Journal of Molecular Structure (Theochem), 205 (1990) 267-211 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

267

AN IMPROVED ITERATIVE MAXIMUM OVERLAP APPROXIMATION METHOD

FANG ZHENG and CHANG-GUO ZHAN*

Department of Chemistry, Central China Normal University, Wuhan (People’s Republic of China)

(Received 4 May 1989)

ABSTRACT

A new formula for calculating bond energy has been suggested in order to improve the iterative maximum overlap approximation method and to describe clearly the physical picture determining molecular geometry according to the principle of hybridization. By use of this formula, one can directly work out the equilibrium internuclear distances on the condition that the sum of bond energies in a molecule is maximum. The improved method has been examined by being applied to some alkanes and silanes, the agreement with experimental results indicates that the idea stated in this paper is reasonable.

INTRODUCTION

Pauling’s idea [ 1 ] about the mixing of pure atomic orbitals was one of the most important steps in the development of the quantum-mechanical descrip- tion of the covalent bond. Many outstanding scientists including Slater [2], Coulson [3], Pople [4] and Mulliken [ 51 have discussed the interpretative power of this description. In the early 196Os, the maximum-overlap method [ 61 appeared. After this, the principle of hybridization was developed and ap- plied to wider fields [ 7-131 of the developed maximum-overlap methods, many were just used to construct the best hybrid orbitals (HAO) for a molecule with known structure, but the iterative maximum-overlap approximation (IMOA) method [lo] proposed by Maksic et al. could be applied not only to construct the best HAOs but also to determine molecular geometries. These authors em- ployed the method to connect the s character of hybrid orbitals and bond- overlap integrals with many physico-chemical properties, and successfully worked out the structural properties such as molecular geometries, strain ener- gies, heats of formation, spin-spin coupling constants across one bond, C-H

*Author to whom correspondence should be addressed.

0166-1280/90/$03.50 0 1990 Elsevier Science Publishers B.V.

268

stretching frequencies, thermodynamic proton acidities and intrinsic bond energies, of a large number of molecules [lo]. All their results are better than those of some semi-empirical methods and are in good agreement with the experimental data [ 14,151.

Using the principle of hybridization to determine molecular geometries, Maksic et al. used ever better formulae to calculate the properties of many molecules and obtained good results [lo]. However, their formulae cannot illustrate the factors affecting bond energy and bond length and cannot indi- cate that the sum of all bond energies in a molecule is maximum at the equilib- rium internuclear distances. In the present paper, the formula for calculating bond energy in the IMOA method is improved and a new formula, based on Anderson’s work, [ 16-181, is suggested. Using this formula, bond lengths can be determined directly on the condition that the sum of the bond energies in a molecule is maximum.

THE ITERATIVE VERSION OF THE IMOA METHOD

In the IMOA method, the following linear equation between the bond en- ergy, EAB, and the bond-overlap integral, SAB, is used [ 191

EAB = kAn Sz+a + LB (1)

where kAB and ZAB are constants and differ only in widely different chemical bonds. Thus, the sum of all bond energies in a molecule can be written as

E total =&Em =&km Sm + A-,L,~ (2)

It is clear that C ZAB is a constant. When C kABSAB is maximum, E,,, takes A-B A-B

its maximum value, and at this point the molecule is in its most stable form. Therefore, the best HAOs for the molecule should satisfy [lo]

&kA& = maximum (3)

In order to calculate bond angles and construct the best HAOs, an assumption is imposed in the IMOA method, i.e., the perfect hybrids for all C-H and C-C bonds were allowed to follow C-H and C-C internuclear lines, respectively. In other words, the angles between the best HAOs forming the “straight” bonds to the same atom are equal to the corresponding bond angles [lo]. By using the orthonormality conditions of HAOs placed on the same atom, the bond angles can be easily determined as soon as the best HAOs are obtained. Again, for an A-B bond SAB can be expressed as a function of the internuclear distance and the components of the bonded pair of hybrids, as can 1 kAB SAB . There-

A-B

fore, for a molecule with known bond lengths, both the bond angles and the

best HAOs corresponding to the maximum E,, can be found through a sys- tematic variation of the independent parameters, i.e. the characters of the HAOs [10,19].

