Chapter 1 Some Important Results in Quantum Mechanics

11 Spectroscopy and Quantum Mechanics

Molecular spectroscopy ndashan experimental subject which concerns with the absorption emission or scattering of electromagnetic (EM) radiation by moleculesQuantum mechanics (QM) ndasha theoretical subject which relates to many aspects of chemistry physics but particularly spectroscopy12 The Evolution of Quantum Theory

In 1885 Balmer used the following empirical formula to fit the emission spectrum of H atom

λ = (nrsquo2G) (nrsquo2 ndash 4) Eq (11)wbt 1where G is a constant and nrsquo = 345hellip

wbt 2

Energy levels and observed transition of the H atom

ν = RH [(1nrdquo2) - (1nrsquo2)]

RH = Rydberg constant

n = 1 Lyman seriesn = 2 Balmern = 3 Paschenn = 4 Brackettn = 5 Pfund

wbt 3

Define wavenumber = 1λν~ Eq (12)

where ν = c λ Eq (13)

Eq (11) may be written asEq (14)ν = RH [(122) - (1nrsquo2)]

= RH [(1nrdquo2) - (1nrsquo2)] nrdquo = 2

RH = Rydberg constantWhen expressing in wavenumber RH = 10967758306 x 105 cm-1

Photoelectric effect ndash (discovered by Hertz 1887)When UV light falls on an alkali metal surface electrons are ejected from the surface only when the frequency (ν) of the radiation reaches the threshold frequency νt for the metal As the frequency is increased the kinetic energy (ke) of the ejected electrons (photoelectrons) increases linearly with ν

Photoelectric effect

wbt 4

In 1900 Planck proposed that the microscopic oscillators have an oscillation frequency ν related to the energy E of the emitted radiation by

E = nhν Eq (15)

h = Planck constant = 6626 x 10-34 Js Eq (16)The energy E is said to be quantized in discrete packets (quanta) each of energy hν In 1906 Einstein applied Planckrsquos theory to the photoelectric effect and showed that

hν = (12)mev2 + I Eq (17)

where hν is the energy associated with a quantum (photon) of the incident radiation (12)mev2 is the ke of the ejected photoelectron ejected with velocity v and I is the ionization energy (work function) of the metal surface

wbt 5

In 1913 Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that theangular momentum pθ for an angle of rotation θ is given by

pθ = nh (2π) Eq (18)where n = 1 2 3 and defines the particular orbit Energy is emitted or absorbed when the electron moves from an orbit with higher n to one of lower n or vice versa The energy En of the electron can be shown by classical mechanics (CM) to be

En = minus[microe4 (8h2ε02)](1n2) Eq (19)

where micro = (memp)(me+mp) is the reduced mass of the system of electron e plus proton p e is the electronic charge and ε0 the permittivity of a vacuum

wbt 6

When the electron transfers from a lower nrdquo to an upper nrsquo orbit the energy ∆E required is [from Eq (19)]

∆En = minus[microe4 (8h2ε02)](1nrdquo2 - 1nrsquo2) Eq (110)

Since ∆En = hν Eq (110) becomes

ν = minus[microe4 (8h3ε02)](1nrdquo2 - 1nrsquo2) Eq (111)

Comparing Eqs (111) and (14) one

RH = [microe4 (8h3ε02)] (in unit of s-1)

When expressing in wavenumber Rydberg constant RH = [microe4 (8h3ε0

2c)] = 10967758306 x 105 cm-1

= 10967758306 x 107 m-1 Eq (112)

wbt 7

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 2

Energy levels and observed transition of the H atom

ν = RH [(1nrdquo2) - (1nrsquo2)]

RH = Rydberg constant

n = 1 Lyman seriesn = 2 Balmern = 3 Paschenn = 4 Brackettn = 5 Pfund

wbt 3

Define wavenumber = 1λν~ Eq (12)

where ν = c λ Eq (13)

Eq (11) may be written asEq (14)ν = RH [(122) - (1nrsquo2)]

= RH [(1nrdquo2) - (1nrsquo2)] nrdquo = 2

RH = Rydberg constantWhen expressing in wavenumber RH = 10967758306 x 105 cm-1

Photoelectric effect ndash (discovered by Hertz 1887)When UV light falls on an alkali metal surface electrons are ejected from the surface only when the frequency (ν) of the radiation reaches the threshold frequency νt for the metal As the frequency is increased the kinetic energy (ke) of the ejected electrons (photoelectrons) increases linearly with ν

