Atomic And Molecular Physics

Front 1

What is the evidence for atoms?

Back 1

Dalton ratios
Periodic table
Brownian motion
Kinetic theory

Front 2

What is the evidence for the existence of the electron?

Back 2

Faraday: equivalence of matter and electricity
Stoney's unit of electricity
Cathode rays

Front 3

Describe Thompsons experiment to calculate e/m.

Back 3

Cathode ray electrons emitted and accelerated.
Deflection between electric plates E=V/d, F=eE
Deflection in the y-direction from eqns of motion and equipment dimensions y α eV/m, zero horizontal acceleration.
Crossed magnetic field F= eE = v0B
v0 = E/B hence possible to deduce e/m from v0/E/B and dimensions of the equipment.

Front 4

Describe Millikan's oil drop experiment.

Back 4

Oil droplets sprayed through a nozzle gain +ve charge.
Ionisation of air molecules by irradiation of the cavity with x-rays and electrons taken up by oil droplets to alter their charge.
. Total force on the oil droplet at terminal velocity F = effective weight inc buoyancy - electrostatic - viscosity = M'g - qe - βv
Zero electric field to equate βv and M'g.
Variable electric field and ionisation of droplets, with multiple measurements => quantisation of charge.

Front 5

What is total collisional cross-section?

Back 5

It is the effective area presented by a target to an incoming projectile normal to the direction of incidence, including electrostatic interaction.
It measures the interaction probability between a projectile and a target.
It is related but not equivalent to the size of the particles.
Loss of intensity I through a target: dI/I =-nσdx where σ = collisional cross-section

Front 6

What is the Beer-Lambert Law and how is it derived?

Back 6

From dI/I =-nσdx where σ = collisional cross-section, I = intensity.
Integrating over target length L:
I(L) = I0 e ^ (-nLσ)

Front 7

How is collisional cross-section related to particle charge?

Back 7

It is larger for charged particles since the couloumb force is long-ranged. Neutral particles interact by short-range Van Der Waals forces which lead to a smaller effective area.

Front 8

How does collisional cross-section vary with incident energy?

Back 8

Static interaction: incident charged particle sees partially screened nucleus, this may be attractive or repulsive.
Polarisation: the approaching particle distorts the electron cloud, this is always attractive.
. At high energy the importance of polarisation decreases as there isn't time for electron cloud distortion. Cross-sections for + and - particles are the same.
At low energy, the electron cloud distorts, so the mechanisms are additive- different cross-sections for + and - particles (larger for - as +ve interactions cancel out)

Front 9

What is positronium?

Back 9

A bound state of an electron and a postitron orbiting each other.

Front 10

What is the positronium cross-section at high energy?

Back 10

The cross-section is the sum of the electron and positron - they interact separately.

Front 11

What is an elastic collision?

Back 11

One where the scattered particles change direction.

Front 12

What is an inelastic collision?

Back 12

One where the internal energy of the target/incident particles is changed by e.g. excitation.

Front 13

What is the total inelastic cross-section?

Back 13

σij is the inelastic cross-section from quantum level i to quantum level j.
The constant of proportionality in dI/I α -ndx.

Front 14

What is threshold energy?

Back 14

The minimum energy for an inelastic process to occur, i.e. the amount of energy needed to excite an electron of the target material.

Front 15

Describe the Frank-Hertz experiment and its findings.

Back 15

Electrons ejected from a heated cathode and accelerated through a potential V1 to a wire grid.
Electrons passing through the grid are collected by a plate and cause a current I to flow.
Vplate < Vgrid - this retarding potential prevents electrons reaching the plate from tunnelling effects.
Tube is filled with Hg vapour, meaurements of I wrt V1
Above a threshold voltage there is a drop in current, where the electron can excite a Hg electron.
This proved that energy levels are quantised.

Front 16

What is the differential cross-section?

Back 16

The particle flux (no. per unit time) scattered by each target nucleus into solid angle Ω divided by the incoming intensity (no. per unit time per unit area)

Front 17

What is the formula for differential cross-section?

Back 17

dN = dσ/dΩ N n l dΩ
where dσ/dΩ = differential cross-section
N = incident flux nl = target density and length

Front 18

How is differential cross-section related to total cross-section and when is this valid?

Back 18

Integrate over solid angle dΩ with dΩ = sin θdθdϕ :
And using cylindrical symmetry, so no ϕ dependence:
2π∫dσ/dΩ (θ) sinθdθ
It is valid below the inelastic threshold.

Front 19

What is the Rutherford formula for differential cross section and when is it valid?

Back 19

dσ/dΩ(θ) = (Z1 Z2 /4πɛ4E)^2 sin (θ/2)^2
It fails at high energy when the finite size of the nucleus becomes important since V→∞ as r→0

Front 20

What is the relationship between dσ/dΩ and θ?

Back 20

It does not vary monotonically - there are diffraction peaks, in semi-quantitative agreement with Airy disks. At high energy, diffraction occurs from nuclei, at low energy from atoms.

Front 21

What is the quantum description of scattering?

Back 21

It uses incoming plane wave and outgoing spherical waves calculated from the TISE.
Ψ(r) = e^ik1z = f(θ,ϕ) e^ik2r/r
This can be applied above the inelastic threshold.

Front 22

What are the two postulates of the Bohr model?

Back 22

The electron moves in certain allwed circular classical orbits about the nucleus without radiating.
Emission or absorption of radiation by an atom is associated with a transition between these states.

Front 23

When is the Bohr model applicable?

Back 23

For hydrogen and hydrogen-like atoms. QM is needed for multi-electron atoms.

Front 24

What are the permitted orbits of the Bohr model?

Back 24

L = nħ

Front 25

How are the energies of the Bohr model calculated?

Back 25

Balanced coloumb/centripetal force Ze^2/4πɛr^2 = mev^2/r
Acceptable radii are a whole number of λ
Allowed velocities => Kinetic energy.
Total energy = KE + PE
E scales as n^-2

Front 26

What is the equation for transitions between quantum states?

Back 26

1/λf = R (1/nf^2 - 1/ni^2)
Corresponds to emission of a photon of energy hʋ = Ef-Ei

Front 27

What is the Lyman series?

Back 27

nf = 1, ni = 2, 3, 4…

Front 28

What is the Balmer series?

Back 28

nf = 2

Front 29

What is the Paschen series?

Back 29

nf = 3

Front 30

What is the α series?

Back 30

Δn = 1

Front 31

What is the β series?

Back 31

Δn = 2

Front 32

What is the ɣ series?

Back 32

Δn = 3

Front 33

What is the correspondence principle?

Back 33

In the limit of large quantum numbers n QM results should tend to classical results.

Front 34

What is reduced mass and why is it used?

Back 34

The Bohr model assumes infinite nuclear mass. Reduced mass is the effective mass of a system once the Centre of Mass motion is separated off.
. Μ = meMp/(me+Mp)

Front 35

What are atomic units?

Back 35

ħ = me = a0 = 4πɛ0 = 1

Front 36

What is the quantum description of angular momentum?

Back 36

p = -iħΔ, position operator is r

Front 37

What is the atomic unit of energy?

Back 37

Hartree: 1 Ha = 1 au = 2x ionisation energy of hydrogen = 2R∞

Front 38

What are wavenumbers?

Back 38

A unit of energy, related to hc/λ and expressed in cm-1

Front 39

What is the energy difference between two states of a hydrogenic atom?

