J . Phys. Chem. 1991,95, 7694-7699
7694
Comparison between Approximate Methods for Calculating Ionization Potentials and the Use of a-Ionizatlon Potentials as a Measure of Relative Basicity of Azdes Issam Jan0 Chemistry Department, Faculty of Science, United Arab Emirates University, P.O.B. 17551, Al- Ain, U.A.E. (Received: February 21, 1991) Approximate methods for calculating valence-shell ionization energies are compared at the INDO level of approximation. These methods are based on Koopmans, many-body Green function, and perturbation theories. The method based on perturbation theory is presented for the first time in this work. Some characteristics and limitations of these methods are pointed out. In addition, a linear relationship between the a-ionization potentials of azole molecules and their protonation energies is found and analyzed on the basis of Mulliken's resonance structure theory of the charge-transfer complexes. It is concluded that the polarization, charge-transfer, and exchange energies are responsible for the stabilization of the protonated systems, whereas the electrostatic energy plays a rather small and destabilizing role.
Introduction Molecular orbital energies of closed-shell molecules present, according to Koopmans' theorem,' approximate values for the ionization potentials. The SCF molecular orbital energies are, in general, larger in absolute values than the correct potentials. In some molecules Koopmans's theorem also fails to produce the correct symmetry type of the ion resulting from the first ionization of the molecule. The nitrogen molecule is, by now, a classical example of this shortcoming of the theorem.2 A more recent example is produced by some azole molecules? The SCF HOMO in these molecules is of ?r-symmetry, whereas the experimental evidence shows that the ionization from the a-orbital requires less energy than from the ?r-orbital. To obtain accurate first ionization energies, one has to calculate the difference (E+ - E O ) between the exact SCF energies of the ion and the corresponding molecule and add to it the difference between the correlation energies of the two species. The value of (E+ - E O ) is usually smaller than the corresponding Koopmans' ionization potential because the calculation of the SCF E+ allows the MOs to relax upon ionization, thus stabilizing the resulting ion. Theoretically, the difference (E+ - E O ) is also expected to be smaller than the exact ionization energy, because the difference in correlation energy between the molecule and the ion stabilizes the parent molecule relative to the ion resulting in an increase in the ionization potential. However, actual calculations show that taking account of the molecular orbital relaxation energy in certain cases is enough to produce the correct type of the first ionization where Koopmans' theorem fails. This is the case, for example, of the azoles? indicating that in heterocyclic compounds, the molecular orbital relaxation is important and should be taken into account in order to obtain the correct order of the ionization energies. The correction on Koopmans' theorem may be obtained by using the many-body Green's function (MBGF) t h e ~ r y . ~The equation of motion (EOM) method has also been used for calculating valence-shell ionization spe~tra.~-6 An approximate method based on the perturbation theory for calculating valence-shell ionization potentials and electron affinities is proposed in this work. Its performance is compared with Koopmans' theorem and the one-particle MBGF method a t the INDO level of approximation. The perturbation method takes account of the relaxation energy of the ion. The results of calculation reveals that it is relatively more successful than Koopmans' theorem in producing the correct order of the valence-shell (1) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab iniiio Molecular Orbital Theory: Wilcy-Interscience: New York, 1986, and references therein. (2) Cade. P. E.; Sales, K. D.;Wahl, A. C. J . Chem. Phys. 1966,44,1973. (3) Mo, 0.;de Paz, J. L.;Yanez, M. J . Phys. Chem. 1986, 90, 5604. (4) S z a b , A.; Ostlund, S.Modern Quantum Chemisiry: Iniroduction io Aduanced Elecfronic Sirucfure Theory; Macmillan Publishing Co.: New York, 1982; Chapter 7. (5) Corriea, N.; Baker, J . J . Phys. Chem. 1985, 89, 3861. (6) Baker, J . Int. J . Quanium Chem. 1985, 27, 145.
ionization energies. It does, however, underestimate the ionization potentials, as is expected. Numerically, the MBGF ionization potentials appear to have higher a correlation coefficient with experimental measurements than do the other two methods. As an application, the ionization potentials of some azole molecules are studied in relation to their protonation energies. A significant correlation is found between the a-ionization potentials of azoles and their protonation energies. This suggests that the u-ionization potentials could be taken as a measure of relative basicity of the azole compounds. More discussion is given below.
