Molecular electrostatic properties using point charges: ab initio hybridization displacement charges combined with bond-centered charges

Molecular electrostatic properties using point charges:

ab initio hybridization displacement charges combined

with bond-centered charges

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

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

Received 3 June 2002; accepted 16 August 2002

Abstract

A new approximate scheme to compute hybridization displacement charges (HDC) in molecules at the ab initio level using

bond-centered charges arising due to overlapping atomic orbitals has been developed. It has been applied at the SCF/6-31G**

and SCF-CIS/6-31G** levels to the ground and excited states, respectively, of several polar and non-polar molecules. It is

found that, statistically speaking, HDC computed using the present approach reproduce SCF dipole moments and surface

molecular electrostatic potential (MEP) patterns quite satisfactorily. In some cases, HDC reproduce MEP features even

somewhat better than the MEP-fitted CHelpG charges, particularly in molecules where MEP values around atoms vary rapidly,

e.g. around the boron atom in certain molecules containing it, and homonuclear diatomic molecules. Some interesting

information about molecular bonding, not available from other charge distributions, is revealed by the present approach, e.g. in

lithium containing molecules. The present approach also appears to describe molecular electrostatic properties and molecular

bonding better than that where the Lowdin’s symmetry orthogonalized atomic orbitals are used.

q 2003 Elsevier Science B.V. All rights reserved.

Keywords: Charge distribution; Hybridization displacement charge; Dipole moment; Molecular electrostatic potential; Molecular bonding

1. Introduction

The concept of point charge distribution in

molecules has played an immensely important role

in the understanding of electrostatic interactions,

hydrogen bonding and chemical reactivity [1,2]. A

comprehensive review covering various aspects in

this regard is available [3]. The potential functions

used in molecular mechanical and molecular

dynamics simulation studies use a term based on

atomic point charges each, and accuracy of results of

such calculations depends significantly on the accu-

racy of point charges used [4,5]. Thus it is desirable

that we have a reliable method to calculate accurate

point charge distributions in molecules. The common

methods to calculate point charges may be classified

in three categories: (i) theoretical methods, (ii)

methods based on the fitting of molecular electrostatic

potentials (MEP) and (iii) experimental methods. In

category (i), there are the following three well-known

methods: (a) Mulliken’s population analysis scheme

0166-1280/03/$ - see front matter q 2003 Elsevier Science B.V. All rights reserved.

PII: S0 16 6 -1 28 0 (0 2) 00 6 42 -5

Journal of Molecular Structure (Theochem) 621 (2003) 261–278

www.elsevier.com/locate/theochem

* Corresponding author. Fax: þ91-542-368468.

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

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

[6], (b) Lowdin’s scheme [7], and (c) Natural

population analysis (NPA) scheme [8]. The

approaches of category (ii) include CHelpG, Merz–

Singh–Kollman and other algorithms [9–11]. The

various methods belonging to category (iii) are based

on NMR, X-ray diffraction, IR intensity and other

experimental measurements [12].

The point charges for a molecule obtained by the

various above-mentioned approaches are usually

widely different. The basic difficulty in this context

is that while a continuous electronic charge distri-

bution is precisely defined, no unique definition can

be given for point charges. In the Mulliken’s scheme,

the charges located in the atomic orbital overlap

region (bond charges) are divided into two equal parts

each, which are placed at the corresponding bonded

atoms [3,6]. Thus this scheme eliminates the charges

located in the bonding region preserving the concept

of point charges located at the atomic sites only. The

Lowdin scheme [3,7] uses symmetry-orthogonalized

orbitals due to which the charges located in the atomic

orbital overlap region vanish formally. The NPA

corresponds to occupancies of the so-called natural

orbitals [3,8], is based on a weighted symmetry

orthogonalization procedure and yields somewhat

improved charges over those obtained by the Lowdin

scheme, say in respect of charge transfer. A serious

drawback of all these methods is that they treat point

charges as essentially located at the atomic sites. Thus

these methods cannot explain lone pairs. The methods

of category (i) preserve only the total electron count

and usually do not preserve the molecular dipole or

higher moments. To be able to place charges correctly

at the atomic sites and away from atoms, one must use

a scheme that at least preserves atomic contributions

to dipole moment.

It appears that a point charge distribution can be

considered reliable only if it satisfies at least the

following two criteria: (a) reproduction of the

contribution of each atom to the total molecular

dipole moment obtained using the continuous (e.g.

SCF) electron density distribution satisfactorily. It

automatically implies a satisfactory reproduction of

the total molecular dipole moment. And, (b) repro-

duction of MEP features on the van der Waals

surfaces of molecules that correlate appropriately with

those obtained using the continuous electron density

distribution. Reproduction of higher electric moments

(e.g. quadrupole moments) by point charges would

also be desirable and may be added to the first

criterion, but it appears to be too stringent a condition

at present. Since point charges can be used to obtain

surface MEP features but not MEP minima that are

uniquely defined and are given by continuous electron

density distributions [13], one cannot compare MEP

values obtained by point charges with those obtained

using continuous electron density distributions quan-

titatively. Thus usually only a qualitative agreement,

particularly one in the statistical sense, can be

expected in this context. The Mulliken, Lowdin and

NPA charges do not satisfy the two criteria mentioned

above satisfactorily and consistently. The charges of

category (ii) like CHelpG are found to preserve total

molecular dipole moments, but they fail to reproduce

surface MEP patterns correctly in some cases e.g. the

molecules of homonuclear diatomic types.

We have defined the charges corresponding to the

hybridization dipole moments of atoms, termed them

as hybridization displacement charges (HDC), and

studied their usefulness in a number of cases earlier

using semiempirical as well as ab initio wavefunc-

tions [1,2,14–21]. The HDC describe charge asym-

metry around atoms in accordance with the individual

atomic dipole moments, preserve not only the total

electron count but also the individual atomic and total

molecular dipole moments and describe surface MEP

features satisfactorily [14–21]. However, in our

previous calculations [19–21], the Lowdin’s sym-

metry orthogonalized atomic orbital basis was

employed while use of an overlapping atomic orbital

basis appears to be more appropriate since it would

conform to the valuable overlap-related concept of

bonding. We would also need a suitable population

analysis scheme that yields point charges resulting

due to orbital overlaps which when combined with

HDC satisfy the two criteria for a reliable point charge

distribution mentioned above satisfactorily. Such a

method has been developed here and its usefulness

examined for molecules involving the first and second

row atoms of the periodic table.

