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11

Finite Size Scaling for Quantum Criticality Using the Finite Element Method

Edwin Antillon,1 Birgit Wehefritz-Kaufmann,2 and Sabre Kais∗1, 3

1Departments of Physics, Purdue University, West Lafayette, IN 479072Departments of Mathematics and Physics, Purdue University, West Lafayette,

IN 47907 and Department of Physics, Princeton University, Princeton NJ 085443Department of Chemistry, Purdue University, West Lafayette, IN 47907

Finite size scaling for the Schrodinger equation, is a systematic approach to calculate the quantumcritical parameters for a given Hamiltonian. This approach was shown (S. Kais and P. Serra, Adv.Chem. Phys. 125, 1 (2003) ) to give very accurate results for critical parameters by using asystematic expansion in a finite basis set: for both Slater and Gaussian type functions. Recently,the finite element method, on the other hand, was shown to be a powerful numerical method forab initio electronic structure calculations (R. Alizadegan, K.J. Hsia and T. J. Martinez, J. Chem.Phys. 132, 034101 (2010)) with a variable real-space resolution. The implementation producessparse matrices since it is implemented in terms of local basis functions, making it ideal for parallelimplementation. Here, we demonstrate how to obtain quantum critical parameters by combiningfinite element method with finite size scaling. The critical parameters could be atomic nuclearcharges, internuclear distances, electron density, disorder, lattice structure, and external fields forstability of atomic, molecular systems and quantum phase transitions of extended systems. Toillustrate this approach we give detailed calculations of applying the finite element method (FEM) toapproximate solutions for the two-electron atom with varying nuclear charge; these include Hartree-Fock (HF), Local Density Approximation (LDA) and an “exact” formulation using FEM. We thenuse a finite size scaling (FSS) approach to determine its critical nuclear charge for stability; here,the size of the system is related to the number of elements used in the calculations. Results are ingood agreement with previous basis set calculations and demonstrate that it is possible to combinefinite size scaling with the finite element method for calculating quantum critical parameters. Thecombined approach provides a promising first principle way to describe quantum phase transitionsfor materials and extended systems.

I. INTRODUCTION

The Finite Element Method (FEM) has proved successful in a variety of areas needing an approximate solution topartial differential equations. The basic idea is to discretize a continuous function to a set of values in sub-domains,called elements with a variable resolution. Its first applications in engineering date back to 1950, in problems suchas elasticity in civil engineering and structural analysis in aeronautical engineering. In the 1970s, some applicationto quantum mechanical problems appeared, but were limited to small systems due to the storage limitation[1, 2].Recently, advances in high-performance computing make FEM a viable alternative to the traditional approach ofelectronic-structure calculations[3, 4].We suggest that any basis approximation to a many body system can be combined with the theory of Finite Size

Scaling (FSS) in statistical mechanics to obtain critical parameters. In the past, FSS calculations for few body systemshave been done based on expanding the wave function in a Slater-type basis sets and Gaussian-type basis functionsobtaining good agreement [5]. However, generalization to larger atomic and molecular systems proved difficult usingsingle basis functions. Recently, we have combined FSS with the finite element method on 1D and 3D sphericalsystems to obtain critical parameters. In a previous article [6] we compared finite element and basis set methodsapplied to finite size scaling and found excellent agreement between the two approaches.In this paper, we would like to show how to combine FEM with FSS for electronic-structure calculations near

criticality. To do so, we examine the two electron system with FEM both exactly and using mean-field equations.The formalism is very general and can be applied similarly to more complex systems using Hartree-Fock equations orKohn-Sham equations in density-functional theory. The paper is organized as follows: in section (II) we discuss FEMformalism, in section (III) we discuss the finite size scaling approach to quantum systems, in section (III) and (IV)we combine FSS with FEM to compute calculations of critical parameters for two electron atoms, lastly in section(V) we discuss the results and application to extended systems.

∗ kais@purdue.edu

2

FIG. 1: Cartoon showing the nodal values of the wave-function in the finite element approximation

II. FINITE ELEMENT METHOD IN ELECTRONIC CALCULATIONS

Computational methods in Chemistry typically make use of Slater or Gaussian basis sets to approximate solutionsin atomic or molecular systems. The advantages of these methods are that some of the integrals involved in thecalculations are analytical and very accurate results can be obtained. There are however problems extrapolating to acomplete basis limit. Plane Waves methods, having a complete basis, present some difficulties if non-periodic boundaryconditions are to be used such as in molecule calculations. A common trait of all these methods is that they employglobal basis. This can result in some difficulties for the formulation of large problems. First, the matrix formulationsbecome dense using global basis and therefore ill-conditioned to parallelism [7]. For some complex objects, there canbe convergence issues if there are large variations of potential terms in small regions which cannot be captured by thetraditional Gaussian basis set that are centered on the nuclei.The Finite Element Method (FEM) presents an alternative to these problems. This method has a local basis

approximation that is general because the bases are composed of polynomials with variable real-space resolution,therefore convergence can be controlled systematically. Furthermore, the locality of the bases in real space resultsin sparse and banded matrices. The number of operations for matrix-vector multiplication can be reduced from N

