PHYSICAL CHEMISTRY I. Chemical Bonds
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1 PHYSICAL CHEMISTRY I Chemical Bonds
2 Review The QM description of bonds is quite good Capable of correctly calculating bond energies and reaction enthalpies However it is quite complicated and sometime unintuitive Chemical Bonds To help understand and simplify complex QM there are two major approaches to the calculation of molecular structure Valence bond theory (VB theory) molecular orbital theory (MO theory) Almost all modern computational work makes use of MO theory
3 Review The Born Oppenheimer is the most common approximation used when dealing with molecules All major QM software uses Born Oppenheimer, making its use very prevalent Born Oppenheimer Without Born Oppenheimer most calulated systems are limited to a few atoms in size Born Oppenheimer states: Electrons move much faster then the heavier nuclei We can therefore ignore the nuclei movement and treat them parametrically solving the nuclei and electrons SEs individually Here s an example of non-born Oppenheimer work: A. W. King, F. Longford and H. Cox, The stability of S-states of unit-charge Coulomb three-body systems: From H to H2+, The Journal of Chemical Physics, 2013, 139,
4 Review Valence-bond theory, although not as prevalent as MO theory has left an imprint on the language of chemistry Specifically it has given us the concepts of hybridization and promotion Valance Bond In VB theory Bonds are pairs of electrons located BETWEEN the atoms An atom has the capability to make a bond if it has unpaired electrons An atoms can promote electrons to gain more unpaired electrons An atom can hybridize (mix) atomic orbitals to make hybrid orbitals that point along a bond
5 MOLECULAR ORBITAL THEORY
6 Basic MO Theory VB In MO theory, unlike in VB theory, electrons can be anywhere H H They are not limited to just between the bonds The one electron atom was taken as the fundamental basis for describing electron behaviour and then extended to many electrons and nuclei H MO H
7 Starting from H 2 + H 2+ is the simplest molecule, with only a single electron p $ & R p " & Under the Born Oppenheimer the distance R is fixed and the electron is solved with the SE r $# r "# This has the following Hamiltonian e # ( H* = ħ. 2m r $# 1 r "5 + 1 R
8 Starting from H 2 + The green terms are attractive interactions between electrons and nuclei (therefore the negative sign) p $ & R p " & The red term is the repulsive term between the like charged nuclei and although R does not change under the BO approximation it must still be accounted for r $# e # ( r "# H* = ħ. 2m r $# 1 r "5 + 1 R
9 Starting from H 2 + The wavefunctions the solution to this SE gives are called molecular orbitals (MO) p $ & R p " & There are like atomic orbitals (AO) but spread over the entire molcules r $# r "# e # (
10 Starting from H 2 + Under the BO approximation this is actually a solvable solution however: p $ & R p " & The wavefunctions are very complicated r $# r "# They cannot be extended to other molecules easily We therefore adopt a simpler approach to extend to molecules e # (
11 Linear Combination H H This simple approach is called Linear Combination of Atomic Orbitals (LCAO) It works on a theory that if an electron can be found in an atomic orbital of atom A then it could also be found on an atomic orbital of atom B These two atomic orbitals are added together in a superposition (linearly combined) H H H H
12 Linear Combination H H In math terms this is ψ ± = N A ± B N is a normalization A denotes atomic wavefunction for atom A, and B for the wavefunction of atom B H H H H
13 Molecular Orbital Theory In molecular orbital (MO) theory molecular orbitals are treated as a linear combination of atomic orbitals This way of forming molecular orbitals from atomic is called LCAO-MO Going beyond a simple A and B model to a more general case the molecular wave function ψ is made up of combinations of atomic orbitals ψ = = c? φ?? This is a sum of atomic orbitals φ? for ith atom multipled by some constant c? which gives the relative contribution of this AO to the MO
14 MO Labels σ AO have labels of s, p, d etc For MO we use σ, π, δ, (sigma, pi, delta) A MO is of type σ if it looks like a s orbital when viewed along a bond A MO is of type π if it looks like a p orbital when viewed along a bond A MO is of type δ if it looks like a d orbital when viewed along a bond H+ H+ H π H
