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From YouTube: VASP Workshop at NERSC: Basics: DFT, plane waves, PAW method, electronic minimization, Part 1
Description
Presented by Martijn Marsman, University of Vienna
Published on December 18, 2016
Slides are available here http://www.nersc.gov/assets/Uploads/VASP-lecture-Basics.pdf
Presented at the 3-day VASP workshop at NERSC, November 9-11, 2016
A
Good
morning,
everyone,
my
name,
is
Martin
marksman
from
the
University
of
Vienna
and
I'm,
one
of
the
developers
of
the
vast
code.
It's
nice
to
see
so
many
people
here
so
nice
to
see
the
interest
and
first
off
I,
of
course,
as
mentioned
in
introduction,
I
mean
I
cannot
help
you
do
better.
Science,
obviously
right
that
is
up
to
you,
but
I
hope
that
I
will
be
able
to
to
clear
up,
maybe
things
that
are
unclear
about
a
program
or
help
you
work
with
with
our
program
more
efficiently.
A
So
that
is
something
that
I
that
I
would
like
to
achieve.
For
this.
It's
always
a
bit
difficult
because
I
know
well
the
general
level
of
expertise
that
people
bring
when
they
come
to
a
workshop.
Some
people
are
very
experienced
users.
Other
people
might
be
complete
beginners,
so
the
lectures
are
sort
of
structured
well.
A
I
would
like
you
to
ask
questions
so
so
that
would
make
the
whole
workshop
for
me
a
more
enjoyable
experience,
because
then
I
have
a
feeling
whether
I'm
conveying
the
information
that
you
that
you're,
actually
interested
in
and
and
I
hope
that
that,
in
that
sense,
I
will
get
some
feedback
from
your
part.
So
the
first
lecture
is
is
really
about
the
basics.
So
about
there's
the
functional
theory
about
the
way
we
solve
the
equations
that
are
involved
in
this
arms
out
yeah.
A
We'll
do
basics
and
we'll
have
the
tutorials
in
connection
to
this,
which,
for
many
of
you,
especially
if
you
have
been
working
with
the
program
with
VASP
already
for
quite
a
while
the
tutorials
of
today
might
not
be
so
interesting.
I
could
see
that,
then
you
can
obviously
start
on
the
material
that
that
I
envisioned
for
tomorrow
I
mean
there's
no,
no
pressure
on
you
to
to
run
these
examples
that
I've
prepared,
but
they
they
are
the
ones
that
are
most
suitable
for
people
that
are
sort
of
starting
off
in
this
area.
A
Same
thing
for
the
cell.
For
the
last
day,
we'll
talk
about
some
high
performance
computing
issues.
So
the
way
the
program
is
structured
way
stuff
is
parallelized
things
that
you
can
do
to
to
run
efficiently
on
on
HBC
hardware.
It's
there's
no
magic
in
that.
So
so
it's
not
something
that
necessarily
takes
a
very
long
time
to
convey.
A
A
A
Simple-Looking
equation
actually
and
for
all
its
simplicity,
it
is
thoroughly
unsolvable,
so
we
have
here
and
Hamiltonian
working
on
our
many-body
wave
function,
and
that
equates
the
energy
that
also
energy
of
the
system
times
this
wave
function,
so
a
kinetic
energy
part
or
local
potential,
and
here
the
source
of
all
our
trouble
and
that's
the
electron-electron
interaction.
So
why
is
this
unsolvable?
Well,
it's
unsolvable
on
on
many
levels,
but,
for
instance,
one
of
the
first
things
that
you
might
imagine
is
just
storing
this
many-body
wave
function.
A
That
essentially
depends
on
all
the
electronic
spatial
coordinates
if
I
would
have
a
wave
function
as
it
is
depicted
here
and
I
would
discretize
this
on
some
grid.
Let's
say
in
a
cubicle
simulation
box
with
10
by
10
by
10
points
I
had
then
storing
an
object
of
this
size
for
five
electron
already
amounts
to
10
petabytes
of
storage
spacing
and
complex
words,
so
that
is
completely
impossible
for
any,
but
for
anything,
but
the
smallest
systems
and
the
solution
that
we
take
at
the
root
that
has
been
taken
is
to
map
this.
A
This
many-body
object,
and
so
this
is
this
wave
function.
That
depends
on
all
the
spatial
coordinates
of
the
electrons
in
our
system
onto
a
one
electron
theory,
and
that
map
is
depicted
here
at
the
bottom.
And
so
basically,
we
would
like
to
cast
this
object
into
a
group
of
functions
that
depends
only
on
one
electronic
coordinates,
so
a
single
function
for
each
electron
and
that's
why?
That's?
Where
texts
name
the
one
electron
theory
and
density
functional
theory
is
one
of
these
theories.
A
A
This
particular
complicated
object
per
se,
this
many-body
wave
function.
It
is
essentially
to
know
the
total
energy
of
a
system.
It
is,
it
suffices
to
know
the
electronic
density,
which
is
a
much
nicer
object,
obviously,
because
it
depends
only
on
one
single
spatial
coordinate
so
and
there
proved
there.
Their
density.
Functional
theory
is
essentially
an
existence
proof,
so
they
they
have
shown
that
and
that
we
can
write
the
total
energy
as
a
function
of
the
density
and
it
then
it
consists
of
a
few
terms:
there's
a
kinetic
energy
term
that
derives
from
this
one.
A
One
electron
orbitals
there
as
a
Hartree
energy
and
so
electrostatic
energy
that
depends
on
the
density.
Obviously,
there
is
a
density
interacting
with
the
ions
that
is
an
energy
term,
the
ions
amongst
each
other,
and
everything
that
we
do
not
know
about
the
system.
We
lump
into
another
energy
contribution
called
exchange
correlation.
That
is
also
a
functional
of
the
density
and
I
have
shown
in
there
proof
that
this
exchange
correlation
functional,
actually
exists,
and
so,
which
is
very
nice
but
Texas
only
part
of
the
way.
A
Obviously,
because
existence
doesn't
mean
that
we
know
it
to
exist,
but
we
don't
know
what
it
is
exactly
right.
So
the
density
is
computed
from
the
one
electron
orbitals
in
a
very
straightforward
way-
and
here
you
see
this
so
much
more
tractable
than
then
storing
such
an
object
would
be
storing
these
objects
and
from
konchem
density,
functional
theory.
A
We
know
how
to
compute
this
one
electron
orbitals
there,
the
solution
to
what
is
what
looks
like
a
swirling,
an
equation,
but
isn't
it
is
a
contamination,
but
it
looks
very
similar
so
where
we
see
again
kinetic
energy
interaction
with
the
nuclei,
electrostatic,
electrostatics
and
well
corresponding
to
this
exchange.
Correlation
energy.
There's
an
exchange
correlation
potential
as
well
and
hohenberg,
confirm
density.
Functional
theory
tells
us
that
both
these
objects
exists.
B
A
A
So
essentially,
all
quantum
mechanics
have
been
packed
into
these
two
objects
that
are
that
we
know
to
exist,
but
we
don't
know
what
they
are
so
and
that
is
where
so
up
to
this
point.
Actually
everything
is
completely
exact
yeah.
So
it's
an
exact
map
of
the
of
the
many-body
theory
on
to
one
electron
theory.
So
approximation
start
at
this
point
now,
because
these
exchange
correlation
functions
well,
we
have
to
assume
something
for
them,
and
one
of
the
first
assumptions
made
was
to
model
these.
A
These
quantities
on
the
uniform,
electron
gas
physicists
really
like
this
uniform
electron
Gossett's,
is
one
of
their
favorite
toys
systems
and
for
this
uniform
electron
gas,
the
correlation
energy
and
enter
potential
have
been
calculated
by
means
of
Monte
Carlo
methods
for
a
wide
range
of
densities,
and
they
have
done
been
parameterised
to
you
to
yield
functionals
for
both
these
quantities
and
in
the
most
well.
One
of
the
first
and
most
common
approximations
in
the
local
density
approximation.