Theoretical Considerations The ionization energy of a valence-shell electron represents a small fraction of the total energy of the molecule, and it is tempting to treat such ionization as a perturbation. If H+a is the Hamiltonian of the ion resulting from a removal of an electron from the molecular orbital 4aof the neutral molecule, then according to Rayleigh-Schrodinger perturbation theory one may write
H"Ca=Ho+XH6 A=-1 (1) where H o is the Hamiltonian of the parent molecule. The perturbation H i may be defined as follows: N
Hi
= hC,
+ Ilf,(i) I
(2)
I=
hCais the energy of the electron in orbital 9, in the field of the core of the molecule. N is the number of electrons in the molecule. The one-electron operator f,(i) is defined in terms of the SCF coulomb and exchange operators corresponding to orbital
fdi) = J a ( i ) - Ka(i)
(3)
It is clear that H i represents the S C F energy of an electron in orbital within the neutral molecule. It will be assumed that the orbital 4a from which an electron is ejected is a frozen orbital. The energy of the ion, E+,, is then equal to E+, = Eo + XE('), + XZE(z)a (4) Eo is the energy of the neutral molecule, and E(')aand E(2)aare first- and second-order perturbation energies, respectively. Putting X = -1, the energy of ionizing an electron from orbital 4., Ia, is given by I, e E+, - E O = + E(2), The first-order energy E(')ais calculated from
(5)
N
E(')a =
(+oIH'aI+o)
=
hCa + ( + d v a c i ) I + o )
=
ocf
hCa + C[Ju6- Ka61 = b
ea
(6)
where +o is the S C F function of the unperturbed molecule, and
0022-365419 112095-7694%02.50/0 0 1991 American Chemical Societv
The Journal of Physical Chemistry, Vol. 95, No. 20, 1991 7695
Relative Basicity of Azoles e, is the energy of the orbital r$a. Equations 5 and 6 show that Koopmans' theorem is, in fact, a first-order perturbation approximation for the ionization potential within the framework of the SCF theory. The second-order term E'2)ais calculated from
(7)
where $n is an excited state of the neutral molecule. The summation in eq 7 is over all possible excited states of the unperturbed molecule. However, since Hb involves only one-electron operators, only single states may give nonvanishing contributions to Therefore, we will consider only spin-adapted singly excited states: = 'pb,where It& is a singlet wave function obtained from the proper combination of Slater determinants arising from promoting one electron from an occupied orbital, r$b, to an unoccupied (virtual) one, 4,. Equation 7 then becomes
+"
TABLE I: Parameters for INDO Calculations (au)' param H C N 0 F ~~
.$
1.24
I, Ir
0.50
tr
-Us
0.4712
+r
+
($'Olwbo')l'vb) I
(9)
Since the excited wave functions are orthogonal to the ground-state function +o, the first term on the right-hand side of the previous equation is zero. With use of Slater's rules for calculating the matrix elements, eq 8 may be expressed in terms of integrals over the molecular orbitals. Combining the results with eqs 5 and 6, one finally gets 1 occ
I , = -fa
vir
12(aa/br) - (ab/ar)12
+ 2 eb cr cb - e, + (rr/bb) - 2(rb/rb)
(10)
The summation term is always negative as can be seen directly from eq 8 where (EO - Elb) is negative. This means that eq 10 gives an ionization potential that is smaller in numerical value than the molecular orbital energy. Since it is known that the SCF MOs's energies are usually larger (in absolute values) than the corresponding ionization energies, this is a step in the right direction. Equation 10 is rigorous within the framework of the perturbation theory, but since it involves only singly excited-state wave functions, it does not take account of the correlation energy difference between the molecule and the ion. The exact ionization potential may be expressed as
I = (E'SCF + E+,,,) - (E'SCF + Eo,) = (E'SCF - EOSCF) + ( E + , - Eocor) (1 1 ) where E+corand E O , , are the correlation energies of the ion and of the molecule, respectively. When a large enough molecular orbital basis is used, eq 10 would be a good approximation of ( P X-F However, since (E+,,,, - EO,) is usually positive, the ionization potential calculated from eq 10 will evidently be smaller than the exact value. It is hoped, however, that eq 10 would predict the correct symmetry type of the first ionization in cyclic compounds, resulting in a qualitative improvement over Koopmans' theorem! An expression for the electron affinity, A, may be derived in a similar way. If one puts X = + I in eq 1 and goes through a similar derivation process, one obtains the following approximate formula: 1 vir 12(rr/bs) - (rb/rs)12 -Ar = E , - Eo = t, + 6 I Cb - C, + ( s s / b b ) - 2(Sb/Sb) (12) r refers to the molecular orbital 4, to which the electron is attached. In the following discussion, we will concentrate our attention on the energies of ionization from the upper valence orbitals. For the purpose of comparison, we recall here the one-particle
2.679 2.322 0.4778 0.3896 3.1000 2.6000
+
r
s
€0
occ occ vir
CCC c
h'a($'Ol'v6)
2.550 2.550 1.1860 0.6373 8.0522 5.8264
many-body Green function (GF) formula in its simplest form for calculating the ionization potential4 -za = ea p a . (13) where the correction term is given by occ vir vir (ra/sb)[2(ar/bs)- (ns/br)] C(2)aa= C C C
The expansion of the matrix N
($OIHbllvb)
2.250 2.250 1.1100 0.4980 4.8300 4.5200
1.950 1.950 0.9870 0.5322 3.5800 3.1100
fi Slater orbital exponent. I ionization potential. U: energy of the electron in the field of the corresponding atomic core. s, r : s-type and r-type Slater orbitals.
