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CHM2S1-A Introduction to Quantum Mechanics Dr R. L. Johnston II: Quantum Mechanics of Atoms and Molecules 5. Electronic Structures of Atoms 5.1 The hydrogen atom and hydrogenic ions. 5.2 Quantum numbers. 5.3 Orbital angular momentum. 5.4 Atomic orbitals. 5.5 Electron spin. 5.6 Properties of atomic orbitals. 5.7 Many-electron atoms. THE UNIVERSITY OF BIRMINGHAM

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6.Bonding in Molecules 6.1 The Born-Oppenheimer approximation. 6.2 Potential energy curves. 6.3 Molecular orbital (MO) theory. 6.4 MO diagrams. 6.5 MOs for 2 nd row diatomic molecules. 6.6 Molecular electronic configurations. 6.7 Bond order. 6.8 Paramagnetic molecules. 6.9 Heteronuclear diatomic molecules.

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5. Electronic Structures of Atoms 5.1The Hydrogen Atom and Hydrogenic Ions The series of atoms/ions H, He +, Li 2+, Be 3+ … all have:1 electron(charge –e) nucleus(charge +Ze) To determine the electronic wavefunction ( = e ) and allowed electronic energy levels (E), we must set up and solve the Schrödinger Equation for a single electron in an atom.

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Consider 3-D motion of the electron (-e) relative to the nucleus (+Ze): Hamiltonian Operator Kinetic Energy where Potential Energy where (electrostatic attraction between electron and nucleus). +Ze mNmN -e meme r

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We must solve the 2 nd order differential equation: Because of the spherical symmetry of the atom, it is convenient to describe the position of the electron in spherical polar coordinates (r, , ), rather than Cartesians (x,y,z). Due to this spherical symmetry, the wavefunction can be separated into a product of radial and angular components: (r, , ) = R(r).Y( , ) Imposing boundary conditions 3 quantum numbers (n,,m ) n,,m (r, , ) = R n, (r).Y,m ( , ) radial wavefunction angular wavefunction (spherical harmonic)

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5.2Quantum Numbers n,,m (r, , ) = R n, (r).Y,m ( , ) depends on 3 quantum numbers (n,,m ). 1. Principal Quantum Number (n) Positive integern = 1, 2, 3 … For H and 1-e ions, the electron energy depends only on n (this is the same result as from the Bohr model). For many-electron atoms E depends on n and.

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2. Orbital Angular Momentum Quantum Number ( ) For a given value of n, can take the integer values: = 0, 1, 2 … (n-1) e.g.n = 1 = 0 n = 2 = 0, 1 n = 3 = 0, 1, 2 3. Orbital Magnetic Quantum Number (m ) For a given value of, m can take the integer values: m = 0, 1, 2 … e.g. = 0m = 0 = 1m = 0, 1 There are (2 +1) allowed values of m.

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5.3Orbital Angular Momentum Q.Nos. and m arise from the angular part of the wavefunction and they relate to the angular momentum of the electron due to its motion around the nucleus (orbital angular momentum). – determines magnitude of angular momentum vector (J) Because is quantized, so is |J|: 0123 |J| 0 22 66 12

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The orientation of the angular momentum vector (J) is also quantized – as it depends on m. The component of angular momentum along a reference direction (e.g. the z-axis): J z = m e.g. for = 2 (m = 0, 1, 2) angular momentum vector J has magnitude |J| = 6 and 5 allowed orientations: J z = 0, , 2 . m

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5.4Atomic Orbitals The wavefunctions ( n,,m ) describing an electron in an atom are known as atomic orbitals. Each atomic orbital (1-e wavefunction) is uniquely defined by 3 quantum numbers (n,,m ). The orbital ( ) gives the spatial distribution of the electron (via the Born interpretation of | | 2 ). Orbitals are often drawn as 3-D surfaces which enclose approx. 90% of the probability of finding the electron.

