General Physical Chemistry II

Transcription

1 General Physical Chemistry II Lecture 10 Aleksey Kocherzhenko October 7, 2014"

2 Last time "

3 promotion" Promotion and hybridization" [He] 2s 2 2p x 1 2p y 1 2p z0 " 2 unpaired electrons" [He] 2s 1 2p x 1 2p y 1 2p z 1? " 4 unpaired electrons!" H" Energy required for promotion is recovered by the atom s ability to form 4 bonds instead of 2" H" C" Valence electron can be promoted from full to empty atomic orbital (AO)" H" s-orbitals and p-orbitals are very different, but:" Ø All bonds in CH 4 are of equal length" Ø All angles between bonds are the same" H" Hybrid orbitals are interference patterns formed by orbitals of different symmetry" Mathematically, hybrid orbitals are linear combinations of s- / p- / d-orbitals" Electrons in a promoted atom occupy hybrid orbitals" Common hybridizations: sp 3, sp 2, sp, "

4 Valence bond theory" Ø A bond is formed when an electron in an atomic orbital on one atom pairs its spin with that of an electron in an atomic orbital on another atom" Ø Orbitals on the same atom are combined to generate a set of directed hybrid atomic orbitals (AOs)" Ø Bonds are described as formed by hybrid AOs on different atoms" Ø Hybrid AOs assumed to contribute independently to electron density and molecular energy"

5 Molecular orbital theory"

6 Molecular orbital vs. valence bond theory" Valence bond theory:" Ø Orbitals on the same atom are combined to generate a set of directed hybrid atomic orbitals (AOs)" Ø Bonds are described as formed by hybrid AOs on different atoms" Ø Hybrid AOs assumed to contribute independently to electron density and molecular energy" Molecular orbital theory:" Ø Electronic wavefunctions (molecular orbitals, MOs) extend across the entire molecule: every electron contributes to every bond" Ø MOs are complex; approximated as linear combinations of atomic orbitals (LCAO) in quantum chemical calculations" Ø All atomic orbitals of appropriate symmetry contribute to MOs " Quantum chemical calculations are overwhelmingly based on MO theory"

7 Linear combination of atomic orbitals (LCAO)" Molecular wavefunction for a diatomic molecule A B:" =c A A + c B B Relative contributions of atomic orbitals:" c A 2 and" c B 2 Atomic orbitals on atoms A and B" AOs interfere constructively " " "à bonding MO:" 1 = p 1 2 ( A + B ) AOs interfere destructively " "à antibonding MO:" 1 = p 1 2 ( A B ) Lowest energy bonding σ-mo " Lowest energy antibonding σ-mo "

8 Overlap integrals" Accumulation of probability density to find electrons in the internuclear region is measured by the overlap integral:" ZZZ S = V A BdV S =0 no overlap" S =1 complete overlap (identical orbitals)" For two 1s orbitals on H nuclei separated by distance :" S = 1+ RrB + R2 3r 2 B R exp R r B R/r B"

9 Bonding vs. antibonding MOs" Bonding orbital: nuclei attracted to electron density between them" Antibonding orbital: nuclei attracted to electron density outside the internuclear region" A" B" The antibonding MO is more antibonding than the bonding orbital is bonding "

10 Inversion symmetry" Projecting any point of orbital through center of molecule to equal distance on the other side" à same value for wavefunction" Gerade (even) symmetry" à wavefunction changes sign" Ungerade (odd) symmetry"

11 Constructing and filling MOs" 1. Construct molecular orbitals by forming linear combinations of all suitable valence atomic orbitals supplied by the atoms." (N atomic orbitals result in N molecular orbitals)" 2. Accommodate the valence electrons supplied by the atoms so as to achieve the lowest overall energy subject to the constraint " of the Pauli exclusion principle." (no more than two electrons may occupy a single orbital, and then they must have opposite spins)" 3. If more than one molecular orbital of the same energy is available, add electrons to different orbitals before doubly occupying any one orbital. " (this minimizes electron-electron repulsion)" 4. Take note of Hund s rule: if electrons occupy different degenerate orbitals," then they do so with parallel spins."

