J . Phys. Chem. 1989, 93, 4422-4429
4422
polaron ground state has been argued for by Nagaoka. Does it cross over to an antiferromagnet for smaller U and larger 6, or to a spin liquid, or both? Is the spin liquid similar in nature to that described by the RVB picture, or is it just a Fermi liquid with unusual parameters, or something else again? In which part of the parameter space is superconductivity favored? These are some of the questions needing answers. Regarding two-band models, even less is known. Here, the basic parameters are (cp - e,,), 6 , and the d-p interaction U,, ( U is assumed large always). In the context of the mixed-valent semiconductor SmS, as (eP - td) decreases, a first-order transition to a low excitation gap, mixed-valent semiconductor occurs. Does this continue to be true for the spin cuprates? What is the effect of doping on the phenomena in either case? Is Udp crucial for superconductivity? For negative charge-transfer energy ep t d , the Anderson lattice model is used for magnetic moments in metals and for heavy fermions. Is there a smooth crossover in properties as a function of charge transfer? It is quite likely that exploring these questions seriously will lead to a better understanding of the rich variety of electronic phenomena in transition-metal oxides. It may also point the way to new classes of systems which, because of optimal pairing, are really high-temperature superconductors. Given the kind of the experimental and theoretical attention the oxides are getting, this does not seem an unrealistic prospect. Many strategies for the synthesis of new families of high T, superconductors seem possible. In the oxide family alone, there
are many possibilities. One possibility is to look for oxides with low 02--cation charge-transfer energy (e.g., layered nickelates), favoring formation of oxygen holes. One need not restrict oneself to two-dimension,al oxides alone. The discovery of Bal-,KXBiO3 ( T , 30 K) suggests the possibility of other three-dimensional oxides exhibiting high T,. Ba1-,K,BiO3 itself is interesting, showing a large isotope effect,E4but no static magnetic orders5 (unlike in the two-dimensional cuprates).
-
Note Added in Proof. There have been a few developments of interest in the past few months. Among them, mention must be made of the discovery (i) of a series of P b cupratesS6of the type Pb2Cal,Ln~r2Cu308,containing CUI+in high proportion^,^^ (ii) of the electron superconductorE8Nd2,Ce,Cu04, and (iii) of a series of cupratesg9of the type T1Cal,Ln,Sr2Cu207+6 showing electron or hole superconductivity depending on x. (84) Hinks, D. G.; Richards, D. R.; Dabrowski, B.; Marx, D. T.; Mitchell, A . W. Nature 1988, 335, 419. (85) Uemura, Y. J.; Sternlieb, B. J.; Cox, D. E.; Brewer, J. H.; Kadono,
R.; Kempton, J. R.; Kiefl, R. F.; Kreitzman, S. R.; Luke, G.M.; Mulhern, P.; Riseman, T.;Williams. D. L.: Kossler. W. J.: Yu.X. H.: Stronach. C. E.; Subramanian, M. A.; Gopalakrishnan, J.; Sleight, A. W. Nature 1988, 335, 151 (86) (87) (88) (89)
Cava, R. J.; et al. Nature 1988, 336, 21 1. Rao, C. N . R.; et al. Phys. Reu., in press. Tokura, Y . ;Takagi, H.; Uchida, S. Nature 1989, 337, 345. Rao, C. N. R.; Ganguli, A. K.; Vijayaraghavan, R. Phys. Reo. B, in
press.
ARTICLES Bond Orders from ab Initio Calculations and a Test of the Principle of Bond Order Conservation G . Lendvay Central Research Institute for Chemistry, Hungarian Academy of Sciences, P.O. Box 17, H-1525 Budapest, Hungary (Received: April 27, 1988: In Final Form: October 31, 1988)
Bond orders and valence indexes of atoms were calculated from ab initio wave functions using the definition suggested by Mayer. Changes of these quantities were investigated under conditions occurring in chemical reactions. Calculations were performed to reveal the dependence of bond orders on bond length. The results are compared with Pauling’s bond order-bond length relation. Bond orders were also calculated in triatomic metathesis reactions. Bond orders are able to describe the similarity of the transition state of reactants or products. Along the minimum-energy path the principle of conservation of bond order is valid to a good approximation. The obtained correlations are applied to a bond energy-bond order type model to estimate trends in reaction series.
I. Introduction Methods of computational quantum chemistry are able to produce physical properties of molecules with increasing accuracy and reliability. It is not possible however to calculate simple qualitative properties like valence of atoms or the bond order between two atoms in a molecule. The quantity calculable from a b initio wave functions which is often related to the strength of bonds in molecules is Mulliken’s overlap population CpEA~vEBPauSpu. but i t is not close to the integer bond order values expected. The recent formulation of valence indexes of atoms and bond orders between atoms in molecules suggested by M a ~ e r ’ can - ~ bridge the gap between the “overdetailed” infor0022-3654/89/2093-4422$01.50/0
mation supplied in the form of large matrices by theoretical chemistry and the need of chemists for qualitative information. Mayer’s method has been successfully applied to a large number of stable molecule^,^^^-^ and a preliminary study of bond orders ( I ) Mayer, I. Chem. Phys. Lett. 1983, 97, 210; 1985, 117, 396 (addendum). (2) Mayer, 1. Int. J . Quantum Chem. 1986, 29, 73. (3) Mayer, I. Int. J . Quantum Chem. 1986, 29, 477. (4) Baker, J . Theor. Chim. Acta 1985, 68, 221. (5) yillar, H. 0.; Dupuis, M. Chem. Phys. Lett. 1987, 142, 59. (6) Angyin, J. G.;Pokier, R. A. Manuscript in preparation. (7) Mayer, I. In Modelling of Structure and Properties of Molecules; Maksif, Z. B., Ed.; Ellis Harwood: Chichester, 1987; p 145.
