Presentation on theme: "CHM2S1-A Introduction to Quantum Mechanics Dr R. L. Johnston"— Presentation transcript:
1CHM2S1-A Introduction to Quantum Mechanics Dr R. L. Johnston THE UNIVERSITYOF BIRMINGHAMII: Quantum Mechanics of Atoms and Molecules5. Electronic Structures of Atoms5.1 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.
26. Bonding in Molecules6.1 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.
35. Electronic Structures of Atoms 5.1 The Hydrogen Atom and Hydrogenic IonsThe 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.
4Consider 3-D motion of the electron (-e) relative to the nucleus (+Ze): Hamiltonian OperatorKinetic Energy wherePotential Energy where(electrostatic attraction between electron and nucleus).+ZemN-emer
5We 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(,)radialwavefunctionangular(spherical harmonic)
6n,,m(r,,) = Rn,(r).Y,m(,) 5.2 Quantum Numbersn,,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 .
72. 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 = 0n = 2 = 0, 1n = 3 = 0, 1, 23. 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, 1There are (2+1) allowed values of m.
81 2 3 2 6 12 5.3 Orbital Angular Momentum |J| 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|:123|J|2612
9angular momentum vector J has magnitude |J| = 6 and 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 = me.g. for = 2 (m = 0, 1, 2)angular momentum vector Jhas magnitude |J| = 6 and5 allowed orientations:Jz = 0, , 2.m
105.4 Atomic OrbitalsThe 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.
11Electronic 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.n1234ShellKLMN123Sub-shellspdfNo. orbitals57
12Possible combinations of shells and sub-shells. Total number of orbitals in shell n is n2. n123…1s2s2p3s3p3d44s4p4d4f
13Electronic Shells, Sub-shells and Orbitals ml =2ml =1ml =0ml =1ml =2n=33p orbitalsl=1ml =1ml =0ml =13s orbitall=0ml =0shellsub-shellsorbitals
14Atomic Orbital Energy Diagram for H Orbital energies depend only on theprincipal quantum number n.For a given n value (shell), all sub-shells() and orbitals (m) have the same energy– i.e. they are degenerate.
15Quantum 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 nthe 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
16Shapes of Atomic Orbitals s orbitals have l = 0 and ml = 0p orbitals have l = 1 and ml = 1 , 0, 1pzpxpyd x2 y2dz2dxydxzdyzd orbitals have l = 2 and ml = 2, 1, 0 ,1, 2
175.5 Electron SpinStern 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 electronElectron spin is characterised by 2 quantum numbers:– spin angular momentum q. no. s ( = ½ for all electrons)– spin magnetic q. no. ms ( = ½).
18Spin 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.
19+ 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 orbitalAs n, orbitals expand – average radius r+
20+ - 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+-
212. Nodal Propertiesa) Angular Nodess 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 Nodess 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 = n1 1s 02s p 03s p d 0Total number of nodes (angular + radial) = n1
22Angular and Radial Nodes in Angular NodesRadial Nodes+-3s orbital = 0-+ = 02pz orbital
233. 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:
24Example: 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 = a0.4r22a0r
25Hydrogenic Ions 1e ions with nuclear charge Z. Max. in 1s RDF at As Z, orbitals contract (rmax).Orbital EnergiesAs Z, orbitals more tightly bound (En – more negative).
265.7 Many-Electron AtomsFor 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 in3 terms inS.E.whereZ = +2e-e(1)r1r2r12-e(2)
27N = (r1,r2,r3 … rN) = (r1)(r2)(r3)…(rN) For N electrons: N = (r1,r2,r3 … rN)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 ApproximationThe 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)
28Orbital 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) < E(np) < E(nd) …Why ?
29Effective Nuclear Charge An electron at distance r from nucleus experiences an effective nuclear charge:Zeff < 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).
30 is cylindrically symmetrical about the internuclear (A-B) axis 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 atomicwavefunctions (AOs) A and B depletionof electron density between the nuclei(decrease of and 2).The depletion of electron density betweenthe 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).ABRA-B+
31Electrons 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, butlower , have peaks in RDF which arecloser to the nucleus better at shieldingother electrons and better at penetratingshielding:Zeff ns > np > nd …E ns < np < nd …Orbitals with same n and are stilldegeneratee.g. E(2px) = E(2py) = E(2pz)
32Electronic 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.