Back 39

E = hc/λ = Z^2 R∞ (1/m^2 - 1/n^2)
OR
1/λ Z^2 R∞ (1/m^2 - 1/n^2) where R is in wavenumber units.

Front 40

What is the energy of the nth state?

Back 40

En = Z^2 R∞/n^2

Front 41

What is R∞ in wavenumbers?

Back 41

109737 cm-1

Front 42

What are the simultaneous eigenfunctions of Lz and L^2?

Back 42

Spherical harmonics Ylm (θ, ϕ)
L^2 Ylm (θ, ϕ) = l(l+1) Ylm (θ, ϕ)
Lz Ylm (θ, ϕ) = mħ Ylm (θ, ϕ)
Where Ylm (θ, ϕ) are separable.

Front 43

What is the parity of the spherical harmonics?

Back 43

P (Ylm (θ, ϕ)) = (-1)^l

Front 44

What is the magnitude of Lz?

Back 44

m ħ

Front 45

What is the magnitude of L^2

Back 45

l (l+1)

Front 46

What is the wavefunction for hydrogen?

Back 46

Ψnlm = Rnl (r) Ylm (θ, ϕ) with Rnl (p) = Nnl e^-p/2 p^l Lnl (p)
Where p = 2Zr/n and Lnl are the Laguerre polynomials.

Front 47

How does the wavefunction of a one-electron atom vary?

Back 47

The exponential term: e^-p/2 where p = 2Zr/n
This dominates for large p. For larger p and hence larger Z the exponential decay is faster and the electrons are closer to the atom.
For higher n the decay is smaller - excited states are more extended.
For large r, the radial wavefunction Rnl gets smaller.

Front 48

How does the quantum model of angular wavefunction differ from the Bohr radius?

Back 48

The Bohr radius has a fixed value, QM assigns probability densities to radii.

Front 49

What is the probability density of finding an e- a distance r from the nucleus?

Back 49

P = r^2Rnl^2 with n-l peaks.

Front 50

How does the probability of finding the e- near the nucleus change with l?

Back 50

For a given n, the probability of finding the e- near the nucleus decreases as l increases, because the centrifugal barrier pushes the e- out. Low-l orbitals are called penetrating.

Front 51

What is the average radius of an electronic orbit?

Back 51

<r> = 3a0/2, due to the long tail in the radial probability density.

Front 52

What is the dependency of E on n in the quantum model?

Back 52

En α n^2
E is dependent on n only.

Front 53

What values of l are allowed for a given n?

Back 53

l = 0,1,2…. n-1

Front 54

What values of ml are allowed for a given l?

Back 54

ml = -l, -l+1, -l+2….l-1, l

Front 55

What is the total number of energy states with degeneracy with respect to l and ml?

Back 55

Σ(2l+1) = n^2

Front 56

Why do single-electron atoms have degeneracy with respect to l and ml?

Back 56

This is a characteristic of a central potential, and is lifted if the straightforward dependence on 1/r is modified by a small perturbation.

Front 57

What are the spectroscopic notations for l?

Back 57

l = 0,1,2,3,4 = s, p, d, f, g

Front 58

What are the spectroscopic notations for n?

Back 58

n= 1,2,3,4 = K, L, M, N

Front 59

What are the transition restrictions for l and ml?

Back 59

Δl = +/-1 Δm = 0, +/-1

Front 60

What are the simultaneous eigenfunctions of Sz and S^2?

Back 60

S^2 (Χsms) = s(s+1)(Χsms) with s = 1/2
Sz (Χsms) = mħ (Χsms) with m= +/- 1/2

Front 61

What is the total wavefunction for H?

Back 61

Ψnlm = Rnl (r) Ylm (θ, ϕ) Χ(sms)

Front 62

Where does spin arise from?

Back 62

From Dirac's relativistic theory of quantum mechanics, where an extra angular momentum was necessary to allow total angular momentum J=L+S to be conserved in a central force field.

Front 63

What is a positron?

Back 63

The normal state of the vacuum is an infinite density of e-. If an e- is excited such that E>0 it leaves behind a positively charged hole, which manifests as a positron.

Front 64

What is the form of the Hamiltonian for a one-electron atom in atomic units?

Back 64

H = -1/2Δ^2 -Z/r

Front 65

What is the form of a many-electron Hamiltonian?

Back 65

H = Σ h + Σ 1/rij
This is no longer analytically solvable due to the coulombic contributions.

Front 66

What is the form of the Hamiltonian in the independent particle model?

Back 66

H = Σ h
the coulombic interaction 1/rij is neglected.

Front 67

What is the solution of the Hamiltonian in the independent particle model?

Back 67

N hydrogen-like solutions
E = E1 + E2 +…+ En

Front 68

What is the deviation in result from the independent particle model?

Back 68

The calculated energy states are too negative.

Front 69

What is the form of the Hamiltonian in the central field approximation?

Back 69

H = Σ h + Vc
Sum of N 1-electron Hamiltonians each of which depends only on the position of the ith electron.
The potential for each has been formed from averaging out the effect of all the other electrons over a sphere - it is spherically symmetric with no (θ, ϕ) dependence.

Front 70

What is the resulting wavefunction for the central field model?

Back 70

The product of the individual orbitals.
Ψ(r1, r2, .. rn) = ϕ(r1) ϕ(r2)… ϕ(rn)

Front 71

What is the form of the individual orbitals in the central field approximation?

Back 71

The angular part is the same as hydrogen. The radial part has been modified by the additional potentials.

Front 72

When is the central field approximation most useful?

Back 72

For atoms with a single optically active electron and a bound inner core, where the inner core screens the nucleus.

Front 73

How is the interaction of an electron affected by the value of l in a multi-electron atom?

Back 73

Close to the nucleus (low l) the nuclear attraction dominates, away from the nucleus the inner core of electrons appear like a single positive charge. Hence high-l electrons are more hydrogenic. The energy degeneracy wrt l is lifted.

Front 74

How can the energy of a single optically-active electron be described in terms of quantum defect?

Back 74

Enl = - Zeff^2/(2 (n - Δnl)^2) where Δnl is the quantum defect and describes the departure from hydrogenic behaviour.

Front 75

What are the characteristics of the quantum defect?

Back 75

To a good approximation they depend only on l.
They are non-integer.
n-Δnl may be considered a good quantum number.

Front 76

Which electrons have the smallest quantum defect?

Back 76

Those with large l - which see the nucleus and inner core electrons as a single positive charge Zeff

Front 77

What is the alternative to quantum defect?

Back 77

enl = (Z-σnl)^2/2n^2 where σnl is the screening constant.

Front 78

How are indistinguishable particles acted upon by the interchange operator and how does it commute with the Hamiltonian?

Back 78

P12 H (1,2) Ψ (1, 2) = H (2, 1) Ψ (2, 1) = H (1, 2) P12 Ψ (1, 2)
Hence {P12 H(1,2) - H(1,2) P12} Ψ (1,2) = 0

Front 79

What are the eigenvalues of the interchange operator and what are the consequences of this?

Back 79

Operating twice: P12 P12 Ψ(1, 2) = p P12 Ψ (2,1) = p^2 Ψ(1, 2).
Hence p = +/- 1 and a wavefunction representing two indistinguishable particles must be either symmetric or antisymmetrc with respect to exchange.

Front 80

What is the Pauli Exclusion Principle?

