4. Robert S. Mulliken
BS, MIT, 1917
PhD, Chicago, 1923
Nobel Laureate, Chemistry, 1966
5. What is an atomic charge?
Easy to define for isolated atoms or ions.
For atoms in molecules, difficult to quantify rigorously.
The charges of atoms in molecules may be fractional, which
reflects how electrons are redistributed when a bond forms.
The pair of electrons which constitutes the bond may lie between two atomic centers
in such a position that there is no electric polarization, or it may be shifted toward
one or the other atom in order to give to that atom a negative, and consequently to
the other atom a positive charge. But we can no longer speak of any atom as having an
integral number of units of charge, except in the case where one atom takes exclusive
possession of the bonding pair, and forms an ion.
- Lewis, 1923
Lewis, G. N. J. Am. Chem. Soc. 54 (1932), p. 83; quoted in Jensen, W. B. J. Chem. Educ. 86 (2009), 545.
6. What is an atomic charge?
Easy to define for isolated atoms or ions.
For atoms in molecules, difficult to quantify rigorously.
The charges of atoms in molecules may be fractional, which
reflects how electrons are redistributed when a bond forms.
Electrons are indistinguishable; this poses a problem in
apportioning the charge density (or density matrix).
What observable operator (if any) is an atomic charge?
A
? A
B B
7. Charge polarization and chemistry
- 367
Charge polarization arises from
electronegativity differences
Avogadro, 1809; Berzelius, 1819; Abegg, 1899;
Lewis, 1916; Pauling, 1932;...
Related to:
chemical reactivity (Davy, 1806)
oxidation state (Avogadro, 1809)
heats of reaction (Berzelius, 1819)
ionization potentials (Stark, 1913)
electrophilicity (Lewis, 1916)
metal workfunctions (Gordy and Orville
Thomas, 1955)
electrode potentials (Kaputinskii, 1960)
“Gradation of electroaffinity in the Periodic Table”
Abegg, R. Z. Anorg. Chem. 39 (1904), 330.
Historical review:
Jensen, W. B. J. Chem. Ed. 37 (1996), 10; 80 (2003), 279.
8. 1.08 IO0
0 1.49 2.48 later, is also included: 1.44 v. e. for N:N. Under each bond
described2.12
energy is1.98
2.15 given the value for a normal covalent bond, calculated from
1932: Pauling’s electronegativity
Observed bond energy F
additivity, and below that the difference A. It is seen that A is positive in
0.33 ' 0.14
twenty of -37 twenty-one cases. The exception, C:I, may be due to
-58 the
experimental error, and be not real.
2.80 3.82
Normal covalent bond energy Regularities observed in the A-values suggest that it is possible to make
2.63
pt., 1932 THE NATURE OF THE CHEMICAL BOND. IV 3573 a rough assignment of the atoms ta positions along a scale representing
A 1.19
AVl electronegativity, with the assumption that A is a function of the
degree of 1.09
linear separation of the loci of the two atoms on the scale, in the way that
C1 2.468 2.231 2.143
genes are mapped2.215 chromosome from crossover data. It is to be ob-
in a 2.001
-
A / are
''
served that the values of 0.142 approximately additive (these values are
0.018
given directly below those of A ) . For example, the sum of A"9 for H:A
. 13 .38
and A:F isBr 2.05, 2.06, 1.91, and 2.06 for A = C, N , 0, and C1, respectively.
1.962 1.801
We accordingly write 1.748
-: B
AA ( X A - XB)* (1)
0.053
with A measured in volt-electrons, and construct the scale shown in Figs. 3
.23
reliability of the method is indicated by Fig. 3,
and 4 on this basis.I The 1.536
in which four distinct procedures are illustrated. The coordinates of the
I11
TABLE
elements on this scale are given in Table 111.
P
Cl, showing that the bond in COdRDINATES OF ELEMENTS THE ELECTRONEGATIVITY
ON SCALE
4 l 3 W. Swietoslawski, "Therrnochemie," Akademische Verlagsgesellschaft m. b. H.,
orine fluoride is more ionic in H 0.00 Br 0.75
Leipzig, 1928.
aracter than that in hydrogen ei P * 10 c1 .94
’
13 I ,40 N .95
oride. OLO, S .43 0 1.40
t is perhaps desirable to point 0 -Cy ) ,5B F 2.00
h
t that the bond type has no 2 4
These coordinates, introduced in Equation 1, lead to values of A which
ect connection with ease of
agree with those of Table I1 with an average error of 0.09 v. e., excluding
ctrolytic dissociation in aque- .3 5
H:F. The calculated A for H:F is 4.00 v. e., 1.23v. e. higher than observed;
solution. Thus the nearly this indicates that Equation 1 is inaccurate when XA - XB becomes as
O--O
mal covalent molecule H I ion- large as 2. 2
s completely in water, whereas
largely ionic H F is only par- 6
O0
-
ly ionized. * 2.
