Chemical softness in model electronic systems: dependence on temperature and chemical potential

Chemical Physics Chemical Physics 204 (19961429-437

Chemical softness in model electronic systems: dependence on temperature and chemical potential Pratim K. Chattaraj ‘, And& Department

Cedillo 2, Robert G. Parr *

of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USA Received 3 December,

1994

Abstract Discrete and continuous model systems are studied in order to understand the behavior of average softness, for equilibrium states in a grand canonical ensemble, when the bath parameters are changed. While the former model seems to be appropriate for describing an isolated system, the latter could take care of atoms in a molecular framework. Three, four, and five level discrete models, and quadratic- and exponential-type continuous models were chosen for studying the dependence of average number of particles, average softness, average electronic, and Helmholtz free energies on temperature and chemical potential. For the quadratic model, the zero temperature limit of the average softness is the absolute softness of density functional theory. In the other models it generally goes to zero at 0 K, but for some specific values of CL.The high temperature limit of it is zero in all the studied cases. When the average number of particles is an integer or half integer in the discrete model at low temperature, there are local extrema in equilibrium softness and the maximum hardness principle is shown to be valid.

1. Introduction Density functional theory (dft) [l] has importance in providing a theoretical background for well-known qualitative chemical concepts like electronegativity and hardness. Electronegativity ( x0> [2] is the negative of the electronic chemical potential ( po) which appears as a Lagrange multiplier in dft. For the

* Corresponding author. ’Permanent address: Department of Chemistry, Indian Institute of Technology, Kharagpur 721 302, India. * Permanent address: Departamento de Qurmica, Universidad Autonoma Metropolitana-Iztapalapa, Ap. Postal 55-534, Mexico, DF 09340, Mexico. 0301-0104/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0301-0104(95)00276-6

ground state of an N-electron system at 0 K, p0 and absolute hardness (vO) [3] are defined by (1.1)

where u(r) is the external potential, and I, and A, are ionization potential and electron affinity respectively. Softness (So) [41 is the reciprocal of the hardness, (1.3)

P.K. Chatturaj et d/Chemical

430

For the equilibrium state in a grand canonical ensemble with bath parameters F (thermodynamic chemical potential), U(T), and T (temperature), the average softness is defined as [4] (,y)=

i

T!!$

1

Physics 204 (1996) 429-437

and

(2.3)

(S>=p[i+a-(i-a)‘], where e-P(lO+PL)

=p[(N*)

- (Nj2].

(1.4)

i=

u(r),T

ensemble average, Here ( . > denotes equilibrium p = l/kT, and the external potential, U(I), is a generalization of the volume of a classical homogeneous system. Note that S, is a quantum mechanical parameter for the system and the heat bath, while (S) is a thermodynamic property of the system [5]. In the present paper we consider both discrete and continuous models for the energy levels of electronic systems. In the discrete model the allowed energy levels are associated with only integral numbers of electrons. This kind of model has been successfully used [1,6,7] to describe isolated systems, specially in the limit of zero temperature. On the other hand, the continuous models assure the reality of energy levels for fractional numbers of electrons. In order to understand the behavior of an atom (or functional group) embedded in a molecular environment [S], the most natural model will be a continuous one. For the particle-number dependence of the energy levels, we consider two cases, a quadratic function or an exponential function. Note that even for the discrete model, the ensemble average of the number of electrons can be fractional.

2. Discrete model Consider a grand canonical ensemble composed of an electronic system with N, electrons and having electronic energy E,, along with the corresponding species having N, - 1 and N, + 1 electrons with electronic energies E, + T, and E,, -A, respectively. This three level discrete model [1,5-71 was first introduced by Gyftopoulos and Hatsopoulos [6]. For this model we can write [I] (N)=N,-i+a,

(2.1)

(E)=E,,+iZ,-uA,,

(2.2)

(2.4

1 + e-B(b+~‘) + ,P(Ao+lr)

and

1 +

e-P(‘o+~)

+ ,P(.%+F)

(2.5)

’

Models comprising more energy levels and/or cited states can be defined similarly.

