From owner-chemistry@ccl.net Mon Jun 29 06:49:01 2020 From: "Marcel Swart (GMail) marcel.swart{:}gmail.com" To: CCL Subject: CCL: Charge Message-Id: <-54121-200629064157-13990-t/oHOQnvm9gRkQSmFGinuA[-]server.ccl.net> X-Original-From: "Marcel Swart (GMail)" Content-Type: multipart/alternative; boundary="Apple-Mail=_914C46F8-9F5C-4AC9-98A6-518E2C7F5CFB" Date: Mon, 29 Jun 2020 12:41:45 +0200 Mime-Version: 1.0 (Mac OS X Mail 12.4 \(3445.104.14\)) Sent to CCL by: "Marcel Swart (GMail)" [marcel.swart|-|gmail.com] --Apple-Mail=_914C46F8-9F5C-4AC9-98A6-518E2C7F5CFB Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=utf-8 Just to add to the discussion: 1) Mulliken charges are flawed in many aspects, but we know how and why, = and therefore we accept its use, since it is simple and easily computed. But very dependent on the size = of the basis set. 2) Mulliken charges do not reproduce the dipole (or higher) moment, but = can be made to do so easily: Thole, van Duijnen, "A general population analysis preserving the dipole = moment=E2=80=9D Theoret. Chim. Acta 1983, 63, 209=E2=80=93221 www.dx.doi.org/10.1007/BF00569246 3) Based on a multipole expansion (used in ADF for the Coulomb = potential) we have extended this to quadrupoles etc. M. Swart, P.Th. van Duijnen and J.G. Snijders=20 "A charge analysis derived from an atomic multipole expansion" J. Comput. Chem. 2001, 22, 79-88 = http://www.dx.doi.org/10.1002/1096-987X(20010115)22:1%3C79::AID-JCC8%3E3.0= .CO;2-B Note that the multipoles result directly from the charge density, no = fitting needed to the electrostatic potential at some grid outside the molecule (as done by other = electrostatic potential fitted charge analyses). For each atom, its multipoles are represented by redistributed = fractional atomic charges (with a weight function based on distance to keep these as close to the original atom = as possible), the sum of these fractional atomic charges then add up to e.g. MDC-m (when only monopoles = are redistributed), MDC-d (both monopoles and dipoles redistributed), MDC-q (monopoles, dipoles, = quadrupoles redistributed). For the N=C60 quartet state as mentioned by Tom, the MDC-m works best. MDC-m, charge N -0.017, spin-dens. charge N 2.874; charge C 0.0003, = spin-dens. charge C 0.002 MDC-d, charge N 0.136, spin-dens. charge N 0.722; charge C -0.002, = spin-dens. charge C 0.038 MDC-q, charge N 0.062, spin-dens. charge N 0.729; charge C ranging from = +0.05 to -0.05, spin-dens. charge C 0.038 In this case, the only other places where to put fractional charges are = at the cage, hence not a representative sample for the method an sich. I must add here that the Mulliken analysis works excellently: charge N = -0.062, spin-dens. charge N 2.970. Hirshfeld (0.138) and Voronoi (0.259) charges for N are larger (no = spin-density equivalent available within ADF). 4) Finally, see also the following papers: M. Cho, N. Sylvetsky, S. Eshafi, G. Santra, I. Efremenko, J.M.L. Martin The Atomic Partial Charges Arboretum: Trying to See the Forest for the = Trees ChemPhysChem 2020, 21, 688-696 www.dx.doi.org/10.1002/cphc.202000040 G. Aullo=CC=81n, S Alvarez=20 Oxidation states, atomic charges and orbital populations in transition = metal complexes. Theor. Chem. Acc. 2009, 123, 67-73 www.dx.doi.org/10.1007/s00214-009-0537-9 G. Knizia Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and = Chemical Concepts J. Chem. Theory Comp. 2013, 9, 4834-4843 www.dx.doi.org/10.1021/ct400687b MS > On 29 Jun 2020, at 01:58, Thomas Manz thomasamanz++gmail.com = wrote: >=20 > Hi Robert, >=20 > >If you see inconsistency across charge schemes, it is not the fault = of a given scheme, but the notion itself. =20 >=20 > I think this all depends on the situation. There may be some = situations where two different atomic population analysis methods yield = different results for one atom-in-material descriptor (e.g., net atomic = charge, atomic spin moment, etc.) and still both values might be = considered reasonable as long as each method yields chemically = self-consistent results (i.e., that method gave various atom-in-material = descriptors that were internally chemically consistent).=20 >=20 > Yet, we must not forget other situations where the computed result = should closely match a chemically well-defined value. For example, in = the N-x-C60 endohedral complex the electronic state of the N atom is = directly observable from spectroscopy to be a quartet state and it is = clear from spectroscopic measurements that charge transfer between the N = atom and the C60 cage is small or negligible in this material. = Therefore, it is clear in this case that any reasonable atomic = population analysis method should give small or negligible charge = transfer amount between the N atom and C60 cage and should yield atomic = spin moments consistent with a quartet spin state of the N atom. It is = fair to say atomic population analysis methods that do not do this have = given a wrong result. A special aspect of this test system is the N atom = is mostly isolated but not fully isolated by virtue of being located in = the middle of the C60 cage; this yields a chemically well-defined state = for this N atom. >=20 > One can imagine other well-defined systems that can serve as useful = test cases. For these well-defined test cases, it is possible to = construct falsifiable scientific tests for atomic population analysis = methods. In other words, it is possible in some cases to prove whether = an atomic population analysis method yields the right or wrong result. = For this reason, atomic population analysis methods can be falsified = under some circumstances.=20 >=20 > Sincerely, >=20 > Tom >=20 Marcel Swart ICREA Research Professor at University of Girona Director of Institut de Qu=C3=ADmica Computacional i Cat=C3=A0lisi Univ. Girona, Campus Montilivi (Ci=C3=A8ncies) c/ M.A. Capmany 69 17003 Girona, Spain www.marcelswart.eu marcel.swart=gmail.com vCard addressbook://www.marcelswart.eu/MSwart.vcf = --Apple-Mail=_914C46F8-9F5C-4AC9-98A6-518E2C7F5CFB Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=utf-8 Just = to add to the discussion:

1) Mulliken charges are flawed in many aspects, but we know = how and why, and therefore we accept its use,
since = it is simple and easily computed. But very dependent on the size of the = basis set.

2) = Mulliken charges do not reproduce the dipole (or higher) moment, but can = be made to do so easily:
Thole, van Duijnen, "A = general population analysis preserving the dipole moment=E2=80=9D
Theoret. Chim. Acta 1983, 63, 209=E2=80=93221=
www.dx.doi.org/10.1007/BF00569246

3) Based on a multipole expansion (used = in ADF for the Coulomb potential) we have extended this to
quadrupoles etc.
M. Swart, P.Th. van = Duijnen and J.G. Snijders 
"A charge analysis derived = > from an atomic multipole expansion"
J. Comput. = Chem. 2001, 22, 79-88
http://www.dx.doi.org/10.1002/1096-987X(20010115)22:1%3C79::AID= -JCC8%3E3.0.CO;2-B

Note that the multipoles result directly from the charge = density, no fitting needed to the electrostatic
potential at some grid outside the molecule (as done by other = electrostatic potential fitted charge analyses).
For = each atom, its multipoles are represented by redistributed fractional = atomic charges (with a weight
function based on = distance to keep these as close to the original atom as possible), the = sum of these
fractional atomic charges then add up = to e.g. MDC-m (when only monopoles are redistributed), MDC-d
(both monopoles and dipoles redistributed), MDC-q (monopoles, = dipoles, quadrupoles redistributed).

For the N=C60 quartet state as = mentioned by Tom, the MDC-m works best.

MDC-m, charge N -0.017, spin-dens. = charge N 2.874; charge C 0.0003, spin-dens. charge C 0.002
MDC-d, charge N 0.136, spin-dens. charge N 0.722; charge C = -0.002, spin-dens. charge C 0.038
MDC-q, charge N = 0.062, spin-dens. charge N 0.729; charge C ranging from +0.05 to -0.05, = spin-dens. charge C 0.038
In this case, the only = other places where to put fractional charges are at the cage, hence not = a
representative sample for the method an = sich.