2. Method of calculations

The total dipole moment of a molecule (m t) is a

vectorial sum of two components, one (m a) due to net

A.K. Singh et al. / Journal of Molecular Structure (Theochem) 621 (2003) 261–278262

charges usually considered to be located at the atomic

sites, and the other (m h) due to electron densities

shifted from the atomic sites to nearby points due to

the atomic orbital hybridization [22,23]. Let us first

consider the hybridization dipole moment (m h) and

obtain the charges associated with it. The Cartesian x,

y and z components of m h of an atom arising due to

the mixing of its ns and mpi atomic orbitals where n

and m represent principal quantum numbers and i ¼ x,

y, z, are given by

mhi ¼ ðKDiÞðQi=KÞ ði ¼ x; y; zÞ ð1aÞ

where

Di ¼ ðnslilmpiÞ

ðn ¼ 1; 2; 3 etc: and m ¼ 2; 3 etc:Þ

ð1bÞ

and

Qi ¼ 22Pns;mpið1cÞ

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

constant (parameter). Let K be absorbed in writing Di

and Qi according to Eq. (1a). While Di is a distance,

Qi has the dimension of charge and both these can be

varied by adjusting K, keeping Eq. (1a) satisfied. It

may be noted that K as such would be arbitrary in any

charge distribution scheme that preserves the contri-

bution of each atom to the total molecular dipole

moment, and it can be obtained only by employing

additional suitable criteria e.g. a satisfactory repro-

duction of surface MEP patterns around molecules.

Here K was fixed for different atoms using the

criterion of best possible overall agreement between

the MEP values at the van der Waals surfaces of

several molecules obtained using HDC and CHelpG

charges. The value of HDC Q and its displacement R

from the atom under consideration can be obtained as

follows. We can write m h 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 distances Di ði ¼ x; y; zÞ

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]. 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 obtained as

Q ¼ mh=R ¼ ½ðQ2

x þ Q2y þ Q2

z Þ=3�1=2 ð4Þ

The direction of displacement of HDC 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 two approximations were made for

calculating HDC:

Approximation I: If Slater type atomic orbitals

(STO’s) are used as basis functions, the distances Dx,

Dy and Dz in Eq. (1c) would be given by

2az bs z

cp ða!Þ

ffiffi

3p

ffiffiffiffiffiffiffiffiffiffi

ð2nðsÞ!Þp ffiffiffiffiffiffiffiffiffiffiffi

ð2mðpÞ!Þp

daþ1ð7Þ

Here, a ¼ nðsÞ þ mðpÞ þ 1; b ¼ nðsÞ þ 1=2; c ¼ mðpÞ

þ1=2; and d ¼ zs þ zp: In these equations, n(s) and

m(p) are the principal quantum numbers of the s and

p atomic orbitals and zs and zp are the corresponding

Slater exponents. For given n(s) and m(p) values,

Di ði ¼ x; y; zÞ depend, beside the constant multiplier

K, only on the Slater exponents of the orbitals.

Therefore, the values of Di and R are fixed for given

Slater exponents. The Slater exponents of the inner

atomic shells were taken from the Ref. [24] while

those for the valence and higher shells were adjusted

so as to obtain the best possible agreement between

the molecular dipole moments calculated using HDC

and those obtained using the appropriate wavefunc-

tions. The adjustment of valence and higher shell z’s

for STO’s was made only for the HDC calculations

using the density matrix obtained from the corre-

sponding calculation performed using the 6-31G**

basis set, without making any change in the

exponents of gaussians. The adjustment of valence

and higher shell z’s became necessary as we

calculated the HDC displacements, as an approxi-

mation, using STO’s and not the actual gaussian

basis set employed in the density matrix calculation.

A.K. Singh et al. / Journal of Molecular Structure (Theochem) 621 (2003) 261–278 263

Partly due to this approximation, the HDC scheme

does not fully reproduce the dipole moments

obtained using the corresponding wavefunctions.

Obviously, the exact approach to calculate HDC

would be one in which the same basis set and basis

functions or contractions are employed for this

purpose using which the density matrix was obtained.

It may be noted that when HDC as an electronic

charge component is displaced from an atom, an

equivalent amount of positive charge has to be

placed at the atom. This is necessary, since,

otherwise, the overall charge state of the molecule

under consideration (e.g. neutral, anionic or cationic)

would get altered.

Approximation II: Let us now consider the net

charges located at the atomic sites and the dipole

moment (m a) arising due to it. The occurrence of a

negative MEP region near the bond center in many

cases e.g. H2, can be satisfactorily explained only by

considering charges located near the bond center.

Charges on bond centers cannot be obtained using the

Mulliken, Lowdin or NPA schemes. However, these

can be obtained using HDC as shown earlier [19–21]. If

an overlapping atomic orbital basis is to be employed,

as in the present study, a suitably modified form of the

Mulliken population analysis scheme may be adopted.

The overlaps of atomic orbitals involved in a bond

under consideration would be extended continuously

around the bonded atoms. Therefore, one may consider

partitioning the total overlap charge into several

components and place them at different points on the

bond and elsewhere around the atoms. As a simple

alternative to it, a fraction r of the total overlap charge

may be placed at the 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 provision seemed to be qualitatively

acceptable and simple enough, and, therefore, with

r ¼ 0.4, was adopted here. In the Mulliken scheme,

r ¼ 0. The choice of r ¼ 0.4 implies that we leave 40%

of the total overlap charge at the bond center and place

30% of the same at each of the two bonded atoms. The

point charges so obtained were combined with those

calculated employing the HDC scheme to get the full

distribution of point charges. This approximation to

partition the overlap charge would cause some

differences between the dipole moments obtained

using the appropriate wavefunctions and those

obtained using the HDC-based scheme. Similarly, it

would also affect the calculated MEP values. This

approximation seems to work satisfactorily and

consistently for calculating both dipole moments and

MEP values for a variety of molecules involving the

atoms of the first and second row of the periodic table.

Vertical excited states of molecules were generated

using configuration interaction involving all singly

excited determinants (CIS) and the optimized ground

state geometries. Geometries of excited states were

not optimized. The study of excited states was not

aimed to investigate details of electronic spectra or

excited state properties of molecules. We only aimed

to compare the calculated ground and excited state

dipole moments and MEP values obtained by the

different methods or charges so that performance of

the HDC-based approach may be examined. The

Windows versions of the GAUSSIAN 94 and GAUSSIAN

98 programs (G94W and G98W) were used to

calculate the density matrices and molecular proper-

ties at SCF and SCF-CIS levels [25,26].

