Hybridization effect on chemical potential and hardness — a quantum chemical study

CHEMICAL

30 August 1996

PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 259 (1996) 138-14!

Hybridization effect on chemical potential and hardness - a quantum chemical study P. Kolandaivel, S. Arulmozhiraja, R. Bhuvaneswari Department of Physics, Bharathiar University, Coimbatore - 641 046, India

Received 14 July 1995; in final form 25 June 1996

Abstract

The hybridization effect on two important quantities, chemical hardness and chemical potential, has been studied by HF-SCF theory using a 6-31G basis set. The maximum hardness principle has been tested for C-C and C-H symmetric stretching with positive and negative mean amplitudes of vibration. For the above study, three different types of hybrid orbital molecules, CH 4, C2H 6, C2H 4 and C2H 2, have been used.

1. Introduction Density functional theory (DFT) has drawn the attention of many scientists due to the emphasis on the one-electron density function p as the fundamental variable for electronic structure theory. Hohenberg and Kohn [1] proved that the ground state energy of a chemical system is a function of p only. Pearson [2] proposed the maximum hardness principle which states that there seems to be a rule of nature that molecules adjust themselves to be as hard as possible. Parr and Chattaraj [3] have given a proof for this principle of maximum hardness (PMH) using the fluctuation-dissipation theorem of statistical mechanics. They have considered the electronic system of interest as a member of a grand canonical ensemble with bath parameters /z, v and T. It has been shown that any nearby non-equilibrium state of the system will evolve towards the equilibrium state with maximum hardness provided the bath parameters do not change, where /z, v and T, are the chemical potential, the potential due to the nuclei plus any external potential, and the temperature,

respectively [3]. However, some queries have also been raised [4,5]. Sebastian [4], argued that the result of Parr and Chattaraj, i.e. S - ( S ) > 0, can be true, while considering a particular non-equilibrium ensemble, but that does not prove it to be so for any other ensemble, where S and ( S ) are the softness for the non-equilibrium and equilibrium ensembles, respectively, and defined through energy below. Subsequently Chattaraj et al. [6] have used the Gyftopoulos-Hatsopoulos principle and explained the maximum hardness principle in a better way. The electronic chemical potential /z and the chemical hardness '0 are defined as /z = ( S E / ~ N ) o . r,

rI

=

½(81z/~N)v,r,

while the softness S is the reciprocal of the hardness and defined as S = 1/(2"0) = ( S N / 5 t z ) v . r , where E is the total energy and N is the number of electrons [7,8]. Pearson and Palke [9] studied the

000%2614/96/$12.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 ( 9 6 ) 0 0 7 3 9 - 7

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P. Kolandaivel et al. / Chemical Physics Letters 259 (1996) 138-141

chemical potential and hardness for a small displacement from equilibrium geometry along the directions given by vibrational symmetry coordinates for ammonia and ethane molecules and found that in asymmetric distortions in which /z and v remain constant

From DFF, the hardness and chemical potentials are defined as

for small changes, "O is maximum for the equilibrium geometry but for symmetric motions, in which /z and v are not constant, variation of ~ is monotonic, The asymmetric modes differ from the symmetric modes. The first destroy some elements of symmetry, changing the point group to another. In asymmetric vibrations, positive deviations from equilibrium produce a configuration which gives the same average nuclear potential as negative deviations; if we let Q be the symmetry coordinate, then 8 1 x / S Q = 0 and ~ r l / 8 Q = 0 at Qe. Again, if the energy is expanded as a power series in AQ, the non-vanishing term is a quadratic one. From this it can be shown that ~ V e n / S Q and 8 v , , / ~ Q are equal to zero, where Gn is the average potential of the nuclei acting on the electrons, and v,n is the nuclear-nuclear repulsion term [9]. For small amplitudes of vibration, the energy equation becomes

where I is the ionization energy and A is the electron affinity of the system. From Koopmans' theorem these two chemical quantities can be obtained as

where EHOMO and ELumo are the orbital energies of the highest occupied and lowest unoccupied molecular orbitals respectively. The normal modes of vibration shift the average nuclear positions from their equilibrium positions. The r. (the vibrational average internuclear distance) has been obtained with the help of the mean amplitudes of vibration. A Cartesian displacement of a nucleus i from its equilibrium position can be represented in terms of the l matrix defined by Nielson [10]:

E = E o + (ff'ol~U/BQIq'o)AQ...

Ace i = a i - a~ e) = rn 7. l/2ll~s)Qs

where AQ is a small displacement from the equilibrium geometry. Since the potential energy, U, is totally symmetric, ( B U / B Q ) must have the same symmetry as Q. The present work attempts to verify the principle of maximum hardness for the C - H and C - C symmetric stretching vibrations (with positive and negative values of mean amplitudes of vibration) for CH4, C2H6, C2H 4 and C2H 2 molecules through ab initio calculations. Finally, the behaviour of the chemical potential and hardness with the order of hybridization has been discussed.