I must = add here that the Mulliken analysis works excellently: charge N -0.062, = spin-dens. charge N 2.970.
Hirshfeld (0.138) and = Voronoi (0.259) charges for N are larger (no spin-density equivalent = available within ADF).

4) Finally, see also the following papers:

M. Cho, N. = Sylvetsky, S. Eshafi, G. Santra, I. = Efremenko,  J.M.L. Martin
The Atomic = Partial Charges Arboretum: Trying to See the Forest for the = Trees
ChemPhysChem 2020, 21, 688-696
www.dx.doi.org/10.1002/cphc.202000040

G. Aullo=CC=81n, S = Alvarez 
Oxidation states, atomic charges and = orbital populations in transition metal complexes.
Theor. Chem. Acc. 2009, 123, 67-73
www.dx.doi.org/10.1007/s00214-009-0537-9

G. Knizia
Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum = Theory and Chemical Concepts
J. Chem. Theory = Comp. 2013, 9, 4834-4843
www.dx.doi.org/10.1021/ct400687b

MS

On 29 = Jun 2020, at 01:58, Thomas Manz thomasamanz++gmail.com <owner-chemistry=ccl.net> wrote:

Hi Robert,

>If you see inconsistency across charge schemes, it is not = the fault of a given scheme, but the notion itself.  

I = think this all depends on the situation. There may be some situations = where two different atomic population analysis methods yield different = results for one atom-in-material descriptor (e.g., net atomic charge, = atomic spin moment, etc.) and still both values might be considered = reasonable as long as each method yields chemically self-consistent = results (i.e., that method gave various atom-in-material descriptors = that were internally chemically consistent). 

Yet, we must not forget = other situations where the computed result should closely match a = chemically well-defined value. For example, in the N-x-C60 endohedral = complex the electronic state of the N atom is directly observable from = spectroscopy to be a quartet state and it is clear from spectroscopic = measurements that charge transfer between the N atom and the C60 cage is = small or  negligible in this material. Therefore, it is clear in = this case that any reasonable atomic population analysis method should = give small or negligible charge transfer amount between the N atom = and C60 cage and should yield atomic spin moments consistent with a = quartet spin state of the N atom. It is fair to say atomic population = analysis methods that do not do this have given a wrong result. A = special aspect of this test system is the N atom is mostly isolated but = not fully isolated by virtue of being located in the middle of the C60 = cage; this yields a chemically well-defined state for this N = atom.

One can = imagine other well-defined systems that can serve as useful test cases. = For these well-defined test cases, it is possible to construct = falsifiable scientific tests for atomic population analysis methods. In = other words, it is possible in some cases to prove whether an atomic = population analysis method yields the right or wrong result. For = this reason, atomic population analysis methods can be falsified under = some circumstances. 

Sincerely,

Tom


Marcel Swart
ICREA Research = Professor at University of Girona
Director of Institut de Qu=C3=ADmica Computacional = i Cat=C3=A0lisi

Univ. Girona, Campus Montilivi = (Ci=C3=A8ncies)
c/ M.A. = Capmany 69
17003 Girona, Spain