3. Results and discussion

Variations of the lowest MEP values in Li2, Be2, B2,

O2, and F2 with the fraction r of overlap charge placed

at the bond center are presented in Fig. 1. The lowest

MEP values are located near the bond centers in Li2,

Be2 and B2 while in O2, and F2, the lowest MEP values

Fig. 1. Variations of the lowest MEP values (kcal/mol) on the van

der Waals surfaces of Li2, Be2, B2, O2, and F2 molecules with the

fraction r of overlap charge placed at the bond center.


are located near the corresponding atoms, outside the

bond regions. In all these cases, MEP is not very

sensitive to r, and the two quantities are almost linearly

related. In O2, the range of variation of r does not

exceed r ¼ 0.4, since beyond this value of r, the lowest

surface MEP region in this case shifts to near the bond

center, a feature that would not agree with that shown

by the SCF ab initio MEP map. The adjusted values of

Slater exponents (zn) for the first and second row

elements of the periodic table, except He and Ne,

where n stands for the principal quantum number

(n ¼ 2,3) are presented in Table 1. The values of z1

taken from the Ref. [24] are also included in this table.

The value of 0.3 for the parameter K was found to be

satisfactory for all these atoms. It is to be noted that this

parameter had different values for different atoms in

our earlier work [19–21], where the Lowdin symmetry

orthogonalization was adopted. Variations of the

lowest MEP values in Li2, Be2, B2, O2 and F2 with K

are presented in Fig. 2. We find that in F2 and O2, MEP

is not very sensitive to K but in Li2, Be2 and B2, it is

quite sensitive to K, the lowest MEP magnitudes

decreasing with increasing K in all the cases.

3.1. Dipole moments

The calculated dipole moments of certain mol-

ecules in the ground and lowest singlet excited states

(S1) using SCF or SCF-CIS wavefunctions as well as

the corresponding CHelpG, HDC and Mulliken point

charges are presented in Table 2. Optimized geome-

tries at the SCF/6-31G** level were used for both

the ground and vertical excited state calculations. We

make the following observations from Table 2:

(1) The CHelpG charges are known to reproduce

molecular dipole moments accurately [20,27]. It

is usually true here also. However, in some cases,

the differences between the SCF and CHelpG

dipole moments are unusually large. Thus for BH

and LiCN, the SCF and CHelpG dipole moments

are different by 0.4 and 0.3 D, respectively.

MEP-fitted charges like CHelpG can be obtained

with less error if the MEP around an atom

changes slowly than when MEP changes rapidly.

This may be the reason for the appreciable

differences between the SCF and CHelpG dipole

moments mentioned above.

(2) The agreement between the SCF and HDC dipole

moments is quite satisfactory in many cases while

in some other cases, it is not so. On the average, the

SCF and HDC dipole moments are different by

about 11%. This sort of error is not unexpected in

view of the two approximations used in the HDC

calculations. As expected, the molecular dipole

moments obtained using Mulliken charges

usually differ from the SCF ones by large amounts,

the average difference between the two sets being

about 30%. The Mulliken charges fare particu-

larly badly for the molecules that involve the B, Li

and Be atoms. On the other hand, the HDC charges

yield quite satisfactory dipole moment even for

the molecules that involve these atoms.

Table 1

Adjusted Slater exponents zn (n ¼ 2,3) where n is the principal

quantum number. The values of z1 were taken from the literature

and were not adjusted

Atom z1 z2 z3

H 1.14 0.7 12.0

Li 2.69 0.25 0.74

Be 3.68 0.4 1.15

B 4.68 0.7 3.4

C 5.67 1.5 6.0

N 6.67 2.4 16.0

O 7.66 10.0 25.0

F 8.65 10.0 30.0

Ref. [24].

Fig. 2. Variations of the lowest MEP values (kcal/mol) on the van

der Waals surfaces of Li2, Be2, B2, O2, and F2 molecules with the

parameter K.


(3) The changes in dipole moments in going from the

ground to the lowest singlet excited states of six

molecules obtained by SCF-CIS calculations are

qualitatively satisfactorily reproduced by the

CHelpG, HDC and Mulliken charges also.

(4) The linear correlation coefficients between the

dipole moments obtained by the different

methods or charges are presented as a footnote

to Table 2. The correlation coefficient between

the SCF and CHelpG dipole moments is 1.00

while those between the SCF and HDC dipole

moments and between the CHelpG and HDC

dipole moments are 0.98 each. The correlation

coefficients involving the Mulliken charges do

not exceed 0.73 only. Thus we find that,

statistically speaking, HDC charges reproduce

SCF dipole moments quite satisfactorily while

the Mulliken charges are much poorer in this

respect.

3.2. MEP values

The MEP values around the different molecules

calculated at the SCF/6-31G** or SCF-CIS/6-31G**

level employing SCF/6-31G** optimized geometries

using various methods or charges are presented in

Table 3. The minimum SCF or SCF-CIS MEP values

obtained using the continuous electron density are

presented in column (1) while those on the van der

Waals surfaces of molecules obtained using the

different types of point charges are presented in

columns (2–4). In comparing SCF or SCF-CIS MEP

minima with the lowest surface MEP values obtained

using point charges, it should be noted that though a

quantitative agreement between the two cannot be

expected, a linear relationship between them would be

expected, at least from the statistical point of view,

since both of these would serve as measures of the

same property i.e. nucleophilicity. We make the

following observations from table:

(1) The homonuclear diatomic molecules studied

here may be divided into two groups: (a) H2, B2,

Li2 and Be2 and (b) N2, O2 and F2. The CHelpG

and Mulliken charges at the atoms in all these

molecules are zero, and, therefore, the MEP

values around the molecules are zero every-

where. In group (a), MEP minima obtained

using SCF wavefunctions and lowest MEP

values on the van der Waals surfaces obtained

using HDC point charges are located near the

bond center in each case. In group (b), MEP

minima obtained using SCF wavefunctions and

lowest MEP values on the van der Waals

surfaces obtained using HDC point charges are

located outside the bond region in each case.

Thus the MEP features obtained using SCF

wavefunctions and HDC point charges are in a

qualitative agreement.