A
We
simply
assume
that
that
the
actual
density
in
our
system
locally
behaves
like
a
uniform
electron
gas,
but
there's
many
other.
So
of
course,
afterwards
many
many
other
functions
have
been
created
that
taking
into
account
many
other
information
as
well.
So
maybe
not
only
the
local
density
but
also
the
local
gradient
in
the
density
for
gradients,
general,
gradient,
approximations
or
the
second
derivative
of
the
density
with
respect
to
the
spatial
coordinates
in
Metta,
GGAs
or
fun.
Lavas
functionals
will
see
them
them
come
by
in
the
rest
of
the
talks.
A
Okay,
so
that's
the
map-21
election
theory
starting
from
many-body
theory
that
takes
us
a
long
way
right,
because
well
we
had
this
object
that
we
can't
store
well.
This
is
something
that
looks
much
more
tractable,
because
here
we
have
something
to
the
power
of
n,
and
here
we
have
simply
any
times
these
functions,
but
one
big
problem,
obviously,
is
that
n
is
for
any
for
any
realistic
piece
of
material
and
is
huge
yeah.
So
so
this
is
only
part
of
the
solution.
A
Obviously,
this
is
still
not
a
tractable
one,
so
income
for
us
translational,
invariance
and
periodic
boundary
conditions,
so
because
if
we,
if
we
use
translational
invariance
right
so
we
have
here,
we
have
a
block
of
material
with
the
unit
cell.
That
is
repeated
endlessly
in
along
all
crystallographic
directions,
and
we
know
that
our
that
our
our
wave
functions
have
to
be
I
have
to
obey
this
periodic
boundary
conditions
that
we
that
we
apply
well.
A
A
We
call
the
block
function
and
we
we
know
that
under
periodic
boundary
conditions
are
one
electron
functions,
consists
of
a
part
and
it
is
written
you
and
K
here
that
is
self
periodic,
and
then
there
is
a
complex
phase
vector
with
the
period
that
is,
that
is
larger
than
and
dimensions
of
our
unit
cell.
So
we
know
under
periodic
boundary
conditions
that
our
one
electron
functions
have
to
look
like
this,
and
this
is
a
strictly
cell
periodic
part.
A
We
label
these
solutions
now
by
two
two
new
quantities
where
one
is
called
the
Bloch
vector
that
this
is
this
complex
phase,
vector
Bloch,
vector
K
and
a
so-called
band
index,
and
this
Bandhan
indexes
is
of
the
order
of
the
number
of
electrons
per
unit
cell.
So
that
is
quite
a
limited
number
right
and
the
Bloch
vector
we
constrain
to
lie
within
the
first
Brillouin
zone
of
the
reciprocal
space.
Lattice
and
I'll
show
you
what
that
means
for.
A
For
a
particular
lattice,
so
this
is
a
an
FCC
lattice
and
real
space
lattice
and
connected
to
this
real
space.
Lattice,
there's
the
reciprocal
space
lattice
and
that
is
depicted
yourself.
An
FCC
lattice
in
real
space
translates
to
a
BCC
lattice
in
reciprocal
space,
and
these
are
reciprocal
space
lattice
vectors
that
are
shown
here.
They
you
can
compute
them
out
of
the
out
of
the
real
space
lattice
vector
by
means
of
these
relations
and
first
Brillouin
zone
is
actually
this
Vigna
Seitz
cell
and
the
first
weaker
side
cell
in
reciprocal
space.
A
Okay,
so
that
takes
us
a
quite
away,
but
still
not
completely
there.
So
one
index
is
already
runs
over
a
very
limited
range,
but
these
block
vectors
inside
of
these
first
Brillouin
zone.
There
is
essentially
still
an
endless
number
of
possible
block
vectors
that
we
could
take
right.
So
our
density,
then,
in
terms
of
these
functions,
our
density
involves
an
integration
of
this
first
Brillouin
zone
computing,
the
density
and
a
summation
of
event
index.
A
So
that
is
that
is
essentially
written
here,
and
the
thing
is
that,
although
there
there's
impairs
and
an
unlimited
number
of
block
factors
that
we
could
choose
in
the
in
the
first
Brillouin
zone,
there's
actually
no
real
need
to
do
it,
because
these
orbitals
at
add
block
vectors
that
are
close
together,
they're
very
similar.
So
the
thing
is
that
what
what
solves
our
problem,
then
in
the
end,
is
that
we
can
replace
this
integration
of
the
first
Brillouin
zone
by
coarse-grained
sampling
of
this
particular
volume.
A
And
so
we
replace
this
integration
by
a
weighted
sum
of
a
discrete
set
of
points.
And
then
we
have
sort
of
arrived
at
the
point
where
we
can
actually
do
something,
because,
starting
from
this
many-body
wave
function
and
we
for
a
huge
number
of
electrons,
we
have
now
reduced
our
computational
task
to
computing.
These
one
electron
functions
for
a
number
of
bands
that
is,
of
the
order
of
the
electrons
in
our
unit
cell
and
at
a
discrete
set
of
K
points
in
this
first
video.
A
A
This
is
well
here,
it's
sort
of
written
for
a
for
a
simple
cubic
mesh,
and
so
this
is
a
four
times
four
mesh
in
two
dimensions
here
inside
of
this
of
this
first
Brillouin
zone
for
cubic
lattice
and
we'll
do
one
more
trick,
and
that
is
that
will
we
do
not
need
to
compute
one
electron
functions
at
all
these
points.
Only
the
symmetry
in
equivalent
ones,
so
they're
symmetry
operations
of
the
lattice
that
relate
one
electron
orbitals
at
this
point,
for
instance,
to
on
electron
orbitals
at
that
point.
A
A
C
A
Yeah
yeah
exactly
that
is
the
wait.
So
if
one
one
of
the,
if
there's
symmetry
operations
that
relate
one
particular
K
point
to
four
others,
and
then
it
would
count
four
times.
Yes
exactly
and
one
one
important
part.
What
one
shouldn't
forget:
if
one
does
this,
so
if
one
reduces
the
set
of
one-electron
equations
that
one
solves
and
ends
up
with
this,
will
this
reduced
set
of
solutions
then
afterwards,
after
constructing
the
density?
That
thing
has
to
be
symmetrized
again,
so
you
have
to
so.
A
You
reduce
the
number
of
points
that
you
that
you
would
the
number
of
block
vectors
that
you
would
actually
use,
but
you
would
have
to
use
the
same
symmetry
operations
on
all
the
objects
that
you
then
afterwards
construct.
So
that
is
so
the
density,
if
you
construct
the
density
by
means
of
this
of
this
recipe,
but
now
for
a
limited
set
of
points
and
symmetry
reduced,
then
this
thing
here
has
to
be
symmetrized
same
thing
for
all
quantities
that
you
derive
out
out
of
this
function.
A
A
It
does
not
have
to
include
the
gamma
point,
but
there
are
situations
so
this
this
one
doesn't
include
the
gamma
point,
for
instance,
so
commonly
what
we
commonly
do
is
so
you
can.
You
can
create
a
regular,
even
lattice.
That
wouldn't
include
the
gamma
point.
You
might
optionally
then
shift
that
towards
comma.
That
is
one
thing
that
you
can
always
do
and
there's
situations
where
you
would
have
to
do
it
I
think
that's
if
in
the
next
slide.
A
A
So
you
extract
the
irreducible
points,
calculate
the
proper,
the
proper
weights
and
things
like
this,
and
then
we
come
to
your
questions,
so
you
can
essentially
have
two
classes
of
meshes
one
centered
on
gamma
and
the
other
ones
around
gamma
and
the
bottom
ones
the
ones
centered
around
gamma
they
can
for
certain
lettuces
break
the
symmetry,
and
that
is
a
that
is
well
shown
here
so
for
FCC
and
HCP
for
hexagonal
lattices
and
for
FCC
lattices.
This
can
occur.
A
So
if
the
gamma
point
is
not
included,
then
this
regular
lattice
that
I
apply
on
my
hexagonal
cell
does
not
have
the
symmetry
of
the
hexagonal
cell
so
and
if
I
then
apply
the
symmetry
operations,
and
that
is
actually
actually
shown
here
and
I
think
this
is
even
though
these
are
the
true
points.
