A NONLINEAR EIGENSOLVER-BASED ALTERNATIVETO TRADITIONAL SCF METHODS
BRENDAN GAVIN AND ERIC POLIZZI UNIVERSITY OF MASSACHUSETTS, [email protected], [email protected]
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.