15 MO Densities According to the Born interpretation, the probability density of the electron at each point in H 2+ is proportional to the square modulus of its wavefunction at that point This means for a wavefunction of ψ ± = N A ± B We have a square of ψ &. = N. A. + B. + 2AB or ψ (. = N. A. + B. 2AB Focusing on ψ &. for now, we can see that: A. and B. are simply the squares of the atomic orbitals What is new is this +2AB which is the interaction between A and B that adds to the probability Meaning increased electron densities between the atoms This extra term is the OVERLAP integral given the symbol SH
16 MO Densities
17 MO and Bonds Bonds form when electrons accumulate in regions where atomic orbitals overlap constructively Explanation of this observation is based on the notion that Accumulation of electron density between the nuclei puts the electron in a position where it interacts strongly with both nuclei This is not the only explanation and there are doubts to its validity
18 MO and Bonds As electron density accumulates in the inter atomic region the orbital shrinks around the nucleus This shrinkage improves electron nucleus attraction and overall lowers the PE more than was lost due to electron accumulation in the interatomic region We ascribe the strength of chemical bonds to the accumulation of electron density in the internuclear region
19 MO and Bonds This type of positive orbital we have described is called a bonding orbital These orbitals helps to bind two atoms together The lowest energy sigma orbital is given the label 1σ and higher one are 2σ, 3σ etc
20 MO and Bonds Lengths Under the BO the atomic distances are fixed and then we calculate the energy To find this best fixed distance we must vary it many times and recalculate the energy many times
21 MO and Bonds Lengths For something simple like H2+ we can vary the single bong length R and plot the energies we calculate The graph to the right shows experimental and theoretical bond length energies for the ground and excited state
22 MO and Bonds Lengths The optimum bond length is the one that gives us the lowest energy (on the graph this is at about 2) For more complex data with many bonds we use optimisation algorithms
23 MO and Anti Bonds The linear combination ψ ( corresponds to an orbital with higher energy than ψ & For H2+ this is a sigma orbital too and is given the 2σ label The wave function when squared is ψ (. = N. A. + B. 2AB The new term is the 2AB term This term reduces electron density between the nuclei In physical terms, there is destructive interference where the two atomic orbitals overlap This is an example of an antibonding orbital
24 MO and Anti Bonds
25 MO and Anti Bonds If these orbitals are occupied, they contributes to a reduction in the cohesion between two atoms This raises the energy of the molecule relative to the separated atoms Meaning separated atoms is more stable then antibonding
26 MO and Anti Bonds The effect of raised energy is partly due to the decrease of density in the inter nuclei region and the increase in the region away from the bond This in effect acts in an opposite manner to the bonding, drawing the nuclei away from each other
27 MO and Anti Bonds In this graph the lower line is the bonding orbital and the upper line the antibonding The magnitude of the energy of the antibonding is greater than the magnitude of energy of the bonding E ( E K > E & E K
28 MO and Anti Bonds This is something quite important that The antibonding orbital is more antibonding than the bonding orbital is bonding This important conclusion stems in part from the presence of the nucleus nucleus repulsion
29 MO and Anti Bonds Labels The anti-bonding orbital is often labelled with a star or asterisks The hydrogen antibonding becomes 2σ This is read as two sigma star
30 Labels with Symmetry For simple systems that have symmetry we use include the symmetry of the orbital in the label For example homonuclear diatomics have symmetry of inversion If you invert the wavefunction through its centre and it stays the same it is given the gerade symmetry label (g) 1σ N H+ H+