Back 80

The wavefunction of a system of identical fermions must be anti-symmetric with respect to exchange P(1, 2) = - (2, 1)
OR electrons must have different sets of quantum numbers n, l, ml, s, ms

Front 81

What are the symmetric and antisymmetric wavefunctions of two non-interacting particles?

Back 81

Ψ+/- (1,2) = 1/√2 (ϕa(1) ϕb(2) +/- ϕb(1) ϕa(2)) where a and b are different eigenstates.
If 1=2 i.e they have the same quantum numbers then the antisymmetric combination disappears.

Front 82

What is the total wavefunction for He?

Back 82

Ψnlm = ϕ(r1, r2) Χ(s1z, s2z)

Front 83

What is total spin for He and what are the eigenfunctions?

Back 83

S = s1+s2
Sz = s1z + s2z
X is the eigenfunction of total S and Sz

Front 84

What are the eigenvalues and eigenfunctions of total spin S?

Back 84

For individual spin the eigenvalues are ms= + 1/2 eigenfunction α
or ms = -1/2 eigenfunction β
Hence the eigenvalues of total spin S are 0 or 1 depending on whether the spins are parallel or anti-parallel.

Front 85

What are the possible values of Ms for total spin S=1?

Back 85

Ms = +1 X =α(1)α(2) ↑↑
Ms = 0 X = 1/√2 (α(1)β(2) + α(2)β(1)) ↑↓ + ↑↓
Ms = -1 X = β(1)β(2) ↓↓
SYMMETRIC

Front 86

What are the possible values of Ms for total spin S=0?

Back 86

Ms = 0 X = 1/√2 (α(1)β(2) - α(2)β(1)) ↑↓ - ↑↓
ANTISYMMETRIC

Front 87

What are the possible total wavefunctions for He?

Back 87

Space symmetric, spin antisymmetric SINGLET STATE
Space antisymmetric, spin symmetric TRIPLET STATE

Front 88

Is the ground state of He a singlet or triplet state?

Back 88

It is a singlet state. Since the spatial wavefunction has the same quantum numbers for both electrons 1S2 the spin must be antisymmetric.

Front 89

Are excited states of He singlet or triplet states?

Back 89

They can be either. If the spatial wavefunctions have the same quantum numbers for each electron the spin must give the antisymmetric singlet state - PARA helium. If the electrons are in different subshells the spatial wavefunction is antisymmetric and the spin will be the triplet symmetric state ORTHO helium.

Front 90

What is the value of spin multiplicity for He?

Back 90

Using TOTAL spin S
S(S+1) = total number of spin states.
For S = 0 multiplicity = 1
For S = 1 multiplicity = 3

Front 91

What is the exchange force?

Back 91

The PEP has introduced a coupling between the spin states and space states which act as if there is a force whose sign depends on the orientation of spin.

Front 92

What is the qualitative explanation for the exchange interaction?

Back 92

For the symmetric spin triplet, the space part must be anti-symmetric and will disappear if the electrons are in the same place. Hence the electrons tend to stay away from each other and reduce the coulombic replusion, and be more tightly bound to the nucleus. For the anti-symmetric spin singlet, the space part is symmetric and the electrons may be together for some of the time, and experience a greater coulomb repulsion, leading to higher energy and less nuclear binding.

Front 93

How is the Hamiltonian for He modified by the exchange force?

Back 93

The non-central part of the Hamiltonian potential 1/rij is modified into a coulombic term C and and exchange term E. For the singlet case the overall term is C+E. For the triplet case the overall term is C-E. Hence the E term is repulsive for the singlet state and attractive for the triplet state.

Front 94

Summarise specroscopic notation for electrons.

Back 94

n is given as a number 1, 2, 3….
l is given as a letter s, p, d, f, g
There are 2l+1 values of l
s = 1/2 for fermions, there are 2 values of ms
each orbital nl can hold 2l+1 electrons.

Front 95

What defines a shell?

Back 95

A shell is has the same n.

Front 96

What defines a subshell?

Back 96

A subshell has the same n and l, also called an orbital.

Front 97

What is an equivalent electron?

Back 97

It has the same nl.

Front 98

What are optically active electrons?

Back 98

Electrons in an open shell, i.e. in a shell which contains less than 2n^2 electrons.

Front 99

What is ionization energy?

Back 99

It is the energy required to remove one valence electron. It is minimised for elements with only one valence electron.

Front 100

What are the good quantum numbers for a non-central field model?

Back 100

Total angular momentum L
Total spin S
The electronic states are simultaneous eigenfunctions of H, L^2 and S^2

Front 101

What are the L and S terms for a multi-electron atom?

Back 101

For closed shells S=L=0, only valence electrons are considered
2S+1 L

Front 102

What are the possible values of S and L for a two-electron atom?

Back 102

S = |s1-s2|……… s1+s2 in integer steps
L = |l1-l2| …… l1+l2 in integer steps.

Front 103

What are the possible configurations for equivalent electrons (same nl)?

Back 103

They must have different ml and/or ms. For the same ml they must have ms= +/- 1/2. For non-similar ml they can have any combinations of ms and ml and these must be deduced one at a time from all possible ml/ms for 1 and 2 then grouped by overall ML. Then all possible combinations of L and S can be deduced.

Front 104

How can electronic configurations be found more simply?

Back 104

Find all possible values of L. The parity of L is (-1)^L - giving alternate symmetric/anti-symmetric wavefunctions. The corresponding spin functions must be antisymmetric (S=0)/symmetric (S=1) alternately. From which all possible values of 2S+1L can be deduced.

Front 105

What do Hund's rules show?

Back 105

They order the terms of an electronic configuration in terms of L and S into sequential energies. They apply rigourously only to ground state configurations.

Front 106

What are Hund's rules including L/S coupling?

Back 106

1. The term with the largest value of S has the lowest energy.
2. For a given value of S, the term with the highest value of L has the lowest energy.

Front 107

How is energy ordering related physically to the values of S and L?

Back 107

Spin-spin interaction: states with symmetric spins with larger S are forced to be spatially separated and feel greater attraction from the nucleus.
Orbit-orbit interaction: If L is large then the individual l are parallel and the electrons orbit in the same direction. So they encounter each other less and on average experience less shielding effect.

Front 108

What is the atomic Hamiltonian including spin-orbit interaction?

Back 108

HT = H + HLS

Front 109

What is the origin of the spin-orbit interaction?

Back 109

The intrinsic magnetic moment caused by spin or orbital angular momentum.

Front 110

What is the magnetic moment due to an angular momentum X?

Back 110

μx = gx q/2M X where gx is the Lande g-factor.

Front 111

What is the torque of a magnetic moment in a uniform magnetic field?

Back 111

Ƭ = dX/dt = μx x B = gx Q/2M X x B
Hence Ƭ = - ω x X where ω is the Larmor precession frequency.

Front 112

What is the effect of the torque Ƭ due to the magnetic moment on the angular momentum X?

Back 112

Ƭ is perpendicular to X and B and results in a precession of X around B

Front 113

What is the Larmor precession frequency?

Back 113

ω = gx Q/2M B

Front 114

What is the interaction potential energy of a magnetic moment placed in a magnetic field?

Back 114

V = - μx . B

Front 115

What is the magnetic moment of an electron due to L?

Back 115

μL = -gL μB L/ħ
gL =1

Front 116

What is the magnetic moment of an electron due to S?

