Chemistry Reference
In-Depth Information
Dynamics.calculations.typically.require.many.evaluations.of.the.potential.energy.
and.the.associated.nuclear.gradient..Studies.of.large.molecules.similarly.beneit.
from.improvements.in.the.eficiency.of.the.TB.method..However,.studies.of.large.
molecules. are. additionally. hampered. by. the. amount. of. memory. required. which.
grows.as.the.square.of.the.number.of.orbitals..There.are.a.number.of.strategies.
that. may. be. employed. to. attack. the. problems. listed. above.. Of. these,. we. con-
sider.three:.improvements.in.the.performance.of.the.linear.algebra.routines.in.the.
solution.of.the.generalized.eigenvalue.problem,.parallelization.and.sparse.matrix.
techniques.
8.6.1 S
electing
r
outineS
for
S
olving
the
e
igenvalue
P
roBlem
In.general,.the.eficiency.of.a.TB.implementation.depends.strongly.on.the.eficiency.
of.the.method.used.to.solve.the.eigenproblem.of.Equation.8.6..We.rewrite.this.equa-
tion.as
(
H ES C
−
)
= 0
.
,
(8.48)
.
where
H
.is.the.Hamiltonian.matrix
E
.is.the.energy
S
.is.the.overlap.matrix
C
.is.the.matrix.of.molecular.orbital.coeficients
The.matrices.
H
.and.
S
.are.symmetric,.which.makes.the.generalized.eigenvalue.prob-
lem. (Equation. 8.48). easier. to. solve.. We. note. that. if. the. overlap. matrix,.
S
,. is. the.
unit.matrix.(as.in.orthogonal.TB.methods).then.the.eigenvalue.problem.becomes.a.
.standard.eigenvalue.problem.that.is.easier.(i.e.,.faster).to.solve.
The. use. of. optimized. mathematical. libraries. is. well. established. in. quantum.
chemistry..One.of.the.most.highly.optimized.libraries.is.the.BLAS.[82].(basic.linear.
algebra. subprograms). library.. The. BLAS. library. contains. routines. for. vector-vec-
tor,. vector-matrix,. and. matrix-matrix. operations.. Modern. implementations. of. the.
BLAS.are.carefully.optimized.to.exploit.the.multi-level.memory.hierarchy.of.mod-
ern. CPUs.. In. doing. so,. optimized. BLAS. implementations. routinely. achieve. high.
performance.that.is.a.signiicant.fraction.of.the.peak.theoretical.performance.of.a.
particular. CPU.. The. LAPACK. [83]. (linear. algebra. package). libraries. build. upon.
the.functionality.offered.by.the.BLAS.to.deliver.a.wide.variety.of.useful.algorithms.
including. algorithms. for. the. solution. of. eigenvalue. problems. in. an. eficient. man-
ner..Note.that.there.are.a.number.of.other.libraries.offering.functionality.similar.to.
LAPACK.such.as.LINPACK.[84].and.FLAME.[85]..However,.we.restrict.our.atten-
tion.to.the.LAPACK.library.for.the.purposes.of.this.section.
The.LAPACK.library.offers.several.routines.for.the.solution.of.the.generalized.
eigenproblem:.DSYGV.(and.the.closely.related.DSYGVX).and.DSYGVD.(we.restrict.
our.attention.to.the.double.precision.versions.of.these.routines.denoted.by.a.leading.
“D”. in. the. name).. We. assume. that. all. eigenvalues. and. eigenvectors. are. required..
Search WWH ::
Custom Search