How is the length of A-B bond determined in the IMOA method? It is shown by eqn. (1) that the bond energy is related to the bond overlap. For Hz, when the distance between the two H atoms is zero, Sun is maximum, as is the bond energy. This is evidently unsuitable for determining bond length. For this rea- son eqn. (3) is not suitable for determining bond lengths and the IMOA method must contain another linear equation [ 111

dAn = KL SAB + L XB (4)

where KLB and LkB are constants and differ only for widely different chemical bonds. In the IMOA method, bond lengths must satisfy eqn. (4) and are em- pirically constrained not to be smaller than Lb + KaB (KAB c 0). As seen from eqn. (4)) in calculating d AB the value of SAB must be known in advance and in calculating SAz the value of dm and the HAOs must be known. Therefore, eqn. (4) can only be solved in an iterative fashion. The calculation is begun by assuming certain A-B bond lengths and the maximum-overlap approximation procedure is then performed. The resulting bond-overlap integrals SAz are sub- stituted into eqn. (4) and the new bond lengths are deduced. The whole cycle is repeated and continued until self-consistency between the input and output bond lengths is achieved. This is the iterative fashion of the IMOA method.

FACTORS AFFECTING BOND ENERGY AND BOND LENGTH

As stated above, eqn. (3) only embodies the influence of overlapping to bond energy, but does not reflect the fact that when all the internuclear distances in a molecule are equal to the bond lengths the total of bond energies is maximum and the molecule is at its most stable. Therefore, eqn. (3) cannot be used di- rectly to determine the bond lengths, but another empirical relation [ eqn. (4) ] must be assumed.

In fact, chemical bonding symbolizes a competition between two opposite forces: (i) the nuclear repulsion and (ii) the attraction of the nuclei for the electron density concentrated between them. A stable chemical bond is formed as soon as the resultant electron-nuclear attractive forces on the nuclei exactly balance those of the nuclear repulsions. How then can the two factors be re- flected in the formula for calculating the bond energy?

Considering a diatomic molecule A-B, let Zi and Ri be the charge and posi- tion of nucleus i and r be the position of the electron. From the Hellmann- Feynman force theorem [ 201, the force on nucleus A, which is the gradient of the expectation value of the full Hamiltonian WA, ( IRA - RBl ), is given by [I71

270

+ p(r& -RB)V~aiRA - rl-‘dv 1 (5)

where Vn, is the gradient operator and p(r,R,- RB) is the eigendensity. Ap- parently, IRA-RBI =R -. With the origin of the coordinates fixed to the nu- cleus B, the force acting on nucleus A along the direction of the bond axis of A-B can be written as

FA = -2, -‘du 1 (6)

Suppose that pi ( r ) and PA ( RA - r) are atomic electronic charge densities centred on nucleus B and nucleus A, respectively. If the charge densities of A and B did not change when the two atoms approached one other, then p ( r,RA ) would be written as pn (r ) +pA (RA- r). But, in fact, chemical binding must lead to redistribution of the electronic charge densities [ 211, which means that eqn. (6) needs an additional term. Denoting the difference between p(r,RA) ~d~(r)+PA(RA 4 asp,,,(r,RA), then 1171

p(r,RA)=m3(r)+PA(RA_r)+P,,,(r,RA) (7)

Putting eqn. (7) into eqn. (6) then gives

F A=- IRA -r/l’dv

IRA-ri-‘du+~p~~,(r,RA~~IR,-rl-‘d~] (8) AB

where the first, second, third and fourth terms are the interactions of nucleus A with nucleus B, the electrons of atom B, the electrons of atom A, and the correction of the charge densities, respectively. The summation of the first three terms may be regarded as the energy of a system in which the molecule A-B is taken as a superposition of two rigid atoms. For spherically symmetric atom-like PA (RA - r), the third term in eqn. (8) should be zero.

