Appendix A

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods (see refs. 1,2 and refs. cited therein).

A.1 The Schrödinger Equation

Quantum mechanics explains how microscopic entities like electrons have both particle-like and wave-like characteristics. The Schrödinger equation describes the wavefunction of a collection of particles like a molecule:

In this equation, Ψ is the wavefunction, V is the potential field in which the particles are moving, R and r denote respectively the relative positions of the nuclei and of the electrons within the molecule, h is the Planck constant, and the summation is over all particles i (nuclei + electrons) where m_i is the mass of particle i. The differential operator (∇²) on the left side of the equation is known as “del-squared”. The operator ∇ (“del”) is equivalent to partial differentiation with respect to x_i, y_i and z_i components of particle-i position:

The energy and many properties of the system can be obtained by solving the Schrödinger equation for Ψ, subject to the appropriate boundary conditions. Many different wavefunctions are solutions to it, corresponding to different stationary states of the system. If V is not a function of time, the Schrödinger equation can be simplified using the mathematical technique known as separation of variables. We will obtain two equations, one of which depends on the position of the particles independent of time and the other of which is a function of time alone. For the problems we are interested on, this separation is valid, and we focus entirely on the familiar time-independent Schrödinger equation:

where E is the energy of the many particles system and H is the Hamiltonian operator . The various solutions to this equation correspond to different stationary states of the molecule. The one with the lowest energy is called the ground state. This equation is a non-relativistic description of the system which is not valid when the velocities of the particles approach the speed of light. Thus it does not give an accurate description of the core electrons in heavy nuclei (such as transition metals). Anyway, for the systems presented in this thesis relativistic effects are negligible.

The Hamiltonian H, like the energy in classical mechanics, is the sum of kinetic and potential parts:

H = T + V

The kinetic energy operator T is a summation of ∇² over all the particles i (nuclei + electrons) in the molecule:

while the potential energy operator V represent the Coulomb interaction between each pair of charged entities (treating each atomic nucleus as a single charged mass):

where rj_k is the distance between two distinct pair of particles (j , k), and e_j and e_k are their charges. For an electron the charge is -e while for a nucleus the charge is Ze, where Z is the atomic number for the atom. Thus:

The first term corresponds to electron-nuclear attraction, the second to electron-electron repulsion and the third to nuclear-nuclear repulsion.

A.2 The Born-Oppenheimer Approximation

This is the first of several approximations used to simplify the solution of the Schrödinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This is possible since the nuclear masses are much greater than those of the electrons, and, therefore, nuclei move much more slowly. As a consequence, the electrons react essentially instantaneously to changes in nuclear positions. This makes it a reasonable approximation to suppose that the electron distribution depends only on the instantaneous positions of the nuclei and not on their velocities. In other words, the nuclei look fixed to the electrons, and the quantum-mechanical problem of electron motion in the field of fixed nuclei may first be solved, leading to an effective electronic energy E_eff(R) which depends on all of the relative nuclear coordinates (R) and describes the potential energy surface (PES) for the system. (This effective energy is then used as a potential energy for the subsequent study of the nuclear motion). This approximation, which allows the two parts of the problem to be solved independently, is called the adiabatic or Born-Oppenheimer approximation [3] and for this reason the potential energy surface described by E_eff(R) is also called adiabatic surface (because its existence depends on the validity of the adiabatic approximation).

Quantitatively, the Born-Oppenheimer approximation may be formulated by writing down the Schrödinger equation for electrons in the field of fixed nuclei (by neglecting the kinetic energy term for the nuclei):

H Ψ^elec(R,r) E^eff(R) Ψ^elec(R,r)

where Ψ^elec is the electronic wavefunction which depends on the electronic (r) and nuclear (R) coordinates. H^elec is the electronic Hamiltonian:

H^elec = T^elec + V

where T^elec is the electronic kinetic energy (the kinetic energy term for the nuclei has been neglected):

and V is the Coulomb potential energy defined above.

Accordingly, E^eff(R) is also used as the effective potential for the nuclear Hamiltonian:

H^nuc = T^nuc + E^eff(R)

where:

This Hamiltonian is used in the Schrödinger equation for the nuclear motion, describing the vibrational, rotational and translational states of the nuclei. Solving the nuclear Schrödinger equation (at least approximately) is necessary for predicting the vibrational spectra of molecules.