Log N for dense matrices to N for sparse matrices, where N is the dimension of the matrix [7]. The sparsity onthe global matrices resulting from the FEM makes the parallel formulation more affordable. Alizadean et al, haverecently applied a Divide-and-conquer method to the FEM-Hartree-Fock approach for electronic calculations showinga facilitation on parallelization and reduced scaling for larger systems [8].FEM uses a variational method to solve differential equations, in this case the Schrodinger equation. The best

approximation to the solution is the one that minimizes the following constraint on the Hamiltonian

E = minΨ

{〈Ψ|H|Ψ〉〈Ψ|Ψ〉

}

The solution region is discretized into elements and compact basis functions Ni(x) are used to interpolate thesolution between nodes. Figure 1 illustrates some important terms. The discretization of a continuous operator isknown as the Galerkin approximation [1, 2, 9]. Mathematically, we write this approximation as

|Ψ(x)〉 ≈ NTψ =∑

i

Ni(x)|ψi〉 (1)

|ψi〉 represents the undetermined ith nodal values of the wave function and Ni(x) is the interpolation polynomial

3

between nodes i and node i+1 [56].In order to solve for these unknowns, a similarity transformation can take the eigenvalue problem H |Ψ〉 = E|Ψ〉

into an generalized eigenvalue problem in the nodal representation i.e.

NTH(NT−1

)︸ ︷︷ ︸

Sym.Matrix

NTψ︸ ︷︷ ︸

new vector

= E NTψ︸ ︷︷ ︸

new vector

(2)

We would like to consider the expectation value for any operator in this representation. This can be achieved byexpanding the Hamiltonian in the local coordinate representation, (related to the global coordinate rn = rn−1+hn ·x,where the nth element has hn width), and writing this expectation value as

〈Ψ|O|Ψ〉 ↔ 〈ψ|N∗ O NT|ψ〉 (3)

↔ 〈ψi|∑

i,j

∫

i,j∈n

N∗i (x) O Nj(x)r2ndrn

︸ ︷︷ ︸

Oij

|ψj〉 (4)

(5)

The summation above has a slight subtlety, known in FEM formalism as overlapping of the nodes; since for everyelement n there is a left value of the wave-function |ψL(x)〉 and a right value |ψR(x)〉, while for the next element|ψR(x)〉 → |ψL(x)〉 and so on. Therefore, a global matrix representation of an operator will result by overlapping thelocal matrices whenever these share a node, for example:

Onij +On+1

ij + . . .↔

O(n)00 +O

(n−1)11 O

(n)01 0 . . .

O(n)10 O

(n)11 +O

(n+1)22 O

(n+1)12 . . .

0 O(n+1)21 O

(n+1)22 +O

(n+2)33 . . .

......

.... . .

,

where Onij is the nth local matrix and (i,j) are the nodal indices.

For a linear basis the overlapping is minimal, but if the continuity of the derivative or more nodes per elementare used, the local matrix elements will share more nodes. Nonetheless, the structure of the global matrix remainsdiagonally banded and hermitian and sparsity is guaranteed.Multiplying Eq. (2) by 〈Ψ| ≈ 〈ψ|N yields

〈ψi|Hij |ψj〉 = ǫ〈ψi|Sij |ψj〉,

and the problem is reduced to solving the above generalized eigenvalue problem. The formalism can be readilygeneralized to three dimensions, viz,

|Ψ(x, y, z)〉 =∑

ijk

Ni(x)Nj(y)Nk(z)|ψijk〉 (6)

III. FINITE SIZE SCALING

We now discuss the role of finite size scaling in systems exhibiting critical behavior. Phase transitions are associatedwith singularities of the free energy. These singularities occur only in the thermodynamic limit [15, 16] where thedimension of the system approaches infinity. Near criticality, fluctuations of the system become correlated overdistances of the order of the correlation length ξ, thus for systems where ξ ≈ L, we expect significant finite size effectsin thermodynamic quantities.The theory of Finite Size Scaling (FSS) provide us with numerical methods [10, 18–23] capable of obtaining accurate

results for infinite systems by studying the corresponding small systems. The finite-size effectively rounds off thesingular points of physical quantities exhibiting critical behavior where any physical property P is expected to scale

4

to its value at the thermodynamic limit [10] via:

PL(t)

P∞= f(

L

ξ∞(t)), (7)

here t = T − Tc is the critical parameter and f(x) is the scaling function[10].Classical phase transitions are governed by the fluctuation of temperature that tend to minimize the free energy.

