The way the energy of a molecular system varies with small changes in its structure is specified by its potential energy surface (PES). A potential energy surface is a mathematical relationship linking molecular structure and the resultant energy. The concepts of energy surfaces for molecular motion, equilibrium geometries, transition structures and reaction paths depend on the Born-Oppenheimer approximation to treat the motion of the nuclei separately from the motion of the electrons (see Appendix A). Minima on the potential energy surface for the nuclei can then be identified with the classical picture of equilibrium structures of molecules (ie. reactant, product and intermediates); saddle points can be related to transition states and reaction rates (see refs. 1 and 2). If the Born-Oppenheimer approximation is not valid, for example in the vicinity of surface crossings, non-adiabatic coupling effects need to be taken in account to correctly describe the evolution of the molecular system. This is done, for instance, when one needs to describe a jump between two different, energetically closed, PES. In this case one use semi-classical theories (see Lecture 2) and the surface-hopping method.
For a diatomic molecule its potential energy surface is a two dimensional plot with the inter nuclear separation (the only degree of freedom) on the x-axis and the energy at that bond distance on the y-axis, producing a curve (the so-called Morse curve, see Figure 1).
For larger systems, the surface has as many dimensions as there are degrees of freedom within the molecule (the number of degrees of freedom is 3N-6, where N is the number of atoms).
Minima, maxima and saddle points can be characterised by their first and second derivatives [1, 2]. For a function of several variables, the first derivatives with respect to each of the variables form a vector termed the gradient. The second derivatives form a matrix called the Hessian. In classical mechanics, the first derivative of the potential energy for a particle is minus the force on the particle, and the second derivative (for a quadratic potential) is the force constant. Thus, the negative of the components of the gradient are the forces on the atoms or nuclei in a molecule, and the second derivative matrix, or Hessian, is also called the force-constant matrix. For a function of one variable, the first derivative of the function is zero at a local minimum or maximum. In more than one dimension, the first derivatives with respect to all of the 3N-6 internal variables must be zero, ie. the gradient vector must have zero length. Equivalently, all of the forces on the atoms in a molecule must be zero; hence such a point is also known as a stationary point (in topology, these points are termed critical points).
The nature of the stationary point can be determined from the second derivatives. In one dimension, the second derivative is positive for a minimum, negative for maximum and zero for a point of inflection. In a multidimensional PES, the eigenvalues of the Hessian, rather than individual elements of the matrix, determine the characteristics of the stationary point. If all the eigenvalues are positive, the point is a local minimum: a minimum is the bottom of a valley on the potential surface. From such a point, motion in any direction - a physical metaphor corresponding to changing the structure slightly - leads to a higher energy. Minima occur at equilibrium structures for the system, with different minima corresponding to different conformations or structural isomers in the case of a single molecule, or reactant and product molecules in the case of multicomponent systems. If all the eigenvalues are negative, the point is a local maximum (also called a peak): a peak is a maximum in all directions. A (first-order) saddle point has one negative eigenvalue and all the rest are positive: it is a maximum in one direction and a minimum in all the other (perpendicular) directions. Transition structures are first-order saddle points. A nth order saddle point has n negative eigenvalues, ie. is a maximum with respect to n mutually perpendicular directions; these points anyway have no chemical meaning. Figure 2 illustrates all these topological features in a model PES (where only 2 degrees of freedom are considered).