-I.t=(

l + A)/2,

r/= (1- A)/2,

(1)

/x = ( EHOMO+ ELUmO)/2, = ( eujmo -- E n o m o ) / 2 ,

(2)

where rni and Q+ are the atomic mass and the vibrational normal coordinate respectively, s can take values from 1 to 3 N - 6 and l is the normal coordinate transformation matrix. Let the local z axis be taken along the direction of the equilibrium position of nuclei i and j, and the x and y axes are perpendicular to the z axis: + [(AX)2 + ( A Y ) 2 + ( re r, =

..[_(AZ)2)]I/2

=re + (AZ)+

((AX) 2 + ( A Y ) 2) (2r+) + ....

Since the last term is extremely small, the above equation can be approximated as rz=re+(AZ)

2. Methods of computations Optimized geometries of the CH 4, C2H 6, C2H + and CzH 2 molecules have been obtained by using 6-31G basis sets at the HF-SCF level of theory.

( A Z ) is the average mean amplitude of vibration along the vibrational coordinates. The chemical potential and hardness have been computed using equation (2) for different geometries of r z, r e and r e ( A Z ) (say rx).

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P. Kolandaivel et a l . / Chemical Physics Letters 259 (1996) 138-141

Table l Chemical potential (/.D and hardness 07) (in eV) for the CH4, C2H 4 and C2H 2 moleculesin 6-31G C-H Stretch C-C stretch -~

n

-~

n

rc

4.231 4.286

10.114 10.578

rx

4.422

10.997

C2H 6 r~

3.524

9.209

3,689

9.699

re

3.623

r~

3.780

9.627

3.623

9.627

10.028

3,551

9.552

re

2.830 2.775

7.247 7.488

2.780 2.775

7.214 7.488

rx

2.721

7.743

2.778

7.778

r: re

2.748 2.748

8.172 8.415

2.786 2.748

8.156 8.415

rx

2.748

8.669

2.712

8.685

CH 4 r.

C 2H 4 rz

C2H 2

3. Results and discussions Table 1 shows the chemical potential (/z) and hardness (r/) of the four molecules C H 4 , C 2 H 6 , C 2 H 4 and C 2 H 2 at the HF/6-31G level of theory for t h e r e, r z a n d r x bond lengths. For all these four molecules, the hardness value is found to be the higher for the negative deviation ( r x ) and smaller for the positive deviation (r z) of the C - H bond. Therefore, the hardness values show no sign of a maximum or minimum near the equilibrium geometry, This result coincides well with the results of Pearson and Palke [9], who reported that the values of r/and /z increase as the nuclei approach each other. The r/ value increases for negative deviation and keeps on increasing and will reach a maximum value when all the nuclei have coalesced. This does not happen because at some value of Q, the sum (BVen//BQ) -t(Sv~/BQ) is equal to zero when averaged [9]. However, the change in /z value is not found to have a uniform trend for the negative and positive deviations. For example, /x values are found to increase from r z to r x of the C - H bond for the C H 4 and C2H 6 molecules while for the C2H 4 molecule it is found to decrease from r. to r x for the same bond

and constant for the C2H 2 molecule. The C - C symmetric stretch distortions gave different results. T h e change in -q values of the C - C stretch for the C 2 H 2 and C2H 4 molecules is the same as the C - H symmetric stretch but in the case of the C2H 6 molecule, the r/ value is found to decrease for the negative deviation of the C - C bond and increases for the positive deviation. The C - C stretch of the C 2 H 6 molecule has ( 8 / z / B Q ) = 0. The maximum value of r/ is found at a bond distance of 3.01 a 0, somewhat greater than the equilibrium value [9]. Again, at a bond distance of 3.07 a 0 the hardness value is found to decrease. Thus the symmetry of the molecule is determined by the hardness value. However, in the case of the C2H 2 and C2H 4 molecules, due to the presence of rr bonds the hardness value is decreases for positive deviations. Moreover, the equilibrium value of Q is determined by the Hellmann-Feynman theorem of balanced forces and not by the maximum value of /,. This is not a violation of Parr and Chattaraj's proof, since neither of the two quantities /z and Oen is constant. If a vibration occurs so that the total energy of the system increases, and as unbalancing force can take place, then the molecule tries to move towards minimum energy and the net force becomes zero. This factor changes the kinetic energy and the inter-electronic repulsion as p is changed. Changes in p are due to changes in N or to changes in the shape factor of/9 at constant N. Again the inter-electronic repulsion is dominant when the subshell is being filled. When a main shell is filled, there is a change in the kinetic energy for the next electron added. Thus these factors induce a change in r/ values. It is worth mentioning that r / m a y not be a maximum when the total energy is minimum because of neglect of the nuclear repulsion term. Here again the PMH is not violated, because/~ and v are not maintaining constant values [3]. At the same time the chemical potential tz varies more slowly with the change in bond length. It is interesting to compare the values o f / x and t/ for all the four molecules to find a general trend for these two parameters due to hybridization effects. Generally, the hybrid orbitals have a tendency of more orbital overlap in a bond. It is well known that an sp hybrid orbital is stronger than pure s and p orbitals. The strength of the bond, formed by an s

