Topic II Quantum chemistry Chapter 10 Molecular structure and chemical bonds

1. Topic II Quantum chemistryChapter 10 Molecular structure and chemical bonds

2. The chemical bond • A chemical bond is a link between atoms. • Chemical bonds determine the structures of solids and individual molecules. • During a chemical reaction, the chemical bonds are made and broken through. When atoms form chemical bonds, the electron densities of the atoms shift and change. • Two approaches to describe the chemical bond • Valence bond (VB) theory: The properties of a molecule is explained from the wavefunctions of the atoms of a molecule. • Molecular orbital (MO) theory: The electrons in a molecule are treated as spreading throughout the entire molecule.

3. Classification of chemical bonds • Ionic bond: The bond is formed by the transfer of electrons from one atom to another atom, causing an attraction between the ions. • Covalent bond (Lewis in 1916): The bond is formed, when two atoms share of a pair of electrons. • According to the modern theory of chemical bond formation, the ionic and covalent bonds are the two extremes of a single type of bond.

4. Potential energy surface: Born-Oppenheimer approximation • Calculation scheme In electronic part, the nuclei are considered as fixed. Once the wavefunctions of the electrons are solved from the Schrodinger’s equation, the total energy of the molecule can be calculated for the configuration of the nuclei of the molecule. Repeat the procedure for different nuclei configurations by varying the bond lengths and bond angles. So, a potential energy surface of the molecule is got as a function in multidimensional space. • Physical reason: The nuclei are much heavier than an electron. So, they move much slowly relative to the motions of electrons in a molecule. • Results: The nuclei are considered as stationary at some locations, while the electrons move around them. • Benefit: The electronic and nuclear parts in the Schrodinger’s equation of a molecule can be decoupled.

5. Potential energy curve of a diatomic molecule A molecule of two atoms are called a diatomic molecule. The configuration of a diatomic molecule is specified by the distance R between the two atoms. The potential energy curve of a diatomic molecule is a function of R and has a minimum. Parameters to specify the potential energy curve of a diatomic molecule Re: Equilibrium bond length The distance corresponding to the minimum of the curve De: The well depth The depth of the minimum below the energy of the infinitely widely separation between the two atoms k: The stiffness of the bond The curvature of the energy function around the minimum.

6. VB wavefunction of a diatomic molecule (a) As the two atoms in the ground state are separated far apart. We can clearly identify that electron 1 is on atom A and electron 2 is on atom B and the interactions between the two electrons can be neglect. So, the wavefunction of the two electrons is (b) As the two atoms are at a distance close to Re, an electron has an equal probability to appear on each atom. We can not distinguish which atom the electron comes from. So, the wavefunction of the two electrons is The wavefunction with the plus sign has lower energy; the one with the minus sign has higher energy. cA(r) and cB(r): Atomic orbitals of atom A and B, respectively. This is the VB theory for the wavefunction of a diatomic molecule. The wavefunction in VB theory is only an orbital approximation.

7. Hydrogen molecule H2 in the ground state Electron density of H2 in the ground state A(1) and B(2) are the 1s orbital of H atom. rA1 : distance of electron 1 relative to nucleus A rB1 : distance of electron 1 relative to nucleus B rA2 : distance of electron 2 relative to nucleus A rB2 : distance of electron 2 relative to nucleus B

8. Pauli exclusion principle requires that the total wavefunction of the diatomic molecule should be anit-symmetric. That is, if the two electrons are exchanged, the total wavefunction should change sign. Valence-bond theory for the sbond a(1): the spin-up wavefunction of electron 1 b(2): the spin-down wavefunction of electron 2 Pauli exclusion principle requires the two electrons in a bond to be in paired. The s bond in VB theory: A valence-bond wavefunction with cylindrical symmetry around the internuclear axis.

9. Energy variation as a bond is formed • As the two atoms are well separated, the wavefunctions of the two atoms do not overlap. The energy of the “molecule” is just the double energy of an atom. • As the two atoms approach each other, the Coulombic repulsion between the two nuclei increase the potential energy. • There is an accumulation of electron density between the two nuclei. The attraction between the accumulated electrons and the two nuclei compensate the repulsions between the two nuclei and the electron-electron interactions. The kinetic energies of the two electrons are lower due to more space to move as electrons may transfer from one atom to the other. So, the total energy of the molecule gets lower as R is reduced from infinite. • As R is less than Re, the repulsion between the two nuclei get much stronger, the electron density between the two nuclei is reduced and the kinetic energies of the two electrons increase due to limited space to move. So, the total energy of the molecule becomes to increase with decreasing R.