‘01989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 11, 1989 4423
Bond Orders from ab Initio Calculations in chemical reactions also provided promising results.’ In this work we intended to apply the definition of ab initio bond orders and valence indexes when the system is not in equilibrium geometry. Bond orders, obviously, are expected to depend on the distance between atoms. This dependence is described empirically by the formula of Pauling,Io which was originally derived from equilibrium bond order-equilibrium bond length relations. The concept of bond order (defined tentatively by Pauling’s formula) found some fruitful applications in chemical kinetics and in examination of transition-state s t r u c t ~ r e s . ~ l - The ’ ~ principle of conservation of bond order, namely, the assumption that along the minimum-energy path of metathetic reactions the sum of bond orders of the making and breaking bonds is constant and is equal to unity, serves as a basis for the familiar BEBO (bond energybond order) The direct theoretical check of the validity of the principle has not been possible so far, although it was found to be valid in some quantum chemical studies using Pauling’s f0rmu1a.l~ Bond order may serve as a measure of the progress of the reaction in theories of structure-reactivity correlation~,~~-’’ partly to express the similarity of the transition state to reactants or products and partly as a variable in quantitative argumentations. The data related to a b initio bond orders permit one to construct a model of the BEBO type for investigation of reaction series and to obtain a theoretical confirmation of the conclusions of more heuristic models. In this paper we first briefly review the method of calculation of atomic valences and bond orders from ab initio wave functions. Then in section 111 changes of bond orders in stretched diatomic molecules are investigated while in section IV their behavior in reactions is presented. Section V contains the discussion of the conclusions concerning the propensity rules that can be derived based on the findings of the preceding sections. 11. Calculation of Bond Orders and Atomic Valences from ab Initio Wave Functions and Their Properties According to Mayer,2 the definition of the bond order between atoms A and B in a molecule is
+
where S is the overlap matrix, P = Pa Po is the total spinless “density matrix”, and F = P* - Pb is the spin density matrix. (P“ and Pb are the usual ”density matrices” for spins a and (3, respectively.) The total valence of atom A is composed of parts used in the bonds and a part that remains free: VA =
C BAB + F A
BZA
(2)
The free valence of atom A is given by
(3) The physical interpretation of these definitions is well-established:2~20-21 bond order reflects exchange effects as B A B is the (8) (a) Mayer, 1. Int. J . Quantum Chem. 1985, 26, 151. (b) Mayer, I. J . Mol. Srrucr.: THEOCHEM 1987, 149, 81. (9) Lendvay, G. J . Mol. Struct.: THEOCHEM 1988, 167, 331. (10) Pauling, L. J . Am. Chem. SOC.1947, 69, 542. (11) Parr, C.; Johnston, H . S. J . A m . Chem. SOC.1963, 85, 2544. (1 2) (a) Johnston, H. S. Gas Phase Reaction Rate Theory; Ronald Press: New York, 1966. (b) Johnston, H. S. Adu. Chem. Phys. 1960, 3, 131. (13) Wolfe, S.; Mitchell, D. J.; Schlegel, H. B. J . Am. Chem. SOC.1981, 103, 7692. I (14) Dunning, Jr., T. H . J . Phys. Chem. 1984, 88, 2469. (15) (a) Marcus, R. A. J . Phys. Chem. 1968, 72,891. (b) Cohen, A. 0.; Marcus, R. A. Ibid. 1968, 72, 4229. (16) Agmon, N.; Levine, R. D. Isr. J . Chem. 1980, 19, 330. (17) Agmon, N . J . Chem. SOC.,Faraday Trans. 2 1978, 74, 388. (1 8) Levine, R. D. J . Phys. Chem. 1979, 83, 159. (19) Varandas, A. J. C.; Formosinho, S. J. F. J . Chem. SOC.,Faraday Trans. 2 1986, 82, 953. (20) Mayer, 1. Int. J . Quantum Chem. 1983, 23, 341. (21) de Giambiagi, M. S.; Giambiagi, M.; Jorge, F. Theor. Chim. Acta 1985, F8, 337.
two-atom contribution to the normalization of the exchange part of the second-order density matrix. It is also connected to the exchange contribution to the two-atom interaction energy in the chemical Hamiltonian energy partition scheme.2,20On the other hand, B A B has a well-defined statistical meaning, in that for single-determinant wave functions it is the correlation of the fluctuation between the electron densities on atoms A and B.21 The total valence index VA can be written in the form VA =
~ Q A-
C (PS),,(PS),,
(4)
a,v€A
where QA = CPEA(PS), is Mulliken’s gross population on atom A.‘ As shown by several transformations, VA is the sum of bonding abilities of natural hybrid orbitals on atom A.2 The free valence index F A describes the effective number of electrons that belong to A and are unpaired.2 The usefulness of definitions 1-3 has been investigated by several authors,’,“ and they were found to provide chemically meaningful and reasonable values that correlate very well with the more detailed information supplied by the ab initio calculations. The calculated valence and bond order indexes depend on the basis set of the calculation. It was found5s6that the calculations with minimal basis and those using large balanced basis sets (of the 6-31 1G** quality) provide the best values. This is in agreement with Mayer’s reasoning in an early work,’suggesting that the definition of bond order applies to the covalent contribution to the bonding. Ionic contributions do not appear in the indexes defined in eq 1-3 and cause a decrease. Similar effects are expected when the basis set is unbalanced or contains functions extending throughout the molecule. Mayer’s definitions are in line with the earlier definitions valid in semiempirical method^.