Theoretical background

Theoretical background 1. Early orbital concepts for the description of saturated and unsaturated compounds - Quantum Chemistry - Molecular orbital (MO) description of the allyl cation, radical and anion - Resonance descripton of the allyl cation, radical and anion - Theories of conjugative stabilization: butadiene - Electrophilic addition to conjugated dienes 2. The Schrödinger equation and the Hamiltonian operators - The central field Hamiltonians - The molecular Hamiltonian 3. The Born-Oppenheimer approximation and the concept of the potential energy surfaces

Orbital concepts The solutions: the atomic orbitals of the H - atom (or H - like ions) the associated orbital energies (E = ). Example: the 1s orbital of H has the following form r := the distance of the electron measured from the nucleus Orbital concepts for the description of saturated and unsaturated compounds - Quantum Chemistry computational chemistry: the numerical implementation of the principles of quantum mechanics to chemical problems For the “one electron” atom the solution is the corresponding one-electron Schrödinger equation: Y Y, of the H-atom or of H - like ions (He+, Li2+, Be3+, B4+ ,...etc.), is a one-electron wave equation which can be solved analytically. Warning: atomic orbitals are mathematical functions and not physical objects

An example: the hydrogen atom (H)1 electron 1 occupied AO Orbital concepts z x y E(RHF/3-21G)=-0.4962 Hartree 2pz AO 3 (+1.069) 2py AO 3 (+1.069) 2px AO 3 (+1.069) 4 times Degenerated AOs 2s AO 2 (+1.069) 1s AO 1(-0.4962)

In a many electron atom (e.g. Be) the solutions of the corresponding many-electron Schrödinger equation is more complicated constructed from one-electron wave functions, i.e., from orbitals. Orbital concepts Example 1: Be with 4 electrons: 4 - electron state-wave function constructed as two doubly occupied atomic orbitals (1s and 2s) the double occupancy is stipulated to involve one electron with  spin and one with  spin. This double occupancy is frequently denoted in terms of the energy levels 1s and 2s. Example 2: Ne with 10 electrons: state-wave function:  constructed from five doubly occupied atomic orbitals (1s, 2s, 2px, 2py, 2pz) corresponding to the 1s2 2s2 2p6 occupancy scheme

An example: the neon atom (Ne)10 electron 5 occupied AO Orbital concepts z x y E(RHF/3-21G)=-127.8038245 Hartree 3 times degenerated AOs -0.79034 Hartree 2pz AO 5 2px AO 3 2py AO 4 2s AO 2 -1.86515 Hartree 1s AO 1 -32.56471 Hartree

many-electron molecular systems analogous to that of many-electron atoms: i.e. the many-electron state wave functions are constructed from many one-electron wave functions. Orbital concepts For atoms: one-centered atomic orbitals are appropriate For molecules:multicentered one-electron functions (molecular orbitals) are needed Molecular orbitals are constructed by the linear combination of atomic orbitals (LCAO).

An example: the hydrogen molecule (H2) 2 electron 1 double occupied MO,= 0 Debye Orbital concepts EPS -0.007  töltés. 0.007 E(RHF/3-21G)= -1.12295984 Hartree (LUMO) anti-bonding MO * MO 2 (0.26570) (HOMO)  MO 1(-0.59424) bonding MO 1s AO 1(-0.4962) The AO of H atom

Orbital concepts the functional form 2pz = Nz exp(r) An axial orientation yields a  - bonding and a * anti-bonding orbital pair the shape A 2pz atomic orbital a parallel orientation yields a  - bonding and a  * anti-bonding orbital pair. memo: the + and - represent the sign of the mathematical function denoted by 2pz. Two types of MO can be constructed from a pair of 2p type AOs depending on their orientations: a  - type and a  - type

Orbital concepts shape vector model the 2p atomic orbitals. Linear combination is a mathematical process of Linear Algebra where vectors are subjected to linear combination i.e. multiplication with weighing coefficients and subsequent addition and/or subtraction. In this sense, atomic orbitals may be regarded as vectors. This is obvious in the case of px, py and pz orbitals memo: 1s or 2s orbitals have no directionality but can also be regarded as vectors. Therefore, the electron distribution of C, N, O, F and Ne can be represented by 5 vectors: 1s, 2s, 2px, 2py, 2pz. These five vectors will be part of the so-called 'basis vectors' or 'basis sets' used for a molecular orbital calculations.

Example 1: the formaldehyde(H2C=O) The LCAO transformation of the 12 AO basis set to 12 MO 8 doubly occupied an 4 empty (denoted by *) s10* s9 * s8 * 9 * Orbital concepts a number of empty (antibonding or virtual) MO’s anti-bonding MO 8 s7 s6 s5 s4 s3 s2 s1 O c1s, c2s, c2px, c2py, c2pz LCAO C c1s, c2s, c2px, c2py, c2pz H c1s H c1s 1  - electron pair two n – type lone electron pairs (lp) 3  - bonds: a C - O and two C - H bonds 2 atomic cores: 1s2 for C and 1s2 for O

12 AO represent the 'basis set' for the MO calculation for formaldehyde Orbital concepts 8double occupied MOs 12 AO The set of MO’s obtained from the LCAO-MO calculation reflects the symmetry of the molecule. These MO’s are called canonical MO’s (CMO’s); they are fully delocalized They are not directed along the traditional representation of chemical bonds.