Photoelectric effect

wbt 4

In 1900 Planck proposed that the microscopic oscillators have an oscillation frequency ν related to the energy E of the emitted radiation by

E = nhν Eq (15)

h = Planck constant = 6626 x 10-34 Js Eq (16)The energy E is said to be quantized in discrete packets (quanta) each of energy hν In 1906 Einstein applied Planckrsquos theory to the photoelectric effect and showed that

hν = (12)mev2 + I Eq (17)

where hν is the energy associated with a quantum (photon) of the incident radiation (12)mev2 is the ke of the ejected photoelectron ejected with velocity v and I is the ionization energy (work function) of the metal surface

wbt 5

In 1913 Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that theangular momentum pθ for an angle of rotation θ is given by

pθ = nh (2π) Eq (18)where n = 1 2 3 and defines the particular orbit Energy is emitted or absorbed when the electron moves from an orbit with higher n to one of lower n or vice versa The energy En of the electron can be shown by classical mechanics (CM) to be

En = minus[microe4 (8h2ε02)](1n2) Eq (19)

where micro = (memp)(me+mp) is the reduced mass of the system of electron e plus proton p e is the electronic charge and ε0 the permittivity of a vacuum

wbt 6

When the electron transfers from a lower nrdquo to an upper nrsquo orbit the energy ∆E required is [from Eq (19)]

∆En = minus[microe4 (8h2ε02)](1nrdquo2 - 1nrsquo2) Eq (110)

Since ∆En = hν Eq (110) becomes

ν = minus[microe4 (8h3ε02)](1nrdquo2 - 1nrsquo2) Eq (111)

Comparing Eqs (111) and (14) one

RH = [microe4 (8h3ε02)] (in unit of s-1)

When expressing in wavenumber Rydberg constant RH = [microe4 (8h3ε0

2c)] = 10967758306 x 105 cm-1

= 10967758306 x 107 m-1 Eq (112)

wbt 7

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 3

Define wavenumber = 1λν~ Eq (12)

where ν = c λ Eq (13)

Eq (11) may be written asEq (14)ν = RH [(122) - (1nrsquo2)]

= RH [(1nrdquo2) - (1nrsquo2)] nrdquo = 2

RH = Rydberg constantWhen expressing in wavenumber RH = 10967758306 x 105 cm-1

Photoelectric effect ndash (discovered by Hertz 1887)When UV light falls on an alkali metal surface electrons are ejected from the surface only when the frequency (ν) of the radiation reaches the threshold frequency νt for the metal As the frequency is increased the kinetic energy (ke) of the ejected electrons (photoelectrons) increases linearly with ν

Photoelectric effect

wbt 4

In 1900 Planck proposed that the microscopic oscillators have an oscillation frequency ν related to the energy E of the emitted radiation by

E = nhν Eq (15)

h = Planck constant = 6626 x 10-34 Js Eq (16)The energy E is said to be quantized in discrete packets (quanta) each of energy hν In 1906 Einstein applied Planckrsquos theory to the photoelectric effect and showed that

hν = (12)mev2 + I Eq (17)

where hν is the energy associated with a quantum (photon) of the incident radiation (12)mev2 is the ke of the ejected photoelectron ejected with velocity v and I is the ionization energy (work function) of the metal surface

wbt 5

In 1913 Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that theangular momentum pθ for an angle of rotation θ is given by

pθ = nh (2π) Eq (18)where n = 1 2 3 and defines the particular orbit Energy is emitted or absorbed when the electron moves from an orbit with higher n to one of lower n or vice versa The energy En of the electron can be shown by classical mechanics (CM) to be

En = minus[microe4 (8h2ε02)](1n2) Eq (19)

where micro = (memp)(me+mp) is the reduced mass of the system of electron e plus proton p e is the electronic charge and ε0 the permittivity of a vacuum

wbt 6

When the electron transfers from a lower nrdquo to an upper nrsquo orbit the energy ∆E required is [from Eq (19)]

