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Conditions for favorable unimolecular reaction paths
CONDITIONS
FOR
FAVORABLE
UNIMOLECULAR
REACTEON
L. SALEM
Laboratoire de Ckimie Thdorique (490), Facults des Sciences,
91, Orsay,
PATHS’ France
Received 23 December 1968
The conditions relating favorable reaction paths to the nature of low-lying electronic in the framework first proposed by Bader [1,2]. These conditions involve the precise molecular reLaxability towards the different normal modes of vibration.
This work investigates two proposals put forth by Bader [1,2]: a) in a molecule the normal vibrational mode with the smallest force constant is that whose symmetry is identical with the symmetry of the lowest possible electronic excited state; b) assuming reaction paths to be determined by changes to second order of the electronic energy in the corresponding nuclear displacements, and hence by force constants, the lowestlying excited state whose symmetry corresponds to that of a possible normal reaction coordinate determines the favorable reaction path (i.e., that with the lowest activation energy). Conditions (a) and (b), although they form the basis of the theory, are unsatisfactory as they stand and we give simple counterexamples below. Our purpose is to establish more rigorous. yet more general, conditions relating reaction paths to low-lying electronic states. Condition (a) is based on the well-known perturbation expression for the curvature of the electronic energy relative to the coordinate Q :
where V is the total potential energy operator, and 10) and ]k) represent ground and excited state electronic wave functions with respective energies E, and Ek. The corresponding force constant is obtained by evaluating (1) at the equilibrium nuclear configuration. The first term of (1) represents the energy change due to the nuclear motion within a fixed electronic density 131. * Work supported in part by National Health-Grant GE-12343.
Institutes of
states are established properties of the
It can be thought of as the “classical” force constant. The second term represents the decrease in energy due to the rearrangement of the electronic density as the nuclei are displaced [3,4]. It can be defined as the “relaxability” of the morecule along the coordinate Q. The matrix elements in the numerator can be reduced to one-electron integrals over the transition density pok between ground and excited states:
(2) where v is the one-electron nuclear-electron potential operator. For a one-electron transition between two molecular orbitals oak is simply the product of the two orbital amplitudes. The integral in (2) is a “transition force” with the same features as transition moments, etc. It vanishes unless Pok has the same symmetry as Z’v/aQ, i.e. as Q itself. Whence, for spatially symmetric ground states, a low force constant for modes whose symmetry is identical with that of iowlying excited states and - taking into accoht the energy denominator in (2) - Bader’s condition that the smallest force constant occurs for the mode whose symmetry is that of the Lowest possible lying excited state. In CO2, for instance, comparison of the symmetric C i and antisymmetic C& stretching modes Favors tie Iatter, since the lowest-lyizg excited electronic state of CO2 corresponds to a 1Ci - ‘Xi transition. Now of course the bending mgde of CO2, which has U, symmetry, has a force constant which is an order of magnitude smaller than either stretching mode. The reason is obvious from (1): the classical term {Oi Z2V/~Q2]O) is much smalLer for a bending motion than for a stretching motion, So it must be specified, at the outset, that comparisons based on (1) must be restticted &o nzodes 99
Volume 3, number 2
CHEMICAL PHYSICS LETTERS
with identical classical force constar.ts, i.e. modes involving the same set of internal valency coordinates. Howav;_r a much more important, though less obvious restriction concerns the location of the transition density pot relative to the nuclear displacements. A given excited state Iii> may have the same symmetry as the mode Q but contribute little to the molecular relaxability towards that mode if wok is large near nuclei which are not involved in the motion but small near nuclei which do contribute to Q. Then the integral in (2 ) is Inon-zero by symmetry, but still very small. Hence a low-lying state of appropriate syrnntetry does not @arantee a low force constant for the corresponding Q. For example, ethylene has four CH stretching modes with respective symmetries A,, Big, B2, and B3u_ Undcubtedly the Iowest exc%ed state possessing either of these four symmetries is the familiar N - V v - 1~* excited state with B3, symmetry; other competitors are o - u* states. Yet the B2u mode has the lowest force constant f5]t. indeed the B - 8* transition density is located well above and below the molecular plane so that the in-plane components of the transition forces which act on the carbon and hydrogen atoms are rather small. On the other hand the o(lbIg) - o*(Zb3,) transition, with B2u symmetry, although possessing a larger excitation energy, gives a more significant relaxability via its bigger transition force. It is further gratifying that o(lblg) is the top OCcupied (Torbital and o*(3bgU), rather than o*(4ag), the lowest antibonding orbital as shown by the most recent SCF calculations on ethylene [6] tf , it?. Within the u - u * maniLold, the lowest excited state does impose t!ie ‘lowest force constant. It is therefore necessary that the trc#zsition &i+zsity he ZocaEzed in tke region of nuclear mo~“ion. In practice one wiI1 seek out transitions of proper symmetry between orbitals localized as near as possible to the moving nuclei. We now turn to condition (b). Clearly the pret