3. Quadratic

ex-

model

In this model the electronic energy is taken to be a continuous function of N, as the following Taylor series in terms of N - N,,

=E,+&N-NO)

+vo(N-N,,)*,

(3.1)

where N, is the number of electrons in the isolated neutral species. This model has been found [9] helpful in correlating pc, with Mulliken’s electronegativity [2,10]. The grand partition function and the average values of powers of N are given in terms of the following integrals: ,-Pf P(E-pN)

va=o,

1,2,

where A+ and No-A-
dNz

(3.2) A-

define

the domain

(p-ro)2, 4770

of N as

(3.3)

P.K. Chattaraj

et al. / Chemical

and

L(P.,

Physics 204 (1996) 429-437

431

Extension to an infinite limit situation can easily be obtained by putting A+ = x and A- = No in these formulas.

&(A+-+/%,)

PI=/ - ~W+b/27~,,)

No+%+&

4. Exponential

(3.4) Here 6~ is p - pO and u is given as

(3.5) Therefore, for the case of a symmetrical around N,[ A+ = A- = A] we have

range of N

model

Electronic energy in this model exponential function of N, E(N)

= E, + d[e-“‘N-N~)

is written

- 11,

where d and h are parameters. model p. and q. are given as

(4.1) Note that, in this

/A~= -d/i,

(4.2)

A sinh(2b2c)e-bZ(‘+C*) (N)=N,+Ac-

t

(3.6)

as an

and

&I( /J, P) 1 v. = ?dh*.

(s)

so =

1 _

2j,e-bZU +c’)

cosh(2b*c)

- c sinh(26*c) lO( /J, P)

and (E)

= ?(S)

+E((N)),

(4.3)

This model was first used by Parr and Bartolotti [ 121 as a basis for a geometric mean principle of electronegativity equalization. By substitution of E(N) in Eq. (3.2), we obtain formulas for (N >, (S) and (E) in this model, which can be calculated numerically for any value of T and p. For specific limits the important integrals (cf. Eq.(3.2)) can be written as

(3.8) de-“-

where b=@%L

(3.9)

ztx

I)

du,

(4.4)

i.e.,

and cc-

(4.5)

SP 2710 A’

(3.10) where IO can be expressed in terms of the complete and incomplete gamma functions [13] as

For the special case p = E.L~(C= O), (3.11)

(A’) =N,, and W=So[l

(3.12)

-$&].

where erf( X) is the error function erf( x) = &i’e-”

dt.

of x given by [ 1 l] (3.13)

(4.6) Making use of eqs (4.4) to (4.6) one can calculate (N), (E), and (S). Thus, one may calculate any equilibrium ensemble averaged quantity at given temperature and chemical potential, for the various models.

P.K. Chuttaraj et al./ Chemical Physics 204 (1996) 429-437

432

Nonequilibrium softness

5.

softness

versus

equilibrium

In this section we analyze the behavior of softness in nonequilibrium states, generated by perturbing p of an equilibrium state at the same temperature. Consider an equilibrium state of a grand canonical ensemble characterized by bath parameters p, u(r), states and T and all possible nearby nonequilibrium characterized by a distribution with weight factors PN i( ,G;)= exp[ - p(EN,i - ,GN)]. Note that there are eqnilibrium states also characterized by jZ, u(r), and T. Denoting nonequilibrium averages by overbars, we have the softness for the nonequilibrium state at ,G, generated by perturbing the equilibrium state at

and at j.i = p, Eq. (5.4) becomes

+ if (S)

has a minimum for a given value of p,

equilibrium

(5.6a)

state;

or, if ( S) has a maximum for a given value of p, 5 will be minimum at the corresponding equilibrium state when l~2W/+2

that is,

(5-5)

3 will be minimum at the corresponding

P, as [I41

(5.1)

q3w;p.