b
Elb being the energy of the state element in eq 8 leads to
1.720 1.720 0.7500 0.4122 2.4800 2.1000
I
b
r
+ Cb -
-
+
+
- e, - t b
(ca/br)[2(ac/rb)- ( a b / r c ) ] €0
(14)
The three approximate methods, Koopmans, perturbation, and GF, are compared with each other in the following section.
Actual Calculation and Results The proper comparison between the three methods of calculating the ionization potentials should be done at a high level of ab initio calculation. However, at this stage we are interested in approximate procedures that could be used readily in fairly large molecules. Therefore the methods are applied at the INDO level of approximation.' It is to be noted here that the original parametrization of INDO yields, in general, Koopmans ionization potentials that are 2-5 eV. larger than experimental values.* If, however, the parametrization is modified as described below (see ref 9),a partial improvement may be obtained. The H+ matrix elements are given by
where a and b refer to atoms, p to an atomic orbital, and Zb to the core charge of atom b. ud, is treated as a parameter. The penetration integral in ( 1 5 ) is computed with an s-type Slater orbital having the same exponent ( and principal quantum number n as the orbital p. The off-diagonal elements of the H matrix are approximated by
= -y2sfi#b('# + 'U) (16) S, is the overlap integral, and Z, and Zu are the atomic ionization potentials of the respective orbitals. Other approximations are left the same as in the original method. Table I gives the values of the parameters used in the present calculations. A variety of molecules with different types of bonds and geometries are selected for testing. Table I1 lists these molecules and the results of calculation. Some ab initio ionization energies are also included for comparison. The molecular geometries adopted for the calculation are either experimental or ab initio (6-31G) geometries quoted from ref 1. Azoles molecules geometries are a b initio (6-31G) equilibrium geometries given in ref 3. Examination of Table I1 reveals that in comparison with experimental and ab initio (E+ - E o ) values, Koopmans's theorem HP.Y)
(7) Pople, J. A.; Beveridge. D. L.; Dobosh, P.A. J. Chem. Phys. 1%7,47, 2026.
(8) Halgren, T. A.; Kleier, D. A.; Hall, Jr., J. H.; Brown, L. D.; Lipscomb, W. N. J. Am. Chem. Soc. 1978, 100, 6595. (9) El-Hag, 1. H.; Jano, 1. Anal. Chim. Acra 1989, 226, 305. (10) Dea Kyne, C. A.; Liebman, J. F.; Frenking, G.; Koch, W. J. Phys. Chem. 1990, 94, 2306.
R. B.; Jorgensen, W. L.;Allen, L. C. J. Am. Chem. Soc. 1970, (11)92, Davidson, ,49. (12) Hodgman, C. D., Weast, R. C., Selby. S. M.,Eds.; Handbook of Chemistry and Physics; Chemical Rubber Co.: Cleveland, 1960.