Actually
so
I
have
here
my
my
regular
lattice.
That
does
not
include
the
gamma
point
now.
A
A
So
the
thing
is
that
well,
first
of
all,
the
bad
thing
that
occurs
is
that
I
now
end
up
with
a
set
of
block
vectors
that
is
larger
than
what
I
started
with,
which
was
definitely
not
the
point
of
applying
symmetry,
the
other,
and
that
is
even
worse.
The
other
thing
is
that
this
is
not
a
really
nicely
uniform
lattice,
and
that
is
even
worse,
because
these
kind
of
integrations
that
we
do
here
or
these
summations
they
converge
most
rapidly.
A
With
respect
to
the
number
of
K
points,
if
we
sample
the
rear
end
zone,
the
first
Brillouin
zone
uniformly-
and
that
is
well
in
this
case
no
longer
the
case,
so
not
only
do
I
end
up
with
a
huge
set
of
block
vectors,
but
with
respect
to
increasing
my
sampling
density,
my
result
doesn't
doesn't
converge
as
rapidly
so
which
is
truly
bad,
but
we
can
always
take
this
lattice.
So
this
this
is
all
about
three
by
two
lattice
or
or
six
by
four
I.
A
A
A
This
is
sort
of
a
rule
of
thumb,
so,
if
I
would
I
would
go
from
two
to
four
to
six,
for
instance,
sampling
density
or
from
1
to
3
to
5,
because
the
convergence
behavior
of
even
and
odd
lettuces
tends
to
differ
so,
but
it's
always
what
what
you
can
do
right.
So
it's
mostly
it's
not
it's
not
a
question
that
am
I
using
enough
okay
points
to
sample.
It's
more
of
the
question.
A
Can
I
use
enough
K
points
too
simple
if
I
have
a
large
system
right,
the
other
thing
and
that
will
come
to
a
later-
is
that
the
larger
your
system
gets.
Essentially,
the
volume
of
your
reciprocal
space
cell
starts
to
shrink
right.
Sorry,
that
is,
that
is
one
of
these
relationships.
That
I
showed
you
before.
So
as
your
as
your
simulation
box
grows,
they
need
to
use
many
k.
A
Points
becomes
less
because
for
an
infinite
cell,
the
the
size
of
my
reciprocal
space
cell
would
be
essentially
zero,
so
I
would
need
only
one
point,
the
gamma
point.
So
there's
always
this
balance
and
I'll
show
you
some
of
the
things
that
that
that
and
well
I'll
address
one
of
the
factors
that
that
arm,
that
is
relevant
to
choosing
a
sampling
density
later,
and
that
is
the
range
of
the
interactions
in
your
system.
But
essentially
you
cannot
predict
this
beforehand.
You
would
have
to
do
convergence
tests
anyway,
always
right.
Yes,.
A
It's
by
choice,
so
so
so
the
the
standard
recipe
that
we
take
to
determine
the
points
in
our
grid,
so
either
these
factors
are
even
or
they're
old.
If
they're,
if
all
of
them
are
even
then
zero
will
not
be
part
of
the
of
the
grid,
and
if,
if
one
of
the
directions
is
odd,
then
the
gamma
point
will
be
so
if
the
sampling
along
one
of
the
directions
is
an
odd
number
of
points,
then
the
gamma
point
will
automatically
be
included.
Yeah.
E
A
If
I
select
a
poor
mesh
fast
does
not
quit
actually
there's
even
warnings
that
tell
you
that
you
have
selected
a
mesh
that
that
is
not
commensurate
with
the
the
real
space
cell
that
you
have
chosen
and
it
will
still
continue.
It
will
even
say
a
warning.
This
is
very
bad
news,
but
but
most
of
the
time
the
results
are
still
usable.
So
no
it
doesn't.
It
can't
judge
it
can't
really
judge
whether
whether
the
result
will
be
good
or
bad
yeah.
E
A
That
is
so,
why
not
always
use
a
gamma
centered
grade
grid?
That
is
a
very
good
question
and
actually
I
always
use
gamma
centered
gay
grids.
Just
don't
think
about
this.
Just
having
said
that
there
are
situations
where
you're,
where
sampling
is
more
dictated
by
what
you
can
do
than
by
what
you
would
like
to
do
right.
So
it
might
be
that
you,
yes,
but
I
would
I
would
actually
advise
to
use
gamma
Center
trees
that
avoids
all
these
problems.
Yes,.
A
A
So
what
actually
happens
going
from
here
to
here,
so
essentially
what
what
what
destroys
the
so,
what
is
the
problem
here
is
that
rotations
by
sixty
sixty
degrees
are
not
are
not
symmetry
operations
of
this
regular,
even
lettuce.
So
what
happens?
Is
that
I?
Take
my
my
cell
or
I?
Take
this
grid
and
I
rotate
everything
by
sixty
degrees
around
the
origin
and
then
I
end
up
with
something
like
this.
A
A
Yes,
this
is
used
basing
well
your
in
your
in
your
file
in
your
K
points
file.
While
you
specified
the
sampling
of
the
reciprocal
space
there,
you
say:
okay,
I
want
four
points
along
this,
this
direction
in
reciprocal
space
and
six
along
another,
and
things
like
this
right,
then
it
was
set
up
discrete
accordingly
and
use
all
the
symmetry
operations.
It
has
found
to
try
to
reduce
the
number
of
points
in
your
in
your
grid,
but
it
might
end
up
by
by
finding
new
ones
right.
Yeah.
A
Okay,
so
why
not
use
guide
points
entered
meshes?
Yes,
that
is
a
very
good
question.
I
simply
always
use
them.
Gamma
centered
meshes
yeah,
okay.
Well,
this
is
just
a
slide
to
do
sort
of
show
you
that,
although
we
use
periodic
boundary
conditions
for
vast
piece
of
code,
that
comes
out
of
solid
state
physics
or
they're,
using
periodic
boundary
conditions
is
an
obvious
choice
and
because
our
solid
state
material
consists
of
a
of
a
unit
cell
that
is
repeated
infinitely
along
order,
essentially
infinitely
along
all
directions.
A
That
I
do
not
interact
anymore,
and
then
we
assume
that
we
have
a
molecule,
a
free
molecule
in
space,
same
thing,
with
surfaces
for
surfaces
there
and
we
have
well.
This
is
the
surface
were
with
of
some
of
some
oxide,
the
surface
oxide
and
then
there's
vacuum,
and
that
is
the
bottom
of
this
slab
periodically
repeated.
A
So
this
has
to
be
large
enough
so
that
one
surface
doesn't
see
the
other
one,
and
another
thing
is
that
the
number
of
layers
in
our
slab
has
to
be
so
large
that
the
electronic
behavior
in
the
center
is
sort
of
bulk
like
otherwise
we
do
not
have
a
true
representation
of
a
semi-infinite
system.
Okay,
so
yes,.
E
E
A
A
software
pack,
I
wouldn't
know,
actually
maybe
somebody
knows
good
software
with
which
to
represent
meshes,
but
there's
only
a
very
limited
number
of
possible
Bhave
lattices
and
there
is
all
kinds
of
nice
PDFs
in
in
the
you
can
find
on
the
web.
That
show
you,
the
reciprocal
space
cells
related
to
to
all
the
possible
Bhave
lattices.
A
So
because
pictures
like
this
I
think
I
did
use
some
software
to
generate
this
picture
that
was
sort
of
prohibitively
complicated
to
to
to
work
with,
but
it's
possible
to
get
all
this
kind
of
pictures,
but
there's
only
a
few
of
them.
In
fact,
a
few
possible
buffet,
lattices,
yes
and
and
for
the
rest,
it's
I,
don't
know
how
much
is
understanding
and
how
much
is
getting
used
to
it.
At
some
point
you
get
used
to
you
see
a
picture
inside
this
is
an
FCC
cell.
A
You
immediately
know
the
shape
of
the
of
the
reciprocal
space
cell.
It's
not
the
most
important
thing.
