Hybridization displacement charges in molecules involving second and third row atoms: correlation with polarizability

Hybridization displacement charges in molecules involving second

and third row atoms: correlation with polarizability

A.K. Singh, Anil Kumar, P.C. Mishra*

Department of Physics, Banaras Hindu University, Varanasi 221 005, India

Received 12 January 2004; accepted 22 March 2004

Available online 4 August 2004

Abstract

Hybridization displacement charges (HDC) that consist of several point charges associated with each atom in a molecule and arise due to

hybridization of atomic orbitals belonging to the same and different shells were computed at the ab initio SCF/6-31G** level for molecules

involving second and third row elements and for some anions and cations. Bond-centered charges (BCC) arising due to overlapping atomic

orbitals were combined with HDC. It is found that the combination of HDC and BCC reproduces dipole moments and surface molecular

electrostatic potential (MEP) patterns obtained using MEP-derived charges quite satisfactorily. Some interesting and useful information

about molecular bonding, not available from other charge distributions, is revealed by HDC. The HDC point charges associated with H, O

and F atoms are spread much less around the corresponding atomic sites than those associated with the metal atoms Li, Be, Na, Mg and Al. In

this respect, the behavior of the atoms C, N, B and Si lies intermediate between those of the above two groups. A quantity called ‘absolute

atomic HDC moment’ that is a measure of the spread of HDC around atomic sites has been defined, and its nearly linear correlation with

atomic dipole polarizability has been shown.

q 2004 Elsevier B.V. All rights reserved.

Keywords: Point charges; Hybridization displacement charges; Molecular electrostatic potential; Molecular bonding; Polarizability

1. Introduction

The concept of point charge distribution has played an

immensely important role in the understanding about inter

molecular interactions particularly where electrostatic

interactions play an important role, e.g. hydrogen bonding

[1–3]. Molecular mechanical and molecular dynamics

simulation studies use potential functions with a term

based on atomic point charges each and the results obtained

from such calculations depend significantly on the point

charges used [4,5]. Point charges in molecules may be

obtained using methods belonging to three categories: (i)

experimental, (ii) those based on fitting of molecular

electrostatic potentials (MEP) and (iii) theoretical. The

methods belonging to category (i) are based on NMR, X-ray

diffraction, IR intensity and other experimental measure-

ments [6]. The approaches of category (ii) include CHelpG,

Merz–Singh–Kollman and other algorithms [7–9]. The

following three well-known population analysis schemes

belong to category (iii): (a) Mulliken’s approach [10], (b)

Lowdin’s approach [11], and (c) Natural population analysis

(NPA) [12]. A serious shortcoming of the charges obtained

by all these methods is that they treat point charges as

essentially located at the atomic sites. Thus these methods

cannot even explain the existence of lone pairs. The

methods of category (iii) preserve only the total electron

count and usually they even do not describe molecular

dipole moments satisfactorily and consistently.

It seems that a point charge distribution can be

considered satisfactory only if it reproduces: (i) the

contribution of each atom to the total molecular dipole

moment obtained using the continuous (e.g. SCF) electron

density distribution satisfactorily, and (ii) MEP features

near various atoms on the van der Waals surfaces of

molecules obtained using the continuous electron density

distribution. The Mulliken, Lowdin and NPA charges

usually do not satisfy the two criteria mentioned above

consistently and satisfactorily. It is obvious that point

charges cannot be used to obtain MEP minima that are

uniquely defined in terms of continuous electron density

distributions [13]. Usually only a qualitative agreement,

0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2004.03.045

Journal of Molecular Structure (Theochem) 682 (2004) 201–213

www.elsevier.com/locate/theochem

* Corresponding author. Fax: þ91-542-317468.

E-mail address: [email protected] (P.C. Mishra).

http://www.elsevier.com/locate/theochem

in the statistical sense, can be expected between surface

MEP features obtained using point charges and MEP

minima obtained using continuous electron density

distributions.