There are some situations in which the Born-Oppenheimer approximation is not valid; this means that the separability of nuclear and electron motions fails and the total wavefunction is no more well approximated by a simple product of an electronic eigenfunction with a vibrational wavefunction (the eigenfunction of the nuclear part of the Schrödinger equation). In this case non-adiabatic effects are important and the meaning of classical chemical structures becomes less clear. In general we suppose Born-Oppenheimer approximation to be valid; anyway in the vicinity of surface crossings (at degenerate or nearly degenerate states) this approximation fails and molecular motion is described not only by the energy surface of the state but also by the almost degenerate one with its own vibrational motions. In other words, even if we can always calculate (by solving the electronic part of the Schrödinger equation ) the potential energy surfaces for the states, we need also to estimate non-adiabatic coupling effects (such as surface hopping probability) to have a correct description of the molecular motion.

The main task of theoretical studies of electronic structures is to solve, at least approximately, the Schrödinger equation for the electronic wavefunction and hence find the effective nuclear potential function E_eff(R). The potential surface E_eff(R) is fundamental to the quantitative description of chemical structures and reactivity. When explored in function of R, it will generally have a number of points with important chemical meaning (see in Lecture 1).

We have shown that the wavefunction Ψ depends on the relative coordinates of all particles, R and r. The square of the wavefunction, Ψ² (or |Ψ|² if Ψ is complex), is interpreted as a measure of the probability density for the particles it describes. Therefore we require that Ψ be normalised: if we integrate over all space, the probability should be unity (the probability of finding all the particles anywhere in space is unity!). Secondly, Ψ must also be antisymmetric, meaning that it must change sign when two identical particles (such as electrons) are interchanged. For an electronic wavefunction, antisymmetry is a physical requirements following from the fact that the electrons are fermions. More specifically, this requirement means that any valid wavefunction must satisfy the following condition:

Ψ(r₁,...,r_i, ...,r_j,...,r_n) = -Ψ(r₁,...,r_j, ...,r_i,...,r_n)

A.3 Theoretical Models: Ab Initio Computational Approach

An exact solution to the Schrödinger equation is not possible for any but the most trivial systems. However, a number of simplifying assumptions and procedures do make an approximate solution possible for large range of molecules. Such an approach defines a theoretical model: it is a theory (a set of reasonable assumptions and approximations) within which all structures, energies, and other physical properties can be explored once the mathematical procedure has been implemented. A theoretical-model chemistry results. If comparisons of computational results with the available experimental evidences prove favourable, the model acquires some predictive value in situations where experimental data are unavailable. In this section it will be described the theoretical models used to produce the results presented in the thesis.

A theoretical model should posses a number of important characteristics. First, it should be unique and well defined. The procedure to obtain an energy and a wavefunction as an approximate solution of the Schrödinger equation should be completely specified in terms of nuclear positions and the number and spins of electrons in the molecule. A second desirable feature is continuity: all potential surfaces should be continuous with respect to nuclear displacements. A theoretical model should also be unbiased. No appeal to chemical intuition should be made in setting up the details of the calculations (a theory can only be used for the analysis of such concepts as bonding if presuppositions have not been built into its formulation). Another important requirement for a satisfactory theoretical model is size-consistency: relative errors involved in a calculations should increase more or less in proportion to the size of the molecule. This is particularly important if the model is to be used in a comparative manner, relating properties of molecules of different sizes. While it is generally not possible to satisfy this condition fully, it is often possible to construct models that are size-consistent for infinitely separated systems. This means that applications of the model to a system of several molecules at infinite separation will yield properties that equal the sum of these same properties for the individual molecules.

It is also desirable that a theoretical model be variational, that is, yield a total energy that is an upper bound to that which would result from exact solution of the full Schrödinger equation.

A.3.1 The Molecular Orbital Model

The first approximation we will consider comes from the interpretation of |Ψ|² as a probability density for the electrons within the system. Molecular orbital (MO) theory is an approach to molecular quantum mechanics which uses one-electron functions (or orbitals) to approximate the full wavefunction Ψ (the theoretical models discussed in this section are all based on this MO model). This approximate treatment of electron distribution and motion assigns individual electrons to one-electron functions termed spin-orbitals. These comprise a product of a spatial function, termed molecular orbital, φ₁(x,y,z), φ₂(x,y,z), φ₃(x,y,z), ..., and either α or β spin function (describing the spin component of the electron). The spin-orbitals are allowed complete freedom to spread throughout the molecule, their exact form being determined variationally to minimise the total energy. For simplification reasons molecular orbitals are chosen to be orthogonal to each other and normalised (normalisation correspond to the physical requirement that the probability of finding the electron anywhere in space is unity):

∫φ_i^* φ_idτ = 1
∫φ_i^* φ_jdτ = 0

In the simplest version of the theory, as in the restricted Hartree-Fock (RHF) method, a single assignment of electrons to orbitals is made (this single assignment defines the so-called electron configuration). These orbitals are then combined together to form a suitable many-electron wavefunction Ψ which is the simplest MO approximation to the Schrödinger equation.

