Quantum mechanical definitions of atomic charge

Do we rea!y need another one?

Jiahao ChenGroup meeting

2011.03.28

Thanks to:

BS, MIT, 1917PhD, Chicago, 1923

Nobel Laureate, Chemistry, 1966

BS, MIT, 1917PhD, Chicago, 1923

Nobel Laureate, Chemistry, 1966

Robert S. Mulliken

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.

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

BB

?

Charge polarization and chemistry

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.

- 367

Historical review:Jensen, W. B. J. Chem. Ed. 37 (1996), 10; 80 (2003), 279.

1932: Pauling’s electronegativity

Pauling, L. J. Am. Chem. Soc. 54 (1932), 3570.

Sept., 1932 THE NATURE OF THE CHEMICAL BOND. IV 3573

HCl, showing that the bond in chlorine fluoride is more ionic in character than that in hydrogen

4

0 - y )

P ei 1 3

It is perhaps desirable to point ’ chloride. OLO,

out that the bond type has no 2

O--O

O-0

h direct connection with ease of electrolytic dissociation in aque- ous solution. Thus the nearly normal covalent molecule HI ion- izes completely in water, whereas the largely ionic H F is only par- 6

Bond Energies for Light Atoms Fs* tially ionized. *. 2 . .O\d

and Halogens.-In the calcula- 1. “--<\ 0 tion of bond energies from heats

of formation and heats of com- bustion the following energies of F C1 Br I

4

.3 5

2

3

. 2 u

.1

3576 LINUS PAULING VOl. 54

C:N. v e. C;:pU'. v . e .

CHrNHn 2.82 (CHdzNH 2.92 C&WHa 2.87 (CsHdnNH 2 .95 CsHiNHz 2.80 (CH8)aN 2.94

Neglecting the possibility that the bond energy in primary amines be slightly less than in secondary and tertiary amines, we take the average value 2.88 v, e. for C:N.

Cyanogen, acetonitrile and hydrogen cyanide (using Thomsen's value for heats of combustion of the first two) lead to 8.86, 8.98, and 8.74 v. e., respectively, for C:::N. The average of these, 8.86 v. e., is very nearly the mean of C:::C = 8.68 v. e. and N:: :N = 9.10 v. e. (in N2).

Heats of combustion of fluorine-substituted hydrocarbons give C:F'= 5.40 v. e. as the average of eight values, maximum deviation 0.35 v. e. Twelve chlorine compounds give an average of 3.41 v. e. for C:Cl. The same value is obtained from the heat of formation of CCI,. Three bromine compounds (heats of combustion from Thomsen) give C:Br = 2.83 v. e., and two iodine compounds, CHsI and C2H61 (Thomsen), give C:I = 2.2 v. e. The last two values are uncertain. Other data, obtained by Berthelot and by Roth and Macheleldt and quoted by Swietoslawski,13 give the value 2.45 v. e. for C:I.

Bond Energies and the Relative Electronegativity of Atoms.-In Table I1 there are collected the energies of single bonds obtained in the preceding sections. One additional value, obtained by a method to be described later, is also included: 1.44 v. e. for N:N. Under each bond energy is given the value for a normal covalent bond, calculated from additivity, and below that the difference A. It is seen that A is positive in twenty of the twenty-one cases. The exception, C:I, may be due to experimental error, and be not real.

Regularities observed in the A-values suggest that it is possible to make a rough assignment of the atoms ta positions along a scale representing degree of electronegativity, with the assumption that A is a function of the linear separation of the loci of the two atoms on the scale, in the way that genes are mapped in a chromosome from crossover data. It is to be ob- served that the values of A'/' are approximately additive (these values are given directly below those of A ) . For example, the sum of A"9 for H:A and A:F is 2.05, 2.06, 1.91, and 2.06 for A = C, N, 0, and C1, respectively. We accordingly write

with A measured in volt-electrons, and construct the scale shown in Figs. 3 and 4 on this basis. The reliability of the method is indicated by Fig. 3, in which four distinct procedures are illustrated. The coordinates of the elements on this scale are given in Table 111.

Leipzig, 1928.