10. A s bond resulting from two collinear pzorbitals • The VB orbital is called a s bond. • A s bond is formed by spin pairing between two electrons in the 2pz orbitalsalong the internuclear axis. • The wavefunction of the VB orbital is cylindrical symmetric and has a refection plane perpendicular to the internuclear axis. • The cylindrical symmetric axis is parallel to the internuclear axis.

11. A p bond from two perpendicular pxorbitals • The VB orbital is called a p bond. • A p bond is formed by spin paring of two electrons in the two p orbitals perpendicular to the internuclear axis. • The wavefunction of the VB orbital has a reflection plane but do not has a cylindrical symmetry. • The reflection plane is perpendicular to the internuclear axis. • The wavefunciton of the VB orbital has a nodal plane, containing the internuclear axis.

12. Atomic orbitals of atoms in N2 molecule The electronic configuration of a N atom: 2s2 2px1 2py1 2pz1 The z axis is the axis connecting the two N atoms. Each N atom has 2pz orbital pointing along the internuclear axis and 2px and 2py orbitals perpendicular to the internuclear axis.

13. Valence bonds in N2 molecule • The merging of the two 2pz orbitals forms a cylindrical symmetric s bond. • The merging of the two 2px orbitals forms a p bonds, so does the merging of the two 2py orbitals. • N2 molecule has a triple bond, denoted as :N≡N:, with one s bond and two p bonds.

14. Resonance • In VB theory, resonance is the superposition of the wavefunctions representing different electron distribution in the same nuclei of a molecule. Homonuclear diatomic molecules: molecules formed by two atoms of the same element. Example: H2, O2, N2 Heteronuclear diatomic molecules: molecules formed by two atoms of different elements. Example: CO, NO, HCl The wavefunction of HCl in a purely covalent bond This covalent bond describes the equal sharing of electron density by the H atom and the Cl atom. This description excludes the electron transfer between the two atoms. yH : 1s orbital of H atom yCl : 2pz orbital of Cl atom

15. Ionic bond: The electron of H atom transfers to the Cl atom so the HCl molecule is in the ionic form H+Cl- so that both electrons are in the 2pz orbital of Cl atom, with one in spin-up and one in spin-down. The wavefunction of HCl in a purely ionic bond In reality, the HCl molecule is in partially in the covalent bond and partially in the ionic bond. The realistic wavefunction of the HCl molecule is a superposition of the covalent and ionic wavefunctions. The wavefunction is called a resonance hybrid of the covalent and ionic bonds. • l2 gives the relative proportion of the ionic contribution. • l is a parameter between 0 and 1 and its final value is determined by that the wavefunction gives the lowest energy of the HCl molecule. • As l is close to zero, the wavefunction is more covalent-like and as l is close to one, the wavefunction is more ionic-like.

16. Polyatomic molecules: Valence bonds in H2O In VB theory, the bonding in a H2O molecule is predicted by each pairing of the two unpaired 2p electrons in the O atom with the 1s electron of each H atom. But, due to the perpendicularity of two 2P orbitals in an O atom, the VB theory predicts the bond angle of H2O molecule to be 90°, which is in poor agreement with the actual bond angle 104.5°. Two deficiencies of the VB theory: The theory predicts incorrectly the bond angles and the number of bonds that atoms can form.

17. Promotion and hybridization The sp3 hybrid orbital of a C atom A hybrid orbital has a directional character, arising from the constructive interference between the s orbital and the positive lobe of the p orbitals. The 2s and three 2p orbitals of a carbon atom hybridize to form four hybrid orbits pointing toward the corner of a regular tetrahedron. The angle between the axes of the hybrid orbits is 109.47 °. The resulting orbital is called the sp3 hybrid orbital. • An electron is allowed to be promoted from a full atomic orbital to an empty one as a bond is formed. • This promotion results in two unpaired electrons, which can participate in bond formation. • Example: C atom in the ground state: [He], 2s2, 2px1, 2py1 • The valance electrons of C atom in promotion: 2s1, 2px1, 2py1, 2pz1 • There are four unpaired electrons.

18. Tetrahedral structure of CH4 due to the sp3 hybrid orbital of C A sp3 hybrid orbital is a linear conbimation of the s orbital and the three p orbitals. The sp3 hybrid orbitals of C lead CH4 into a tetrahedral molecule containing four equivalent C-H bonds. Each hybrid orbital of C contains a single unpaired electron and the 1s electron of H can paired with each one, giving rise to a s bond pointing in a tetrahedral direction. Wavefunction of two pairing electrons Wavefunction A is the 1s orbital of a H atom.