In diatomic molecules the PES of two states (eg. the ground state and the first excited state) will only intersect if the states have a different (spatial or spin) symmetry. However, an analogous statement is not true of polyatomic systems [3]: two PES 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 can try to give a quantitative analysis of this situation [4]. We imagine that all but two of the solutions of the Schrödinger equation for the electronic wavefunction have been found, and that φ1 and φ2 are any two functions which, together with the found solutions, constitute a complete orthonormal set. Of course the two missing solutions correspond to the two states (whose energy is E1 and E2) whose crossings we are interested in. Then it must be possible to express each of the two remaining electronic eigenfunctions (which describes the states we want to examine) in the form
Ψ = c1 φ1 + c2 φ2
The well known secular eigenvalue equation is obviously expressed as:
and after very simple passages we can write down the expressions for the energies E1 and E2 of the two states as
E1 = [(H11 + H22) - √((H11 - H22)2 + 4H122)]/2
E2 = [(H11 + H22) + √((H11 - H22)2 + 4H122)]/2 (1)
where, for the matrix elements:
H11 = ⟨φ1|H|φ 1⟩
H22 = ⟨φ2|H|φ 2⟩
H12 = ⟨φ1|H|φ 2⟩ = H21
Thus, in order to have degenerate solutions (the radicand must vanish), it is necessary to satisfy two independent conditions
H11 = H22
H12 (=H21) = 0
and this requires the existence of at least two independently variable nuclear coordinates. In a diatomic molecule there is only one variable coordinate - the interatomic distance - so the non-crossing rule follows: for states of different (spatial or spin) symmetry, H12 is always zero and we have to verify the only condition H11 = H22; this is possible for a suitable value of the single variable coordinate. Otherwise, if the two states have the same symmetry, they will not intersect; but in a system of three or more atoms there are enough degrees of freedom for the rule to break down: the two conditions can be simultaneously satisfied by choosing suitable values for two independent variables, while the other n-2 degrees of freedom (n=3N-6) are free to be varied without exiting from the crossing region.
If we denote the two independent coordinates by x1 and x2, and take the origin at the point where H11 = H22 and H12 (=H21) = 0, the secular equations may be cast in the form
or
(2)
where m = ½(h1 + h2), k = ½(h1 - h2)/2. The eigenvalues are
E = W + mx1 ± √(k2 x12 l2 x22)
and this is the equation of a double cone with vertex at the origin (see Scheme 2). For this reason, such crossing points are called conical intersections. Indeed, if we plot the energies of the two intersecting states against the two internal coordinates x1 and x2 (whose values at the origin satisfy the two conditions H11 = H22 and H12 (=H21) = 0) we obtain a typical double cone shape (see Scheme 1a).
Let's try to have a deeper insight into the physical meaning of the two conditions H11 = H22 and H12 (=H21) = 0. If we consider φ1 and φ2 as the diabatic components of the adiabatic electronic eigenfunction (a diabatic function describes the energy of a particular spin-coupling [5a], while the adiabatic function represents the surface of the real state), the crossing condition (real or avoided) is fulfilled when the two diabatic components φ1 and φ2 cross each other, and this happens when H11 = H22, ie. the energy of the two diabatic potentials (H11 is the energy for the diabatic function φ1 and H22 is the energy for the diabatic function φ2) is the same.
At the crossing of the diabatic functions (H11 = H22, the expressions for the energies of the two real states (see equations (1)) become
E1 = H11 - H12
E2 = H11 + H12
and we see that the energy gap between the two real states is
E2 - E1 = 2H12
Thus, if the exchange term H12 is not zero, the crossing will be avoided and the potential surfaces of the two real states will “diverge”, being one of the two energies slower and the other higher then the diabatic energy H11. Moreover, the value of the exchange term H12 determines how deep the avoided crossing minimum is (small values will generate deep minima, big values shallow minima). In general, H12 is zero (and the crossings will be real) when the two electronic states have a different (spatial or spin) symmetry, while it is usually assumed not zero for states of the same symmetry (which will generate avoided crossings). Anyway, we have shown that this rule is true only for diatomic molecules: in a polyatomic system we can have real crossings for suitable values of a pair of independent coordinates (x1 and x2), which will simultaneously satisfy equations (2) (see Appendix B for further details).
In conclusion the following statement is true: for a polyatomic system two states (even with the same symmetry) will intersect along a n-2 dimensional hyperline as the energy is plotted against the n internal nuclear coordinates.