^,^*'*^ Some alternative formulas were suggested by Gopinathan and Jug22 and by Natiello and Med r a n ~ .Discussions ~~ concerning the relations and usefulness of these definitions appeared in ref 4, 6, and 24-26. In the present work ab initio calculations were performed using 0~~ The levels the MONSTERGAUSS~’and G ~ U s s l ~ N - 7packages. of calculations were single-determinant closed-shell R H F and UHF, and in order to include electron correlation, MC-SCF and GVB calculations were also performed. In the latter studies configurations resulting from excitations from the highest occupied u orbital to the lowest unoccupied u orbital were taken into account. Although formulas 1-4 were originally derived for single-determinant wave functions, they can be generalized to correlate wave functions, t 0 0 . ~The ~ ~ calculated bond order and valence indexes based on MC-SCF and GVB wave functions were also reported as r e a ~ o n a b l e . ~ . ~ 111. Bond Length Dependence of Bond Orders and Free Valences in Diatomic Molecules On the basis of chemical insight, one expects that the bond order of a chemical bond in a covalent diatomic molecule decreases from the appropriate integer value at the equilibrium distance down to zero at large internuclear separations. The total valence indexes of the atoms are usually expected to be constant during the bond stretching “process” while free valence indexes of atoms should increase from zero up to the total valences. The behavior of bond orders and valences with increasing bond length is investigated here in different kinds of diatomic species: for the simplest molecule H2, for the heteronuclear HF, for the free radical O H , and for the multiply bonded species C N . The effects of basis set (22) Gopinathan, M . S.; Jug, K. Theor. Chim. Acta 1983, 63, 497. (23) Natiello, M. A,; Medrano, J. A. Chem. Phys. Lett. 1984, 105, 180. (24) Mayer, I. Theor. Chim. Acta 1985, 67, 315. (25) Mayer, I. Chem. Phys. Lett. 1984, 110, 440. (26) Tapas, Kar; Sannigrahi, A. B.; Mukherjee, D. C. J . Mol. Struct.: THEOCHEM 1987, 153, 93. (27) MONSTERGAUSS: Peterson, R. M.; Poirier, R. A . Department of Chemistry, University of Toronto and Memorial University of Newfoundland, St. John’s, Newfoundland, Canada. (28) GAUSSIAN~O:Hehre, W. J.; Lathan, W . A,; Ditchfield, R.; Newton, M. D.; Pople, J. A QCPE No. 23. (29) Gilliom, R. D. J . Chem. Phys. 1976, 65, 5027.
4424
The Journal of Physical Chemistry, Vol. 93, No 11, 1989
m
'
Lendvay
06-
-7L 20
-1 00
--110
10
20 Bond length/A
30
I
10
Figure 1. Bond length dependence of bond order (BHH),total valence ( VH). and free valence (FH) indexes of hydrogen atoms (upper part) and of energy (lower part) in the H2 molecule as obtained in two-configuration MC-SCF calculations: full lines, minimal STO-3G basis; dashed lines, 6-31 1G** basis; dotted line, chemists' bond order ~ z H Hcalculated according to eq 5
I
1
99 7
I
. 3
m
10
I
I
20 Band length/A
I
"
J
30
.I.
I
20 Bond lengthlh
I
I 1
30
2.
998
Y C
1
I\
I
\*/
99 9
/
I
10 Bond
I
1
20 lengthlh
Figure 2. Bond length dependence of bond order ( E H F ) , total valence (VH and VF),and free valence (FH and FF) indexes of atoms (upper part) and of energy (lower part) in the HF molecule as obtained in two-configuration MC-SCF calculations: full lines, minimal basis; dashed lines, 4-3 1G basis; dotted line, chemists' bond order nHFcalculated according to eq 5 .
and the level of calculation were also considered. The bond length of these species was increased from 0.4 to 2.5 or 3.0 8, in 0.1-8, steps. The expectations are satisfied very well for the H2 molecule. Results of the calculations are shown in Figure 1. They were obtained in MC-SCF calculations using minimal STO-3G and extended 6-31 1G** basis sets. The bond order is very close to unity in the region of the equilibrium distance. With increasing internuclear distance the bond order first decreases slowly and then it falls off steeply. When the energy smoothly approaches its limiting value, the bond order also approaches zero slowly. The free valence index of H as a function of bond length is complementary to the bond order-bond length curve and also fulfills the expectations. The same trends were observed with both basis sets, the only remarkable difference being that the bond order descends slightly slower with increasing bond length in the case of the larger basis. Similar observations were made by others as weL6 In the heteronuclear H F molecule the free valence indexes increase according to the expectation for both series of MC-SCF calculations performed using minimal and 4-3 1G basis sets (Figure 2). I n the range of the equilibrium bond length, however, the
Figure 3. Bond length dependence of bond order ( B O H ) , total valence (Vo and VH), and free valence (Fo and FH)indexes of atoms (upper part) and of energy (lower part) in the OH radical as obtained in (a) UHF and (b) two-configuration MC-SCF calculations: full lines, minimal basis;
dashed lines, 4-31G basis; dotted line, chemists' bond order nOH calculated according to eq 5 . bond order and the total valence indexes of H and F behave irregularly. Obviously, at small internuclear separations some highly ionic terms appear in the minimal basis wave functions, decreasing covalency and reducing total valences. This deficiency is transmitted to the bond orders which are smaller than expected by the same amount as total valences are. The shape of the bond order-bond length curve becomes reasonable from a distance of about 1.0 8, where the total bond order approaches unity. The bond orders and total valence indexes obtained in the calculation with split-shell double-{ basis are even more difficult to interpret. Probably, as discussed in ref 1, the appearance of exaggerated ionic terms is responsible for the small values obtained in the range of the equilibrium distance. When the total valence indexes of H and F approach unity, indicating that the calculation can be considered reliable, the decrease of the bond order becomes similar to that obtained with minimal basis but it is slightly slower. At distances above the equilibrium bond length the bond orderbond length dependence in H F is very similar to that in H2 for both types of basis sets; the corresponding curves can actually be shifted to overlap. The OH radical serves as an example for the class of free radicals. The molecule was calculated at the U H F and MC-SCF levels. As it is usually observed, U H F calculations (Figure 3a) give a relatively steep increase of the total energy when the bond length increases from the equilibrium value and turns sharply to the limiting energy level. In accordance with the steep increase of the energy, the bond order falls off steeply. In the region of
Bond Orders from a b Initio Calculations
The Journal of Physical Chemistry, Vol. 93, No. 11, 1989 4425 I
I
I
