Intersection of Potential Energy Surfaces in Polyatomic Molecules BY G. HERZBERG * AND H. C. LONGUET-HIGGINS t Received 28th January, 1963 It is shown that in polyatomic systems the conical intersections described by Teller 4 will occur not only where demanded by symmetry but also in certain non-symmetrical systems. Other kinds of intersection are also described, and it is suggested that “ near-intersections ” are likely to be as important in polyatomic as in diatomic systems. 1. INTRODUCTION In diatomic molecules the potential energy curves of two states will only inter- sect if the states differ in symmetry or in some other essential characteristic.1-3 However, an analogous statement is not true of polyatomic systems : 4 two potential energy surfaces of a polyatomic molecule can in principle intersect even if they belong to states of the same symmetry and spin multiplicity.This sentence leaves open the question whether such intersections ever occur in polyatomic systems. We have therefore tried to find some examples of intersections between states of the same species, and this paper presents some miscellaneous results. 2. GENERAL CONSIDERATIONS We first outline Teller’s analysis of the case in which one may neglect the spin terms in the electronic Hamiltonian, so that the electronic wave function may always be taken in real form. We imagine, following von Neumann and Wigner,3 that all but two of the solutions of the electronic wave equation have been found, and that q11 and q 2 are any two functions which, together with the found solutions, constitute a complete orthonormal set.Then it must be possible to express each of the two remaining electronic wave functions in the form where, in an obvious notation, $ = c1q1 +c2q2, (2.1) all quantities in this equation being real. independent conditions, namely, and this requires the existence of at least two independently variable nuclear co- ordinates. In a diatomic molecule there is only one variable co-ordinate-the interatomic distance-so the non-crossing rule follows; but in a system of three or more atoms there are enough degrees of freedom for the rule to break down. * National Research Council, Ottawa, Canada. p University Chemical Laboratory, Cambridge. 77 In order that (2.2) shall have degenerate solutions it is necessary to satisfy two (2.3) Hll = H229 H,A= H21) = 0,78 POTENTIAL ENERGY SURFACES Following Teller we denote the two independent co-ordinates by x and y , and take the origin at the point where Hl1 = H22 and Hl2 = 0.The secular equations may then be cast in the form : W+h,x-E, zy c1 [ZY, W+h,x-E][c,l = O or W+(m+k)x-E, Zy ly, where m = +(hi + hz), k = 3(h1- h2). The eigenvalues are E = W + mx ,/(k2x2 + 12y2), and this is the equation of a double cone with vertex at the origin. This result was obtained by Teller,4 but he did not draw attention to the following property of the wave function near the origin. Define an angle 6 by the equations kx = R cos 8, l y = R sin 8, (2.7 where R = J(k2x2+ Z2y2)>0. Taking the lower root of (2.5), namely, we deduce that on the lower sheet of the energy surface the coefficients c1 and c2 satisfy E = W-tmx-R, (2.9) R+R cos 8, R sin R sin 8, R+R cos It follows that c1 -sin8 cosB-1 c2 1+cos 8 sin 8 -=-=-- - -tan 30.(2.10) (2.11) Hence, if t,b is to be real, like ~1 and q 2 , we must have c1 = sin $9, c2 = -cos 30, (2.12) or c1 = -sin $8, c2 = cos 38. (2.13) In either case, as we move round the origin keeping R constant and allowing 6 to increase from 0 to 27r, both c1 and cz change sign, and so does t,b. This result is a generalization of one which has been proved 5 in connection with the Jahn-Teller effect, 6 where one also encounters a conically self-intersecting potential surface. It shows that a conically self-intersecting potential surface has a different topolo- gical character from a pair of distinct surfaces which happen to meet at a point.Indeed, if an electronic wave function changes sign when we move round a closed loop in configuration space, we can conclude that somewhere inside the loop there must be a singular point at which the wave function is degenerate; in other words, there must be a genuine conical intersection, leading to an upper or lower sheet of the surface, as the case may be. 3. THREE HYDROGEN-LIKE ATOMS A useful illustration of the above generalizations is a system of three hydrogen atoms near the vertices of an equilateral triangle. If the internuclear distancesG . HERZBERG AND H. C. LONGUET-HIGGINS 79 are a, b and c, a convenient set of internal co-ordinates are x = (b+c-2a)/J6, Y = J2 and z = (a + b+ c)/J3. (3.1) According to both the valence-bond theory and the molecular orbital theory, the ground state is of species 2E‘ in the D3h configuration.It therefore exhibits the Jahn-Teller