tt
constants: Ag = 6.20 x 109 dynes/cm: Big = = 6.15 x lo5 dynes/cm; B2, = 6.01 x 105 dynes/cm; B3u = 6.14 x 105 dynes/cm. Results from ref. 161: Elblg +
0.261 a.~.,
l4a
= - 0.505
a.U.,
E3bsu
= + 0.290 a-u. The symmetry
=
8 designation is that with the molecule in the x-y plane. T. H. Dunning and V. McKay. unpublished calculations with Gaussian orbitals. ReSdtS: EXble = 0.515 a.u.. E3bsu = i 0.261 ax., Qag = + 0.287 a.u. These calculations give an energy of - ‘78L!llO a.u. for the ethylene molecule.
100
diction of potential energy surfaces from the knowledge of the differentials to all orders of the energy at one point alone is an insuperable task. However the second differential alone is certainly insufficient and one should include the third differential which represents the extent of anharmonicity, in order to make reasonably reliable predictions. The information given by the force constant on the energy barrier along a given path can be totally misleading. The fluorine molecule, for instance, has a force constant of 4.73 X X 105 dynes/cm while th: lithium molecule has a force constant oE 0.25 x 105 dynes/cm, almost twenty times smaller. Yet the dissociation energy of F2 (36 kcal/mole) is hardly larger than that of Li2 (25 kcal/mole). Why F2 manages to dissociate almost as readily as Li2, in spite of the important energy curvature at its equilibrium distance, lies in the much greater anharmonicity of the energy in F2 (anharmonic constant 38.23 x 1013 dynes/cm2) than in Li2 (-0.548 X x 1013 dynes/cm2. The third differential of the energy with respect to Q , and thereby the anharmonic constant at the equilibrium configuration, can be obtained by differentiating (1): a3Eo
=(O]~ + aQs aQ3 lo) * 4
c’
E’
(Ek -EoXEn
6
-g
k+O
40)
(Olav/aQIk>(klav/aQla(~lav~a
-Et,)
(olav/aQlk>(kla2v/aQ210) +
Ek-Eo
Force
=
ttt
February 1969
a(~,
+2
C’ k#O
-Ed)
aQ
(01WaQ\k)2 (Ek-E,)"
where the familiar sum-over-states expansion has been used for i3lO>/aQ [‘I]. The first term in equation (3) is the pclassicalW anharmonicity. while the other terms arise from rearrangements of the charge density. For modes which are neither totally symmetric nor degenerate the second, third and Eourth terms vanish. Indeed (OlEJv/8Qlk> and (n 1aV,‘aQ1O} are non-zero only if both 1k) and
Volume 3, number 2
CHEMICAL PHYSICS LETTERS
In) have the symmetry of Q; but then (k Ia V/a Q]n) vanishes. Similarly, if jk) has the symmetry of vanishes because the operator Q, (kpv/a@jo) a2V,,aQ2 is totally symmetric. Only for the totally symmetric mode and certain degenerate modes - for instance the ezg mode in the D6h pointgroup, for which the symmetric direct product [8] ezgxezg contains e2g -will these terms not vanish. However for all vibrational modes the last term does contribute and represents the change in relaxability due to the modification of the excitation energy Ek -EC, as Q increases. We expect this term to be particnlarly important: if Ek - Eo decreases with Q, the relaxability increases and the motion becomes more and more facile. We surmise this to be the case in F2: the 2P(Jg - 3s~ lowest 1~2 excited state probably has an equr*B;brmm _ bond length iarger than that (R. = 1.435 A) in the ground lCg state, or it may even be dissociative. Hence, as the bond is stretched from Ro, the energy difference Ek -E, decreases and there is a large negative contribution to the anharmonicity. On the other hand a large relaxability which decreases rapidly because of an increase in Ek -E, may yet lead to a high energy barrier because of the low anharmonicity. Therelore a reaction path will be favored not just if there is a low-lying state of appropriate symmetry giving a large relaxability, but also if the excitation energy to this excited state decreases as the reaction proceeds. In summary, the molecular relaxability and its characteristics can be used to determine whether a given reaction proceeds with a small energy expenditure. Physically the molecule seeks out excited states which allow the nuclei to move readily in the manner determined by the reaction coordinate. Reasonable predictions on reaction paths can be expected when the fcur following conditions are a22 satisfied:
February 1963
1) there is an excited state of same symmetry as the normal mode corresponding to the reaction coordinate (Bader); 2) the excitation energy is as low as possible (Bader); 3) the transition density is localized in the region of nuclear motion (large transition forces acting on the nuclei); 4) the excitation energy decreases as the reaction coordinate proceeds (large anharmonicity). These condiliocs are valid only for comparison of motions where the classical energy expenditure is the same. Application of these results to organic reaction paths will be found elsewhere [9].
