CHAPTER 6 FUNCTIONAL THEORY

Transcription

1 CHAPTER 6 FUNDAMENTAL CONCEPTS BASED ON DENSITY FUNCTIONAL THEORY IKTRODUCTION 6.1 THE PRINCIPLE OF MAXIMLrM PHYSICAL HARDNESS The Griineisen constant 6.2 THEORETICAL MODEL 6.3 CONCLUSION

2 CHAPTER 6 FUNDAMENTAL CONCEPTS BASED ON DENSITY FUNCTIONAL THEORY INTRODUCTION There are several important concepts of chemistry, such as, electronegativity chemical hardness and softness, physical or mechanical hardness, hard and soft acids and bases, have recently obtained a firm foundation in the density functional theory. This chapter presents the above concepts as seen in simple diatomic molecules using density functional theory. 6.1 THE PRINCIPLE OF MAXIMUM PHYSICAL HARDNESS Porr and Chattaraj gave proof for a principle of maximum chemical hardness Chemical or absolute hardness is the resistance to change the electron cloud density of a chemicnl system and this is defined as where p is the electronic chemical potential. N the number of electrons, and u the potential due to the nuclei The softness u is the reciprocal of the hardness q.

3 The proof of Parr and Chattaraj depends on the properties of grand canonical ensembles. The softness can be written as the fluctuation in N from the average value (N) : where p = -, as usual. Soft systems have large fluctuations [Ill]. kt At equilibrium 0 is a minimum and r) is a maximum. Equation (6.11 and t6 2) have analogous equations in classical thermodynamics, such as where K IS the compressibility = P (t N-(N 11') 16.3) We see, here, that the compressibility is a measure of the physical or mechanical softness is analogous to the chemical softness 0. Parr has shown that K 'V, where V, is the molecular volume for various crystalline solids. is proportional to the crystal hardness as measured by Mohs scale and other such scales [1121. Thus, it seems reasonable to call K'V,, which has the units of energy, the physical hardness and its reciprocal the softness. Thus, it is reasonable

4 to define that chemical hardness is a measure of resistance to change of the electron distribution of a system. Physical hardness measures the resistance to change of the nuclear positions in a system. An equilibrium state should have the greatest resistance to change for both properties. In this section we examine the physical hardness of some element and binary compounds. The required condition for physical hardness is MBV,) = 0, where B is the bulk modulus and V,, = CR,,' where C is a constant and R,,, the equilibrium distance between the modest neighbours at room temperature. Since the hardness is the resistance to change of the nuclear positions, an important test is to see if BV,, has a maximum value at R,,, We have the relationships where ;iv = 3CR6R. The potential energy U, is expanded in a power series of x = tr - R,). where x is small : U,, = U + f - x' + gx' + hx' 2

5 ifg=-p&. For small values of x and h, eqs. (6.4) and (6.5) are compatible only The principle of maximum physical hardness has given a definite relationship between the harmonic constant f and the anharmonic constant g. The anharmonic term in eq.(6.5) determines the co-efficient of thermal expansion and the variation of the compressibility with pressure also depends on g. The volume as a function of pressure can be written as V = V,, 11 - ap t bp2) (6.6) K = a - 2bp + a'p (6.7) From the definition of the bulk modulus we then find Waser and Pauling showed that [I131 2g = i 1-b/a2) E/R, setting g = -W, we find Wa' = 3 nnd db/dp = 5 Therefore, for cubic solids, the pressure derivative of the modulus is equal to 5.00, a constant and dimensionless number Table (6.1).

6 Table 6.1 Crystal Li F LiCl LiBr Li I NaF NaCl NaBr NnI KF KC 1 KBr KI RbF RbCl RbBr RbI Y , aa, - ap

7 Again, if the pressure - volume relationship of a solid is writing using Murnaghan equation I1141 B, is the bulk modulus at zero pressure and B, = db/dp. Expanding the logs in eq ) at small values of AV leads to eq. (6.6) again, with a = 1/B, and b = 11 t B,/2B: putting b/at2 = 3 leads to B, = 5, as before The Griineisen Constant The thermal expansion of a solid depends on the anharmonic term in eq Because of this term, the average value of rr-%) is different from zero and increases w~th temperature. A simple treatment, partly classical and partly quantum mechanical, gives I1151 where a is the linear expansion coefficient, and C,, the heat capacity per mole of atoms. The value of the force constant, can be found from the compressibility. The procedure is to equate the pressure - volume to the energy needed to compress the chemical bonds between nearest-neighbour atoms. The result for a cubic crystal in given by waser and Pauling [ 1.