Maybe
the
very
important
thing
to
realize
is
is
how
these
sizes
relate,
how
that
a
huge
cell
or
a
large
cell
in
real
space
corresponds
to
a
small
cell
in
reciprocal
space.
That's
one
of
the
things
that
is.
That
is
a
that.
It's
good
to
know,
because
that
directly
translates
to
the
number
of
of
K
points.
One
would
use
yes.
A
Actually,
yes,
that's
a
very
good
question,
so,
if
you
use
so
if
you,
if
you
use,
if
you
won't
want
to
do
something
like
this
so
a
molecule
in
free
space,
you
do
not
want
to
spend
effort
describing
the
interaction
between
neighbors.
So
you
would
use
only
the
gamma
point.
Never
anything
else,
because
also
using
a
different
K
point
or
adding
other
ones
only
means
that
you
that
you
are
trying
to
to
get
a
better
description
of
the
interaction
between
the
neighbors.
A
So
actually
your
worsening
your
result,
if
you
use
anything
but
the
K
point
same
thing
here,
so
there's
a
for
instance.
This
direction
right
so
where
we
have
a
vacuum
so
they're
associated
with
this
direction,
there's
also
a
reciprocal
space
vector
and
along
that
particular
direction.
You
would
never
use
more
than
one
k
point,
because
otherwise
you're
just
spending
computational
effort
in
trying
to
describe
the
interaction
between
the
slabs,
the
things
that
you
exactly
want
to
throw
away.
Ng
yeah,
that's
a
good
question.
A
So,
for
a
very
large
cell,
which
you
do,
you
would
like
to
switch
it
off
for
a
very
large
cell.
There's
no
real
need
to
switch
it
off,
although
if
you
have
a
very
large
cell-
and
you
have
only
1k
pointers-
also
no
real
need
to
use
symmetry
because
it
doesn't
bring
you
a
lot,
but
it
might
help
in
some
cases
to
still
even
in
that,
in
that
particular
situation
to
use
the
symmetry,
because
it
will
reduce
noise
in
some
quantities.
A
That's
one
thing,
but
what
there's
one
other
thing,
and
that
is
maybe
very
important
to
realize
that
that
using
symmetry
obviously
also
dictates.
A
Well
it
it
dictates
the
symmetry
that
of
the
solution
you
will
find
so,
for
instance,
consider
you
have
a
cubic
system,
maybe
a
large
surface
and
on
the
surface
you
would
like
to
add
well
in
real
life
on
the
surface.
Something
like
a
polar
bond
exists,
so
two
atoms
moved
to
each
other
and
electron
would
would
localize
in
between
them
on
the
surface
or
on
some
defect,
so
that
that,
actually,
that
physical
situation
might
might
break
the
symmetry
of
the
of
the
of
the
lattice.
A
So
yes,
where
are
where
is
this
important?
This
is
important,
for
instance,
reconstructions
on
surfaces.
They
might
break
the
symmetry
completely.
Where
I,
would
this
conceivably
be
very
important?
Magnetic
systems
magnetic
systems
might
have
them.
The
magnetic
subsystem
might
have
a
lower
symmetry
than
the
underlying
crystalline
system.
A
Don't
think
there
is
a
difference
there
shouldn't
be
so
Mon
crossbuck
grid
or
a
gamma
centered
one
I
I,
don't
actually
know
what
the
auto
great
would
be.
Maybe
it's
something
we
can
I
would
have
to.
I
would
have
to
look
it
up.
So
all
automatic
grids
are
essentially
Mon
crossbar
crates,
so
they
might
additionally
be
gamma
centered.
If
they're
all
there
are,
they
are
gamma.
Centered
might
be
that
there
is
another
option
that
is
called
Auto,
but
that
does
probably
do
exactly
the
same
thing.
But
I
could
come
back
to
this.
A
So
maybe,
if
you
could
advise
people
to
if
they
have
these
kind
of
questions,
they
could
send
it
to
me
by
email.
That
would
be
probably
be
better
so
that
we
could
do
it
offline
yeah,
because
here
we
can
obviously
talk
in
the
breaks,
but
for
offline
people
that
would
be
difficult.
Okay,
so
I
already
see
that
I
probably
need
five
days
for
this
workshop.
A
This
is
the
same
picture
as
before,
just
less
nice,
okay,
so
well,
let's
move
on
yeah.
We
can,
if
there,
if
there's
any
more
questions
with
respect
to
sampling
of
reciprocal
space.
We
please
we'll
simply
talk
about
this
later
or
during
the
hands-on
or
whatever
yeah,
so
that
this
is
the
total
energy
expression
that
that
we
talked
about
right
so,
depending
essentially
on
the
density.
A
Kinetic
energy
is
written
here,
it's
a
straightforwardly,
computable
from
from
our
column
or
bottles.
Our
tree
energy
is
simply
electrostatic,
so
we
have
two
two
charge
distributions
interacting
with
the
R
minus
R,
with
one
over
R,
where
R
s
and
R
minus
R,
prime,
the
distance
between
two
points
in
space
and
well.
This
is
the
electronic,
the
interaction
of
the
electrons
with
the
world.
A
Sorry,
this
is
the
density
consisting
of
the
electronic
density
and
the
nuclei
and
the
density
we
compute
from
our
conscience
or
bottles,
basically
repeat
of
something
that
we
have
seen
before,
and
this
is
the
central
essential
thing
that
we
have
to
solve.
Those
are
these
Cohen
charm
equations
right
and
they
are,
as
you
see
here,
there
are
eigen
value
equations
and
they
depend
obviously
on
their
own
solution
right,
because
our
Hamiltonian
is
basically
determined
by
the
electronic
density
that
is
determined
by
the
solution
of
the
equation.
So
this
is
a
self-consistent
problem.
A
Right
first
thing
that
that
we
do
is
what
we
have
to
make
a
choice:
how
to
express
the
solutions
of
our
equations
and
in
solid
state
physics.
Well,
it
was
sort
of
a
an
obvious
choice
at
some
point
to
use
plane
waves
to
do
this,
so
we
have
to
express
our
solutions
and,
in
essence,
the
cell
periodic
part
of
these
block
functions.
We
have
to
express
in
some
basis
set
to
be
due
to
be
able
to
work
with
it
and
what
we
do
is
essentially
Fourier
analysis,
so
we
expand
all
our
functions.
A
A
So
this
would
be
the
complete
block
function
with
the
additional
block
vector
in
the
in
the
exponent
and
all
other
cell
periodic
quantities
are
expanded
in
plane
waves
as
well,
so
our
density
and
the
potentials
and
we
have
to
make
a
choice-
and
this
is
one
of
the
essential,
essential
parameters
that
we
have
to
set
in
our
calculation.
So
because
this,
of
course
the
set
of
plane
waves
is
again
essentially
infinite
and
we
could
choose
any
any
where.
A
If
you
look
at
one
of
one
of
these
simulation
boxes,
we
could
use
any
sine
or
cosine
that
fits
in
in
this
unit
cell.
So
there's
an
infinite
amount
of
them
now
and
we
decide
on
which
to
include
in
our
basis
based
on
this
criterion,
and
this
is
the
kinetic
energy
associated
with
a
particular
plane
wave
and
what
we
say.
A
Okay,
we
limit
our
basis
set
to
those
plane
waves
with
a
kind
kinetic
energy
below
a
certain
cutoff
energy
that
you
have
to
choose
yes
and
there's
defaults
in
the
program,
because
it
depends
on
on
on
the
kind
of
potentials
that
that
you
use
there's
information
in
those
in
those
files
that
gives
give
some
hint.
What
kind
of
cutoff
would
be
sensible,
but
essentially
this
is
something
that
the
user
that
I
advise
people
to
set.
A
It
means
that
computationally,
it
scales,
favorably
and
so
fast
Fourier,
transform
scales
like
n
times
the
logarithm
of
n.
Where
n
is
the
number
of
plane
waves
in
your
in
your
in
your
unit
cell
okay,
so
that
is
sort
of
recaptured
here.
So
why
did
people
start
out
with
plane
waves?
