Subject: WORDS LOCAL AND NONLOCAL:SUMMARY

From: elewars <[email-masked]>
Date: Thu, 06 Jul 2000 13:13:21 -0400

Thurs, 2000 July 6

Thanks very much to all who responded to my question. The answers,
#1--8, are summarized below. I hope I haven't overlooked anyone.

EL
----

Original question:
Wednesday, 2000 June 21

Hello,

Does any one know why DFT functionals without gradient corrections are
called _local_, while gradient-corrected functionals are called
_nonlocal_? I am not asking about the theory behind these, but why these

terms are used. I know that nonlocal has a certain meaning in quantum
physics; does it have some special meaning in connection with
mathematical functions or functionals?  The term nonlocal applied to
gradient-corrected functional has been said to be mathematically wrong
(A. St-Amant, Reviews in Computational chemistry, vol. 7, chapter 5,
1996; p. 223).

Thanks

E. Lewars
=====
Answers:

Andrew T. Pudzianowski, [email-masked]
#1

Dr. Lewars -
                    The term "local" comes from the local density
approximation
(LDA) in DFT, where it's assumed that the electron density at any given
point in a
molecule is the same as that in an electron gas, a system which has been
solved
exactly. The LDA provides correlation and is more accurate than the
Hartree-Fock
approximation in many cases, but it tends to "overbind" systems, i.e.,
it leads to
overestimates of bond energies and complexation energies. The LDA is
particularly
bad in describing H-bonded systems.
     The term "nonlocal" has been used to refer to anything that goes
beyond the
LDA, including gradient-corrected functionals. Some of these have
"local"
components derived from the electron gas description, but I don't know
if that's
what's objectionable about the use of "nonlocal" terminology with these
functionals.
                                                              Regards,
                                                       Andrew
Pudzianowski

---------
#2
Huub van Dam, [email-masked]

The term nonlocal originates from the fact that in addition to the local

terms you have gradient terms which say something about how the
functional
behaves in the immediate vicinity of the point where you're at. E.g.
compare gradient corrected functionals with a first order Taylor
expansion.

The reason I wouldn't use the term nonlocal is because of the word
"vicinity". In practice the gradient corrected functionals are still so
local that any long range correlation effect (e.g. dispersion) isn't
accounted for at all. So in fact these functionals are much more local
than
the name nonlocal suggests. So if you really don't want to write
"gradient
corrected" all the time than the term "semi-local" wouldn't be too bad
(although strictly mathematically that is probably wrong as well,
because
even gradient corrected functionals contain only quantities evaluated at
1
particular point even though it is a density and the norm of the
gradient).

------------------------
#3
Thomas Bligaard Pedersen,  [email-masked]

Hello,

Locality in physics is often related to closeness in the time-space
continuum -
that a theory should yield its results without taking information into
account
that is not accessible due to information travelling only at the speed
of light or
slower. Accepting general relativity to be an axiom, means that all
correct
theories of nature should be local, which is not yet the case for
quantum
mechanics.

This meaning of locality would then principally be of interest in TDFT,
but nobody
would really care about it, as it is related to extremely small
corrections of
general relativity, while it is plentily difficult to include all
special
relativistic terms.

For many years the most used functionals in solid state physics were LDA
and LSDA.
That's probably why everything else going beyond the LSDA got to be
called
non-local, including the GGAs which are mathematically local.
Interestingly Walter
Kohn calls GGAs quasi-local when he speaks about them.

Best regards

Thomas B. Pedersen

-------
#4
Lou Noodleman  [email-masked]

Dear Dr. Lewars,
Yes, words can cause big problems. DFT functionals without gradient
corrections are called local, precisely because there are no gradient
terms (and gradient terms in some sense sample the surroundings of
a given point); gradient-corrected functionals are called nonlocal
for exactly this reason. There is some confusion attending this,
since in quantum chemistry, "nonlocal" is also used for situations
where each orbital sees a different exchange potential (or exchange-
correlation potential). The latter meaning is true for Hartree-Fock (and
also
probably in a number of explicitly correlated methods like
MC-SCF and CI methods), but not in DFT methods either LDA or gradient-
corrected (GGA) methods. Hybrid Hartree-Fock-DFT methods like B3LYP
should be like Hartree-Fock, nonlocal in that each orbital sees a
different exchange-correlation potential in the one-electron SCF
equations.
Nonlocal terms cause the worse scaling behavior of Hartree-Fock (n**4)
versus
DFT (n**3) as a function of the number of active electrons in the
molecule.
However, there are fitting tricks (and threshold tricks)
 that can help to control this worse
scaling behavior in ab initio methods (as implemented in local MP2
methods
in Jaguar), and also there are other tricks in Turbomole (I hear).
A general rule in quantum chemistry is: "be sure you know how a
technical
term is being used in a particular context, what the author really
means."
Further, realize that the same term can be used to mean different things

by different people in different contexts. For example, "correlation"
can
mean whatever is left out of Hartree-Fock
(but is this spin restricted or unrestricted or broken symmetry?), but
there
are other equally valid frameworks.
 "Fermi correlation" has a statistical meaning (see McWeeney), and in my

view, "correlation" as defined in a statistical context as in the theory

of reduced density matrices has a much deeper meaning than the usual
"correction to Hartree-Fock" meaning.
Lou Noodleman
Molecular Biology TPC15
The Scripps Research Institute
La Jolla, CA 92037
.
-------------------
#5
Ross Dickson, [email-masked]