www.marcelswart.eu
marcel.swart=gmail.com

vCard
addressbook://www.marcelswart.eu/MSwart.vcf

= --Apple-Mail=_914C46F8-9F5C-4AC9-98A6-518E2C7F5CFB-- From owner-chemistry@ccl.net Mon Jun 29 09:49:00 2020 From: "Stefan Grimme grimme**thch.uni-bonn.de" To: CCL Subject: CCL: Charge Message-Id: <-54122-200629094728-23161-KLAy3OmbOr8MCGhjZBh7Ww ~ server.ccl.net> X-Original-From: "Stefan Grimme" Date: Mon, 29 Jun 2020 09:47:27 -0400 Sent to CCL by: "Stefan Grimme" [grimme..thch.uni-bonn.de] One more comment to the Mulliken charge discussion: even methods without a well-defined basis set limit can be useful as already mentioned by Marcel Swart. This holds for the Mulliken atomic charge partitioning in compact MB/DZ basis sets (even TZ is often reasonable). For example the DFTB and GFN-xTB tight-binding methods are fundamentally based on a Mulliken analysis of the density matrix and yield physically very reasonable electrostatic energies. In GFN2-xTB this also works well up to quadrupole moments. Its clear that the Mulliken scheme breaks down for AO basis sets containing diffuse components but I really would like to see a differentiated view on the topic (and not as in a recent general statement of a reviewer something like "I do not think Mulliken charges are trustworthy"). Best Stefan Grimme From owner-chemistry@ccl.net Mon Jun 29 11:32:01 2020 From: "Thomas Manz thomasamanz*|*gmail.com" To: CCL Subject: CCL: Charge Message-Id: <-54123-200629112745-20760-oHIyc6HhGLZaiQ9YcCPTqQ||server.ccl.net> X-Original-From: Thomas Manz Content-Type: multipart/alternative; boundary="000000000000e4b01405a93ab27e" Date: Mon, 29 Jun 2020 09:27:28 -0600 MIME-Version: 1.0 Sent to CCL by: Thomas Manz [thomasamanz,,gmail.com] --000000000000e4b01405a93ab27e Content-Type: text/plain; charset="UTF-8" Hi Stefan, I think the confusion arises, because the Mulliken populations are sometimes confused with net atomic charges. The expansion of the electron density can be performed using any desired basis. In your application, the Mulliken partitioning is just a basis representation for expanding the electron density in terms of a distributed multipole expansion (e.g., up to quadrupole order). Yes, the Mulliken partitioning can be a mechanism to formulate a distributed multipole expansion of the electron density which can be a useful computational algorithm for computing electrostatic interactions during a quantum chemistry calculation. This is somewhat related to the fast multipole moments expansion of the Coulomb operator in quantum chemistry calculations. But, this is an entirely different topic than extracting chemically meaningful atom-in-material descriptors from a quantum chemistry calculation. Extracting chemically meaningful atom-in-material descriptors (net atomic charges, atomic spin moments, bond orders, s-p-d-f-g populations, etc.) carries with it the extra requirements of exhibiting correlations to experimental observables and of having well-defined mathematical values (including a complete basis set limit) and of exhibiting chemical consistency between various chemical descriptors. A multipole expansion of the Coulomb operator (such as the Mulliken-based multipole expansion you mentioned) has nothing to do with chemically meaningful descriptors, it is simply a trick to re-write the density matrix using a different basis representation to simplify the calculation of Coulomb integrals. In other words, it is merely algorithmic. The great confusion regarding Mulliken populations, which are simply mathematical artifices and not chemical properties, is that they have historically been confused with chemical properties like net atomic charges. Just like basis set overlap integrals, Mulliken populations can be a useful ingredient for expanding the Coulomb operator, as your example illustrates, but they are no more chemical properties of a material than basis set overlap integrals are chemical properties of material. In other words, not everything used in a quantum chemistry calculation is a chemical property of a material: some are just mathematical constructs whose utility resides in the algorithmic computation of another quantity (e.g., electrostatic interaction). The origin of this great confusion is that for small basis sets the Mulliken populations often resemble the net atomic charges computed by other methods, but this is somewhat coincidental because the correlation fails to hold when the basis set is improved. The reason this often confuses people is because there are actually two opposite ways to construct a polyatomic multipole expansion: (a) using quantities that are merely algorithmic (e.g., Mulliken populations) in the sense they have no complete basis set limit but none-the-less can be used as a basis representation to expand the Coulomb potential and (b) using chemically well-defined quantities (e.g., DDEC6 or QTAIM or Hirshfeld NACs and atomic multipoles) that have well-defined complete basis set limits and can be used as a basis representation to expand the Coulomb potential People often fail to recognize the distinction between these two cases, which have a day and night difference between them. Sincerely, Tom On Mon, Jun 29, 2020 at 8:43 AM Stefan Grimme grimme**thch.uni-bonn.de < owner-chemistry-$-ccl.net> wrote: > > Sent to CCL by: "Stefan Grimme" [grimme..thch.uni-bonn.de] > One more comment to the Mulliken charge discussion: > even methods without a well-defined basis set limit can be useful > as already mentioned by Marcel Swart. This holds > for the Mulliken atomic charge partitioning in compact MB/DZ basis sets > (even TZ is often reasonable). For example the DFTB and GFN-xTB > tight-binding methods are fundamentally based on a Mulliken analysis of the > density matrix and yield physically very reasonable electrostatic energies. > In GFN2-xTB this also works well up to quadrupole moments. > Its clear that the Mulliken scheme breaks down for AO basis sets containing > diffuse components but I really would like to see a differentiated view on > the topic (and not as in a recent general statement of a reviewer something > like "I do not think Mulliken charges are trustworthy"). > Best > Stefan Grimme> > > --000000000000e4b01405a93ab27e Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Hi Stefan,

I think the confusion arises= , because the Mulliken populations are sometimes confused with net atomic c= harges.