∆En = minus[microe4 (8h2ε02)](1nrdquo2 - 1nrsquo2) Eq (110)

Since ∆En = hν Eq (110) becomes

ν = minus[microe4 (8h3ε02)](1nrdquo2 - 1nrsquo2) Eq (111)

Comparing Eqs (111) and (14) one

RH = [microe4 (8h3ε02)] (in unit of s-1)

When expressing in wavenumber Rydberg constant RH = [microe4 (8h3ε0

2c)] = 10967758306 x 105 cm-1

= 10967758306 x 107 m-1 Eq (112)

wbt 7

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

Photoelectric effect

wbt 4

In 1900 Planck proposed that the microscopic oscillators have an oscillation frequency ν related to the energy E of the emitted radiation by

E = nhν Eq (15)

h = Planck constant = 6626 x 10-34 Js Eq (16)The energy E is said to be quantized in discrete packets (quanta) each of energy hν In 1906 Einstein applied Planckrsquos theory to the photoelectric effect and showed that

hν = (12)mev2 + I Eq (17)

where hν is the energy associated with a quantum (photon) of the incident radiation (12)mev2 is the ke of the ejected photoelectron ejected with velocity v and I is the ionization energy (work function) of the metal surface

wbt 5

In 1913 Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that theangular momentum pθ for an angle of rotation θ is given by

pθ = nh (2π) Eq (18)where n = 1 2 3 and defines the particular orbit Energy is emitted or absorbed when the electron moves from an orbit with higher n to one of lower n or vice versa The energy En of the electron can be shown by classical mechanics (CM) to be

En = minus[microe4 (8h2ε02)](1n2) Eq (19)

where micro = (memp)(me+mp) is the reduced mass of the system of electron e plus proton p e is the electronic charge and ε0 the permittivity of a vacuum

wbt 6

When the electron transfers from a lower nrdquo to an upper nrsquo orbit the energy ∆E required is [from Eq (19)]

∆En = minus[microe4 (8h2ε02)](1nrdquo2 - 1nrsquo2) Eq (110)

Since ∆En = hν Eq (110) becomes

ν = minus[microe4 (8h3ε02)](1nrdquo2 - 1nrsquo2) Eq (111)

Comparing Eqs (111) and (14) one

RH = [microe4 (8h3ε02)] (in unit of s-1)

When expressing in wavenumber Rydberg constant RH = [microe4 (8h3ε0

2c)] = 10967758306 x 105 cm-1

= 10967758306 x 107 m-1 Eq (112)

wbt 7

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

In 1900 Planck proposed that the microscopic oscillators have an oscillation frequency ν related to the energy E of the emitted radiation by

E = nhν Eq (15)

h = Planck constant = 6626 x 10-34 Js Eq (16)The energy E is said to be quantized in discrete packets (quanta) each of energy hν In 1906 Einstein applied Planckrsquos theory to the photoelectric effect and showed that

hν = (12)mev2 + I Eq (17)

where hν is the energy associated with a quantum (photon) of the incident radiation (12)mev2 is the ke of the ejected photoelectron ejected with velocity v and I is the ionization energy (work function) of the metal surface

wbt 5

In 1913 Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that theangular momentum pθ for an angle of rotation θ is given by

pθ = nh (2π) Eq (18)where n = 1 2 3 and defines the particular orbit Energy is emitted or absorbed when the electron moves from an orbit with higher n to one of lower n or vice versa The energy En of the electron can be shown by classical mechanics (CM) to be

En = minus[microe4 (8h2ε02)](1n2) Eq (19)

where micro = (memp)(me+mp) is the reduced mass of the system of electron e plus proton p e is the electronic charge and ε0 the permittivity of a vacuum

wbt 6

When the electron transfers from a lower nrdquo to an upper nrsquo orbit the energy ∆E required is [from Eq (19)]

∆En = minus[microe4 (8h2ε02)](1nrdquo2 - 1nrsquo2) Eq (110)

Since ∆En = hν Eq (110) becomes

ν = minus[microe4 (8h3ε02)](1nrdquo2 - 1nrsquo2) Eq (111)

Comparing Eqs (111) and (14) one

RH = [microe4 (8h3ε02)] (in unit of s-1)