1696 The Journal of Physical Chemistry, Vol. 95, No. 20, 1991
Jan0
TABLE II: Ionization Potentials (au) Obtained at INDO Level of Approximation: Corresponding ab Initio or Exptrimentr) Values Are Listed0 no. molecule Koopmans GF pert. (E+ - E o ) (ab initio)* 1 N2 -44 0.5267 0.5373 0.5265 0.573 -4r) 0.5779 0.6277 0.5778 0.624 --t.(d 0.5595 0.5146 0.4736 2 H2O 0.567 1 0.501 1 0.4396 0.463 -tbl(a) 3 NH3 -%I 0.4595 0.4104 0.3622 0.400 0.4913 -to 0.5475 0.5314 4 FH -e(*) 0.6089 0.5351 0.4713 0.580 0.581 5 CH4 -612 0.5023 0.4824 0.4405 0.529 6 C6H6 -%I 0.3462 0.2890 0.2782 0.353 7 C2H4 ( D 2 h ) -4*) 0.385 0.3839 0.3112 0.4484 0.4060 -4o) 0.4477 0.4492 0.3728 0.4263 8 C2Hz (D-h) -4r) 0.4140 0.4335 0.4863 -40) 0.4961 0.5098 0.3490 9 C2H6 ( 0 3 6 ) -%lg 0.4179 0.4023 0.4202 0.41 I6 -cog 0.4614 0.41 IO IO HC=N -44 0.4807 0.488 1 0.4697 0.4388 0.5439 -44 0.4946 0.5091 11 12 -4.) 0.3760 0.3559 0.3760 0.356 0.4496 -44 0.4496 0.4224 12 cyclobutadiene -e(*) 0.3434 0.3020 0.2786 0.2903 0.3446 0.4193 -40) 0.3723 0.3330 13 cyclopropane -ee# 0.3745 0.3439 0.2868 0.40-0.38 -€e" 0.4656 0.3777 0.3826 14 imidazole -44 0.3755 0.3374 0.2957 0.2885 0.27 14 0.3131 -44 0.3934 0.3004 15 pyrazole -e(*) 0.3777 0.3104 0.2918 0.3061 0.3075 0.3366 -40) 0.3946 0.3185 16 4H-1,2,4-triazole -4*) 0.41 20 0.3311 0.3302 0.3491 0.2189 0.3377 -40) 0.4106 0.2879 17 1 H-1,2,4-triazole -44 0.4144 0.3309 0.3010 0.3458 0.3999 0.3115 0.3087 0.3462 -44 18 2H-tetrazole -4r) 0.4209 0.3705 0.3594 0.3973 0.3422 0.3800 -40) 0.4255 0.3343 19 1H-tetrazole -44 0.4284 0.3661 0.3537 0.4046 0.3327 0.3767 -40) 0.4200 0.3207 20 1H-pentazole -4d 0.4473 0.3463 0.3942 0.4318 0.4381 0.3482 0.3370 0.4237 -4.1 'GF: Green function method (eqs 13 and 14). Pert: perturbation method (eq 10). (E+ - E O ) : (6-31G*)//(6-31G) calculations from references cited. le.& experimental ionization potentials from references cited. *Ab initio potentials are quoted from refs 3, 5, IO, and 18. eExperimental potentials are quoted from refs 4, 8, 1 I , and 12.
produces the wrong type of the first ionization in 25% of the cases studied, and the G F and the perturbation methods fail in 15% and 10% of the cases, respectively. In the azole molecules, the perturbation method agrees with the (E+ - E O ) ab initio calculation in all cases except one (imidazole). This discrepancy must be due to the nature of the INDO approximations used in the calculation because eq 10, when used with ab initio HF S C F molecular orbitals, is expected to give ionization potentials close to the (E+ - E O ) values. It is also interesting to note here that although the azole molecules have *-electron systems, the first ionization in most of them is of u-type rather than *-type. This is in agreement with the fact that azoles are, in general, u-electron donors rather than *-donors. If the order of the MOs is disregarded and the calculated ionization energies are compared with the corresponding measured values (12 cases), the following correlation coefficients are obtained: r = 0.94, 0.86, and 0.84 for the GF, perturbation, and Koopmans's methods, respectively. The regression lines of these correlations are shown together in Figure I , the regression expressions being as follows: GF: I,, sz 0.0646 + 0.903610~, SEE = 0.0301 au (17a) perturbation: Iapz 0.1097 + 0.88141Per,, SEE = 0.0442 au (17b) Koopmans:
I,,
10.0599
+ 0.87 I 31Kwp, SEE = 0.474 au
(17c) where Is, is the approximate value of the ionization energy. This is to be compared with the experimental value I,!,. SEE is the standard error of estimate. The highest deviations from the regression lines are 1.20, 1.86, and 2.50 eV, respectively. Judging