12 H 2 + and H 2" 1. Construct molecular orbitals by forming linear combinations of all suitable valence atomic orbitals supplied by the atoms" 2. Accommodate the valence electrons supplied by the atoms so as to achieve the lowest overall energy subject to the constraint of Pauli exclusion principle" Ground state of H 2+ : 1σ 1" Ground state of H 2 : 1σ 2" 1 = 1 p 2 ( A1s B1s ) Bond order: " b = 1 2 (N N ) H H # of e in bonding orbital" # of e in antibonding orbital" For H 2 : " b =1 1 = 1 p 2 ( A1s + B1s ) The greater the bond order, the shorter and stronger the bond "

13 He or He 2? " Bond order: " b = 1 2 (N N ) # of e in antibonding orbital" For He # of e in bonding orbital" 2 : b =0à He " 1σ 2 g 1σ 2" u Antibonding orbital is higher in energy than bonding orbital (nuclear repulsion destabilizes the latter) " E ab > E b Configuration: " 1 = 1 p 2 ( A1s B1s ) E He " ab He " He 2 has higher energy than 2 He atoms à He is a monoatomic gas" E b 1 = 1 p 2 ( A1s + B1s )

14 Homonuclear diatomic molecules: σ-bonds" Ø In period 2 atoms the valence orbitals are 2s and 2p" Ø σ orbitals are built from all valence orbitals of appropriate symmetry (symmetric with respect to the z-axis): for period 2, these are 2s and 2pz" 4 MOs of σ-symmetry are formed by an appropriate choice of coefficients " = ca2s A2s + cb2s B2s + ca2pz A2pz + cb2pz B2pz

15 Constructing σ-bonds " =c A2s A2s + c B2s B2s + c A2pz A2p z + c B2pz B2p z Ø 2s and 2p z orbitals have distinctly different energy à treat them separately " Ø 2s orbitals on each atom overlap to form bonding and antibonding orbitals: 1σ and 1σ* (1σ g and 1σ u ) " 1 = A2s ± B2s 1σ *" 1σ" Ø 2p z orbitals lie along the internuclear axis and can form cylindrically symmetric bonds 2σ and 2σ* (2σ g and 2σ u ) " 2 = A2p z ± B2p z

16 Homonuclear diatomic molecules: π-bonds " Ø 2p x and 2p y form bonding and antibonding orbitals when arranged side by side (4 AOs à 4 MOs) " Inversion symmetry for π-bonds " 2p x and 2p y have the same energy à 2 doubly degenerate bonding and 2 doubly degenerate antibonding orbitals "

17 Diatomic molecules: Orbital mixing " Ø Real σ MOs are a mixture of 2s orbitals and 2p z orbitals that participate both in 2σ g and in 1σ u MOs:" =c A2s A2s + c B2s B2s + c A2pz A2p z + c B2pz B2p z Ø Note the location of 1π u relative to 2σ g (true for period 2 diatoics up to and including N 2 ) " For O 2 and F 2 : "

18 Energy level ordering for 2 nd period diatomics " 2σ u 2π g 2π u 2σ g 1σ u MO diagrams can be used to determine bond order and whether molecule is diamagnetic or paramagnetic" 1σ g

19 Summary" Ø Like valence bond theory, molecular orbital theory uses atomic orbitals (s / p / d / f) to construct another type of orbitals" Ø However, in valence bond theory the new set of orbitals are hybrid atomic orbitals localized on a single atom that interact to form bonds" Ø On the other hand, in molecular orbital theory the new set of orbitals are molecular orbitals that extend throughout the entire molecules" Ø Bonds are understood in terms of the electron distribution in the molecular orbitals: in bonding orbitals, most of the electron density is in the region between nuclei, in antibonding orbitals, most of the electron density is outside this region" Ø N atomic orbitals result in the formation of N molecular orbitals; only orbitals in the shell containing valence electrons are used" Ø The order in which molecular orbitals are filled is such as to minimize the total energy of the molecule and, for degenerate orbitals, such as to maximize the total spin" Ø An important characteristic of molecular orbitals is the type of their inversion symmetry: gerade or ungerade"