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On the non-crossing rule for potential energy surfaces of polyatomic molecules
Volume 15, number 4
CHEMICAL PHYSICS LETTERS
ON THE NON-CROSSING
RULE FOR POTENTIAL
1 September 1972
ENERGY SURFACES
OF POLYATOMIC MOLECULES K. Razi NAQVI
Department of Chemistry, The University, Sheffield, $3 7HF, UK Received 23 May 1972
It is shown that the non-crossing rule for the intersection of potential energy curves of diatomic molecules applies also to potential energy surfaces of polyatomic molecules.
The non-crossing rule is generally proved [ 1 - 3 ] by showing that two conditions must be met if crossing is to occur. In a diatomic molecule, only one parameter - the internuclear distance, R - is available for variation; potential energy curves will, therefore, not cross unless they belong to different irreducible representations of the symmetry group of the hamiltonian, because one of the conditions is then alatomatically satisfied at all R. It has been argued [1, 2, 4] that an equivalent statement cannot be made for polyatomic molecules; for, by varying more than one parameter, the two ineluctable requirements can easily be satisfied. It will be shownhere that for a molecule with n variable parameters, (n + 1) conditions must be fulfilled for crossing; hence the non-crossing rule applies to polyatomic molecules as well as to diatomic molecules. The proof presented earlier [3] can easily be extended to polyatomic molecules. As the discussion is concerned with the symmetry properties of the adiabatic potential surfaces, it will be convenient to use normal mode displacements as the variable parameters. Consider a contact of two surfaces, so that E l ( 0 ) = E2(0 ) ,
(1)
where the argument 0 designates the complete nuclear configuration in which the contact occurs. As pointed out earlier [3], this condition does not by itself guarantee contact, for the two surfaces may be degenerate in all nuclear configurations. A contact of the two surfaces in the configuration 0 means that an additional demand, viz., 634
E I ( Q I ' Q2 ..... Qn) 4=E2(Q1 , Q2 ..... Qn)'
(2)
must be met; i n t h e above inequality the Q's stand for normal mode displacements from the contact configuration, and at least one Q must be non-vanishing. Repetition of the analysis carried out in ref. [3] leads to the equation
(@°I@HI@°)=O,
(3)
where
6H = H(q; QI' Q2 ..... Qn) - H(q; O) and q~°,2 are the eigenfunctions of the two surfaces in the contact configuration; H is the hamiltonian, and q denotes the complete set of electronic coordinates. H(q; Q1, Q2, "", Qn) and H(q; O) will henceforth be abbreviated as H(Q) and H(O), respectively. By expanding 6 H of eq. (3) in a power series and retaining only the firstrorder terms, one obtains n
(~ OI(~ H/~Qi)o I ~ o) Qi = 0 .
(4)
i=1 Since each term in the sum depends only on one Qi, all Qi's being independent variables, the sum can be zero only if each term vanishes. Thus eq. (4) embodies n conditions and, together with eq. (1) forms a set of (n + 1) conditions which must be full'died if the two surfaces are to have a contact. The essential difference between a diatomic and a polyatomic molecule is that, in the former, aH/bR is
Volume 15, n u m b e r 4
CHEMICAL PHYSICS LETTERS
1 S e p t e m b e r 1972
of necessity totally symmetric, but in the latter,
belonging to different irreducible representations of
aH/aQi call be symmetric or antisymmetric. Were
H(o).