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Electronic Shells, Sub-shells and Orbitals All orbitals with the same value of n together constitute an electronic shell. For each n, orbitals with the same value of together constitute an electronic sub-shell. Each sub-shell consists of (2 +1) orbitals, each with a different m value. n 1234 Shell KLMN 0123 Sub-shell spdf No. orbitals 1357

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Possible combinations of shells and sub-shells. Total number of orbitals in shell n is n 2. n 0123… 1 1s 2 2s2p 3 3s3p3d 4 4s4p4d4f …

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Electronic Shells, Sub-shells and Orbitals n=3 l =0 l =1 l =2 m l =2m l =1m l =0 ml =1ml =1ml =2ml =2 3d orbitals m l =1m l =0 ml =1ml =1 3p orbitals m l =0 3s orbital shellsub-shellsorbitals

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Atomic Orbital Energy Diagram for H Orbital energies depend only on the principal quantum number n. For a given n value (shell), all sub-shells ( ) and orbitals (m ) have the same energy – i.e. they are degenerate.

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Quantum Numbers and Atomic Orbitals Quantum theory postulates that electrons in atoms are fixed in regions of space corresponding to atomic orbitals. The probability of finding an electron at a particular point in space (r) is proportional to | (r)| 2 (Born Interpretation). Orbitals differ from each other in size, shape and orientation: –the orbital size is defined by the principle quantum number n –the type of orbital (it’s shape) is defined by the angular momentum quantum number –the orientation of the orbital is defined by the orbital magnetic quantum number m

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Shapes of Atomic Orbitals s orbitals have l = 0 and m l = 0 d x 2 y 2 dz2dz2 d xy d xz d yz d orbitals have l = 2 and m l = 2, 1, 0, 1, 2 pzpz pxpx pypy p orbitals have l = 1 and m l = 1, 0, 1

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5.5Electron Spin Stern and Gerlach (1921) observed that a beam of Ag atoms is split into 2 beams by an inhomogeneous magnetic field. Dirac (1930) introduced relativistic effects into Quantum Mechanics and showed that to completely describe the state of an electron we must specify: 1. The orbital ( n,,m ) 2. The spin state of the electron Electron spin is characterised by 2 quantum numbers: – spin angular momentum q. no. s ( = ½ for all electrons) – spin magnetic q. no. m s ( = ½).

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Spin angular momentum (S) has magnitude: The projection of S on the z-axis: S z = m s = ½ There are two possible electron spin states: m s = +½“spin up” m s = -½“spin down” Spin is an intrinsic property of the electron and is not connected with orbital motion. Complete specification of an electron in an atom requires 4 quantum numbers (n,,m,m s ) – as s is fixed.

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5.6Properties of Atomic Orbitals 1. Shape – determined by the angular wavefunction Y,m ( , ) 1s orbital(n = 1, = 0, |J| = 0) Normalized wavefunction: Has no angular dependence (spherically symmetric, depends only on r). All s orbitals are spherically symmetrical. 2s orbitalns orbital As n , orbitals expand – average radius r +

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2p orbitals(n = 2, = 1, |J| = 2 ) There are 3 degenerate orbitals, with 3 different m values (0, 1). e.g. m = 0 J z = m = 0 2p z orbital (pointing along z- axis) m = 1 J z = 2p x, 2p y orbitals

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2. Nodal Properties a) Angular Nodes s orbitals – have no angular nodes (spherically symmetric) p orbitals – have one angular node (a nodal plane, where = 0) No. of angular nodes = (s = 0, p = 1, d = 2 …) b) Radial Nodes s orbitals – have a maximum in at nucleus (r = 0) For other orbitals (p, d, f …), = 0 at nucleus due to angular nodes. All orbitals decay exponentially at large r values. Radial nodes – nodal surfaces where = 0, with changing sign either side of the node. No. of radial nodes = n 11s 0 2s 1 2p 0 3s 2 3p 1 3d 0 Total number of nodes (angular + radial) = n 1

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Angular and Radial Nodes in Angular Nodes Radial Nodes - + = 0 2p z orbital + - 3s orbital = 0 +

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3. Radial Distribution Functions Electron density at point (r) in space: Probability of finding electron at point r: Radial Distribution Function – probability density of finding electron at distance r from nucleus: P(r)dr – probability of finding electron in a shell of thickness dr at distance r from the nucleus. For non-spherical orbitals (p, d, f …), use:

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Example: H(1s) Peak in RDF at r = a 0 (the Bohr radius for n = 1). The delocalized electron is represented by a wave which has maximum probability at r = a 0. r 4r224r22 a0a0

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Hydrogenic Ions 1e ions with nuclear charge Z. Max. in 1s RDF at As Z , orbitals contract (r max ). Orbital Energies As Z , orbitals more tightly bound (E n – more negative).