P. Kolandaivel et al. / Chemical Physics Letters 259 (1996) 138-141

orbital is 1.00, and the relative strength of p, sp, sp 2 and sp 3 are 1.77, 1.93, 1.99 and 2.00 respectively [ll]. The CH 4 molecule has been formed by the overlap of the s orbitals of the hydrogen atom with the sp 3 hybrid orbitals of the carbon atom. These overlaps are effective, since the charge clouds of hybrid orbitals are characterized by extended axis concentration. This is one of the reasons that the bonds in CH 4 are strong. Thus, sp 3 hybrids with 75% p character, would form stronger bonds. Thus the hardness value is higher for CH 4 than for the other three molecules. In the case of the ethane molecule, the unequal distribution of electrons among all the atoms and the lower bond energy of C - H compared with CH 4, cause a lower hardness value than CH 4. In the case of the ethylene molecule, the s orbital and two p orbitals of each carbon atom in ethylene are hybridized and form three coplanar sp 2 hybrid orbitals. Thus the C - C bond in C2H 4 consists of one (r bond and one nv bond (double bond). It is worth noting that the ethylene molecule has a smaller hardness value than the other three molecules. When the hardness value of a molecule is small then it has a higher reactivity. The donation of two electrons from the highest occupied molecular orbital to the lowest unoccupied molecular orbital of the carbonhydrogen bond makes a complete transfer of the original bonding pair to the hydrogen. The "rr orbitals have equal contributions from the two carbon atomic orbitals. Further, the ionization potentials of the -rr electrons are smaller than those of cr electrons and in large aromatic hydrocarbons are as low as 6

141

ness is exclusive of hybridization order. From this study, it is clear that the absence of "rr bonds makes the molecule have a higher hardness value, but the number of rr bonds in a molecule does not influence the hardness value.

4. Conclusion From the present study, it has been concluded that the chemical hardness and chemical potential of the molecules do not depend upon the order of the hybridization or on the C - H stretching mode but is due to the electron overlap of the C - C bond. The hardness should be maximum at equilibrium provided both /~ and /)en are constant. The hardness values are found to be in the order 9(sp 3) > "q(sp) > ~?(sp2).

Acknowledgements SA would like to thank CSIR, New Delhi for the award of a senior research fellowship. We thank the referees for valuable comments on the manuscript.

References [11 P. Hohenberg and W. Kohn, Phys. Rev. (B) 136 (1964) 864. [2] R.G. Pearson, J. Chem. Educ. 64 (1987) 561. [3] R.G. Parr and P.K. Chattaraj, J. Am. Chem. Soc. 113(1991) 1854.

eV [12]. Due to the weak bonding of the -rr electrons of unsaturated hydrocarbons, chemical reactivity is high for the ethylene molecule, Its hardness value is minimum amongst the four molecules. In the C2H 2 molecule, there are cr bonds between the C - H atoms

[4] K.L. Sebastian, Chem. Phys, Lett. 231 (1994) 40. [5] K.L. Sebastian, Chem. Phys, Lett. 236 (1995) 621. [6] P.K. Chattaraj, G.H. Liu and R.G. Parr, Chem. Phys. Lett. 237 (1995) 171. [7] R.G. Parr, R.A. Donnelly, M. Levy and W.E. Palke, J.

and between the C - C atoms. The two unchanged p orbitals of carbon form two "rr bonds between the C - C atoms, by lateral overlap. Thus the C - C bond in C2H 2 has a triple bond (one tr and two rr bonds). The bond energies of the C - C double and triple bonds are 146 and 200 k c a l / m o l , respectively. This

[8] R.G. Parr and R.G. Pearson, J. Am. Chem. Soc. 105 (1983) 7512. [9] R.G. Pearson and W.E. Palke, J. Phys. Chem. 96 (1992) 3283. [10] H.H. Nielsen, Rev. Mod. Phys. 23 (1951)90. [11] M. Clyde Day and J. Selbin, Theoretical inorganic chemistry

may be the reason for the C2H 2 molecule having a higher hardness value than C2H 4. Hence the hard-

Chem. Phys. 68 (1978) 3801.

(East-West, New Delhi, 1985). [12] J.N. Murrell, S.F.A. Kettle and J.M. Tedder, The chemical bond (Wiley, New York, 1978).