. O
.
d
3
Fs*
“- .2u
Bond Energies for Light Atoms
d Halogens.-In the calcula- -<
Pauling developed his concept of electronegativity as
1.
an empirical additive correction to reaction enthalpies.
n of bond energies from heats
formation and heats of com-
0
.1
stion the following energies of F C1 Br I
Pauling, L. J. Am. Chem. Soc. 54 (1932), 3570.
9. 1934: Mulliken’s electronegativity
*
+
*in energies
+A − B |H| A+ B − ≈ A − B |H| A− B +
Downloaded 13 Sep 2005 to 130.126.230.252. Redistribution subject to A
The first serious attempt to justify an electronegativity
scale using quantum mechanical arguments.
Mulliken. R. S. J. Chem. Phys. 2 (1934), 782.
aded 13 Sep 2005 to 130.126.230.252. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright
11. 1935: Mulliken’s charges for diatomics
i.e. in modern terms, (McWeeny, 1951)
2
2 1
a ab 1 S a 2 ab 1 S
tr = tr 1
ab b2 S 1 2 ab 0 S 1
density overlap 1
0 2 ab 1 S
matrix matrix + tr 1 2
2 ab b S 1
N = NA + NB
McWeeny, R. J. Chem. Phys. 19 (1951) 1614.
The general (polyatomic) case
was worked out by Mulliken in
1949.
Mulliken, R. S. J. Chim. Phys. 46 (1949) 675.
canonical reference:
Mulliken, R. S. J. Chem. Phys. 23 (1955) 1833; 1841; 2338; 2343.
12. A problem with Mulliken charges
Consider the wavefunction of minimal basis HeH2+
ψ = cos γ φHe + sin γ φH
Then
1 1 + cos 2γ
NHe =
2 1 + S sin 2γ
localized
more antibonding more bonding
Attributed to K. Ruedenberg in
Mulliken, R.S; Ermler, W. C. Diatomic molecules: results of ab initio calculations.
Academic Press, NY, 1977, pp. 33-38.
13. A problem with Mulliken charges
Consider the wavefunction of minimal basis HeH2+
ψ = cos γ φHe + sin γ φH
Then
1 1 + cos 2γ
NHe =
2 1 + S sin 2γ
physically
unreasonable
region
localized
more antibonding more bonding
Attributed to K. Ruedenberg in
Mulliken, R.S; Ermler, W. C. Diatomic molecules: results of ab initio calculations.
Academic Press, NY, 1977, pp. 33-38.
14. Other charge definitions
• Other population analysis schemes: Coulson, Löwdin, natural
bond order (NBO/NPA),...
• Density fitting: distributed multipole analysis (DMA),...
• Density partitioning: Hirshfeld, Bader, Voronoi-deformed...
• Electrostatic potential (ESP) fitting: CHELP, RESP,...
• Experimentally derived charges: Szegeti, ESCA, Born,...
• Empirical charge models: Gasteiger-Marsili, QEq, fluc-q, ...
Some useful reviews:
Bachrach, S. M. Rev. Comp. Chem. 5 (1994), 171
Meister, J.; Schwarz, W. H. E. J. Phys. Chem. 98 (1994), 8245
Francl, M. M.; Chirlian, L. E. Rev. Comp. Chem. 14 (2000), 1 - ESP charges
Rick. S.W.; Stuart, S. J. Rev. Comp. Chem., 18 (2002), 89 - empirical models
15. Can we do better?
Want a definition that has a clear physical basis
Other desiderata:
Should “give stable and rational results”
Does not require additional fragment calculation
Works well with constrained DFT
Based purely on the density, so that the charges are true
density functionals
17. ρ(RA , RB ) ρA (RA ) + ρB (RB )
A A
B B
v(RA , RB ) → v (RA , RB ) v → v
ρ(RA , RB ) ρA (RA ) + ρB (RB )
A A
A’ A’
B B
18. ρ(RA , RB ) ρA (RA ) + ρB (RB )
A A
B B
v(RA , RB ) → v (RA , RB ) v → v
ρ(RA , RB ) ρA (RA ) + ρB (RB )
A A
A’ A’
B B
Rigid response to perturbation
19. A density matrix ρ (r, r ) = Pij φi (r) φj (r )
in general, has the derivative ij
A A
ρ (r, r ) = Pij φi (r) φj (r )
ij
A
+ Pij φi (r) φj (r )
ij
+ Pij φi (r) φA (r )
j
ij
Introducing the projection matrix ΠA for the basis functions on
atom (fragment) A, can rewrite the derivative in block matrix form
†
A φ (r) PA ΠA P φ (r )
ρ (r, r ) =
φx (r) PΠA 0 φx (r )
We want to find the nearest density matrix that has the form
†
A φ (r) 0 PA ˜ φ (r )
ρ (r, r ) =
˜ ˜A 0
φx (r) P φx (r )
20. We can find such a density that minimizes
2
˜
f PA = ρA (r, r ) − ρA (r, r ) drdr
˜
†
S Sx PA ˜
ΠA P − PA
= tr x xx ˜
−S S PΠA − PA 0
S S x
P A ˜A
ΠA P − P
× ˜
−Sx Sxx PΠA − PA 0
˜
:= tr SZ PA† SZ PA ˜
†
where S = φφ
x x†
S = φφ
xx x x†
S = φ φ
21. The density matrix A that minimizes
†
f (A) = tr SZ A SZ (A)
df † † † †
satisfies = −2 U SZ SV + V SZ SU
dA
I
U=
0
where are projection matrices
0
V=
I
Recover the population from the fragment density NA = tr AS
and finally calculate the charge as qA = ZA − NA
22. The density matrix A that minimizes
†
f (A) = tr SZ A SZ (A)
df † † † †
satisfies = −2 U SZ SV + V SZ SU
dA
I
U=
0
where are projection matrices
0
V=
I
Recover the population from the fragment density NA = tr AS
and finally calculate the charge as qA = ZA − NA
If we neglect Sx and Sxx, then this reduces to Mulliken charges!
