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302 Cards in this Set

  • Front
  • Back
What is the evidence for atoms?
Dalton ratios
Periodic table
Brownian motion
Kinetic theory
What is the evidence for the existence of the electron?
Faraday: equivalence of matter and electricity
Stoney's unit of electricity
Cathode rays
Describe Thompsons experiment to calculate e/m.
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.
Describe Millikan's oil drop experiment.
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.
What is total collisional cross-section?
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
What is the Beer-Lambert Law and how is it derived?
From dI/I =-nσdx where σ = collisional cross-section, I = intensity.
Integrating over target length L:
I(L) = I0 e ^ (-nLσ)
How is collisional cross-section related to particle charge?
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.
How does collisional cross-section vary with incident energy?
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)
What is positronium?
A bound state of an electron and a postitron orbiting each other.
What is the positronium cross-section at high energy?
The cross-section is the sum of the electron and positron - they interact separately.
What is an elastic collision?
One where the scattered particles change direction.
What is an inelastic collision?
One where the internal energy of the target/incident particles is changed by e.g. excitation.
What is the total inelastic cross-section?
σij is the inelastic cross-section from quantum level i to quantum level j.
The constant of proportionality in dI/I α -ndx.
What is threshold energy?
The minimum energy for an inelastic process to occur, i.e. the amount of energy needed to excite an electron of the target material.
Describe the Frank-Hertz experiment and its findings.
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.
What is the differential cross-section?
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)
What is the formula for differential cross-section?
dN = dσ/dΩ N n l dΩ
where dσ/dΩ = differential cross-section
N = incident flux nl = target density and length
How is differential cross-section related to total cross-section and when is this valid?
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.
What is the Rutherford formula for differential cross section and when is it valid?
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
What is the relationship between dσ/dΩ and θ?
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.
What is the quantum description of scattering?
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.
What are the two postulates of the Bohr model?
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.
When is the Bohr model applicable?
For hydrogen and hydrogen-like atoms. QM is needed for multi-electron atoms.
What are the permitted orbits of the Bohr model?
L = nħ
How are the energies of the Bohr model calculated?
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
What is the equation for transitions between quantum states?
1/λf = R (1/nf^2 - 1/ni^2)
Corresponds to emission of a photon of energy hʋ = Ef-Ei
What is the Lyman series?
nf = 1, ni = 2, 3, 4…
What is the Balmer series?
nf = 2
What is the Paschen series?
nf = 3
What is the α series?
Δn = 1
What is the β series?
Δn = 2
What is the ɣ series?
Δn = 3
What is the correspondence principle?
In the limit of large quantum numbers n QM results should tend to classical results.
What is reduced mass and why is it used?
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)
What are atomic units?
ħ = me = a0 = 4πɛ0 = 1
What is the quantum description of angular momentum?
p = -iħΔ, position operator is r
What is the atomic unit of energy?
Hartree: 1 Ha = 1 au = 2x ionisation energy of hydrogen = 2R∞
What are wavenumbers?
A unit of energy, related to hc/λ and expressed in cm-1
What is the energy difference between two states of a hydrogenic atom?
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.
What is the energy of the nth state?
En = Z^2 R∞/n^2
What is R∞ in wavenumbers?
109737 cm-1
What are the simultaneous eigenfunctions of Lz and L^2?
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.
What is the parity of the spherical harmonics?
P (Ylm (θ, ϕ)) = (-1)^l
What is the magnitude of Lz?
m ħ
What is the magnitude of L^2
l (l+1)
What is the wavefunction for hydrogen?
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.
How does the wavefunction of a one-electron atom vary?
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.
How does the quantum model of angular wavefunction differ from the Bohr radius?
The Bohr radius has a fixed value, QM assigns probability densities to radii.
What is the probability density of finding an e- a distance r from the nucleus?
P = r^2Rnl^2 with n-l peaks.
How does the probability of finding the e- near the nucleus change with l?
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.
What is the average radius of an electronic orbit?
<r> = 3a0/2, due to the long tail in the radial probability density.