effect,6 and for given z the surface E(x, y ) is a double cone with vertex at the origin. It may be argued that this example does not really contravene the non-crossing rule, because the degeneracy arises from symmetry; but we can use it for constructing a non-trivial example, in the following way. Let (PA be the valence-bond wave function for a situation in which the electron on A has spin “ up ” and the electrons on B and C are spin-paired ; let c p ~ and cpc be similarly defined, so that (This is a standard result of the simple valence-bond theory.) We now take the system round a continuous loop in (x, y ) space, starting with B close to C and A far away (see fig.1). From the arguments of Q 2 the wave function must have changed sign on completion of the loop ; in fact it evolves as shown in fig. 1 : V A + ~ P B + ( P C = 0. (3.2) A A A A A A ,/ J3 B---C B-C - (PA ,A, / J3 B-C B---C B C B C B C - cpB \ A \\\ J3 VA-YB -(PB+VC VC-V-A (PC FIG. 1. SDA We have now confirmed the change of sign required by the general arguments of $2. However, the valence-bond theory requires the wave function to evolve in this way even when the atoms A, B and C are not chemically identical; they might, for example, be atoms of Li, Na and K respectively. The conical intersection implied by the London-Eyring formula 77 8 must therefore be a real one unless, which is most unlikely, the theory is qualitatively in error as to the way in which the sign of the wave function is affected by taking the system round the cycle depicted in fig.1. We conclude that in a triangular system of three hydrogen-like atoms the lowest doublet state is linked with an excited doublet by a conical intersection even when all three atoms are dissimilar. Both states are symmetric with respect to the mole- cular plane (species ZA’), so in this case at least symmetry cannot be held responsible for the existence of the intersection! E = Q * JCHJAB- Jd2 + H J B c - Jcd2 + ~ ( J c A - J,4d21 (3.3) 4. OTHER TRIATOMIC SYSTEMS In linear molecules all the electronic states are orbitally degenerate except the C states, but this degeneracy is removed when the molecule is bent.9 For example, a ll state of a linear molecule splits into one state which is symmetric and another which is antisymmetric about the plane of the bent molecule.It is therefore not unusual to find that the ground state of a bent molecule is adiabatically correlated with an excited state of different symmetry, obtained by straightening the molecule80 POTENTIAL ENERGY SURFACES out and bending it again in a different plane. We might speak of a “glancing intersection ” between the potential functions of the two states. It is also possible, however, for an A’ (or A”) state of the bent molecule to be adiabatically linked to another A’ (or A”) state, through a conical intersection in the linear configuration. An example is provided by HNO, which we now consider briefly.Bancroft, Hollas and Ramsay 10 have shown that the ground state of HNO is of species 1A’, symmetric with respect to the molecular plane. When the molecule is straightened out, this state will pass adiabatically into the lowest singlet state of the linear molecule ; analogy with the isoelectronic molecule 0 2 strongly suggests that the lowest singlet state of linear HNO is a 1A state. If the H atom is now pulled away along the NO axis, the adiabatic correlation rules imply that the products are a 2s hydrogen atom and a 2A NO molecule. But the ground state of NO is 2lI not 2A, so the linear dissociation curve must somewhere cross the In curve for H(2S)++O(zII), at a point P, say. 2 ‘d FIG. 2. We now consider what happens when we take the HNO molecule at the point P and bend it out of a straight line.The resulting perturbation is necessarily sym- metric with respect to the plane of bending; it can mix together the A’ components of the In and 1A states, or their A” components, but cannot mix the A‘ component of one with the A” component of the other. Restricting attention to the A‘ com- ponents, and denoting the bending co-ordinate by y , we arrive at a situation of exactly the type described in $2 ; the states 401 and cp2 are the A’ components of the 1l-I and 1A states, and x is an in-line co-ordinate measured from the point P. If we extend the potential energy functions for the linear situation by introducing y as a co-ordinate perpendicular to the plane of the paper, we can depict the resulting conical intersection in the manner shown in fig.2 (which is not intended to be drawn to scale). This diagram shows that there is no difficulty in forming HNO adiabatically from H and NO in their ground states, provided that the H atom does not approach the NO molecule along its axis. 5. NEAR-INTERSECTIONS So far we have considered only genuine