19. The sp2 hybrid orbital of a C atom • The interference of a s orbital and two p orbitals forms three sp2 hybrid orbitals. • The three sp2 orbitals lie in a plane and has a bond angle 120°. • The third p orbital that is not involved in the hybridization has a axis perpendicular to the plane.

20. Valence bonds in C2H4 • Each C atom and two H atoms form a CH2 group by two of the sp2orbitals of the C atom and the 1s orbital of the two H atoms. • Each of the sp2 orbital and the 1s orbital forms a s bond. • The third sp2 orbital of the two CH2 groups form a s bond to connect the groups. • The unhybridized p orbitals of the two C atoms forms a p bond that is perpendicular to the plane composed of by the two CH2 groups. • The two CH2 groups are connected by one s bond and one p bond.

21. Valence bonds in C2H2 The sp hybrid orbital of a C atom • The interference of a s orbital and one p orbital forms two sp hybrid orbitals. • The two sp orbitals lie along a axis. • The two p orbital that is not involved in the hybridization have a axis perpendicular to the sp axis. • Each C atom and one H atom form a CH group by one of the sporbitals of the C atom and the 1s orbital of the H atom. • Each of the sp orbital and the 1s orbital forms a s bond. • The other sp orbital of the two CH groups form a s bond to connect the two CH groups. • The two unhybridized p orbitals of the two C atoms forms two p bonds connecting the two CHgroups. • The two CH groups are connected by one s bond and two p bonds.

22. Hybrid orbital and molecular geometry Bond angles between pure p orbitals are 90°. Bond angles between sp orbitals are 180°. Bond angles between sp2 orbitals are 120°. Bond angles between sp3 orbitals are 109.5°. Intermediate hybrid orbitals are obtained by changing the relative ratio l between the p orbitals and the s orbital that are involved in the hybridization. So, the bond angle between the intermediate hybrid orbitals may change with the relative ratio l.

23. Molecular orbitals • Basic concept: Electrons are treated as spreading throughout the entire molecule: every electron contributes to the strength of every bond. • Example of the simplest molecular system Hydrogen molecular ion H2+ : one electron and two nuclei R is the distance between the two nuclei. rA is the distance of the electron from nuclear A. rB is the distance of the electron from nuclear B.

24. LCAO approximation of molecular orbitals • Molecular wavefunctions are formed from a linear combination of atomic orbitals (LCAO). The molecular wavefunctions in this construction are called the LCAO approximation. cAand cBare the atomic orbitals centered on nuclear A and B, respectively. For homonuclear diatomic molecules, the molecular wavefunctions should describe equal probabilities for finding the electron around nuclear A and nuclear B. This requirement gives the following condition for the coefficients of cAand cB. There are two possible choices for the coefficients of cAand cB. Choice I, with cA = cB, y+ =cA+ cB (Symmetric molecular orbital) Choice II, with cA = -cB, y - =cA- cB (Antistymmetric molecular orbital)

25. Bonding and antiboning molecular orbitals • The two wavefunctions are called the s molecular orbital (MO), since the wavefunctions are cylindrically symmetric around the internuclear axis. • The bonding wavefunction in choice I is the 1s MO. • The antibonding wavefunction in choice II is the 1s* MO. Bonding MO Antibonding MO

26. Formula for 1 s and 1 s* orbitals cA and cB are 1s atomic orbitals. N+ and N- are normalization constants. S : The overlap integral R is the distance between the two nuclei.

27. The overlap integral • The atomic wavefunctions cAand cBarecentered at the nuclei A and B, respectively. • The overlap integral S of cAand cB is defined as the integration over all space for the product of the two atomic wavefunctions. V(R) of the 1s MO has a minimum at 2a0. V(R) of the 1s* MO has no minimum. • The overlap integral between two s atomic orbitals decreases with increasing R.

28. Energies of molecular orbitals • In the bonding MO, the wavefunction between the two nuclei is constructive, so the electron density is accumulated in the intermediate region and the molecular orbital has a lower energy than either of the two atomic orbitals due to the electron-nuclear attractions. • In the antibonding MO, the wavefunction between the two nuclei is destructive and has a node plane at the half-way between the two nuclei and perpendicular to the internuclear asix , so the electron density is excluded in the intermediate region and the molecular orbital has a higher energy than either of the two atomic orbitals. • The energies of the bonding and antibonding MOs vary with R, the distance between the two nuclei and depend on the overlap integral. Bonding MO Antibonding MO

29. Symmetry of molecular orbitals • Homonuclear diatomic molecules have an inversion center. • An inversion center is a point that the system is invariant if the coordinates relative to the point are changed from to . • The wavefunctions of the homogeneous diatomic molecules are either even or odd symmetry relative to the inversion center. • The bonding MO is gerade (even ) symmetry, denoted as sg. • The anitbonding MO is ungerade (odd) symmetry, denoted as su.

