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Lecture 15 Molecular Bonding Theories 1) Molecular Orbital Theory • Considers all electrons in the field of all atoms constituting a polyatomic species, so that all molecular orbitals (MO’s) are multicenter. • The most common way to build MO’s, yMO,j, is by using a linear combination of all available atomic orbitals yAO, i (LCAO): y MO, j cijy AO,i i • Consider dihydrogen molecule ion H2+ as an example. We can form two molecular orbitals from two hydrogen’s atomic orbitals yA and yB : yA yB • yg = y A + y B yu = y A - y B yA yB The probability to find an electron on the yg orbital is then N 2 y g2 dv N 2 ( y A2 dv 2 y Ay B dv y B2 dv ) N 2 (1 2S 1) 2 N 2 (1 S ) 1 where S is the overlap integral for atomic orbitals yA and yB and N is normalizing constant. S > 0 corresponds to a bonding (yg) and S < 0 (yu) – to antibonding situations. 2) Molecular Orbital Theory. Dihydrogen molecule • In dihydrogen molecule we have 2 electrons, 1 and 2. Assuming that 1 and 2 are independent one from another (one-electron approximation), we can get: yg = [y A(1) + y B(1)][y A(2) + y B(2)] yg = [y A(1)y A(2) + y B(1)y B(2)] + [y A(1)y B(2) + y A(2)y B(1)] yu = [y A(1)y A(2) + y B(1)y B(2)] - [y A(1)y B(2) + y A(2)y B(1)] • The first term corresponds to ionic contribution and the second one – to covalent contribution to the bonding (compare with VB theory). AMPLITUDE (A.U.) yu yA u g yB yg g < u INTERNUCLEAR DISTANCE, BOHR 3) Atomic orbital overlap and covalent bonding • • • Interaction between atomic orbitals leads to formation of bonds only if the orbitals: – 1) are of the same molecular symmetry; – 2) can overlap well (see explanation of the overlap integral below); – 3) are of similar energy (less than 20 eV energy gap). Any two orbitals yA and yB can be characterized by the overlap integral, S y Ay B dV Depending on the symmetry and the distance between two orbitals, their overlap integral S may be positive (bonding), negative (antibonding) or zero (non-bonding interaction). S>0 S<0 s g) s u) * p u) *p g) H H P H CC P M P C S=0 * p g) *p u) * (no symmetry match) 4) Simplified MO diagram for homonuclear diatomic molecules (second period) • Combination of 1s, 2s, 2px, 2py and 2pz atomic orbitals of two atoms A of second row elements leads to ten molecular orbitals. 10) 9) 8) 7) 6) 5) 4) 3) 2) 1) *2p = y2pA – y2pB *2py = y2pyA – y2pyB *2px = y2pxA – y2pxB 2py = y2pyA + y2pyB 2px = y2pxA + y2pxB 2p = y2pA + y2pB *2s = y2sA – y2sB 2s = y2sA + y2sB *1s = y1sA – y1sB 1s = y1sA + y1sB (3u) (g) (g) (u) (u) (3g) (2u) (2g) (1u) (1g) 3u g 2p 2p u 3g 2u 2s 2s 2g 1u 1s 1s 1g x2 x1 A A y1 y2 z 5) Orbital mixing and level inversion in homonuclear diatomic molecules • • Orbitals belonging to the same atom mix if all of the following is true: 1) they are of the same symmetry; 2) they are of similar energy (less than 20 eV difference). Note that there is no -orbitals of the same symmetry in diatomic homonuclear molecules (g and u only). So, they energy levels will remain unaffected by mixing. Before s-p mixing After s-p mixing 3u 3u mix 3g mix 3g 1u less bonding, lone pair like 1u 2u 2u 2g more antibonding 2g less antibonding, lone pair like more bonding 6) MO diagrams of homonuclear diatomic molecules • Filling the resulting MO’s of homonuclear diatomic molecules with electrons leads to the following results: A 3u B g AB # of e’s Bond order # unpair. e’s Bond energy, eV Li2 6 1 0 1.1 Be2 8 0 0 - B2 10 1 2 3.0 C2 12 2 0 6.4 N2 14 3 0 9.9 O2 16 2 2 5.2 F2 18 1 0 1.4 Ne2 20 0 0 - Bond order = ½ (#Bonding e’s - #Antibonding e’s) 2p 3g 2p u 2s 2u 2s 2g 1u 1s 1g 1s 7) Molecular Orbital Theory. Energy levels in N2 molecule • Photoelectron spectroscopy of simple molecules is an invaluable source of the information about their electronic structure. • The He-I photoelectron spectrum of gaseous N2 below proves that there is the - level inversion in this molecule. It also allows identify bonding (peaks with fine vibronic structure) and non-bonding MO (simple peaks) in it. N 6u N 2g 15.6 2p 16.7 5g 2p 1u 18.8 4u 2s 2s 3g 2u E, eV 1s 1g 1s