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5.7Many-Electron Atoms For atoms with 2 or more electrons, the Schrödinger Eqn. must include KE terms for all electrons and PE terms for all e-e and e-n interactions. Example: He (2 electrons) 2 terms in 3 terms in S.E. where Z = +2e -e(1) r1r1 r2r2 r 12 -e(2)

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For N electrons: N = (r 1,r 2,r 3 … r N ) For > 1 e, S.E. cannot be solved analytically – though good numerical solutions may be obtained from computer calculations. Approximations must be made. The Orbital Approximation The total wavefunction ( N ) for the N-electron atom is approximated by the product of N 1-e orbitals similar to those of hydrogenic ions: N = (r 1,r 2,r 3 … r N ) = (r 1 ) (r 2 ) (r 3 )… (r N )

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Orbital Energies for Many-Electron Atoms For H and hydrogenic ions, E depends only on n. –e.g. 3s, 3p and 3d orbitals are degenerate. For many-electron atoms, E depends on n and. In shell n, E n as i.e. E(ns) < E(np) < E(nd) … Why ?

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Effective Nuclear Charge An electron at distance r from nucleus experiences an effective nuclear charge: Z eff < Z (Z = actual charge on nucleus = atomic number) Electrons inside a sphere of radius r repel the electron and shield (screen) it from the nucleus: Z eff = Z ( = shielding constant).

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Properties of For large R, behaves like 2 independent H(1s) AOs. For small R, there is significant overlap between the AOs. Destructive interference between atomic wavefunctions (AOs) A and B depletion of electron density between the nuclei (decrease of and 2 ). The depletion of electron density between the nuclei leads to decreased e-n attraction: E( ) > E( A, B ) is an antibonding MO. is cylindrically symmetrical about the internuclear (A-B) axis labelled as a MO (or sometimes *, as it is antibonding). A B R AA -B-B +

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Electrons in orbitals with low n (core orbitals) have greater electron density (higher probability of finding the electron) close to the nucleus experience higher Zeff lower energy. Electrons in orbitals with same n, but lower, have peaks in RDF which are closer to the nucleus better at shielding other electrons and better at penetrating shielding: Z eff ns > np > nd … Ens < np < nd … Orbitals with same n and are still degenerate e.g.E(2p x ) = E(2p y ) = E(2p z )

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Electronic Configurations To determine the electronic configuration of an atom (how the electrons are distributed among the atomic orbitals), we need to know: –The available orbitals and their relative energies. –The number of electrons. –The rules for filling the orbitals.

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Rules for Filling Orbitals 1. The Pauli Exclusion Principle No two electrons in a particular atom or ion can have the same values of all 4 quantum numbers (n,,m,m s ). No more than 2e can occupy a given orbital If 2e are in the same orbital (same n,,m ), they must have opposite (= paired) spins. H1s 1 He1s 2 Li1s 2 2s 1 X

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2. The Aufbau (Building-up) Principle Add available electrons into orbitals – starting from the lowest in energy, put maximum of 2e in each orbital. General order of orbital occupation: 1s < 2s < 2p < 3s < 3p < 4s ~ 3d < 4p < 5s < 4d … Usually completely fill each sub-shell ( ) before starting to fill another. –Exceptions – e.g. in the 1 st row transition metals: Cr[Ar]4s 1 3d 5 Cu[Ar]4s 1 3d 10

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The Aufbau Principle n

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The 3d and 4s orbitals are close in energy for the first transition metal series. For heavier atoms (higher atomic number Z), E(3d) < E(4s). As Z , the sub-shells of the inner (lower n) orbitals become approx. degenerate Variation of Orbital Energy with Z

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3. Spatial Separation Electrons occupy different orbitals (m ) of a given sub- shell ( ) before starting to pair electrons. e.g.N[He]2s 2 2p 3 = [He]2s 2 2p x 1 2p y 1 2p z 1 Electrons in different orbitals on average are further apart lower e-e repulsion lower E.

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4. Hund’s Rule (of Maximum Multiplicity) Provided rules 1 and 2 are satisfied, an atom in its ground state adopts the configuration with the maximum number of unpaired (= parallel) spins. e.g. C[He]2s 2 2p 2 = [He]2s 2 2p x 1 2p y 1 (equivalent to … 2p x 1 2p z 1 and 2p y 1 2p z 1 ). Parallel Spins Antiparallel Spins Electrons with parallel spins ( ) tend to avoid each other better than those with antiparallel spins ( ) reduced e-e repulsion lower E. This is known as spin correlation.