In our earlier work [1,2,14–22], we introduced hybrid-

ization displacement charges (HDC) in molecules and

studied their influence on other molecular electrostatic

properties using semiempirical as well as ab initio

wavefunctions. In the most recent work of this series [22],

we improved the methodology by combining HDC with

bond-centered charges (BCC) arising due to atomic orbital

overlaps. It is found that HDC describe individual atomic

contributions to molecular dipole moments and hence total

molecular dipole moments satisfactorily [14–22]. Thus the

charge asymmetry around atoms in molecules is described

by HDC in accordance with the individual atomic

contributions to dipole moments [14–22]. It is obvious

that this asymmetry cannot be described by the atomic site-

based point charges. Further, HDC describe surface MEP

features satisfactorily [14–22]. It was found that HDC

belonging to certain atoms, e.g. lithium and beryllium are

highly spread in comparison to those near certain other

atoms, e.g. hydrogen and oxygen. There are a few important

questions in this context that need to be answered: (i)

Whether the spread of HDC is related to any other intrinsic

atomic or molecular property? (ii) How the HDC distri-

bution associated with an atom is modified when it is

bonded with different other atoms? (iii) In which other

atoms, HDC are strongly spread like those in lithium and

beryllium and whether any generalization is possible in this

context? And (iv) only electrically neutral molecules were

studied earlier [22]. Therefore, we need to understand how

HDC vary in going from electrically neutral molecules to

the corresponding anions and cations. We have addressed

ourselves these questions in the present study.

2. Method of calculations

The total dipole moment of a molecule ðmtÞ has two com-

ponents: (i) ma due to net charges located at the atomic sites,

and (ii) mh due to electron densities shifted from the atomic

sites due to atomic orbital hybridization [23]. The Cartesian x;

y and z components ofmh arise due to mixing of the ns and mpi

atomic orbitals of an atom where n and m represent principal

quantum numbers and i ¼ x; y; z: Therefore

mhi ¼ ðKDiÞðQi=KÞ ði ¼ x; y; zÞ ð1aÞ

where

Di ¼ ðnslilmpiÞ ðn ¼ 1; 2; 3;… and m ¼ 2; 3;…Þ ð1bÞ

and

Qi ¼ 22Pns;mpi ð1cÞ

where P represents density matrix. In Eq. (1a), K is a

constant (parameter). While Qi has the dimension of charge,

Di is a distance along the x; y or z direction. The two factors

ðKDiÞ and ðQi=KÞ of Eq. (1a) can be varied by adjusting K:

The parameter K as such would be arbitrary, and it can be

obtained using additional suitable criteria, e.g. a satisfactory

reproduction of surface MEP patterns. We obtained K for

different atoms using the criterion of best possible overall

agreement between the surface MEP values at the van der

Waals surfaces of several molecules obtained using MEP-

derived CHelpG charges [7] and HDC. The value of HDC

represented by Q and its displacement R from the atom under

consideration can be obtained as follows. Let us write mh and

R in terms of their Cartesian components:

mh ¼ ðmh2x þ mh2

y þ mh2z Þ1=2 ð2Þ

and

R ¼ ðD2x þ D2

y þ D2z Þ

1=2 ð3Þ

As discussed further later, the components of displace-

ment Di ði ¼ x; y; zÞ from atoms depend only on Slater

exponents that are usually considered to be the same for the

s and pi ði ¼ x; y; zÞ atomic orbitals corresponding to the

same principal quantum number [24,25]. Then Dx ¼ Dy ¼

Dz: Now, for each combination of shells (e.g. when a 1s or

2s orbital hybridizes with the 2px; 2py; 2pz or 3px; 3py; 3pz

orbitals of an atom), HDC is given by

Q ¼ mh=R ¼ ½ðQ2

x þ Q2y þ Q2

z Þ=3�1=2 ð4Þ

The direction of displacement of Q from the atom under

consideration is given in spherical polar coordinates by the

following angles

w ¼ tan21ðmy=mxÞ ð5Þ

and

u ¼ cos21ðmz=mhÞ ð6Þ

The following three approximations were made for

calculating HDC:

Approximation I. For Slater type atomic orbitals (STO’s)

used as basis functions, the distances Dx; Dy and Dz

occurring in Eq. (3) would be given by

2az bs z

cp ða!Þ

p3pð2nðsÞ!Þ

pð2mðpÞ!Þdaþ1

ð7Þ

where a ¼ nðsÞ þ mðpÞ þ 1; b ¼ nðsÞ þ 1=2; c ¼ mðpÞþ 1=2;

and d ¼ zs þ zp; where nðsÞ and mðpÞ are the principal

quantum numbers of the s and p atomic orbitals respectively

while zs and zp are the corresponding Slater exponents. For

given nðsÞ and mðpÞ values, Di ði ¼ x; y; zÞ depend only on

the exponents of STO’s. Thus Di ði ¼ x; y; zÞ and R are

fixed for given Slater exponents. In the present work,

Slater exponents of innermost atomic shells ðn ¼ 1Þ were

taken from the literature [25] and were kept fixed at the

same values while those for the higher shells were adjusted

so as to obtain best possible agreement between molecular

dipole moments obtained using HDC and Hartree–Fock

A.K. Singh et al. / Journal of Molecular Structure (Theochem) 682 (2004) 201–213202