Well,
it
was
historical
reasons
and
it
we
don't
have
to
go
in
to
now.
A
Practical
reasons
is
that
many
of
the
expressions
that
we
saw
before
are
easy
to
implement
if
we
are
able
in
terms
of
plane,
waves
and
computational
reason
is
that
we
can
use
these
FFTs
to
efficiently
do
a
lot
of
the
operations
that
we
are,
that
we
have
to
do
solving
the
column
equations
of
which
the
computing,
the
action
of
the
Hamiltonian
on
the
orbit.
Osis
is
the
key
one
and
the
action
I
don't
know.
I
will
probably
use
this
this
term
quite
a
lot
so
computing.
A
A
A
We
discretize
our
real
space
in
this
case
place
the
cubic
lattice,
and
we
have
a
number
of
points
here
and
it's
the
N
by
n
grid
in
in
this
in
this,
in
our
real
space
simulation
box
and
with
this
associated
is
a
as
an
N
by
n
grid
in
reciprocal
space
as
well
and,
for
instance,
one
of
these,
this
design
function
along
a
certain
direction
in
our
real
space
cell,
corresponds
to
a
certain
point
in
our
reciprocal
space
cell.
Reciprocal
space
grid
connected
by
fast
Fourier
transforms-
and
this
is
this-
this
cut
off
energy.
A
A
Okay.
So
that
means
that
what
this
one
was
sell,
periodic
part
of
the
part
of
the
basis
another
one
that
is
also
so
must
be
so
periodic,
always
another
plane,
waves
and
still
part
of
our
basis
and
at
some
point,
obviously
we'll
get
higher
frequency
components
that
are
that
lie
outside
of
of
the
area
that
we
consider
part
of
our
basis.
D
C
D
A
Actually,
this
would
be
well.
This
could
be
the
density.
Yes,
yes,
although
negative
values
would
be
bad
in
the
density.
So
sorry,
so
let's
say,
let's
say
that
this
is
an
orbital
yeah
so
and
both
of
these
directions
are
actually
real
space
directions.
So
in
that
sense,
this
this
sign
is
badly
chosen
right.
So
this
this
is
just
sort
of
a
hand,
waving
explanation.
So
there's
you
shouldn't
you
shouldn't
associate
anything
with
this
particular
direction.
Yeah,
yes,
yeah
I
should
do
more
fancy
graphics
to
do
this.
A
A
A
We
would
multiply
this
function
essentially
by
its
complex
conjugate,
which
means
that
that
if
well,
we
have
to
dislike
this
sort
of
Gaussian
shape
here
times
itself
we'll
end
up
with
another
Gaussian,
but
that
will
be
more
strongly
peaked,
yeah
so
and
ending
in
Fourier
analysis.
That
means
that
the
frequency
components
had
to
describe
this
one.
Truly,
you
need
higher
frequency
components.
A
So
essentially,
if
I
have
here,
at
the
boundary
a
certain
a
certain
sine
function
or
an
exponential
function
and
with
a
certain
that
is
a
certain
frequency
contribution
to
this
one,
then
this
one
will
have
contributions
with
with
reciprocal
space,
vector
twice
the
size
yeah
so
consider
the
size
of
this
particular
particular
basis.
How
large
does
my
does
my
grid?
Have
to
be
in
reciprocal
space
well
to
represent
the
density,
which
is
a
product
of
these
one.
A
Electron
orbitals
I
need
twice
as
many
points
along
a
certain
direction
right,
so
that's
that's
one
one
thing
and
let's,
if
I
wouldn't
do
that,
then
I
would
end
up
with
well.
I
would
have
contributions
that
are
outside
of
my
grid
and
in
signal
theory
that
means
aliasing.
It
means
that
you
have
contributions
with
a
certain
width,
a
higher
frequency
that
are
picked
up
at
a
lower
one,
so
it
would
start
to
contact
if
I
have
points.
A
So,
if
my
grid
is
not
large
enough
to
include
this
particular
circle,
then
I
will
have
points
at
higher
frequency
that
will
contaminate
my
fully
egg
components
at
lower
frequency,
and
that's
one
of
the
things
that
you
will
always
see.
If
the
program
the
program
will
allow
for
a
certain
amount
of
aliasing
in
standard
position,
it
will
allow
for
for
for
a
certain
amount
of
aliasing
and
will
write
out
a
warning.
A
But
if
you,
if
you
want
to
do
except
in
a
very
accurate
calculation
and
you
set
the
precision
to
accurate,
you
will
essentially
end
up
in
this
situation.
So
you
have
a
basis,
consists
of
a
bunch
of
of
plane
waves
to
represent
the
orbitals,
and
the
grid
will
be
large
enough
to
essentially
incorporate
a
reciprocal
space,
vector
of
twice
the
size
of
the
largest
one
that
is
used
to
represent
your
orbitals.
A
A
Let's
more
strongly
peaked
and,
of
course,
more
localized,
it
looks
only
more
localized
in
this
sense
of
the
example
that
of
Charleston
here
right,
because
you
have
completely
delocalized
electronic
states
as
well
that
that
my
and
if
you
don't
look
at
the
charge
density
that
arise
from
a
completely
delocalized
one
electron
orbital
the
density
that
is
associated
with
it.
This
is
also
delocalized,
but
it
will
have
features
with
higher
frequency
components.
That's
that's
the
way
to
look
at
it.
So.
A
C
E
A
Good
question
so
so
the
cutoff
determines
this
area
right,
so
I
say
that
I
want
to
have
I
want
to
have
basis
vectors
with.
So
yes,
let's,
let's
go
to
this
oops.
Oh
sorry,
so
the
distance
between
these
points
here
is
essentially
determined
by
the
size
of
of
my
unit
cell,
and
so
that
is
that
is
written
here.
You
see
here
b2,
that
is
the
reciprocal
space
cell.
So
it's
one
of
the
factors,
one
of
the
the
basis
vectors
of
my
reciprocal
space
cell
yeah.
A
So
if
I
say:
okay,
I'll,
let's
start
in
real
space,
I
said:
I
want
to
use
a
10
by
10
grid
in
in
real
space.
Let's
say
start
with
this,
then
in
a
reciprocal
space,
I
will
have
will
have
vectors
that
are
that
are
along.
This
direction
are
10
times
the
size
of
the
reciprocal
space
basis,
vector
yeah
and
my
my
salt
2
for
myself
periodic
part
that
is
G
vectors
that
express
the
fully
a
the
fully
a
components
of
my
self
periodic
part.
A
There
are
always
integer
multiples
of
my
reciprocal
space,
vectors
yeah,
so
I
choose
10
points
in
real
space.
I
end
up
with
with
10
points
in
reciprocal
space,
the
distance
between
them
is
well.
This
is
1
times
G,
2
times,
G
and
so
on
and
so
forth.
So
that
means
that
that
starting
from
zero
to
some
point,
there
is
going
to
be
a
maximal,
a
maximum
size,
G
vector
that
I
can
incorporate
in
this
lattice.
A
Yes,
so
the
maximum
size
with
this
maximum
size,
there's
a
kinetic
energy
criterion
connected
yeah,
so
if
I
have,
if
I
can
at
max,
have
a
G
vector
of
this
size,
that
means
that
I
can
detect
that
I
can
represent
my
wave
function
with
with
respect
to
this
kinetic
energy
criterion
with
respect
to
a
certain,
let's
say:
100
evey.
If
I
choose
more
points
in
in
real
space,
then
this
will
incorporate
larger
G
vectors,
which
means
that
I
can
now
represent
my
wave
function
with
the
cutoff
energy
of
let's
say,
200
evey.
A
So,
by
setting
this
particular
kinetic
energy
katrien
going
the
other
way
right,
I
sort
of
start
to
dictate
how
many
points
I
need
to
sample
in
real
space
and
how
many
points
do
I
need
to
sample,
and
that
is
given
by
the
fact
that
I
not
only
have
to
be
able
to
represent
my
one
electron
orbital,
but
the
associated
density
yeah.