31 Labels with Symmetry For simple systems that have symmetry we use include the symmetry of the orbital in the label For example homonuclear diatomics have symmetry of inversion If you invert the wavefunction through its centre and it is not the same it is given the ungerade symmetry label (u) 1σ O H+ H+
32 Labels with Symmetry The exact labelling differs according to symmetry Hetronuclear elements for example don t have inversion symmetry and so are labelled differently Each symmetry label has its own number so for H2+ starting at 1 for the lowest energy 1σ 1σ N 2σ 1σ O H+ H+
33 Molecular Orbital Theory MO theory does not treat electron as existing solely between a bond but instead the electron can be anywhere in the molecule, in a probabilistic manner H H H C C C H H H H H H H H H
34 Molecular Orbital Theory Orbital Bonding In MO theory there is the concept of bonding, non binding and antibonding orbitals In phase so these interact constructively Wrong symmetry so no interaction Out of phase so these interact destructively
35 HOMO NUCLEAR DIATOMICS Math
36 Homonuclear Diatomics For He and Li we built atomic orbitals by combining hydrogenic wave functions For many electron molecules we do the same but use H2+ as a basis for discussion To construct a MO we observe that 1. The electrons supplied by the atoms are accommodated in the orbitals so as to achieve the lowest overall energy subject to the constraint of the Pauli exclusion principle 2. If several degenerate molecular orbitals are available, electrons are added singly to each individual orbital before doubly occupying any one orbital 3. According to Hund s maximum multiplicity rule, if two electrons do occupy different degenerate orbitals, then a lower energy is obtained if they do so with parallel spins
37 A Simple Diatomic H2+ is a simple diatomic that we will use to build up our example energy levels p $ & R p " & It has two nuclei and one electron r $# r "# e # (
38 A Simple Diatomic For the case of a simple diatomic of atom A and B which has only 1s orbitals for their atomic orbitals ψ = C $ φ $ + C " φ " φ $, φ " are the atomic orbitals of atoms A and B C $, C " are the molecular orbital coefficients ψ is the molecular orbital
39 A Simple Diatomic Since A and B are the same atom C $. = C ". C $ = ±C " = C ψ ± = C φ A ± φ B There are two combinations (in phase + and out of phase -)
40 A Simple Diatomic This leads to two energies E & = φ & H* φ & E ( = φ ( H* φ ( Which can be evaluated as: ψ & H*ψ & dτ = C φ $ + φ " H*C φ $ + φ " dτ ψ & H*ψ & dτ = C. φ $ H*φ $ dτ + φ $ H*φ " dτ + φ " H*φ $ dτ + φ " H*φ " dτ
41 A Simple Diatomic If we assign these atomic Hamiltonians to H?,[ and realise that H $$ = H "" and that H $" = H "$ this simplifies to ψ & H*ψ & dτ = C. φ $ H*φ $ dτ + φ $ H*φ " dτ + φ " H*φ $ dτ + φ " H*φ " dτ ψ & H*ψ & dτ = 2C 2 H AA + H AB
42 A Simple Diatomic We do a similar treatment for the normalisation constant ψ & ψ & ψ & ψ & dτ = C. φ $ φ $ dτ + φ $ φ " dτ + φ " φ $ dτ + φ " φ " dτ In this case since φ? φ? dτ = 1 and we assign φ $ φ " dτ = φ " φ $ dτ = S. Where S means overlap integral ψ & ψ & dτ = 2C. 1 + S
43 A Simple Diatomic Putting these together: E & = φ & H* φ & = 2C. H $$ + H "$ ψ & ψ & 2C. 1 + S E ( = φ ( H* φ ( = 2C. H $$ H "$ ψ ( ψ ( 2C. 1 S = H $$ + H "$ 1 + S = H $$ H "$ 1 S
44 A Simple Diatomic Since H?,[ is always negative then E & < E ( In a normalised wave function dψ ψdτ = 2C. 1 + S = 1 1 C = S
45 A Simple Diatomic If the two atoms are infinitely far away then S = 0 C = 1 2
46 HOMO NUCLEAR DIATOMICS MO Diagram
47 Energy Levels Combing 2 atomic orbitals gives us two MO with different energies (if they interact that is) We can plot these in a MO diagram The new MO that forms the bond is shown in the middle Atomic Orbitals that make the bond are shown on the left and right
48 MO Diagram We can plot these two energy we calculate before in an MO diagram σ E ( ψ ( = 1 2 ψ $ ψ " + - E σ E & ψ & = 1 2 ψ $ + ψ " + +
49 MO Diagram Depending on the electrons available we fill up the diagram with electrons similar to Hund s rules E σ E ( ψ ( = 1 2 ψ $ ψ " 2 electrons as in H 2 σ E & ψ & = 1 2 ψ $ + ψ "
50 MO Diagram Bond Order We calculate the bond order with this formula b = 1 2 Bonding Electrons Antibonding Electrons σ E ( ψ ( = 1 2 ψ $ ψ " E 2 electrons as in H 2 σ E & ψ & = 1 2 ψ $ + ψ " b = = 1
51 MO Diagram Bond Order Non integer bond orders are possible E σ σ E ( E & ψ ( = 1 2 ψ $ ψ " ψ & = 1 2 ψ $ + ψ " 1 electron as in H 2 + b = #. 1 0 = #.