When the internuclear distance is infinite, the molecule A-B does not exist. The energy of the system is equal to the sum of the energies of two separated atoms. If the separation is zero, then the energy of molecule A-B with the internuclear distance RAB (relative to separated atoms) can be calculated using

W,(R,,)=S_R*BF,dR,,=W,(R,,)+Wcar(RAB)

where

271

It is clear that Wn (RAB) is repulsive and positive. As RAB tends to infinity, WB (RAB ) approaches zero. The smaller the RAB, the greater is WB (RAB). The WC,, ( RAB ) term is the energy due to the electronic-charge-density redistribu- tion that occurs as the atoms come together to form the molecule; it should reduce the energy of the system. Thus, W,,,(R,,) must be attractive and negative.

How is the term W,,, ( RAB ) determined? If p_,, ( r,RA) is known, the problem is solved. Unfortunately, this value is not generally known: pcor (r,RA) corre- sponds to a “density difference” which has been well studied only for Hartree- Fock calculations near the equilibrium internuclear distance [ 201. Thus find- ing WC,, ( RAB ) from pcor (1: ,RA ) is an extremely difficult problem [ 171. We would like to evaluate this term using an approximate model. As is well known, the closer the nuclei are to one another, the more electronic charge is attracted between them, and the larger the orbital overlap becomes and the lower the energy W,,,(R,,) falls. In addition, a stable chemical bond must have more charge built up between the two nuclei so that it will “cement” them together. From the view of the mixing of pure atomic orbitals, this action might be car- ried out by the positive overlap of a bonded pair of hybrids. Connecting W,,,(RAB) with the bond-overlap integral SAB and RAB, and making the fol- lowing equation hold (as a hypothesis )

WC,, (RAB ) = -KABSAB~~~[-LAR(RAB-~OAB)I (11)

where doAB is the standard bond length of A-B and the parameters KAs and LAB differ only to widely different chemical bonds. These parameters can be determined from the single bond energies and standard bond lengths. It has been shown [eqn. (11) ] that W,,,(R,) is attractive and increases as RAB increases. Summing eqns. (9) and (11)) the molecular energy of the system can be written as

~AB(RAB)=WB(RAB)-KABSAB~~~[-LAB(RAB-~OAB)I (1.2)

Because the internuclear distance, RAB, is equal to the equilibrium bond length, &n, then - WA, (CRAB) is in fact the bond energy EAB (dAB ) . The values of dAB and EAa ( dAB ) can be worked out directly from eqn. (12) because the equilib- rium bond length is just the internuclear distance at the minimum WA, (RAB) and only the following equation need be solved

(13)

The above discussion is valid only for a diatomic molecule A-B. For the case of polyatomic molecules, the energy of the system can similarly be expressed as the sum of two terms of which one is repulsive and the other is attractive. The latter is also evaluated using eqn. (11)) but the repulsive term contains the interaction not only between bonded atoms but also between non-bonded atoms. For example, in a molecule C-A-B there is interaction between atoms B and C as well as the interactions between A and B, and C and A. However, from the calculation, it can be seen that the interactions among rigid atoms become weak rapidly and fall to zero as the internuclear distances increase, i.e. the interactions between non-bonded atoms can be ignored. Therefore, the energy of a polyatomic molecule system is approximately

Wtota,= C WBWABJ- C K S A_B AB AB~W[-LAB(RAB-CEOAEJ) 1 B<A

(15)

OUTLINE OF METHOD FOR DETERMINING MOLECULAR CONFIGURATION

From eqn. (15), the formula for calculating the bond energy is approxi- mately given by

E total = ATBE~~ (RAB )

(16)

which corresponds to eqn. (2). It is better to maximize Etotal than to minimize Wtotal so that the procedure is consistent with the IMOA method. In principle, E total should be evaluated for various internuclear distances and various bond lengths. Then, those values corresponding to the maximum E,,, are chosen as the equilibrium bond lengths and the equilibrium bond angles, respectively. However, for a molecule with given internuclear distances the first term in eqn. (16) is only a constant because it has nothing to do with the bond angles and

HAOs. Therefore, requiring E,,, to be maximum that

~~KABSABexp[--L,,(RAB-d~AB)l=maximum

If we let

k AB=KAB~~P[-LAB(RAB-~oAB) 1

213

is tantamount to requiring

(17)