So
it's
it's
all
connected,
but
these
these
distances
here
between
the
points
in
reciprocal
space
they're
given
by
the
size
of
my
reciprocal
space
basis,
vectors
so
asking
about
K
points.
A
C
D
A
It
has
to
do
with
so
so
essentially.
Well,
the
kinetic
energy
criterion
you
have
to
choose,
but-
and
you
have
to
sort
of
choose
by
hand,
but
but
obviously
it
will
be
associated
with
so
essentially
you
control.
How
many
fine
features
in
my
in
my
orbital
can
I
describe.
So
if
I
have
a
very
s
like
delocalized,
orbital
I
wouldn't
need
so
many
plane
waves
to
describe
it.
If
I
go
to
a
system
where
I
have
D
electrons
that
are
strongly
localized,
I
would
need
many
more
Fourier
components.
D
A
A
Otherwise
we
would
be
dead
completely,
but
essentially
a
D
electronic
orbital
doesn't
have
this
right,
so
the
3d
orbital
doesn't
have
nodal
features
per
se,
but
it's
strongly
peaked
closer
to
the
nucleus,
so
it
would
have
a
lot
of
would
have
a
lot
more
Fourier
components
than
a
1s
orbital
that
is
spread
out
much
more
strongly.
Yeah,
yes,
so
essentially,
the
kind
of
cutoff
you
have
to
choose
depends
on
the
physics
and
all
the
grids
are
then
associate
are
done
derived
from
this
particular
particular
cutoff.
A
A
Is
it
acceptable
that
will
depend
on
the
thing
that
you
are
doing
mostly
obviously
on
the
kind
of
energy
differences
that
you
are
trying
to
resolve
if
you're,
if
you're,
comparing
two
situations
and
there's
like
a
few
million
view
of
energy
difference
between
them,
then
probably
you
wouldn't
accept
any
alias
yeah,
so,
okay,
good!
So
yes,
so
the
action
of
the
Hamiltonian,
like
I,
said
before
this
point-by-point
product
of
this
object
with
the
one
electron
orbitals.
A
Yet
to
compute
this-
and
it
is
nice
to
do
some
stuff
in
reciprocal
space
and
some
stuff
in
real
space,
as
I
mentioned
before.
So,
for
instance,
the
kinetic
energy
is
easily
evaluated
in
reciprocal
space,
so
this
would
be
this
would.
This
is
exactly
this
is
the
the
representation
of
our
orbital
in
in
reciprocal
space.
Those
are
essentially
these
coefficients
right.
These
Fourier
coefficients
that
we
talked
about
and
the
kinetic
energy
of
such
of
such
an
orbital
in
that
particular
representation,
is
a
simple
product
of
the
kinetic
energy
of
the
plane
wave.
A
It
consists
of
with
these
Fourier
components,
so
that
is
easily
calculated
with
a
with
a
computational
effort
or
that
scales
with
the
number
of
plane
waves.
Something
like
the
local
potential
is
more
easily
calculated
in
real
space
right.
So
this
one
point-by-point
product
in
real
space,
which
is
a
salt,
so
we
need
to
be
able
to
shuttle
back
and
forth.
We
need
to
be
able
to
represent
our
orbital
in
reciprocal
and
real
space,
kinetic
energy.
We
do
in
reciprocal
space.
Local
potential
will
do
in
real
space.
A
On
the
other
hand,
solving
the
Poisson
equation
for
the
Hartree
contribution
to
the
to
the
local
potential
is
again
better
done
in
in
reciprocal
space.
That's
mentioned
here,
so
exchange
correlation.
We
can
do
in
real
space,
but
for
the
Hartree
potential,
where
we
need
to
know
the
potential
arising
from
the
electrostatic,
the
electrostatic
potential
arising
from
our
from
our
density,
and
for
that
we
it's
more
more
easily
done
in
reciprocal
space.
We
transform
our
density
to
reciprocal
space
and
the
potential.
A
So
the
solution
of
the
Poisson
equation
is
then
the
simple
product
of
our
foodie
I
components
of
the
density
with
1
over
G
squared
solving
the
Poisson
equation
in
real
space
is
far
from
trivial,
but
do
it
doing
it
in
reciprocal
space
is
actually
trivial.
So
we
get
our
potential
in
reciprocal
space
and
then
we
do
another
fully
a
transformation
to
get
it
into
real
space,
and
then
we
have
our
total
local
potential
in
real
space
and
do
a
point-by-point
multiplication.
A
The
cost
of
that
is
to
actually
scales
as
the
cost
of
doing
these
bouvier
transforms,
and
that
is
n
log
n.
Where
n
is
the
number
of
plane
waves
in
our
FFT
grid,
so
not
in
the
basis
but
in
the
FFT
grid
right?
So
the
number
of
plane
waves
in
our
basis
is
much
smaller
than
the
one
in
our
FFT
grid.
Yeah,
ok,
so
the
worst
scaling
part
of
such
an
operation
is
actually
n
log
N
and
that
that
is
due
to
the
fact
that
we
are
able
to
use
this
fast.
Fourier
transforms.
A
So
we
might
have
another
look
at
this
situation
that
we
saw
before
and
saw
the
action
of
our
Hamiltonian
and
disguise
the
action
of
the
local
potential
on
our
orbitals.
We
can
have
a
quick
look
at
what
that
means
in
terms
of
of
cut-offs
in
in
reciprocal
space.
So,
as
we
said
before,
we
have
this
large
sphere
that
just
about
fits
in
inside
of
our
FFT
grid
in
reciprocal
space,
that
that
represents
the
basis
for
our
density.
A
So
from
this
we
get
our
potential
by
dividing
by
G
squared
at
all
the
points
where
we
have
data,
and
that
gives
us
our
local
potential
so
and
essentially
what
we
then
do.
We
take
it
to
real
space.
So
we
have
some
object,
and
this
is
now
our
local
potential
in
real
space
and
we'll
multiply
it
again
by
our
orbital.
So
again
we
have
a
multiplication
which,
which
would
mean
obviously
that
if,
if
we
had
a
basis
for
our
orbitals
of
of
radius
G
cut,
then
this
would
be
2
times
G
cut.
A
If
we
then
multiply
the
potential
with
an
orbital,
we
would
end
up
with
fully
egg
components
up
to
3
times.
G
cut
actually
right
so
I,
let
this
what
is
what
is
written
here,
yeah
and
and
this
this
picture
tries
to
show
you
why
it's
not
necessary
to
use
an
FFT
grid
that
incorporates
this
whole
huge
circle.
So
essentially,
what
we
accept
here
now
is
aliasing.
We
don't,
we
don't
increase
our
grid.
So
all
these
all
these
areas
that
are
outside
of
our
outside
of
our
grid
fold
back
into
the
grid.
A
A
So
when
we
compute
this
product-
and
we
have
all
kinds
of
fully
air
components
that
are
strongly
contaminated
by
by
aliasing,
the
part
of
our
representation
that
is
unaffected
by
aliasing
is
exactly
the
part
that
that
makes
up
the
basis
of
our
orbitals
and
H
sy
equals
pi.
That's
essentially
the
kind
of
equation
that
we
solve.
So
these
are
the
only
components
that
will
keep
after
computing
the
action
of
the
Hamiltonian
on
the
orbital.
A
H
A
So
we
go
from
2g
to
3G
yeah,
and
that
is
the
radius
of
this
large
circle
yeah
and
this
and
this
sort
of
trice
you
very
handwriting,
Lee,
tries
to
tries
to
show
you
that
we
do
not
need
to
choose
our
fft
grids
so
large
that
we
that
we
actually
incorporate
is
this
three
times
G
cut.
It
is
sufficient
to
incorporate
two
times
two
times:
G
cut
yeah.
A
A
The
number
of
plane
waves
to
describe
tightly
bound
states
so
strongly
localized
state
or
the
rapid
oscillations
of
the
wave
function
near
the
nucleus
would
still
be
prohibitively
large
right.
So
we
have
this.
We
have
this
way
of
representing
our
are
our
orbitals,
but
in
practice
anything
beyond
lithium
or
hydrogen.