52 MO Diagram Bond Order Filling an antibonding reduces the bond order From this we learn that non valence orbitals do not contribute to bonding σ ψ ( = 1 2 ψ $ ψ " E ( E 4 electron as in He 2 b = #. 2 2 = 0 σ E & ψ & = 1 2 ψ $ + ψ " He 2 does not exist
53 MO Diagram Higher Orbitals Including the 2s and 2p orbitals we get s diagram like this σ.o π.o π.o E σ.o σ.n σ.m σ #m σ #n
54 MO Diagram Higher Orbitals When we represent the constituents of a bond we place them to the left or right These can be AO or other MOs E σ.o π.o π.o σ.o σ.n σ.m σ #m σ #n 2p 2s 1s X X. X
55 MO O2 Here is an example of O2 σ.o π.o π.o 2p E σ.o σ.n σ.m 2s b = = 2 σ #m σ #n 1s X X. X
56 MO Switched Order For diatomics less that O it is found that sp mixing or hybridisation alters the energy levels switching the 2s and 2p bonding orbitals E σ.o π.o σ.o π.o σ.n σ.m 2p 2s σ #m σ #n 1s X X. X
57 MO N2 Here s an N2 example E σ.o π.o σ.o π.o σ.n σ.m 2p 2s b = = 3 σ #m σ #n 1s X X. X
58 MO Magnetism The number of of unpaired electrons gives the multiplicity (2S + 1) Unpaired electrons also predicts paramagnetism σ.o E π.o π.o 2p σ.o
59 HETRO NUCLEAR DIATOMICS
60 MO - Hetronuclear For a hetronuclear atomic the energy levels aren't so symmetric HF H: 1s1 F: 1s2 2s2 2p5 E 1s 2p 2s
61 MO - Hetronuclear One of the AO will be higher, indicating it contributes less to the bonding orbital. This AO will be smaller in the bonding MO, and larger in the antibonding E 1s 2p 2s
62 MO - Hetronuclear Non bonding orbitals do not contribute typically due to symmetry E 1s 2p 2s
63 MO HOMO LUMO MOs are sometimes labelled using the terms highest occupied molecular orbital (HOMO) or lowest unoccupied molecular orbital (LUMO) LUMO+1 E LUMO HOMO HOMO-1
64 Polyatomic Molecules For the more complex systems we name orbitals differently Instead of labelling them as σ, π, δ (for the diatomic) that correspond to the s,p d or the atomics we use the symmetry to label the orbitals This typically involves finding the point group of the whole molecule Each orbital will correspond to a particular symmetry operation of that point group However in real life often there is no symmetry
65 Molecular Orbital Theory Uses MO theory is good at calculating ionisation energies and excitations in an intuitive manner In excitation we move an electron to a different orbital The total energy is the sum of the filled orbitals energies (times 2 if they are doubly filled) In ionisation and electron affinities we either add or remove an electron
66 HUCKLE MO
67 Huckle MO Theory Huckle MO is a theory that is simple enough to do in class It is very similar to MO theory but with an added approximation that is valid when dealing with conjugated systems
68 Huckle MO Theory In Huckle theory the assumption is made that the σ and π parts can be separated in the case of conjugated organic molecules Overlap in the xy plane p orbitals with the s lead to sp. hybridisation which forms the conjugated backbone The p t orbitals are said to be left to form π bonding The result is a delocalised π system that extends over the conjugated system
69 Huckle MO Theory - Butadiene The MOs for the π orbitals can be written as a linear combination of 2p t orbitals, as in: u ψ = = c? φ? = c # φ # + c. φ. + c w φ w + c u φ u?v# The C? s are coefficients that need to be optimised. We use the variaitional principle to find the optimal values
70 Huckle MO Theory - Butadiene This leads to a set of secular equations c # H ## S ## E + c. H #. S #. E + c w H #w S #w E + c u H #u S #u E = 0 c # H.# S.# E + c. H.. S.. E + c w H #w S.w E + c u H.u S.u E = 0 c # H w# S w# E + c. H w. S w. E + c w H #w S ww E + c u H wu S wu E = 0 c # H u# S u# E + c. H u. S u. E + c w H #w S uw E + c u H uu S uu E = 0