I
I
I
q
=,
-0.10
-0.1s
I:
Bond length/A Figure 4. Bond length dependence of bond order (BcN),total valence (VC
and VN), and free valence (Fc and F N ) indexes of carbon and nitrogen (upper part) and of energy (lower part) in the CN radical as obtained in UHF calculations using minimal basis. the equilibrium bond length irregularities of the type observed in the case of the H F molecule appear: if minimal basis is used, the bond orders and total valence indexes are smaller than they could be expected by the same amount. The split-shell basis results in a well on the total valence-bond length curve near the equilibrium bond length which is transmitted to the bond order values, too. (This behavior seems to be a pecularity of the 4-31G basis set.) The free valences of 0 and H atoms increase in a monotonous way, which is more or less reasonable. The results of the MC-SCF treatment of the same molecule (Figure 3b) are in agreement with those of the U H F calculations, the only exception being that the changes in both the energy and bond order and valence indexes are slower and smoother. The C N radical with the multiple bond between C and N does not show significantly different behavior, as it can be Seen in Figure 4. Some characteristics that are worth mentioning are that the 4-31G basis set overestimates the bond order and the total valence indexes of the multiply bonded atoms at small interatomic distances and causes a wavy decrease of the bond order. The valence of carbon decreases from four at small to two at large interatomic distances. This reflects that carbon enters the molecule in the promoted valence state of valence four as opposed to valence two characterizing the free atom. As a conclusion, one can state that for the calculation of bond order and valence indexes the minimal basis set is appropriate if one is interested in the values of these quantities in the equilibrium geometry of a molecule. The bond orders and valences calculated by using this basis are not very sensitive to small changes in the geometry around the equilibrium one. So, if an equilibrium geometry of good quality is obtained by optimization using an extended, but unbalanced, basis set that provides good energies but not reliable bond indexes, a calculation of these indexes using the same geometry and STO-3G basis set will give chemically more meaningful results. If the basis set applied in the geometry optimization is balanced like the 3-21G** or 6-31 1G** sets, the bond order and valence indexes are also reasonable. The calculated trends in bond orders can be compared with the predictions obtained using Pauling’s definition of “chemists’ bond order”1%12 AB = exP[-b(RAB - ROAB)] (5) where RAB is the actual interatomic distance, ROAB is the equilibrium bond length of the single bond, and b = 3.85 is a universal constant. ( b is sometimes treated as an adjustable parameter.i5.i6v28)This relation was suggested as appropriate to describe the correlation of equilibrium bond lengths corresponding to single, double, and triple bonds with bond multiplicities. So, strictly speaking, one would not expect them to be appropriate for the description of bond order-bond length relations within a
In(BHH)
Figure 5. (a) Binding energy ( VHH)as a function of bond order in H1: full line, results of two-configuration MC-SCF treatment in 6-31 1G** basis; dashed line, least-squares fit according to eq 6 (p = 2.07). (b) log-log plot of the binding energy-bond order relation.
given molecule. We compared ab initio and chemists’ bond orders in Figures 1-3 (dotted lines) and found several useful features of the a b initio definition. It can be seen from the data on H2 (Figure 1) that, at bond lengths where the bonding energy is half the dissociation energy of the molecule, the bond order calculated from q 5 is less than 0.1 while that provided by Mayer’s definition is 0.7. Although the value of 0.7 may be an overestimation, it does better reflect the fact that a considerable bonding interaction is still in effect between the atoms. It seems that the soft decrease of the ab initio bond orders is more reasonable for the bond rupture “process”. Similar conclusions can be drawn from the results on the other molecules studied. There is another advantage of Mayer’s definition, namely, that bond orders do not increase rapidly when the bond is compressed but rather they change only slowly. If one calculates bond orders from eq 5 for bond lengths smaller than the equilibrium distance, one obtains values significantly exceeding unity. The appearance of so large bond orders is difficult to explain, e.g., for the H2 molecule having altogether two electrons. The increase of bond orders expected from eq 5 at very small interatomic distances where the total energy exceeds the dissociation energy is unreasonable, too. The trends obtained in the a b initio calculations seem to be more reasonable: in molecules consisting of single-valence atoms the bond order turns slowly to unity with decreasing bond length in the vicinity of the equilibrium bond length. The calculations discussed above also provide some semiquantitative information concerning the bond energy-bond order relationship in free molecules. For the H2 molecule, this connection is represented in Figure 5a which is constructed from the results of a series of MC-SCF calculations using the 6-31 1G** basis set. As expected, at low bond orders (corresponding to large interatomic separation) the binding energy of the molecule is small. When the bond order increases (the bond length decreases), the energy of the molecule decreases toward the equilibrium value. At large bond orders (very short bond lengths) a discrepancy is
4426
TABLE I: Optimized Exponents in the Fitting Equation (6) for ab Initio Bond Order-Bond Length Data of Diatomic Molecules
molecule
method
H2 HZ HF HF OH OH OH
MC-SCF MC-SCF MC-SCF MC-SCF MC-SCF MC-SCF UHF UHF
OH
basis set 6-31 IC** STO-3G 4-31G STO-3G 4-31G STO-3G 4-3 1 G STO-3G
+
-
X F
CI Br
BHX 0.50 0.45 0.49
BHH 0.12 0.10 0.08
VH
1.0 1.0 1.0
FH 0.38 0.45 0.42
Fx 0.09 0.14 0.14
+
F CI
Br
0.08 0.50 0.72
BX-H2
0.08 0.14 0.11
vH2 FH~ 1.00 0.03 1.00 0.42 1.00 0.65
"The geometries were taken from ref 14. Results of U H F calculations using STO-3G basis.