This implies that,

(5.6b)

I
Therefore it becomes apparent that the behavior (S) will dictate the behavior of s.

of

6. Results and discussion

2

min{(S&}.

(5.2)

P

Now, for the extremal behavior of s as a function of ,G, at a given temperature and equilibrium chemical potential, consider its derivative,

+ 2P[@

- (N)pJ
Therefore, SPP( F> is an extremum at fi = p if (S)PF has an extremum at this value of p. Differentiating Eq. (5.3) with respect to jZ. we get

Before discussing the overall temperature dependence of (S) in various models, we analyze the zero temperature behavior of (N >, (E), and (5) for all possible values of p since this limiting behavior had been found to be important in the past [1,7]. We summarize the results for a discrete three level model in Table 1, with the explicit assumption that E is a convex function of N, which fixes the domain allowed for p. As we can see, (N > and ( E) can not have arbitrary values but only five allowed values. Therefore, at 0 K, (E) is discrete and not a continuous series of straight line segments as a function of ( N >, [ 1,7] ( S> vanishes at 0 K for integral numbers of particles but diverges for (N > = N, + 3 ( t.~= -I,,A,) because the fluctuation has a finite value of f in these cases. Note that hardness behaves [l] like a delta function near 0 K; a fact in conformity Table 1 Zero temperature limits of some average three level model

quantities

in a discrete

(N)

(E)

(G/P

lJ<-Io

No- 1 N,,-f

4 + 4, E,,+I,/2

0

/L=-I, -I,,
N,,

;

p=-A0

N,, + +

EO E,-A,/2

II>--Ar,

N, + 1

Eo-

CL

Ao

L L ;

P.K. Chattaraj et al./

Chemical Physics 204 (1996) 429-437

with the zero temperature limit of (S). Inclusion of more levels or excited states does not change this behavior of (S) except for additional divergences, e.g. at (N) = N, + 2 for a five level model. It is clear from the nature of the exponential function and error functions [ 1I] that, in the quadratic model (both finite and infinite intervals), the zero temperature limit of (S) is S,, independent of the value of p. This shows the connection between the thermodynamic and quantum mechanical properties and assures that the ground state of a system can be conceived [l] as the zero temperature limit of the equilibrium state in a grand canonical ensemble. In this limit (N) is a linear function of 6~ = CL- p0 and (E) is a quadratic function of 6~. At p = p0 corresponding limits are N, and E, respectively. As in discrete models, (S) vanishes at 0 K for exponential model in both finite and infinite limits, except for special values of Al.for which the number fluctuation is nonzero at that temperature. Fig. 1 depicts the variation of (S)/S, as a function of T, for p = p0 = - (1, + A,)/2, for various models. (S) goes to zero at high temperature in all cases. Unless the temperature is very high (above lo4 K) curves for discrete models with any number of levels coincide. We have taken the oxygen atom and its ions as test cases, with the corresponding energy values from Refs. [15] and [16]. It has been proved recently that [5], for a three level model at a given temperature, softness is minimum for the equilibrium state, at p = pO, when lI

/

0

20

40

1

60

60

433

1.6,

0.9

1 -1

1

46

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 2. $,$ CL,,+ Aw)/( S)p,, as a function of PAP for the discrete models. Continuous curve corresponds to 10000 K, and is the same for three, four, and five level models, as well as three level model including excited states. Dot and dash curves repmsent three level model at 30000 K and 100000 K.

compared to the corresponding values for all nearby nonequilibrium states, characterized by a distribution corresponding to a different CL, but at the same temperature. This accords with the maximum hardness principle [5,17], which states that ‘there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible’. In order to understand the situation more completely, we have calculated s and (S) for three (O+, 0, O-), four CO+, 0, O-, 02->, and five level (02’, O+, 0, O-, 02- ) models with experimental ionization potential and electron affinity values from Ref. [15], and also by including excited states [16] of O+ and 0. The results are presented in Fig. 2, where A p = jTi- pO. Curves for all these models are indistinguishable unless the temperature is extremely high. For any temperature up to lo4 K we get a common curve for three, four, five level models and also using excites states. These curves show a minimum for the equilibrium state. 3 For higher temperatures, results for only three level model are presented. The minimum is easily discernible. This analysis demonstrates that for temperature up to lo4 K, a three level model [1,5-71 is sufficient. For brevity, we do not present