this not so (i.e., if only symmetry-conserving displacements were possible in polyatomic molecules) one could easily formulate the following rule for polyatomic molecules: potential energy surfaces of the same symmetry cannot touch as the various internuclear distances are varied. We now take account of the non-totally symmetric displacements and discuss the contact of potential energy surfaces in a real polyatomic molecule. The adiabatic surfaces and the displacements will be classified by means of the point group of H(O). It will be convenient to rewrite eq. (4) as
Let us now return to case 1 and examine the contact of two potential energy surfaces in the configuration 0. To discuss whether this degeneracy contravenes the non-crossing rule, we must decide whether the group of H(O) or H(Q) is more suitable for our purpose. Now H(Q) belongs to a group (say B) of lower symmetry than that of the group (say A) of H(O), and B will be a subgroup of A. As we are interested in the behaviour of surfaces for non-vanishing Q's, it will be more expedient (and illuminating) to classify the surfaces in terms of the irreducible representations of B (rather than A as we have been doing so far). Two eigenvalues which are degenerate in A split, under the action of Qa (which would now transform according to the totally symmetric representation of B), into two levels belonging to two different irreducible representations of B. Case 1 now represents accidental contact (at Qa = 0) of two surfaces of different symmetry and does not really violate the non-crossing rule. It is worth remarking here that when one confines attention to a particular displacement, say Qk, one essentially reduces the polyatomic molecule under consideration to the status of a diatomic molecule; that the molecule possesses other variable displacements Ql(l :k k) becomes irrelevant, for variations in al cannot influence the integral (~ OI(a H/~ak)OI ~ 0) which must vanish for contact to occur. In their discussion of intersection of potential energy surfaces of polyatomic molecules, Herzberg and Longuet-Higgins [4] constructed an example to demonstrate that crossing can take place even if it is not demanded by symmetry. They considered an equilateral triangular system of three univalent atoms (A, B, and C) and argued that "the conical intersection implied by the ... formula
m
o H/Oe?ol s=l n
+ ~ a=m
(~b°lOH/aQa)ol~b°)Qa=O,
(5)
+l
where the subscripts s and a denote the m totally symmetric displacements and the (n-m) non-totally symmetric displacements, respectively. Now if0 and ~00 can have either the same or different symmetry properties; we investigate each possibility in turn. Case 1:if0 and ~00 have the same symmetry. When if0 and if0 belong to the same irreducible representation of the symmetry group of H(O),we can follow Wigner [5] and choose the "correct linear combinations" of the two-fold degenerate eigenfunctions of H(O) so as to make the off-diagonal matrix elements (~b01(aH/aaa)ol ~°2) vanish. Eq. (5) can then be satisfied (in general) only if all the totally symmetric displacements are zero. Case I therefore subsumes both the Jahn-Teller effect and the Renner effect; the contact is enforced by symmetry, and the degeneracy can be removed by lowering the symmetry of H(O). We will shortly return for a more detailed discussion of case 1. Case 2: ffl0 and if20 have different symmetry properties. When ff 01 and if20 belong to different onedimensional representations of H(O), the integral (~011(a H/aQs)ol~kO)will be identically zero, and the sum involving Qa on the left-hand side of eq. (5) can be made to vanish by setting all the non-totally symmetric displacements equal to zero. Thus case 2 represents accidental contact of two potential surfaces
E=
(6) + i
2
x
2
1
Q-[~(JAB--JBc) +~(JBc--JcA) +~(JcA--/AB)
2 1/2
]
must therefore be a real one ...". Unfortunately they did not realise t~at the two values of E given by eq. (6) can become equal only if
&. --JBc =JCA' In other words, the London-Eyring formula (6) does not imply an intersection except when the three 635
Volume 15, number 4
CHEMICAL PHYSICS LETTERS
atoms are identical; but then the crossing will be enforced by symmetry and will not contravene the noncrossing rule. The foregoing considerations apply to the intersection of only two surfaces. They will remain valid for the intersection of a p-fold degenerate surface if0 with a q-fold degenerate surface ~0 (p > q ~ 1) only if the components into which ~0 decomposes in the point group of H(Q)are all different (in symmetry) from the corresponding components into which ~0 splits. The work described here was carried out during the tenure of an S.R.C. Fellowship which is sincerely appreciated; helpful discussions with Dr. J.G. Baker and
636
1 September 1972
Professor G.J. Hoytink are also gratefully acknowledged.
References
[1] J. von Neumann and E.P. Wigner, Z. Physik 30 (1929) 467. [21 E. Teller, J. Phys. Chem. 41 (1937) 109. [3] K. Razi Naqvi and W. Byers Brown, Intern. J. Quantum Chem. 6 (1972) 271. [4] G. Herzberg and H.C. Longuet-Higgins, Discussions Faraday Soc. 35 (1963) 77. [5] E.P. Wigner, Group theory (Academic Press, New York, 1959) pp. 46, 122, 123; V. Heine, Group theory in quantum mechanics (Pergamon Press, London, 1960) pp. 106, 107.