REFERENCES R. F. W. Bader, Mol. Phys. 3 (1960) 137. R. F. W. Bader, Can. J. Chem.40 (1962) 1164. L. Salem, J. Chem. Phys. 38 (1963) 1227. R. F. W. Bader and A. D. Brandrauk, J. Chem. Phys. 49 (i968) 1666. [5] B. L. Crawford, J. E. Lancaster and R. G. Inskeep. 3. Chem. Phys. 21 (1963) 678. IS) _ _ M. B. Robin, A. Basch. N. A. Kuebler, B. E. Kaplan and J. Meinwald. J. Chem. Phys. 48 (i96S) 593?. 171 _ _ W. Byers Brown, Proc. Camb. Phil. Sot. 54 (1958) 251: [S] R. M. Hochstrasser, Molecular Aspects of Symmetq (W. A. Eenjamin, New York, 1966) pp. 172-173. [9] L. Salem, J. Am. Chem. Sot., submitted for publication. [l] [2] [3] [4]
101
FOR
FAVORABLE
UNIMOLECULAR
REACTEON
L. SALEM
Laboratoire de Ckimie Thdorique (490), Facults des Sciences,
91, Orsay,
PATHS’ France
Received 23 December 1968
The conditions relating favorable reaction paths to the nature of low-lying electronic in the framework first proposed by Bader [1,2]. These conditions involve the precise molecular reLaxability towards the different normal modes of vibration.
This work investigates two proposals put forth by Bader [1,2]: a) in a molecule the normal vibrational mode with the smallest force constant is that whose symmetry is identical with the symmetry of the lowest possible electronic excited state; b) assuming reaction paths to be determined by changes to second order of the electronic energy in the corresponding nuclear displacements, and hence by force constants, the lowestlying excited state whose symmetry corresponds to that of a possible normal reaction coordinate determines the favorable reaction path (i.e., that with the lowest activation energy). Conditions (a) and (b), although they form the basis of the theory, are unsatisfactory as they stand and we give simple counterexamples below. Our purpose is to establish more rigorous. yet more general, conditions relating reaction paths to low-lying electronic states. Condition (a) is based on the well-known perturbation expression for the curvature of the electronic energy relative to the coordinate Q :
where V is the total potential energy operator, and 10) and ]k) represent ground and excited state electronic wave functions with respective energies E, and Ek. The corresponding force constant is obtained by evaluating (1) at the equilibrium nuclear configuration. The first term of (1) represents the energy change due to the nuclear motion within a fixed electronic density 131. * Work supported in part by National Health-Grant GE-12343.