What is the dependency of E on n in the quantum model?
En α n^2
E is dependent on n only.
What values of l are allowed for a given n?
l = 0,1,2…. n-1
What values of ml are allowed for a given l?
ml = -l, -l+1, -l+2….l-1, l
What is the total number of energy states with degeneracy with respect to l and ml?
Σ(2l+1) = n^2
Why do single-electron atoms have degeneracy with respect to l and ml?
This is a characteristic of a central potential, and is lifted if the straightforward dependence on 1/r is modified by a small perturbation.
What are the spectroscopic notations for l?
l = 0,1,2,3,4 = s, p, d, f, g
What are the spectroscopic notations for n?
n= 1,2,3,4 = K, L, M, N
What are the transition restrictions for l and ml?
Δl = +/-1 Δm = 0, +/-1
What are the simultaneous eigenfunctions of Sz and S^2?
S^2 (Χsms) = s(s+1)(Χsms) with s = 1/2
Sz (Χsms) = mħ (Χsms) with m= +/- 1/2
What is the total wavefunction for H?
Ψnlm = Rnl (r) Ylm (θ, ϕ) Χ(sms)
Where does spin arise from?
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.
What is a positron?
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.
What is the form of the Hamiltonian for a one-electron atom in atomic units?
H = -1/2Δ^2 -Z/r
What is the form of a many-electron Hamiltonian?
H = Σ h + Σ 1/rij
This is no longer analytically solvable due to the coulombic contributions.
What is the form of the Hamiltonian in the independent particle model?
H = Σ h
the coulombic interaction 1/rij is neglected.
What is the solution of the Hamiltonian in the independent particle model?
N hydrogen-like solutions
E = E1 + E2 +…+ En
What is the deviation in result from the independent particle model?
The calculated energy states are too negative.
What is the form of the Hamiltonian in the central field approximation?
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.
What is the resulting wavefunction for the central field model?
The product of the individual orbitals.
Ψ(r1, r2, .. rn) = ϕ(r1) ϕ(r2)… ϕ(rn)
What is the form of the individual orbitals in the central field approximation?
The angular part is the same as hydrogen. The radial part has been modified by the additional potentials.
When is the central field approximation most useful?
For atoms with a single optically active electron and a bound inner core, where the inner core screens the nucleus.
How is the interaction of an electron affected by the value of l in a multi-electron atom?
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.
How can the energy of a single optically-active electron be described in terms of quantum defect?
Enl = - Zeff^2/(2 (n - Δnl)^2) where Δnl is the quantum defect and describes the departure from hydrogenic behaviour.
What are the characteristics of the quantum defect?
To a good approximation they depend only on l.
They are non-integer.
n-Δnl may be considered a good quantum number.
Which electrons have the smallest quantum defect?
Those with large l - which see the nucleus and inner core electrons as a single positive charge Zeff
What is the alternative to quantum defect?
enl = (Z-σnl)^2/2n^2 where σnl is the screening constant.
How are indistinguishable particles acted upon by the interchange operator and how does it commute with the Hamiltonian?
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
What are the eigenvalues of the interchange operator and what are the consequences of this?
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.
What is the Pauli Exclusion Principle?
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
What are the symmetric and antisymmetric wavefunctions of two non-interacting particles?
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.
What is the total wavefunction for He?
Ψnlm = ϕ(r1, r2) Χ(s1z, s2z)
What is total spin for He and what are the eigenfunctions?
S = s1+s2
Sz = s1z + s2z
X is the eigenfunction of total S and Sz
What are the eigenvalues and eigenfunctions of total spin S?
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.
What are the possible values of Ms for total spin S=1?
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
What are the possible values of Ms for total spin S=0?
Ms = 0 X = 1/√2 (α(1)β(2) - α(2)β(1)) ↑↓ - ↑↓
ANTISYMMETRIC
What are the possible total wavefunctions for He?
Space symmetric, spin antisymmetric SINGLET STATE
Space antisymmetric, spin symmetric TRIPLET STATE
Is the ground state of He a singlet or triplet state?