30. Electronic states of homonuclear diatomic molecules Hydrogen molecule Hydrogen molecular ion H2+ : single electron Electronic ground state: 1sg Hydrogen molecule H2 : two electrons Electronic ground state: 1sg2 Helium diatomic molecule He2 : four electrons Electronic ground state: 1sg2 1su2 Since the antibonding orbital is more antibonding than the bonding of a bonding orbital, the energy of the helium molecule has a energy higher than the energy as the two helium atoms are separated apart. So, the helium molecule is unstable in energy and helium is a monatomic gas. Energetic stable Helium molecule Energetic unstable

31. Homonuclear diatomic molecules for Periord 2 elements • Consider only atomic orbitals in the valence shell and ignore any core atomic orbitals. • For Period 2 elements, the atomic orbitals in the valence shell are 2s and 2p. • But, the energy of 2s oribital is lower than those of 2p orbitals so that they are treated separately. • The atomic orbitals 2s and 2pz have cylindrical symmetry about the internuclear axis.

32. Bounding and antibounding s molecular orbitals • The 2s atomic orbitals on the two atoms give the bonding and antibonding MOs, labelled as 1sg and 1su, respectively. • The 2pz atomic orbitals on the two atoms give rise to the bonding and antibonding Mos labelled as 2sg and 2su, respectively • The bonding and anitbonding s MOs have the cylindrical symmetry around the internuclear axis. su: Odd symmetry sg: Even symmetry

33. Bounding and antibounding p molecular orbitals • The 2px and 2py atomic orbitals of each atom are perpendicular to the internuclear axis. • The overlap of the 2px orbitals on the two atoms leads to the bonding and antibonding p MOs, and so do the 2py orbitals. • So both bonding and antibonding p MOs are double degenerated in energy. • The bonding effect of the electrons in a p MO is weaker than that in a s MO, because the electron density for bonding does not lie between the two nuclei. p molecular orbitals pu: Odd symmetry pg: Evensymmetry

34. The overlap integral • The overlap integral S is a measure for the extent to which two atomic orbitals on different atoms overlap. • The overlap integralis a function of the distance R between the two atoms and decreases with increasing the internuclear distance R. • The overlap integral of a s orbital and a px orbital is zero.

35. Symmetry and overlap I. One s and one pz atomic orbitals may form one s MO, and two pz atomic orbitals may also form one s MO, II. Two px atomic orbitals may form one p MO. III. One s and one px atomic orbitals can not form a MO. Reasons: Both s and pz atomic orbitals have cylindrical symmetry along the internuclear axis, so that the overlap integral S of the two atomic orbitals is not zero. But, due to different symmetries of the s and px orbitals, the overlap integrals of the s wavefunction with the upper half of the px wavefunction and with the lower half of the px orbital have an equal magnitude but an opposite sign so that the overall overlap integral of the two atomic orbitals is zero.

36. Energy levels of N2 and O2 molecules N atom: 2s2 2px1 2py1 2pz1 Total number of valance electrons: 10 Ordering of the MOs of N2: 1sg , 1su , 1pu , 2sg , 1pg , 2su Ground-state electronic configuration: 1sg2 1su2 1pu4 2sg2 O atom: 2s2 2px2 2py1 2pz1 Total number of valance electrons: 12 Ordering of the MOs of O2: 1sg , 1su , 2sg , 1pu ,1pg ,2su Ground-state electronic configuration: 1sg2 1su2 2sg2 1pu4 1pg2 A reactive component with a net spin S = 1 An unreactive component without net spin

37. Energy levels of homonuclear diatomic molecules for Period 2 elements

38. Bond length and bond dissociation energy Bond order b N : the number of electrons in the bonding orbitals N* : the number of electrons in the antibonding orbitals • The greater the bond order, the shorter the bond length. • The greater the bond order, the greater the bond strength.

39. Photoelectron spectroscopy • Photoelectron spectroscopy (PES) measures the ionization energies of molecules when electrons are ejected from orbitals by absorption of a photon of known energy.