(HF) SCF wavefunctions. Adjustment of z-0s for STO’s was

made only for HDC calculations. Density matrices obtained

from SCF calculations performed using the 6-31G** basis

set, without making any change in the exponents of

gaussians, were used. The adjustment of valence and higher

shell z0s became necessary as we calculated the HDC

displacements, as an approximation, using STO’s and not

the actual gaussian basis set employed in the density matrix

calculation. Partly due to this approximation, the HDC

scheme does not fully reproduce the dipole moments

obtained using the corresponding, e.g. SCF, wavefunctions.

It is to be noted that when the negative electronic charge Q

(HDC) is displaced from an atom, an equivalent amount of

positive charge has to be placed at the atom so that the

overall charge state (e.g. neutral, anionic or cationic) of the

molecule under study remains unchanged.

Approximation II. A negative MEP region occurs near

the bond center in some cases, e.g. H2 that suggests the

occurrence of a negative charge near the bond center. The

Mulliken, Lowdin or NPA schemes do not yield charges

located at bond centers. However, these can be obtained

using HDC as shown earlier where orthogonalized atomic

orbitals are considered [19–21]. If an overlapping atomic

orbital basis is employed, as in the present study, a suitably

modified form of the Mulliken population analysis scheme

may be adopted. One may partition the total overlap charge

into several components that may be places at different

points on bonds and elsewhere around atoms. Alternatively,

a fraction r of the total overlap charge may be placed at a

bond center and the remaining (1 2 r) fraction of the same

may be partitioned into two equal parts that may be placed

at the two bonded atoms. This simple provision is

computationally convenient and, therefore, was adopted

here with r ¼ 0:4: In the Mulliken scheme, r ¼ 0: The

choice of r ¼ 0:4 corresponds to 40% of the total overlap

charge being placed at the bond center while 30% of the

same is placed at each of the two bonded atoms. The point

charges arising due to atomic orbital overlaps so obtained

were combined with those calculated employing the HDC

scheme. Thus we get the full distribution of point charges.

This approximation to partition the overlap charge into three

components each would have some effect on the calculated

dipole moments and MEP using point charges.

Approximation III. To calculate HDC from SCF density

matrices, only the mixing of ns and mpi atomic orbitals

was considered (n;m ¼ principal quantum numbers, and

i ¼ x; y; z). Mixing of d orbitals with s and p orbitals, for the

sake of computational convenience, was not considered in

the present calculations though the methodology applied

here can be extended to the same. The effect of d orbitals

may be quite significant for some of the third row atoms of

the periodic table.

The Windows versions of the GAUSSIAN 94 and GAUS-

SIAN 98 programs (G94W and G98W) were used to

calculate density matrices and molecular properties [26,

27]. Molecular structures were visualized using

the GaussView program [28]. HDC and surface MEP

were studied using locally developed softwares.

3. Results and discussion

The adjusted values of Slater exponents and parameter K

are presented in Table 1. van der Waal radii of the various

atoms [30] are also included in this table. The structures and

atomic numbering schemes in some of the molecules

studied here are shown in Fig. 1. The calculated dipole

moments of all the molecules studied here using the SCF/6-

31G** method are presented in Table 2. The lowest MEP

values on the van der Waal surfaces near the different atoms

of the molecules obtained using the HDC and CHelpG point

charges along with the corresponding MEP minima

obtained using SCF/6-31G** wavefunctions are presented

in Table 3. Locations and values of HDC corresponding to

mixing of different orbitals of the same or different shells in

some selected molecules are presented in Tables 4 and 5

while such details in some cases are shown in Figs. 2–4.

3.1. Dipole moments

The dipole moments obtained using the SCF/6-31G**

wavefunctions, CHelpG and HDC point charges (Table 2)

are usually in a satisfactory agreement. In some cases, the

SCF and HDC dipole moments are significantly different,

e.g. for MgHF, AlFO and AlH2F. This can be understood in

view of the approximations made in the calculations. In

some other cases, the dipole moments obtained by HDC are

in a significantly better agreement with the SCF/6-31G**

dipole moments than those obtained using the CHelpG point

charges. Thus in the cases of FSiN and SiF2O, the dipole

moments obtained using CHelpG charges are less than

those obtained using the SCF wave functions by 1.2

Table 1

Adjusted Slater exponents zn ðn ¼ 2; 3Þ where n is the principal quantum

number, values of parameter K and van der Waals radii of different atoms

Atom z1 z2 z3 z4 K van der Waalsa

radii (A)

