Subject: CCL:DFT:local-nl, SUMMARY

From: Steven Creve <[email-masked]>
Date: Mon, 23 Nov 1998 16:57:44 +0100 (NFT)

Hi,

concerning the discussion "local <-> non-local in DFT", my first 
impression has been that there seems to be a lot of confusion on
definitions in this area. I think that the best definitions and
viewpoints are those offered by Keith Refson, Rene Fournier and Mark
Casida, the first three answers given below. For the sake of
completeness, I have also includede the other answers I received. I'd
wish to thank all people who contributed to this interesting
discussion on the CCL.

Steven
--------------------------------------------------------------------------
Steven Creve                       [email-masked]
Labo Quantumchemie
Celestijnenlaan 200F
3001-HEVERLEE                      tel: (32) (16) 32 73 93
BELGIUM                            fax: (32) (16) 32 79 92

=================================================================
A N S W E R S  (Refson, Fournier and Casida)
=================================================================

 > is the following correct:
 > 
 > non-local means that the xc potential in point r depends not only on the
 > density in r (and also gradients of rho in r, ...) but also on the density
 > in other points r'.

Exactly. in DFT, the LDA - local density approximation -- is precisely the
statement that 

Exc[n(r)] = integral exc(n(r)) dr

where on the LHS Exc is a functional of the charge density, ie it
depends on the functional form of n(r) and exc(n) is a pure function
of the charge density n at any point r.  Nonlocality corresponds to
increasing the *dimensionality* of the RHS so strictly, generalised
gradient functionals which also depend on grad n are still local.
Hartree-Fock exchange which depends (in the DFT formalism) on

integral ex(n1(r1), n2(r2)) dr1 dr2

is nonlocal.

Email: Keith.Refson@  | Tel: +44 1865 272026 | Dr Keith Refson,              |
       earth.ox.ac.uk | Fax: +44 1865 272072 | Dept of Earth Sciences        |
Spam:            [email-masked]         | Parks Road, Oxford OX1 3PR, UK|

------------------------------------------------------------------
   Local or nonlocal?

   Good question.  "Gradient-corrected" functionals (Becke88, Perdew86,
Perdew-Wang91, etc...) are often called "non-local" but in my
opinion that's an unfortunate misuse of the term.  I suppose they
came to be called "non-local" because they are NOT the usual local
density approximation (LDA).  But "NOT the USUAL LDA" is not
synonymous with "nonlocal".
   All functionals of the form

              b
   J[f(x)] = S  ( F(x,f,f',f'',...,f(n)) ) dx
              a

are local in the mathematical sense explained in
I.M. Gelf'and S.V. Fomin, "Calculus of Variations"
(Prentice-Hall, Englewood Cliffs, 1963).
J[f(x)] is an integral between a and b; it can be broken
into any number "n" of integrals on contiguous subdomains [a,c_1],
[c_1,c_2], ... [c_(n-1),b], and the subdomains can be chosen as
small as you wish.  So you can decompose J into a sum of local
contributions, irrespective of the order of derivatives in the
integrand F.

    An example of nonlocal functional given by Gelf'and and
Fomin is:

              S x sqrt(1+y'*y') dx
   J[y(x)] = ----------------------
               S sqrt(1+y'*y') dx

which gives the abscissa of the center of mass of a curve
y(x).  You can break the numerator and denominator into 
sums of local contributions, but you can not directly break
J itself into sums of local contributions.
   I think the class of functionals

   J[f] = S S ( f(x) K(x,y) f(y) dx dy )          

should also be called nonlocal (provided  K(x,y) is
"non-diagonal", i.e., K(x,y) is NOT delta(x,y)*K(x,x) )

    If you go by the definition of "local and nonlocal" above,
most functionals in common use (including LYP?) are local.

 --------------------------------------------------------------------
 | Rene Fournier                  | Bureau/Office  303 Petrie       |
 | Chemistry Dpt, York University |      (416) 736 2100 Ext. 30687  |
 | 4700 Keele Street,  Toronto    | FAX: (416) 736-5936             |
 | Ontario, CANADA   M3J 1P3      | e-mail: [email-masked]          |
 --------------------------------------------------------------------

------------------------------------------------------------------
Exchange-correlation functionals may be classified as:

1) Local --- the exchange-correlation energy density functional
   at r, e_xc(r), depends only on the density at r.

2) Semi-local --- e_xc(r) depends only on the density and a finite
   number of its derivatives at r.