It is a singlet state. Since the spatial wavefunction has the same quantum numbers for both electrons 1S2 the spin must be antisymmetric.
Are excited states of He singlet or triplet states?
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.
What is the value of spin multiplicity for He?
Using TOTAL spin S
S(S+1) = total number of spin states.
For S = 0 multiplicity = 1
For S = 1 multiplicity = 3
What is the exchange force?
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.
What is the qualitative explanation for the exchange interaction?
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.
How is the Hamiltonian for He modified by the exchange force?
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.
Summarise specroscopic notation for electrons.
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.
What defines a shell?
A shell is has the same n.
What defines a subshell?
A subshell has the same n and l, also called an orbital.
What is an equivalent electron?
It has the same nl.
What are optically active electrons?
Electrons in an open shell, i.e. in a shell which contains less than 2n^2 electrons.
What is ionization energy?
It is the energy required to remove one valence electron. It is minimised for elements with only one valence electron.
What are the good quantum numbers for a non-central field model?
Total angular momentum L
Total spin S
The electronic states are simultaneous eigenfunctions of H, L^2 and S^2
What are the L and S terms for a multi-electron atom?
For closed shells S=L=0, only valence electrons are considered
2S+1 L
What are the possible values of S and L for a two-electron atom?
S = |s1-s2|……… s1+s2 in integer steps
L = |l1-l2| …… l1+l2 in integer steps.
What are the possible configurations for equivalent electrons (same nl)?
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.
How can electronic configurations be found more simply?
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.
What do Hund's rules show?
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.
What are Hund's rules including L/S coupling?
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.
How is energy ordering related physically to the values of S and L?
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.
What is the atomic Hamiltonian including spin-orbit interaction?
HT = H + HLS
What is the origin of the spin-orbit interaction?
The intrinsic magnetic moment caused by spin or orbital angular momentum.
What is the magnetic moment due to an angular momentum X?
μx = gx q/2M X where gx is the Lande g-factor.
What is the torque of a magnetic moment in a uniform magnetic field?
Ƭ = dX/dt = μx x B = gx Q/2M X x B
Hence Ƭ = - ω x X where ω is the Larmor precession frequency.
What is the effect of the torque Ƭ due to the magnetic moment on the angular momentum X?
Ƭ is perpendicular to X and B and results in a precession of X around B
What is the Larmor precession frequency?
ω = gx Q/2M B
What is the interaction potential energy of a magnetic moment placed in a magnetic field?
V = - μx . B
What is the magnetic moment of an electron due to L?
μL = -gL μB L/ħ
gL =1
What is the magnetic moment of an electron due to S?
μS = -gS μB S/ħ
gS =2
What do the values of gL and gS tell us about the electron?
That it is a structureless fermion. Other fermions (e.g the proton) do not have these values and hence are not structureless.
What is the energy of the spin-orbit interaction?
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∞
What is the effect of spin-orbit coupling on quantum numbers?
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.
How is total angular momentum J and Jz defined?
J = L+S
Jz = Lz + Sz
What is the effect of spin-orbit coupling on energy levels?
They are no longer eigenstates of Lz and Sz but are eigenstates of J^2, Jz, L^2 and S^2.
What is the derivation of energy levels for spin-orbit splitting?
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)]
What is the Lande Interval Rule?
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.
What are the eigenvalues of J^2 and Jz?
J^2 Ψ = j(J+1) Ψ
Jz Ψ = m ħ Ψ
What are the allowed values of total angular momentum j?
j = |l-s|…….. l+s in integer steps
What are the allowed values of total angular momentum magnetic quantum number mj for single electrons?
mj = -j, -j+1 …… j-1, j in integer steps.
What are the good quantum numbers including spin-orbit coupling for a single electron?
n, l, j, mj
What is the spectroscopic term notation for energy levels including spin-orbit coupling for a single electron?
n 2s+1 Lj
What is the selection rule for transitions between j states for a single electron?
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.
What are the two methods for calculating J for the multi-electron case and when are they applied?
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.