The expansion of the electron density can = be performed using any desired basis. In your application, the Mulliken par= titioning is just a basis representation for expanding the electron=C2=A0de= nsity in terms of a distributed multipole expansion (e.g., up to quadrupole= order). Yes, the Mulliken partitioning can be a mechanism to formulate a d= istributed multipole expansion of the electron density which can be a usefu= l computational algorithm for computing electrostatic interactions during a= quantum chemistry calculation. This is somewhat related to the fast multip= ole moments expansion of the Coulomb operator in quantum chemistry calculat= ions. But, this is an entirely different topic than extracting chemically m= eaningful atom-in-material descriptors from a quantum chemistry calculation= .

Extracting=20 chemically meaningful atom-in-material descriptors (net atomic charges, ato= mic spin moments, bond orders, s-p-d-f-g populations, etc.) carries with it= the extra requirements of exhibiting correlations to experimental observab= les and of having well-defined mathematical values (including a complete ba= sis set limit) and of exhibiting chemical consistency between various chemi= cal descriptors. A multipole expansion of the Coulomb operator (such as the= Mulliken-based multipole expansion you mentioned) has nothing to do with c= hemically meaningful descriptors, it is simply a trick to re-write the dens= ity matrix using a different basis representation to simplify the calculati= on of Coulomb integrals. In other words, it is merely algorithmic.

The great confusion regarding Mulliken populations, which = are simply mathematical artifices and not chemical properties, is that they= have historically been confused with chemical properties like net atomic c= harges. Just like basis set overlap integrals, Mulliken populations can be = a useful ingredient for expanding the Coulomb operator, as your example ill= ustrates, but they are no more chemical properties of a material than basis= set overlap integrals are chemical properties of material. In other words,= not everything used in a quantum chemistry calculation is a chemical prope= rty of a material: some are just mathematical constructs whose utility resi= des in the algorithmic computation of another quantity (e.g., electrostatic= interaction). The origin of this great confusion is that for small basis s= ets the Mulliken populations often resemble the net atomic charges computed= by other methods, but this is somewhat coincidental because the correlatio= n fails to hold when the basis set is improved.

Th= e reason this often confuses people is because there are actually two oppos= ite ways to construct a polyatomic multipole expansion:=C2=A0
(a) using quantities that are merely algorithmic (e.g., Mullike= n populations) in the sense they have no complete basis set limit but none-= the-less can be used as a basis representation to expand the Coulomb potent= ial and=C2=A0

(b) using chemically well-defined qu= antities (e.g., DDEC6 or QTAIM or Hirshfeld NACs and atomic multipoles) tha= t have well-defined complete basis set limits and can be used as a basis re= presentation to expand the Coulomb potential=20

People often fail to recognize the distinction be= tween these two cases, which have a day and night difference between them.<= br>

Sincerely,

Tom
<= div>=C2=A0
On Mon, Jun 29, 2020 at 8:43 AM Stefan Grimme grimme**thch.uni-bonn.de <owner-chemistry-$-ccl.net> wrote:

Sent to CCL by: "Stefan=C2=A0 Grimme" [grimme..thch.uni-bonn.de= ]
One more comment to the Mulliken charge discussion:
even methods without a well-defined basis set limit can be useful
as already mentioned by Marcel Swart. This holds
for the Mulliken atomic charge partitioning in compact MB/DZ basis sets
(even TZ is often reasonable). For example the DFTB and GFN-xTB tight-bindi= ng methods are fundamentally based on a Mulliken analysis of the density ma= trix and yield physically very reasonable electrostatic energies. In GFN2-x= TB this also works well up to quadrupole moments.
Its clear that the Mulliken scheme breaks down for AO basis sets containing=
diffuse components but I really would like to see a differentiated view on = the topic (and not as in a recent general statement of a reviewer something= like "I do not think Mulliken charges are trustworthy").
Best
Stefan Grimme