3) Nonlocal --- e_xc(r) depends on properties of the density at
   several values of r, typically involving an integral.  (The
   restriction to a finite number of derivatives in (2) is
   because some integrals can be represented as a gradient
   expansion in an infinite number of derivatives.)

The distinction between (2) and (3) is often not made, so that
both are called "nonlocal".  On the other hand, it is probably
worth noting that the generalized gradient approximations (GGAs)
fall under (2) and the older weighted density approximation falls
under (3) and not (2), so the distinction has some value.

                        Hope this helps,
                              [email-masked]

==================================================================
O T H E R   A N S W E R S :
==================================================================

> 1) what are the *exact* definitions of local vs. non-local?
>

This is an critical point in DFT. Nowadays, there is a great interest in
properties of the exchange and correlation functionals like homogeneity
and locality. From the DFT point of view, a functional is local when all
the terms in the functional derivative, with the exception of the first,
vanish. What does it means?. It can be presumed that we are dealing with
non-local functionals, so we can suppose several non-local dependences
(succesive derivatives of the density) in the exchange or correlation
integrand. From here, the corresponding contribution to the functional
derivative is obtained. The complete corrections to the first, local,
term would reproduce the exact potential if DFT is exact.

>
> 2) is the LYP correlation functional, which incorporates gradient
> corrections, local or non-local?
>

Following the previous discussion:non-local, since it will provide
higher order corrections, non-local, to the local term in the
correlation potential.

Juan Carlos
[email-masked]
----------------------------------------------------------------------

>1) what are the *exact* definitions of local vs. non-local?

	The term "local" is used in dft to indicate that only the density is being
considered to build the functionals. The term "non-local" is often
(mis)used to indicate that gradient corrections to the density are also
considered to write down the functionals. I prefer the term "gradient
corrections to the density" since "non-local" actually means that one is
modeling the exchange-correlation hole without any connection with the
gradients just mentioned.

>2) is the LYP correlation functional, which incorporates gradient
>corrections, local or non-local?

	Well, the lyp c functional is both local and non-local because it has
terms including both the density and its gradient.

------------------------------------------------------------------

In Hartree-Fock theory, the exchange potential is non-local,
i.e. you need to know the wave function at point r and r' to
be able to compute the exchange energy.
In DFT theory, the exchange (or exchange-correlation) potential
is local! It depends only on the properties of the electron density
at point r where you want to compute the exchange energy.
Words "local" and "non-local" frequently used in DFT are misleading
because they do not refer to the potential but to the density! 
A local functional use only the value of the electron density at point r.
A non-local functional use the electron density, but also its 
first, second, ... derivatives at point r.
Hope this helps.

                                      ...Xav

Ast. Pr. Xavier Assfeld             [email-masked]
Laboratoire de Chimie theorique     (T) 33 3 83 91 21 49
Universite Henri Poincare           (F) 33 3 83 91 25 30
F-54506 Nancy BP 239                http://www.lctn.u-nancy.fr
------------------------------------------------------------------
	I missed your original question. However, the definition of
nonlocal depends upon who is using it.  Some people consider gradient
functionals as nonlocal, some call them quasilocal.  I prefer to consider
as local  any functional whose density (i.e integrand) does not involve an
integral. This would include gradient functionals.
	Nonlocal functionals are rare -- besides the classical coulomb
energy  J[[p(r)]. I've done some for magnetic responses, and for xc in the
presence of a field, but not for the field free case.

Hope this helps 

Fred Salsbury
[email-masked]
http://waugh.cchem.berkeley.edu/~sals
------------------------------------------------------------------
> is the following correct:
> 
> non-local means that the xc potential in point r depends not only on the
> density in r (and also gradients of rho in r, ...) but also on the density
> in other points r'.

No, that's incorrect. The xc potential of nonlocal functionals depends
only on the density, gradient, laplacian at r. The reason why they are
called non-local might be because to know gradient and laplacian of
density at point r, you need to know their behaviors at other points, r',
r", ....

Shubin
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> is the following correct:
> 
> non-local means that the xc potential in point r depends not only on the
> density in r (and also gradients of rho in r, ...) but also on the density
> in other points r'.

I don't believe so (but others can answer this better than I). I believe that non-local means
that the energy functional depends on the density at r and the gradient of the density at r.
The functional is independent of density at other points in space.

Alan

Alan Shusterman
Dept of Chemistry
Reed College
Portland, OR
www.reed.edu/~alan
------------------------------------------------------------------
> 1) what are the *exact* definitions of local vs. non-local?