observed: the energies turn up with increasing bond order. The reason is that the a b initio bond order does not stop increasing when the bond length decreases beyond the equilibrium value where energy starts to increase along the repulsive wing of the potential energy curve. This obviously results from the fact that the bonding nature of the electronic wave function (a measure of which bond order is) does not stop increasing at the equilibrium bond length, but the increase in nuclear repulsion overcompensates this electronic bonding contribution. This observation, however, does not depreciate the reasonable behavior at extended bond lengths. These general trends were found to be the same in other molecules, too. Considering the extended bond length wing of these curves, one concludes that the energy decreases in a convex manner with significant curvature. The binding energy is often taken to be a power function of bond order V = -DBP
0.796 0.856 0.925 1.004 1.098 1.213 1.361 1.927
-
+
-
Fx VHl FHI 1.02 0.88 1.00 0.01 0.99 0.35 0.99 0.06 1.01 0.18 1.00 0.05 vx
1.436 1.280 1.188 1.123 1.073 1.032 0.997 0.941
0.07 0.16 0.28 0.45 0.63 0.79 0.88 0.95
0.92 0.80 0.64 0.45 0.27 0.14 0.07 0.01
F,
FHI FH2
0.01 0.04 0.08 0.10 0.09 0.05 0.02 0.00
0.89 0.78 0.63 0.45 0.27 0.12 0.04 0.00
0.03 0.1 1 0.24 0.43 0.62 0.78 0.88 0.98
* Variable (I of
TABLE V Bond Orders and Free Valences along the MEP of the ClH' H2" Reaction CI H'H2
TABLE 111: Bond Orders and Valences at the Saddle Point of Collinear Triatomic Abstraction Reactions X H'H2 XH' + H2"
BHI-H2 0.92 0.44 0.23
+
Results of GVB calculations using STO-3G basis. the bond strength-bond length method; ref 34.
"The geometries were taken from ref 14. The calculations were performed at the U H F level using STO-3G basis.
BX-HI
+
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9
+
Vx 1.09 1.04 1.13
-
TABLE IV: Bond Orders and Free Valences along the MEP of the Reaction F H*H2 FH' H2"
progress variableb R F - H I8,, R H I - H 8, ~ , PF-HI B#-H2
P 2.07 1.84 1.46 1.24 1.53 1.30 1.55 1.25
TABLE 11: Bond Orders and Valences at the Saddle Point of Collinear Triatomic Exchange Reactions H XH HX H"
x
Lendvay
The Journal of Physical Chemistry, Vol. 93, No. 11, 1989
(6)
where Vis the binding energy at bond order B, D is the dissociation energy of the molecule, and p is a constant. This relation was suggested by Johnston,I2 who studied dissociation energies of a series of stable compounds. Parr and Johnston1' found that p is often close to unity. Figure 5b shows a log-log plot of the binding energy-bond order pairs obtained in our a b initio calculations. The plot is curved in contrast to the straight line expected from eq 6, showing that a power function can describe this dependence only approximately. To have a qualitative impression, we applied eq 6 as a nonlinear fitting function for the data at extended bond lengths and obtained p values that significantly differ from unity (see Table I ) . The reason for the difference may be that Parr and Johnston studied stable molecules and not the same molecule with different bond lengths. IV. Bond Orders in Chemical Reactions Bond order is often considered as a measure of the progress of the r e a c t i ~ n ' ~and J ~ *is~used ~ in several approximate theories of structure-reactivity relationship^.^^-'^.^^,^^ We calculated bond orders at the saddle point of some thermoneutral, exothermic, and (30) Basilevsky, M . V.; Weinberg, N. N.; Zhulin, V . M . I n t . J . Chem. Kine!. 1979, I I , 853 (31) Agmon, N . I n t . J . Chem. Kinet. 1981, 13, 333.
progress variableb 0.2 0.3 0.4 0.5 0.6 0.7 0.8
+
8, R H I - H ~ A, 1.706 0.799 1.597 0.833 1.520 0.873 1.460 0.920 1.411 0.977 1.370 1.051 1.334 1.156
Fcl
&-HI
&-HI,
0.14 0.22 0.33 0.44 0.55 0.66 0.77
0.85 0.74 0.63 0.51 0.39 0.29 0.19
FHI F H ~
0.78 0.02 0.06 0.66 0.03 0.14 0.54 0.05 0.23 0.42 0.05 0.34 0.30 0.05 0.46 0.19 0.04 0.59 0.100.020.71
Results of U H F calculations using STO-3G basis. *Variablea of the bond strength-bond length method, ref 34. endothermic atom-transfer reactions (see Tables I1 and 111) to reveal the applicability of a b initio bond orders for this purpose. For thermoneutral reactions, as expected, the bond order of both the forming and breaking bond is close to 0.5. For highly exoH2) the forming bond has not yet thermic reactions (as F developed when the reaction achieved the barrier while the breaking bond is almost intact; Le., "the transition state is reactant-like". For endothermic reactions the reverse applies; the transition state resembles the products. So ab initio bond orders are able to reflect the trends predicted by Hammond's principle.32 In order to establish the change of bond orders during metathesis reactions, we performed MC-SCF and U H F calculations on the
+
HI
+ H2H3
+
H 1 H 2+ H 3
(R1)
reaction using the 6-31 1G** and STO-3G basis sets. Bond order and valence indexes were calculated at geometric arrangements corresponding to the minimum-energy path (MEP) taken from L ~ u Figure . ~ ~ 6a shows the energy, bond orders, and free valences as a function of the arc length along the MEP obtained at the MC-SCF level. The right-hand side of the figure shows the results calculated with 6-3 1 lG** basis while the left-hand side shows those obtained by using minimal basis. Figure 6b is arranged similarly with the results of U H F calculations. All four series of calculations provided the expected results. Bond order and valence indexes obtained by taking into account electron correlation seem to be more reliable as the bond orders of both bonds at the barrier are very close to 0.5 in the MC-SCF calculations while UHF treatment of the system resulted in somewhat lower values (0.45 with 6-31 1G** and 0.46 with STO-3G). A rather important feature of all the calculations is that the sum of bond orders of the making and breaking bonds is very close to unity. The difference is indicated by the small values of the free valence of the central H atom (FH2)which remains below 0.05 in each calculation except for one. This observation can be considered as a theoretical verification of the principle of conservation of bond order. We obtained similar results when we used U H F and the GVB treatment suggested by DunningI4 for reactions X
+ H'H2
-
X H 1 + H2
(32) Hammond, G. S. J . A m . Chem. SOC.1955, 77, 334 (33) Liu, B . J . Chem. Phys. 1973, 58, 1925.