When expressing in wavenumber Rydberg constant RH = [microe4 (8h3ε0

2c)] = 10967758306 x 105 cm-1

= 10967758306 x 107 m-1 Eq (112)

wbt 7

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

In 1913 Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that theangular momentum pθ for an angle of rotation θ is given by

pθ = nh (2π) Eq (18)where n = 1 2 3 and defines the particular orbit Energy is emitted or absorbed when the electron moves from an orbit with higher n to one of lower n or vice versa The energy En of the electron can be shown by classical mechanics (CM) to be

En = minus[microe4 (8h2ε02)](1n2) Eq (19)

where micro = (memp)(me+mp) is the reduced mass of the system of electron e plus proton p e is the electronic charge and ε0 the permittivity of a vacuum

wbt 6

When the electron transfers from a lower nrdquo to an upper nrsquo orbit the energy ∆E required is [from Eq (19)]

∆En = minus[microe4 (8h2ε02)](1nrdquo2 - 1nrsquo2) Eq (110)

Since ∆En = hν Eq (110) becomes

ν = minus[microe4 (8h3ε02)](1nrdquo2 - 1nrsquo2) Eq (111)

Comparing Eqs (111) and (14) one

RH = [microe4 (8h3ε02)] (in unit of s-1)

When expressing in wavenumber Rydberg constant RH = [microe4 (8h3ε0

2c)] = 10967758306 x 105 cm-1

= 10967758306 x 107 m-1 Eq (112)

wbt 7

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

When the electron transfers from a lower nrdquo to an upper nrsquo orbit the energy ∆E required is [from Eq (19)]

∆En = minus[microe4 (8h2ε02)](1nrdquo2 - 1nrsquo2) Eq (110)

Since ∆En = hν Eq (110) becomes

ν = minus[microe4 (8h3ε02)](1nrdquo2 - 1nrsquo2) Eq (111)

Comparing Eqs (111) and (14) one

RH = [microe4 (8h3ε02)] (in unit of s-1)

When expressing in wavenumber Rydberg constant RH = [microe4 (8h3ε0

2c)] = 10967758306 x 105 cm-1

= 10967758306 x 107 m-1 Eq (112)

wbt 7

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

In 1924 de Broglie proposed that p = hλ Eq (113)which relates the momentum p in the particle picture to the wavelength λ in the wave picture The wave and particle pictures can be used to explain why the electron in the H atom may be only in particular orbits with angular momentum given by Eq (18) This is the circumference 2πr of an orbit of radius r must contain an integer number of wavelengths nλ = 2πr Eq (114)where n = 1 2 3 infin for a standing wave to be set up

The de Broglie picture of the electron in an orbit as a standing wave indicates that the exact location of the electron is not specified

wbt 8

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 9

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 10

In Fig 14(a) the momentum px of the electron is certain but the position x of the electron is completely uncertain whereas in Fig 14(b) x is certain but the wavelength and therefore px is uncertain

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

In 1927 Heisenberg proposed that the uncertainties ∆px and ∆xin px and x are related by

∆px∆x ge ħ = h(2π) Eq (115)

which is known as the Heisenberg uncertainty principle Eq (115) shows that in the extreme wave picture ∆px = 0 and ∆x = infin and in the extreme particle picture ∆x = 0 and ∆px = infin

Another important form of the uncertainty principle is

∆t∆E ge ħ = h(2π) Eq (116)

relating the uncertainties in time t and energy E

Eq (116) shows that if one knows the energy of a state exactly ∆E = 0 and ∆t = infin Such a state does not change with time and is known as a stationary state

wbt 11

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

13 The Schoumldinger Equation and Some of its Solutions

131 The Schoumldinger EquationWave function Ψ(xyzt) ndasha function of position and time describing the amplitude of an electron wave The probability of finding a particle is ΨΨ It follows that

wbt 12

intΨΨ dτ = 1 Eq (117)part(intΨΨ dτ ) partt = 0 Eq (118)

This indicates that the probability is independent of time on the basis of the non-relativistic QMIn 1928 Dirac showed that relativity needs to be taken into account and the wave function may be expressed as

Ψ = b exp(iAh)where b is a constant and A is the action related to the ke (T) and the potential energy (pe) (V)

Eq (119)

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

-partApartt = T + V = H Eq (120)where H known as the hamiltonian in CMFrom Eqs (119) and (120) we obtain