A
We
couldn't
represent
two
orbitals
in
any
tractable
number
of
plane
wave
size
as
a
because,
if
you,
if
you
have
I,
don't
know
you
go
from
2
s,
2
3
s,
you
get
one
node
close
to
the
nucleus
had
to
keep
those
to
keep
those
states
orthogonal
to
each
other.
They
have
to
have
to
have
an
additional
node
and
describing
these
fine
features
takes
a
lot
of
plane
waves.
A
So
there's
two
things
that
that
are
commonly
done
there.
One
is
to
introduce
the
frozen
core
approximation,
and
that
is
because
there
is,
if
you
have
an
atom
with
quite
a
few
electrons,
there's
a
whole
bunch
of
them
that
are
that
don't
really
contribute
to
the
to
the
chemical
binding
right:
they
don't
they
don't
notice
whether
they
are
a
free
atom
or
they're
inside
some
material
they're
always
the
same.
A
So
they
don't
they're,
not
important
for
the
physics
or
the
not
important
for
the
for
the
chemistry
of
the
situation,
and
we
can
compute
them
one
time
and
then
compute
the
density
associated
with
them
and
keep
that
frozen.
So
that's
already
quite
good,
because
that
gets
rid
of
these
deep
lying
States
that
there
are
generally
quite
contracted
in
in
real
space,
so
they
would
need
a
lot
of
plane
waves
to
represent.
A
Yes,
so
pseudo
potential
methods
there's
a
whole
bunch
of
them.
So
starting
from
northern
conserving
pseudo
potentials,
one
step
beyond
ultra
soft
pseudo
potentials
and
I'll
I
will
show
you
something
about
projected
augmented,
projector,
augmented,
wave
method,
because
that
is
the
flavor
of
pseudo
potential
theory
that
that
we
actually
use
in
in
vasp.
A
So
the
general
idea
is
like
I
said
if
we
have
features
around
here,
so
we
have
an
orbital
with
a
few
notable
features
near
the
core.
Then
getting
these
rights
to
express
this
part
of
the
orbital
in
plane
waves
is
expensive
and
often
prohibitively
expensive.
The
thing
is,
though,
that
that
the
part
of
the
orbital
that
we
have
to
get
right
is
the
part
where,
if
I
put
two
orbital,
two
atoms
close
together,
their
orbitals
will
feel
each
other,
and
that
will
happen
at
some
distance
away
from
the
nucleus.
A
If
I,
if
I
describe
this
part
of
the
orbital
correctly,
let's
say
the
part
where
chemical
bonding
occurs
and
that
would
be
sufficient,
so
I
don't
need
to
to
represent
these
nodal
features
per
se.
So,
instead
of
using
the
exact
potential,
I'll
use
a
pseudo
potential
that,
at
some
cutoff
radius
resembles
the
true
potential.
Exactly
I'll
use
this
potential
instead
of
the
of
the
true
one
and
I'll
end
up
with
with
an
orbital.
That
is
much
more
well
behaved
if
I
choose
this
potential
wisely
I
end
up
with
a
poodle.
A
A
A
So
if
I
say,
okay,
I
have
an
aluminum
and
that
normally
well
the
states
that
are
considered
to
be
part
of
my
of
my
valence,
and
so
this
is
frozen
core
and
then
these
would
be
states
that
I
consider
part
of
my
of
my
valence
many
folds
or
3p
and
3's.
They
would
need
to
represent
using
this
to
the
potential
theory.
I
get
rid
of
the
nodes,
so
my
effective
aluminum
atom
is
of
the
states
that
I
represent
are
effectively
of
1
s
and
2
P
nature
that
is
sort
of
depicted
here.
A
They
are
no
less
but
they're
not
guaranteed
to
be
orthogonal
to
the
core
anymore
right.
So
I
check
this
out
and
using
this
effective,
total
potential
I
end
up
with
states
that
not
necessarily
orthogonal
to
these
core
States
anymore,
the
piw.
We
do
things
slightly
differently
and
implicitly,
we'll
use
in
the
formalism
implicitly
we'll
be
working
with
three
B
and
3s
States
or
not
with
nodelist
ones.
We
have
a
way
of
conserving
the
nodes
and
a
way
of
conserving
the
orthogonality
to
the
core
state.
A
So
in
that
sense
it
is
founded,
and
that
is
what
is
what
is
written
here.
The
P
R
W,
as
opposed
to
to
a
conventional
tool
of
potential
methods,
conserves
the
nodal
structure
in
a
certain
way
without
having
to
represent
them
in
plane
waves.
That
is,
that
is
the
essence
of
the
method.
So,
on
the
other
hand,
this
this
picture
does
actually
apply.
So
we
have.
A
We
have
a
part
around
our
around
our
ionic
sites,
where
we
do
stuff
to
the
wave
function,
and
then
we
have
an
interstitial
part
where
we
commonly
represent
the
wave
function
exactly
so
that
is
sort
of
depicted
here.
Sorry,
the
essence
of
the
PW
method
is
that
we
represent
a
true
orbital
in
two
different
bases,
and
so
we
have
a
part,
and
that
is
well.
This
is
the
site
tilde.
This
is
called
two
pseudo
part
of
our
of
our
orbital
of
our
baw
orbital.
A
The
pseudo
part
is
expressed
in
plane,
waves
and
then
locally
in
spheres
around
around
our
atomic
sites.
We
we
use
an
additional
basis,
and
that
is
this.
These
are
localized
functions
that
are
not
expressed
in
plane
waves.
They
are
expressed
on
radial
grids
on
radio,
logarithmic
grids
and
they're
there.
You
can
retain
the
nodal
structure,
because
these
radio
logarithmic
grids
they
can
represent
very
fine
features
without
without
too
much
computational
effort.
So
this
Pluto,
this
poodle
part
it
is
expressed
in
plane,
waves
that
doesn't
contain
nodal
features.
A
They
are
sort
of
being
brought
back
in
inside
spheres
around
the
atomic
sites
and
these
localized
functions
essentially.
How
much
of
these
localized
functions
we
add
mix
to
our
plane.
Wave
solution
is
determined
by
a
projection
of
our
plane,
wave
solution
onto
a
projection
operator.
So
these
functions
they
are
called
the
partial
waves.
These
are
the
only
electron
partial
waves
to
to
low
partial
waves
and
the
PW
projectors.
These
functions
you
can
recompute
beforehand,
so
they
are
stored
on
these
potential
files.
Yeah,
the
variational
quantities
are
for
which
we
will
actually
solve
the
equations.
A
Are
these
plane
wave
parts?
So
in
that
sense
we
augment
this
plane
wave
with
these
localized
functions
and
what
that
sort
of
ends
up
with
is
depicted
here.
So
this
is
our
Pluto
orbital
expressed
in
plain
waves
and
apart
from
zero
at
the
origin,
where
an
atom
sits,
it
doesn't
have
nodal
features.
Then
we
subtract
apart,
that
the
artist
despised
partial
waves
that
have
been
pre
computed
we
subtract
apart
and
assuming
that
that
this
set
of
functions
is
a
complete
basis.
B
A
A
Actually,
that
is
guaranteed
because,
because
of
the
relationship
between
these,
these
all
electron,
partial
waves
and
and
absurd
iced
ones,
so
at
all,
electric
partial
eyes
and
I'll
show
you
examples
of
these:
their
solutions
to
the
radial
atomic
problem
and
the
two
dyes
once
they
match
on
to
the
at
a
certain
cutoff
they
match
continuously
on
to
the
old
electron
ones.
So
this
particular
this
this
doesn't
introduce
a
kink
yeah,
so
you
have
to
you
have
to
be
well
continuous
up
to
the
second
derivative.
A
A
No
there's
no
effort
in
vasp
tool
to
allow
for
mixing
these
types.
There
are
other
programs
that
use
don't
know
exactly
one
that
uses
Gaussian
type,
orbitals
sort
of
a
Gaussian,
Augmented
plane,
wave
formalism.
Something
like
that
does
exist.