The simplest possible way of making Ψ for the description of an n-electron molecular system would be in the form of a simple product of spin orbitals (the so called Hartree product). However, such a wavefunction is not acceptable, as it does not have the property of antisymmetry. To ensure antisymmetry, the spin orbitals may be arranged in a determinantal wavefunction (often referred to as Slater determinant). For a molecule with n (even) electrons doubly occupying n/2 molecular orbitals φ_i, the many-electron wavefunction is:

where the initial factor is necessary for normalisation.

The determinantal wavefunction does have the property of antisymmetry. Swapping two electrons correspond to interchanging two rows of the determinant, which will have the effect of changing its sign. Moreover, it is not possible for a molecular orbital φ_i, to be occupied by two electrons of the same spin. This is the Pauli exclusion principle [4], which follows because the determinantal wavefunction vanishes if two columns are identical. Hence orbitals may be classified as doubly occupied, singly occupied or empty. Since each orbital φ_i is later associated with an energy, this assignation of electrons is usually referred to as an electron configuration and is often represented by an electron configuration diagram such as shown in Figure 1.

Figure A.1. Electron configuration diagram for φ_1α φ_1β φ_2α φ_2β φ_3α

Most molecules have an even number of electrons and their ground states may be represented by closed-shell wavefunction with orbitals either doubly occupied or empty. This is the case of one of the most commonly used methods: the RHF method. Here the many-electron wavefunction Ψ is represented by a single electron configuration corresponding to a single closed-shell type Slater determinant (see, as example, the single-determinant wavefunction written above), where α and β coupled electrons are constrained to move in the same space (their spatial functions, i.e. molecular orbitals φ_i, are identical).

A.3.2 Basis Sets: The LCAO Approximation

A further approximation involves expressing the molecular orbitals φ_i as linear combination of a pre-defined finite set of N one-electron functions known as basis functions.

where the coefficients c_μi are known as the molecular orbital expansion coefficients.

The basis functions (also chosen to be normalised) constitute the basis set. These basis functions (which are defined in the specification of the model) are usually centred on the atomic nuclei and so bear some resemblance to atomic orbitals. If the basis functions are the atomic orbitals for the atoms making up the molecule, the previous equation is often described as the linear combination of atomic orbitals (LCAO) approximation, and is frequently used in qualitative descriptions of electronic structure. However, the actual mathematical treatment is more general than this, and any set of appropriately defined functions may be used.

To provide a basis set that is well defined for any nuclear configuration and therefore useful for theoretical model, it is convenient to define a particular set of basis functions associated with each nucleus, depending only on the charge on that nucleus. Such functions may have the symmetry properties of atomic orbitals, and may be classified as s, p, d, f,... types according to their angular properties. Two types of basis functions have received widespread use: Slater-type functions [5] and gaussian-type functions [6]. The former provide a reasonable representations to atomic orbitals but are not well suited to numerical work, and their use in practical molecular orbital computations has been limited. The latter are less satisfactory as representation of atomic orbitals (particularly because they do not have a cusp at the origin), nevertheless they have the important advantage that all integrals in the computations can be evaluated explicitly (analytically) without recourse to numerical integration.

Actually, linear combination of gaussian functions are used to define the basis functions (for example an s-type basis function χ_p can be expanded in terms of s-type gaussians ζ_i):

where the d coefficients (called contraction coefficients) are fixed constants within a given basis set. Basis functions of this type are called contracted gaussians, the individual ζ being termed primitive gaussians.

All of the construction result in the following expansion for molecular orbitals:

where m represents the generic angular type (s, p, d, f,...).