What is the Russell-Saunders term for low-Z atoms?
Combine individual spins to get total spin S, similarly for total L.
Then, J = |L-S|……. |L+S| in integer steps.
What is the third Hund's rule with respect to J-values?
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.
What is the jj-coupling term for high Z atoms?
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.
What is parity?
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
What is an electric dipole moment?
P = er
What is the Hamiltonian for a one-electron atom exposed to an oscillating electric field?
H = H0 - P.E where E(t) = E0 exp (iωt)
What is the time-dependent perturbation?
V = P.E with z component V = er cosθ E0 exp(iωt)
What is Fermi's golden rule?
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.
From what part of a wavefunction is the transition probability Tif formed?
The radial part is constant and does not yield any selection rule. The angular part (θ, ϕ) yield selection rules.
What are the electric dipole selection rules?
Δl = +/-1
Δml = 0, +/-1
Δs=0
initial and final states have opposite parity.
How are the selection rules gained from Fermi's golden rule?
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.
How does the electric dipole selection rule differ if spin-orbit coupling is significant?
Selection rules apply instead to j
Δj = 0, +/- 1 i.e. with a transition from j=0 to j'=0 forbidden
What are the selection rules for multi-electron atoms?
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.
What are the 3 ways by which an atom can interact with a radiation field?
Absorption, sponteneous emission and stimulated emission.
What is the rate of absorption when an atom interacts with a radiation field?
The rate of absorption is proportional to the population of the bround state and energy density α N1 U(ʋ12)
What is the rate of decay in spontaneous emission?
Rate of decay is α the excited state population N2
What is the rate of emission in stimulated decay?
Rate of emission is α the excited state population and the energy density N2 U(ʋ12)
What is stimulated emission?
A photo causes an excited atom to decay, emitting a photon that is coherent with the stimulating photon. (same phase and amplitude)
What are the constants of proportionality for interaction of an atom in a radiation field?
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.
What is the probability of a spontaneous decay?
Pif = Aif
What is the lifetime of a spontaneous decay?
Obtained by summing over all possible decay paths:
Δƭi = 1/ Σaif
What is the energy uncertainty for a given decay?
From Heisenberg:
ΔƬΔE > ħ/2
What is the intrinsic energy width for a decay?
ΔE ~ ħ/2ΔƬ leading to spectra frequency width Δʋ - the natural line-width.
What is a metastable energy level?
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.
What are the populations of excited states to lower states in two-level systems?
A Boltzmann distribution:
N1/N2 = exp (E2-E1)/kT
There are more atoms in state 1 than state 2
What is the most likely radiation interaction in a normal two-level system?
Absorption is most likely, since there are more atoms in lower states than in excited states, normally.
What is population inversion?
Population inversion is when there are more atoms in an excited state than in a lower state.
What is the most likely radiation interaction in a population inverted system?
Spontaneous or stimulated emission is likely in population inversion as there are more atoms in the excited state than in the ground state.
Why can population inversion not be attained in a two-level system?
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.
How is population inversion attained in a 3-level system?
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.
How can a laser be used to cool atoms?
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
How are X-ray spectra produced?
By bombardment of inner electrons with high energy electrons from a cathode, with kinetic energy eV where V is the accelerating voltage.
How do X-ray spectra differ from normal atomic spectra?
They are transitions of inner tightly bound electrons rather than outer optical electrons.
What are the two features of an X-ray spectrum?
A continuous spectrum - Brehmsstrahlung - and characteristic X-ray spectra peaks.
What is the Brehmsstrahlung spectrum of an X-ray spectrum?
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
What are the characteristic X-ray spectra peaks?
They are produced by transitions of inner electrons, and then decay of higher electrons into the holes left behind and emission of a photon.
What is the frequency of X-ray emission?
ʋif = (Ei - Ef)/ 2 = R Zeff^2 (1/nf^2 + 1/ni^2) where Zeff = Z-S and S is the screening constant.
What are transitions to different final states nf denoted as in X-ray spectra?
nf = 1: K-series
nf = 2: L-series
nf = 3: M-series.