33X 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 orbitalIf 2e are in the same orbital (same n,,m), they must have opposite (= paired) spins.H 1s1He 1s2Li 1s22s1X
342. 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 1st row transition metals:Cr [Ar]4s13d5Cu [Ar]4s13d10
36Variation of Orbital Energy with Z The 3d and 4s orbitals are close inenergy for the first transition metalseries.For heavier atoms (higher atomicnumber Z), E(3d) < E(4s).As Z, the sub-shells of the inner(lower n) orbitals become approx.degenerate
373. Spatial SeparationElectrons occupy different orbitals (m) of a given sub-shell () before starting to pair electrons.e.g. N [He]2s22p3 = [He]2s22px12py12pz1Electrons in different orbitals on average are further apart lower e-e repulsion lower E.
384. 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]2s22p2 = [He]2s22px12py1(equivalent to … 2px12pz1 and 2py12pz1).Parallel Spins Antiparallel SpinsElectrons 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.
396. Bonding in Molecules PE KE Exact solution of the Schrödinger Equation is not possible for any molecule – even the simplest molecule H2+.Full Hamiltonian operator for H2+ :eHAHBRABrArBPEKEnuclear K.E.e-n attractionn-n repulsionelectron K.E.
406.1 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
41Schrödinger Equation for electronic motion: Example 1: H2+Fix RAB ( R).Schrödinger Equation for electronicmotion:Note: although the electronic SE is solved for fixed RAB (= R), the solutions (e,Ee) depend on the value of R.eHAHBRrArBelectron K.E.e-n attractionn-n repulsion(constant)
436.2 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 lengthDe = 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,).V(R)RRe-DeR1R2
446.3 Molecular Orbital Theory 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.6.3 Molecular Orbital TheoryElectrons 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
45Linear 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.JustificationWhen 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).
46+ = N+(A + B) in-phase (bonding) Example: H2+ and H2MOs 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)AABBR
47+ A B Properties of + R For large R, + behaves like 2 independent H(1s) AOs.For small R, there is significant overlap between the AOs.Constructive interference between atomicwavefunctions (AOs) A and B build-upof electron density between the nuclei(increase of and 2).RABRAB+
48 Covalent bonding – due to sharing of electrons. 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
49 -B A 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 atomicwavefunctions (AOs) A and B depleteof electron density between the nuclei(decrease of and 2).-BABRAnode
50 is cylindrically symmetrical about the internuclear (A-B) axis 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
51Normalization 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:
52Similarly, for we get: Since 0 < SAB < 1, this means that N > 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
536.4 Molecular Orbital Diagrams Example 1. H2+Ground state configuration: (+)11e in bonding orbital bound state.Excited state configuration: ()11e in antibonding orbital unbound state.Energies defined relative to dissociationH2+ H + H+A=H(1sA)B=H(1sB)+ = 1 = 2*EnergyV(R)Re-De+R
54A=H(1sA) B=H(1sB) + = 1 = 2* Energy Example 2. H2 Ground state configuration: (+)22e in bonding orbital bound state.H2 has a shorter stronger bond than H2+ (more bonding electrons).Note: De(H2) < 2De(H2+) due to e-e repulsion.Re/pmDe/kJ mol-1H2+106255H274.1430
55A=He(1sA) B=He(1sB) + = 1 = 2* Energy Example 3. He2 Ground state configuration: (+)2()2No 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.
566.5 MOs for 2nd Row Diatomic Molecules Valence AOs = 2s, 2px, 2py, 2pzCore 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”:+A(2s) + B (2px)+A(2px) + B (2pz)
57The 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 (*,*).34*12*2p4*12*32pz-2pz overlap > overlap 3-4* splitting > 1-2*.
586.6 Molecular 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:Due to 2s-2p mixing(hybridization) whichraises 3 and lowers2*.As Z, the 2s-2p separationincreases, so s-p mixingis weaker.3 > 1 (B2, C2, N2)3 < 1 (O2, F2)
596.7 Bond OrderThe 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 MOsN* = number of electrons in antibonding MOsExamplesNBN*bH2+10.5H22He2N283O24F26F2+51.5
606.8 Paramagnetic Molecules Even with even numbers of electrons, certain molecules are paramagnetic (i.e. they have unpaired electron spins).e.g. O2Ground state electronic configuration: (1)2 (2*)2 (3)2 (1)4 (2*)2There 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.34*12*2p
616.9 Heteronuclear Diatomic Molecules Generally AOs of different atoms have different energies – depending on relative electronegativities of the atoms.The MO closest in energy to anAO has more character(greater LCAO coefficient) ofthat AO. bonding and antibonding orbitalsusually have opposite characters.e.g. HF = 0.19H(1s)+0.98F(2pz)* = 0.98H(1s)0.19F(2pz)(1s+2p)*(1s-2p)HFnon-bondingF(2px,2py)AOs.