(R2)
The Journal of Physical Chemistry, Vol. 93, No. 11, 1989 4427
Bond Orders from a b Initio Calculations
r-- - -_-
-155
;3
k-1605
--160
d - - E(STO-3GJ - . J ! 6 - 3 1 1 G x x )
,
,
,
I
I
,
,
. 2 B
-:--,-I65
I
Arc length along the M W d
0.2
'
06 OL
c8 0 2
--160
I
-10
3 x
+r
--160;
-08
, -06
,
I
I
I
I
-06 - 0 2 0 02 04 Arc length along the MEPI&
I
06
08
-
I .-I65 10
Figure 6. Energy E , bond orders ( B H I - ~and z 8 H 1 4 ) and free valence ( F H i and FHz)along the MEP of reaction R1. Level of calculation: (a) two-configuration MC-SCF, (b) UHF. Basis: 6-31 1G**on the righthand side and STO-3G on the left-hand side of the figures.
\-
1
07
10
15 Bond length/&
1
I
i
0.8
Figure 8. Potential profile of a series of reactions as a function of the bond order of the making bond. Calculated from eq 10 with DBc = 80 kcal mol-', p = 1.5.
6
3 -1 55-
06 BAB
Chemists' bond order
.G
0.k
I
20
Figure 7. Comparison of ab initio H-H bond orders as a function of bond length in free H2and in reactions R2. Chemists' bond order nHH is also
plotted (dotted line). with X = F, CI. The results are collected in Tables IV and V. In the calculations the coordinates of the M E P were estimated from a semiempirical treatment.34 (Cf. ref 9; the system was taken to be collinear.) As the principle of bond order conservation was originally formulated by using chemists' bond order defined by eq 5, it was interesting to compare the a b initio and chemists' bond orders in the reacting system. In Figure 7 , a b initio bond orders are plotted as functions of bond length in the H2 molecule, along the MEP of reactions R1 and R2 (X = F, C1) and chemists' bond order from eq 5. The agreement of a b initio and chemists' bond orders is surprising in light of our earlier conclusion that in free molecules a b initio and chemists' bond orders differ significantly. The resolution of this contradiction is that bond orders decrease with increasing bond length much faster in reactive systems than in free molecules (Figure 7). The reason for the faster decrease is (34) Lendvay, G.; LQszld, B.; Berces, T. Chem. Phys. Lett. 1987,137,175. The calculated MEP coordinates agree very well with available ab initio calculations of ref 33 and Bender et al. (Bender, C. F.; O'Neal, S. V.;Pearson, P. K.; Schafer. H. F. Science 1972, 176, 1412).
that in reacting systems there is a possibility of redistribution of electrons from the given bond to the other. The resemblance of the curves corresponding to chemists' bond order and a b initio bond order in reactions accounts for the fact that the M E P suggested by the BEBO method is so successful (see e.g. ref 35 and 36) and conclusions drawn for reaction series by the use of eq 6 are so c o n ~ i s t e n t . ' ~ ~ ' ~
V. Structure-Reactivity Correlations In the previous sections we have shown that the principle of conservation of bond order is satisfied in the cases we studied. We also obtained some semiquantitative relations between bond energy and bond order. This is just the information needed to construct a model of the potential profile of metathesis reactions based on the principles of the BEBO method. The model is used here to obtain some insight into the features governing the rules manifested in a series of related reactions. We expect that if the parameter measuring the reaction series is changed, the reaction * ' ~the ~ ~properties ~ rate, or rather the free energy of a ~ t i v a t i o n , ' ~and of the transition state32change systematically (see later). If the only parameter governing these changes is reaction free energy or the closely related reaction heat, the series is called a Bransted series.16J8 In BEBO type models, the potential energy profile of an atom-transfer reaction A
+ BC-.
AB
+C
(R3)
is composed of several terms, the most important being the energy contribution of the forming and breaking bonds. (As a first approximation, we neglect the terms resulting from the interaction of A and C as it is done in ref 15-1 9.) If one assumes that the bond energy is in one-to-one correspondence with bond order, irrespective of whether the bond is forming or breaking or connects a pair of atoms in a free molecule, one can use eq 6 with the value of p obtained for a free molecule. By virtue of the principle of bond order conservation, the "degree of reaction" is uniquely determined by the order of the forming bond BAB, and one obtains VMEP= -DABBAB' - DBCBBC'+ DBC
where D, are the dissociation energies of the reactant and product molecules, and the energy is now measured from the level of the reactants. The merits of this equation were analyzed by Marcusi5 in the limit when the exponents p and q are close to one. The equation was extensively used in investigations of reaction se(35) Truhlar, D. G. J . A m . Chem. SOC.1972,94, 7584. (36) Berces, T.; Dombi, J. In?. J . G e m . Kinef. 1980,12, 123. (37) Levine, R. D. Isr. J . Chem. 1985,26, 320.