intersections, such as the conical inter- sections described by Teller or the glancing intersections associated with the RennerG . HERZBERG AND H . C. LONGUET-HIGGINS 81 effect. But almost equally important are " near-intersections " where two potential surfaces nearly meet and only just avoid crossing because of a weak interaction at the point of closest approach. Near-intersections of this kind are well known in diatomic molecules ; a classic example is the near-intersection between the lowest " ionic " and " covalent '' states of an alkali halide molecule.In NaCl, for example, this occurs at a rather large interatomic distance, at which the resonance integral for electron transfer between the two atoms is very small. In essence, the small- ness of H12 is due to the considerable difference in electron distribution between 'pl and cp2. We shall now examine briefly a somewhat analogous case in a poly- atomic system. Douglas 11 has recently studied in detail the strong first ultra-violet absorption system of NH3, whose upper state is 12 a 1Ai state of the planar molecule (sym- metry group D3h). In molecular orbital terms this state results from exciting one of the unshared a:2p electrons of planar NH3 to the Rydberg orbital 3sa;.The absorption system shows strong signs of predissociation, but in the first few members of the main vibrational progression of ND3 rotational structure is clearly discernible. Now it seems likely that the predissociation of excited NH3 is produced by inter- action with a state arising from a hydrogen atom and an NH2 radical in their ground states. We therefore consider the approach of an H atom to an NH2 radical to form a planar NH3 molecule. The ground state of NH2 is 2 8 1 , antisymmetric with respect to the plane of the bent radical. There are two electrons in the non-bonding hybrid orbital on the N atom, and these will impede the approach of the third H atom. The singlet state formed by the combination of H(2S) and NH@Br) should therefore be repulsive, at least in the early stages of formation.But this state has the same multiplicity and symmetry (with respect to the group CzV) as the observed 1A[; state of NH3. Why, then, is the interaction between these two states so weak that the latter is only predissociated and not completely disrupted by the former? The answer, possibly, is that although the two states have the same symmetry and multiplicity, a substantial electronic rearrangement is needed to convert one into the other: the electron on the approaching H atom must be transferred to a Rydberg 3s orbital on the NH2 radical before the lone pair on the N atom can form a satisfactory bond to the H nucleus. It seems likely, therefore, that the matrix element H12 between the observed 1A'; state of NH3 and the singlet state arising from H(2S) + NHz(2Bl) is very small, so that the potential surfaces for these two states, while not actually intersecting, approach so closely that all the usual conditions are satisfied for a predissociated absorption spectrum.6. GENERAL CONCLUSIONS Our conclusions are somewhat diverse, like the facts of molecular spectroscopy. First, the conical intersections described by Teller 4 will occur not only in situations where symmetry demands them, but in asymmetrical systems such as a set of three dissimilar hydrogen-like atoms. This follows from the fact that conical intersections differ topologically from accidental meetings of potential surfaces, and cannot be " abolished " by making infinitesimal changes in the electronic Hamiltonian. Secondly, in non-linear triatomic molecules two states of the same species are sometimes connected through a conical intersection occurring in the linear con- figuration; this kind of situation is likely to occur in many cases of interest. Lastly, narrowly avoided crossings may well be as important in polyatomic as in diatomic systems; their occurrence has nothing to do with symmetry, but is due to a substantial difference in electronic character between the two states involved.82 POTENTIAL ENERGY SURFACES We are indebted to Dr. A. E. Douglas for kindly allowing us to quote his unpub- lished results. 1 Hund, 2. Physik., 1927, 40, 742. 2 Franck and Haber, Berliner Akademieberichte XIII, 193 1. 3 von Neuinann and Wigner, Physik. Z., 1927, 30,467. 4 Teller, J. Physic. Qern., 1937, 41, 109. 5 Longuet-Higgins, Opik, Pryce and Sack, Proc. Roy. Soc. A , 1958,244, 1. 6 Jahn and Teller, Proc. Roy. SOC. A, 1937, 161, 220. 7 London, 2. Elektrochem., 1929, 35, 522. 8 Glasstone, Laidler and Eyring, The Theory of Rate Processes (McGraw-Hill, New York, 1941). 9 Renner, 2. Physik, 1934, 92, 172. 10 Bancroft, Hollas and Ramsay, Can. J. Physics, 1962, 40, 377. 11 Douglas, this discussion. 12 Walsh and Worsop, Trans. Faraday SOC., 1961,57, 345.