(18)

the forms of eqn. (17) and (3) are the same, then the optimization procedure [ 111 of the IMOA method can also be applied to the process maximizing Etotal of a molecule with given internuclear distances, and the IMOA method may clearly be improved by using eqn. (16) instead of eqns, (2) and (4). The cal- culation is begun with an arbitrary set of internuclear distances and optimizes the sum of weighted bond-overlap integrals. The resulting HAOs are used to calculate Etotal with various internuclear distances in order to find out those which maximize E,,,. The set of internuclear distances is the first-improved bond lengths, and also the new ones for the following cycle calculation. The same procedure is repeated until full consistency, compatible with precon- ceived tolerance between the pth and (p - 1) th bond lengths is attained.

In order to test whether the proposed method is reasonable, calculations were made for some alkanes and silanes. The initial lengths of C-C, C-H, Si- Si and Si-H were taken as 1.539, 1.093, 2.300 and 1.480 A, respectively, and the Slater AOs were adopted (the orbital exponents of H, C and Si atoms are 1.20, 1.625 and 1.383, respectively [ 221). Let B be a C or Si atom, then eqn. (9) can be rewritten in the forms [ 181

(19) =P(--k&c)

+?!!R .+75 45 & AS’ 24% ‘2GiRAsi

eW(-%RAsi) (20)

where A is a C, H or Si atom. The other parameters used are:

K cc = 174.233, KCH = 209.04, Ksisi = 113.27, KsiH = 190.22 kcal mol-l;

and Lee = 0.565, Len = 0.755, LSiSi = 1.5094, LS* = l-0239.

A simple BASIC program was written in accordance with the steps expressed above. The calculated results are listed in Table 1, and compared with the experimental data and the results of IMOA and MINDO/S calculations.

TA

BL

E 1

Th

e re

sult

s cal

cula

ted

wit

h v

ario

us

met

hod

s an

d th

e ob

serv

ed o

nes

for

som

e al

kan

es a

nd

sila

nes

Mol

ecu

le

Bon

d le

ngt

h (

A)

Bon

d an

gle

( ’ )

E

,,i

(kca

l m

ol-‘)

Bon

d C

alc.

IM

OA

b M

IND

0/3”

O

bs.

An

gle

Cal

c.

IMO

Ab

MIN

DO

/3’

Oh

s.a

Cal

c.

Obs

.d

CH

, C

-H

1.09

5 1.

098

1.10

2 1.

093

HC

H

109.

47

109.

47

109.

47

109.

47

392.

8 39

7.2

CH

,CH

, C

-H

1.09

3 1.

102

1.10

8 1.

093

HC

H

110.

90

109.

96

109.

75

675.

8 67

5.4

c-c

1.53

9 1.

542

1.48

6 1.

534

HC

C

107.

96

112.

8

‘CH

:CH

,CH

, ‘C

-‘C

1.53

7 1.

538

1.49

7 1.

54 +

0.0

2 ‘C

’C’C

10

6.81

10

8.45

11

9.6

111.

5f3

‘C-H

1.

094

1.09

9 1.

110

1.09

H

’CH

11

0.85

10

9.96

95

8.8

955.

5 *C

-H

1.09

2 1.

096

1.11

8 1.

09

H*C

H

112.

37

110.

54

102.

5

‘CH

;CH

, ‘C

-Y?

1.53

7 1.

501

1.53

9 kO

.003

‘C

*CH

10

9.29

10

8.8

110.

37 f

0.25

I 2c

-2c

1.53

5 1.

524

1.53

9 f

0.00

3 ‘C

*C*C

10

6.84

11

9.5

1241

.7

1236

.1

CH

,CH

, ‘C

-H

1.09

4 1.

111

l.lO

f0.0

03

H’C

’C

108.

00

113.

2 11

0.37

to.2

5 ‘C

-H

1.09

2 1.

121

1.10

+ 0

.003

‘CH

(‘C

H,)

, ‘C

-2c

1.53

5 1.

535

1.51

3 1.

54

H’C

H

110.

81

110.

03

108.

5 ‘C

-H

1.09

4 1.