Back 116

μS = -gS μB S/ħ
gS =2

Front 117

What do the values of gL and gS tell us about the electron?

Back 117

That it is a structureless fermion. Other fermions (e.g the proton) do not have these values and hence are not structureless.

Front 118

What is the energy of the spin-orbit interaction?

Back 118

VLS = Zαħ/2m^2c (L.S/r^3) where α is the fine structure constant.
If r ~ a0, <L, S> ~ ħ and Z=1 then VLS ~ α^2 R∞

Front 119

What is the effect of spin-orbit coupling on quantum numbers?

Back 119

ms and ml no longer have fixed z components and are not good quantum numbers. However, total AM J=L+S does have a constant z component, hence mj is a good quantum number.

Front 120

How is total angular momentum J and Jz defined?

Back 120

J = L+S
Jz = Lz + Sz

Front 121

What is the effect of spin-orbit coupling on energy levels?

Back 121

They are no longer eigenstates of Lz and Sz but are eigenstates of J^2, Jz, L^2 and S^2.

Front 122

What is the derivation of energy levels for spin-orbit splitting?

Back 122

ΔEnl = <n, l, j, mj | VLS | n, l, j, mj> where VLS α L.S
Using J^2 = (L+S)^2, multiplying out, substituting for VLS in the first expresssion we get
ΔEnl = 1/2 A(L,S) [J(J+1) - L(L+1) - S(S+1)]

Front 123

What is the Lande Interval Rule?

Back 123

Δenl (J) -ΔEnl (J-1) = A (L,S) J
i.e. The separation between adjacent energy levels is proportional to the larger of the two J values.

Front 124

What are the eigenvalues of J^2 and Jz?

Back 124

J^2 Ψ = j(J+1) Ψ
Jz Ψ = m ħ Ψ

Front 125

What are the allowed values of total angular momentum j?

Back 125

j = |l-s|…….. l+s in integer steps

Front 126

What are the allowed values of total angular momentum magnetic quantum number mj for single electrons?

Back 126

mj = -j, -j+1 …… j-1, j in integer steps.

Front 127

What are the good quantum numbers including spin-orbit coupling for a single electron?

Back 127

n, l, j, mj

Front 128

What is the spectroscopic term notation for energy levels including spin-orbit coupling for a single electron?

Back 128

n 2s+1 Lj

Front 129

What is the selection rule for transitions between j states for a single electron?

Back 129

Δj = 0, +/- 1 with a transition from j=0 to j'=0 forbidden.
ie, in an orbital transition, there cannot be a jump of more than an integer step of j.

Front 130

What are the two methods for calculating J for the multi-electron case and when are they applied?

Back 130

The Russell-Saunders coupling is used for low-Z atoms when the spin-orbit coupling is much less than the interaction between electrons.
j-j coupling is used for high-Z atoms when the spin-orbit coupling is strong.
Neither gives a complete description of AM especially for medium-Z atoms.

Front 131

What is the Russell-Saunders term for low-Z atoms?

Back 131

Combine individual spins to get total spin S, similarly for total L.
Then, J = |L-S|……. |L+S| in integer steps.

Front 132

What is the third Hund's rule with respect to J-values?

Back 132

a) if the outer shell is less than half-full the lowest energy in the lowest energy terms corresponds to the smallest J-value
b) If the outer shell is more than half-full the lowest energy in the lowest energy term corresponds to the largest J-value.
c) If the subshell is half full there is no multiplet splitting.

Front 133

What is the jj-coupling term for high Z atoms?

Back 133

The s and l are combined for each individual electron to give each an individual j term. The total J is then the sum of the individual j terms.
J = Σji (min) => Σji (max) in integer steps.

Front 134

What is parity?

Back 134

Parity describes the behaviour of Ψ under reflection at the origin (nucleus).
The parity of N electrons is (-1) ^l1 (-1) ^l2…. (-1) ^li = (-1) ^ Σli

Front 135

What is an electric dipole moment?

Back 135

P = er

Front 136

What is the Hamiltonian for a one-electron atom exposed to an oscillating electric field?

Back 136

H = H0 - P.E where E(t) = E0 exp (iωt)

Front 137

What is the time-dependent perturbation?

Back 137

V = P.E with z component V = er cosθ E0 exp(iωt)

Front 138

What is Fermi's golden rule?

Back 138

Transitions occur between two quantum states i→f with a probability Tif which is given by the square of the matrix element of the perturbation.
Tif α |<Ψf | V | Ψi>|^2 α |<Ψf | rcosθ | Ψi>|^2 for the z-component.

Front 139

From what part of a wavefunction is the transition probability Tif formed?

Back 139

The radial part is constant and does not yield any selection rule. The angular part (θ, ϕ) yield selection rules.

Front 140

What are the electric dipole selection rules?

Back 140

Δl = +/-1
Δml = 0, +/-1
Δs=0
initial and final states have opposite parity.

Front 141

How are the selection rules gained from Fermi's golden rule?

Back 141

The spherical harmonics are orthogonal, hence the integral forming the transition probability is zero unless l=+/-l' and m=m'. m = +/-m' comes from the x and y parts of the integral. The parity of initial and final states must be odd or the integral will be zero.

Front 142

How does the electric dipole selection rule differ if spin-orbit coupling is significant?

Back 142

Selection rules apply instead to j
Δj = 0, +/- 1 i.e. with a transition from j=0 to j'=0 forbidden

Front 143

What are the selection rules for multi-electron atoms?

Back 143

ΔJ = 0, +/- 1 with a transition from J=0 to J'=0 forbidden.
ΔL = +/-1 if spin-orbit coupling is weak.
ΔMJ = 0, +/-1
ΔS=0
initial and final states have opposite parity.

Front 144

What are the 3 ways by which an atom can interact with a radiation field?

Back 144

Absorption, sponteneous emission and stimulated emission.

Front 145

What is the rate of absorption when an atom interacts with a radiation field?

Back 145

The rate of absorption is proportional to the population of the bround state and energy density α N1 U(ʋ12)

Front 146

What is the rate of decay in spontaneous emission?

Back 146

Rate of decay is α the excited state population N2

Front 147

What is the rate of emission in stimulated decay?

Back 147

Rate of emission is α the excited state population and the energy density N2 U(ʋ12)

Front 148

What is stimulated emission?

Back 148

A photo causes an excited atom to decay, emitting a photon that is coherent with the stimulating photon. (same phase and amplitude)

Front 149

What are the constants of proportionality for interaction of an atom in a radiation field?

Back 149

The Einstein co-efficients:
A for sponteneous emission.
B for absorption or stimulated emission.
These co-efficients are related to the transition probabilities from Fermi's golden rule.

Front 150

What is the probability of a spontaneous decay?

Back 150

Pif = Aif

Front 151

What is the lifetime of a spontaneous decay?

Back 151

Obtained by summing over all possible decay paths:
Δƭi = 1/ Σaif

Front 152

What is the energy uncertainty for a given decay?

Back 152

From Heisenberg:
ΔƬΔE > ħ/2

Front 153

What is the intrinsic energy width for a decay?

Back 153

ΔE ~ ħ/2ΔƬ leading to spectra frequency width Δʋ - the natural line-width.

Front 154

What is a metastable energy level?

Back 154

If the probability of decay Aif = 0 for all final states f the state is metastable.
Other transitions are possible (magnetic dipole, electric quadrupole) but with much smaller probability.