H 1.14 0.7 012.0 – 0.30 1.20

Li 2.69 0.25 0.54b – 0.30 2.10

Be 3.68 0.4 01.15 – 0.30 1.45

B 4.68 0.70 3.40 – 0.30 1.45

C 5.67 1.50 6.00 – 0.30 1.77

N 6.67 2.40 16.0 – 0.30 1.60

O 7.66 10.0 25.0 – 0.30 1.50

F 8.65 10.0 30.0 – 0.30 1.47

Na 10.61 0.70 0.10 0.020 0.20 2.27

Mg 11.59 1.30 0.40 0.021 0.25 1.73

Al 12.56 1.60 0.74 0.020 1.00 1.24

Si 13.53 1.00 0.56 0.023 0.10 2.10

The values of z1 were taken from Ref. [24].a From Ref. [30].b Previous value 0.74 in Ref. [22].

A.K. Singh et al. / Journal of Molecular Structure (Theochem) 682 (2004) 201–213 203

and 0.8 Debye, respectively. Though such a situation is rare

as is evident from the other cases presented in Table 2, since

CHelpG charges are obtained by fitting them to SCF MEP

[7,29], yet it is noteworthy that in some cases it does

happen. The linear correlation coefficients between the SCF

and HDC dipole moments and that between the SCF and

CHelpG dipole moments considering all the molecules

(Table 2) are both 0.99 while the correlation coefficient

between the CHelpG and HDC dipole moments is 0.98. It

shows that, statistically speaking, the HDC point charges are

quite reliable to calculate molecular dipole moments.

3.2. MEP values

The minimum MEP values obtained using SCF wave

functions and the surface MEP values on the van der Waals

surfaces of the molecules obtained using CHelpG charges

and HDC presented in Table 3 reveal the following

information. In all homonuclear diatomic molecules

included in this table, SCF MEP minima as well as lowest

surface MEP values obtained using HDC are located near

the bond center. The MEP values obtained using CHelpG

charges in these cases are all zero. This shortcoming of

surface MEP patterns obtained using CHelpG charges

arises due to the point charges being located at the atomic

sites only. It may be noted that in some homonuclear

diatomic molecules, e.g. O2, N2 and F2 studied earlier,

lowest surface MEP values obtained using HDC as well as

SCF MEP minima are located near the corresponding

atoms [22]. The linear correlation coefficient between the

SCF MEP minima and the surface MEP values obtained

using HDC is also 0.91 while that between the former and

the surface MEP values obtained using CHelpG charges is

0.98. Thus trends contained in the minimum SCF MEP

values and the minimum surface MEP values obtained

using CHelpG charges and HDC near the various atomic

sites of the molecule are similar. Again, it is not surprising

that the surface MEP values obtained using CHelpG

charges are somewhat better in this respect, as these are

obtained by fitting to the MEP values. The minimum SCF

MEP values and surface MEP values are not expected to be

in agreement numerically or to vary exactly in the same

way in going from one molecule to another. The MEP

values around the pyruvate and glycine anions are much

larger than the MEP values around the corresponding

neutral molecules. This feature is revealed by the MEP

minima and surface MEP values obtained using HDC and

CHelpG point charges also.

Fig. 1. Structures of some molecules and atomic numbering scheme.


3.3. HDC values and locations

The amounts (in the unit of magnitude of lel; e ¼

electronic charge) and locations of HDC associated with the

different atoms in some selected molecules, arising due to

the mixing of orbitals of the same or different shells, are

presented in Table 4. The locations of HDC point charges in

glycine, glycine anion and glycine cation are presented in

Table 5. The HDC locations associated with the atoms of

Li2O, BLiO2 and BeHF are shown in Fig. 2, those for Na2

and Mg2 are presented in Fig. 3 while those for Al2 and Si2are presented in Fig. 4. In these figures, the HDC are marked