The word "local" is from the "local density approximations", which refers to a
formulation of the density functional theory by Kohn and Sham, Phys. Rev. 140,
A1133 (1965). It assumes that the exchange (and correlation) potential is a
local operator as opposed to the nonlocal operator in the Hartree-Fock theory.
This greatly simplifies the calculation of the exchange (correlation) energy
 but trimmed off subtle physical properties of the exchange-correlation holes
of a realistic electron system.

"Nonlocal" really means "not a local" and in practice, the "generalized
gradient approximations". For the latter, the igonored properties of the
exchange-correlation hole in the "local desnity approximations" have been
recovered to a certain degree. Mainly, the rate of change of the density in the
vicinity of the spatial point where the potential is calculated has been
included: the gradient corrections. However, it is not a simple Taylor
expansion! The spectra in wavelength space were critically examined and some
spurious errors removed.

These jargons have been used by physicists for more than two decades and their
meanings becomes well known after hundreds of papers published debating the
theory and the approximations in the '70s and '80s.  As it is now popular in
the computational chemistry world, probably, less confussion would be caused if
these jargon words could be replaced with more self-interpreting phrases like
"local..." or "local ... with gradient corrections", or "a hybrid of the exact
Hartree exchange and the gradient corrected local density exchange-correlation
potentials".

> 2) is the LYP correlation functional, which incorporates gradient
> corrections, local or non-local?
>

LYP is "nonlocal".

Liang Lou
Wavefunction, Inc.
------------------------------------------------------------------

Hi, Steve. This term "non-local" is not correct in general because
gradient potentials STILL local. LYP corr. correction includes both
local and non-local terms. So, the "non-local" is used only to distinct
from pure local potentials (S, VWN) in LDA,
-- 
Galina Orlova
Vis. Res. Professor
[email-masked]
Department of Chemistry and Biochemistry
University of Guelph
Guelph, ON, N1G 2W1, Canada

ph. (519) 824 4120 ext 3061
Fax (519) 766-1499
------------------------------------------------------------------
	I have always heard that local functionals are ones that use the
electron density only  and nonlocal functionals are those that use a
gradient of electron density.  It is not necessary to take into account
the density at other points because the gradient gives an extrapolation to
nearby points (to the extent that the density is approximately linear in a
localized region).

                                Dave Young
                                [email-masked]
------------------------------------------------------------------

---

From: <[email-masked]>
Date: Tue, 24 Nov 1998 13:27:56 -0800

> definitions in this area.

-------------------------------
> synonymous with "nonlocal".

-------------------------------
> nearby points (to the extent that the density is approximately linear in a

I guess there is a misunderstanding (of the terms local vs. nonlocal) in other
terms: physics vs. math.

The Fock exchange potential is a nonlocal operator, with a nonlocality
mathematically extending to infinity and physically to the point where the
magnitude of an atomic wave function vanishes.

Going with every electron there is a hole created by the Pauli exchange rules
and the repulsions missing in the Hartree potential. The shape and range of
this hole is determined not only by the carrying electron itself but all the
surrounding electrons. This is where the nonlocality comes from. The electron
system is highly responsive and the perturbations along the trajectory can be
quickly damped (screened) out. The effective range of the xc hole is limited
(may be described by a characteristic length). This may be regarded as a "short
sightedness" of electron systems. And in fact, it has been shown that
much of the semiquantitative success of the LDA can be explained by a spherical
average made to the skewed and singularity plagued xc hole (a luck?).

Assuming a "slowly varying" density for an atomic system is too crude. There
have been many efforts to go beyond this restrictive framework. The goal is to
take into account the nonlocality, up to a finite distance in accordance with
the short sightedness of electrons in a pool. This can be done with the help of
a generalized Taylor expansion for the density around the very point r. The r'
dependence is absorbed in the derivative terms of r.  The term "generalized"
refers to a treatment of the singular points (cusps) within the xc hole (which
is a function of r and r').  The gradient corrections as they stand involve
local operators depending on r only. But physically, the properties of the xc
hole in the vicinity of the spatial point r have been described correctly, as
approved by the success of GGAs. Also, without an explicit r' dependence, much
computational cost can be saved.

In my perhaps misrepresented vocabulary, the word "nonlocal" is a physical
term, with a meaning which can be read as "it takes some nonlocality into
account and goes beyond the local density approximations".

There may be misunderstaninding of the physics in the above descriptions. Any
oppositions (fires), please, direct to my e-mail address [email-masked]

Thanks,
Liang Lou