from the correlation coefficient, the slope of the regression, the SEE, and the highest deviation from the regression line, the GF ionization potentials appear to parallel the experimental values the best, followed by the perturbation potentials. On the other hand, the intercepts in the relations 17, as well as the regression lines in Figure 1, show that the perturbation method underestimates the ionization potentials the most, by an average of 0.1 1 au (2.99 eV). The G F and Koopmans' ionization energies are, on the average, lower than the experimental energies by 0.065 au (1.77 eV) and 0.060 au (1.63 eV), respectively. Improved approximate ionization potentials, I, ,may eventually be obtained from the regression relations 17. Tbese relations give ionization energies that are within about 0.82, 1.20, and 1.30 eV, respectively, from the experimental energies. These deviations are the SEE values corresponding to the regressions 17. It is to be remarked here that even though the number of the experimental ionization energies used in the above analysis is modest, the spread of these energies between about 9.60 and 16.98 eV warrants confidence in the conclusions reached. In fact, the results of calculation are in agreement with what might be expected on a purely theoretical basis as discussed in the previous section. To summarize, the GF method gives ionization potentials that are numerically more satisfactory than the potentials obtained from the other two methods. Koopmans' theorem, when combined with a regression correction (eq 17c), may yield satisfactory results on the average, but in individual cases the deviation from the exact ionization potential may be large. The perturbation method has an advantage of producing a more plausible order of the ionization types in heterocyclic compounds than the other two methods. If this method is used at a high-level ab initio calculation, it may be comparable to calculating (E+ - E O ) .
The Journal of Physical Chemistry, Vol. 95, No. 20, 1991 7697
Relative Basicity of Azoles 'C~I 1a.u. 0.6
TABLE 111: Calculated Protonation Ewreies of Azole Molecules
0.4
/yx
0.3
7 /
0.5
no.
molecule imidazole
1 2 3 4 5
pyrazole 4H-1,2,4-triazole lH-l,2,4-triazole 2H-tetrazole 1H-tetrazole 1H-pentazole
6 7
protonation energy, kcal/mol ab initio INDO (6-3 IG*//6-3 1G)" Ab 318 312 307 300 294
293 275
240.1
78
227.0
85
231.8
75 15 84
225.1 209.6 213.4 191.6
80 83 av 80
'Reference 3. The protonation energy corresponds to the most stable protonated species. b A is the difference between the INDO and ab initio protonation energies.
"i X
0
5 0.5
-
0.:
-
0.2 I
I
.
I
0.4
0.5
0.6
I
exp
-
(a.u.)
Figure 1. Regression lines between calculated, la,, and experimental ionization potentials, 1: Koopmans (V). 2 Green function method (O), 3: Perturbation method (X). The symbol R indicates two super-
imposed points, v and X.
As an application, the correlation between the ionization potentials and the protonation energies of the azole molecules is examined in the following section. The contributions to the protonation energy are also analyzed on the basis of Mulliken's resonance structure theorm of the charge-transfer complexes.
Correlation between the Ionization Potentials and Protonation Energies of Azoles Protonation energy in the gas phase is usually utilized as a measure of basicity. Theories have also been advanced to correlate the protonation energy with some properties of the base such as the Is orbital energy of the protonation site.I3J4 It is also suggested that the hybridization of the valence atomic orbitals of the protonation center may be a determining factor of basicity of the azoles.' NMR studies have also been used to establish the order of relative basicities of triazole rings.I5 All of such theories imply that basicity is mainly determined by structural factors. In the present work, the possible relation between the energy of ionization from the a-framework (a-ionization) and the basicity of azoles is investigated. Such a relation may be expected to exist on the ground that azoles are a-electron donors, and the base-acid reaction involves a charge transfer. Using the terminology of Mulliken's resonance structure theory of the charge-transfer complexes,16the protonation energy is the difference between the energy of the protonated or dative system and the energy of the separate azole molecule and proton or nebond system. According to the theory, this energy Eprolmay be expressed as where I,, and A, are the ionization potential of the base and the (13) Martin, R. L.;Shirley, D. A. J . Am. Chem. Soc. 1974,96, 5299. (14)Brown, R. S.;Tse, A. Can. J . Chcm. 1980,58, 694. (IS)Paul, 0.S.;Ford, T. A. S. Afr. J . Chem. 1988,11(3)108. (16)(a) Mulliken, R. S.;Person, W. B. Molecular Complexes; WileyInterscience: New York, 1969. (b) Mulliken, R.S.J. Am. Chem. Soc. 1952, 74, 811.
ab-lnltto
(Lt-IYl
d.".