What are the transition differences Δn denoted as in X-ray spectra?
Δn = 1: α
Δn = 2: β
Δn = 3: ɣ
What is the full Hamiltonian for multi-electron atoms in external fields?
H = H0 + HLS + VB + VE
What is the perturbative case of interactions between atoms and internal/external fields?
H0>> VB, VE, HLS
What is the Zeeman effect?
Splitting of spectral lines in a magnetic field.
What is the weak field Zeeman effect?
When VB<< HLS
When does the normal Zeeman effect occur?
For total spin S=0 and the magnetic moment is entirely due to the orbital angular momentum.
What is the potential energy relating to a magnetic moment?
V= μx.B where X is any angular momentum.
μx = -gx μB X/ħ
What is the potential of a magnetic field along the Z-direction for the normal Zeeman effect and what is the consequence?
V = μB/ħ L.B = BzμB Bz ml
Hence the spectral line due to l is split into 2l+1 equally spaced levels.
What do the dipole selection rules tell us about the weak normal Zeeman effect?
The spectral line is split into 3
l => 2l+1 => ml => 3 transitions allowed since Δl = +/-1
What is the effect of the anomalous Zeeman effect for a weak field?
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.
What is the interaction potential energy for the weak field anomalous Zeeman effect?
This must be expressed in good quantum numbers j, l, s, mj
VB = μB/ħ B.(L+2S) - μB B.(J+S)
How is the energy splitting for the weak field anomalous Zeeman effect derived?
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.
What is the effect of the weak field anomalous Zeeman effect?
The degeneracy with respect to mj is lifted - there is a spectral line for each value of mj.
What is the strong field Zeeman effect and what is it called?
VB>> HLS - the Paschen-Beck limit.
What is the effect of a strong magnetic field on an atom (the Paschen-Beck limit)?
L and S decouple and both precess independently about the magnetic B-field direction. J is not constant.
What is the torque of a magnetic moment in a uniform magnetic field?
Ƭ α L x B
What is the total magnetic moment in the strong field Paschen-Beck limit?
μ = μL + μS
What are the good quantum numbers in the Paschen-Beck limit?
l, s, ml and ms.
What is the potential energy in the Paschen-Beck limit (strong field)?
VB = -μ.B = μB/ħ B/ (L+2S) = μB/ħ B/ (Lz+2Sz)
What is the energy splitting in the Paschen-Beck limit?
ΔE = μB B (ML+2MS)
What are the projections of L and S on the Z-axis in the Paschen-Beck limit?
mlħ and msħ
BOTH CONSTANT
What effect does the Paschen-Beck limit have on the spectral lines?
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.
What are the possible spectral splits due to the Zeeman effect?
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.
What is the potential and the force in the Stern-Gerlach experiment?
VB = -μS.B = 2μB B.S
F = -dV/dz = -2Sz μB dB/dz
What does the Stern-Gerlach experiment show?
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.
What does the Stern-Gerlach experiment show with regards to atoms?
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.
What is the nuclear spin of an atom?
It is the net spin of the protons and neutrons, given by I.
What is the magnetic moment of the nuclear spin I?
μI = gN μN I/ħ where μN is the nuclear Bohr magneton.
What are the hyperfine sublevels?
F = J + I where J is (S+L) and I is the nuclear net spin.
It is analogous to spin-orbit coupling.
What is the characteristic of hyperfine sublevels?
They are very tiny with small energy differences.
What is the Stark effect?
The splitting of energy levels of an atom in static electric fields.
What is the field strength assumption for the Stark effect?
VE<< Z/r
What is the perturbation to the Hamiltonian for the Stark effect?
VE = - μ.E
What is the potential along the Z-axis in the Stark effect?
VE = - (-er.E) = ez.E
What is the effect of the potential in the Stark effect?
The potential VE is positive - ie repulsive, and it decreases the binding energy.
When does the quadratic Stark effect occur?
It occurs in atoms which have no intrinsic dipole moment.
What is the effect of a an electric field in an atom with no intrinsic dipole moment?