8 Here, n is the number of nearest neighbors or the coordination number CN divided by two or the number of bonds per atom. V, is the volume per mole of atoms. Equations and (6.13) can be combined to calculate a. However, it is more interesting to calculate the Griineisen constant, y, which is defined as 3a V,, y = - C.K Using eqn ) and , the value of y can be calculated. The result is that y=n/3, and whose value is for crystals with the diamond structure and 2.00 for cubic close packlng. and so on Crunelsen constant y for the number of covalent and partly Ionic solids are shown in Table (6.2). The number can be compared with the predicted Although the agreement is not quantitative, the trend is encouraging. Equation (6.13) is derived on a model of covalent bonds between nearest neighbours. It is not strictly applicable to ionic solids. The repulsive part of the potential energy must be similar for ionic and covalent cases, but the attractive part for ionic solids must also include the sum of the columbic interaction with the remainder of the lattice. In effect, the number of bonds is increased.

9 Table 6.2 Values used for. Griineisen constant y, and a comparison of e~perimehtal'~ and calculated equilibrium atomic volumes

10 A simple correction can be made by multiplying the CN by the Madelung constant, A, so that n becomes A,,. When this is done, g d results are obtained for ionic solids with a coordination number of 6 or 8 ( THEORETICAL MODEL First it is worth defining the chemical hardness and the chemical potential. It is well known that the chemical potential and the chemical hardness of a system can be expressed as first and second derivatives of the energ?i with respect to the number of electrons, that is where E IS the energy of the system. N 1s the number of electronics and d is the externnl potential. Most of the concepts in DFT are built on the basis of Sanderson's principle of electronegativity equalization [12l]. 'when atoms (or other combining groups) of different chemical potentials unite to form a molecule with its own characteristics chemical potential, to the extent that the atoms (groups) retain their identify, their chemical potentials must equalize'. So. on the formation of n molecule, for example, a diatomic molecule AB from

11 atoms A and B, electrons flow from the atom B to A if p, ) pa to equalize the chemical potential, where pa and p, are the atomic chemical potentials of A and B atoms respectively. potential is So, on the formation of a molecule AB, the resulting chemical where p', and p', are the chemical potentials of atoms A and B respectively in the molecule AB and p, represents the molecular chemical potential. p',, and p', are given by where qa and q, are the chemical harnesses of atoms A and B respectively and AN is the amount of charge transfer. Consequently, to first order. and the energy change, that is binding energy 121. is

12 This is the basic equation for the present study. From eq. (6.18), it is obvious that, if the chemical potential difference is high, then the charge transfer is also high. This is true for the case of binding energy also (see eq. (6.19)), that is the binding energy increases if the chemical potential difference increases. More importantly, these two parameters, charge transfer and the binding energy, are greater if the harnesses of both the partners are small. Hence both the atomic chemical potential and the chemical hardness play a vltal role in forming the molecule. To relate the molecular electro-negativity to its constituent atoms, Sanderson (1211 introduced the postulate of geometric mean; to a certain accuracy, the molecular electronegativ~ty is defined through its atomic scale as I1211 Because p = -x. In this equation, the product has to be taken over the M constituent atoms and p, refers to the original chemical potential of the atom i. Through earlier studies. the molecular hardness is defined as [122,1231 where q, is the atomic hardness.