Eq (121)HΨ = iħ(partΨpartt)By replacing the ke [T = frac12(mv2)]

Eq (122)HΨ = -(ħ22m)nabla2 + V

The symbol nabla is called ldquodelrdquo and in cartesian coordinates nabla2 known as the laplacian is given by

Eq (123)nabla2 = part2partx2 + part2party2 + part2partz2

From Eqs (121) and (122) we obtain the t-dept Sch eq

-(ħ22m)nabla2Ψ + VΨ = iħ(partΨpartt) Eq (124)

wbt 13

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

Consider a wave traveling in the x direction and assume thatΨ(xt) can be factored into a t-dept part θ(t) and a time-independent Ψ(x) giving

Eq (125)Ψ(xt) = Ψ(x)θ(t)Substitute Eq (125) into Eq (124) one gets

-[ħ22mΨ(x)][part2Ψ(x)partx2] + V(x) = [iħθ(t)][partθ(t)partt] Eq (126)

Let the right side of Eq (126) equal to E then one gets

-[ħ22m][part2Ψ(x)partx2] + V(x)Ψ(x) = EΨ(x) Eq (127)

Eq (127) is the one-dimensional (1-d) t-dept Sch Eq or simply the wave equation It can be written as

HΨ = EΨ(x) Eq (128)

wbt 14

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

132 The Hydrogen Atom

wbt 15

In the QM picture of the H atom the total energy En is quantized having exactly the same values as in Eq (19) derived from CM The angular momentum of the electron in an orbital may take onlydiscrete values This orbital angular momentum p is a vector and is defined by its magnitude and direction as shown in Fig 15 If an external field (eg magnetic field M) is introduced p can only take certain orientations with respect to that direction so that thecomponent of p in that direction can take only certain discrete values This is referred to as space quantization The effect caused by M is known as the Zeeman effect

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 16

For the H atom the hamiltonian of Eq (122) becomesHΨ = - (ħ22micro)nabla2 - e2(4πε0r) Eq (130)

The second term on the right-hand side is the coulombic potential energy for the attraction between charges ndashe and +e with a distance r apart nabla2 = (12rsinθ) sinθ(partpartr)[r2(partpartr)] + (partpartθ)[sinθ(partpartθ)]

+ (1sinφ)[part2partφ2] Eq (131)The wave function expressed in spherical polar coordinates may be factorized as follows

Ψ(rθφ) = Rnℓ(r)Yℓmℓ(θφ) Eq (132)

The Yℓmℓ(θφ) functions are known as the angular wave functions (wfs) or spherical harmonics (because it describes the distribution of Ψ over the surface of a sphere of radius r) ℓ is the azimuthalquantum number (qt no) associated with the discrete orbital angular momentum values and mℓ is known as the magnetic qt no which results from the space quantization of the orbital qt no

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 17

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

ℓ = 0 1 2 (n-1) Eq (133)Eq (134)mℓ = 0 plusmn1 plusmn2 plusmnℓ

The Yℓmℓ(θφ) can be factorized to be

Yℓmℓ(θφ) = (2π)-12Θℓmℓ(θ)middotexp(imℓφ)

wbt 18

Eq (135)

The radius of the Bohr orbit for n = 1 is given bya0 = (ħ24πε0)(microe2Z) Eq (136)

For H a0 = 0529 Aring and a useful quantity ρ is related to r by

ρ = (Zr) a0Eq (137)

The Θℓmℓ(θ) functions are the associated Legendre polynomials which are independent of the nuclear charge number Z and thus the same for all one-electron atomsThe Rnℓ(r) functions are referred to as the radial wfs since they are only funcions of r They Rnℓ(r) are also known as the Laguerrefunctions

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 19

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

Orbitals are labeled according to the values of n and ℓ

There are 3 useful ways of representing Rnℓ graphically1 Plot Rnℓ against ρ (or r)2 Plot Rnℓ

2 against ρ (or r) The quantity Rnℓ2dr is the

probability of finding the electron between r and r+dr This plot represents the radial probability distribution of the electron3 Plot 4πr2Rnℓ

2 against ρ (or r) The quantity 4πr2Rnℓ2 is called

the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr radius r and the volume 4πr2 dr