Front 155

What are the populations of excited states to lower states in two-level systems?

Back 155

A Boltzmann distribution:
N1/N2 = exp (E2-E1)/kT
There are more atoms in state 1 than state 2

Front 156

What is the most likely radiation interaction in a normal two-level system?

Back 156

Absorption is most likely, since there are more atoms in lower states than in excited states, normally.

Front 157

What is population inversion?

Back 157

Population inversion is when there are more atoms in an excited state than in a lower state.

Front 158

What is the most likely radiation interaction in a population inverted system?

Back 158

Spontaneous or stimulated emission is likely in population inversion as there are more atoms in the excited state than in the ground state.

Front 159

Why can population inversion not be attained in a two-level system?

Back 159

As atoms are excited to level 2, the probability of emission to level 1 increases until N1 = N2 (saturated transmission). There cannot be more atoms in level 2 than level 1.

Front 160

How is population inversion attained in a 3-level system?

Back 160

Atoms are pumped with photons of frequency ʋ13.
Atoms accumulate in level 3 which decays spontaneously to level 2.
Atoms accumulate in metastable level 2.
Levels 1 and 2 now have a population inversion.
A beam of ʋ12 photons can cause stimulated emission from 2→1.

Front 161

How can a laser be used to cool atoms?

Back 161

An atom is hit with many photons which are scattered and slow down the atom.
The photons have frequency ʋL which is slightly detuned from the resonant frequency of an atomic transition ʋif > ʋL
The likelyhood of scattering (ie absorption and re-emission) is related to the closeness to resonant frequency.
Atoms travelling in the same direction as the laser will see a red-shift and decreased scattering probability.
Atoms travelling away will see a blue-shift and increased scattering probability.
So mainly atoms travelling opposite to the direction of the laser will undergo scattering and be slowed down.
The scattering re-emission direction is random, so the net recoil is zero.
Net frictional force F = - βv

Front 162

How are X-ray spectra produced?

Back 162

By bombardment of inner electrons with high energy electrons from a cathode, with kinetic energy eV where V is the accelerating voltage.

Front 163

How do X-ray spectra differ from normal atomic spectra?

Back 163

They are transitions of inner tightly bound electrons rather than outer optical electrons.

Front 164

What are the two features of an X-ray spectrum?

Back 164

A continuous spectrum - Brehmsstrahlung - and characteristic X-ray spectra peaks.

Front 165

What is the Brehmsstrahlung spectrum of an X-ray spectrum?

Back 165

A continuous spectrum caused by fast moving electrons deflected and slowed down in the coulomb field of an atom.
There is a cutoff wavelength where an electron gives up all of its energy λmin = hc/Ei or ʋmax = Ei/h

Front 166

What are the characteristic X-ray spectra peaks?

Back 166

They are produced by transitions of inner electrons, and then decay of higher electrons into the holes left behind and emission of a photon.

Front 167

What is the frequency of X-ray emission?

Back 167

ʋif = (Ei - Ef)/ 2 = R Zeff^2 (1/nf^2 + 1/ni^2) where Zeff = Z-S and S is the screening constant.

Front 168

What are transitions to different final states nf denoted as in X-ray spectra?

Back 168

nf = 1: K-series
nf = 2: L-series
nf = 3: M-series.

Front 169

What are the transition differences Δn denoted as in X-ray spectra?

Back 169

Δn = 1: α
Δn = 2: β
Δn = 3: ɣ

Front 170

What is the full Hamiltonian for multi-electron atoms in external fields?

Back 170

H = H0 + HLS + VB + VE

Front 171

What is the perturbative case of interactions between atoms and internal/external fields?

Back 171

H0>> VB, VE, HLS

Front 172

What is the Zeeman effect?

Back 172

Splitting of spectral lines in a magnetic field.

Front 173

What is the weak field Zeeman effect?

Back 173

When VB<< HLS

Front 174

When does the normal Zeeman effect occur?

Back 174

For total spin S=0 and the magnetic moment is entirely due to the orbital angular momentum.

Front 175

What is the potential energy relating to a magnetic moment?

Back 175

V= μx.B where X is any angular momentum.
μx = -gx μB X/ħ

Front 176

What is the potential of a magnetic field along the Z-direction for the normal Zeeman effect and what is the consequence?

Back 176

V = μB/ħ L.B = BzμB Bz ml
Hence the spectral line due to l is split into 2l+1 equally spaced levels.

Front 177

What do the dipole selection rules tell us about the weak normal Zeeman effect?

Back 177

The spectral line is split into 3
l => 2l+1 => ml => 3 transitions allowed since Δl = +/-1

Front 178

What is the effect of the anomalous Zeeman effect for a weak field?

Back 178

The magnetic field is too weak to uncouple L and S, and these precess rapidly around J with constant projection. J precesses slowly around B.
As a result ml and ms are not good quantum numbers as L and S do not have constant z-components.

Front 179

What is the interaction potential energy for the weak field anomalous Zeeman effect?

Back 179

This must be expressed in good quantum numbers j, l, s, mj
VB = μB/ħ B.(L+2S) - μB B.(J+S)

Front 180

How is the energy splitting for the weak field anomalous Zeeman effect derived?

Back 180

Take the Z-component of B.L to find ΔEL. Take the Z-component of B.S and note that S precesses around J with constant component Sj. We can find the component Sj to get the relationship between S and J. Using the relationship J.S and J = L+S we get J.S = 1/2 [J^2 +S^2 +L^2].
Substituting back into the original expression for ΔES we get the total energy difference due to spin.
Adding the contributions we get ΔE = μB B mj gj where gj is the Lande-g-factor due to j.

Front 181

What is the effect of the weak field anomalous Zeeman effect?

Back 181

The degeneracy with respect to mj is lifted - there is a spectral line for each value of mj.

Front 182

What is the strong field Zeeman effect and what is it called?

Back 182

VB>> HLS - the Paschen-Beck limit.

Front 183

What is the effect of a strong magnetic field on an atom (the Paschen-Beck limit)?

Back 183

L and S decouple and both precess independently about the magnetic B-field direction. J is not constant.

Front 184

What is the torque of a magnetic moment in a uniform magnetic field?

Back 184

Ƭ α L x B

Front 185

What is the total magnetic moment in the strong field Paschen-Beck limit?

Back 185

μ = μL + μS

Front 186

What are the good quantum numbers in the Paschen-Beck limit?

Back 186

l, s, ml and ms.

Front 187

What is the potential energy in the Paschen-Beck limit (strong field)?

Back 187

VB = -μ.B = μB/ħ B/ (L+2S) = μB/ħ B/ (Lz+2Sz)

Front 188

What is the energy splitting in the Paschen-Beck limit?

Back 188

ΔE = μB B (ML+2MS)

Front 189

What are the projections of L and S on the Z-axis in the Paschen-Beck limit?

Back 189

mlħ and msħ
BOTH CONSTANT

Front 190

What effect does the Paschen-Beck limit have on the spectral lines?

Back 190

Δhʋ = (ml-ml') μB B + 2(ms - ms') μB B.
However, due to the selection rules, Δml = 0, +/-1 and Δms = 0
Therefore there are 3 spectral lines.

Front 191

What are the possible spectral splits due to the Zeeman effect?

Back 191

Weak field - no overall spin - normal Zeeman effect - 3 spectral lines.
Weak field - spin splitting - anomalous Zeeman effect - splitting of each spectral line by each term in the overall Lande g-factor.
Strong field - spin-orbit decoupling - 3 spectral lines.