13 Yang et al. [I241 have modified this formula to Thus, both the geometric mean principle of chemical potential and the arithmetic mean principle of chemical hardness depend only on composition and do not differentiate between different structures of a molecule. From eqs ) and , AE for a diatomic molecule becomes, Equation (6.23) implies that the binding energy may be expressed in terms of molecular chemical hardness. So if the binding energy of a molecule is h~gh; then the molecule should have a higher hardness value. The dependence of the bond energy on the depth of a potential energy well nnd the dependence of the force constant on the curvature of that well suggest that there is no theoretical basis for a simple relationship. In contrast, when the size of the force constant, that is, the stiffness of the molecular spring, represents the strength of the chemical bond in some way, there should be a correlation between the force constant, the bond disassociation energies and the bond lengths. In nddition Pearson [I251 has studied some diatomic hydride molecules HX and concluded that there is a correlation between the dissociation energy D, and both k" tk is the force

14 constant) and (the electronegativity), from the above facts, it is obvious that there is some correlation between the force constant and the chemical hardness of a molecule. It has already been proved that in a molecular softness o is proportional to the bond length re, and the molecular electronegatinty to its Inverse square. where D is a constant. Pearson I1251 deduced a formula relating the force constant and the atomic electroactivity x, in a diatomic covalent molecule as kr, = 77NxA (6.251 It has already been mentioned that the molecular electronegativity of a diatom~c molecule can be expressed zah = (xa ~,,l'*~, (6.26) where x, and X, are the atomic electronegativities and x,, is the molecular electronegativity of the molecule AB. However, the molecular electronegativity is defined on an energy scale as

15 I-A I+A q = -,p= where I and A are the ionization potential and the electronic affinity respectively of a system. So x=n+a. (15) Thus from equations (6.25),(6.26) and (6.29) kr, a n and finally K n n. A once agaln confirms our earlier assumption that there is a relation exlsting between the force constant and the chemical hardness. The chemical hardness and the chemical potential are defined through the orbital energies. where E, and E,,,.,, are the energies of the highest occupied and lowest unoccupied molecular orbitals respectively. To test the above relation, more than 35 atomic molecules have been considered. All the molecules have been

16 optimized at the HF16-31G' level of theory using the Micromol program [1261. The chemical hardness and the chemical potential have been calculated using the above equation (6.30) for all the molecules. The dissociation energies have been taken from the literature. The force constants have been calculated from the wave numbers which have been taken from the literature. 6.3 CONCLUSIONS This chapter work is focussed on examining whether the hardness principle is useful to define a chemical bond. Since all the three parameters, chemical hardness, association energy and force constant, taken for the analysis represent different behaviours of the system, a similarity between the bonds. atoms or groups should be maintained for comparison. Meanwhile. this study has been made not to extract the accurate relationship between the force constant and the chemical hardness, in to examlne whether there exists any correlation between these two quantities. So, all the molecules under investigation have been taken as a single group and have been examined for the correlation between different chemical parameters. The calculated bond length, the chemical hardness and the chemical potential at HFI6-31G' level of theory have been listed with the dissociation energy and the calculated force constant in the table (6.3). The calculated chemical hardness values are found to be slightly higher than the available experimental values. This is not surprising, since both the ionization

17 Table 6.8 Bond lengths re, force constants k, binding energies D,, chemical hardness n and chemical potentials p

18 potential and the electron &nity calculated from the molecular orbital method are somewhat different from the experimental values, owing to the incompleteness of the basis set and the exclusion of electron correlation effect. The molecules N,, F,, H, and HF have chemical hardness values well above 10 ev. The hardness values for the molecules Al,, BCI, C1, and Li, are found to be below 3 ev. It is obvious that the chemical hardness is higher for stable molecules and very low for unstable molecules. So the chemical hardness is a useful tool for scaling the stability of the molecules. Except for the N2 molecule, the expected higher values of force constant of the abovementioned four molecules of higher chemical hardness are not observed. Among the molecules investigated only these molecules are found to disobey greatly the expected correlation. Hence, it is mandatory to explain the reason for this behaviour Let us consider the molecules in which fluorine is one of the atoms. It has already been observed that there is a good correlation between the force constant and the bond dissociation energy (increasing bond dissociation energy results in increasing force constant) for the hydrogen halides, except for the low dissociation energy of F,, which is not matched by a correspondingly lower force constant. This is not too surprising. It seems to require quite a stretch of the imagination to be able to predict the behaviour of potential energy V (r) at a position away from re, given the curvature of V(r) at r=r, (1271.