A A
f (A) = tr SP SP + tr S (ΠA P − A) S (ΠA P − A)
which is clearly minimized by
A = ΠA P
This yields Mulliken populations
NA = tr AS = tr ΠA PS
30. Reviewing our design criteria
Want a definition that has a clear physical basis
Other desiderata:
Should “give stable and rational results”
Does not require additional fragment calculation
Works well with constrained DFT
Based purely on the density, so that the charges are true
density functionals
31. Reviewing our design criteria
Want a definition that has a clear physical basis ☺
Other desiderata:
Should “give stable and rational results” ■
Does not require additional fragment calculation ☺
Works well with constrained DFT ?
Based purely on the density, so that the charges are true ■
density functionals
Possible to write down such a variant; haven’t done it
32. Reviewing our design criteria
Want a definition that has a clear physical basis ☺
Other desiderata:
Should “give stable and rational results” ■
Does not require additional fragment calculation ☺
Works well with constrained DFT ?
Based purely on the density, so that the charges are true ■
density functionals
Possible to write down such a variant; haven’t done it
Disadvantages
More costly: need density and overlap derivatives
Don’t know analytic formula, must solve numerically
Notes de l'éditeur
This story begins mostly with one person, who has ties to MIT.
Mulliken was born in Newburyport, MA. His father was a professor of organic chemistry at MIT. Mulliken is perhaps most famous for his population analysis scheme, which is perhaps the most widely used method of deriving charges from quantum chemical calculations today.
There is is a long and fascinating history to the problem of defining atomic charges, some of which predates quantum mechanics. The possibility that atoms in molecules, in contrast with isolated atoms/ions, may have fractional charges can be traced back to Lewis. The first quantitative use of partial charges that I am aware of is in Bernal and Fowler’s 1933 paper describing their water model.
It is precisely in bonding configurations that the spatial distribution of electrons, and hence the values of atomic charges, are the most interesting; these are also precisely the cases where the definition of atomic charges becomes the most ambiguous and problematic.
In a sense, the easiest way to define atomic charges is by their effect on the properties of atoms and molecules. As a quantitative measure of charge polarization (imbalance), the atomic charges are vitally related to many chemical concepts, most notably that of electronegativity. Shown here, for fun, is a picture from Abegg’s 1904 paper showing the periodic trends in electronegativity.
It seems impossible to discuss electronegativity in any detail without paying homage to Linus Pauling. Pauling’s scale was initially proposed as an empirical quantity designed to explain why the observation of bond additivity broke down, i.e. why E(H-H) + E(F-F) =/= 2 E(H-F). Although Pauling was the first chemist to popularize the use of valence bond theory, and hence quantum mechanics, it is more instructive to study the contributions of Robert Mulliken to the theory of electronegativity and charge analysis.
In a follow-up paper (JCP 3 (1935), 573), Mulliken discusses in detail the errors introduced by the LCAO approximation in assuming that the orbitals of atoms in molecules are the same as the orbitals, and relates the difference in orbitals to charge screening. Mulliken also develops the theory for calculating electronegativities from spectroscopic data and quantum mechanical grounds, with particular emphasis that the states (electronic configurations) that ought to go into the scale are not isolated atom states, but those of atoms in molecules.
Mulliken’s early work on defining an electronegativity scale played a pivotal role in his later development of his famous population analysis scheme. In this 1935 paper, the fundamental idea behind Mulliken population analysis had already been described.
The general case appears to have been independently rediscovered by Chirgwin and Coulson in 1950, and McWeeny in 1951.
In practice, this problem does not crop up often, which could be because most population analyses are performed on the ground state. Sometimes the population of individual orbitals can become negative, but this is usually explained away as an artifact of the analysis scheme that does not affect the interpretation of the gross atomic charge.
The new density will not in general transform this simply. Nevertheless we can find the nearest such density that will behave like this. We expect that in the complete basis set limit, the two densities will coincide almost everywhere (except possibly at points of degeneracies, e.g. conical intersections).
Here the matrices in blackboard bold are the block supermatrices and dagger is transposition.
To understand this formula, first note that 2 S Z S is what would be obtained from the derivative d/dZ (up to a factor of 2). The application of the projection matrices extract the upper right and lower left quadrants, those being the only quadrants which contain A. The overall minus sign comes from the chain rule.
