Manifestations of stability interplay between potential-hypersurface multiple energy minima in heat-capapcity enchancement

Journal of Molecular Structure (Theo&em), 233 (1991) 89-98

89

Elsevier Science Publishers B.V., Amsterdam

MANIFESTATIONS OF STABILITY INTERPLAY BETWEEN POTENTIAL-HYPERSURFACE MULTIPLE ENERGY MINIMA IN HEAT-CAPAPCITY ENCHANCEMENT”J

ZDEN&lK SLANINA’

Max-Planck-Znstitut fir Chemie (Otto-Hahn-Znstitut), Mainz (Germany) (Received 16 July 1990)

Abstract The location of several different local energy minima on a potential hypersurface represents a quite frequent event in contemporary quantum-chemical practice. The prevailing approach to the multiplicity problem consists in eliminating all structures with the exception of the global minimum. It is shown that the approach can be completely misleading as in some cases all minima, or their part, can coexist (i.e. be present in comparable amounts) under conditions of thermodynamic equilibrium. Then, the global minimum is not a sufficient representation of the system and a pseudospecies composed of the equilibrium isomeric mixture is to be applied. The approach can have substantial consequences for evaluating various characteristics of such systems. Particular attention is paid to the heat capacity term as this quantity can exhibit a qualitatively different temperature course compared to the case without structure multiplicity, viz. a course with a temperature maximum. The behaviour is potentially useful for the experimental proof or discovery of isomerism in systems of very varied types. The concept is illustrated using isomers of Cg, and using the isomerism in the associates HDO-HDO or HF-HCl.

INTRODUCTION

Characterization and representation of potential energy hypersurfaces by their stationary (critical) points has become a prevailing approach in the computational treatment of organic chemistry species (see for example refs. 1-3). In an infinite-time limit the approach is to be replaced, firstly, by the whole hypersurface in an analytical representation, and then by the total, molecular wavefunction going beyond the conventional Born-Oppenheimer framework. Qsdicated to Professor William A. Steele on the occasion of his 60th birthday. &rt LVIII in the series Multimolecular Clusters and Their Isomerism; for Part LVII, see ref. 21. ‘Permanent address: The J. Heyrovsk$ Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, Dolejgkova 3, CS-18223 Prague 8-Kobylisy, Czech and Slovak Federal Republic.

0166-1280/91/$03.50

0 1991 Elsevier Science Publishers B.V. All rights reserved.

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However, in our times the stationary-point description of hypersurfaces of species of chemical interest represents in fact the state of the art. There is an increasing amount of systems (mostly relatively simple ones) for which a manifold occurrence of stationary points has been encountered in computational treatments (for a survey, see refs. 2 and 4). In fact, the multiplicity of stationary points can be considered as a rather common feature of potential hypersurfaces. Quite typically, the complex situation is straightforwardly simplified by neglecting all stationary points higher in potential energy than the global minimum. In this paper we do not follow the conventional approach; rather, all the minima are considered according to their proportions under thermodynamic equilibrium and their effects on observed values are studied in terms of the heat capacity at constant pressure. COMPUTATIONAL TREATMENT

An isomeric mixture can be characterized by the values of the mole fractions, Wi,of the individual isomers. Of course, there would be other terms applicable, but the mole fractions in an equilibrium isomeric mixture do not depend on the total pressure imposed on the system but only on the temperature T. The isomers are to a first approximation described by their potential energy terms, AEi, especially in quantum-chemical praxis where the term represents a primary computational output. However, if their rotational-vibrational motions are considered, then relevant quantities are standard enthalpy changes at the absolute zero, A& pi,and the isomeric partition functions, qi. Under conditions of inter-isomer thermodynamic equilibrium the mole fractions are given [ 2,4,5 ] by qi

Wi=

exp 1-AfG,iI (RT) 1

n

(1)

jzl qj expI- AfG,j/ (W 1 where n is the number of isomers and R denotes the gas constant. Equation (1) was primarily designed for gas-phase conditions; however, it can be applied [ 61 to other environments too, if the partition functions are adjusted accordingly. It is useful to mention a critical difference from rather widespread treatments based on so-called simple Boltzmann configurational or steric factors [7-111:

w;= elrp[-Wl(RT)l n

j& em [ -mjl

(2)