To understand the relationship between the surface crossing and photochemical reactivity, it is useful to draw a parallel between the role of a transition state in thermal reactivity and that of a conical intersection in photochemical reactivity [5b].
In a thermal reaction, the transition state (TS) forms a bottleneck through which the reaction must pass on its way from reactants (R) to products (P) (Scheme 2a). A transition state separates the reactant and product energy wells along the reaction path. An accessible conical intersection (CI) (Scheme 2b) also forms a bottleneck that separates the excited state branch of the reaction path from the ground state branch. The crucial difference between conical intersections and transition states is that, while the transition state must connect the reactant energy well to a single product well via a single reaction path, an intersection is a “spike” on the ground state energy surface (see inset in Scheme 2b) and thus connects the excited state reactant to two or more products (P and P') on the ground state via a branching of the excited reaction path into several ground state relaxation channels. The nature of the products generated following decay at a surface crossing will depend on the ground state valleys (relaxation paths) that can be accessed from that particular structure.
Theoretical investigations of surface crossings have required new theoretical techniques based upon the “mathematical” description of conical intersections and we now briefly review the central theoretical aspects. The double cone shape of the two intersecting potential energy surfaces can only be seen if the energies are plotted against two special internal geometric co-ordinates of the molecule (x1 and x2 in Scheme 1a, see also Lecture 2). The co-ordinate x1 is the gradient difference vector
x1 = ∂(E1 - E2)/∂q
while x2 is the gradient of the interstate coupling vector
x2 = ⟨ C1* (∂H/∂q)C2⟩
where C1 and C2 are the configuration interaction (CI) eigenvectors in a CI problem and H is the CI Hamiltonian. The vector x2 is parallel to the non-adiabatic coupling vector
ζ(q) = ⟨Ψ1| (∂Ψ2/∂q)⟨
These geometric co-ordinates form the so called “branching space”. As we move in this plane, away from the apex of the cone, the degeneracy is lifted (see Scheme 2a), and ground state valleys must develop on the lower cone. In contrast, if we move from the apex of the cone along any of the remaining n-2 internal co-ordinates (where n is the number of degrees of freedom of the molecule), the degeneracy is not lifted. This n-2 dimensional space, called “intersection space”, is a hyperline consisting of an infinite number of conical intersection points (see Scheme 1b).
Nonadiabatic events (transition from the excited state to the ground state at the CI) pose a serious challenge because the non-adiabatic transition is rigorously quantum mechanical without a well-defined classical analogue. At a crude level of theory [11] (we return to a better treatment in Lecture 2) the probability of a surface hop is given as
P = exp[-(π/4)ξ]
where the Massey parameter is
ξ = ΔE(Q)/hQ.&#zeta;(Q)
Thus this simple theory predicts that radiationless transitions will occur when the energy gap ΔE(Q) is small and the scalar product between the velocity vector and the non-adiabatic coupling Q.ρ(Q) is large.
Often the chemically relevant conical intersection point is located along a valley on the excited state potential energy surface. Figure 3 illustrates a two-dimensional model example. Here two potential energy surfaces are connected via a conical intersection. This intersection appears as a single point (CI) since the surfaces are plotted along the branching space (x1, x2). The intermediate M* is reached by relaxation from the Franck-Condon region (FC) and it is separated from the intersection point by a transition state (TS). In this case, the molecular structure of the intersection and the reaction pathway leading to it can be studied by computing the minimum energy path (MEP) connecting FC to M* and M* to CI using the standard intrinsic reaction co-ordinate (IRC) method [6] and these will be discussed in Lecture 2. However, there are situations where there is no transition state connecting M* to the intersection point or where an excited state intermediate on the upper energy surface does not exist. In such situations, mechanistic information must be obtained by locating the lowest lying intersection point along the n-2 intersection space of the molecule. The practical computation of the molecular structure of a conical intersection energy minimum [7] will be illustrated in Lecture 2.