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--000000000000e4b01405a93ab27e-- From owner-chemistry@ccl.net Mon Jun 29 12:06:00 2020 From: "Thomas Manz thomasamanz%gmail.com" To: CCL Subject: CCL: Charge Message-Id: <-54124-200629115429-31746-5vKyCiQ684sKE273CwxLpQ-$-server.ccl.net> X-Original-From: Thomas Manz Content-Type: multipart/alternative; boundary="00000000000087bfdf05a93b122c" Date: Mon, 29 Jun 2020 09:54:12 -0600 MIME-Version: 1.0 Sent to CCL by: Thomas Manz [thomasamanz ~~ gmail.com] --00000000000087bfdf05a93b122c Content-Type: text/plain; charset="UTF-8" Hi Stefan, I wanted to further clarify one aspect of my earlier response. Suppose that one has a NaCl crystal, for example. Using a Mulliken population analysis, depending on the basis set, the populations of the Na and Cl atoms could take on any values that sum to 11 + 17 = 28 electrons. For example, one could have 11.3488 electrons on the Na atom and 28 - 11.3488 = 16.6512 electrons on the Cl atom. Using a different basis set, the Mulliken population analysis might yield 10.2342 electrons on the Na atom and 28 - 10.2342 = 17.7658 electrons on the Cl atom. Using another basis set, you might get 11.0000 electrons on the Na atom and 28 - 11.0000 = 17.0000 electrons on the Cl atom. For any chosen basis set, the Mulliken populations (or any other kind of populations) and their corresponding atomic multipoles and charge penetration (i.e., Coulombic electrostatic interaction between overlapping functions) integrals could be used as a representation to expand the Coulomb operator (i.e., to calculate the Coulomb interaction between electrons in the quantum chemistry calculation). If the expansion is carried out to high enough order, then its precision could be arbitrarily high (e.g., reproduce the Coulomb interaction to machine precision). It is very clear to see this has nothing to do with chemically meaningful atom-in-material descriptors, because in case 1 the Na atom in the NaCl crystal would be assigned [sic] as an anion and the Cl atom as a cation; in case 2 the Na atom in the NaCl crystal would be assigned [sic] as an cation and the Cl atom as a anion; and in case 3 the Na atom in the NaCl crystal would be assigned [sic] as neutral (i.e., bearing no net charge) and the Cl atom also as neutral. Hence, the Mulliken populations cannot be net atomic charges in general. Sincerely, Tom On Mon, Jun 29, 2020 at 9:27 AM Thomas Manz wrote: > Hi Stefan, > > I think the confusion arises, because the Mulliken populations are > sometimes confused with net atomic charges. > > The expansion of the electron density can be performed using any desired > basis. In your application, the Mulliken partitioning is just a basis > representation for expanding the electron density in terms of a distributed > multipole expansion (e.g., up to quadrupole order). Yes, the Mulliken > partitioning can be a mechanism to formulate a distributed multipole > expansion of the electron density which can be a useful computational > algorithm for computing electrostatic interactions during a quantum > chemistry calculation. This is somewhat related to the fast multipole > moments expansion of the Coulomb operator in quantum chemistry > calculations. But, this is an entirely different topic than extracting > chemically meaningful atom-in-material descriptors from a quantum chemistry > calculation. > > Extracting chemically meaningful atom-in-material descriptors (net atomic > charges, atomic spin moments, bond orders, s-p-d-f-g populations, etc.) > carries with it the extra requirements of exhibiting correlations to > experimental observables and of having well-defined mathematical values > (including a complete basis set limit) and of exhibiting chemical > consistency between various chemical descriptors. A multipole expansion of > the Coulomb operator (such as the Mulliken-based multipole expansion you > mentioned) has nothing to do with chemically meaningful descriptors, it is > simply a trick to re-write the density matrix using a different basis > representation to simplify the calculation of Coulomb integrals. In other > words, it is merely algorithmic. > > The great confusion regarding Mulliken populations, which are simply > mathematical artifices and not chemical properties, is that they have > historically been confused with chemical properties like net atomic > charges. Just like basis set overlap integrals, Mulliken populations can be > a useful ingredient for expanding the Coulomb operator, as your example > illustrates, but they are no more chemical properties of a material than > basis set overlap integrals are chemical properties of material. In other > words, not everything used in a quantum chemistry calculation is a chemical > property of a material: some are just mathematical constructs whose utility > resides in the algorithmic computation of another quantity (e.g., > electrostatic interaction). The origin of this great confusion is that for > small basis sets the Mulliken populations often resemble the net atomic > charges computed by other methods, but this is somewhat coincidental > because the correlation fails to hold when the basis set is improved. > > The reason this often confuses people is because there are actually two > opposite ways to construct a polyatomic multipole expansion: > > (a) using quantities that are merely algorithmic (e.g., Mulliken > populations) in the sense they have no complete basis set limit but > none-the-less can be used as a basis representation to expand the Coulomb > potential and > > (b) using chemically well-defined quantities (e.g., DDEC6 or QTAIM or > Hirshfeld NACs and atomic multipoles) that have well-defined complete basis > set limits and can be used as a basis representation to expand the Coulomb > potential > > People often fail to recognize the distinction between these two cases, > which have a day and night difference between them. > > Sincerely, > > Tom > > > On Mon, Jun 29, 2020 at 8:43 AM Stefan Grimme grimme**thch.uni-bonn.de < > owner-chemistry.-$-.ccl.net> wrote: > >> >> Sent to CCL by: "Stefan Grimme" [grimme..thch.uni-bonn.de] >> One more comment to the Mulliken charge discussion: >> even methods without a well-defined basis set limit can be useful >> as already mentioned by Marcel Swart. This holds >> for the Mulliken atomic charge partitioning in compact MB/DZ basis sets >> (even TZ is often reasonable). For example the DFTB and GFN-xTB >> tight-binding methods are fundamentally based on a Mulliken analysis of the >> density matrix and yield physically very reasonable electrostatic energies. >> In GFN2-xTB this also works well up to quadrupole moments. >> Its clear that the Mulliken scheme breaks down for AO basis sets >> containing >> diffuse components but I really would like to see a differentiated view >> on the topic (and not as in a recent general statement of a reviewer >> something like "I do not think Mulliken charges are trustworthy"). >> Best >> Stefan Grimme>> >> >> --00000000000087bfdf05a93b122c Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Hi Stefan,