with arrows. The HDC are denoted in Figs. 2–4 using the

symbol Xij where X represents an atom, i represents its

atomic numbering in a molecule and j; which varies from 1

to 2 for hydrogen atoms, from 1 to 6 for atoms of the second

row of the periodic table and from 1 to 12 for atoms of the

third row of the periodic table, represents the different HDC

components. At the SCF/6-31G** level, for hydrogen

atoms, the values of j correspond to mixing of orbitals as

follows: 1:(1s, 3p), 2:(2s, 3p). For atoms of the second row,

the different values of j correspond to the mixing of orbitals

as follows: 1:(1s, 2p), 2:(2s, 2p), 3:(1s, 3p), 4:(2s, 3p),

5:(3s, 2p) and 6:(3s, 3p) while for atoms of the third row,

the different values of j correspond to mixing of orbitals

as follows: 1:(1s, 2p), 2:(2s, 2p), 3:(1s, 3p), 4:(2s, 3p),

Table 2

Dipole moments (Debyes) of molecules obtained using various methods or

charges calculated at the SCF/6-31G** level employing the corresponding

SCF optimized geometries

S. no. Molecule SCF (1) CHelpG (2) HDC (3)

1. BF2NH2 2.7 2.7 2.5

2. B3F3N3H3 0.0 0.0 0.0

3. BF2OH 2.1 2.0 2.0

4. B3N4H7 1.8 1.8 1.5

5. B2H2O3 1.2 1.2 1.5

6. B3H3O3 0.0 0.0 0.0

7. BLiO2 10.7 10.8 10.0

8. B2O3 1.2 1.3 1.0

9. Li2O 0.2 0.2 0.2

10. LiNO3 10.2 10.4 9.9

11 BeH(OH) 1.6 1.6 1.5

12. Be(OH)2 2.6 2.3 2.7

13. BeHF 0.7 0.7 0.5

14. Na2 0.0 0.0 0.0

15. Na2O 0.1 0.2 0.1

16. NaF 7.8 7.8 8.6

17. NaOH 6.2 6.1 7.1

18. Mg2 0.0 0.0 0.0

19. Mg(OH)2 0.0 0.0 0.0

20. MgHF 1.1 1.2 2.4

21. MgO 7.9 8.1 7.4

22. MgNH 4.1 4.7 4.1

23. AlHO 5.2 5.3 5.0

24. Al2 0.0 0.0 0.0

25. AlH2F 1.4 1.5 2.3

26. AlF3 0.0 0.0 0.0

27. AlFO 3.8 3.9 2.9

28. Si2 0.0 0.0 0.0

29. Si2H6 0.0 0.0 0.0

30. FSiN 3.9 2.7 3.8

31. SiF2O 2.6 1.8 2.7

32. SiNOH 7.1 7.1 6.9

33. SiO2 0.0 0.0 0.0

34. Pyruvic acid 4.2 4.2 4.4

35. Glycine 5.7 5.7 5.7

Table 3

MEP values (kcal/mol) around molecules using various methods or charges

calculated at the SCF/6-31G** level employing the corresponding SCF

optimized geometries

S. no. Molecule Near atom MEP

V scf (min)

(1)

V s(CHelpG)

(2)

V s(HDC)

(3)

1. BF2NH2 F 227.8 230.4 227.7

2. B3F3N3H3 F 222.8 228.2 226.1

3. BF2OH O 237.3 235.8 231.0

F 220.0 223.6 220.0

4. B3N4H7 N (NH2) 234.7 233.5 221.8

5. B2H2O3 O2, O3 233.0 249.4 244.3

O5 230.2 227.7 220.7

6. B3H3O3 O 234.6 235.8 223.8

7. BLiO2 O4 293.8 296.4 280.6

8. B2O3 O1, O5 244.8 251.8 241.7

9. Li2O O 2125.3 2126.9 2107.5

10. LiNO3 O4, O5 271.5 265.0 264.4

11. BeH(OH) O 249.2 242.3 229.5

12. Be(OH)2 O 257.8 248.3 238.3

13. BeHF F 234.2 237.3 226.3

14. Na2 –a 211.6 0.00 210.5

15. Na2O O 2151.6 2141.1 297.8

16. NaF F 2122.9 2109.5 2196.0

17. NaOH O 2113.4 296.5 2101.2

18. Mg2 –a 258.3 0.00 269.6

19. Mg(OH)2 O 255.5 250.2 237.6

20. MgHF F 264.7 259.3 264.3

21. MgO O 2102.2 2115.0 2104.1

22. MgNH N 282.4 266.2 278.5

23. AlHO O 293.8 297.4 277.7

24. Al2 –a 239.1 0.00 261.2

25. AlH2F F 241.9 241.8 257.1

26. AlF3 F 229.8 0.00 236.1

27. AlFO O 283.6 294.2 2142.0

28. Si2 –a 213.9 0.00 28.5

29. Si2H6 H 22.5 29.5 26.3

30. FSiN N 253.4 264.1 274.9

31. SiF2O O 259.2 271.9 277.3

32. SiNOH N 270.8 282.5 284.3

33. SiO2 O 244.6 254.6 248.6

34. Pyruvic acid O3 252.7 248.0 250.5

O8 (OH) 213.5 215.2 216.6

O9 254.2 250.7 248.8

35. Pyruvate anion O8, O9 2187.7 2176.0 2175.4

36. Glycine O(CyO) 266.8 260.0 264.4

O(OH) 248.6 251.5 251.4

37. Glycine anion N 2198.9 2189.6 2187.9

In column (1), minimum SCF MEP values ðV scfÞ and in columns (2) and

(3), MEP values on the van der Waals surfaces ðVsÞ are presented. For

atomic numbering, see Fig. 1.a Near middle of the bond.