Institutes of
states are established properties of the
It can be thought of as the “classical” force constant. The second term represents the decrease in energy due to the rearrangement of the electronic density as the nuclei are displaced [3,4]. It can be defined as the “relaxability” of the morecule along the coordinate Q. The matrix elements in the numerator can be reduced to one-electron integrals over the transition density pok between ground and excited states:
(2) where v is the one-electron nuclear-electron potential operator. For a one-electron transition between two molecular orbitals oak is simply the product of the two orbital amplitudes. The integral in (2) is a “transition force” with the same features as transition moments, etc. It vanishes unless Pok has the same symmetry as Z’v/aQ, i.e. as Q itself. Whence, for spatially symmetric ground states, a low force constant for modes whose symmetry is identical with that of iowlying excited states and - taking into accoht the energy denominator in (2) - Bader’s condition that the smallest force constant occurs for the mode whose symmetry is that of the Lowest possible lying excited state. In CO2, for instance, comparison of the symmetric C i and antisymmetic C& stretching modes Favors tie Iatter, since the lowest-lyizg excited electronic state of CO2 corresponds to a 1Ci - ‘Xi transition. Now of course the bending mgde of CO2, which has U, symmetry, has a force constant which is an order of magnitude smaller than either stretching mode. The reason is obvious from (1): the classical term {Oi Z2V/~Q2]O) is much smalLer for a bending motion than for a stretching motion, So it must be specified, at the outset, that comparisons based on (1) must be restticted &o nzodes 99
Volume 3, number 2
CHEMICAL PHYSICS LETTERS
with identical classical force constar.ts, i.e. modes involving the same set of internal valency coordinates. Howav;_r a much more important, though less obvious restriction concerns the location of the transition density pot relative to the nuclear displacements. A given excited state Iii> may have the same symmetry as the mode Q but contribute little to the molecular relaxability towards that mode if wok is large near nuclei which are not involved in the motion but small near nuclei which do contribute to Q. Then the integral in (2 ) is Inon-zero by symmetry, but still very small. Hence a low-lying state of appropriate syrnntetry does not @arantee a low force constant for the corresponding Q. For example, ethylene has four CH stretching modes with respective symmetries A,, Big, B2, and B3u_ Undcubtedly the Iowest exc%ed state possessing either of these four symmetries is the familiar N - V v - 1~* excited state with B3, symmetry; other competitors are o - u* states. Yet the B2u mode has the lowest force constant f5]t. indeed the B - 8* transition density is located well above and below the molecular plane so that the in-plane components of the transition forces which act on the carbon and hydrogen atoms are rather small. On the other hand the o(lbIg) - o*(Zb3,) transition, with B2u symmetry, although possessing a larger excitation energy, gives a more significant relaxability via its bigger transition force. It is further gratifying that o(lblg) is the top OCcupied (Torbital and o*(3bgU), rather than o*(4ag), the lowest antibonding orbital as shown by the most recent SCF calculations on ethylene [6] tf , it?. Within the u - u * maniLold, the lowest excited state does impose t!ie ‘lowest force constant. It is therefore necessary that the trc#zsition &i+zsity he ZocaEzed in tke region of nuclear mo~“ion. In practice one wiI1 seek out transitions of proper symmetry between orbitals localized as near as possible to the moving nuclei. We now turn to condition (b). Clearly the pret
tt
constants: Ag = 6.20 x 109 dynes/cm: Big = = 6.15 x lo5 dynes/cm; B2, = 6.01 x 105 dynes/cm; B3u = 6.14 x 105 dynes/cm. Results from ref. 161: Elblg +
0.261 a.~.,
l4a
= - 0.505
a.U.,
E3bsu
= + 0.290 a-u. The symmetry
=
8 designation is that with the molecule in the x-y plane. T. H. Dunning and V. McKay. unpublished calculations with Gaussian orbitals. ReSdtS: EXble = 0.515 a.u.. E3bsu = i 0.261 ax., Qag = + 0.287 a.u. These calculations give an energy of - ‘78L!llO a.u. for the ethylene molecule.