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6. Bonding in Molecules Exact solution of the Schrödinger Equation is not possible for any molecule – even the simplest molecule H 2 +. Full Hamiltonian operator for H 2 + : e HAHA HBHB R AB rArA rBrB nuclear K.E.e-n attractionn-n repulsionelectron K.E. KE PE

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6.1The Born-Oppenheimer Approximation Nuclei are much heavier (thousands of times) than electrons. they move much more slowly. In the Born-Oppenheimer approximation, nuclei are treated as being stationary. Consider motion of electrons relative to fixed nuclei. Total wavefunction ( ) is split into the product of electronic and nuclear wavefunctions: = e. n

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Example 1: H 2 + Fix R AB ( R). Schrödinger Equation for electronic motion: Note: although the electronic SE is solved for fixed R AB (= R), the solutions ( e,E e ) depend on the value of R. e HAHA HBHB R rArA rBrB electron K.E.e-n attractionn-n repulsion (constant)

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Example 2: H 2 Electronic Hamiltonian: SE for molecules containing other elements (atomic no. Z): –e-n attraction terms e 2 Ze 2 –n-n repulsion terms +e 2 +Z 2 e 2. electron K.E.e-n attractionn-n repulsione-e repulsion e1e1 HAHA HBHB R r 1A r 1B e2e2 r 2B r 2A r 12

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6.2Potential Energy Curves For a diatomic molecule, solving the electronic SE for different fixed positions of the nuclei (i.e. fixed inter-nuclear distances, R) gives the molecular potential energy curve V(R). R e = equilibrium bond length D e = PE well depth = dissociation energy. As R , V(R) 0 (dissociation limit). This can be extended to larger molecules: –e.g. for a general triatomic molecule we get a potential energy surface V(R 1,R 2, ). R1R1 R2R2 V(R) R ReRe -D e 0

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How can we determine and E for an electron in a molecule? SE can be solved exactly for (within the B-O approximation) but it is complicated and SE cannot be solved exactly for > 1e. We need to make more approximations Molecular Orbital theory. 6.3Molecular Orbital Theory Electrons in molecules have spatial distributions which are described by 1e wavefunctions called molecular orbitals (MOs) – analogous to atomic orbitals (AOs). Let represent a MO and an AO. MOs are spatially delocalised over the molecule. Probability of finding electron at point r in space in MO : P(r) = | (r)| 2 d

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Linear Combination of Atomic Orbitals (LCAO) Approximation Construct MOs ( ) as linear combination of AOs ( ): c i = coefficients (numbers) = contribution of i th AO to the MO. N AOs N MOs. Justification –When electron is close to one nucleus (A) it experiences an electrostatic (Coulomb) attraction that is greater than that to B. – MO wavefunction ( ) close to A, resembles an atomic orbital centred on A ( A ).

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Example: H 2 + and H 2 MOs formed as linear combinations of H(1s) AOs. 2 AOs ( A, B ) 2 MOs ( +, ) + = N + ( A + B )in-phase (bonding) = N ( A B )out-of-phase (antibonding) A AA B BB R

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Properties of + For large R, + behaves like 2 independent H(1s) AOs. For small R, there is significant overlap between the AOs. Constructive interference between atomic wavefunctions (AOs) A and B build-up of electron density between the nuclei (increase of and 2 ). R AB R AA BB ++

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The accumulation of electron density between the nuclei leads to increased e-n attraction (electrons interact strongly with both nuclei): E( + ) < E( A, B ) + is a bonding MO. Covalent bonding – due to sharing of electrons. + is cylindrically symmetrical about the internuclear (A-B) axis labelled as a MO: + side view + + cross section

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Properties of For large R, behaves like 2 independent H(1s) AOs. For small R, there is significant overlap between the AOs. Destructive interference between atomic wavefunctions (AOs) A and B deplete of electron density between the nuclei (decrease of and 2 ). -B-B AB R AA node

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The depletion of electron density between the nuclei leads to decreased e-n attraction. E( ) > E( A, B ) is an antibonding MO. is cylindrically symmetrical about the internuclear (A-B) axis labelled as a * MO (* denotes antibonding character): + nodal plane

