CHM2S1AIntroduction to Quantum Mechanics Dr R' L' Johnston

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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.
•  

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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 2nd 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

• 5.1 The Hydrogen Atom and Hydrogenic Ions
• The series of atoms/ions H, He, Li2, Be3
• 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).

r
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• We must solve the 2nd 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,?,?) Rn,?(r).Y?,m?(?,?)

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• 5.2 Quantum Numbers
• ?n,?,m?(r,?,?) Rn,?(r).Y?,m?(?,?)
• ? depends on 3 quantum numbers (n,?,m?).
• 1. Principal Quantum Number (n)
• Positive integer n 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. ? 0 m? 0
• ? 1 m? 0, ?1

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• 5.3 Orbital 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

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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)
• Jz m??
• e.g. for ? 2 (m? 0, ?1, ?2)
• angular momentum vector J
• has magnitude J ?6? and
• 5 allowed orientations
• Jz 0, ??, ?2?.

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• 5.4 Atomic 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.

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• Possible combinations of shells and sub-shells.
• Total number of orbitals in shell n is n2.

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Electronic Shells, Sub-shells and Orbitals
3d orbitals
l2
ml ?1
ml ?2
ml 2
ml 1
ml 0
n3
3p orbitals
l1
ml ?1
ml 1
ml 0
3s orbital
l0
ml 0
shell
sub-shells
orbitals
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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 (its 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

• p orbitals have l 1 and ml 1 , 0, ?1

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• 5.5 Electron 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. ms ( ?½).

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• Spin angular momentum (S) has magnitude
• The projection of S on the z-axis
• Sz ms? ?½ ?
• There are two possible electron spin states
• ms ½ spin up ?
• ms -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?,ms) as s is
  fixed.

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• 5.6 Properties 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 orbital ns 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 ? Jz m? ? 0 ? 2pz orbital
  (pointing along z-axis)
• m? ?1 ? Jz ? ? ? 2px, 2py 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???1 1s ? 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
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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 a0 (the Bohr radius for n
  1).
• The delocalized electron is represented by a wave
  which has maximum probability at r a0.

4?r2?2
r
a0
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• Hydrogenic Ions
• 1e ions with nuclear charge Z.
• Max. in 1s RDF at
• As Z?, orbitals contract (rmax?).
• Orbital Energies
• As Z?, orbitals more tightly bound (En? more
  negative).

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• 5.7 Many-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

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• For N electrons ?N ?(r1,r2,r3 rN)
• For gt 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 ?(r1,r2,r3 rN) ?(r1)?(r2)?(r3)?(rN)

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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, En?? as ?? i.e. E(ns) lt E(np) lt
  E(nd)
• Why ?

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• Effective Nuclear Charge
• An electron at distance r from nucleus
  experiences an effective nuclear charge
• Zeff lt 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
• Zeff 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(??) gt 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).

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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
• Zeff ns gt np gt nd
• E ns lt np lt nd
• Orbitals with same n and ? are still
• degenerate
• e.g. E(2px) E(2py) E(2pz)

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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?,ms).
• 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.
• H 1s1
• He 1s2
• Li 1s22s1

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 lt 2s lt 2p lt 3s lt 3p lt 4s 3d lt 4p lt 5s lt 4d
• Usually completely fill each sub-shell (?) before
  starting to fill another.
• Exceptions e.g. in the 1st row transition
  metals
• Cr Ar4s13d5
• Cu Ar4s13d10

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The Aufbau Principle
n
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Variation of Orbital Energy with Z
The 3d and 4s orbitals are close in energy for
the first transition metal series. For heavier
atoms (higher atomic number Z), E(3d) lt
E(4s). As Z?, the ? sub-shells of the inner
(lower n) orbitals become approx. degenerate
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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 He2s22p3 He2s22px12py12pz1
• Electrons in different orbitals on average are
  further apart ? lower e-e repulsion ? lower E.

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• 4. Hunds 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 He2s22p2 He2s22px12py1
• (equivalent to 2px12pz1 and 2py12pz1).
• 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 H2.
• Full Hamiltonian operator for H2

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• 6.1 The 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 H2
• Fix RAB ( ? R).
• Schrödinger Equation for electronic
• motion
• Note although the electronic SE is solved for
  fixed RAB ( R), the solutions (?e,Ee) depend on
  the value of R.

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• Example 2 H2
• Electronic Hamiltonian
• SE for molecules containing other elements
• (atomic no. Z)
• e-n attraction terms ? ?e2 ? ?Ze2
• n-n repulsion terms e2 ? Z2e2.

electron K.E.
e-n attraction
n-n repulsion
e-e repulsion
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• 6.2 Potential 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).
• Re equilibrium bond length
• De 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(R1,R2,?).

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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 gt 1e.
• We need to make more approximations ? Molecular
  Orbital theory.
• 6.3 Molecular 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)2d?

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• Linear Combination of Atomic Orbitals (LCAO)
  Approximation
• Construct MOs (?) as linear combination of AOs
  (?)
• ci coefficients (numbers) contribution of ith
  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 H2 and H2
• 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)

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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).

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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(?) lt 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

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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).

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

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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 lt SAB lt 1, this means that N? gt 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, then N N? 1/?2

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• 6.4 Molecular Orbital Diagrams
• Example 1. H2
• 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
• H2 ? H H

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• Example 2. H2
• Ground state configuration (?)2
• 2e in bonding orbital ? bound state.
• H2 has a shorter stronger bond than H2 (more
  bonding electrons).
• Note De(H2) lt 2De(H2) due to e-e repulsion.

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• Example 3. He2
• 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).
• He2 has the configuration (?)2(??)1 and does
  have net covalent bonding.

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• 6.5 MOs for 2nd Row Diatomic Molecules
• Valence AOs 2s, 2px, 2py, 2pz
• Core AOs 1s (not involved in bonding)
• Linear combinations of 2s orbitals
• ?? N? (?A(2s) ? ?B (2s)) ? 1?(2s) and
  2? (2s) (as for H2)
• Some combinations are not allowed zero net
  overlap orthogonal orbitals

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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 (?,?).

2pz-2pz ? overlap gt ? overlap ? 3?-4? splitting
gt 1?-2?.
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• 6.6 Molecular Electronic Configurations
• Follow same rules as for atomic electronic
  configurations (Aufbau principle, Hunds rule
  etc.).
• Note the ordering of MOs can vary e.g. the
  3?(2p) and 1?(2p) MOs are sometimes reversed

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.
3? lt 1? (O2, F2)
3? gt 1? (B2, C2, N2)
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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(NB ? N)
• NB number of electrons in bonding MOs
• N number of electrons in antibonding MOs
• Examples

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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 Hunds rule the lowest energy
  configuration has the most unpaired spins 2.
• The magnetic effects of these 2 electrons do not
  cancel out.

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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(2pz)
• ? 0.98?H(1s)?0.19?F(2pz)