100
diction of potential energy surfaces from the knowledge of the differentials to all orders of the energy at one point alone is an insuperable task. However the second differential alone is certainly insufficient and one should include the third differential which represents the extent of anharmonicity, in order to make reasonably reliable predictions. The information given by the force constant on the energy barrier along a given path can be totally misleading. The fluorine molecule, for instance, has a force constant of 4.73 X X 105 dynes/cm while th: lithium molecule has a force constant oE 0.25 x 105 dynes/cm, almost twenty times smaller. Yet the dissociation energy of F2 (36 kcal/mole) is hardly larger than that of Li2 (25 kcal/mole). Why F2 manages to dissociate almost as readily as Li2, in spite of the important energy curvature at its equilibrium distance, lies in the much greater anharmonicity of the energy in F2 (anharmonic constant 38.23 x 1013 dynes/cm2) than in Li2 (-0.548 X x 1013 dynes/cm2. The third differential of the energy with respect to Q , and thereby the anharmonic constant at the equilibrium configuration, can be obtained by differentiating (1): a3Eo
=(O]~ + aQs aQ3 lo) * 4
c’
E’
(Ek -EoXEn
6
-g
k+O
40)
(Olav/aQIk>(klav/aQla(~lav~a
-Et,)
(olav/aQlk>(kla2v/aQ210) +
Ek-Eo
Force
=
ttt
February 1969
a(~,
+2
C’ k#O
-Ed)
aQ
(01WaQ\k)2 (Ek-E,)"
where the familiar sum-over-states expansion has been used for i3lO>/aQ [‘I]. The first term in equation (3) is the pclassicalW anharmonicity. while the other terms arise from rearrangements of the charge density. For modes which are neither totally symmetric nor degenerate the second, third and Eourth terms vanish. Indeed (OlEJv/8Qlk> and (n 1aV,‘aQ1O} are non-zero only if both 1k) and
Volume 3, number 2
CHEMICAL PHYSICS LETTERS
In) have the symmetry of Q; but then (k Ia V/a Q]n) vanishes. Similarly, if jk) has the symmetry of vanishes because the operator Q, (kpv/a@jo) a2V,,aQ2 is totally symmetric. Only for the totally symmetric mode and certain degenerate modes - for instance the ezg mode in the D6h pointgroup, for which the symmetric direct product [8] ezgxezg contains e2g -will these terms not vanish. However for all vibrational modes the last term does contribute and represents the change in relaxability due to the modification of the excitation energy Ek -EC, as Q increases. We expect this term to be particnlarly important: if Ek - Eo decreases with Q, the relaxability increases and the motion becomes more and more facile. We surmise this to be the case in F2: the 2P(Jg - 3s~ lowest 1~2 excited state probably has an equr*B;brmm _ bond length iarger than that (R. = 1.435 A) in the ground lCg state, or it may even be dissociative. Hence, as the bond is stretched from Ro, the energy difference Ek -E, decreases and there is a large negative contribution to the anharmonicity. On the other hand a large relaxability which decreases rapidly because of an increase in Ek -E, may yet lead to a high energy barrier because of the low anharmonicity. Therelore a reaction path will be favored not just if there is a low-lying state of appropriate symmetry giving a large relaxability, but also if the excitation energy to this excited state decreases as the reaction proceeds. In summary, the molecular relaxability and its characteristics can be used to determine whether a given reaction proceeds with a small energy expenditure. Physically the molecule seeks out excited states which allow the nuclei to move readily in the manner determined by the reaction coordinate. Reasonable predictions on reaction paths can be expected when the fcur following conditions are a22 satisfied:
February 1963
1) there is an excited state of same symmetry as the normal mode corresponding to the reaction coordinate (Bader); 2) the excitation energy is as low as possible (Bader); 3) the transition density is localized in the region of nuclear motion (large transition forces acting on the nuclei); 4) the excitation energy decreases as the reaction coordinate proceeds (large anharmonicity). These condiliocs are valid only for comparison of motions where the classical energy expenditure is the same. Application of these results to organic reaction paths will be found elsewhere [9].
REFERENCES R. F. W. Bader, Mol. Phys. 3 (1960) 137. R. F. W. Bader, Can. J. Chem.40 (1962) 1164. L. Salem, J. Chem. Phys. 38 (1963) 1227. R. F. W. Bader and A. D. Brandrauk, J. Chem. Phys. 49 (i968) 1666. [5] B. L. Crawford, J. E. Lancaster and R. G. Inskeep. 3. Chem. Phys. 21 (1963) 678. IS) _ _ M. B. Robin, A. Basch. N. A. Kuebler, B. E. Kaplan and J. Meinwald. J. Chem. Phys. 48 (i96S) 593?. 171 _ _ W. Byers Brown, Proc. Camb. Phil. Sot. 54 (1958) 251: [S] R. M. Hochstrasser, Molecular Aspects of Symmetq (W. A. Eenjamin, New York, 1966) pp. 172-173. [9] L. Salem, J. Am. Chem. Sot., submitted for publication. [l] [2] [3] [4]
101