Given the basis set, the unknown coefficients c_pi are determined so that the total electronic energy calculated from the many-electron wavefunction is minimized and, according to the variational theorem, is as close as possible (an upper bound) to the energy corresponding to the exact solution of the Schrödinger equation. The Self Consistent Field (SCF) procedure is the iterative methods through which expansion coefficients c_pi are optimized to gain the minimum value of the energy (a threshold in the energy gap between consecutive iterations is chosen, under which the procedure is considered “convergent”). This energy and the corresponding wavefunction represent the best that can be done within the chosen approximation, that is, the best given the constrains imposed by: (a) the use of a limited (not infinite) basis set, and (b) the constraint in the wavefunction extension (for the RHF methods being the use of a single-determinant wavefunction).

A.3.3 Post-SCF and Dynamic Correlation Methods: CAS-SCF, CIS and CAS-PT2.

The main deficiency in HF theory stands in its single-determinant wavefunction. Even with a large and completely flexible basis set, the full solution of the Schrödinger equation cannot be expressed in terms of a single electron configuration. Moreover a chemical system undergoing bond-breaking/bond-forming processes (and in general transition states) cannot be expressed by a single-determinant wavefunction, and single-excitation states (as the spectroscopic excited state in polyenes) are open-shell electron configurations (not closed-shell as in the RHF methods). To correct for such a deficiency, it is necessary to use wavefunctions that go beyond the HF level, that is, that represent more than a single electron configuration. This is done in post-SCF methods.

If Φ₀ is the HF many-electron wavefunction, the extended approximate form for the more accurate wavefunction Φ is

Φ = a₀Φ₀ + a₁Φ₁ + a₂Φ₂ + a₃Φ₃ + ...

Here Φ₁, Φ₂, Φ₃, ... are wavefunctions (Slater determinants) corresponding to other electron configurations, and the linear coefficients a₀, a₁, a₂, a₃, ..., are to be determined (optimised through a variational SCF procedure) together with the expansion coefficients defining molecular orbitals, in order to get the minimum energy value. Inclusion of wavefunctions for all possible alternative configurations (within the framework of a given basis set) is term full configuration interaction (full-CI). It represents the best that can be done using that basis set. Practical methods seek to limit the number of configurations or to approximate the effect which their inclusion has on the total wavefunction.

The CAS-SCF Method:

In the CAS-SCF method [7] all the electron configurations arising within a chosen active space of orbitals (a set of chemically-important orbitals which is chosen depending on the chemical process we are interested in) are considered into the multiconfigurational expansion for Ψ. Thus the orbital space is divided into three subsets: a core space (with all doubly occupied orbitals), a valence or active space (where all possible configurations are considered in) and a virtual space (with all empty orbitals).

The CIS Method:

In the CIS method [8] only single-excitation-type Slater determinants are added into the expansion for Ψ. Moreover the molecular orbitals are determined through a RHF computation on the system. This methods allows us to correctly describe only single-excitation electronic states (as it is the case for the spectroscopic ionic states of polyenes) at a lower computation effort than with CAS-SCF.

Perturbation Theory Methods:

CAS-SCF is certainly a suitable method for topological studies on potential energy surfaces both in the ground and in the excited states, and it gives accurate descriptions of chemical reactivity problems. Nevertheless to have the correct energetics in chemical and photochemical problems, contributions due to dynamic electron correlation have to be considered. These depends on the inadequacy of CAS-SCF to describe the instantaneous interelectronic repulsion due to the 1/r_ij terms in the Hamiltonian. The correction due to this terms is necessary to have correct (comparable to experimental data) energy gaps and activation barriers. Second order perturbation theory methods compute these contributions perturbatively [9]. The CAS-SCF wavefunction is perturbed by adding more terms to the multiconfigurational expansion of Ψ, including also Slater determinants of core-virtual excitation type, valence-virtual excitation type, and so on. The energetic effect due to this perturbation is then computed and added to the energy (to gain a second order corrected value) and to the wavefunction (to gain a first order corrected function). To take in account these effects we have used the CAS-PT2 [10] method. It computes dynamic correlation contribution by adding all possible double-excitation configurations to CAS-SCF wavefunction.