The field polarises the electron distribution, inducing a separation of charge and dipole moment which is proportional to E
What is the interaction energy for the quadratic Stark effect?
VE = -1/2 α E^2
a term varying quadratially with the field, where α is the dipole polarisability.
What is the magnitude of the induced dipole moment of an atom in an electric field?
μ = dVE/dE = αE
When does the linear Stark effect occur?
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.
What is the linear Stark effect?
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.
What is the Hamiltonian for a molecule?
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.
What is the coulombic potential term for a molecule?
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.
Why is the Born-Oppenheimer approximation needed?
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.
What is the starting assumption of the Born-Oppenheimer approximation?
That the nuclei move much more slowly than electrons and the Hamiltonian can be written using the centre of mass motion.
What is the Hamiltonian for a diatomic molecule at the centre of mass?
H (R, r) = -ħ^2/2μ ΔR^2 + Σ-ħ^2/wm Δi^2 + V(R, ri)
What are the 6 steps of the Born-Oppenheimer approximation?
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.
What is the consequence of the Born-Oppenheimer approximation?
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.
What is the probability of the electron being around nucleus 1 or nucleus 2 in H2+?
|C1|^2 = |C2|^2
equal probability.
What is the electronic Hamiltonian Hel for H2+?
Hel = -1/2 Δr^2 -1/rA -1/rB +1/R
What are the wavefunctions for H2+ in terms of the coefficients C1 C2?
C1 = +/- C2
There is a symmetric (gerade) and antisymmetric (ungerade) wavefunction.
Ψ± = C [ϕ1s(rA) ± ϕ1s(rB)]
What is LCAO?
Linear Comination of Atomic Orbitals
Building a molecular wavefunction from a superposition of atomic orbitals.
What do the gerade and ungerade wavefunctions represent?
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.
What is the electronic energy for H2+?
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.
What is the normalised expectation value of the Hamiltonian for a diatomic molecule?
<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.
What is the normalisation constant for the electronic energy for H2+?
N is given by 1 ± I(R)
I(R) = <ϕA|ϕB> - the overlap integral between the two electronic/nuclear wavefunction.
What is the expectation value of the electronic Hamiltonian, A±?
A± = 1/2 [HAA + HBB] ± HAB
What is the HAA or HBB term of the electronic Hamiltonian for H2+?
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.
What is the HAB term of the electronic Hamiltonian for H2+?
HAB = ± (E1s +1/R) I(R) Ŧ K(R)
Where the terms are:
energy of H atom and nuclear repulsion - overlap integral - exchange integral
What are the terms, I, J and K in the Hamiltonian of H2+?
I = overlap integral = (1+R+ R^2/3) exp -R
J = coulomb integral = -exp -2R
K = exchange integral = (1+R) exp -R
What is the characteristic of the overlap integral I in the molecular Hamiltonian?
0 < I < 1 hence 1 ± I > 0 and 1 ± I^2 >0
What is R in molecular forms?
The nuclear separation (for diatomic molecules)
How does the energy of H2+ change as R→∞?
As R→∞ J, K, I → 0 so E- ~ E+ → E1s +1/R
What effect does the splitting have on the molecular energy of H2+?
The splitting term 2 K(R) / (1 ± I) lowers E+ and raises E-
How does the energy of H2+ change as R→0?
As R→0 E± → ∞ because of the repulsion between nuclei.
What are the characteristics of E± which lead to bonding or anti-bonding orbitals?
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.
What is the bonding orbital operator equation for H2+?
Hel Ψ+ = E+ Ψ+ where H is the electronic Hamiltonian and E is the electronic energy.
What is the antibonding orbital operator equation for H2+?
Hel Ψ- = E- Ψ- where H is the electronic Hamiltonian and E is the electronic energy.
What is De and Re for a stable molecule?
De = disassociation energy
Re = Equilibrium distance of nuclei.
How does the wavefunction for H2+ differ from that of H2?
H2 has two electrons so the Pauli Exclusion Principle must be considered.