At zero temperature, however, these fluctuations are driven by Heisenberg’s uncertainty principle [40]. A signature ofa phase transition is non-analicity of the ground state energy E(λ) with respect to a parameter λ in the Hamiltonian.The non-analicity can be the result of avoided crossing, or actual level crossing. In the first case, the configurationof the system that minimizes the energy maintains the energy gap (∆Eλ) between the first excited energy state andthe ground state as the parameter λ is varied. In the latter case, the energy levels cross so that the energy is nowminimized by a configuration that previously corresponded to the first excited state of the system. As in statisticalmechanics, we can define some critical exponents related to the asymptotic behavior of different quantities near thecritical point. The energy gap behaves near the critical point λc can be written as:

∆Eλ ≈ (λ − λc)α (8)

where α is a universal critical exponent. In addition to the vanishing energy scale, a diverging length scale also occurs,which can be associated with the exponential decay equal-time correlations in the ground state[40], in our case weshall focus on the behavior of the wavefunction at threshold.

ξ ∼ |λ− λc|−ν

To apply finite size scaling method to quantum system, we assume a Hamiltonian of the form

H = Ho + Vλ (9)

where λ is the critical parameter showing a phase transition (λc).For potentials linear in λ ( Vλ = λV ), Simon[24] showed that the critical exponent is equal to one if and only if

the Hamiltonian H(λc) has a normalizable eigenfunction with eigenvalue equal to zero. The existence or absence ofa bound state at the critical point is related to the type of the singularity in the energy. Using statistical mechanicsterminology, we can associate first order phase transitions with the existence of a normalizable eigenfunction at thecritical point. The absence of such a function could be related to continuous phase transitions.In statistical mechanics, the singularities in thermodynamic functions associated with a critical point occur only in

the thermodynamic limit. Kais ed al have developed a FSS theory to quantum systems [25–34]. In this approach, thefinite size corresponds not to the spatial dimension but to the number of elements (N) in a complete basis set usedto expand the exact eigenfunction of a given Hamiltonian [35–39].In the variational approach singularities in the different mean values will occur only in the limit of infinite basisfunction,

〈O〉∞λ ∼ (λ − λc)µo

we will assume that the following ansatz applies to the operators with truncated magnitudes. The FSS ansatz inthe Hilbert Space assumes the existence of a scaling function F (x) such that

〈O〉(N)λ ∼ 〈O〉∞λ FO(N |λ− λc|ν) (10)

with a different scaling function FO for each different operator but with a unique scaling exponent. At the critical

point, the expectation value is related to N as a power law, 〈O〉(N) ∼ N−µo/ν .To obtain the critical parameters, we define the following function that will remove the N dependence at a critical

point

△O(λ;N,N′) =

ln(〈O〉Nλ / 〈O〉N′

λ )

ln(N ′/N)(11)

5

For the energy operator O = H , and using the customary α Greek letter for the corresponding exponent µo wehave

△H(λc;N,N′) =

α

ν(12)

In order to isolate the critical exponent α from numerical calculations, it is convenient to define another function:

Γα(λ,N,N′) =

△H(λ;N,N ′)

△H(λ;N,N ′)−△ ∂Vλ∂λ

(λ;N,N ′)(13)

which at the critical point is independent of N and N ′ and takes the value of α. Namely, for λ = λc and any valuesof N and N ′ we have

Γα(λc, N,N′) = α (14)

IV. IMPLEMENTATION

We illustrate the combined approach on the two-electron atom, however the method is general and can be appliedto more complex atomic and molecular systems. We will seek the minimum charge (critical charge) needed to bindboth electrons to the nucleus with charge Z. It is well known that ignoring the electron-electron energy correlationwill not predict the stability of H− [42], since the critical charge obtained by such assumption is Zc ≈ 1.02. Stillinger(1974) [43] took into account the e-e correlation by using non-linear wavefunction e−ar1−br2 + e−ar1−br2 in order tofind the minimum energy as the nuclear charge Z was varied, he found a critical charge around Zc ≈ .9537, and Baker(1990)at al [44], performed a thorough 401-order perturbation calculation to resolve the controversy for the criticalcharge, they obtained a critical charge of Zc ≈ .91103, this is the value we will use as reference.In the following sections we illustrate the mean-field equations and exact formulations for the ground state of the

two-electron system, while in the next section we apply FSS to these approximations.