References

1. Hehre, W. J.; Radom, L.; Schleyer, P. v.R.; Pople, J. A. Ab Initio Molecula Orbital Theory; Wiley: New York, 1986; Chapters 1-2 and references cited therein.
2. Foresman, J. B.; Aeleen, F. Exploring Chemistry with Electronic Structure Methods; Gaussian, Inc.: Pittsburgh, 1996; Appendix A and references cited therein.
3. Born, M.; Oppeheimer, J. R. Ann. Physik 1927, 84, 457.
4. Pauli, W. Z. Physik 1925, 31, 765.
5. Slater, I. J. Phys. Rev. 1930, 36, 57.
6. Boys, S. F. Proc. Roy. Soc. (London) 1950, A200, 542.
7. (a) Roos, B. O. Adv. Chem. Phys. 1987, 69, 399-446.
   (b) The MC-SCF programme we used is implemented in Gaussian 94, Revision B.2, M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsburgh PA, (1995).
8. Foresman, J. B.; Head-Gordon, M.; Pople, J. A.; Frish, M. J. J. Chem. Phys. 1992, 96, 135.
9. Møller, C.; Plesset, M. S. Phys. Rev. 1934, 46, 618.
10. (a) K. Andersson, P-A Malmqvist, B. O. Ross J. Chem. Phys. 1992, 96, 1218.
    (b) MOLCAS, Version 3, K. Andersson, M. R. A. Blomberg, M. Fülscher, V. Kellö, R. Lindh, P.-A. Malmqvist, J. Noga, J. Olsen, B. O. Roos, A. J. Sadlej, P. E. M. Siegbahn, M. Urban, P. O. Widmark, University of Lund, Sweden (1994).

Appendix B

B.1 Conical Intersections in Polyenes: A VB-Theory Model for Covalent S₁/S₀ Crossings

All the discussions following in this section are based on equations (giving the energy of the involved covalent states: ground state S₀ and excited state S₁) which have been developed according to a VB-theory formalism [1] (we will see that it is quite simple to formulate the problem according to a VB formalism, and that very simple equations follow).

In general a chemical reaction between organic molecules involves a many-electrons treatment, and it is very difficult to gain exact expressions for the potential energy of the system. On the other hand only few electrons and orbitals are involved into the bond-breaking/bond-forming processes occurring in the chemical transformations. Thus the problem may be simplified by separating the electrons and the orbitals of the system into a valence block and a core block. Valence electrons/orbitals are the ones directly involved in the bond-breaking/bond-forming processes while core orbitals are not involved in the chemical process and remain doubly occupied all along the reaction path (core electrons are involved in the bonds characterising the molecular structure which remains unchanged during the chemical process, as for example the C-C and C-H σ-bonds of ethylenes in the ethylene plus ethylene cycloaddition reaction). The formalism used in this treatment is based on the generalisation to polyelectronic systems of the Heitler-London method (HL-VB) [1,2] and it follows that we will handle a valence space (the set of valence electrons and orbitals) formed by singly occupied atomic orbitals. Therefore only covalent-type valence states will be treated in this way and this method becomes a tool to rationalise the photochemistry of the studied polyenes where crossings between covalent states occur.

The distinction between core and valence electrons/orbitals makes it possible to treat the many-electrons organic reactions as reactions involving systems with a number of electrons equal to the n valence electrons which move in a field of shielded nuclei. In particular it will be possible to use for a n valence electrons/orbitals system the formulae of the potential energy developed for systems with n total electrons/orbitals [2]. Thus the PES for 2e/2o (2 valence electrons in 2 valence orbitals), 3e/3o, and 4e/4o can be expressed using the same HL-VB formalism used to describe the PES for a 2, 3 or 4 total electrons system (in 2, 3 or 4 total orbitals). In particular for the 2, 3 and 4 electrons case it exists an analytical expression for the energies of the ground and excited covalent states [2]:

E_GS = Q - T
E_IE = Q + T (B.1)

where

T = √[(K_P - K_R)² + (K_P - K_E)² + (K_R - K_E)²] (B.2)

E_GS describes the PES of the singlet (doublet) ground state, while E_IE describes the PES of the singlet (doublet) covalent excited state for the 2 and 4 (3) valence electrons problem.

The Coulomb term Q includes two different contributes to the total energy E: (i) Coulomb effects (due to the presence of charges inside the system); (ii) steric effects (due to the fact that the energetic effect of nuclei and core electrons is included in this term).