Front 192

What is the potential and the force in the Stern-Gerlach experiment?

Back 192

VB = -μS.B = 2μB B.S
F = -dV/dz = -2Sz μB dB/dz

Front 193

What does the Stern-Gerlach experiment show?

Back 193

The spin magnetic moment is quantised - it may only take two values. However a general angular momentum can take 2X+1 values so the spin must be fractional.

Front 194

What does the Stern-Gerlach experiment show with regards to atoms?

Back 194

The deflection force may be experienced by μL or μJ as well as μS. Hence a beam of atoms will be split into 2 x (2L+1) beams or (2J+1) beams.

Front 195

What is the nuclear spin of an atom?

Back 195

It is the net spin of the protons and neutrons, given by I.

Front 196

What is the magnetic moment of the nuclear spin I?

Back 196

μI = gN μN I/ħ where μN is the nuclear Bohr magneton.

Front 197

What are the hyperfine sublevels?

Back 197

F = J + I where J is (S+L) and I is the nuclear net spin.
It is analogous to spin-orbit coupling.

Front 198

What is the characteristic of hyperfine sublevels?

Back 198

They are very tiny with small energy differences.

Front 199

What is the Stark effect?

Back 199

The splitting of energy levels of an atom in static electric fields.

Front 200

What is the field strength assumption for the Stark effect?

Back 200

VE<< Z/r

Front 201

What is the perturbation to the Hamiltonian for the Stark effect?

Back 201

VE = - μ.E

Front 202

What is the potential along the Z-axis in the Stark effect?

Back 202

VE = - (-er.E) = ez.E

Front 203

What is the effect of the potential in the Stark effect?

Back 203

The potential VE is positive - ie repulsive, and it decreases the binding energy.

Front 204

When does the quadratic Stark effect occur?

Back 204

It occurs in atoms which have no intrinsic dipole moment.

Front 205

What is the effect of a an electric field in an atom with no intrinsic dipole moment?

Back 205

The field polarises the electron distribution, inducing a separation of charge and dipole moment which is proportional to E

Front 206

What is the interaction energy for the quadratic Stark effect?

Back 206

VE = -1/2 α E^2
a term varying quadratially with the field, where α is the dipole polarisability.

Front 207

What is the magnitude of the induced dipole moment of an atom in an electric field?

Back 207

μ = dVE/dE = αE

Front 208

When does the linear Stark effect occur?

Back 208

For atoms with an intrinsic dipole moment μE such as excited states of hydrogenic atoms. These atoms have l-degeneracy: all of the s, p, d levels have the same energy and the eigenstates are superpositions of l-orbitals.

Front 209

What is the linear Stark effect?

Back 209

The applied electric field mixes the states and connects states with Δl = +/-1 and Δm = 0 because the electric field is in the z-direction only.
The resulting energy shift is linear in E.

Front 210

What is the Hamiltonian for a molecule?

Back 210

H (R1, R2,…. r1, r2….) = Σ-ħ^2/2Mn + Σ-ħ^2/2m Δi^2 + V(RN, rn)
This is a combined nuclear (R) and electronic (r) form.
The terms are KE of nuclei - KE of electrons - Coulomb interactions.

Front 211

What is the coulombic potential term for a molecule?

Back 211

V (RN, rn) = Σ 1/|ri - rj|+ Σ 1/|RN-RM| - Σ 1/|Rn-ri|
The terms are e- e- repulsion - nuclear repulsion - nuclear/e- attraction.

Front 212

Why is the Born-Oppenheimer approximation needed?

Back 212

The molecular Hamiltonian is not separable due to the electronic-nuclear attraction term in the potential. An approximate separability is introduced by the Born-Oppenheimer approximation.

Front 213

What is the starting assumption of the Born-Oppenheimer approximation?

Back 213

That the nuclei move much more slowly than electrons and the Hamiltonian can be written using the centre of mass motion.

Front 214

What is the Hamiltonian for a diatomic molecule at the centre of mass?

Back 214

H (R, r) = -ħ^2/2μ ΔR^2 + Σ-ħ^2/wm Δi^2 + V(R, ri)

Front 215

What are the 6 steps of the Born-Oppenheimer approximation?

Back 215

1. Fix the nuclei in space by setting the internuclear co-ordinate to a constant value. We can then neglect nuclear K.E.
2. Solve the SE for electronic motion with fixed nuclei. H(rR)ϕ(rR) = E(R) ϕ(rR)
3.Assume a solution of form Ψ(rR) = ʋ(R)ϕ(rR)
4. Plug this into the full original Hamiltonian.
5. Electronic wavefunctions are insensitive to changes in nuclear positions so ΔR terms can be ignored. 6. Simplify the Hamiltonian using this approximation.

Front 216

What is the consequence of the Born-Oppenheimer approximation?

Back 216

Electron part depends on the position of the nuclei, not their motion. Electrons adjust instantaneously to changes in nuclei positions. The electronic energies at different internuclear distances provide a potential in which nuclei move.

Front 217

What is the probability of the electron being around nucleus 1 or nucleus 2 in H2+?

Back 217

|C1|^2 = |C2|^2
equal probability.

Front 218

What is the electronic Hamiltonian Hel for H2+?

Back 218

Hel = -1/2 Δr^2 -1/rA -1/rB +1/R

Front 219

What are the wavefunctions for H2+ in terms of the coefficients C1 C2?

Back 219

C1 = +/- C2
There is a symmetric (gerade) and antisymmetric (ungerade) wavefunction.
Ψ± = C [ϕ1s(rA) ± ϕ1s(rB)]

Front 220

What is LCAO?

Back 220

Linear Comination of Atomic Orbitals
Building a molecular wavefunction from a superposition of atomic orbitals.

Front 221

What do the gerade and ungerade wavefunctions represent?

Back 221

Gerade is symmetric and stable, and represents a bonding orbital, as the presence of the electrons between the nuclei helps to stablise the coulombic repulsion.
Ungerade is antisymmetric and unstable, represents an anti-bonding orbital as there is little probability of finding the electron between the nuclie.

Front 222

What is the electronic energy for H2+?

Back 222

Electronic energies are given by
<Ψ±|Hel|Ψ±> = E±<Ψ±|Ψ±> = A±/N± where A is the expectation value of the electronic Hamiltonian and N is a normalisation constant.

Front 223

What is the normalised expectation value of the Hamiltonian for a diatomic molecule?

Back 223

<Hel> = <Ψ1|Hel|Ψ2> / <Ψ1|Ψ2> where the denominator is a normalisation constant.
It isn't zero as the two wavefunctions are not around the same origin - they are around the centre of each nucleus.

Front 224

What is the normalisation constant for the electronic energy for H2+?

Back 224

N is given by 1 ± I(R)
I(R) = <ϕA|ϕB> - the overlap integral between the two electronic/nuclear wavefunction.

Front 225

What is the expectation value of the electronic Hamiltonian, A±?

Back 225

A± = 1/2 [HAA + HBB] ± HAB

Front 226

What is the HAA or HBB term of the electronic Hamiltonian for H2+?

Back 226

HAA/BB = E1s + 1/R - J(R)
Where the terms are:
energy of H atom - internuclear repulsion - coulomb integral: of interaction with the other atom.
Two equal terms for identical nuclei.

Front 227

What is the HAB term of the electronic Hamiltonian for H2+?