Table 4

Amounts of HDC (in the unit of lel; e ¼ electronic charge) associated with different atoms in some molecules and their locations in terms of distances (A) and

directions of displacements from the corresponding atoms owing to the mixing of orbitals of the same or different shells

S. no. Molecule/atoma Mixing of orbitalsb Amount of HDC and other

point charges

Distance of HDC from

the atom

HDC location and

displacement direction

1. Li2O, Li 1 (1s, 2p) 20.123 0.018 Along bond, outside

2 (2s, 2p) 20.446 1.588 Towards O

3 (1s, 3p) 20.084 0.024 Towards O

4 (2s, 3p) 20.303 1.048 Along bond, outside


6 (3s, 3p) 20.201 1.029 Towards O

Total HDC 21.451

ASC 1.993

BCC 20.058

O 1 (1s, 2p) 20.000 0.114 Perpendicular to bond

2 (2s, 2p) 20.003 0.040 Close to the atom

3 (1s, 3p) 20.000 0.073 Perpendicular to bond. Also

perpendicular to component (1s, 2p)

4(2s, 3p) 20.001 0.021 Close to the atom



Total HDC 20.015

ASC 20.836

BCC 20.116

2. BeHF, Be 1 (1s, 2p) 20.019 0.019 Towards F

2 (2s, 2p) 20.005 0.992 Towards H

3 (1s, 3p) 20.036 0.049 Towards F

4 (2s, 3p) 20.171 0.427 Towards H

5 (3s, 2p) 20.379 0.427 Towards H

6 (3s, 3p) 20.100 0.483 Towards H

Total HDC 20.710

ASC 1.342

BCC 20.164

F 1 (1s, 2p) 20.029 0.105 Towards B


3 (1s, 3p) 20.022 0.060 Towards B




Total HDC 21.948

ASC 21.667

BCC 20.079

3. BLiO2, Li 1 (1s, 2p) 20.071 0.018 Along bond, outside

2 (2s, 2p) 20.174 1.588 Along bond, inside





Total HDC 20.421

ASC 1.192

BCC 20.421

B 1 (1s, 2p) 20.025 0.027 Towards O2

2 (2s, 2p) 20.125 0.567 Towards O4

3 (1s, 3p) 20.025 0.141 Towards O4

4 (2s, 3p) 20.136 0.085 Towards O2

5 (3s, 2p) 20.056 0.085 Towards O4

6 (3s, 3p) 20.067 0.163 Towards O2

Total HDC 20.434

ASC 1.248

BCC 20.234

4. Na2, Na 1 (1s, 2p) 20.002 0.002 Along bond, outside

2 (2s, 2p) 20.063 0.378 Towards other Na

3 (1s, 3p) 20.004 0.000 At the atomic site(continued on next page)


Table 4 (continued)


point charges

Distance of HDC from

the atom

HDC location and

displacement direction




7 (1s, 4p) 20.000 0.000 At the atomic site






Total HDC 20.578

ASC 0.655

BCC 20.076

5. Mg2, Mg 1 (1s, 2p) 20.005 0.006 Along bond, outside

2 (2s, 2p) 20.107 0.254 Towards other Mg











Total HDC 20.953

ASC 1.103

BCC 20.141

6. Al2, Al 1 (1s, 2p) 20.007 0.007 Along bond, outside

2 (2s, 2p) 20.035 0.827 Towards other Al











Total HDC 20.386

ASC 0.585

BCC 20.198

7. Si2, Si 1 (1s, 2p) 20.000 0.002 Along bond, outside












Total HDC 20.346

ASC 0.498

BCC 20.152

ASC stands for atomic site charge while BCC stands for bond-centered charge.a HDC locations in the molecules are shown in Figs. 2–4.b HDC are numbered from 1 to 6 for second row atoms and from 1 to 12 for third row atoms (Figs. 2–4).


