Theor Chem Acc (2015) 134:117
1 3
As the IRC proceeds from reactants to product, molecular
radial density is seen to move from the π bond systems
of the ethene and butadiene molecules into the regions of
space between the ethene and butadiene carbon atoms to
form sigma bonds. In the transition state, there is molecular
radial density between these carbon atoms that is about half
of the molecular radial density found in the product of the
reaction. However, the ethene bond in the transition state
has maintained much of its double bond character in the
transition state as compared to the reactant complex, which
is consistent with an early transition state for the reaction
as expected through the Hammond postulate.
4 Conclusions
We have demonstrated that the molecular electron density
can be partitioned into atomic and bonding contributions
using radial density and electron density. The Becke weight
was employed to partition the molecular space. Although
the Becke weight works well in a number of cases, there
are several drawbacks. First, the reliance on fi xed Bragg–
Slater radii makes the Becke weight less versatile in dealing
with AIM that can have different charged states. Another
problem arises when there is a large difference between
the Bragg–Slater radii of the atoms. In the fi rst step of the
Becke weight calculation, if the ratio of the Bragg–Slater
radii of two atoms is larger than 2.4 (or smaller than 0.41),
the ratio will be capped to these extremes. For example,
the ratio of the Bragg–Slater radii for LiH and LiF is 4.15
(=2.74/0.66) and 2.91 (=2.74/0.94), respectively, and both
become capped to 2.4. Consequently, H and F are effectively treated as the same size when paired with Li. The
restriction on the range of the ratio is a problem of the
Becke weight because the properties of AIM and BIM can
change signifi cantly depending on the ratio of the Bragg–
Slater radii. This is illustrated in Tables S3–S5, where
changing the radius of Li from its Bragg Slater radii (2.74)
to r of Li
+ (1.15) changes its AIM and BIM properties.
This motivates the need to explore or develop alternative
partition functions to replace or modify the Becke weight.
The topology of ABIM is rich in features such as critical points, rings, and spheres, which can be identifi ed and
characterized using the gradient and diagonalized Hessian
of radial density, respectively. ABIM is able to quickly and
easily determine the properties of atoms and bonds, including shape, volume, dipole, and expectation values and link
them to chemically intuitive properties such as electron
density distortion and bond orders. Two of the most important features arising from the ABIM model is that the bonding region is explicitly defi ned and the number of electrons
in the bonding region can be calculated.
Acknowledgments We gratefully acknowledge the support of the
Natural Sciences and Engineering Council of Canada and the Atlantic Excellence Network (ACEnet) and Compute Canada for the computer time. We would also like to dedicate this paper to Marco Häser,
who was the fi rst to defi ne molecular radial density [ 21 ]. The authors
express their best wishes to Peter Surjan on the occasion of his 60th
birthday.
References
1. Warburton PL, Poirier RA, Nippard D (2011) J Phys Chem A
115:852
2. Pauling L (1960) The nature of the chemical bond, 3rd edn. Cornell University Press, Ithaca, NY
3. Bader RFW (1990) Atoms in molecules: a quantum theory. Clarendon, Oxford
4. Yáñez M, Stewart RF, Pople JA (1978) Acta Cryst A34:641
5. Hirshfeld FL (1977) Theor Chim Acta 44:129
6. Moffi tt W (1951) Proc R Soc Lond Ser A 210:245
7. Bader RFW (1985) Acc Chem Res 18:9
8. Bader RFW, Larouche A, Gatti C, Carroll MT, MacDougall PJ,
Wiberg KB (1987) J Chem Phys 87(2):1142
9. Bader RFW, Matta CF (2004) J Phys Chem A 108:8385
10. Bader RFW (2010) J Phys Chem A 114:7431
11. Stewart RF (1970) J Chem Phys 53(1):205
12. Politzer P, Parr RG (1976) J Chem Phys 64:4634
13. Boyd RJ (1977) J Chem Phys 66:356
14. Schmider H, Sagar RP, Smith VH (1991) J Chem Phys 94:8627
15. Becke AD (1988) J Chem Phys 88(4):2547
16. Poirier RA, Hollett JW, Warburton PL, MUNgauss, Memorial
University, Chemistry Department, St. John’s, NL A1B 3X7.
With contributions from Alrawashdeh A, Becker J-P, Besaw JE,
Bungay SD, Colonna F, El-Sherbiny A, Gosse T, Keefe D, Kelly
A, Nippard D, Pye CC, Reid D, Saputantri K, Shaw M, Staveley
M, Stueker O, Wang Y, Xidos J
17. Gill PMW, Johnson BG, Pople JA (1993) Chem Phys Lett
209:506
18. Simas AM, Sagar RP, Ku ACT, Smith VH (1988) Can J Chem
66:1923
19. Hollett JW, Kelly A, Poirier RA (2006) J Phys Chem A
110(51):13884
20. Blair SA, Thakkar AJ (2014) Chem Phys Lett 609:113
21. Häser M (1996) JACS 118:7311
71
Reprinted from the journal
1 3
As the IRC proceeds from reactants to product, molecular
radial density is seen to move from the π bond systems
of the ethene and butadiene molecules into the regions of
space between the ethene and butadiene carbon atoms to
form sigma bonds. In the transition state, there is molecular
radial density between these carbon atoms that is about half
of the molecular radial density found in the product of the
reaction. However, the ethene bond in the transition state
has maintained much of its double bond character in the
transition state as compared to the reactant complex, which
is consistent with an early transition state for the reaction
as expected through the Hammond postulate.