The exchange term T describes the non classic energy of the system which depends on the particular spin coupling between valence electrons. Valence bonds formation, occurring when two electrons in two distinct orbitals have a singlet spin coupling, depends upon this term. Therefore the existence of minima, intermediates and products on the PES is in general due to this quantity. Equation B.2 shows that the exchange term is a function of the variables K_P, K_R and K_E. K_P, K_R and K_E are terms coming from the sum of bicentric exchange integrals (i.e. exchange integrals computed between couples of distinct atomic orbitals), and their expression is different depending on the number n of valence electrons/orbitals characterising the ne/no system (remember that we have the same number n of valence electrons and valence orbitals). In particular:

2e/2o case:

K_P = K₁₂
K_R = 0
K_E = 0 (B.3)

3e/3o case:

K_P = K₁₂
K_R = K₂₃
K_E = K₁₃ (B.4)

4e/4o case:

K_P = K₁₂ + K₃₄
K_R = K₁₄ + K₂₃
K_E = K₁₃ + K₂₄ (B.5)

K_ij are the exchange integrals between all the different pairs (i, j) of atomic orbitals centred on the atoms which have been schematically represented above. Even if the expressions (B.3, B.4, and B.5) for the terms K_P, K_R and K_E depend on the number n of valence electrons/orbitals, nevertheless it is possible to give an identical chemical meaning to the terms K_P, K_R and K_E. In fact they are functions, respectively, of the exchange integrals between the electrons coupled as in the bond situation of the products, or of the reactants, or between electrons not at all involved in bonds (neither in the products, nor in the reactants).

According to HL-VB formalism, exchange bicentric integrals K_ij between the two centres i and j are formally identical to the ones occurring in the VB treatment of the H₂ molecule [2], and their expression is:

K_ij = ⟨ij|ji⟩ + 2S_ij⟨i|j⟩ (B.6)

where ⟨ij|ji⟩ and ⟨i|j⟩ are respectively the bi-electronic and mono-electronic exchange integrals between the orbitals i and j, while the term S_ij is the overlap integral between the same pair of atomic orbitals. The second term of equation B.6 (which is negative in sign) is usually much greater then the first term and thus dominate equation B.6: since ⟨i|j⟩ is proportional to the overlap S_ij we can state that every bicentric exchange integral K_ij is proportional to S_ij². This assumption implies that the behaviour of the three terms K_P, K_R and K_E depends on the position of the different atomic centres involved in the reaction (where valence electrons/orbitals have been localised) and therefore on the geometry of the reacting system.

For example in ethylene (which represents a model system for 2e/2o π-interactions) K_ij is proportional to the overlap S_ij between the two involved p-type atomic orbitals (in the VB treatment π-MO have been localised on each atom thus generating the two p-type carbon centred atomic orbitals used in the HL-VB model): in fact K_ij vanishes for long distances between the two carbon centres, or when the two p-type atomic orbitals are orthogonal to each other (i.e. when the overlap is zero). The same happens for the exchange integrals K_ij in the ethane case (a model for 2e/2o σ-interactions) which vanish as well when the overlap integral S_ij becomes zero (this happens at long distances between the two interacting carbon atoms) and is still (qualitatively) proportional to this term. Obviously, due to the fact that the overlap between two p-type orbitals is less then between two σ-type orbitals (at the same distance and orientation conditions), K_ij in ethane will be greater then in ethylene.

The set of variables Q, K_P, K_R and K_E represent the basis of this VB model. The following properties may be attributed to these terms:

1. The magnitude of the term Q may be qualitatively guessed and rationalised based on the Coulomb and steric effects in the system.
2. The behaviour of the terms K_P, K_R and K_E may be guessed and rationalised based on the overlap between valence orbitals.
3. The term T may be computed using equation B.2.
4. The energies (E_GS and E _EX) of the ground and excited PES may be computed using equations B.1.

Comparing equations 4.1 and equations B.1 we get:

Q_IE = (H₁₁ H₂₂)/2 T = √[(K_P - K_R)² + (K_P - K_E)² + (K_R - K_E)²] = ½ √[(H₁₁ - H₂₂)² + 4H₁₂²] (B.7)

and the expression for the exchange term T clarifies its chemical meaning: this term depends on the stability of the canonical bond structures (represented by the terms H₁₁ and H₂₂) and on the magnitude of the resonance energy among them (represented by the term H₁₂). In conclusion, while Q may be evaluated following the steric and Coulomb effects acting in the system, T may be explained on the basis of the resonance-theory concepts.

In Lecture 1 we have shown that, to have degenerate solutions (see equations 1 in Lecture 1), the radicand must vanish and (for this to occur) the two independent conditions given in equations 2 of Lecture 1 have to be satisfied. Under the VB model used here to develop the expressions for the energies (E_GS and E_EX) of covalent states (equations B.1 and B.2), these two conditions transform into the two equivalent conditions:

K_P = K_R
K_P = K_E (B.8)

which makes the exchange term T zero (the radicand in equation B.2 vanishes), and the two energies (E_GS and E_EX) identical (these are the conditions for degenerate solutions and real crossings, i.e. conical intersections between covalent states).