A. Hartree Fock (HF) Approximation

In this approximation the electron-electron interaction of the Hamiltonian are included only in the average sense.We start with the Hamiltonian for the two-electron atoms

H = −1

2∇2

1 −1

2∇2

2 +e2

|r1 − r2|− (

Z

r1+Z

r2), (15)

where r1 and r2 are the coordinates of the two electrons referenced from the nuclei, and Z is the charge of the nucleus,this will play the role of the critical parameter.In the Hartree approximation the total wave-function is approximated as a product of single particle wavefunctions

is Ψ(r1, r2) = ψ1(r1)ψ2(r2). We can arrive at a self-consistent form by multiplying the Hamiltonian by the complexconjugate wave-function above and integrating one coordinate out [12].

[−1

2∇2

1 −Z

r1+

∫

d3r21

|r1 − r2||ψ2(r2)|2]|ψ1(r1)〉 = E|ψ1(r1)〉 (16)

Using the FEM ansatz

|ψ(x)〉 ≈∑

i

Ni(x)|ψi〉 = NT|ψ〉 (17)

6

the generalized eigenvalue problem Hklψl = ǫSklψl is obtained; with the above terms defined as:

Hkl =

∫

[

hkl︷ ︸︸ ︷

1

2∇Nk ∗ ∇NT

l +1

rNkN

Tl +

Jkl︷ ︸︸ ︷∑

iv

Qijklψiψj ]d3r1

Qijkl ≡∫ ∫

d3r1d3r2Ni(r2)

∗Nj(r2)1

|r1 − r2|Nk(r1)

∗Nl(r1)

Skl =

∫

NkNTl d

3r1

The operator hkl in the FEM basis is the single electron energy, Jkl is the coulomb term, and Skl is the energy overlapterm. Since the electrons are fermions, the overall product of the single particle states needs to be antisymmetric.This last requirement gives rise to the exchange coulomb term Kkl that repels states of parallel spin. We can get thisterm by noting that the indices (i, j) ∈ ψ2 and similarly indices (k, l) ∈ ψ1, then the operator Kkl is obtained fromJkl by by switching indices (j ↔ l), thereby making the operator non-local.The ground state energy is determined from the smallest eigenfunction |ψs〉 by computing the expectation value

EG =

Ne∑

i

〈ψs|hi|ψs〉 − 1

2

Ne∑

i

〈ψs|Ji −Ki|ψs〉

The ground state for the Helium atom (Ne = 2) does not have parallel spins, therefore the the exchange term doesnot arise. Using FEM with 500 elements we obtain EG = −2.859110, this compares well with the reference valueEG = −2.861680 [11].

B. Total Energy

To improve upon the Hartree-Fock approximation we take into account for the correlation Energy, using the freeelectron gas approximation due to Wigner. Here the “correlation” potential is given by [13].

Vc(r) =0.29e2

5.1 + rs(r)(18)

where rs is the local density parameter for the Helium atom.

ρ(r) =1

43πr

3s

(19)

ρ(r) = ρ(r)↑ + ρ(r)↓ (20)

(21)

The last expression follows for Helium since in the ground state the two electrons have opposite spin, but share thesame spatial orbital. Hence the correlation potential is included self consistently since the density is obtained from thesum of densities for each electron i.e., ρ(r) = 2|ψ(r)|2. The correlation energy in the FEM representation becomes:

E(corr)kl =

∫

NkVc(r1)NTl d

3r1

The value for the total energy obtained is:

Etotal = EHF + E(corr) (22)

Using FEM with 500 elements we obtain Etotal = −2.90924, this compares well with the reference value Etotal =−2.90372 Ref [14].

7

C. Density Functional Theory

In the Hartree-Fock approximation, the many-body wavefunction is written in the form of a Slater determinantwhile the correlation energy is treated separately. This can be done for small systems, but for larger systems this isnot an easy task.The theory Kohn and Sham developed in 1965 simplifies both the exchange and correlation effect of electrons by

reducing their effect to a effective potential VKS that is in principle exact. In their seminal paper, they showed that todetermine the ground state energy it is sufficient to determine the ground state electron density ρ(r). By constructinga simple system of non-interacting electrons which yields the same density as the “real” fully interacting system, theground state energy can be obtained.The Kohn-Sham equations for two electrons are:

[−1

2∇2

1 −Z

r1+

∫

d3r21

|r1 − r2|ρ(r2) + Vxc(ρ)]ψi(r1) = ǫψi(r1) (23)

The equations are self-consistent in the density and differ from the Hartree-Fock equations only in the exchange-

correlation potential Vxc[ρ] which is a function of the total local density: ρ(r) =∑N

i |ψi(r)|2

Vxc[ρ](r) =δ

δρ(r)Exc[ρ] (24)

While the exact form of the exchange-correlation functional Exc[ρ] is not known, the simplest approximation is theLocal Density Approximation (LDA).