Example 2: water(H2O) The LCAO transformation of the 7 AO basis set to 5 MO Orbital concepts Orbital concepts Instead, an anti-symmetric (σantisym) and a symmetric (σsym) delocalized -type CMO No canonical MO’s, obtained from the LCAO calculation, correspond to the two equivalent bonds directed along the O - H lines. the linear combination (i.e. addition and subtraction) of these CMO’s leads to localized molecular orbitals (LMO’s) which show the directionality of the two O - H bonds

Orbital concepts they are directly computable from the AO basis set and they faithfully represent the symmetry of the molecule The CMO representations of molecules have many advantages: The CMO representations of molecules have disadvantages: they do not have a directionality, which coincides with the direction of the chemical bonds. The three most important structural motifs in organic chemistry Cartesian coordinates do not match all three structural motifs The directionality of the AO is not a prerequisite for the formation of a MO, for a qualitative or pictorial description of bonding, chemists frequently hybridize the Cartesian AO. As a result, chemical problems are often discussed in terms of hybrid atomic orbitals (HAO).

Orbital concepts hybridization is an orthogonal transformation of a given basis set (1 = 2s; 2 = 2px; 3 = 2py; 4 = 2pz) to an equivalent basis set {i}. Since the hybridization may involve the mixing (i.e. linear combination) of two (2s, 2px), three (2s, 2px, 2py) or four (2s, 2px, 2py, 2pz) AO’s, it is possible to derive 3 types of hybridization, referred to as sp, sp2 and sp3 respectively.

The computed CMO’s are delocalized over the whole molecule whether they belong to the  or the  representations. CMO’s are symmetry adapted. The geometrical equivalents of localized molecular orbitals (LMO) are governed by the stereochemistry of the molecular bonding. . Orbital concepts Example: Methane may be used to illustrate this point A schematic representation of the shapes and orientation of the four valence CMO of CH4. Approximate shapes of hybrid atomic orbitals (HAO) of carbon

EPS -0.007  charge. 0.007 Orbital concepts Orbital concepts Computed CMOs Of CH4 Degenerated MOs RHF/3-21G closed shell MO 4 (-0. 54483) MO 2 (- 0.94587) MO 1 (-11.14448) MO 5(-0. 54483) MO 3 (-0.54483) schematic representation of the change of energy levels upon localization of CH4

Orbital concepts Localization involves the addition or subtraction of the four  (CMO) to obtain the four  (LMO) The shapes and orientation of the four valence LMO of CH4 obtained by localization of the four valence CMO’s

Orbital concepts For CH4, the four C - H bonds { I }, may also be formed from the four sp3 hybrid AO {i} and the 1s orbitals { Hi } of the four hydrogen atoms. 1 = a1 + bH1 2 = a2 + bH2 3 = a3 + bH3 4 = a4 + bH4 The linear combination of carbon HAO and hydrogen AO to form methane LMO. However, { i }, the HAOs also have their own expression in terms of the unhybridized AO of carbon. Take for example the first of the series  (remember that 1 = 2s, 2 = 2px, 3 = 2py, 4 = 2pz and H1 = 1sH1). In order to see the set of y(LMO) expressed in terms of the AO we have to specify the expressions for the set of (CMO). 1 = C11 + C2 (H1 + H2 + H3 + H4) 2 = C12 + C2 (H1 - H2 + H3 - H4) 3 = C13 + C2 (H1 - H2- H3 + H4) 4 = C14 + C2 (H1 + H2 - H3 - H4)

The first LMO (y1) may be written as the following linear combination of the four CMO: and Orbital concepts y 1 = ½ 1 + ½ 2 + ½ 3 + ½ 4 = ½ {1 + 2 + 3 + 4} = ½ {C11 + C2 (H1 + H2 + H3 + H4) + C12 + C2 (H1 - H2 + H3 - H4) + C13 + C2 (H1 - H2 - H3 + H4) + C14 + C2 (H1 + H2 - H3 - H4)} When the addition is performed, the terms involving H2, H3 and H4 cancel and we are left with the expression The hybrid structure is clearly recognizable in the first term of the above expression and we may rewrite it conveniently in the form Compare this to the equation given at the beginning of this section for the first LMO, i.e. one of the C – H, Conclusion: the CMO and LMO representations are equivalent

Consider H2O again. The LMO picture of the 4 valance electron pairs related to an sp3 hybridized oxygen Orbital concepts The two lone electron pairs (lp) point to the back and the two bond pairs (O - H) point towards the front. The two bond pairs have been discussed earlier in this chapter LMO for the water: in the ideal case a = 109.5o = b; in reality a < 109.5o < b.

Orbital concepts Fact is that both LMO and CMO representations of the 2 lone pairs of oxygen in H2O lead to the same 4-electron density. Two orthogonal planes containing the 4 bond electron density and the 4 lone-pair electron density.

Molecular Orbital (MO) Description of the Allyl Cation, Radical and Anion Orbital concepts

Orbital concepts

Schrödinger equ. & Hamiltonian

Born-Oppenheimer approx.

Theoretical background