Figure 2. Regression line between ab initio protonation energies and ab
initio ionization energies (E+ - E o ) (in au).
electron affinity of the acid, respectively. G is an energy term that involves coulomb and exchange parts.I6 In this work, the protonation energy is calculated as the difference between the energy of the azole molecule and the energy of the protonated species using INDO method with the parameters given in Table 1. The geometry of the protonated system is taken simply as the same geometry of the unprotonated molecule with a proton attached to the basic nitrogen atom with a bond length N-H equal to 0.996 A. Table I11 contains the protonation energies obtained. (6-31G*)//(6-31G) a b initio energies from ref 3 are also included for comparison. A in Table 111 is the excess of the INDO protonation energy E rot above the a b initio value. It is clear from Table 111 that IdDO method exaggerates the protonation energy systematically by about 80 kcal/mol, but it does reflect the general trend of the basicity variation. The relative order of basicities of the azole molecules produced by INDO method is slightly different from the order obtained from the ab initio calculation. In both cases, however, the order of basicities is almost the reverse order of the u-ionization potentials. The correlation coefficient between the GF a-ionization potentials (Table 11) and the INDO protonation energies is equal to r = -0.80. When the ab initio protonation energies are used, r becomes equal to -0.91. To check the validity of this result, the correlation coefficient between the a b initio a-ionization energies, (E+ - E O ) (Table 11) and the a b initio protonation energies (Table Ill) is also calculated. It was found equal to -0.995! Figure 2 shows the corresponding correlation line. This result leaves little doubt that the protonation energy of the azoles is related to the energy of ionizing an electron from the a-framework of the molecule. Since the slope of the correlation line in Figure 2 (-0.70) is different from -1, it may be concluded that the term G in eq 18 is also linearly related to the a-ionization potential in the azoles' family. There is no simple explanation to this relationship. A close look at the problem reveals that the protonation process causes considerable perturbation in the entire basic molecule. For
7698 The Journal of Physical Chemistry, Vol. 95, No. 20, 1991
Jan0
TABLE I V Compodtions of Charge Densities in Protonated pYrazoleo
atom 1 qor
qo. q+.. q+o
bq, bq,
1.336 3.707 1.726 3.465 0.39 -0.242
2 0.8173 3.1737 0.4769 3.3171 -0.3404 0.1434
3 1.235 2.932 1.347 2.822 0.112 -0.1100
4 0.8 135 3.1285 0.6039 3.2331 -0.2096 0.1046
5
1.798 3.457 1.847 3.315 0.049 -0.142
6
I
8
9
0.8637
0.9240
0.9004
0.9149
0.7502
0.7957
0.7625
0.8150
-0.1135
-0.1283
-0.1379
-0,0999
10
0.7232
“Atoms are numbered according to Figure 3; o and + refer to neutral and protonated pyrazole, respectively; bq is the increment of charge density.
\
1
fU.057‘10)
/
’
pJ
I).2 7 6 7 9
l(1
/
I’i 260
24u
2711
1
r
-
U.90
}
6
Figure 3. Atomic charge densities of protonated pyrazole. Numbers in parentheses are the densities in the unprotonated pyrazole.
example, a detailed study shows that, upon protonation, the entire set of the occupied molecular orbital energy levels of pyrazole is stabilized by about 7-8 eV for each orbital. At the same time, the core repulsion energy is increased because of the addition of a proton to the molecule. This is also accompanied by a remarkable charge density rearrangement resulting in a net charge transfer and polarization. Figure 3 displays the net charge densities on the different atoms of the protonated species. The numbers in parentheses are the densities in the unprotonated molecule. The acid hydrogen Hlocarries a net charge 0.2768 equal to 1 - 6, where 6 = 0.7232 is the electronic charge transferred from the base molecule to the proton. This charge is withdrawn from the a-structure of the pyrazole molecule. The compositions of the electronic charge densities and their changes are given in Table IV,where qr and q., refer to ?r and u electron densities, and the superscripts o and to the neutral and protonated systems respectively. 6q is the change in density brought about by the protonation reaction. The nitrogen atom, N,, to which the acid proton is bonded suffers a decrease in its a-electron density, and so do the hydrogen atoms of the molecule. There is also an alternate decrease and increase of a-densities on the atoms in the ring. It can also be seen that the ?r-electron densities are readjusted according to the variations of the a-densities. Where there is a decrease of a-density in the ring, there is an increase of *-density and vice versa. All these data indicate that the significance of correlating the basicity to a single atomic parameter within the basic molecule, such as the energy of the 1s orbital or the hybridization on the nitrogen atom a t the site of protonation, is questionable. No correlation whatsoever is found between the protonation energy and the a-charge density on the basic nitrogen atom of the neutral azoles. A significant correlation is found, however, between the protonation energy Epro,and the charge 6 transferred from the base to the acid proton. Figure 4 shows the corresponding correlation line. But the amount of the charge 6 depends not only on the characteristic of the base but also on the type of interaction with the acid probe. Such an interaction involves, in general, coulomb, polarization, charge-transfer, and exchange energies.”