4428
The Journal of Physical Chemistry, Vol. 93, No. I 1 1989
Lendvay
~
ries.16*17*19,30,31 According to our results, however, the exponents p and q are usually far from unity so that the expansion into a Taylor series used in ref 15 is not needed (and is not valid). The appearance of the VMEp vs BAB profiles is reasonable even if p and q are far from one. A series of profiles are shown in Figure 8 where DBc, the dissociation energy of the breaking bond, is kept constant. Increasing the dissociation energy of the forming AB molecule causes the potential barrier to shift toward the reactant limit. For very endothermic reactions the barrier is close to the product limit, just as formulated by Hammond's principle.32 The height of the barrier also follows the propensity rules being higher for more endothermic reactions. Expressing the potential energy in terms of the reaction heat AEO = DBC -DAB, one obtains the form
s
0 8-
'0 0 6 -
:I,,
I
/
/
,
20
-80
0
-40
40
Reaction heat / kcal mol-' which is similar to that used by Agmon and L e ~ i n e . ~The ~,~~ mixing term is now Figure 9. Approximate structure-reactivity plots based on eq 10. Upper:
Agmon and LevineI6 showed that M(BAB) must be a concave function with limiting values M ( 0 ) = M(1) = 0 and must be symmetric in BAB and 1 - BAB. The first two requirements are met by the form (9), but M(BAB) is symmetric only if p = q. This means that a reaction series can be expected to be a Brtansted series if the exponents of the bond energy-bond order relations are close to each other for each member of the series. In this case (p = q ) the potential profile is VMEp = AEOBAg
+ DBC( 1 - BABP - ( 1 - BAB)P)
(IO)
This form makes it possible to derive analytical formulas describing the propensity rules expected (see the Appendix). When p > 1 as we found in section 111, the following conclusions can be drawn: (i) The location of the barrier B*AB is an increasing function of the reaction heat. (ii) The barrier height also increases with increasing endothermicity, and the (reaction heat dependent) Brtansted coefficient is closely related to the degree of reaction at the barrier. (iii) The barrier height is a concave function of AEO and is quasi-linear at both the very exothermic (AEo > 0) limits. The concave nature of the barrier height-reaction heat plots conforms with the reactivity-selectivity principle. This principle of physical-organic chemistry states that within a series of reactions the difference in the reactivity of highly exothermic members of the family is smaller than the difference in the reactivity of very endothermic members even if the thermodynamic "driving force" differs by the same amount.37,39-42It is worthwhile emphasizing that this principle is expected to be only valid for a Bransted series. The conclusions discussed above are illustrated in Figure 9 where, as functions OC reaction heat, the barrier heights are plotted in the lower part, and the location of the barrier B*AB together with the Bransted slope a is shown in the upper part. The data were calculated for a realistic set: DBC = 80 kcal mol-' and p = 1.5. The change of the Brtansted slope CY = B'AB' is parallel to that of the position of the barrier so that it can serve as a measure of the "degree of reaction" at the "transition state" of the system. VI. Conclusion The definitions of bond orders and atomic valences given by Mayer seem to be useful in the treatment of chemical reactions. The behavior of free molecules and reacting systems is reasonable. It seems to be important from the point of view of chemical kinetics that the principle of bond order conservation is justified to a good (38) Agmon, N.; Levine, R. D.J . Chem. Phys. 1979, 71, 3034. (39) Dewar, M . J . S . The Molecular Orbital Theory of Organic Chemistry; McGraw-Hill: New York, 1969. (40) Jencks, W. P. Chem. Reo. 1985, 85, 511. (41) Pross, A. Adu. Phys. Org. Chem. 1977, 14, 69. (42) Buncel, E.;Wilson, H. J . Chem. Educ. 1987, 64, 475.
location of the barrier (B',,) (full line) and the slope of the barrier height-reaction heat curve (dashed line). Lower: increase of barrier height with increasing reaction heat from exotherm to endotherm limits (DBc = 80 kcal mol-', p = 1.5).
approximation from a b initio theoretical calculations. Since a proper partition of the energy contributions in the ab initio theory is not available, the use of models similar to BEBO is reasonable at least for prediction of trends. We applied a very approximate relation for the description of changes of energies with bond orders in reactions. The very simple model based on this relation is far from complete but provides insight into the factors governing structure-reactivity relationships, the understanding of which is very important for chemists. Acknowledgment. The author is indebted to Dr. I. Mayer for many helpful discussions, to Prof. T. Birces for critical reading of the manuscript, and to Drs. J. G. Angyan and P. R. Surjiin for some helpful pieces of advice. Appendix The propensity rules of reaction series can be transparently investigated analytically when the exponents p and q for the making and breaking bonds are equal to each other. In this case the potential profile along the MEP is - BABY+ DBC
VMEP= -DABBAB~ -
(All
which is equivalent to eq 10. At the top of the barrier the derivative of VMEp with respect to BAB is zero, and one gets from this requirement that the location of the barrier is B'AB = 1/(1
+ r)
(A21
where
At this point the second derivative
is negative if p > 1, so that we have really a maximum. (The results of the a b initio calculations show that p > 1; see Table I). For very exothermic reactions ( A E O is large and negative and DAB is large and positive), BiAB 0, the barrier is in an early position while for endothermic reactions (DAB decreases, AEO is 1, the barrier is late. positive and increases), BAB The derivative of BiABwith respect to AEO shows the direction and speed of shift of the location of the barrier when the reaction heat is changed in the reaction series. It is easy to show that a t a fixed value of parameter DBC the derivative
-
-
4429
J. Phys. Chem. 1989, 93, 4429-4435 is positive if p > 1 so that the barrier is shifted toward the product limit when the reaction heat increases. In the limiting cases of very exothermic and very endothermic reactions the shift of the barrier location is slow as in these limits dBtAB/dAEo 0. The height of the barrier
Therefore,
vb
is a concave function of reaction heat because
-+
vb
= AEoB*Ag + DBCM(B*AB)
(A6)
also increases with reaction heat. (Recall that this quantity has a sign and not only a magnitude.) The Brernsted coefficient a is closely related to the location of the barrier a = dVb/dAEO = BJA#
(447)
and, according to eq A2, it is an increasing function of
AEO.