ELDAxc [ρ](r) = Ex[ρ](r) + Ec[ρ](r) (25)

Ex[ρ](r) = −(3

4)(3

π)1/3

∫

ρ(r)4/3d3r (26)

Ec[ρ](r) =

∫ −0.29ρ(r)

5.1 + rs(r)d3r (27)

The potential are

Vx[ρ](r) = −(3

π)1/3ρ(r)1/3 (28)

Vc[ρ](r) =−0.29

5.1 + rs(r)− 1

3

0.29rs(r)

(5.1 + rs(r))2(29)

Finally the ground state energy is obtained from:

EG =

Ne∑

i

ǫi −1

2

∫ ∫

d3r2d3r1ρ(r1)

1

|r1 − r2|ρ(r2) + Exc[ρ]−

∫

d3rVxc[ρ](r) (30)

The Kohn-Sham equations are in general simpler that HF equations, while including correlation effects beyond theHF approximation. Furthermore their computing time grows as a power of the number of the N-electrons, whereasthe exact solution of the N-electron Schrodinger equations is exponential in N [41].

D. Exact Formulation

In 1930 Breit showed that the Schrodinger equation for the two-electron atom becomes separable if one of thecoordinate axes is aligned with one of the radius vector [45]. For the s state, the exact wavefunction depending on six

8

FIG. 2: example of an adaptive mesh used in approximating the solution in 3 dimensions, the number of elements is Nelements =15× 15× 3 = 675

.

total variables Ψ(~r1, ~r2) can be reduced to only three variables: the positions of the two electrons r1 and r2, and theangle between them θ12 [46], viz.

∫ ∫ ∫1

2

∂Ψ∗

∂r1

∂Ψ

∂r1+

1

2

∂Ψ∗

∂r2

∂Ψ

∂r2+

1

2(1

r21+

1

r22)

∂Ψ

∂ cos θ12

∂Ψ

∂ cos θ12+Ψ∗(−Z

r1− Z

r1+

1

r12− E)Ψr21r

22dr1dr2d cos θ12 = 0

where

r12 =√

r12 + r22 − r1r2 cos θ12

We approximate the solution to the above Hamiltonian using FEM by discretizing each variable independently.The following ranges were used: −1 ≤ cos θ12 ≤ 1, cutoff radius at r1,2 < 40 a.u., also the adaptive grid was refinedcloser to the nuclei, where each element is 1.3 times larger than the previous. We followed a formulation very closely

as in reference [46]. For the mesh shown in figure 2, we obtain a ground state energy of E(L)exact = −2.7577664 using

linear interpolation basis and E(H)exact = −2.8994022 using Hermite polynomial as interpolation basis.

V. RESULTS AND DISCUSSION

In order to find the critical parameter we approximate the energy gap in the ground state numerically as

∆E0(Z) ≈ E2e0 (Z)− E

(e)0 (Z),

where E(2e)0 is the binding energy obtained for two electrons and E

(e)0 is binding energy for a single electron.

In order to apply the theory of finite size scaling to the Helium Hamiltonian of the form,

H = Ho + VZ ; VZ = −Z( 1r1

+1

r2) (31)

the following general procedure is used

General Procedure:

(1) Compute the expectation value of 〈H〉(N) and potential 〈VZ〉(N) for various values of N.(2) Construct the gamma function Γα(Z,N,N

′) (Eqn. 13) for several values of N, N’ vs Z. At criticality, the gamma

9

0.98

1

1.02

1.04

1.06

1.08

1.1

1.028 1.03 1.032 1.034 1.036 1.038 1.04 1.042 1.044

Γ α

Z

HFΓ(99,100)

Γ(199,200)Γ(299,300)Γ(399,400)Γ(499,500)Γ(599,600)Γ(699,700)Γ(799,800)

0.9995

1

1.0005

1.001

1.0015

1.002

0.91 0.9105 0.911 0.9115 0.912 0.9125 0.913 0.9135 0.914

Γ α

Z

Total EnergyΓ(149,150)Γ(199,200)Γ(249,250)Γ(299,300)Γ(349,350)Γ(399,400)Γ(449,450)Γ(499,500)Γ(549,550)Γ(559,600)Γ(649,650)Γ(699,700)Γ(749,750)Γ(799,800)