(RT) 1

In the latter case no reference is made to the rotational-vibrational motions. In terms of partition functions and energetics the whole system thermodynamics can be described. For isomeric systems it has become customary to

91

distinguish [ 2,4-61 between two categories of quantities: the standard partial terms, AX:, belonging to the individual isomers, and for practical applications, the standard overall terms, AX& into which all the isomers contribute accordingly, i.e. according to their mole fractions or weighting factors, Wi.Finally, in addition to the partial and overall terms a third quantity has been introduced, the so-called isomerism contributions to thermodynamic terms SX, defined as sx,=AXo,-AX;

(3)

Clearly enough, the values of 6X, generally depend on the choice of the reference isomer labelled with i= 1. It is convenient that the most stable species (in the low temperature region) be chosen as the reference structure. Here, X denotes a standard thermodynamic term (enthalpy X= H; entropy X= S; or heat capacity at constant pressure X=C,); however, the treatment can, in principle, be considered for any structure-dependent quantity. It holds for X= C, that (4) where Ep,W,l stands for the so-called isofractional isomerism contribution the heat capacity (see below) :

to

(5) and the isomerism contribution to enthalpy is simply given by 6Hl= i Wi(AHp-AH;) i=l

(6)

The SC,,, term in eqn. (4) considers the effects of changes in composition upon a temperature change and it is therefore called the relaxation isomerism contribution to the heat capacity. The isofractional contribution &&W,ldoes not allow for such changes-the relaxation term is reduced in the latter contribution if the Witerms are considered to be temperature independent (the situation is practically reached in both high and low temperature limits). In fact, eqn. (4) follows from eqn. (6) by temperature differentiation with a due respect paid to the fact that all the terms involved are temperature dependent. DESCRIPTION OF THE ILLUSTRATIVE SYSTEMS

C, aggregates

Altogether six stationary points were found [ 121 on the system potential hypersurfaces, four of them being local energy minima. A cyclic structure with

92 TABLE 1 Energetic5 of the isomeric gas-phase systems studied System

Isomer

AE; (kJmol-‘)

glib (kJ mol’ ’)

Energy approximation

G

Dm Dcohf3c, ) Dmh(“XT)

0.0 46.4 157 335

0.0 41.8 146 328

“Best” in ref. 12

D 6h HDO-HDO

HOD.OHD DOH.OHD

HF-HCl

HF.HCl HCl*HF HF.HCl HCl.HF HF.HCl HCl.HF HF.HCl HCl*HF

HF-HCl HF-HCl HF-HCI

[ 14,151

0.0 0.0

0.0 0.77

BJH/G

0.0 -0.42 0.0 -0.42 0.0 0.25 0.0 0.25

0.0 0.53 0.6 0.51 0.0 1.20 0.0 1.18

SCF [16]” SCF [161d SCF and BSSE [ 161’ SCF and BSSE [ 161d

“Potential energy relative to the lowest structure. ‘The enthalpy change at absolute zero or the ground state energy change, i.e. AEi corrected for zero-point vibrations. “Complete harmonic-frequency set considered. dTorsionaI mode replaced by free internal rotation.

DSh symmetry was the lowest in potential energy, being followed by a linear Dcahstructure with a triplet (3X; ) electronic state. In the most sophisticated approximation of energy used in ref. 12 (“Best”), the energy difference between the structures was about 46.4 kJ mol - ’ (Table 1) . The third isomer was again a linear D cobspecies; however, this had “Zf electronic state symmetry. Finally, the structure having the highest potential energy was another cyclic structure of DGhsymmetry. HDO-HDO dimers Typically, but not exclusively, different isomers are related to different local energy minima. A contra-example can be served by isomers originating from symmetry properties of the wavefunctions of rotation and nuclear spin (“nuclear spin isomerism”: orth- and paru-Hz). Isomers belonging to the same energy minimum can also be produced by isotopic labelling. An interesting example is the dimer (HDO ) 2.Within the usual C,*structureof the water dimer [ 131 there are two possible arrangements of the donor molecule, HOD*OHD and DOH-OHD (the donor molecule is on the left in the schemes). For our

93

purposes the isotopomers were described by means of the flexible BJH/G potential [ 14,151. The deuterium-bonded dimer was found to be lower in the zero-point energy than the hydrogen-bonded one by about 0.77 kJ mol-l (Table 1). (Clearly, within the Born-Oppenheimer approximation the potential energy term for both species is exactly the same.)