is positive at any AEo. Both if Dm increases (the reaction becomes more and more exothermic) and if it decreases (the reaction gets more endothermic), da/dAEo tends to zero; Le., the barrier height-reaction heat relation (the "free energy relation") becomes linear in the limits of very exothermic and very endothermic reactions. Registry No. H2, 1333-74-0; HF, 7664-39-3; O H , 3352-57-6; H, 12385-13-6; HCI, 7647-01-0; HBr, 10035-10-6; F, 14762-94-8; CI, 22537-15-1; Br, 10097-32-2.
Vibrational Studies of Reactive Intermediates of Aromatic Amines. 2. Free-Radical Cation and Dication Resonance Raman Spectroscopy of N ,N ,N',N'-Tetramet hylbenzidine and N ,N ,N',N'-Tetraethyibenzidine V. Guichard, A. Bourkba, 0. Poizat,* Laboratoire de Spectrochimie Infrarouge et Raman, CNRS, 2 rue Henri Dunant. 94320 Thiais. France
and G. Buntinx Laboratoire de Spectrochimie Infrarouge et Raman, CNRS, USTLFA, BLit. CS, 59655 Villeneuve d'Ascq Cedex, France (Received: June 13, 1988; In Final Form: December 19, 1988)
The resonance Raman spectra of the radical cation and dication are reported for various isotopic derivatives of the N,N,N',N'-tetramethylbenzidine (TMB) and N,N,N',N'-tetraethylbenzidine (TEB). Complete vibrational assignments are proposed. The spectra are consistent with the expected quinoidal conformation of the framework in both ions. This conformation is characterized by significant frequency increases of the v(inter-ring) and v8(N-ring) modes with respect to the ground-state spectra, and by the strong resonance enhancement of various bands due to vibrations of the N(alkyl)* groups. The radical cation is very comparable to the biphenyl radical cation concerning the ring-ring configuration, but the quinoidal distortion is extended to the amino groups by reason of the conjugation of the nitrogen n-pairs of electrons with the ring *-electrons. The Raman excitation profiles in the contour of the visible absorption have been measured for the radical cation TEB" and satisfactorily fitted by using a theoretical model based on a single Franck-Condon process.
Introduction Aromatic amines are efficient photoreductors which are model compounds for the investigation of photochemical electron-transfer processes.14 Extensive work on the excited-state reactivity of aromatic amines has been performed using time-resolved methods of transient analysis, such as absorption and emission spectroscopies, chemically induced dynamic nuclear polarization (CIDNP), or ESR spectroscopy. However, resonance Raman spectroscopy is certainly an unequaled technique to obtain detailed, specific information on molecular structures, and the recent development of time-resolved Raman spectroscopy makes possible the direct investigation of transient structures. The first paper of this series5 was devoted to the resonance Raman study of N,N,N',N'-tetramethyl- and N,N,N',N'-tetraethyl-p-phenylenediamine derivatives (TMPD/TEPD) in the radical cation (S") and first triplet (TI) states. Complete vibrational assignments have been achieved and revealed significant analogies between the radical and triplet structures: in both cases quinoidal-type distortions with strengthening of the N-ring bonds ~
~~~
(1) Kavarnos, G. J.; Turro, N. J. Chem. Rev. 1986, 86,401. (2) Cohen, S. G.; Parola, A.; Parsons, G. H. Chem. Rev. 1973, 73, 141. (3) Masuhara, H.; Mataga, N. Acc. Chem. Res. 1981, 14, 312. (4) Malkin, Y. N., Kuz'min, V. A. Russ. Chem. Rev. 1985, 54, 1041. ( 5 ) Poizat, 0.;Bourkba, A.; Buntinx, G.; Deffontaine, A.; Bridoux, M. J . Chem. Phys. 1987, 87, 6379.
0022-3654189 12093-4429$01.50/0
-
-
take place, in agreement with the resemblance of the T, T, So+ absorption spectra. These results were based and So+* mainly on two types of information: the observation of isotopic effects, and the comparison with rigorous assignments previously established for the ground state. Both arguments appeared essential for interpreting the Raman results. We are now interested by the N,N,N',N'-tetramethylbenzidine (TMB) and -tetraethylbemidine (TEB) molecules. The following
TMB: X = C H 3 T E B : X =C2H5
paper6 will be concerned with the first triplet states TI T M B and TI TEB. This paper deals with the radical cation ( S o + )and the dication (S2+)of these compounds. Both species are known t o be produced by chemical oxydation,'J by electr~lysis,~ or by photolysis.l"12
Resonance Raman studies of the photolytic
(6) Guichard, V.; Buntinx, G.; Poizat, O., following paper in this issue. (7) Kratochvil, B.; Zatko, D. A. Anal. Chem. 1968, 40, 422. (8) Takemoto, K.; Matsusaka, H.; Nakayama, S.; Suzuki, K.; Ooshika, Y. Bull. Chem. SOC.Jpn. 1968, 41, 164. (9) Fritsch, J. M.; Adams, R. N. J . Chem. Phys. 1965, 43, 1887. (10) Alkaitis, S. A.; Gratzel, M. J . Am. Chem. SOC.1976, 98, 3549. (11) Narayana, P. A,; Li, A. S. W.; Kevan, L. J. Am. Chem. SOC.1981, 103. 3603. 0 1989 American Chemical Societv