I wanted to further clarify = one aspect of my earlier response.

Suppose that on= e has a NaCl crystal, for example. Using a Mulliken population analysis, de= pending on the basis set, the populations of the Na and Cl atoms could take= on any values that sum to 11=C2=A0+ 17=C2=A0 =3D 28 electrons. For example= , one could have 11.3488 electrons on the Na atom and 28=C2=A0 -=C2=A0 11.3488 =3D=C2=A0 16.6512 electrons on the Cl atom. Using a different basis= set, the Mulliken population analysis might yield 10.2342 electrons on the= Na atom and 28 -=20 10.2342=20 =3D 17.7658 electrons on the Cl atom. Using another basis set, you might get 11.0000 el= ectrons on the Na atom and 28 - 11.0000 =3D 17.0000 electrons on the Cl ato= m. For any chosen basis set, the Mulliken populations (or any other kind of= populations) and their corresponding atomic multipoles and charge penetrat= ion (i.e., Coulombic electrostatic interaction between overlapping function= s) integrals could be used as a representation to expand the Coulomb operat= or (i.e., to calculate the Coulomb=C2=A0interaction between electrons in th= e quantum chemistry calculation). If the expansion is carried out to high e= nough order, then its precision could be arbitrarily high (e.g., reproduce = the Coulomb interaction to machine precision). It is very clear to see this= has nothing to do with chemically meaningful atom-in-material descriptors,= because in case 1 the Na atom in the NaCl crystal would be assigned [sic] = as an anion and the Cl atom as a cation; in case 2 the Na atom in the NaCl = crystal would be assigned [sic] as an cation and the Cl atom as a anion; an= d in case 3 the Na atom in the NaCl crystal would be assigned [sic] as neutra= l (i.e., bearing no net charge) and the Cl atom also as neutral. Hence, the= Mulliken populations cannot be net atomic charges in general.
Sincerely,

Tom

On Mon, Jun 29, 2= 020 at 9:27 AM Thomas Manz <tho= masamanz.-$-.gmail.com> wrote:
Hi Stefan,

I think th= e confusion arises, because the Mulliken populations are sometimes confused= with net atomic charges.