HF-HCZ hetero-dimers The third example is again related to different local energy minima, now on the interaction hypersurface of HF and HCl. Two minima were located [ 161 on the hypersurface, viz. the structures HF*HCl and HCl- HF. Their energetics were described within an SCF approach corrected at the MP2 level for electron correlation, and also within the SCF data corrected for basis set superposition error (BSSE). Moreover, two different approaches were considered with the system vibrational motion, too. Either a full six-membered set of frequencies was applied, or the torsional motion involved was treated as free internal rotation. Thus, four different levels of the 1IH”,iterms can be distinguished (Table 1). In all four treatments the HF*HCl isomer is lower in the ground state energy term. RESULTS AND DISCUSSION

The above three isomeric systems were treated in a unified way. In terms of the calculated energetics listed in Table 1 and the rigid-rotor and harmonicoscillator partition functions the mole fractions wi were calculated. Then, the isomerism contributions to the system thermodynamics were evaluated. For methodical and demonstration purposes rather wide temperature intervals were considered. It should be noted, however, that the plots start at a low temperature but not at absolute zero; this is due to the limiting behaviour of some partition functions involved. Observational conditions [ 17,181 for the Ce species are rather extreme, i.e. a temperature of 2500 K. Thus it is relevant to study the isomeric interplay over a rather wide temperature interval (Figs. 1 and 2). The relative stability of the & species systematically decreases (Fig. 1) so that both linear structures finally become relatively more stable. The other cyclic species is insignificant throughout. It is clear that the simple Boltzmann factors are practically useless in understanding the behaviour of this system. Figure 2 shows the isomerism contributions to enthalpy, entropy and heat capacity of C&,all three terms reaching substantially high values. The rather fast mutual stability interchange in the relatively low temperature region is reflected in a sharp maximum in the relaxation SC,,, term (while its isofractional component is quite negligible). The maximum in the SC,,, term is located at a temperature of about 1007 K and amounts to about 78.5 J K-’ mol- ‘.

1000

2000

3000

4000

5000 -T(K)

Fig. 1. Temperature dependence of the weighting factors Wi (-) and simple Boltzmann factors w: ( --- ) for Cs (g ) isomers. The order of the ZUiterms in the high temperature region is (from top (the population of the Debspecies is so small that to bottom): Dmh (?L; ), Dmh ("Z,' ),and D3,, it cannot be seen in the figure ) .

--------

0 1000 2000

3000 4000

5000

-T(K)

Fig. 2. Temperature dependence of the isomerism contributions to enthalpy 6H,, entropy 6&, and heat capacity at constant pressure SC,,, (relaxation term) or 6CP,,,1 (isofractional term (---) ) of the C,(g) isomeric system (the contributions are related to the D3,, species as the reference structure).

95

In fact, about 40.6% of the overall heat capacity at constant pressure of the Cs isomeric system is thus formed at the latter temperature by the isomerism contribution. Although it is a considerably high value it is still not the highest known heat capacity enhancement by isomeric interplay. The highest enhancement has so far been found [ 191 with the SisHs isomeric system with a maximum in the SC,,, term as high as 179.8 J K-’ mol-l (which is about 51.6% of the corresponding overall term). The results for the (HDO), isomeric system (Figs. 3 and 4) are not as impressive as in the previous case. There is again a crossing point of the relative stabilities of both isomers (Fig. 3)) viz. at about 638 K; however, with a further temperature increase the relative populations remain practically unchanged. Within the Born-Oppenheimer approximation the simple Boltzmann factors are exactly equal to 50% each (so that they certainly represent a poor approximation to the rigorous Wivalues). It is evident from Fig. 4 that all the three types of isomerism contributions exhibit a temperature maximum. For the 6H, term the maximum is located at about 157 K and reaches some 0.31 kJ mol-l. According to the rigorous rule [20] the corresponding entropy contribution must have its maximum at exactly the same temperature, and its maximum is about 6.5 J K-’ mol-‘. In a high temperature limit the latter term approaches the value relevant for two optical isomers, i.e. R In 2. Finally, the maximum in the isomerism contribution to heat capacity is reached at about 39 K and amounts to about 4.6 J K-’ mol-‘. The third example studied, isomers in the HF-HCl associating system, exhibits (Figs. 5 and 6) some features similar to those of the water-dimer isomerit isotopomers, this being in relation to rather small isomeric energy separations. Figure 5 shows that the wi weighting factors can be quite sensitive to moderate changes in both energetics as well as in partition functions. In con-