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

Γ α

Z

LDAΓ(45,50)Γ(75,80)

Γ(95,100)Γ(195,200)Γ(295,300)Γ(395,400)Γ(495,500)

0.9

0.95

1

1.05

1.1

1.15

1.2

0.914 0.915 0.916 0.917 0.918 0.919 0.92

Γ α

Z

ExactΓ(192,243)Γ(243,300)Γ(300,363)Γ(363,432)Γ(432,507)Γ(507,588)Γ(588,675)Γ(675,768)Γ(768,867)

FIG. 3: Plot of Γα obtained by using FSS method, as a function of the charge Z for increasing number of elements (N), forHatree-Fock and Total Energy (HF + correlation energy) (top), and LDA approximation and exact formulation (bottom)

function Γα(Z,N,N′) becomes independent of N and N’ and the curves will cross. This is shown in figure 3, the

gamma function obtained by using different number of elements begins to cross for various approximations to theEnergy.(3) In order to go to the complete basis set limit 〈H〉∞, we have to let (N,N ′) → ∞. The best choice is to keep

N − N ′ minimal [20]. Then, we proceed as follows: we solve 〈H〉(N)for (N-1, N , N+1). This provide us with two

gamma curves: Γα(Z,N − 1, N) and Γα(Z,N,N + 1), with minimum N − N ′. The Crossing of these two curves

gives us the pseudo-critical parameters α(N) and Z(N)c with a truncated basis set number. Figure 4 illustrates the N

dependence of the crossing.

(4) In order to extrapolate the values α(N) and Z(N)c to the infinite N limit, we use the algorithm of Burlish and

Stoer. Figures 5 and 6 show their behavior by systematically increasing N for the mean-field approximations andexact formulation respectively.In our calculations we use linear interpolation functions, NL→R(x) = x and NL←R(x) = 1−x, , where the function

NL→R(x) interpolates the wavefunction from the left node to the right node and similarly for NL←R(x). For theexact 3D formulation we improved calculations using Hermite polynomials to interpolate the wavefunction usingNL→R(x) = 1− 3x2+2x3 and NL←R(x) = 3x2− 2x3, and additionally interpolate the derivatives of the wavefunctionwith NL→R(x) = hx(x − 2x2 + x3) and NL←R(x) = hx(x

3 − x2), where hx is the element width. The continuityof the wavefunction is thereby enforced by using Hermite polynomials, but the local overlap matrices double in sizeincreasing the memory cost as well as the computing time. To solve the generalized eigenvalue problem we have useda fortran subroutine for dense matrices EWEVGE [52] for 1D equations and a sparse matrix solver ARPACK++[53]

10

0.99

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

0.91 0.915 0.92 0.925 0.93Γ α

Z

Γ(99,100)Γ(100,101)Γ(199,200)Γ(200,201)Γ(499,500)Γ(500,501)

FIG. 4: The crossing of Γα(Z,N − 1, N) and Γα(Z,N,N + 1) as N is increased using Total Energy approximation in HF. Thecrossing has been slightly enhanced in order to demonstrate the dependence versus N.

0 0.002 0.004 0.006 0.008 0.01

1/N

1.032

1.034

1.036

1.038

1.04

1.04

2

Zc

HF

0 0.002 0.004 0.006 0.008 0.01

1/N0.95

0.96

0.97

0.98

0.99

1

Zc

LDA

0 0.002 0.004 0.0061/N

0.91

0.912

0.914

0.916

0.918

0.92

Zc

Total Energy

0 0.002 0.004 0.006 0.008 0.01

1/N

1

1.02

1.04

1.06

1.08

α

HF

0 0.002 0.004 0.006 0.008 0.011/N

1

1.01

1.02

1.03

1.04

α

LDA

0 0.002 0.004 0.006 0.0081/N

1

1.002

1.004

1.006

1.008

1.01

1.012

α

Total Energy

FIG. 5: Extrapolated values for the critical exponent α using HF approximation, LDA approximation, and Total energy. Thefilled dots are the extrapolated values at 1/N → 0 limit

for 3D equations.In extrapolating the values α(N) and Z(N) in the 1/N → 0 limit some numerical fluctuations can arise depending

on the approximation used. We found several ways to deal with these fluctuations. One way is to use more accurateinterpolation polynomial (ie Hermite instead of linear basis), another method is to construct a gamma function from apotential in the Hamiltonian that explicitly takes into account all variables, for example in the 3D case we found that

it is better to use the potential VZ = 1/Z√r21+r2

2−r1r2 cos θ12

rather than VZ = Zr1

+ Zr2, since the first takes into account

all three variables when the expectation value 〈VZ〉 is computed whereas the latter introduces numerical errors inthe expectation value. Lastly, another method is to increase the minimum difference |N −N ′| ≥ δ used in finding acrossing between Γα(Z,N − δ,N) and Γα(Z,N,N + δ) until a systematic crossing is obtained for all values of N; In

the LDA approximations we used |N −N ′| = 5, whereas in the 3D formulation we used |√