4 Conclusions
We have demonstrated that the molecular electron density
can be partitioned into atomic and bonding contributions
using radial density and electron density. The Becke weight
was employed to partition the molecular space. Although
the Becke weight works well in a number of cases, there
are several drawbacks. First, the reliance on fi xed Bragg–
Slater radii makes the Becke weight less versatile in dealing
with AIM that can have different charged states. Another
problem arises when there is a large difference between
the Bragg–Slater radii of the atoms. In the fi rst step of the
Becke weight calculation, if the ratio of the Bragg–Slater
radii of two atoms is larger than 2.4 (or smaller than 0.41),
the ratio will be capped to these extremes. For example,
the ratio of the Bragg–Slater radii for LiH and LiF is 4.15
(=2.74/0.66) and 2.91 (=2.74/0.94), respectively, and both
become capped to 2.4. Consequently, H and F are effectively treated as the same size when paired with Li. The
restriction on the range of the ratio is a problem of the
Becke weight because the properties of AIM and BIM can
change signifi cantly depending on the ratio of the Bragg–
Slater radii. This is illustrated in Tables S3–S5, where
changing the radius of Li from its Bragg Slater radii (2.74)
to r of Li
+ (1.15) changes its AIM and BIM properties.
This motivates the need to explore or develop alternative
partition functions to replace or modify the Becke weight.
The topology of ABIM is rich in features such as critical points, rings, and spheres, which can be identifi ed and
characterized using the gradient and diagonalized Hessian
of radial density, respectively. ABIM is able to quickly and
easily determine the properties of atoms and bonds, including shape, volume, dipole, and expectation values and link
them to chemically intuitive properties such as electron
density distortion and bond orders. Two of the most important features arising from the ABIM model is that the bonding region is explicitly defi ned and the number of electrons
in the bonding region can be calculated.
Acknowledgments We gratefully acknowledge the support of the
Natural Sciences and Engineering Council of Canada and the Atlantic Excellence Network (ACEnet) and Compute Canada for the computer time. We would also like to dedicate this paper to Marco Häser,
who was the fi rst to defi ne molecular radial density [ 21 ]. The authors
express their best wishes to Peter Surjan on the occasion of his 60th
birthday.
References
1. Warburton PL, Poirier RA, Nippard D (2011) J Phys Chem A
115:852
2. Pauling L (1960) The nature of the chemical bond, 3rd edn. Cornell University Press, Ithaca, NY
3. Bader RFW (1990) Atoms in molecules: a quantum theory. Clarendon, Oxford
4. Yáñez M, Stewart RF, Pople JA (1978) Acta Cryst A34:641
5. Hirshfeld FL (1977) Theor Chim Acta 44:129
6. Moffi tt W (1951) Proc R Soc Lond Ser A 210:245
7. Bader RFW (1985) Acc Chem Res 18:9
8. Bader RFW, Larouche A, Gatti C, Carroll MT, MacDougall PJ,
Wiberg KB (1987) J Chem Phys 87(2):1142
9. Bader RFW, Matta CF (2004) J Phys Chem A 108:8385
10. Bader RFW (2010) J Phys Chem A 114:7431
11. Stewart RF (1970) J Chem Phys 53(1):205
12. Politzer P, Parr RG (1976) J Chem Phys 64:4634
13. Boyd RJ (1977) J Chem Phys 66:356
14. Schmider H, Sagar RP, Smith VH (1991) J Chem Phys 94:8627
15. Becke AD (1988) J Chem Phys 88(4):2547
16. Poirier RA, Hollett JW, Warburton PL, MUNgauss, Memorial
University, Chemistry Department, St. John’s, NL A1B 3X7.
With contributions from Alrawashdeh A, Becker J-P, Besaw JE,
Bungay SD, Colonna F, El-Sherbiny A, Gosse T, Keefe D, Kelly
A, Nippard D, Pye CC, Reid D, Saputantri K, Shaw M, Staveley
M, Stueker O, Wang Y, Xidos J
17. Gill PMW, Johnson BG, Pople JA (1993) Chem Phys Lett
209:506
18. Simas AM, Sagar RP, Ku ACT, Smith VH (1988) Can J Chem
66:1923
19. Hollett JW, Kelly A, Poirier RA (2006) J Phys Chem A
110(51):13884
20. Blair SA, Thakkar AJ (2014) Chem Phys Lett 609:113
21. Häser M (1996) JACS 118:7311
71
Reprinted from the journal