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Normalization of the MO Wavefunction What are the normalization constants ( N + and N )? + = N + ( A + B ) = N ( A B ) Normalization condition: where = + or e.g. for + atomic orbitals are normalized: define overlap integral between orbitals A and B : normalization constant:

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Similarly, for we get: Since 0 N +. Similar arguments can be used to show that the antibonding orbital ( ) is raised in energy by more than the bonding orbital ( + ) is lowered in energy when a bond is formed. Note: if we ignore overlap, thenN + = N = 1/ 2

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6.4Molecular Orbital Diagrams Example 1. H 2 + Ground state configuration: ( + ) 1 1e in bonding orbital bound state. Excited state configuration: ( ) 1 1e in antibonding orbital unbound state. Energies defined relative to dissociation H 2 + H + H + A =H(1s A ) B =H(1s B ) + = 1 = 2 * Energy V(R) ReRe -D e 0 ++ R

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Example 2. H 2 Ground state configuration: ( + ) 2 2e in bonding orbital bound state. H 2 has a shorter stronger bond than H 2 + (more bonding electrons). Note: D e (H 2 ) < 2D e (H 2 + ) due to e-e repulsion. A =H(1s A ) B =H(1s B ) + = 1 = 2 * Energy R e /pmD e /kJ mol -1 H2+H H2H

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Example 3. He 2 Ground state configuration: ( + ) 2 ( ) 2 No net covalent bonding (bonding and a-b contributions cancel out). Only weak dispersion forces hold He atoms together (see Intermolecular Forces lectures). He 2 + has the configuration ( + ) 2 ( ) 1 and does have net covalent bonding. A =He(1s A ) B =He(1s B ) + = 1 = 2 * Energy

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6.5MOs for 2 nd Row Diatomic Molecules Valence AOs = 2s, 2p x, 2p y, 2p z Core AOs = 1s (not involved in bonding) Linear combinations of 2s orbitals: = N ( A (2s) B (2s)) 1 (2s) and 2 *(2s) (as for H 2 ) Some combinations are not allowed – zero net overlap = “orthogonal orbitals”: + + A (2s) + B (2p x ) + + A (2p x ) + B (2p z )

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The 2p orbitals interact to give -type (0 angular nodes with respect to the molecular axis) and -type (1 angular node) MOs, which can be bonding ( , ) or antibonding ( *, *). 33 4*4* 11 2*2* 2p 4*4* 11 2*2* 33 2p z -2p z overlap > overlap 3 -4 * splitting > 1 -2 *.

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6.6Molecular Electronic Configurations Follow same rules as for atomic electronic configurations (Aufbau principle, Hund’s rule etc.). Note: the ordering of MOs can vary – e.g. the 3 (2p) and 1 (2p) MOs are sometimes reversed: 3 < 1 (O 2, F 2 )3 > 1 (B 2, C 2, N 2 ) Due to 2s-2p mixing (hybridization) which raises 3 and lowers 2 *. As Z , the 2s-2p separation increases, so s-p mixing is weaker.

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6.7 Bond Order The strength of a covalent bond is the net outcome of occupying bonding and antibonding orbitals. Bond Order b = ½(N B N*) N B = number of electrons in bonding MOs N* = number of electrons in antibonding MOs Examples NBNB N*N*b H2+H H2H2 201 He N2N2 823 O2O2 842 F2F2 861 F2+F

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6.8 Paramagnetic Molecules Even with even numbers of electrons, certain molecules are paramagnetic (i.e. they have unpaired electron spins). e.g. O2 –Ground state electronic configuration: (1 ) 2 (2 *) 2 (3 ) 2 (1 ) 4 (2 *) 2 –There are 2 electrons in the antibonding pair of 2 * orbitals. –From Hund’s rule – the lowest energy configuration has the most unpaired spins = 2. –The magnetic effects of these 2 electrons do not cancel out. 33 4*4* 11 2*2* 2p

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6.9 Heteronuclear Diatomic Molecules Generally AOs of different atoms have different energies – depending on relative electronegativities of the atoms. The MO closest in energy to an AO has more character (greater LCAO coefficient) of that AO. bonding and antibonding orbitals usually have opposite characters. e.g. HF = 0.19 H (1s)+0.98 F (2p z ) * = 0.98 H (1s) 0.19 F (2p z ) (1s+2p) *(1s-2p) HF non-bonding F(2p x,2p y ) AOs.

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