100

Fig. 1. (5)/S, as a function of T/a, at p=pa, where a= nc/(wV,)*. For oxygen, a= 1.103X103. D3L, QLl, QL2, ELI. and EL2 denote discrete three-level model, quadratic model with finite and mfmite limits, and exponential model with finite and infmite limits, respectively.

3 The maximum hardness principle is obeyed if we choose ~a in such a way that conditions (5.6) are satisfied. Otherwise no minimum for 5 would be observed at the equilibrium state. See Ref. [18], where no extremum in 3 was found because pLo= 0 was taken, for which conditions (5.6) are not satisfied.

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et d/Chemical

Physics 204 (1996) 429-437

Fig. 3. Surface plot of (N) - NO as a function of T and p. for a three level model.

Fig. 5. Surface plot of (F) three level model.

here results for other models. However, we have verified that for the quadratic model with finite limits The same is true. For the quadratic model with infinite limits and exponential mode1 with finite limits, minimum in S occurs at ,G = puo only when the temperature is low. Note that p0 is defined for the exponential model in FQ. (4.2). For the exponential model with infinite limits we do not get a minimum at this value of p. A possible reason for that is given later. Figs. 3-6 depict the surface plots of ( N) - N,, (E),,, (F) -E,, and (S) for a discrete three level model. Basal rectangular mesh designates 0 < T G IO5 and - 20 G Al.,< 5 and F denotes Helmoltz free energy. When temperature is very small, three-step structures in the first three plots are as expected from Table 1. In the plot of (N) - N,,, the three steps correspond to ( N) = N,, N,, ZL1. As temperature increases, the plot becomes continuous making any N,-l~(N)~N,+lallowed.Intheplotof(E) - E,, corresponding steps at low temperature correspondto(E)=E,, E,+Z,, E,--A,.Thisplotalso becomes continuous with increase in temperature,

implying possibilities of any value in the range E, - A, < (E) < E, + I,. Note that the relatively large Z, value and small A, value of oxygen are properly reflected in this plot. At low temperatures (E) and (F) plots look alike, as expected. The difference becomes conspicuous as temperature increases. The most remarkable of these plots is of (S). At low temperature, two peaks correspond to p = -I,,A,. Since two maxima in a continuous function must be separated by a minimum, we get one for Al.= pO. As temperature increases two maxima come closer, coalesce and form a single maximum at p = pO. For the whole temperature range ( N) = N,,, for Al.= p,,. From the analysis in Section 5, it is clear that since (S) is always an extremum for p = pO, 5 is also an extremum there. As has been proved [5] and numerically verified here, s> (S). Here arises the important question of characterizing pO. Considering p0 = (aE/aN), I No, the definition of p0 is intimately linked with that of N,,. Note that for the three level mode1 we have considered here N, is the number of particles in the neutral

Fig. 4. Surface plot of (E)three level model.

Fig. 6. Surface plot of (S) as a function level model.