+
0.611
0.62
0.64
0.66
0.08
0.70
0.72
0.74
~ m o i t n t of r h a r a e t r a n s f c r f c
Figure 4. Correlation between the protonation energy and the charge transfer in the azoles family.
Looking back at eq 18, it may be assumed that the term G comprises coulomb (or electrostatic) E,, polarization Epol,and exchange E,,, energies, the charge-transfer energy E,, being implicitly included to a great extent in the term (I - A ) . We may therefore write Since the protonated azoles are positively charged, the electrostatic interaction energy E, between the acidic hydrogen and the rest of the complex molecule at the equilibrium structure is positive (repulsive). If the net atomic charges are considered as point charges, then E,, for the azole systems studied, falls in the range 0.013-0.023 au (8.2-14.5 kcal/mol). The electron affinity of the proton AH+ may be set equal to the ionization potential of the hydrogen atom, 0.5 au, and the term (I, - AH*) lies in the range -0.2 to -0.1 au (-1 25.7 to -62.9 kcal/mol). It is obvious, therefore, that the sum of the charge-transfer and the electrostatic energies ( I , - AH+ E,) accounts for less than half of the protonation energy. These considerations lead to the conclusion that the polarization and exchange energy term ( E + E,,,) constitutes the main part of the protonation energy o c h e azole molecules. Finally, it is to be noted that since E, varies relatively little, as compared with the variation of the protonation energy Eprot the linear relationship between E, and I , implies that the energy (Epl + E,,,) must also be either a slowly varying term or practically constant within the azoles’ family. A similar conclusion was reached by Alcami et al.18 ‘These authors found a linear relationship between the protonation energies and lithiation energies of azoles. They also demonstrated that the bond azole-H+ is basically covalent, whereas the bond azole-Li’ is ionic. They consequently concluded that “the trend in relative basicities is
+
(17) Morokuma, K. J . Chem. Phys. 1971, 55. 1236. (18) Alcami, M.; Mo, 0.: Yanez, M . J . Phys. Chem. 1989, 93, 3929.
7699
J . Phys. Chem. 1991, 95, 7699-7702
Conclusion
are pointed out. In addition, a relation between the a-ionization potentials and protonation energies of azole molecules is established. The nature of the protonation energy is analyzed on the basis of Mulliken's resonance structure theory. It has become quite evident that the stability of the protonated systems of azoles is mainly due to polarization, charge-transfer, and exchange energies. The electrostatic energy, at the equilibrium structure, plays a rather small and destabilizing (repulsive) role. It is also found that the trend in basicity is mainly determined by the u-ionization potentials of the bases considered. This, of course, suggests the use of the a-ionization potentials of the heterocyclic azole compounds as a measure of their relative basicities.
Three approximate methods based on Koopmans, perturbation, and many-body Green function theories for calculating the ionization potentials of electrons from the upper occupied valence-shell molecular orbitals are compared at the INDO level of approximations. Different characteristics and limitations of these methods
Acknowledgment. I gratefully acknowledge the generous support from the Computer Centre of the UAE University, where this research was carried out. Registry No. Imidazole, 288-32-4; pyrazole, 288-1 3- 1; 4H- 1,2,4-
mainly determined by ion-multipole interactions", while the covalent interactions in H+-azoles must be practically constant. The analysis of the protonation energy based on Mulliken's theory, presented above, shows, however, that the trend of basicities is determined mainly by the a-ionization potential rather than by the electrostatic interaction. Besides, the example of pyrazole given above points out the importance of polarization and charge transfer in the formation of the protonated azoles. This conclusion is consistent with what has been found in a previous study on the stability of iodine complexes with ammonia and pyridine.19
(19) Jano, 1. Theor. Chim. Acta 1987, 71, 305
triazole, 63598-7 1-0; lH-I,2,4-triazole, 288-88-0; 2H-tetrazole, 100043-29-6; IH-tetrazole, 288-94-8; 1H-pentazole, 289-1 9-0; pyrazole conjugate acid, 17009-91-5.