71 Huckle MO Theory - Butadiene The solutions to a set of secular equations is only non-trivial is the determinate is zero c # H ## S ## E + c. H #. S #. E + c w H #w S #w E + c u H #u S #u E c # H.# S.# E + c. H.. S.. E + c w H #w S.w E + c u H.u S.u E = 0 c # H w# S w# E + c. H w. S w. E + c w H #w S ww E + c u H wu S wu E c # H u# S u# E + c. H u. S u. E + c w H #w S uw E + c u H uu S uu E This leads to a quadratic equation with 4 roots (4 answers)
72 Huckle MO Theory To simplify this greatly Huckle introduces these approximations All overlap integrals S?,[ are set to zero All coulomb integrals H?,[ with the same i and j e.g. H ##, H.. are given the value α All resonance integrals between neighbouring carbons atoms e.g. H #,. are given the value β All resonance integrals between non neighbouring carbon atoms are zero Although the last approximation is reasonable the other introduce significant error
73 Huckle MO Theory With these approximations the secular determinate becomes: α E β 0 0 β α E β β 0 α E β β α E = 0
74 Huckle MO Theory This can be further simplified with dividing through by β and setting x = (} ~ x x x 1 1 x = 0
75 Huckle MO Theory We expand the determinate and solve the quadratic equation with to find the four roots of the equation These are x. = 2.62 and x. = Which leads to the four roots x = ±1.62, x = ±0.62
76 Huckle MO Theory The four energy levels are therefore: E # = α β E. = α β E w = α 0.62β E u = α 1.62β Since β is negative the lowest energy is E #
77 Huckle MO Theory To find the wavefunction we return to our secular equations c # H ## S ## E + c. H #. S #. E + c w H #w S #w E + c u H #u S #u E = 0 c # H.# S.# E + c. H.. S.. E + c w H #w S.w E + c u H.u S.u E = 0 c # H w# S w# E + c. H w. S w. E + c w H #w S ww E + c u H wu S wu E = 0 c # H u# S u# E + c. H u. S u. E + c w H #w S uw E + c u H uu S uu E = 0
78 Huckle MO Theory Which look like this after applying out approximations c # x + c. = 0 c # + c. x + c w = 0 c. + c w x + c u = 0 c w + c u x = 0 First lets find the lowest energy ψ E # E # = α β x = 1.62
79 Huckle MO Theory We substitute x into the equations and then express all the coefficients in terms of c # x = 1.62 Therefore since c # = ƒ c # x + c. = 0 c # = c. x c # + c. x + c. = 0 c # = c. x + c w 1 c. x x = c w c. = c w c. = 1.62c # = c w
80 Huckle MO Theory Therfore since c w + c u x = c # + c u 1.62 = 0 c u = c # To obtain the numerical values we normalise c #. + c.. + c ẇ + c u. = 1 c # c # c #. + c #. = 1 c # = 0.37 Therefore the wave function is: ψ # = 0.37φ # φ φ w φ u
81 Huckle MO Theory The other wave functions are obtained the same way In Butadiene there are four π electrons Two electrons in E # and two in E. The total energy is therefore E = 2E # + 2E. = 4α β In Huckle MO Theory the answers should be left in terms of α and β
82 Summary MO Theory is an approximation to extend AO from hydrogen and molecular orbitals from H2+ to molecules in general AOs are combined linearly to give MO There are 2 solutions for these linear combinations, + and The + form increases density between the bond The form decreases density between the bond Symmetry is used to label MOs Energy level diagrams can be calculated from combinations of AO (or other MOs) Care should be taken with the order of MOs for diatomics below O2 due to a swapped MO order Heteronuclear atoms have asymmetric AO contributions, with some orbitals contributing more to the MO bonding than the antibonding Huckle theory is an approximation for conjugate systems that is simple to calculate
83 Homework 1. Use Huckle theory to find the energies of a two and 3 carbon aromatic chain 2. Use Huckle theory to find the energies of a 3 carbon aromatic ring
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