What is the electronic Hamiltonian Hel for H2?
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.
What is the possible spin for H2?
S = 0 or S = 1 as there are 2 electrons. The spin functions are the same as for He.
What condition is placed on the wavefunction for H2?
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.
What is the spin triplet state for H2?
ΨT ~ 1/√2 [ϕ1s(rA1)ϕ1s(rb2) - ϕ1s(rA2)ϕ1s(rB2)] XT
An antisymmetric superposition of the 1s wavefunctions.
What is the spin singlet state for H2?
ΨS~ 1/√2 [ϕ1s(rA1)ϕ1s(rb2) + ϕ1s(rA2)ϕ1s(rB2)] XS
A symmetric superposition of the 1s wavefunctions.
What is the difference between H2 wavefunctions and He wavefunctions?
The H2 wavefunctions are approximate, derived from the Born-Oppenheimer approximation. They are not exactly the same as the He wavefunctions.
Which energy eigenfunctions correspond to the energy eigenvalues of H2?
E+ → ΨS → symmetric space
E- → ΨT → antisymmetric space
What is the normalised expectation value of the Hamiltonian for H2?
E± = <Ψ(S/T) |Hel| Ψ(S/T)> / <Ψ(S/T) | Ψ(S/T)>
What is the energy of H2, accounting for the triplet and singlet states?
E± = 2E1s + 1/R + J/(1±I^2) ± K/(1±I^2)
where + = spin singlet
symmetric space part, - = spin triplet, antisymmetric space part.
What are the terms for the energy of H2?
Two hydrogen atoms - nuclear repulsion - overlap/coulomb/exchange integral composite terms.
What is the characteristic of the coulomb integral J in the molecular Hamiltonian?
J is a positive contribution to the energy.
What is the characteristic of the exchange integral K in the molecular Hamiltonian for H2?
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.
What are the relative energies of the singlet and triplet states for He?
Singlet - HIGHER
Triplet - LOWER
What are the relative energies of the singlet and triplet states for H2?
Singlet - LOWER
Triplet - HIGHER
How do the energy levels of H2 compare with those of He?
In He the triplet state is lower in energy than the spin singlet. In H2, the triplet state is higher in energy.
Why do the singlet and triplet energy levels for He and H2 vary?
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.
What are the degrees of freedom available to an N-nuclei molecule?
3N
What are the three types of motion for a molecule?
Translation, rotation, vibration.
How many degrees of freedom does translational motion have?
3 - translation in 3 dimensions.
How many degrees of freedom does rotational motion have?
Rotation about the centre of mass-
3 DoF for a solid body
2 DoF for a linear molecule.
How many degrees of freedom does vibrational motion have?
Vibration around the equilibrium position
3N - 6 DoF for a solid body
3n - 5 DoF for a linear molecule.
What do spherical harmonics describe in relation to a diatomic molecule?
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.
What is the moment of inertia of a diatomic molecule?
If the centre of mass is at the origin of co-ordinates then Ie = μRe^2
What are the eigenvalues associated with rotational motion of a diatomic molecule?
Erot = BJ (J+1) where B is the rotational constant = ħ^2/2Ie.
If J =0 then Erot = 0
What is the angular frequency of rotation of a diatomic molecule?
ωrot α 1/μ
What are the eigenstates of vibrational motion?
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.
What is the significance of the eigenvalues of vibrational motion?
Eel has a minimum at R = Re. The nuclear motion is generally well confied to a small region around Re.
How is the QHO term of vibrational motion obtained from the electronic eigenstates of a diatomic molecule?
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.
What is the vibrational energy for a diatomic molecule in the QHO formulation?
Ev = ħω (v +1/2) with ω = √k/μ and v = vibrational number 1, 2, 3….
What is the zero-point energy of the vibrational energy of a diatomic molecule?
E = 1/2 ħω
What is the total energy sum of an ideal diatomic molecule?
E ~ Eel + BJ (J+1) + (v+1/2)ħω
equilibrium electronic energy + rotational energy + vibrational energy
What are the assumptions made in characterising an ideal diatomic molecule?