I believe the term "local" derives from the "local density
approximation"
(LDA), of Kohn & Sham (1965, Phys. Rev. A 1133).  Although the idea
is theirs, I can't say for certain that the term originated with them
also,
as I haven't the paper easily at hand.  I further suspect that the term
"non-local" arose as a casual or inexact means of distinguishing the
non-(local density approximation) functionals from the at-the-time
dominant LDA.

(I hope TeX notation is clear enough for this.)
A consequence of the Hohenberg-Kohn theorem is that
    e_{xc}(x) = e_{xc}[\rho]:
The exchange-correlation energy density at a point x is a _functional_
of the charge density at all points.  The local density approximation
amounts to supposing that
    e_{xc}(x) = e_{xc}(\rho(x)),
i.e. that the exchange-correlation energy density at a point x can be
adequately approximated by a _function_ of the charge density at that
point x.  It's local in the very concrete sense of "doesn't depend on
anything far away in coordinate space."

The various degrees of gradient correction can be characterized as
    e_{xc}(x) = e_{xc}(\rho(x), \rho'(x), \rho''(x), ...)
and so on:  You're still using a function, not a functional, for e_{xc},

but employing more and more derivatives of the charge density as
arguments (signified by ', '', etc.)  This, I think, is why Alain
St.-Amant
suggested that "non-local" is a bad term for this family of functionals,

and I am inclined to agree.  One is still working with the charge
density in the locality of the point at which one is evaluating the
energy
density (or the exchange-correlation potential), even though one is no
longer using the local density approximation.

Cheers,
Ross Dickson,
no longer a quantum chemist...

-----------
#6
Per-Ola Norrby, [email-masked]

       Hi,

        I think the term "local" in this context means that the value
of the function in a mathematical point depends on that point only.
The gradient is by definition the difference between the function
values for two points separated by an infinitesimal distance.  That
is, the gradient and any functional depending on it can never be
exactly "local".  In principle, there could also be other types of
"non-local" functionals, but I believe that the gradient correction
is the first "non-local" fix that anybody ever introduces.  I'm not
aware of any non-local functional without gradient correction, but
there could well be.

        Caveat: I'm a chemist, so my language is probably not
mathematically correct...

        Per-Ola Norrby
--

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  *  Per-Ola Norrby, Associate Professor
  *  The Royal Danish School of Pharmacy, Dept. of Med. Chem.
  *  Universitetsparken 2, DK 2100 Copenhagen, Denmark
  *  Tel.   +45-35306506, fax +45-35306040
  *  Email: [email-masked] (preferred), [email-masked]
  *  WWW:   http://compchem.dfh.dk/PeO/
-------------------
#7
Herbert Fruchtl, [email-masked]

There is a mathematical theorem  which states that if you know all
derivatives of an 'analytical function' (i.e. one that can be expressed
in a power series) in one point, you know the function in its complete
definition range.

Nonlocal functionals depend not only on the value of the density, but
also its derivatives (mostly only the first one, the gradient). The
gradient contains information about the surrounding area, not only the
point you look at. So the expression nonlocal my be not very intuitive,
but I don't think it's wrong.

HTH,

  Herbert

--------------
#8
Bert te Velde, [email-masked]

Hi,

The 'local' functionals all use (only) the value rho(r), i.e. the
density in a particular point in space. The formula for the
density-functional potential (and/or) the energy-density in a given
point in space is a more or less simple formula of rho(r), the charge
density in the same point.

The term 'non-local' has been used to stress that the (original)
non-local gradient corrections used not only rho(r) but also its
derivatives, so you could see it as implying that not only the local
value of the density went into the formula but also (implicitly,
through the derivatives) values at other (nearby) points in space:
non only the LOCAL value of rho.

Critics of the phrase 'non-local' refer to the fact that the
potential (or energy-function) is not in itself a non-local function:
it still is a 'simple' function v(r), respectively e(r).
Consequently, the term 'nonlocal' has gradually disappeared and has
been replaced by the more appropriate phrase 'gradient-corrected', or
'generalized gradient approximation (GGA)'.

Best regards
Bert
============
=========