Table 2

Dipole moments (Debye) of molecules obtained using various

methods or charges calculated at the SCF/6-31G** or SCF-CIS/6-

31G** level employing the corresponding SCF optimized

geometries

S.No. Molecule SCF

(1)

CHelpG

(2)

HDC

(3)

Mulliken

(4)

1. H2O 2.1 2.2 2.1 1.8

2. NH 1.8 1.7 1.8 1.4

3. NH3 1.8 1.9 1.8 1.4

4. HCN 3.2 3.2 3.7 3.5

5. HCN(S1) 2.5 2.4 2.5 2.4

6. HF 1.9 2.0 1.9 1.7

7. HOF 2.1 2.2 2.5 2.4

8. HNO 2.0 1.9 2.5 2.3

9. HNO(S1) 2.2 2.1 2.7 2.6

10. CH3OH 1.8 1.8 2.4 2.4

11. H2CO 2.7 2.7 3.0 3.0

12. H2CO(S1) 2.9 2.9 3.4 3.4

13. NH2OH 3.4 3.3 3.7 3.1

14. NH2CHO 4.1 4.1 4.2 3.9

15. NH2CHO(S1) 1.8 1.9 2.0 2.3

16. NH2CN 4.6 4.6 4.0 3.5

17. NH2CN(S1) 4.0 3.8 3.4 2.9

18. NH2NC 3.8 3.6 3.8 2.5

19. NH2NC(S1) 3.0 2.8 2.6 1.9

20. BH 1.5 1.1 1.2 0.7

21. BN 4.4 4.5 3.6 1.8

22. BH2F 0.9 0.9 0.9 1.3

23. BHF2 1.0 1.0 0.7 1.3

24. LiH 6.0 5.9 5.4 1.5

25. LiF 6.2 6.2 5.7 4.9

26. LiCN 9.4 9.1 9.4 7.6

27. BeO 6.8 6.8 6.8 2.5

28. BeCO 4.8 4.8 4.4 1.7

29. BeCH2 3.7 3.7 3.7 0.8

Linear correlation coefficients C(i,j ) between the different

columns (i,j ) are as follows: C(1,2) ¼ 1.00; C(1,3) ¼ 0.98;

C(1,4) ¼ 0.67; C(2,3) ¼ 0.98; C(2,4) ¼ 0.67; C(3,4) ¼ 0.73.


Table 3

MEP values (kcal/mol) around molecules using various methods or charges calculated at the SCF/6-31G** or SCF-CIS/6-31G** level

employing the corresponding SCF optimized geometries. In column (1), minimum SCF MEP values and in columns (2–4), MEP values on van

der Waals surfaces are presented

S.No. Molecule Atom/site MEP

V scf(min) (1) V s(CHelpG) (2) V s(HDC) (3) V s(Mulliken) (4)

1. H2 –a 22.7 00.0 22.3 00.0

2. H2O O 262.4 256.7 255.4 247.3

3. NH N 253.4 228.5 234.0 228.5

4. NH3 N 285.7 258.8 257.3 243.3

5. HCN N 248.7 240.2 250.6 242.3

6. HCN(S1) N 235.0 226.6 233.8 227.9

7. HF F 234.2 239.0 237.8 233.2

8. HOF O 228.4 236.8b 234.0b 232.8

F 225.5 236.0b 233.7b 232.6

9. HNO O 239.1 227.3c 235.1 227.3

N 236.0 227.4c 227.9c 227.4

10. HNO(S1) O 236.5 230.4 237.6 234.9

N 244.0 229.8 230.1 224.7

11. CH3OH O 265.6 251.8 261.0 256.2

12. H2CO O 248.5 246.1 249.6 246.0

13. H2CO(S1) O 252.4 247.7 254.3 252.2

14. NH2OH O 258.0 277.3 277.2 265.3

N 274.9 277.5 277.2 265.6

15. NH2CHO O 265.2 262.8 264.1 260.3

N 24.6 212.4 213.7 215.3

16. NH2CHO(S1) O 255.6 253.5 251.6 250.8

N 260.2 238.1 229.7 230.8

17. NH2CN N(NH2) 215.3 220.0 224.0 218.8

N 261.1 256.1 252.0 241.5

18. NH2CN(S1) N(NH2) 25.1 26.3d 25.3d 215.2d

N 248.2 238.3 241.4 234.0

19. NH2NC C 257.4 234.2 240.0 233.0

N(NH2) 244.8 229.5 235.2 243.6

20. NH2NC(S1) C 240.8 223.2e 225.4e 227.5e

N(NC) 235.6 224.7 230.7 237.6

21. B2 –a 255.3 00.0 234.7 00.0

22. BH B 262.6 219.5 262.8 217.4f

23. BH3 H 25.8 216.4 21.5 25.4

24. BF3 F 211.9 219.0 216.4 215.8

25. BN N 244.7 268.2 250.8 228.1

26. BH2F F 221.6 224.7 224.7 227.4

27. BHF2 F 218.2 222.6 219.3 221.6

28. LiH H 268.7 2119.7 293.5 230.8

29. Li2 –a 216.2 00.0 232.3 00.0

30. LiF F 2101.0 297.1 292.6 276.7

31. LiCN N 288.0 272.1 278.3 267.0

32. BeO O 297.7 2114.6 2103.1 241.4

33. BeCH2 –g 262.3 264.2 279.0 217.2

34. Be2 –a 278.8 00.0 265.4 00.0

35. BeF2 F 231.7 237.2 226.6 225.8

36. BeCO O 244.5 244.8 248.3 234.9

37. BeH2 H 213.3 236.8 25.5 210.9

(continued on next page)


(2) Let us now consider the effect of electronic

excitation on MEP to the lowest singlet excited

state (S1). In HCN, H2CO, NH2CHO and

NH2CN, changes in the MEP values due to

excitation near the various sites obtained by the

different methods are in a qualitative agreement.

Following electronic excitation of HNO to the

lowest singlet excited state, the magnitude of the

SCF MEP minimum near the oxygen atom is

decreased while that near the nitrogen atom is

increased. The changes in the surface MEP

values obtained using HDC and CHelpG

charges near both the oxygen and nitrogen

atoms of HNO are somewhat increased due to

excitation. Such differences between the beha-

vior of MEP minima obtained directly using

wave functions and surface MEP values

obtained using point charges are not unex-

pected, and may occur particularly where the

differences in MEP values are small.

(3) We noted that CHelpG charges do not reproduce

SCF dipole moments in some cases, e.g. BH, as

accurately as in other cases. A similar difficulty

is evident in surface MEP values obtained using

CHelpG charges also. Thus in BH, the magni-

tude of MEP calculated using CHelpG charges

is less than one third of each of the SCF MEP

minimum and the surface MEP value obtained

using HDC. In BH3, the reverse trend is

observed and the surface MEP magnitude

obtained using CHelpG charges is much larger

than those of the SCF MEP minimum and the

surface MEP value obtained using HDC.