19 Most of the behavioural differences in inorganic chemistry between fluorine and the other halogens can be attributed to the restriction of the valence shell of the fluorine to an octet of the electrons, the relatively small sizes of the fluorine atom and the low dissociation energy of the F, molecule, but it is sometimes also necessary to invoke the less rigidly defined power of attracting an electron within the stable molecule of fluorine. The weakness of the bond in F, is generally attributed to repulsion between the pairs of unshared electrons on each atom. So, for these reasons, the F, has a lower bond energy and a lower force constant even though its stability is very high. The chemical hardness of F2 has a lower bond energy and a lower force constant even though its stability is high. The chemical hardness of F, molecule is ev, and this parameter correctly predicts its hgher stability The smaller size of the H2 molecule also makes its bonding energy and force constant low but its chemical hardness indicates its higher chemical stability. When fluorine is bonded to other elements, the bond in the fluoride is always stronger than that in the corresponding chloride, when there are no unshared pairs of electrons on the combined atom. The smaller size of the fluorine atom not only permits high co-ordination numbers in molecular fluorides but also leads to a better overlap of atomic orbitals and hence to shorter and stronger bonds. The calculated bond length is just A. The high electronegativity of fluorine is involved in the strong hydrogen bonding in hydrogen fluoride. When comparing HF and HCI molecules. both the force constant and the chemicnl hardness higher for fluoride than for chloride

20 molecules also true between the molecules A1F and AlCI,, BC1, and LiF and LiC1. All these above facts confirm that the chemical hardness is proportional the force constant. It is clear force constant is proportional to the binding. This support our theoretical model. It is worthwhile mentioning that, even in a study on the relationship between the electro-negativity and the force constant, to obtain the linear relationship between these two parameters, Pearson I1251 was restricted to some molecules within the same group. Similarly, if selection is restr~cted to some molecules in the present work, a linear relationship may also be obtained. The best-known relationship between bond energies and chemical potential lelectronegativity) comes from Paulilng's concept of ionic energies. The bond stabilization depends on the differences between the electro activities of the bonded atoms. So the bond energy, that is, binding energy In a diatomic molecule is high when molecular electronegativity or, in other words, molecular chemical potential is a maximum since the molecular chemical potential has already been defined geometric mean of the atomic chemical potentials. This electronegative atoms hold on to all their electrons more strongly, including the electrons in their bonds and its makes the bond stronger. This feature is implicit in Mulliken I1281 theory of the chemical bond. Both coulomb and exchange integrals are assumed to be proportional to the ionization potentials of the atoms solved. The ionization potential indirectly involves in chemical hardness as well as the chemical potential. Pearson 1151 studied some hydride molecules and showed that there is a

21 linear relationship between the force constant and the electronegativity and obtained a correlation coefficient as The chemical hardness is a good indicator of chemical stability of the molecules, but chemical potential is a poor indicator. This is true for the case of isomers also [129,1301. For example, the chemical potential for the unstable Al, molecule is ev while, for the stable molecule F,, it is ev. One point has to be mentioned that. in general, the variation in the values of chemical potential for molecules is less than that in the chemical hardness, which In turn, easily makes the chemical potential a poor indicator for the stability of the molecules. When analysing the table, it is worth noting that in very few cases, does the chemical potential also look good as a measure of the stability of the molecules. On correlating the force constant with other molecular parameters, caution must be exercised on choos~ng the parameters, since the force constant represents the curvature at the bottom of the potentlal well which is a very important factor. From thls present study, the following points have been concluded. 1. For diatomlc molecules, an increment in either the force constant or the binding energy with increasing chemical hardness has been observed. 2. If a molecule has a high binding energy and a high force constant, then its chemical potential will also be high. 3. The chemical hnrdness is a better indicator of the chemical stability of the molecules than is the chemical potential.