N/3−√

N ′/3| = 1 sincein this case the number of elements increases as Nelements = Nr1 ×Nr2 ×Ncos θ, and Ncos θ was kept fixed at 3It is well known that using mean field theory (for dimensionality ≤ 4) does not give correct values near a critical

point, this is indeed the case by using only the Hartree-Fock approximation (see table I ), but we see that incorporating

11

0.918

0.9185

0.919

0.9195

0.92

0.001 0.002 0.003 0.004 0.005 0.006

Zc

1/N

Exact

0.99

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

0.001 0.002 0.003 0.004 0.005 0.006

α

1/N

Exact

FIG. 6: Extrapolated values for the critical exponent Zc and α in the 1/N → 0 limit solving the system exactly in 3D. Weconstruct the Γα function using Hermite interpolation polynomials for two potentials: VZ = −Z( 1

r1+ 1

r2) (denoted by �) and

VZ = 1/Z√r21+r2

2−r1r2 cos θ12

(denoted by ©); the latter is obtained by making the transformation r → r/Z on the Hamiltonian.

We checked that the fluctuations seen on the figure vanish by replacing the numerical approximation to the binding energywith the analytical form, E(e) = Z2/2, while the extrapolated value remains unchanged.

TABLE I: Results for Critical Parameters

HF LDA Total Energy Exact Reference [5, 44]

critical charge

Zc 1.03114 0.95361 0.90946 0.91857 0.91103

critical exponent

α 0.99971 0.99960 1.00009 1.00082 1.00000

the correlation energy with LDA approximation improves the estimate of the critical charge within 5%, this is closeto the value obtained by Stillinger using a non-linear wavefunction. Both Total Energy approximation and exactformulation bring the estimate close within 1% of the accepted value. It is interesting to note that all approaches givea consistent estimate on the critical exponent, very close to the accepted value.This study demonstrates that the use of the Finite Element Method is a viable tool that can be combined with

Finite Size Scaling to obtain critical values for electronic calculations if the electron-electron interactions are describedaccurately. It is important to check our scaling ansatz by looking at an independent expression describing the finitesize scaling. Recall that near criticality we have the following form

EZ ∼ (Z − Zc)α ⇒ dMMZ

dZm∼ (Z − Zc)

α−m (32)

Therefore, we have a similar relation for the truncated expression. i.e.

E(N)Z ∼ N−α/ν ⇒ dMI

(N)Z

dZm∼ N−(α−m)/ν (33)

Since E(N)Z is a regular function, it has a Taylor expansion around |Z − Zc| ≤ 1.

12

-10 -5 0 5 10 15

(Z - Zc) N

1/ν

-6

-4

-2

0

2

4

E0 N

−α/ν

N = 100N = 200N = 300N = 400N = 500N = 600

HF

-5 0 5

(Z - Zc) N

1/ν

-2

-1

0

1

2

3

E0 N

α/ν

N = 100N = 300N = 500N = 600N = 700N = 800N =900

Total Energy

0 5 10 15

(Z -Zc) N

1/ν

-5

-4

-3

-2

-1

0

1

E0 N

α/ν

N=100N=200N=300N=400N=500N=600N=700

LDA

-20 -10 0 10

(Z - Zc) N

1/ν

-2

0

2

4

6

E0 N

−α/ν

N = 432N = 507N = 588N = 675N = 768N = 867

Exact

FIG. 7: Data collapse for the HF and Total Energy (top), and LDA approximation and exact formulation (bottom), for variousN. The value from this collapse study that give us the best correlation length exponent ν ≈ .85± .05

E(N)Z =

∑

m

dmE(N)Z

dZm· (Z − Zc)

m

m!(34)

=∑

m

N (−α+m)/ν (Z − Zc)m

m!(35)

= N−α/ν∑

m

[N1/ν(Z − Zc)]m

m!︸ ︷︷ ︸

(36)

= N−α/νG0(N1/ν(Z − Zc)) (37)

Here G0 is a universal scaling function. By plotting E(N)Nα/ν vs the argument of the function G0, collapses allquantities from different N into one curve.Figure 7 shows that all curves do indeed collapse into a single curve, this gives strong support for the results of

FSS with FEM. The correlation length exponent was varied until the best collapse curve was found, this occurredaround ν ≈ .85± .05 for all implementations.