A A : B (XA - XB)* (1)

l3 W. Swietoslawski, "Therrnochemie," Akademische Verlagsgesellschaft m. b. H.,

Sept., 1932 THE NATURE OF THE CHEMICAL BOND. IV

TABLE I1 H C N

H 4.44 4.34 3.89 4.02 2.94

0.32 0.95 .57 .98

C 3.60 2.88 2.52

_ _ ~

0 4.75 2.99

1.76 1.33 3.55 2.55

-

F

6.39 3.62

2.77 1.67 5.40 3.20

-

Cl

4.38 3.45

0.93 .97

3.41 3.03

__

Rr 3.74 3.20

0 .54 .74

2.83 2.78

~

0.36 .60

N 1.44

0

Observed bond energy Normal covalent bond energy

A A V l

1.00 1 .00

1.49

F

2.20 1.48 3.29 2.12

0.38 * 62

1.95 1.95

1.17 1.08 2.48 2.15

0.00 IO0

2.12 1.98

0.33 ' 0.14 -58 -37

2.80 3.82 2.63

1.19 1.09

C1 2.468

Br

TABLE I11

0.05 .22

2.231 2.215

0.018 . 13

1.962

I

3577

I 3.07 2.99

0.08 .28

2.45 2.57

-0.12

_.

-

2.143 2.001

0.142 .38

1.801 1.748

0.053 .23

1.536

-

-

COdRDINATES OF ELEMENTS ON THE ELECTRONEGATIVITY SCALE

H 0.00 Br 0.75 P * 10 c1 .94 I ,40 N .95 S .43 0 1.40 C , 5 B F 2.00

These coordinates, introduced in Equation 1, lead to values of A which agree with those of Table I1 with an average error of 0.09 v. e., excluding H:F. The calculated A for H:F is 4.00 v. e., 1.23 v. e. higher than observed; this indicates that Equation 1 is inaccurate when XA - XB becomes as large as 2.

Pauling developed his concept of electronegativity as an empirical additive correction to reaction enthalpies.

1934: Mulliken’s electronegativity

Mulliken. R. S. J. Chem. Phys. 2 (1934), 782.

The first serious attempt to justify an electronegativity scale using quantum mechanical arguments.

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*in energies+�A−B |H|A+

B−� ≈

�A−B |H|A−

B+�

*

+

1935: Mulliken’s charges for diatomics

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Mulliken. R. S. J. Chem. Phys. 3 (1935), 573.

1935: Mulliken’s charges for diatomics

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i.e. in modern terms, (McWeeny, 1951)

densitymatrix

overlapmatrix

tr

�a2 abab b2

��1 SS 1

�= tr

�a2 1

2ab12ab 0

��1 SS 1

�

+ tr

�0 1

2ab12ab b2

��1 SS 1

�

N = NA +NB

Mulliken, R. S. J. Chim. Phys. 46 (1949) 675.

canonical reference:Mulliken, R. S. J. Chem. Phys. 23 (1955) 1833; 1841; 2338; 2343.

The general (polyatomic) case was worked out by Mulliken in 1949.

McWeeny, R. J. Chem. Phys. 19 (1951) 1614.

A problem with Mulliken chargesConsider the wavefunction of minimal basis HeH2+

Then

ψ = cos γ φHe + sin γ φH

NHe =1

2

1 + cos 2γ

1 + S sin 2γ

localized

more bondingmore antibonding

Attributed to K. Ruedenberg inMulliken, R.S; Ermler, W. C. Diatomic molecules: results of ab initio calculations. Academic Press, NY, 1977, pp. 33-38.

A problem with Mulliken chargesConsider the wavefunction of minimal basis HeH2+

Then

ψ = cos γ φHe + sin γ φH

NHe =1

2

1 + cos 2γ

1 + S sin 2γ

localized

more bondingmore antibonding

Attributed to K. Ruedenberg inMulliken, R.S; Ermler, W. C. Diatomic molecules: results of ab initio calculations. Academic Press, NY, 1977, pp. 33-38.

physicallyunreasonable

region

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), 171Meister, J.; Schwarz, W. H. E. J. Phys. Chem. 98 (1994), 8245Francl, M. M.; Chirlian, L. E. Rev. Comp. Chem. 14 (2000), 1 - ESP chargesRick. S.W.; Stuart, S. J. Rev. Comp. Chem., 18 (2002), 89 - empirical models