CHEMICAL PHYSICS LETTERS
ON THE NON-CROSSING
RULE FOR POTENTIAL
1 September 1972
ENERGY SURFACES
OF POLYATOMIC MOLECULES K. Razi NAQVI
Department of Chemistry, The University, Sheffield, $3 7HF, UK Received 23 May 1972
It is shown that the non-crossing rule for the intersection of potential energy curves of diatomic molecules applies also to potential energy surfaces of polyatomic molecules.
The non-crossing rule is generally proved [ 1 - 3 ] by showing that two conditions must be met if crossing is to occur. In a diatomic molecule, only one parameter - the internuclear distance, R - is available for variation; potential energy curves will, therefore, not cross unless they belong to different irreducible representations of the symmetry group of the hamiltonian, because one of the conditions is then alatomatically satisfied at all R. It has been argued [1, 2, 4] that an equivalent statement cannot be made for polyatomic molecules; for, by varying more than one parameter, the two ineluctable requirements can easily be satisfied. It will be shownhere that for a molecule with n variable parameters, (n + 1) conditions must be fulfilled for crossing; hence the non-crossing rule applies to polyatomic molecules as well as to diatomic molecules. The proof presented earlier [3] can easily be extended to polyatomic molecules. As the discussion is concerned with the symmetry properties of the adiabatic potential surfaces, it will be convenient to use normal mode displacements as the variable parameters. Consider a contact of two surfaces, so that E l ( 0 ) = E2(0 ) ,
(1)
where the argument 0 designates the complete nuclear configuration in which the contact occurs. As pointed out earlier [3], this condition does not by itself guarantee contact, for the two surfaces may be degenerate in all nuclear configurations. A contact of the two surfaces in the configuration 0 means that an additional demand, viz., 634
E I ( Q I ' Q2 ..... Qn) 4=E2(Q1 , Q2 ..... Qn)'
(2)
must be met; i n t h e above inequality the Q's stand for normal mode displacements from the contact configuration, and at least one Q must be non-vanishing. Repetition of the analysis carried out in ref. [3] leads to the equation
(@°I@HI@°)=O,
(3)
where
6H = H(q; QI' Q2 ..... Qn) - H(q; O) and q~°,2 are the eigenfunctions of the two surfaces in the contact configuration; H is the hamiltonian, and q denotes the complete set of electronic coordinates. H(q; Q1, Q2, "", Qn) and H(q; O) will henceforth be abbreviated as H(Q) and H(O), respectively. By expanding 6 H of eq. (3) in a power series and retaining only the firstrorder terms, one obtains n
(~ OI(~ H/~Qi)o I ~ o) Qi = 0 .
(4)
i=1 Since each term in the sum depends only on one Qi, all Qi's being independent variables, the sum can be zero only if each term vanishes. Thus eq. (4) embodies n conditions and, together with eq. (1) forms a set of (n + 1) conditions which must be full'died if the two surfaces are to have a contact. The essential difference between a diatomic and a polyatomic molecule is that, in the former, aH/bR is
Volume 15, n u m b e r 4
CHEMICAL PHYSICS LETTERS
1 S e p t e m b e r 1972
of necessity totally symmetric, but in the latter,
belonging to different irreducible representations of
aH/aQi call be symmetric or antisymmetric. Were
H(o).