Table 5

Amounts of HDC (in the unit of lel; e ¼ electronic charge) associated with some atoms in neutral glycine, its anion and cation, and their locations in terms of

distances (A) and directions of displacements from the corresponding atoms owing to the mixing of orbitals of the same or different shells


point charges

Distance of HDC

from the atom

HDC location and displacement direction

1. Glycine, O1 1(1s, 2p) 20.047 0.114 Inside the H1O1C2 angle

2(2s, 2p) 20.406 0.040 Opposite to H1O1C2 angle

3(1s, 3p) 20.035 0.073 Inside the H1O1C2 angle



6(3s, 3p) 20.503 0.022 close to the atom

Total HDC 22.023

ASC 1.546

BCC 20.185

N4 1(1s, 2p) 20.040 0.073 Inside the C3N4H4H40 pyramid

2(2s, 2p) 20.283 0.165 Above N, outside the C3N4H4H40 pyramid

3(1s, 3p) 20.043 0.111 Inside the C3N4H4H40 pyramid




Total HDC 22.317

ASC 1.742

BCC 20.183

2. Glycine, O1 anion 1(1s, 2p) 20.045 0.114 Along O1C2, outside the bond

2(2s, 2p) 20.326 0.040 Along O1C2, inside the bond

3(1s, 3p) 20.040 0.073 Along O1C2, outside the bond angle




Total HDC 22.100

ASC 1.441

BCC 20.100

N4 1(1s, 2p) 20.046 0.073 Inside the C3N4H4H40 pyramid






Total HDC 22.327

ASC 1.790

BCC 20.186

3. Glycine, O1 cation 1(1s, 2p) 20.045 0.114 Inside the H1O1C2 angle


3(1s, 3p) 20.040 0.073 Inside the H1O1C2 angle




Total HDC 22.053

ASC 1.549

BCC 20.119

N4 1(1s, 2p) 20.004 0.073 Inside the H4N4H40H400 pyramid


3(1s, 3p) 20.017 0.111 Inside the H4N4H40H400 pyramid



6(3s, 3p) 20.280 0.035 Along C3N4 bond

Total HDC 20.663

ASC 0.265

BCC 20.225

ASC stands for atomic site charge while BCC stands for bond-center charge.a Atomic numbering in glycine is given in Fig. 1.b HDC are numbered from 1 to 6 as in Table 4.


5:(3s, 2p) and 6:(3s, 3p), 7:(1s, 4p), 8:(2s, 4p), 9:(3s, 4p),

10:(4s, 2p), 11:(4s, 3p) and 12:(4s, 4p).

The locations of some of the HDC point charges

associated with the atoms of the second and third rows are

not resolved due to being coincident or nearly coincident

in Figs. 2–4. We find that all the HDC point charges

associated with H, O and F atoms (set I) are located much

closer to the corresponding atomic sites than some of the

HDC point charges associated with the metal atoms Li, Be,

Na, Mg and Al (set II) (Tables 4 and 5, Figs. 2–4). Thus in

the atoms of set II, the HDC point charges are much more

spread than those in the atoms of set I. The HDC distribution

Fig. 2. Location of HDC components in Li2O, BLiO2, and BeHF molecules.


around C atom was studied earlier [22]. The atoms C, N, B

and Si (set III), from the point of view of the spread of HDC

around the corresponding atoms (Tables 4 and 5), appear to

lie between the atoms of sets I and II. One HDC point charge

belonging to each Si atom is located far away from the

atomic site while the other HDC charges are located near the

same (Fig. 4). The role of the HDC point charge associated

with Si and located far away from the corresponding atomic

site was found to be important in reproducing the surface

MEP feature near Si in the molecules containing this atom

obtained using CHelpG charges, and, therefore, its location

appears to be significant. Thus from the point of view of

spread of HDC, Si is partly similar to the atoms of both the

sets I and II. It is interesting since Si in bulk is a

semiconductor while the elements of set II in bulk are

good conductors.

A comparison of HDC distributions in the corresponding

anions, cations and neutral molecules appears to be

interesting. Due to this reason, HDC distributions in neutral

glycine, glycine anion and glycine cation are presented in

Table 5. We find that the HDC values associated with the N4

site of glycine cation to which the additional proton is

attached are usually much smaller in magnitude than the

corresponding ones in the neutral and anionic forms of the

molecule. The HDC values associated with the O1 site of

glycine which is deprotonated on anion formation, are

modified only by small amounts in going from the neutral

form of the molecule to the anionic form. Similar results

were found for pyruvic acid, its cation and anion also

(Tables 2 and 3). These results show that a stronger

rehybridization of atomic orbitals takes place on an atom if

it is bonded to the additional proton leading to cation

formation than that on an atom which is deprotonated

leading to anion formation.

