<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
Lebedev	B-Algorithm
quadrature	I-Algorithm
,	O
named	O
after	O
Vyacheslav	O
Ivanovich	O
Lebedev	O
,	O
is	O
an	O
approximation	O
to	O
the	O
surface	O
integral	O
of	O
a	O
function	O
over	O
a	O
three-dimensional	O
sphere	O
.	O
</s>
<s>
The	O
number	O
and	O
location	O
of	O
the	O
grid	O
points	O
together	O
with	O
a	O
corresponding	O
set	O
of	O
integration	O
weights	O
are	O
determined	O
by	O
enforcing	O
the	O
exact	O
integration	O
of	O
polynomials	O
(	O
or	O
equivalently	O
,	O
spherical	O
harmonics	O
)	O
up	O
to	O
a	O
given	O
order	O
,	O
leading	O
to	O
a	O
sequence	O
of	O
increasingly	O
dense	O
grids	O
analogous	O
to	O
the	O
one-dimensional	O
Gauss-Legendre	B-Algorithm
scheme	O
.	O
</s>
<s>
The	O
Lebedev	B-Algorithm
grid	I-Algorithm
is	O
often	O
employed	O
in	O
the	O
numerical	B-General_Concept
evaluation	I-General_Concept
of	O
volume	O
integrals	O
in	O
the	O
spherical	O
coordinate	O
system	O
,	O
where	O
it	O
is	O
combined	O
with	O
a	O
one-dimensional	O
integration	O
scheme	O
for	O
the	O
radial	O
coordinate	O
.	O
</s>
<s>
Despite	O
this	O
,	O
the	O
Lebedev	B-Algorithm
grid	I-Algorithm
still	O
outperforms	O
product	O
grids	O
.	O
</s>
<s>
The	O
Lebedev	B-Algorithm
grid	I-Algorithm
points	O
are	O
constructed	O
so	O
as	O
to	O
lie	O
on	O
the	O
surface	O
of	O
the	O
three-dimensional	O
unit	O
sphere	O
and	O
to	O
be	O
invariant	O
under	O
the	O
octahedral	O
rotation	O
group	O
with	O
inversion	O
.	O
</s>
<s>
Two	O
more	O
sets	O
of	O
points	O
,	O
corresponding	O
to	O
the	O
centers	O
and	O
vertices	O
of	O
the	O
octahedron	O
,	O
are	O
all	O
eight	O
uncorrelated	O
permutations	B-Algorithm
of	O
(	O
denoted	O
as	O
a3	O
)	O
,	O
and	O
all	O
twelve	O
permutations	B-Algorithm
of	O
(	O
denoted	O
as	O
a2	O
)	O
.	O
</s>