this not so (i.e., if only symmetry-conserving displacements were possible in polyatomic molecules) one could easily formulate the following rule for polyatomic molecules: potential energy surfaces of the same symmetry cannot touch as the various internuclear distances are varied. We now take account of the non-totally symmetric displacements and discuss the contact of potential energy surfaces in a real polyatomic molecule. The adiabatic surfaces and the displacements will be classified by means of the point group of H(O). It will be convenient to rewrite eq. (4) as
Let us now return to case 1 and examine the contact of two potential energy surfaces in the configuration 0. To discuss whether this degeneracy contravenes the non-crossing rule, we must decide whether the group of H(O) or H(Q) is more suitable for our purpose. Now H(Q) belongs to a group (say B) of lower symmetry than that of the group (say A) of H(O), and B will be a subgroup of A. As we are interested in the behaviour of surfaces for non-vanishing Q's, it will be more expedient (and illuminating) to classify the surfaces in terms of the irreducible representations of B (rather than A as we have been doing so far). Two eigenvalues which are degenerate in A split, under the action of Qa (which would now transform according to the totally symmetric representation of B), into two levels belonging to two different irreducible representations of B. Case 1 now represents accidental contact (at Qa = 0) of two surfaces of different symmetry and does not really violate the non-crossing rule. It is worth remarking here that when one confines attention to a particular displacement, say Qk, one essentially reduces the polyatomic molecule under consideration to the status of a diatomic molecule; that the molecule possesses other variable displacements Ql(l :k k) becomes irrelevant, for variations in al cannot influence the integral (~ OI(a H/~ak)OI ~ 0) which must vanish for contact to occur. In their discussion of intersection of potential energy surfaces of polyatomic molecules, Herzberg and Longuet-Higgins [4] constructed an example to demonstrate that crossing can take place even if it is not demanded by symmetry. They considered an equilateral triangular system of three univalent atoms (A, B, and C) and argued that "the conical intersection implied by the ... formula
m
o H/Oe?ol s=l n
+ ~ a=m
(~b°lOH/aQa)ol~b°)Qa=O,
(5)
+l
where the subscripts s and a denote the m totally symmetric displacements and the (n-m) non-totally symmetric displacements, respectively. Now if0 and ~00 can have either the same or different symmetry properties; we investigate each possibility in turn. Case 1:if0 and ~00 have the same symmetry. When if0 and if0 belong to the same irreducible representation of the symmetry group of H(O),we can follow Wigner [5] and choose the "correct linear combinations" of the two-fold degenerate eigenfunctions of H(O) so as to make the off-diagonal matrix elements (~b01(aH/aaa)ol ~°2) vanish. Eq. (5) can then be satisfied (in general) only if all the totally symmetric displacements are zero. Case I therefore subsumes both the Jahn-Teller effect and the Renner effect; the contact is enforced by symmetry, and the degeneracy can be removed by lowering the symmetry of H(O). We will shortly return for a more detailed discussion of case 1. Case 2: ffl0 and if20 have different symmetry properties. When ff 01 and if20 belong to different onedimensional representations of H(O), the integral (~011(a H/aQs)ol~kO)will be identically zero, and the sum involving Qa on the left-hand side of eq. (5) can be made to vanish by setting all the non-totally symmetric displacements equal to zero. Thus case 2 represents accidental contact of two potential surfaces
E=
(6) + i
2
x
2
1
Q-[~(JAB--JBc) +~(JBc--JcA) +~(JcA--/AB)
2 1/2
]
must therefore be a real one ...". Unfortunately they did not realise t~at the two values of E given by eq. (6) can become equal only if
&. --JBc =JCA' In other words, the London-Eyring formula (6) does not imply an intersection except when the three 635
Volume 15, number 4
CHEMICAL PHYSICS LETTERS
atoms are identical; but then the crossing will be enforced by symmetry and will not contravene the noncrossing rule. The foregoing considerations apply to the intersection of only two surfaces. They will remain valid for the intersection of a p-fold degenerate surface if0 with a q-fold degenerate surface ~0 (p > q ~ 1) only if the components into which ~0 decomposes in the point group of H(Q)are all different (in symmetry) from the corresponding components into which ~0 splits. The work described here was carried out during the tenure of an S.R.C. Fellowship which is sincerely appreciated; helpful discussions with Dr. J.G. Baker and
636
1 September 1972
Professor G.J. Hoytink are also gratefully acknowledged.
References
[1] J. von Neumann and E.P. Wigner, Z. Physik 30 (1929) 467. [21 E. Teller, J. Phys. Chem. 41 (1937) 109. [3] K. Razi Naqvi and W. Byers Brown, Intern. J. Quantum Chem. 6 (1972) 271. [4] G. Herzberg and H.C. Longuet-Higgins, Discussions Faraday Soc. 35 (1963) 77. [5] E.P. Wigner, Group theory (Academic Press, New York, 1959) pp. 46, 122, 123; V. Heine, Group theory in quantum mechanics (Pergamon Press, London, 1960) pp. 106, 107.