Computational Study of the Concerted Gas-Phase Triple Dissociations of 1,3,5-Triasacyclohexane and Its 1,3,5-Trinitro Derivative (RDX) Dariush Habibollahzadeh, Michael Grodzicki, Jorge M. Seminario, and Peter Politzer* Department of Chemistry, University of New Orleans, New Orleans, Louisiana 70148 (Received: March 7, 1991)
Concerted triple dissociation reactions of 1,3,5-triazacyclohexane (I) and 1,3,5-trinitro-1,3,5-triazacyclohexane(11, RDX) were investigated by using semiempirical, ab initio, and local density functional (LDF) methods. Gaseous phase structures and energies for the ground and the transition states of I and I1 are presented and analyzed. The activation barriers for both processes are predicted to be in the range 72-75 kcal/mol. The LDF results are comparable to those obtained by the highest level ab initio procedure [MW(SDTQ)/6-31G*//3-21G]. The dipole moments and the electrostatic potentials associated with the amine nitrogen lone pairs have been computed and are used to interpret the structural and electronic factors involved in the formation of the transition states.
Introduction
As part of a continuing study of the thermal decomposition mechanisms of energetic m ~ l e c u l e s , we ' ~ have calculated the activation energy barriers for the concerted ring decompositions of 1 and 11 in the gaseous phase.
investigate the concerted gas-phase ring fragmentations of I and 11, as given in eqs 1 and 2. I 3H,C=NH (2) +
Methods and Procedure
We optimized geometries through a b initio SCF-MO (Hartree-Fock, HF) calculations with GAUSSIAN 88 and at the 9
1,3,5-lriazacyclohexane,I
1,3,5-trinilro-1,3,5-triaracyclohexane, I1 (RDX)
Several experimental studies of the decomposition of RDX (11) have been reported.s-'O On the basis of these experiments, several competing primary dissociation channels have been suggested, including the scission of N-N02 bonds and the concerted symmetric ring-fragmentation shown in eq 1. In an earlier study of
I1
+
3H$=N-N02
(1)
diazacyclobutane and its dinitro derivative, the energetic requirements for N-N02 bond rupture and for ring decomposition (both in the gaseous phase) were found to be approximately comparab1e.I~~It is accordingly of interest to determine the situation in the case of RDX. Our present objectives are to To whom correspondence should be addressed.
0
~
~
9
~
( I ) Grodzicki, M.; Seminario, J. M.; Politzer, P. Theor. Chim. Acta 1990, 77, 359. (2) Murray, J. S.;Politzer. P. Chemistry and Physics of Energetic Materials; Bulusu, S. N., Ed.; Kluwer: Dordrecht, The Netherlands, 1990; Chapter 9. (3) Grodzicki. M.; Seminario, J. M.; Politzer, P. J . Chem. Phys. 1991, 94, 1668. (4) Seminario, J . M.; Grodzicki, M.; Politzer, P. Density Functional Methods in Chemistry; Labanowski, J. K., Andzelm, J. W., Eds.; SpringerVerlag: New York, 1991. ( 5 ) Farber, M.; Srivastava, R. D. Chem. Phys. feu. 1979.64, 307. (6) Dubovitskii, F. I.; Korsunskii, B. L. Russ. Chem. Reu. 1981,50,958. (7) Zuckermann, H.; Greenblatt, G.D.; Haas, Y. J . Phys. Chem. 1987, 91, 5159. (8) Zhao, X.;Hintsa, E. J.; Lee, Y. T. J . Chem. Phys. 1987. 88, 801. (9) Oyumi, Y.;Brill, T. B. Propellants, Explos., Pyrotech. 1988, 13.69. (IO) Capellos, C.; Papagiannakopoulos,P.; Liang. Y.-L. Chem. Phys. Lett. 1989, 164, 533. ( I I ) Frisch, M. J.; Head-Gordon, M.; Schlegel, H. B.; Raghavachari, K.; Binkley, J . S.;Gonzalez, C.; Defrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R.; Kahn, L. R.; Stewart, J . J . P.; Fluder, E. M.; Topiol, S.; Pople, J. A. GAUSSIAN 88; Gaussian Inc.: Pittsburgh, PA, 1988.
0022-3654/91/2095-7699%02.50/00 I99 1 American Chemical Society
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