Incluing the fourth orer term improves the total KE, although this unerestimates the exact KE for η = 2. The sixth orer GE is the worst because this unerestimates the exact KE completely where the sixth orer term subtracts the total KE. This inicates that the sixth orer term is sensitive to the graients of the ensity, causing the KE to iverge way from the exact KE. The ownfall of the sixth orer GE comes from having strong graients of the ensity where the ensity becomes rapily-varying, causing the sixth orer to behave unpreictably. The GE can only perform better for small V 0, particularly at the sixth orer. Looking back at Fig. 8, 9,12, an 13, one can safely say that the GE is asymptotic an systematically improves results as p grow for higher η an not much from smaller η, i.e. η = 1. One might come up with an appealing iea that expaning to the eight orer can increase the accuracy of the KE base using the results to back up this claim. However, that notion is uncertain since no one has expane the GE to the eight orer. Computational cost using the recursion formulas from [7] become so immense that the eight orer expansion becomes not feasable. If that is the case, one may not be sure to fin out if the coefficients of the sixth orer GE from (7 is correct since one must nee to expan at higher orer to check if the lower terms hol to be true, i.e. the sixth orer expansion is consistent by expaning to the eight orer an that is consistent if a tenth orer expansion is accomplishe. One may believe that increasing the orer of the GE may further improve accuracy, but this iea can be true if the expansion to higher orers can be accomplishe. The extrapolation approach is only an approximation to get the KE at the thermoynamic limit because this approach selects finite values of N c to generate a quaratic fit that obtains the y-intercept where 1/N c = 0 an N c. However, computational costs become severe for choosing high values of N c to generate a quaratic fit since the number of terms of the ensity becomes high, causing the calculation KE by the GE to slow own significantly. However, the KE calculations for various 1/N c o not always behave quaratically as inicate from Fig. 7, but the fits perform better when 1/N c is extremely small. Perhaps, the extrapolation approach is a useful shortcut approach to etermine the KE within the thermoynamic limit, which is useful to stuy perioic systems that moel solis. Overall, the extrapolation approach can become more accurate for choosing higher N c while sacrificing computational cost an time. VIII. CONCLUSION The calculations of the KE for the cosine potential reveal particular trens about the performance of the GE for various number of electrons per unit cell η, the unit cell length, the potential V 0, an the number of unit cells N c. The basis of these KE calculations involve on using the extrapolation approach to obtain the KE within the thermoynamic limit, which consequently, choosing high N c values traes off between accuracy an more computational costs. The GE oes fail for large V 0 since the ensity becomes rapily-varying an the graients of the ensity becomes bigger. The sixth orer GE falls apart miserably where the sixth orer term subtracts the total KE, which unerestimates the exact KE. For small V 0 an large η, the aition of successive orers of the GE obtains the KE closer to the exact results. The sixth orer GE performs the best because total KE is closer to the exact KE. However, the performance of the sixth orer correction varies where the sixth orer term corrects the KE by a slight fraction for smaller V 0 an small η. But for larger η, the sixth orer terms corrects the total KE by larger fraction, improving the total KE. The improve performance of the GE for each successive orer inicates that the GE can perform best when the graient of the ensity is small when η is large an V 0 is small. Due to this tren, one may believe that expaning

10 10 towars the eight orer can improve the accuracy of the KE, but this notion may or may not be true. If the eight orer expansion is not carrie, one may be uncertain if the sixth orer expansion is correct. But espite of these short comings, the results an finings presente in this paper reveal fascinating trens an behaviors about the GE at least at the sixth orer an the investigation of obtaining higher orers of the GE can be the subject for future investigation. IX. ACKNOWLEDGEMENTS I thank John Syner for his programming techniques on generating interactive plots an figures an coes for expaning the GE of the KE an helpful iscussions about the GE. I thank Lucas Wagner for his contributions on unerstaning the formalisms of constructing eigenfunctions, ensities, an the KE of the cosine potential, iscussions an insights about the GE an the TF moel, an his work for comparing his numerical results to the analytical results presente on this paper. I thank Prof. Burke for the opportunity to tackle the cosine potential problem an the GE project an his knowlegeable an insightful iscussions an comments about the behavior of the GE, ensities of the cosine potential, an proceures for analyzing results. I thank the Chem-SURP program for funing this project uring the summer of 2011 an the Unergrauate Research Opportunities Program (UROP for their support for unergrauate research. I am greatful to thank Prof. A.J. Shaka for his attentive support for the Honors Program in Chemistry in UC Irvine, unergrauate research an eucation, an his support for aspiring unegrauate researchers in the program. [1] W. Kohn an L. J. Sham, Self-consistent equations incluing exchange an correlation effects, Phys. Rev. 140, A1133 A1138 (1965. [2] P. Hohenberg an W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 B871 (1964. [3] E. Engel an R. M. Driezler, Density Functional Theory: An Avance Course (Springer, New York, NY, [4] L. H. Thomas, The calculation of atomic fiels, Math. Proc. Camb. Phil. Soc. 23, (1927. [5] E. Fermi, Eine statistische Methoe zur Bestimmung einiger Eigenschaften es Atoms un ihre Anwenung auf ie Theorie es perioischen Systems er Elemente (a statistical metho for the etermination of some atomic properties an the application of this metho to the theory of the perioic system of elements, Zeitschrift für Physik A Harons an Nuclei 48, (1928. [6] F. Jansen, Introuction to Computational Chemistry, 2n E. (Wiley, West Sussex, Englan, [7] L. Samaj an J. K. Percus, Recursion representation of graient expansion for free fermion groun state in one imension, The Journal of Chemical Physics 111, (1999. [8] Estle T. L. Lane N. F. Morrison, M. A., Quantum States of Atoms, Molecules, an Solis (Prentice Hall, Englewoo Cliffs, NJ, [9] D. J. Griffiths, Introuction to Quantum Mechanics, 2n E. (Pearson Prentice Hall, Upper Sale River, NJ, [10] R S. A. Ross J. Berry, R. S., Physical Chemistry, 2n E. (Oxfor University Press, New York, NY, [11] T. R. Carver, Mathieu s functions an electrons in a perioic lattice, Amer. J. Phys. 39, 1225 (1971. [12] K. Burke an L. Wagner, DFT in a nutshell, (2011.