E, as a function of T and p, for a

- E, as a function

of T and I_L.for a

of T and p, for a three

P.K. Chatturaj et d/Chemical

species at 0 K implying p0 = --(I,, + A,)/2. However, if we prepare our ensemble to be composed of, let us say, both singly and doubly charged negative ions and the neutral atom, p0 will be different as has been already discussed by Gyftopoulos and Hatsopoulos [6,10]. Therefore, for a four level or five level models, we have different p0 values unless the temperature is small (< lo4 K) where the contribution from other levels are negligible. At any given temperature, all such p0 values in a given discrete model can be calculated easily in terms of the respective ionization potentials and electron affinities. It has been found for a discrete model that for all such pr, values, (S) is a local extremum, implying S > (S), provided the conditions discussed in Section 5 are met. For models containing more than three levels, we have extrema in addition to those for -A,,, and --(I, + A,)/2, at low temperaP= -I,, ture. In general, for any discrete model at low temperature, local extrema appear at a p such that max(N) = No + n/2, n = 0, AI 1, f 2,. . . . The ima appear for odd values of n, for which the number fluctuation does not vanish at 0 K and the minima appear when n is even. It is observed that for all such points, 5 > (S). For other models we present only plots of (S), for brevity. Other quantities are discussed whenever necessary. Figs. 7 and 8 present the surface plots of (S)/S,,, for quadratic model with finite and infinite limits respectively. In the first case, at low temperature, (S)/S, is a maximum and equal to one for k = p,,, and decreases to zero for high temperature, as well as for p values far removed from pO. It has been observed that the conditions of Section 5 are met and S z (S). In the second case, at low temper-

43s

Physics 204 (1996) 429-437

1

c

Fig. 8. Surface plot of (S)S, as a function of T/a and c, for quadratic model with infinite limits. See caption for Fig. 1 and Eq. (3.10) for definition of LJand c respectively.

ature, (S)/S, forms a terrace of unit value and becomes zero for small values of EL.As the temperature increases the terrace vanishes. At low temperature and for any p on the terrace, s 2 (S). Note that even at very high temperature, (S) does not vanish for large p values. This peculiar behavior of (S) is a consequence of unphysical unlimited growth of (N) and (E) with the increase in p, at any given temperature. Since, in exponential models, (S) is calculated numerically, instead of presenting the surface plots, we present in Figs. 9 and 10 the plots of (S) as a function of p, for various temperatures for this model with finite and infinite limits respectively. There are many local extrema in the former case, whereas mainly two maxima are discernible in the latter. In the former case, at low temperature for p = p0 = ~dh, we observe a local maximum in (S), and S > (S) is valid. In this case N, - 1 <

0.4 ,

1

0.3

0.2

-20

Fig. 7. Surface plot of (S)/.S, as a function of T/a and c, for quadratic model with finite limits. See caption for Fig. 1 and Eq. (3.10) for definition of a and c respectively.

-15

-10

-5

0

5

10

Fig. 9. (S) as a function of F for the exponential model with finite limits. Continuous, dot and dash curves correspond to 1000 K, 3000 K and 10000 K, respectively.

436

P.K. Chattaraj

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a

-6

-4

-2

et al./

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Chemical Physics 204 (1996) 429-437

2

Fig. 10. (S) as a function of /.I for the exponential model with infinite limits. Continuous, dash and dot curves correspond to 3000 K, 10000 K and 100000 K, respectively.

model gives the average softness equal to the absolute softness. In other cases, it generally goes to zero, except for the discrete model with chemical potential the same as the negative of an ionization potential or electron affinity, which gives a nonzero number fluctuation at 0 K. Average softness goes to zero at high temperature for all models. Local extrema in equilibrium averaged softness are special points in c(S), T, p) space and are important in understanding the maximum hardness principle. This principle is valid for all local minima in (S), and is also valid for those local maxima in (S), where

a2(s>

1I a/J*

(N) d N, + 1, at all temperatures. However, in the second case there is an unlimited growth of (N ), and there is no structure in (S) for p = -dh, and consequently, (S) is not necessarily a lower bound for 5. In both the cases, (N) and (E) behave like step functions of /J at low temperatures, which gradually become continuous with an increase in temperature.

7. Summary Chemical softness is a very important property of atoms and molecules and related to the polarizability. Its variation with temperature and chemical potential is useful to study and can be accomplished by defining it as a thermodynamic property within a grand canonical ensemble framework. The present work attempts to unveil some of the related truths. The maximum hardness principle is also better understood by manipulating the average softness as a number fluctuation. At very low temperature, the maximum hardness principle has been found to be valid in all models for those equilibrium states which satisfy conditions (5.6). In most cases these states are associated with integral number of particles. Variation of average softness, for the equilibrium state of a grand canonical ensemble, associated with changes in temperature and chemical potential has been studied for discrete, quadratic, and exponential models. In the limit of zero temperature, the quadratic

< 2pw;p.