3.4. HDC and atomic polarizability

The values of different HDC components ðQijÞ and their

locations ðRijÞ with respect to the corresponding atoms i in

molecules represent the manner and extent to which

Fig. 3. Locations of HDC components in Na2 and Mg2 molecules.


electronic charges are polarized on molecular formation.

The quantity

mi ¼X

j

QijRij

is the HDC contribution of atom i to the total molecular

dipole moment while the quantity

Ai ¼X

j

lQijllRijl

can be defined as the ‘absolute atomic HDC moment’

corresponding to atom i: This quantity Ai would represent

the extent to which electronic charges belonging to atom i

are polarized on molecular formation that cannot be

represented by either lQijl or lRijl alone. Using the

calculated HDC magnitudes and displacements (Tables 4

and 5), we obtained Ai for the atoms of the following

homonuclear diatomic molecules: H2, Li2, Be2, B2, C2, N2,

O2, F2, Na2, Mg2, Al2 and Si2. We selected homonuclear

diatomic molecules so that effects of molecular dipole

moments arising due to electronegativity differences

between the bonded atoms are not involved. The polari-

zation of electronic charges as measured by Ai may be

considered to be related to atomic dipole polarizability. In

other words, it may be conjectured that the extents of

polarization of electronic charges of the atoms represented

by HDC, associated with formation of the molecules, would

be related to the corresponding atomic dipole polarizabil-

ities. Keeping this possibility in mind, we sought a

correlation between the Ai values and atomic dipole

polarizabilities ðaiÞ computed using the HF/6-31G** and

B3LYP/6-31G** methods. The HF/6-31G** method is

particularly suitable in this context as the HDC were

obtained using it. A comparison of the HF/6-31G** values

of ai with those obtained at the B3LYP/6-31G** level

provides information about effects of electron correlation on

this atomic property. The calculated values of Ai and ai are

presented in Table 6. We find that calculated Ai values for

the HDC point charges associated with the Li, Be, Na, Mg

Fig. 4. Locations of HDC components in Al2 and Si2 molecules.


and Al atoms are much more than those for the other atoms.

A plot between Ai and ai is given in Fig. 5. The linear

correlation coefficients between the Ai values and atomic

dipole polarizabilities calculated at the HF/6-31G** and

B3LYP/6-31G** levels are both 0.93. These results show

that the extent of polarization of electronic charges on

molecular formation follows polarizabilities of molecules.

The above discussion, particularly that of Sections 3.3 and

3.4, shows that HDC provide a great deal of useful and

interesting information about bonding and properties of

molecules that is not available from the atomic site-based

point charge distributions.

4. Conclusion

The present study leads us to the following important

conclusion. Using HDC, from the statistical point of view,

surface MEP and dipole moments can be obtained with an

accuracy comparable to that achieved using the CHelpG

point charges. HDC around the metal atoms Li, Be, Na, Mg

and Al are spread much more than those around H, O and F

while the behavior of C, N, B and Si, in this respect, lies

intermediate between those of the above two groups. The

spread of HDC around atoms follows their dipole

polarizabilities. HDC can be used profitably in place of

the conventional atomic site-based point charge distri-

butions in molecules getting a deeper insight about

molecular bonding and properties with a small additional

computational effort.

Acknowledgements

The authors are thankful to the University Grants

Commission (New Delhi) and the Council of Scientific

and Industrial Research (New Delhi) for financial support.

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Table 6

Absoloute atomic HDC moments ðAiÞ in homonuclear diatomic molecules

and corresponding calculated atomic dipole polarizabilities ðaiÞ arranged

according to decreasing calculated ai values

S. no. Atom ai (Bohr3) Ai (Debye)

HF/6-31G** B3LYP/6-31G**

1. Na 186.4 145.6 3.103

2. Li 153.1 130.9 2.695

3. Mg 79.4 71.6 1.945

4. Al 42.3 38.5 1.763

5. Be 37.8 35.9 1.691

6. Si 19.0 19.7 0.432

7. B 14.4 12.4 1.052

8. C 6.7 6.7 0.615

9. N 2.5 2.4 0.567

10. O 2.0 2.0 0.168a

11. F 1.6 1.6 0.192

12. H 0.3 0.3 0.005

a For 3O2, HDC were calculated in Ref. [22].

Fig. 5. Variation of absolute atomic HDC moment (Ai) with polarizability

ðaiÞ:


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Documents

Hybridization displacement charges in molecules involving second and third row atoms: correlation with polarizability