40. Polar bond of heteronuclear diatomic molecules • The electronic distribution in a heteronuclear diatomic molecule is asymmetric between the two atoms of the molecule, because it is energetically favorable for the electron pair to be found closer to one atom than to the other. • This asymmetry results in a polar bond, which is a covalent bond with an electron pair unequally shared by the two atoms. • An example is the HF molecule, in which the electron pair between the two atoms results in the F atom having a partial negative charge d- and the H atom having a partial positive charge d+. An atom of the diatomic molecule has a partial negative charge d- and the other has a partial positive charge d+. The molecule has a permanent electric dipole moment p= d |d-|, where d is the bond length of the diatomic molecule.

41. Molecular orbitals of heteronuclear diatomic molecules Specail case I: For the unpolar bond, |cA|2 = |cB|2 Special case II: For the ionic bond, cA= 0 and cB = 1. General case: For a polar bond, cA ≠ cB. The coefficiants cA and cB are determined by the variation principle.

42. Variation principle for molecular orbitals • The two coefficients cA and cB are considered as two variables. We vary the two coefficients until the lowest energy of the molecular orbital is achieved. Coulomb Integral Resonance Integral Overlap Integral

43. Molecular orbitals of variation principle Case I: Homonuclear diatomic molecules (aA = aB = a) Because b< 0, so E+ < E-. E+: Energy of bonding MO E-: Energy of antibonding MO Case II: Heternuclear diatomic molecules (aA ≠ aB) If the overlap integral S between the atoms is approximately zero. If |aA - aB | >> 2|b|, (aB < aA < 0)

44. Molecular-orbital energy levels of HF The atomic orbital yH is 1s and yF is 2pz. The elctronic ionization energy I of H is 13.6 eV and that of F is 17.4 eV. The electron affinity Eea of H is 0.75 eV and that of F is 3.34 eV. The estimated energy of atomic orbital in diatomic molecule is – (I + Eea )/2.

45. MO energy levels of CO and NO molecules Ground-state electronic configuration of CO molecule: 1s2 2s2 1p4 3s2 The HOMO is 3s and the LUMO is 2p. Ground-state electronic configuration of NO molecule: 1s2 2s2 3s2 1p4 2p1 The HOMO is 2p and the LUMO is 4s. HOMO: The highest occupied molecular orbital LUMO: The lowest unoccupied molecular orbital

46. Molecular orbitals of polyatomic molecules • In the LCAO approximation, a molecular orbital of a polyatomic molecule is a linear combination of atomic orbitals of valance shells of all atoms in the molecule so that the molecular orbital spreads over the entire molecule. Example: H2O Atomic orbitals of two H atoms: 1s Atomic orbitals of the O atom: one 2s and three 2p • There are total six atomic orbitals so there are six molecular orbitals. • The lowest-energy MO has the least number of nodes between adjacent atoms and is the most strongly bonding orbital. • The highest-energy MO has the greatest number of nodes between adjacent atoms and is the most strongly antibonding orbital.

47. Theoretical method • Born-Oppenheimer approximation and LCAO approximation ym:Molecular wavefunction ci: Atomic orbital The self-consistent field (SCF) procedure is to find the coefficients cmi by satisfying the Schrodinger’s equation with a repetition in tried calculations. Semi-empricial method: Some parameters are chosen by the experimental data, such as the spectroscopic data and ionization energies. Ab initio method: calculations from first principles, avoiding the need by appearing the experimental data and using the number of the atoms present. Density functional theory: The energy of the ground-state MO is a function of the electron density.

48. Electronic configuration for C and structure of benzene For each C, the p orbital that is not participated in the sp2 hybridization is perpendicular to the benzene ring. The six p orbitals form the p molecular orbitals.

49. Molecular orbitals of p bonds in benzene The lowest-energy MO is the most strongly bonding orbital ,with all cn = 1 for n = 1 to 6, and has no internuclear nodes, for the p-electron density can accumulated between the nuclei. The highest-energy MO is the most strongly antibonding orbital and has a node plane between two neighboring C atoms, so the p-electron density is reduced between the nuclei. 2b = -460 kJ mol-1. Total energy E = 6a +8b The four intemediate MOs, 1e1g and 1e2u, are two doubly degenerate pairs. Ground-state electronic configuration: a2u2, e1g4 The six electrons are not localized on atoms but delocalized on the benzene ring. The HOMO is 1e1g and the LUMO is 1e2u.

50. Computational chemistry • Researches in chemistry by computational calculations • Tools: computational resources and software packages • Theory: fundamental principles of quantum mechanics • Topics: studies of electronic structure and reactivity of complex molecular processes • Application: pharmaceutical chemistry for drug design • Perspective: “Computational Chemistry in 25 years”