I
do
not
necessarily
agree
that
it's
that
it's
better
to
use
gaussians
for
molecules
per
se
and
I
have
I
have
a
slide
on
the
in
the
in
the
second
lecture
as
a
slide
where
we
compared
against
gaussian
description
of,
I
think
it
is
CL,
so
the
glory
glory
and
dimer
we're.
A
Actually
you
see
that
the
number
to
be
really
accurate.
You
see
that
the
number
of
functions
that
you
would
have
to
include
in
your
Gaussian
basis
is
huge
and
it
at
some
point.
It
will
even
be
larger
than
the
number
of
plane
waves
that
that
are
being
used
there,
so
because
that
would
be
the
measure
right.
So
how
large
does
my
basis
have
to
be
in
terms
of
gaussians
with
respect
to
how
many
basis
functions?
A
Having
said
that,
there
are
excellent
Gauss's
Gaussian
bases
around
right,
I
mean
people
have
been
using
them
in
chemistry
for
a
long
time,
so
there's
very
good
basis
sets
around
that
are
known
to
do
work.
Well,
it
is
hard
to,
let's
say
how
to
say
it
is
with
plain
waves,
there's
a
natural
way
of
reaching
completeness
of
your
basis
set.
You
simply
increase
the
the
cutoff
energy
and,
as
you
increase
the
cut
of
energy,
there's
a
there's,
a
controllable
measure
by
which
your
basis
set
becomes
more
complete.
A
This
is
much
more
difficult
with
Gaussian
basis
functions.
Having
said
there's
so
much
effort
have
been
has
been
put
into
Gaussian
basis
said
that
there
are
really
truly
excellent
basis
sets
around
you.
I,
don't
want
to
knock
them
but
from
from
a
conceptual
point
of
view,
there's
there's
difficulties
there.
Yes,.
I
A
A
J
A
I
think
we
should
do
that
and
I
think
that
yes,
I,
will
simply
continue
sort
of
we'll
see
where
we
end
today
and
where
I
continue
tomorrow.
Right,
yes,
I
mean
I,
like
it
I
like
the
way.
It
is
going
now
with
lots
of
questions,
but
it
might
mean
that
I
can't
do
each
and
every
slide
that
I've
brought
with
me,
but
I
think
that
that
is
less
important
than
then
answering
the
questions.
So
I
would
like
to
continue
as
we
are
doing
now,.
A
Okay,
that's
good!
So
then
I'll
I'll
squeeze
in
another
slide
right.
A
A
So
and
these
basis
functions,
how
are
they
computed?
They
are
the
solutions
to
the
radial
scalar
relativistic
non
spin-polarized
Schrodinger
equation
for
the
atom,
so,
if
I
have
a
helium
atom
in
my
system
at
some
point
in
the
creation
of
these
Patkar
files
that
are
associated
these
potential
files,
the
this
particular
radial
scalar
relativistic
non
spin-polarized
Schrodinger
equation
was
solved
for
the
helium
atom
in
some
particular
configuration
that
is
written
here
and
from
these
states,
which
are
the
eigenstates
of
this
atomic
problem,
and
they
are,
they
are
on.
They
live
on
radial,
logarithmic
grids.
A
So
from
these
from
these
states,
we
then
use
a
to
the
Z
realization
procedure
to
get
the
relevant
civilized
counterparts.
So
we
have
eigenstates
for
this
particular
for
this
particular
atomic
problem
and
from
the
from
this
whole
electron
eigenstates,
we
can
create
two
realized
versions.
We
can
sua
dies
the
the
effective
potential
of
the
atomic
problem.
We
can
construct
these
projector
functions
and
these
projector
functions.
The
only
thing
that
we
that
we
need
to
ensure
is
that
our
jewel
to
this
absolute
iced
States.
A
Unless
there's
procedures
to
do
this,
how
to
construct
them
I,
don't
go,
don't
want
to
go
into
that
too
much
so,
which
functions
do
we
do
we
take
into
our
bases?
Well,
the
natural
choices
that
we
that
we
make
is
that
we
would
include
all
the
bound
States
for
the
atomic
problem,
all
the
bound
states
of
the
valence
and
not
the
core
states.
A
They
are
put
into
this
to
this
frozen
core
approximation,
but
for
the
valence
electrons
of
the
atom,
we
include
all
the
bound
States
and
then
we
compute
some
additional
solutions
at
some
energies,
a
bit
away
from
the
bound
state.
So
I
did
this
yields
some
additional
variational
freedom
inside
of
these
local
basis.
A
For
that
we
construct
these
disputed
eyes,
dibs
or
eyes
counterparts
and
the
projector
functions,
and
a
lot
of
the
of
let's
say
of
the
technical
expertise,
goes
into
making
a
wise
choice
for
these
projector
functions,
because
if
you
look
at
this,
the
only
place
where
radial
logarithmic
quantities-
and
those
are
those
guys
they
live
on.
Radio
logarithmic
reads
where
they
meet
is
here
and
we
have
to
project
our
our
our
object
in
plane.
Waves
onto
this
projector
function,
so
these
projector
functions.
A
They
have
to
be
representable
in
in
plane,
waves
as
well,
and
everything
is
very
presentable
in
plane
wave.
Obviously,
but
we
want
to
do
this
with
a
limited
number.
Otherwise
we
end
up
with
an
operation
that
here
that
is
prohibitively
expensive.
So
there
we
have
to
make
a
wise
choice
and
well.
That
is
one
of
the.
What
we
consider
to
be
a
strong
point
of
our
program
is
that
that
geoguessr
has
lots
of
expertise
in
doing
this
and
has
clearly
created
this
set
of
pseudo
potentials.
A
That
is
quite
robust,
because
this
all
applies
obviously
for
the
atomic
problem,
but
but
then
we
put
it
into
some
chemical
environment
and
the
transferability
of
the
piw
method
is
very
high.
So
if
you
would,
if
you
would
ask
how
does
this
compare
to
other
suitor
potential
methods?
It's
in
that
sense,
it's
incomparable
it's
much
more
transferable,
that
you
can
take
the
atomic
problem
and
put
it
into
a
chemical
environment,
and
it
will
still
work.
Yes,.
A
Yes,
that
is
a
very
good
question,
so
this
choice
of
basis
related
a
scalar
relativistic
basis.
How
does
this
affect
spin
orbit
coupling?
It
does
affect
spin
orbit
coupling.
Obviously
software
heavy
elements,
the
spin
orbit
splitting
is
I
think
for
lead.
It
is
off
by
something
like
10%
because
of
the
fact
that
that
these
local
functions
that
you
include
are
not
there
are
not
j
functions.
There
l
functions
right,
yes,
so,
yes,
that
is
a
which
one
could
one
could
extend
the
code
to
use
different
functions,
obviously
to
a
children
for
the
code.
A
It
shouldn't
matter
what
it
gets
here.
So
it
could.
You
could
work
with
with
functions
that
were
created
for
for
an
equation
where
spin
orbit
coupling
was
was
included
in
some
other
way,
but
that
we
haven't
done
and
actually,
unfortunately,
that
would
be
quite
a
lot
of
work
to
to
include
that
if
you
look
at
other
methods
like
like
FL
APW,
where
you
can
include
local
orbitals
or
things
like
this,
this
is
typically
what
they
would
include.
A
Absent
because
there's
always
a
displaying
wave
part
that
carries
more
freedom,
then
then
what
is
removed
by
this
particular?
So
so
we
don't
reach.
We
don't
particularly
ever
reach
this
situation,
where
this
is
completely
canceled
out.
The
plane
wave
part
survives
partly
in
this
area,
and
that
is
that,
would
you
would
say,
okay,
that
is.
That
is
a
bad
thing.
Actually,
it
is
a
good
thing,
because
some
some
degrees
of
freedom
in
your
plane,
waves
set,
they
still
survive
here
and
they
the
help.
A
Well
anyway,
it
is
so
this
is
this
clean
separation
where
you
cancel
this
out
completely,
we
don't
want
to
achieve
that
always,
and
there
are
situations
where,
having
the
additional
degrees
of
freedom
of
the
plane
wave
survive
in
the
spheres
actually
helps
you
yeah.