(4) Linear correlation coefficients Cði; jÞ between

the different columns ði; jÞ; excluding as well

as including the cases with zero MEP in

columns 2 and 4, are presented in the footnote

to Table 3. The linear correlation coefficient

between the SCF or SCF-CIS MEP minima

and the surface MEP values obtained using

CHelpG charges excluding the molecules with

zero MEP in the latter case is 0.79 that

reduces to 0.74 when the molecules with zero

MEP in the latter case are included. The

correlation coefficient between the SCF or

SCF-CIS MEP minima and the surface MEP

values obtained using HDC is 0.92. These

correlation coefficients indicate that, statisti-

cally speaking, HDC, in these cases, yield

somewhat better surface MEP maps than the

CHelpG charges.

MEP maps for the LiH and BN molecules, taken as

representative cases, obtained by three different

methods, may be compared as follows. The SCF

MEP map of LiH obtained using the continuous

electron density distribution in terms of isopotential

contours is shown in Fig. 3(a) while the MEP maps of

the same molecule on the van der Waals surface

obtained using CHelpG and HDC point charges are

Table 3 (continued)

S.No. Molecule Atom/site MEP


38. 3O2 O 21.7 00.0 20.5 00.0

39. 1O2(S1) O 214.7 00.0 24.9 00.0

40. N2 N 214.6 00.0 27.0 00.0

41 F2 F 23.2 00.0 22.8 00.0

V s stands for surface MEP; Linear correlation coefficients Cði; jÞ between the different columns ði; jÞ; excluding the cases with zero MEP in

column (2) and (4) are as follows. (The Cði; jÞ values obtained including zero MEP in column (2) and (4) are given in parentheses)

C(1,2) ¼ 0.79(0.74); C(1,3) ¼ 0.91(0.91); C(1,4) ¼ 0.74(0.70); C(2,3) ¼ 0.90(0.81); C(2,4) ¼ 0.60(0.60); C(3,4) ¼ 0.73(0.65).a Near the bond-center in both V scf(min) and V s(HDC) MEP maps.b Near the middle point of OF bond.c Near the middle point of NO bond.d Near the middle point of NC bond.e Near the middle point of NC bond.f Near the hydrogen atom. In other methods it is near the boron atom.g In BeCH2, V s(HDC) and V scf(min) are near the BeC bond center while V s(CHelpG) is near the carbon atom.


shown using different shades in Fig. 3(b) and (c),

respectively. The MEP minimum is located near the H

atom in Fig. 3(a) and the corresponding lowest MEP

points are also located near the same atom in Fig. 3(b)

and (c). The zero MEP contour in Fig. 3(a) passes by

the region close to the middle portion of the LiH bond

and the zero MEP value also lies in the same region of

the van der Waals surfaces in Fig. 3(b) and (c). Thus

the MEP distributions in Fig. 3(a), (b) and (c) are

properly correlated. The MEP results for the BN

molecule corresponding to the above results for LiH

are shown in Fig. 4(a), (b) and (c), respectively. In

these maps, the MEP minimum or the lowest MEP

values are located near the nitrogen atom. The MEP

distributions in Fig. 4(b) and (c) are properly

correlated with those in Fig. 4(a). These results

show reliability of HDC obtained using the approach

adopted here.

3.3. Basis set effect

As mentioned earlier the values of 0.3 and 0.4

were used for K and r, respectively, to obtain the

results presented in Tables 2 and 3. These results

were obtained using the 6-31G** basis set. A

question arises if these choices for K and r would

also be valid for other basis sets. For this purpose,

ground state dipole moments of certain molecules

were calculated using the above mentioned values

of K and r, SCF/6-31G** optimized geometries and

the 6-31G and 4-31G basis sets at the SCF level.

Dipole moments of the molecules were also

Fig. 3. (a) Isopotential contour map of LiH obtained using the SCF wave function showing the MEP minimum (kcal/mol) as a dark patch near

the hydrogen atom. (b) MEP map on the van der Waals surface of LiH obtained using CHelpG point charges. (c) MEP map on the van der Waals

surface of LiH obtained using HDC point charges combined with BCC.


calculated using the CHelpG, HDC and Mulliken

point charges. These results are presented in

Table 4. It is to be noted that HDC do not occur

on hydrogen atoms at the level of 6-31G and 4-

31G basis sets, as these atoms have only a 1s

orbital each. Therefore, molecules made up of only

non-hydrogen atoms are included in Table 4. We

find that usually there is a small difference between

the dipole moments obtained using the 6-31G**

(Table 2), 6-31G and 4-31G (Table 4) basis sets

when SCF wavefunctions, CHelpG or HDC point

charges are used. However, the dipole moments

obtained using the Mulliken charges are, in some

cases, quite sensitive to the choice of the basis set.

MEP values around certain molecules obtained

using the 6-31G and 4-31G basis sets and

molecular geometries optimized at the SCF/6-

31G** level are presented in Table 5. MEP (Tables

3 and 5) is seen to be much more sensitive to the

choice of basis set than dipole moment (Tables 2

and 4). However, our aim here is limited only to

examining if the dipole moment and MEP values

computed using HDC with K ¼ 0.3 and r ¼ 0.4 and

the different basis sets correlate consistently

satisfactorily with the corresponding SCF values.

The correlation coefficients presented as footnotes

to Tables 4 and 5, particularly the values of C(1,3)

and C(2,3), are usually larger than 0.95 while one

Fig. 4. (a) Isopotential contour map of BN obtained using the SCF wave function showing the MEP minimum (kcal/mol) as a dark patch near the

nitrogen atom. (b) MEP map on the van der Waals surface of BN obtained using CHelpG point charges. (c) MEP map on the van der Waals

surface of BN obtained using HDC point charges combined with BCC.


value i.e. C(2,3) in Table 5 is 0.91. It shows that

the above-mentioned values of K and r are

satisfactory even for other basis sets than 6-31G**.