Now we have all the information necessary for locating the geometries of conical intersections inside the reacting system.

The simplest molecular system showing a CI between the ground and the excited covalent states is the H₃ molecule [4] (see Scheme B.1a). The advantage of this system (which is a model for 3e/3o σ-interactions) is that there are only 3 independent degrees of freedom (the three interatomic distances shown in Scheme B.1a) and only s-type atomic orbital: this allows us to neglect the orientation of the orbitals (important for p-type orbitals occurring in ethylene and, more in general, in polyenes) due to the fact that the bicentric exchange integrals K_ij depend only on the distance between the two involved atomic centres i and j.

From equations B.8 and B.4 we know that we will have real crossings (CI) when the two conditions below are satisfied:

K₁₂ = K₁₃ = K₂₃ (B.9)

and since the exchange integrals K_ij depend only on the distance between the two involved atomic centres i and j, identical distances between the three H atoms will give identical integrals, thus fulfilling the conditions B.9 for degenerate states: a CI will exist for every geometry where atoms are placed at the vertex of an equilater triangle, whatever the distance between pairs of H atoms is (see Scheme B.1a). Therefore we can arbitrarily vary one variable without exiting from the crossing region and this subspace (called intersection space) is mono-dimensional as we should already know from Lecture 1 where it has been shown that the intersection space is an hyperline (i.e. a n-2 dimensional subspace, where n=3 in this case).

The energy of the two degenerate states (T = 0) will be given (see equations B.1) by the only Coulomb term Q: if we plot this energy along the hyperline (where the H₃ geometries have all equilater triangle structures) starting from infinite and progressively approaching atoms, we get a curve (see Figure B.1) where the energy progressively slows down, reaches a (shallow) minimum, and rises up quickly as the atoms approach to close each other due to the big internuclear repulsion effects. The distance (between the H atoms in the equilater triangle geometry) giving the minimum corresponds to the so called “minimum along the hyperline” [5] (see Lecture 1). Moreover, for H atoms at an infinite distance the three integrals becomes zero because the overlap between s-type atomic orbitals vanishes (of course we have again a crossing). In polyenes this situation may be reached not only for infinite distances between the atomic centres, but also when the p-type orbitals involved are orthogonal to each other, thus making the overlap zero.

Figure B.1. Hyperline energy (Q) for the H₃ molecule as a function of the internuclear distance r (side of the equilater triangle).

We can now face this problem for polyenes; in particular, to simplify the situation, we will examine the allyl radical which is a model for 3e/3o π-interactions (the conditions for crossing states shown in the H₃ systems are the same here): in fact we have seen in Lecture 1 that benzene and polyenes photochemistry may be explained on the basis of -(CH)₃- “kinked” conical intersections which are all similar to the ones found in the shorter allyl radical system.

Figure B.2 shows the localised valence π-orbitals for the allyl radical. In this case, as told above, also the orientations of the three p-type orbitals (together with the internuclear distances ) are important to determine the values of the K_ij.

Figure B.2. π-Orbital system in the allyl radical along the disrotatory and conrotatory path.

For the relaxed planar geometry of the system, K₁₂ and K₂₃ are identical but much greater then K₁₃ (due to the longer distance between these centres). A way to increase the 1-3 overlap (and thus the value of the integral K₁₃) is to reduce the value of the binding angle C-C-C, but this leads the system to a very high energy crossing region (of very low chemical interest) on the PES due to the strong steric interactions between the terminal methylenes. Then, to increase the value of K₁₃ reducing simultaneously the values of K₁₂ and K₂₃, we can rotate the methylenes about the σ C-C bonds in a conrotatory or disrotatory way as shown in Figure B.2. As the rotation goes on the values of K₁₂ and K₂₃ reduce (the overlap between the pairs of the corresponding orbitals decreases as in ethylene) while K₁₃ (which is developing a σ-bond character) increases proportionally to the increased overlap between the two terminal p-type orbitals. Thus, along the rotation, the energy gap between the two states decreases till to the crossing point which is reached when the three integrals are identical. Of the two (conrotatory or disrotatory) ways to reach the crossing, the disrotatory rotation is more favoured (reaching easily the CI) since in this motion the terminal p-type atomic orbitals face each other while in the conrotatory motion they stay parallel and doesn’t cross: the overlap increases much more quickly following a disrotatory motion for the terminal methylenes and along this way the crossing will be reached first and easily.