Back 227

HAB = ± (E1s +1/R) I(R) Ŧ K(R)
Where the terms are:
energy of H atom and nuclear repulsion - overlap integral - exchange integral

Front 228

What are the terms, I, J and K in the Hamiltonian of H2+?

Back 228

I = overlap integral = (1+R+ R^2/3) exp -R
J = coulomb integral = -exp -2R
K = exchange integral = (1+R) exp -R

Front 229

What is the characteristic of the overlap integral I in the molecular Hamiltonian?

Back 229

0 < I < 1 hence 1 ± I > 0 and 1 ± I^2 >0

Front 230

What is R in molecular forms?

Back 230

The nuclear separation (for diatomic molecules)

Front 231

How does the energy of H2+ change as R→∞?

Back 231

As R→∞ J, K, I → 0 so E- ~ E+ → E1s +1/R

Front 232

What effect does the splitting have on the molecular energy of H2+?

Back 232

The splitting term 2 K(R) / (1 ± I) lowers E+ and raises E-

Front 233

How does the energy of H2+ change as R→0?

Back 233

As R→0 E± → ∞ because of the repulsion between nuclei.

Front 234

What are the characteristics of E± which lead to bonding or anti-bonding orbitals?

Back 234

E+(R) has a minimum which leads to a stable molecule- bonding.
E-(R) is higher than the energy of the separated atoms - anti-bonding. If a molecule is excited to this state it falls apart.

Front 235

What is the bonding orbital operator equation for H2+?

Back 235

Hel Ψ+ = E+ Ψ+ where H is the electronic Hamiltonian and E is the electronic energy.

Front 236

What is the antibonding orbital operator equation for H2+?

Back 236

Hel Ψ- = E- Ψ- where H is the electronic Hamiltonian and E is the electronic energy.

Front 237

What is De and Re for a stable molecule?

Back 237

De = disassociation energy
Re = Equilibrium distance of nuclei.

Front 238

How does the wavefunction for H2+ differ from that of H2?

Back 238

H2 has two electrons so the Pauli Exclusion Principle must be considered.

Front 239

What is the electronic Hamiltonian Hel for H2?

Back 239

Hel = -1/2 Δr1^2 -1/2 Δr2^2 -1/rA1 -1/rA2 -1/rB1 -1/rB2 +1/R + 1/r12
Terms are:
KE of electrons 1 and 2 - nucleus/electron attraction - nuclear and electronic coulombic repulsion.

Front 240

What is the possible spin for H2?

Back 240

S = 0 or S = 1 as there are 2 electrons. The spin functions are the same as for He.

Front 241

What condition is placed on the wavefunction for H2?

Back 241

if Ψ(r, R) is antisymmetric then spin must be symmetric - triplet state.
If Ψ(r, R) is symmetric then spin must be antisymmetric - singlet state.

Front 242

What is the spin triplet state for H2?

Back 242

ΨT ~ 1/√2 [ϕ1s(rA1)ϕ1s(rb2) - ϕ1s(rA2)ϕ1s(rB2)] XT
An antisymmetric superposition of the 1s wavefunctions.

Front 243

What is the spin singlet state for H2?

Back 243

ΨS~ 1/√2 [ϕ1s(rA1)ϕ1s(rb2) + ϕ1s(rA2)ϕ1s(rB2)] XS
A symmetric superposition of the 1s wavefunctions.

Front 244

What is the difference between H2 wavefunctions and He wavefunctions?

Back 244

The H2 wavefunctions are approximate, derived from the Born-Oppenheimer approximation. They are not exactly the same as the He wavefunctions.

Front 245

Which energy eigenfunctions correspond to the energy eigenvalues of H2?

Back 245

E+ → ΨS → symmetric space
E- → ΨT → antisymmetric space

Front 246

What is the normalised expectation value of the Hamiltonian for H2?

Back 246

E± = <Ψ(S/T) |Hel| Ψ(S/T)> / <Ψ(S/T) | Ψ(S/T)>

Front 247

What is the energy of H2, accounting for the triplet and singlet states?

Back 247

E± = 2E1s + 1/R + J/(1±I^2) ± K/(1±I^2)
where + = spin singlet
symmetric space part, - = spin triplet, antisymmetric space part.

Front 248

What are the terms for the energy of H2?

Back 248

Two hydrogen atoms - nuclear repulsion - overlap/coulomb/exchange integral composite terms.

Front 249

What is the characteristic of the coulomb integral J in the molecular Hamiltonian?

Back 249

J is a positive contribution to the energy.

Front 250

What is the characteristic of the exchange integral K in the molecular Hamiltonian for H2?

Back 250

K generally represents a negative contribution to the energy. Hence, the term (J±K)/(1±I^2) raises the energy for the spin triplet and lowers it for the spin singlet.

Front 251

What are the relative energies of the singlet and triplet states for He?

Back 251

Singlet - HIGHER
Triplet - LOWER

Front 252

What are the relative energies of the singlet and triplet states for H2?

Back 252

Singlet - LOWER
Triplet - HIGHER

Front 253

How do the energy levels of H2 compare with those of He?

Back 253

In He the triplet state is lower in energy than the spin singlet. In H2, the triplet state is higher in energy.

Front 254

Why do the singlet and triplet energy levels for He and H2 vary?

Back 254

In He, the triplet is lower energetically because the electrons avoid each other in this state and reduce their mutual coulombic repulsion. In H2 this translates to a lower probability of the electrons being found between the two nuclei, and hence a low probability of them neutralising the coulombic repulsion between the nuclei. Hence the energy for the triplet state is higher in H2.

Front 255

What are the degrees of freedom available to an N-nuclei molecule?

Back 255

3N

Front 256

What are the three types of motion for a molecule?

Back 256

Translation, rotation, vibration.

Front 257

How many degrees of freedom does translational motion have?

Back 257

3 - translation in 3 dimensions.

Front 258

How many degrees of freedom does rotational motion have?

Back 258

Rotation about the centre of mass-
3 DoF for a solid body
2 DoF for a linear molecule.

Front 259

How many degrees of freedom does vibrational motion have?

Back 259

Vibration around the equilibrium position
3N - 6 DoF for a solid body
3n - 5 DoF for a linear molecule.

Front 260

What do spherical harmonics describe in relation to a diatomic molecule?

Back 260

The Yjmj describe a rotation motion of the molecule with angular momentum quantum numbers J and Mj. The molecule behave like a rigid rotor that rotates about an axis perpendicular to the internuclear axis through the centre of mass.

Front 261

What is the moment of inertia of a diatomic molecule?

Back 261

If the centre of mass is at the origin of co-ordinates then Ie = μRe^2

Front 262

What are the eigenvalues associated with rotational motion of a diatomic molecule?

Back 262

Erot = BJ (J+1) where B is the rotational constant = ħ^2/2Ie.
If J =0 then Erot = 0

Front 263

What is the angular frequency of rotation of a diatomic molecule?

Back 263

ωrot α 1/μ

Front 264

What are the eigenstates of vibrational motion?

Back 264

In the nuclear Hamiltonian:
[KE + rotational + electronic] F(R) = E F(R)
where E is the total energy and F(R) are the vibrational eigenstates.

Front 265

What is the significance of the eigenvalues of vibrational motion?

Back 265

Eel has a minimum at R = Re. The nuclear motion is generally well confied to a small region around Re.