Different models used in the present work are supposed to mimic various real situations. An isolated atom or a molecule possesses an integral number of electrons and a description of such a species may be given in terms of the discrete model [l&7]. Within this model, an isolated species is considered to be a thermodynamic system and represented in terms of a grand canonical ensemble at some given temperature and chemical potential. Although most of the mechanical properties of this species assume only discrete values, corresponding thermodynamic quantities, expressed as ensemble averages, are in general continuous functions of the bath parameters. Unless the temperature is very high (> lo4 K), a three level model, comprising the atom or molecule and its positive and negative ions, is adequate for most purposes. For any temperature below lo4 K, the softness of the equilibrium state, with an integral number of particles, is lower than that of the corresponding nonequilibrium states at the same temperature but having different chemical potentials. The zero temperature limit of this model perhaps best is pictured as associated with certain specific discrete energy values and not a continuous series of straight-line segments. An atom (or a functional group) inside a molecule, however, behaves more or less like an open system [8] and can have nonintegral number of electrons. Consequently, a continuous model may represent the atom in a molecule better. In the quadratic model, the zero temperature limit of (S) is S,, which ensures that the quantum mechanical system can be

P.K. Chatraraj et al. / Chemical Physics 204 (1996) 429-437

retrieved as the respective at zero temperature.

thermodynamic

ensemble

Acknowledgement Grants to UNC from the National Science Foundation, the Petroleum Research Fund of the American Chemical Society, and Exxon Education Foundation are gratefully acknowledged. Andres Cedillo appreciates support from the Secretarfa de Education Publica, Mexico. We are pleased to dedicate this paper to Bernard Pullman, one of those who first dared to apply quantum chemistry to molecules of biological interest, and one who has long labored on behalf of quantum chemists everywhere.

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M.K. Harbola, P.K. Chattaraj and R.G. Parr, Israel J. Chem. sot. 31 (1991) 395. [51 P.K. Chattaraj, G.H. Liu and R.G. Parr, Chem. Phys. Letters 237 (1995) 171. t61 E.P. Gyftopoulos and G.N. Hatsopoulos, Proc. Natl. Acad. Sci. USA 60 (1965) 786. 171 J.P. Perdew, R.G. Parr, M. Levy and J.L. Balduz Jr., Phys. Rev. Letters 49 (1982) 1691. Advan. Quantum b31 R.F.W. Bader and T.T. Nguyen-Dang, Chem. 14 (1981) 63. 191 H.O. Pritchard and F.H. Sumner, hoc. Roy. Sot. A 235 (1956) 136; R.P. Iczkowski and J.L. Margrave, J. Am. Chem. Sot. 83 (1961) 3547. I101 R.S. Mulliken, J. Chem. Phys. 21 (1934) 782; 3 (1935) 573. [ill M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1972) p. 297. [121 R.G. Parr and L.J. Bartolotti, J. Am. Chem. Sot. 104 (1982) 3801. [131 M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1972) p. 260. [141 D. Chandler, Introduction to modem statistical mechanics (Oxford Univ. Press, New York, 1987). [151 D.F. Shriver, P.W. Atkins and C.H. Langford, Inorganic chemistry (Freeman, New York, 1990). 1161J.C. Slater, Quantum theory of atomic structure, Vol. I (McGraw-Hill, New York, 1960) pp. 341-342. iI71 R.G. Pearson, J. Chem. Ed. 64 (1987) 561; R.G. Parr and P.K. Chattaraj, J. Am. Chem. Sot. 113 (1991) 1854; R.G. Parr and J.L. G%zquez, J. Phys. Chem. 97 (1993) 3939. 1181K.L. Sebastian, Chem. Phys. Letters 231 (1994) 40; 236 (1995) 621.