3.4. HDC magnitudes and locations

Amounts of HDC (in the unit of magnitude of

electronic charge) associated with different atoms in

some representative cases, their locations with respect

to the corresponding atoms in terms of distances and

directions of displacements where necessary and the

mixing of orbitals of the same or different atomic

shells due to which the HDC arise are presented in

Table 6. The HDC values obtained using the Lowdin’s

symmetry orthogonalization procedure are given in

parentheses for a few cases for comparison. A

difference between the present approach that includes

bond-centered charges and the earlier one that uses the

Lowdin’s procedure where the bond-centered charges

are formally zero, is to be noted. That is, in the present

approach, for a given atom, the sum of total HDC and

ASC (atomic site charge) would correspond to the

atomic net charge. As the molecules considered here

are electrically neutral, the sum of both the atomic net

charges (HDC and ASC values) and the BCC (bond-

centered charges) in each case would be zero. If the

Lowdin charges are considered, since BCC are zero,

the sum of the atomic net charges would be zero. We

make the following observations from Table 6:

(1) The total HDC values on the N, F, Li and Li

atoms in N2, F2, Li2 and LiH molecules,

respectively, obtained using the present bond-

centered charge-based and the Lowdin’s sym-

metry orthogonalization-based procedures differ

usually only by small amounts. However, the

internal distributions of HDC values among the

different components in the two schemes are

quite different. For example, the HDC values

arising due to the mixing of (2s,3p) orbitals in the

above mentioned molecules are quite different in

the two approaches. This is not surprising since

the Lowdin’s symmetry orthogonalization pro-

cedure would modify the original character of

atomic orbitals significantly.

(2) We find that the different HDC components are

displaced either inside or outside the bonds.

Those displaced inside the bond region would

play a similar role in bonding as BCC while those

displaced outside the bond region would corre-

spond to lone pairs. However, at the ab initio

level, as in the present study, there would usually

be more than one HDC components correspond-

ing to lone pairs in each case. In semiempirical

calculations that include only valence electrons,

there is only one HDC component corresponding

to lone pairs in each case [14–18]. In this

context, it may be noted that in ab initio

calculations, if we use different basis sets that

comprise different atomic orbitals, the magni-

tudes and the total number of HDC components

associated with an atom would also become

different. It does not imply any sort of short-

coming of the HDC concept since the calculated

charge distribution is known to depend on the

basis set used. The reliability of a charge

distribution should ultimately be judged on the

basis of the physical properties like dipole

moment and MEP distribution obtained using

it, as done in the present study.

(3) The values of different HDC components as well

as the total HDC associated with the same type of

Table 4

Dipole moments (Debye) of molecules obtained using various

methods or charges calculated at the SCF/6-31G and SCF/4-31G

levels employing SCF/6-31G** optimized geometries. In each case,

the upper value was obtained using the 6-31G basis set while the

lower value was obtained using the 4-31G basis set

S.No. Molecule Dipole moment

SCF (1) CHelpG (2) HDC (3) Mulliken (4)

1. BN 4.5 4.6 3.9 1.9

4.5 4.6 3.9 2.4

2. LiF 6.5 6.5 6.4 5.5

6.4 6.4 6.3 5.4

3. LiCN 9.5 9.1 9.3 8.4

9.4 9.0 9.3 8.3

4. BeO 6.9 7.1 6.8 3.8

6.9 7.0 6.9 4.2

5. BeCO 5.3 5.2 5.2 3.2

5.4 5.3 5.5 4.3

Linear correlation coefficients Cði; jÞ between the different

columns ði; jÞ are as follows: At 6-31G level: C(1,2) ¼ 1.00;

C(1,3) ¼ 1.00; C(1,4) ¼ 0.95; C(2,3) ¼ 0.99; C(2,4) ¼ 0.93;

C(3,4) ¼ 0.95. At 4-31G level: C(1,2) ¼ 1.00; C(1,3) ¼ 0.98;

C(1,4) ¼ 0.94; C(2,3) ¼ 0.96; C(2,4) ¼ 0.94; C(3,4) ¼ 0.98.


atom in different molecules may some times be

quite different. For example, the total HDC

values associated with the fluorine atom in F2,

BF3 and LiF are 21.688, 21.768 and 20.593,

respectively. The behavior of boron in this

respect is quite strange. Thus in B2, the total

HDC associated with the boron atom is 20.780

while in each of BH3 and BF3, the various HDC

components as well as the total HDC values

associated with B are almost zero each. Let us

examine if the charges associated with B and F in

BF3, taken as an example, are meaningful. The

sum of total HDC and ASC (or only ASC) on

boron in BF3 is 1.235 while the corresponding

sum on each of the fluorines is 20.209. There are

three BCC values of value 20.202 each. If we

want to generate here a distribution of atomic net

charges, we can divide each of this BCC value, as

an additional approximation, as done in the

Mulliken scheme, by 2 and the resulting value of

20.101 may be added to each sum of total HDC

and ASC on fluorine. Correspondingly, we must

add 20.303 to the total (ASC þHDC) or only

ASC value on boron. This gives the net charges

on boron and each fluorine in BF3 as about 0.93

and 20.31, respectively. These charges on boron

and fluorine in BF3 are not too different from the

values of CHelpG charges that were found to be

about 1.11 and 20.37, respectively.

Zero HDC on boron in BF3 and BH3 would imply

that hybridization of atomic orbitals does not occur on

boron due to interaction with the other atoms. It

suggests that the covalent character of the BH and BF

bonds in these molecules would be negligible and the

bonds would be dominantly ionic. Further, boron

Table 5

MEP values (kcal/mol) around molecules obtained using various methods or charges calculated at the SCF/6-31G and SCF/4-31G levels

employing SCF/6-31G** optimized geometries. In column (1), minimum SCF MEP values (V scf) and in columns (2–4), MEP values on van der

Waals surfaces (V s) obtained using different charges are presented. In each case, the upper value was obtained using the 6-31G basis set while

the lower value was obtained using the 4-31G basis set

S.No. Molecule Atomic site MEP


1. B2 –a 251.0 00.0 232.3 00.0

–a 252.9 00.0 228.4 00.0

2. BF3 F 220.5 225.2 222.2 220.0

F 221.4 225.1 223.7 221.8

3. BN N 254.0 270.0 255.8 229.7

N 255.3 270.0 256.2 236.4

4. Li2 –a 216.6 00.0 228.7 00.0

–a 217.3 00.0 226.6 00.0

5. LiF F 2119.0 2101.7 2105.8 286.3

F 2118.3 2100.5 2105.2 283.5

6. LiCN N 299.2 271.4 277.0 270.1

N 2100.4 271.0 278.4 271.9

7. BeO O 2111.3 2116.8 2104.0 263.3

O 2112.7 2115.3 287.1 269.4

8. Be2 –a 276.2 00.0 262.3 00.0

–a 280.2 00.0 259.1 00.0

9. BeF2 F 246.4 245.2 237.3 236.4

F 246.9 244.4 238.3 237.7

10. BeCO O 258.7 247.5 257.6 245.4

O 261.8 248.2 263.9 256.3

Linear correlation coefficients C(i,j ) between the different columns (i,j ) are as follows: At 6-31G level: C(1,2) ¼ 0.92; C(1,3) ¼ 0.95;

C(1,4) ¼ 0.96; C(2,3) ¼ 0.96; C(2,4) ¼ 0.77; C(3,4) ¼ 0.89. At 4-31G level: C(1,2) ¼ 0.92; C(1,3) ¼ 0.95; C(1,4) ¼ 0.80; C(2,3) ¼ 0.91;

C(2,4) ¼ 0.80; C(3,4) ¼ 0.96.a Near the bond center.