096

1.11

1 1.

09

‘C’C

’C

108.

13

109.

14

103.

8 10

9.9

1241

.6

1237

.6

‘C-H

1.

091

1.10

1 1.

141

1.09

*CW

Ha)

.,

(W-I

,)%

(‘C

H,)

s~

‘C-H

1.

094

1.10

2 1.

098

1.09

H

’CH

11

0.79

10

9.96

10

9.5(

ass)

15

24.3

15

21.3

‘C

-2c

1.53

4 1.

539

1.30

8 1.

54 +

0.0

2 V

C’C

10

9.47

10

9.47

10

9.5(

ass)

‘C-H

1.

094

1.10

3 1.

09

H’C

H

110.

79

109.

96

*c-2

c ‘C

-*C

1.

534

1.53

0 1.

538

1.54

0 1.

54

1.58

‘c

*c’c

10

9.39

10

9.37

11

1&2

2372

.7

Cyc

le-c

&H

,2

C-H

1.

092

c-c

1.53

5

SiH

r Si

-H

1.47

7

SiH

aSiH

s Si

-Si

2.30

2 Si

-H

I.47

7

‘SiH

~SiH

~SiH

~ ‘S

i-2S

i 2.

301

‘Si-

H

1.48

4 2S

i-H

1.

484

1.12

3 1.

09

HC

H

112.

29

109.

5 (a

ss)

1697

.8

1.51

7 1.

53+

0.03

1.47

7 X

L 0.00

3 H

SiH

10

9.47

10

9.47

28

1.4

2.32

i

0.03

H

SiH

10

9.49

10

9.7

+ 0

.2

506.

4 1.

480

i 0.

008

H’S

iH

109.

48

647,

l H

’SiH

10

9.35

“Ref

. 23

. b

Ref

. 24

. T

he

angl

es

of t

he

alka

nes

and

the

bond

le

ngth

s of

the

m

olec

ules

w

ere

calc

ulat

ed

acco

rdin

g to

th

e m

etbo

d ex

pres

sed

by

the

auth

ors.

T

he

adop

ted

hybr

idiz

atio

n in

dice

s an

d ov

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ps

wer

e ta

ken

from

T

able

1

in r

ef.

24.

e R

ef.

25.

d R

ef.

26.

276

DISCUSSION

In order to interpret the formula for calculating the bond energy more clearly CH, was taken as an example. Let the four RCH values in CH, change simul- taneously in the same step. The molecular energy can then be calculated using the relation

W total=4WcH(RcH)=4WB(RcH)-4KcHScHexp[--cH(RcH-dOH)l

=wn+w*

The W,, WA and W,,, curves are plotted in Fig. 1. As shown in the figure, part of the total energy is repulsive, W,, which decreases steeply as RCH increases. The other part of the energy is attractive, WA which increases smoothly as RCH increases. When RCH decreases (starting from RCH = GO), W,,, decreases be- cause of the electron cloud-nuclei attraction and then increases because of rigid atom-atom repulsion. In the W,, curve there is an extreme point. The equilibrium bond lengths are the distances at the minimum W,,,.

From the data in Table 1 it can be seen that the calculated results are in agreement with the experimental ones, which suggests that the method pro- posed in this paper is reasonable. It is the repulsive and attractive factors that affect the determination of bond length and bond energy.

Comparing eqns. (14) and (l), it can be shown that the first and the last terms in eqn. (14) correspond to 1 An and kA, in eqn. (1). This means that lAB

Energy

(Kcal mo?)

Fig. 1. Energy curves: I W,; II WA; III Wbu

277

stands for the repulsion between the rigid atoms A and B. However, lAB is not a constant but a function of the internuclear distance RCH.

In summary, the formula for calculating the bond energies and the manner of obtaining the bond lengths from the formula without eqn. (4) have been used to improve the IMOA method. The physical picture determining the mo- lecular geometry in view of the principle of hybridization has thus been made more clear and more easily understandable.

ACKNOWLEDGEMENT

The authors express their sincere thanks to Prof. Frank Liu for his support and directorial instructions.

REFERENCES

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22

23

24 25

26

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