IJ

I

200

400

600

1

800

I

-TlKl

Fig. 3. Temperature dependence of the weighting factors Wi (-) and simple Boltzmann factors w: (--- ) for (HDO ), (g ) isomers. The order of the Witerms in the low temperature region is (from top to bottom): HOD.OHD and DOH-OHD.

96

-1 200

400

600

800 -T(K)

Fig. 4. Temperature dependence of the isomerism contributions to enthalpy 6H,, entropy 6S1, and heat capacity at constant pressure SC,,, (relaxation term) or 6CP,,,1 (isofractional term (---) ) of the (HDO),(g) isomeric system (the contributions are related to the HOD-OHD species as the reference structure).

trast to previous cases, there is no crossing point on the wi temperature dependences; however, there is a point of closest approach of the two curves (with the exception of the case of the BSSE corrected energetics and the full harmonic vibrational set). This point in fact represents the highest degree of coexistence of both species. The positions of the points in the three treatments are 249 K (52.4% of HF*HCl), 181 K (52.7% of HFaHCl), or 274 K (61.4% of HF*HCl) in the approaches without the BSSE contribution combined with the full vibrational set, without the BSSE combined with one free internal rotation, or with the BSSE correction combined with the internal rotation, respectively. The temperature dependence of the SC,,, term presented in Fig. 6 exhibits a sharp maximum in the low temperature region (its height being around 5 J K-l mol-’ ) followed by a minimum and in the treatments involving the partition function of free internal rotation also by a second local maximum. The isofractional term &‘P,W,l,in all three illustrative systems, is negligible, compared with the relaxation one, in the lower temperature region. However, in the high temperature region the terms finally become quite close. A most

97

300

600

904 1200

300

600

900 1200 --T(K)

-T(K)

Fig. 5. Temperature dependence of the weighting factors wi for HF-HCl (g) isomers. The order of the wi terms is (from top to bottom): HF*HCl and HCl. HF. Energetics results without the BSSE correction combined with the full vibrational set, for energetics without the BSSE combined with a free internal rotation, for energetics with the BSSE combined with the full vibrational set, or for energetics with the BSSE combined with a free internal rotation are presented in the upper left, upper right, lower left or lower right parts, respectively.

-2

s -2 B 2 4 \ 2. " 2

4 2 2: u" r 0 LQ

La 1

t = s

0

-2 300

600

900

1200

-T(K)

300

600

900

t

1200 --T(K)

Fig. 6. Temperature dependence of the isomerism contributions to heat capacity at constant pres(isofractional term ( ) ) of the HF-HCl (g ) isomeric syssure SC,,, (relaxation term) or GCP,W,l tem (the contributions are related to the HFeHCl species as the reference structure). For a description of the figure parts see Fig. 5.

---

interesting feature of the relaxation isomerism contribution to the heat capacity is the presence of a sharp temperature maximum rather close to the region of the isomeric relative stability crossing or highest coexistence (i.e. close to the region of relatively fast temperature changes in the Wiweighting factors). Although the isomerism contributions to the thermodynamic terms are not primarily measured quantities, their extreme behaviour can in some cases be

98

conserved in the overall terms and thus observed. Therefore, heat capacity may play a similar role in isomerism phenomena as in phase transitions. ACKNOWLEDGEMENTS

This study was carried out during a research stay at the Max-Planck-Institut fur Chemie (Otto-Hahn-Institut ) supported by the Alexander von HumboldtStiftung. The support of and the valuable discussions with Professor Karl Heinzinger and his kind hospitality, and that of his group, and of the Institute are gratefully acknowledged.

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