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

A A

BB

ρA(RA) + ρB(RB)ρ(RA, RB)

A A

BB

ρA(RA) + ρB(RB)ρ(RA, RB)

v �→ v�v(RA, RB) �→ v�(R�A, RB)

ρ(R�A, RB) ρA(R

�A) + ρB(RB)

A A

BB

A’ A’

A A

BB

ρA(RA) + ρB(RB)ρ(RA, RB)

Rigid response to perturbation

v �→ v�v(RA, RB) �→ v�(R�A, RB)

ρ(R�A, RB) ρA(R

�A) + ρB(RB)

A A

BB

A’ A’

ρ (r, r�) =�

ij

Pijφi (r)φj (r�)

ρA (r, r�) =�

ij

PAijφi (r)φj (r

�)

+�

ij

PijφAi (r)φj (r

�)

+�

ij

Pijφi (r)φAj (r�)

A density matrix in general, has the derivative

Introducing the projection matrix for the basis functions on atom (fragment) A, can rewrite the derivative in block matrix form

ΠA

ρA (r, r�) =

�φ (r)φx (r)

�† �PA ΠAPPΠA 0

��φ (r�)φx (r�)

�

ρ̃A (r, r�) =

�φ (r)φx (r)

�† �0 P̃A

P̃A 0

��φ (r�)φx (r�)

�We want to find the nearest density matrix that has the form

We can find such a density that minimizesf�P̃A

�=

�� ��ρA (r, r�)− ρ̃A (r, r�)��2 drdr�

= tr

�S Sx

−Sx Sxx

��PA ΠAP− P̃A

PΠA − P̃A 0

�†

�

S Sx

−Sx Sxx

��PA ΠAP− P̃A

PΠA − P̃A 0

�

:= tr SZ�P̃A†

�SZ

�P̃A

�

S =

�φφ†

Sx =

�φφx†

Sxx =

�φxφx†

where

The density matrix A that minimizes

satisfies df

dA= −2

�U†SZ†SV+ V†SZ†SU

�f (A) = tr SZ

�A†� SZ (A)

U =

�I0

�

V =

�0I

�where are projection matrices

Recover the population from the fragment densityand finally calculate the charge as

NA = tr ASqA = ZA −NA

The density matrix A that minimizes

satisfies df

dA= −2

�U†SZ†SV+ V†SZ†SU

�f (A) = tr SZ

�A†� SZ (A)

U =

�I0

�

V =

�0I

�where are projection matrices

which is clearly minimized by

This yields Mulliken populations

If we neglect Sx and Sxx, then this reduces to Mulliken charges!f (A) = tr SPASPA + tr S (ΠAP−A)S (ΠAP−A)

A = ΠAP

NA = tr AS = tr ΠAPS

Recover the population from the fragment densityand finally calculate the charge as

NA = tr ASqA = ZA −NA

STO-3G

6-31G**

++**

++

3-21G

[HeH]+, R=0.7882, B3LYP

STO-3G

6-311G

++

**

++**

6-31G**

++**

++

3-21G

[HeH]+, R=0.7882, B3LYP

STO-3G

6-311G

++

**

++**

6-31G**

++**

++

3-21G

cc-pVDZ

aug-

d-aug-

aug- d-aug- aug- d-aug-

cc-pVTZ

cc-pVQZ

[HeH]+, R=0.7882, B3LYP

essentially converged

essentially converged

convergence unclear

STO-3G

6-311G

++

**

++**

6-31G**

++**

++

3-21G

cc-pVDZ

aug-

d-aug-

aug- d-aug- aug- d-aug-

cc-pVTZ

cc-pVQZ

[HeH]+, R=0.7882, B3LYP

Reviewing our design criteriaWant 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

Reviewing our design criteriaWant 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

■☺

Reviewing our design criteriaWant 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

DisadvantagesMore costly: need density and overlap derivativesDon’t know analytic formula, must solve numerically

■

☺

?

Possible to write down such a variant; haven’t done it

■☺