Table 6

Amounts of HDC (in the unit of lel, e ¼ electronic charge) associated with different atoms in some representative 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. ASC stands for atomic site charge while BCC stands for bond center charge. HDC obtained using the Lowdin’s symmetry

orthogonalization are given in parentheses

S.No. Molecule/atom Mixing of orbitals Amount of HDC and

other point charges

Distance of HDC

from the atom

HDC location and

displacement directiona

1. Li2, Li (1s,2p) 20.048(20.052) 0.018 Along bond, outside

(2s,2p) 20.088(20.117) 1.588 Towards other Li


(2s,3p) 20.017(20.108) 0.650 Along bond, outside



Total HDC 20.554(20.618)

ASC 0.631(0.618)

BCC 20.154(0.0)

2. LiH, Li (1s,2p) 20.099(20.099) 0.018 Along bond, outside

(2s,2p) 20.376(20.250) 1.588 Towards H

(1s,3p) 20.013(20.043) 0.051 Towards H

(2s,3p) 20.005(20.137) 0.650 Towards H

(3s,2p) 20.322(20.258) 0.650 Towards H

(3s,3p) 20.004(20.142) 0.751 Towards H

Total HDC 20.819(20.929)

ASC 1.093(0.958)

BCC 20.160(0.000)

H (1s,2p) 20.007(20.165) 0.082 Towards Li

(2s,2p) 20.031(20.072) 0.003 Towards Li

Total HDC 20.038(20.088)

ASC 20.076(0.060)

3. LiF, Li (1s,2p) 20.069(20.065) 0.018 Along bond, outside

(2s,2p) 20.140(20.188) 1.588 Towards F

(1s,3p) 20.047(20.015) 0.051 Towards F



(3s,3p) 20.066(20.054) 0.751 Towards F

Total HDC 20.513(20.436)

BCC 20.065(0.0)

ASC 1.206(0.845)

F (1s,2p) 20.014(20.004) 0.105 Towards Li


(1s,3p) 20.005(20.005) 0.060 Towards Li




Total HDC 20.593(20.543)

ASC 20.034(0.133)

4. Be2, Be (1s,2p) 20.065 0.019 Along bond, outside

(2s,2p) 20.187 0.992 Towards other Be

(1s,3p) 20.003 0.049 Along bond, outside




Total HDC 20.640

ASC 0.820

(continued on next page)


Table 6 (continued)


other point charges

Distance of HDC

from the atom

HDC location and


BCC 20.360

5. B2, B (1s,2p) 20.082 0.027 Along bond, outside

(2s,2p) 20.323 0.567 Towards other B





Total HDC 20.780

ASC 1.040

BCC 20.520

6. N2, N (1s,2p) 20.056(20.005) 0.073 Towards other N


(1s,3p) 20.039(20.029) 0.111 Towards other N




Total HDC 23.244(23.080) ASC 3.410(3.080)

BCC 20.332(0.000)

7. F2, F (1s,2p) 20.045(20.020) 0.105 Towards other F


(1s,3p) 20.023(20.020) 0.060 Towards other F




Total HDC 21.688(21.860)

ASC 1.706(1.860)

BCC 20.037(0.000)

8. BN, B (1s,2p) 20.101 0.027 Along bond, outside

(2s,2p) 20.493 0.567 Towards N

(1s,3p) 20.031 0.141 Towards N


(3s,2p) 20.306 0.085 Towards N


Total HDC 21.271

BCC 20.404

ASC 1.783

N (1s,2p) 20.045 0.073 Towards B


(1s,3p) 20.050 0.111 Towards B




Total HDC 22.759

ASC 2.652

9. BH3, B (1s,2p) 0.000 0.027

(Planar) (2s,2p) 0.000 0.567

(1s,3p) 0.000 0.141

(2s,3p) 0.000 0.085

(3s,2p) 0.000 0.085

(3s,3p) 0.000 0.163

Total HDC 0.000

BCC 20.207


would behave as a highly electron deficient atom in

these molecules and would get readily involved in

donor–acceptor complexes with other electron donat-

ing atoms in molecules. Complexation of BF3 with

NH3 where B binds to N is a well-known experimental

fact [28,29]. To be able to understand the behavior of

boron in this respect better, we studied HDC and BCC

in the dianions of BF3 and BH3 (BF223 and BH22

3 ). The

different HDC components were found to be non-zero

and appreciable in these cases the total HDC values

being as follows: 24.87 on B and 20.96 on F in

BF223 ; and 24.59 on B and 20.04 on H in BH22

3 :

Thus in these dianions, boron would be strongly

involved in covalent bonding with the other atoms. It

is clear that zero HDC on boron in BF3 and BH3 is due

to lack of hybridization or polarization of its orbitals

and its electron deficient nature in these molecules.

Locations of the different HDC components in

Li2, F2 and LiF are shown in Fig. 5(a), (b), and (c)

while those of B2, N2 and BN are shown in Fig.

6(a), (b) and (c), respectively. These were taken as

representative cases and are discussed here in order

to show the usefulness of HDC from the point of

view of molecular bonding. In Li2(Fig. 5(a)), two

HDC components of each lithium atom are located

appreciably outside the bond region, two are

located close to the lithium atom, and two are

appreciably displaced into the bond region, one of

these going beyond BCC. In F2(Fig. 5(b)), four

HDC components associated with each fluorine

atom are located outside the bond region while two

are located inside the bond region. All these HDC

components of fluorine are located near the

corresponding atom and even those located inside

the bond region remain appreciably away from the

BCC, towards the corresponding fluorine atom.

