A NONLINEAR EIGENSOLVER-BASED ALTERNATIVETO TRADITIONAL SCF METHODS

BRENDAN GAVIN AND ERIC POLIZZI UNIVERSITY OF MASSACHUSETTS, AMHERSTbgavin@ecs.umass.edu, epolizzi@ecs.umass.edu

PROBLEMSOLVING THE KOHN-SHAM EQUATIONS

H[n]xi = λiSxi , n =∑i

|xi|2

Where H[n] is the typical Kohn-Sham Hamiltonianwith (for our work here) spinless LDA approxima-tion, xi, λi are eigenvector/eigenvalue pairs, n isthe electron density, and S is the mass matrix re-sulting from a finite elements discretization.

TRADITIONAL SCFThe Kohn-Sham equations are traditionally solvedusing self-consistent field methods that follow thegeneral procedure:

1. Find good initial guess n1 (e.g. with Thomas-Fermi model)

2. Get n′ = f(ni), where f(ni) is the procedureconsisting of:

(a) Forming the Kohn-Sham hamiltonianH[ni] by solving poisson’s equation, cal-culating the exchange-correlation opera-tor, etc.

(b) Solving the eigenvalue problemH[ni]xi = λiSxi

(c) Calculating the new density from the oc-cupied states xi

3. Form ni+1 from some function of n′ and ni. If‖ni+1 − f(ni+1)‖ < ε then stop; otherwiseGOTO step 2.

SIMPLE MIXING:

ni+1 = αni + (1− α)n′ , 0 ≤ α ≤ 1

The value of α is determined heuristically. Typicalα = 0.1. Convergence is usually very slow.

DIIS:

A list of m densities is first produced using someother procedure (e.g. simple mixing), and thensubsequent ni+1 are given by[2]:

ni+1 =m∑j=1

cjni−j + cjri−j , ri = ni − f(ni)

where cj are the coefficients that minimize‖∑cjrj‖ such that

∑cj = 1 . Convergence is

known to be super linear.

ADVANTAGESComputational efficiency: The primary computa-

tional costs of our method are mat-vec mul-tiplication and boundary condition assign-ment in solving Poisson’s equation, both ofwhich can easily be parallelized.

No Initial Guess: Our method converges on thecorrect result for arbitrary initial guess ofdensity and subspace

High Convergence rate: Our method convergessignificantly faster than DIIS, and the conver-gence rate remains high regardless of systemsize.

NONLINEAR EIGENSOLVEROur method is based on the FEAST algorithm[1],a method for solving linear eigenvalue problems.The nonlinear Kohn-Sham equations are solved inan approximate subspace Q, which is updated ateach iteration:

1. Get initial guess for X , the matrix whosecolumns are the eigenvectors of interest xi,and n, either or both of which can be com-pletely random

2. Get density matrix ρ via numerical complexcontour integration:

ρ =∮C(zS −H[n])−1dz, z ∈ C

where C is a contour around the part of thereal axis where the eigenvalues of the occu-pied states are expected to be found.

3. Form the subspace Q = ρSX

4. Solve the projected, reduced nonlinear eigen-vector problem

QTH[n]Qxq = λqQTSQxq

by iterating over the density n. This pro-cecure is computationally inexpensive com-pared to performing the full SCF iteration.

5. Calculate xi = Qxq, λi = λq , and thenew density. If

∑λi has converged, exit.

Otherwise, GOTO step 2.

RESULTSThe following is the result of testing done with LDA all-electron DFT using a third order finite elementsdiscretization. The total energy residual, the relative difference between the calculated total energy atsubsequent iterations, is used as the means of gauging convergence.

0 1 2 3 4 5 6 7 8 910

-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Tota

l E

ner

gy R

esid

ual

H2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Iteration

CO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Nonlinear SolverDIIS

C6H6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Iteration

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Tota

l E

ner

gy R

esid

ual

DIIS (Good guess)

Good Initial GuessInitial Guess n=0.0

C6H6Nonlinear Solver: Good Initial Guess Vs. Poor Initial guess

C60 (Buckminsterfullerine):

FUTURE WORK• Parallelize the solution of the subspace prob-

lem QTH[n]Qxq = λqQTSQxq . This will be-

come the primary bottleneck for much largersystems.

• Update the FEAST package to provide ablack-box interface for the algorithm de-scribed above. FEAST 2.0 currently providesblack-box fuctionality for steps 2 and 3 de-scribed in "Nonlinear Eigensolver.

http://www.ecs.umass.edu/∼polizzi/feast/

REFERENCES[1] E. Polizzi, "Density-matrix-based algorithm forsolving eigenvalue problems," Phys. Rev. B., vol 79115112, Mar. 2009.

[2] T. Rohwedder, R. Schneider, "An analysis for theDIIS acceleration method used in quantum chem-istry calculations," J. Math Chem.,Volume 49, Issue9, pp 1889-1914, Oct2011.