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.
What are the corrections to idealness for a real diatomic molecule?
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.
What is the correction to the rotational energy for real diatomic molecules?
EJ = BvJ (J+1) - Dv J^2 (J+1)^2 where the second term is the centrifugal distortion.
What is the Morse potential?
Vmorse (R) = De [exp -2α(R-Re) - 2exp -α(R-Re)]
where Re = equilibrium bond length and De = potential minimum.
What is the expansion of the Morse potential in powers of (R-Re)?
Vmorse (R) ~ De (-1 + α^2 (R-Re)^2 +……) For small displacements.
How is the Morse potential expansion related to the electronic energy Eel of the ideal diatomic molecule?
To 2nd order α = √(k/2De)
How is the rotational spectrum of a diatomic molecule affected by its components?
Only molecules with a permanent dipole moment can undergo rotation of the oscillating electric field - PURE rotational transitions.
What are the selection rules for pure rotational transitions in diatomic molecules?
ΔJ = ± 1 where J is the nuclear angular momentum.
Homonuclear diatomic molecules have a weak spectrum of ΔJ= 2 transitions instead.
What energy region do pure rotational transitions occur in?
The microwave region.
What are the frequencies of transition lines in pure rotational spectra?
Erot(J) - Erot (J-1)/h = 2BJ/h
What energy region do ro-vib transitions occur in?
The infra-red region.
What are the selection rules for rotational-vibrational transitions in diatomic molecules?
Δv = ± 1
|J - J'| = 1
What are the spectral lines for ro-vib transitions?
There are two branches, the R-branch ΔJ = 1 and the P-branch ΔJ = -1
What are the energy transitions for the R-branch in a ro-vib transition?
E (v+1, J+1) - E(v, J) = 2BJ + ħω0
J = 0, 1, 2...
What are the energy transitions for the P-branch in a ro-vib transition?
E (v+1, J-1) - E(v, J) = -2BJ + ħω0
J = 1, 2, 3...
What is the missing transition in a ro-vib transition?
There is no line at ħω due to ΔJ = 0 being forbidden.
What does the line spacing in a ro-vib transition tell us?
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.
What defines the eigenfunctions of a diatomic molecule?
They have axial symmetry, not spherical, hence the the eigenfunctions are of Hel and Lz.
What are the good quantum numbers for electronic transitions in diatomic molecules?
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.
What are the angular momentum letters for molecules?
Λ is analogous to L
Λ = 0 → Σ
Λ = 1 → Π
Λ = 2 → Δ
Λ = 3 → Φ
What is the spectroscopic term notation for a molecule?
2S+1 Λ u, g
where u, g is gerade or ungerade and applies to homonuclear molecules only.
What are the selection rules for molecular electronic transitions?
ΔΛ = 0, ±1
g → u ONLY
ΔS = 0
What is the energy difference for an elec-ro-vib transition?
The energy difference between electronic levels is much higher than for rotational or vibrational transitions, so they are usually higher energy (UV)
What are the selection rules for an elec-ro-vib transition?
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.
What is the Franck-Condon principle?
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.
What is an ionic bond in a molecule?
One wherein the electron is much more likely to be found near one of the atoms in the molecule.
What is electron affinity A?
The binding energy of an additional electron, eg F → F-
Electron affinities are negative - energy is released.
What is ionization energy I?
It is the energy cost of removing an outer electron, eg Na → Na+
What is the energy cost of forming a pair of +/- ions from two atoms?
The ionisation energy of one + the electron affinity of the other
I(1) + A(2) remembering that A is negative
For what nuclear separation R does the Coulomb attraction overcome the energy loss due to ionisation in the formation of an ionic bond?
E = 1/R where E is the difference between the ionization energy I and the electron affinity A for the respective atoms.
Why is the dissociation energy required to separate an ionic bond slightly higher than that calculated using the bond energy and coulombic attraction term?
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.
What is the dissociation energy required to separate an ionic bond?
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.
What is the fractional ionic character of a bond?
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.