13

The solution to a set of Hartree-Fock or Kohn-Sham equations for large number of atoms becomes the limitingstep, and it is here that the Finite Element Method can extend the range of sizes that can be investigated currentlyby other means. The efficiency of FEM in the context of large-scale calculations derives from the strict locality of theFEM basis in real space, thus minimizing the need for extensive interprocessor communications on massively parallelcomputational architectures[7].The work of Pask et al[7] has formulated a real-space DFT approach for condensed matter applications without the

need of using expensive Fourier transforms. Impressive results are obtained for GaAs calculations using this approachon DFT+FEM that systematically converges, as the number of elements is increased, to the values obtained usingthe conventional Plane Wave calculations, employing Fourier transforms and Ewald summations.The scope of the dimension attainable with FEM has also been demonstrated by Sterne et al [50]. They have

employed FEM to compute positron distributions and lifetimes for various defect structures. Because the distributionis non-electron-like (concentrating away from the nuclei rather than around them), the generality of the FEM basismakes it particularly well suited for such problems. In order to study defects they have modeled a 5488 atom cell;this is one of the largest such calculations reported by any method to date.Another applications of FEM to large-scale ab initio calculations is the work of Tsuchida et al on molecular

dynamics simulations of liquids [51]. They apply FEM study of liquid formamide (HCONH2) using a generalizedgradient approximation to exchange and correlation to improve modeling of hydrogen bonds. The use of adaptivecoordinate transformations helped to double the resolution of the basis at the C, N and O centers. The liquid wasmodeled by 100 molecules in a cubic supercell of side 35.5 a.u., for a total of 600 atoms. The scope of this simulationis on the order of the largest performed by standard PW methods to date and demonstrates the capacity of FEM forsuch large-scale computations already.A new promising approach developed recently by Mazziotti [54] is the large-scale semidefinite programming for

many-electron quantum mechanics. In this approach, the energy of a many-electron quantum system can be ap-proximated by a constrained optimization of the two-electron reduced density matrix (2-RDM) that is solvable inpolynomial time by semidefinite programming (SDP). The developed SDP method for computing strongly corre-lated 2-RDMs is 10-20 times faster than previous methods[55]. This approach was illustrated for metal-to-insulatortransition of H50, with about 107 variables.These examples along with our current work, suggest that we can combine the Finite Element Method with Finite

Size Scaling to obtain critical parameters in extended systems. One possibility, is to calculate the phase transition atzero temperature in the 2 dimensional electron gas. The ground state energy of the system favors a crystallize stateas the density of the electrons is lowered. Tanatar et al [47] performed a variational Monte Carlo simulations to thetotal energy using Hartree-Fock approach.

E0(rs) = EHF (rs) + Ec(rs) (38)

They find the gas-to-crystal transition at rs = 37 ± 5. This system would be ideal to test FEM with Finite SizeScaling for quantum criticality in extended systems.

VI. CONCLUSIONS

The Finite Element Method can be combined with Finite Size Scaling for electron structure in atoms, molecules,and extended systems. Despite it being in its developmental state, the FEM has already shown great promise forextended systems. Because its bases used are arbitrary polynomials, they can be systematically varied for optimalresolution. Furthermore, the local nature of the bases lends the method easily to parallel architectures.We have presented detailed calculations showing this combined approach for two-electron atoms using an exact

formulation and the mean-field methods: Hartree-Fock approximation with correlation terms included and density-functional Theory (DFT) using LDA approximation. We demonstrate that more accurate representation of potentialterms yields estimates closer to the accepted result. Application of this method to extended systems follows withoutloss of generality, where this combined approach can shed some light into the nature of phase transitions.

14

Acknowledgments

S.K. would like to acknowledge the financial support of ARO and B. W.-K. thankfully acknowledges support formthe NSF under the grant PHY-0969689.

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15

[52] Dirk Porezag <http://tfy.tkk.fi/ asf/physics/lectures/codes/mathsub.f>[53] Francisco M. Gomes <http://www.ime.unicamp.br/ chico/arpack++>[54] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011).[55] D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004).[56] We can demand a more rigorous approximation by extending the nodal space to include the derivative of the wave-function

at the node : |Ψ〉 = ∑iNi(r)|ψi〉+ ∂xNi(r)|∂xψi〉 therefore ensuing continuity of the wavefunction between the nodes.