Front 266

How is the QHO term of vibrational motion obtained from the electronic eigenstates of a diatomic molecule?

Back 266

Expand Eel as a Taylor series around Re and neglect terms higher than quadratic:
Eel (R) = Eel (Re) + (R - Re) dEel/dR| R=Re + (R - Re)^2/2 d2Eel/dR2| R=Re
To 2nd order Eel (R) = Eel (Re) + 1/2 K(R-Re)^2 with k = d2Eel/dR2
Substitute back into the total KE+rotational+electronic Hamiltonian and rearrange.

Front 267

What is the vibrational energy for a diatomic molecule in the QHO formulation?

Back 267

Ev = ħω (v +1/2) with ω = √k/μ and v = vibrational number 1, 2, 3….

Front 268

What is the zero-point energy of the vibrational energy of a diatomic molecule?

Back 268

E = 1/2 ħω

Front 269

What is the total energy sum of an ideal diatomic molecule?

Back 269

E ~ Eel + BJ (J+1) + (v+1/2)ħω
equilibrium electronic energy + rotational energy + vibrational energy

Front 270

What are the assumptions made in characterising an ideal diatomic molecule?

Back 270

Vibration v is small - the harmonic approximation is best at the bottom of the well.
J is small, centrifugal distortion tends to stretch the molecule and lower the rotational energy.

Front 271

What are the corrections to idealness for a real diatomic molecule?

Back 271

1. Centrifugal distortion: as J increases internuclear distance stretches. J>10 is important.
2. Rotational constant B depends on v. Since molecules vibrate effective bondlength is not Re. 3. Potential Eel is not really parabolic, only approximately for low v. For high v, anharmonicity corrections are needed.

Front 272

What is the correction to the rotational energy for real diatomic molecules?

Back 272

EJ = BvJ (J+1) - Dv J^2 (J+1)^2 where the second term is the centrifugal distortion.

Front 273

What is the Morse potential?

Back 273

Vmorse (R) = De [exp -2α(R-Re) - 2exp -α(R-Re)]
where Re = equilibrium bond length and De = potential minimum.

Front 274

What is the expansion of the Morse potential in powers of (R-Re)?

Back 274

Vmorse (R) ~ De (-1 + α^2 (R-Re)^2 +……) For small displacements.

Front 275

How is the Morse potential expansion related to the electronic energy Eel of the ideal diatomic molecule?

Back 275

To 2nd order α = √(k/2De)

Front 276

How is the rotational spectrum of a diatomic molecule affected by its components?

Back 276

Only molecules with a permanent dipole moment can undergo rotation of the oscillating electric field - PURE rotational transitions.

Front 277

What are the selection rules for pure rotational transitions in diatomic molecules?

Back 277

ΔJ = ± 1 where J is the nuclear angular momentum.
Homonuclear diatomic molecules have a weak spectrum of ΔJ= 2 transitions instead.

Front 278

What energy region do pure rotational transitions occur in?

Back 278

The microwave region.

Front 279

What are the frequencies of transition lines in pure rotational spectra?

Back 279

Erot(J) - Erot (J-1)/h = 2BJ/h

Front 280

What energy region do ro-vib transitions occur in?

Back 280

The infra-red region.

Front 281

What are the selection rules for rotational-vibrational transitions in diatomic molecules?

Back 281

Δv = ± 1
|J - J'| = 1

Front 282

What are the spectral lines for ro-vib transitions?

Back 282

There are two branches, the R-branch ΔJ = 1 and the P-branch ΔJ = -1

Front 283

What are the energy transitions for the R-branch in a ro-vib transition?

Back 283

E (v+1, J+1) - E(v, J) = 2BJ + ħω0
J = 0, 1, 2...

Front 284

What are the energy transitions for the P-branch in a ro-vib transition?

Back 284

E (v+1, J-1) - E(v, J) = -2BJ + ħω0
J = 1, 2, 3...

Front 285

What is the missing transition in a ro-vib transition?

Back 285

There is no line at ħω due to ΔJ = 0 being forbidden.

Front 286

What does the line spacing in a ro-vib transition tell us?

Back 286

We can get the rotational constant B and hence the moment of inertia Ie and Re.
The missing line can give us ω0 and hence the spring constant.

Front 287

What defines the eigenfunctions of a diatomic molecule?

Back 287

They have axial symmetry, not spherical, hence the the eigenfunctions are of Hel and Lz.

Front 288

What are the good quantum numbers for electronic transitions in diatomic molecules?

Back 288

Eigenstates of Lz and Hel:
so ML ħ = ± Λ ħ
where Λ is the projection of the TOTAL electronic angular momentum on the internuclear -z axis.

Front 289

What are the angular momentum letters for molecules?

Back 289

Λ is analogous to L
Λ = 0 → Σ
Λ = 1 → Π
Λ = 2 → Δ
Λ = 3 → Φ

Front 290

What is the spectroscopic term notation for a molecule?

Back 290

2S+1 Λ u, g
where u, g is gerade or ungerade and applies to homonuclear molecules only.

Front 291

What are the selection rules for molecular electronic transitions?

Back 291

ΔΛ = 0, ±1
g → u ONLY
ΔS = 0

Front 292

What is the energy difference for an elec-ro-vib transition?

Back 292

The energy difference between electronic levels is much higher than for rotational or vibrational transitions, so they are usually higher energy (UV)

Front 293

What are the selection rules for an elec-ro-vib transition?

Back 293

The electronic part has selection rules but the rotational constant B can differ for the initial and final states.
Hence ΔEJ = [BJ (J+1) + B'J'(J'+1)] does not lead to evenly spaced lines.
Since En, Ev>> Erot this spreads the elec-vib frequency into a band of closely spaced lines.

Front 294

What is the Franck-Condon principle?

Back 294

A change from one vibrational level to another in an elec-vib transition will be more likely to happen if the two vibrational levels overlap significantly.

Front 295

What is an ionic bond in a molecule?

Back 295

One wherein the electron is much more likely to be found near one of the atoms in the molecule.

Front 296

What is electron affinity A?

Back 296

The binding energy of an additional electron, eg F → F-
Electron affinities are negative - energy is released.

Front 297

What is ionization energy I?

Back 297

It is the energy cost of removing an outer electron, eg Na → Na+

Front 298

What is the energy cost of forming a pair of +/- ions from two atoms?

Back 298

The ionisation energy of one + the electron affinity of the other
I(1) + A(2) remembering that A is negative

Front 299

For what nuclear separation R does the Coulomb attraction overcome the energy loss due to ionisation in the formation of an ionic bond?

Back 299

E = 1/R where E is the difference between the ionization energy I and the electron affinity A for the respective atoms.

Front 300

Why is the dissociation energy required to separate an ionic bond slightly higher than that calculated using the bond energy and coulombic attraction term?

Back 300

The electron clouds overlap slightly to add an additional repulsion which raises the energy. There is also a small correction for the zero-point energy.

Front 301

What is the dissociation energy required to separate an ionic bond?

Back 301

E(R) = E∞ - 1/R + be (cR)
where the terms are:
I(1)+A(2) - coulomb repulsion - overlap of electron clouds
b and c are characteristic constants.

Front 302

What is the fractional ionic character of a bond?

Back 302

Fractional ionic character = p/eR0
p = dipole moment of the molecule
R0 is the bond length
p/eR0 = 1 → completely ionic
p/eR0 = 0 → completely covalent.