The expansion of the ele= ctron density can be performed using any desired basis. In your application= , the Mulliken partitioning is just a basis representation for expanding th= e electron=C2=A0density in terms of a distributed multipole expansion (e.g.= , up to quadrupole order). Yes, the Mulliken partitioning can be a mechanis= m to formulate a distributed multipole expansion of the electron density wh= ich can be a useful computational algorithm for computing electrostatic int= eractions during a quantum chemistry calculation. This is somewhat related = to the fast multipole moments expansion of the Coulomb operator in quantum = chemistry calculations. But, this is an entirely different topic than extra= cting chemically meaningful atom-in-material descriptors from a quantum che= mistry calculation.

Extracting=20 chemically meaningful atom-in-material descriptors (net atomic charges, ato= mic spin moments, bond orders, s-p-d-f-g populations, etc.) carries with it= the extra requirements of exhibiting correlations to experimental observab= les and of having well-defined mathematical values (including a complete ba= sis set limit) and of exhibiting chemical consistency between various chemi= cal descriptors. A multipole expansion of the Coulomb operator (such as the= Mulliken-based multipole expansion you mentioned) has nothing to do with c= hemically meaningful descriptors, it is simply a trick to re-write the dens= ity matrix using a different basis representation to simplify the calculati= on of Coulomb integrals. In other words, it is merely algorithmic.

The great confusion regarding Mulliken populations, which = are simply mathematical artifices and not chemical properties, is that they= have historically been confused with chemical properties like net atomic c= harges. Just like basis set overlap integrals, Mulliken populations can be = a useful ingredient for expanding the Coulomb operator, as your example ill= ustrates, but they are no more chemical properties of a material than basis= set overlap integrals are chemical properties of material. In other words,= not everything used in a quantum chemistry calculation is a chemical prope= rty of a material: some are just mathematical constructs whose utility resi= des in the algorithmic computation of another quantity (e.g., electrostatic= interaction). The origin of this great confusion is that for small basis s= ets the Mulliken populations often resemble the net atomic charges computed= by other methods, but this is somewhat coincidental because the correlatio= n fails to hold when the basis set is improved.

Th= e reason this often confuses people is because there are actually two oppos= ite ways to construct a polyatomic multipole expansion:=C2=A0
(a) using quantities that are merely algorithmic (e.g., Mullike= n populations) in the sense they have no complete basis set limit but none-= the-less can be used as a basis representation to expand the Coulomb potent= ial and=C2=A0

(b) using chemically well-defined qu= antities (e.g., DDEC6 or QTAIM or Hirshfeld NACs and atomic multipoles) tha= t have well-defined complete basis set limits and can be used as a basis re= presentation to expand the Coulomb potential=20

People often fail to recognize the distinction be= tween these two cases, which have a day and night difference between them.<= br>

Sincerely,

Tom
<= div>=C2=A0

On Mon, Jun 29, 2020 at 8:43 AM Stefan Grimme grimme**thch.uni-bonn.de <<= a href=3D"mailto:owner-chemistry.-$-.ccl.net" target=3D"_blank">owner-chemistry= .-$-.ccl.net> wrote:

Sent to CCL by: "Stefan=C2=A0 Grimme" [grimme..thch.uni-bonn.de= ]
One more comment to the Mulliken charge discussion:
even methods without a well-defined basis set limit can be useful
as already mentioned by Marcel Swart. This holds
for the Mulliken atomic charge partitioning in compact MB/DZ basis sets
(even TZ is often reasonable). For example the DFTB and GFN-xTB tight-bindi= ng methods are fundamentally based on a Mulliken analysis of the density ma= trix and yield physically very reasonable electrostatic energies. In GFN2-x= TB this also works well up to quadrupole moments.
Its clear that the Mulliken scheme breaks down for AO basis sets containing=
diffuse components but I really would like to see a differentiated view on = the topic (and not as in a recent general statement of a reviewer something= like "I do not think Mulliken charges are trustworthy").
Best
Stefan Grimme



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