All these considerations are based upon equations B.1-B.2 which give exactly the energies for the ground and excited covalent states as a function of the exchange integrals and a Coulomb term (these equations are true in the 2e/2o, 3e/3o, and 4e/4o situations). Unlikely, similar expressions for a ne/no system with n>4 are not yet known. This means that if we apply the above considerations to such systems we will identify again allyl-like crossing points. However, in our computational studies on polyenes (see the results presented in Lecture 3), we have been able to locate -(CH)₃- “kinked” conical intersections which are indeed very similar to the (conrotatory or disrotatory) allyl crossings presented above (Scheme B.1b). Moreover, it has been shown that these points (which are all characterised by these 3-C-atoms subunit, see Scheme B.1b) may explain very well the photochemical behaviour observed in polyenes, rationalising the experimental evidences of their photochemistry.

Scheme B.1

B.2 Conical Intersections in Protonated Schiff Bases: A Qualitative Model for Ionic S₁/Covalent S₀ Real Crossings

In Lecture 3 we show that the computed CI in PSB involve a crossing between the ionic excited state and the covalent ground state. These points have a 90° twisted C-C “central” double bond and correspond to a type of twisted intramolecular charge transfer (TICT) state [6] (this has been proved in particular for the short PSB presented in Lecture 3, where a net one-electron transfer between the twisted fragments results at the crossing). The geometrical and electronic structures of the conical intersections computed for the shorter PSB (see Lecture 3) are shown in Scheme B.2: their electronic configurations show clearly a net one-electron transfer from the polyene to the allyl-like -CH-CH-NH₂ twisted molecular fragments, going from the covalent ground state (S₀) to the ionic excited (S₁) state (Lecture 3 for further details).

Scheme B.2

We can now focus our attention on the minimal retinal model, the pentadieniminium cation. The existence of a CI point and the charge motion observed along the computed isomerization co-ordinate (see Figure 8 and 9 in Lecture 3) can be rationalized using the “two-electron two-orbital model” of Michl, Bonacic-Koutecky et al. [7]. According to this theory the twisted CI structure corresponds to a “critically heterosymmetric biradicaloid”. An heterosymmetric biradicaloid is a structure where two localised orbitals have different energies but do not interact. This is the situation found in the CI structure presented in Scheme B.2b where one has that the SOMO π-orbital of the allyl -CH-CH-CH₂ fragment (SOMO A in Scheme B.3a) and the SOMO π-orbital of the allyl-like -CH-CH-NH₂ fragment (SOMO B in Scheme B.3a) are not overlapping, (they are 90° twisted and, therefore, orthogonal to each other, see Scheme B.3a).

Scheme B.3

Thus this CI can be understood on the basis of a 2e/2o VB model using two (a covalent and a ionic charge-transfer) configurations constructed on the basis of the (fragment) SOMO orbitals of the two allyl-like fragments, as shown in Scheme B.3b. In this condition the energy separation of the “ionic” S₁ and “covalent” S₀ states depends on the difference between the electron affinity (EA) of the allyl SOMO (SOMO A) and the ionisation potential (IP) of the -CH-CH-NH₂ SOMO (SOMO B): one has that a conical intersection occurs when the EA of SOMO A and the IP of SOMO B are equal (i.e. the two configurations shown in Scheme B.3b have exactly the same energy).

In particular we know that two independent conditions, given in equations 4.2, have to be satisfied in order to have degenerate solutions. When the two allyl fragments are 90° twisted, i.e. the two (fragment) SOMO are not overlapping being orthogonal to each other, H₁₂ must be 0 and thus one must search for a particular twisted structure where H₁₁=H₂₂ (i.e. EA is equal to IP) to find degeneration between the two states. These quantities (EA and IP) can be indeed changed as a function of the fragment structure. Thus along the last part of the S₁ reaction co-ordinate the geometry of the two fragments is such that these energies become equal. Consequently the S₁ energy is lowered and ultimately the S₁ surface crosses with the S₀ surface. This interpretation is strongly supported by the π electron densities for the degenerate S₀ and S₁ states, and by the evolution of the charge distribution along the excited state isomerization path (Figure 8 and 9 in Lecture 3): the fact that the two degenerate states match with the two configurations given in Scheme B.3b is consistent with these computed quantities (charge distribution and π electron densities).

References

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