This techniques provide information on the structure and accessibility of the intersection point which controls the locus and efficiency of internal conversion (IC). The evolution of ground state photoproducts following decay via such an IC channel requires a study of the possible ground state relaxation process. Observe the shape of the ground state surface in the region of the conical intersection in Figure 3. The double cone in this case is “elliptic” and two sides are steeper than the others. This situation is typical of many cases and relaxation valleys develop more quickly in these directions. We have recently implemented a method to locate and characterise all the relaxation directions that originate at the lower vertex of the CI cone [8]. This is the initial relaxation direction (IRD) method and will be illustrated in Lecture 2. The MEP starting along these relaxation directions define the ground state valleys which determine the possible relaxation paths and ultimately the type of photoproducts which can be generated by decay. This information is structural (ie. non-dynamical) and provides insight into the mechanism of photoproduct formation from vibrationally “cold” excited state reactants such as those encountered in many experiments where slow excited state motion or/and thermal equilibration is possible (in cool jets, in cold matrices and in solution).
In many cases, such structural or static information is not sufficient. The excited state may not decay at the minimum of the conical intersection line. Alternatively, the momentum developed on the excited state branch of the reaction co-ordinate may be sufficient to drive the ground state reactive trajectory along paths that are far from the ground state valleys. In such cases, a dynamics treatment of the excited state/ground state motion is required [9]. These technique will be illustrated in Lecture 2.
In S1 benzene there is a ~3000 cm-1 threshold for the disappearance of S1 fluorescence (see ref. 10). This observation is assigned to the opening of a very efficient, radiationless decay channel (termed “channel 3”) leading to the production of fulvene and benzvalene. Ab initio CAS-SCF and multi-reference MP2 computations [10] show that the topology of the first (π-π*) excited state energy surface is consistent with that of Figure 3. Thus the observed energy threshold (which is reproduced) corresponds to the energy barrier which separate S1 benzene from a S1/ S0 conical intersection point.
Conical intersections are intrinsically transient entities. Thus, the molecular structure can only be derived only from theoretical computations. The optimized conical intersection structure for S1 benzene is shown in Figure 4. The structure contains a triangular arrangement of three carbon centres corresponding to a -(CH)3- kink of the carbon skeleton. The electronic structure corresponds to three weakly interacting electrons in a triangular arrangement which are loosely coupled to an isolated radical centre (this is delocalised on an allyl fragment). This type of conical intersection structure appears to be a general feature conjugated systems and has been documented in a series of polyene and polyene radicals. We will illustrate this point in Lecture 3. The electronic origin of this feature can be understood by comparison with H3 where any equilateral triangle configuration corresponds to a point on the D1/ D0 conical intersection in which the three H electrons have identical pairwise interactions. A deeper insight in the nature of this molecular structure can be gained by applying qualitative VB theory. This point is discussed in Appendix B.
By examining the molecular structure of the conical intersection, its electronic distribution and the directions indicated by the x1 and x2 vectors we can derive information on the photochemical reaction path. For instance it is obvious from Figure 4 that the kink feature suggest the formation of a cyclopropyl ring upon decay to the ground state. Thus the “primary” photoproduct of the the action is predicted to be a diradical species characterized by a 1,3-transanular bond. Indeed this structure corresponds with that of pre-fulvene which was previously proposed as the intermediate in fulvene photochemical production (see Scheme 3). The same conclusion can be reached by looking at the shape of the non-adiabatic coupling and gradient difference vectors in Figure 5b and 5c respectively. These corresponds, by definition, to the deformation which lift the degeneracy and that therefore causes a fast decrease in S0 energy. Thus, in turn they represent, loosely, the possible directions of relaxation along the ground state potential energy surface. There is little doubt that the gradient difference vector, which is almost parallel to the S1 transition vector (see Figure 5a), suggest a relaxation towards the pre-fulvene diradical in the positive direction and a planar ground state benzene in the negative direction (ie. reversing the arrows). On the other hand the non-adiabatic coupling vector describes the simultaneous double bond re-construction which occurs upon relaxation.
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