Thus while the charge cloud of lithium in Li2 gets

highly polarized, that of fluorine in F2 is much less

affected. In LiF(Fig. 5(c)), two HDC components

of lithium are located appreciably away from the

LiF bond region, two components are located very

close to the lithium atom, one of these being

inside the bond region while the other is outside

the bond region, one component is located close to

the BCC while one HDC component of lithium that

is largest in magnitude among all the HDC

Table 6 (continued)


other point charges

Distance of HDC

from the atom

HDC location and


ASC 0.541

H (1s,2p) 20.011 0.082 Towards B


Total HDC 20.049

ASC 0.075

10. BF3, B (1s,2p) 0.000 0.027

(Planar) (2s,2p) 0.000 0.567

(1s,3p) 0.000 0.141

(2s,3p) 0.000 0.085

(3s,2p) 0.000 0.085

(3s,3p) 0.000 0.163

Total HDC 0.000

BCC 20.202

ASC 1.235

F (1s,2p) 20.035 0.105 Towards B


(1s,3p) 20.028 0.060 Towards B




Total HDC 21.768

ASC 1.559

a Locations of HDC of boron in BH3 and BF3 would have no meaning since all the corresponding HDC components are zero.


components of the atom is located beyond the

fluorine atom. The HDC components of fluorine are

located close to the fluorine atom. The locations of

HDC of the lithium atom, in going from Li2 to LiF

(Fig. 5(c)), are rearranged such that some HDC

components that were located outside the Li2 bond

region get shifted into the LiF bond region and

vice versa. However, no such rearrangement of the

locations of HDC of fluorine takes place. Thus

while the charge cloud of Li in LiF is highly

polarized that of fluorine is comparatively only

mildly polarized. If we sum up total HDC and ASC

on each of Li and F, we find that approximately

0.6 electronic charge is transferred from the former

atom to the latter. These are not too different from

the magnitudes of the CHelpG charges on the Li

and F atoms that were found to be about 0.8 each.

Thus, according to the present HDC-based

approach, the properties of LiF would be controlled

by two factors: first, transfer of about 0.6 electronic

charge from Li to F and second, polarization of

electronic charge cloud of the Li atom that extends

partly even beyond the F atom. The former effect

(charge transfer) would give ionic character to the

LiF bond while the second effect (polarization)

would give partly covalent (due to the HDC

components of Li and F located on the LiF bond)

and partly ionic (due to the HDC component of Li

going beyond the F atom) character to the same.

Thus, on the basis of the HDC-based results, the

LiF bond appears to be partly ionic and partly

covalent.

In B2 (Fig. 6(a)), two HDC components of each

atom are located outside the bond region, one being

close to the corresponding atom and other being at an

appreciably larger distance, and four HDC com-

ponents are in the bond region, one of them being

close to BCC while the other three are comparatively

closer to the corresponding atom. In N2 (Fig. 6(b)),

four HDC components of each nitrogen atom are

located outside the bond region, three components

being close to the corresponding atom while the fourth

is at a comparatively larger distance from the atom.

Two HDC components of each nitrogen atom in N2

are located inside the bond region but both these are

quite away from BCC. Thus the charge cloud of boron

in B2 is much more polarized than that of nitrogen in

N2. The HDC locations of boron, in going from B2 to

BN (Fig. 6(c)), are rearranged such that some

components that were located outside the bond region

in B2 get shifted into the bond region in BN and vice

versa. No such rearrangement of the locations of HDC

Fig. 5. Arrows indicate the locations of the different atoms, HDC

components and BCC in (a) Li2, (b) F2 and (c) LiF. Atoms are

shown by open circles and the numbering scheme is as follows. The

atoms are numbered as i (i ¼ 1, 2) while the HDC components are

numbered as i, j where i refers to an atom and j ¼ 1–6 for the

different HDC components. The HDC components are numbered as

follows; 1: (1s,2p), 2: (2s,2p), 3: (1s,3p), 4: (2s,3p), 5: (3s,2p), 6:

(3s,3p).


of nitrogen takes place in BN. The above discussion

shows that lithium and boron atoms are much more

polarizable than fluorine and nitrogen atoms. The

HDC components located in the bond region would

play a similar role in bonding as BCC. Thus HDC

provide useful information about molecular bonding

that is not available from other charge distributions.

4. Conclusion

We arrive at the following conclusions from the

present study. Statistically speaking, HDC computed

using the present approach that includes overlapping

atomic orbitals and the resulting bond-centered

charges reproduce SCF dipole moments and surface

MEP patterns quite satisfactorily. In some cases, HDC

reproduce MEP values even somewhat better than the

MEP-fitted CHelpG charges, particularly in homo-

nuclear diatomic molecules and other molecules

where MEP values around a given atom vary rapidly,

e.g. around boron in certain molecules containing it.

The present approach yields interesting information

about molecular bonding that is not available from

other charge distributions. It is particularly evident in

molecules containing lithium and boron atoms. The

present HDC-based approach including BCC also

appears to describe molecular bonding and molecular

electrostatic properties better than the HDC-based

approach where the Lowdin’s symmetry orthogona-

lized atomic orbitals were used.

Acknowledgements

The authors are thankful to the Council of

Scientific and Industrial Research, New Delhi and

the University Grants Commission, New Delhi for

financial support.

References

[1] P.C. Mishra, A. Kumar, in: J.S. Murray, K.D. Sen (Eds.),

Molecular Electrostatic Potentials: Concepts and Appli-

cations, Theoretical and Computational Chemistry Book

Series, vol. 3, Elsevier, Amsterdam, 1996.

[2] A. Kumar, C.G. Mohan, P.C. Mishra, J. Mol. Struct.

(Theochem) 361 (1996) 135.

[3] K. Jug, Z.B. Maksic, Theoretical Models of Chemical

Bonding, Part III: Molecular Spectroscopy, in: Z.B. Maksic

(Ed.), Electronic Structure and Intermolecular Interactions,

Springer, Berlin, 1991, pp. 235, and references cited therein.

Fig. 6. Arrows indicate the locations of the different atoms, HDC

components and BCC in (a) B2, (b) N2 and (c) BN. Atoms are shown

by open circles and the numbering scheme is as follows. The atoms

are numbered as i (i ¼ 1, 2) while the HDC components are

numbered as i, j where i refers to an atom and j ¼ 1–6 for the

different HDC components. The HDC components are numbered as

follows; 1: (1s,2p), 2: (2s,2p), 3: (1s,3p), 4: (2s,3p), 5: (3s,2p), 6:

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Molecular electrostatic properties using point charges: ab initio hybridization displacement charges combined with bond-centered charges