In the absence of an electric (E) or magnetic (M) field all the Yℓmℓfunctions with ℓne0 are (2ℓ+1)-fold degenerate This means that there are (2ℓ+1) functions each having one of the (2ℓ+1) possible values of mℓwith the same energy

wbt 20

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 21

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

Linear combinations of degenerate functions are solutions of theSch Eq eg Ψ2p1 and Ψ2p-1 are solutions so are

Ψ2px = 2-12(Ψ2p1 + Ψ2p-1 ) Eq (138)Ψ2py = -2-12(Ψ2p1 + Ψ2p-1 )

From Eqs (132) (135) and (138) together with the Θℓmℓfunction one getsΨ2px = 1[2(4π)12]R21(r)312sinθ[exp(iφ) + exp(-iφ)] Eq (139)

Ψ2py = 1[2(4π)12]iR21(r)312sinθ[exp(iφ) - exp(-iφ)]Recall exp(iφ) + exp(-iφ) = 2 cosφ Eq (140)

exp(iφ) - exp(-iφ) = 2i sinφ

Eqs (139) becomeEq (141)Ψ2px = 1[(4π)12]R21(r)312sinθcos(φ)

wbt 22Eq (142)Ψ2py = 1[(4π)12]iR21(r)312sinθsin(φ)

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

In addition the third degenerate Ψ2p0 wf is always real and is labeled Ψ2pz where

Ψ2pz = 1[(4π)12]R21(r)312cosθ Eq (143)

All the Ψns wfs are always real and so are the Ψnd Ψnf etc wave functions for mℓ = 0 However for Ψnd with mℓ = plusmn1 or plusmn2 it is necessary to form linear combinations of the imaginary wave functions Ψnd1 and Ψnd-1 or Ψnd2 and Ψnd-2 to obtain real functions The Ψnd orbital wave functions for any ngt2 are distinguished by subscripts ndz2 (mℓ = 0) ndxz and ndyz (mℓ = plusmn1) and ndxy and ndx2-y2 (mℓ = plusmn2)

Fig 18 shows polar diagrams for all 1s 2p and 3d orbitals For all (except 1s) there are regions in space whereΨ(rθφ) = 0 because either Yℓmℓ =0 or Rnℓ = 0 In these regions the electron density is zero and we call them nodal surfaces or simply nodes

wbt 23

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 24

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

Unlike the total energy the QM value Pℓ of the orbital angular momentum is significantly different from that in the Bohr theorygiven in Eq (18) It is given by

Pℓ = [ℓ(ℓ + 1)]12ħ Eq (144)

where ℓ = 0 1 2 (n - 1) as in Eq (133)When a magnetic field is introduced say along the z axis the total angular momentum P of magnitude Pℓ may take up only certain orientations such that the component (Pℓ)z is given by

(Pℓ)z = mℓħ Eq (145)

where mℓ = 0 plusmn1 plusmn2 plusmnℓ as in Eq (134)

wbt 25

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 26

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

133 Electron Spin and Nuclear Spin Angular MomentumFrom Diracrsquos relativistic QM treatment the magnitude of the angular momentum Ps resulting from the spin of one electron is given by

Ps = [s(s + 1)]12ħ s = frac12 Eq (146)

Eq (147)Space quantization of this angular momentum results in

(Ps)z = msħ ms = plusmn12

wbt 27

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

Similarly the magnitude of the angular momentum PI due to nuclear spin is given by

PI = [I(I + 1)]12ħ Eq (148)

The nuclear spin qt no I may be zero half-integer or integer depending on the nucleus concerned

wbt 28

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

134 The Born-Oppenheimer ApproximationThe hamiltonian H for a molecule is given by

H = Te + Tn + Ven + Vee + VnnEq (149)

For fixed nuclei Tn = 0 and Vnn is constant there is a set of electronic wave function Ψe which satisfy

HeΨe = EeΨe Eq (150)where He = Te + Ven + Vee Eq (151)Born-Oppenheimer (BO) approximation ndashBecause electrons move much faster than nuclei their motions may be treated separately The hamiltonian for nuclei may be written as

Hn = Tn + Vnn + Ee Eq (152)

satisfying HnΨn = EnΨnEq (153)

wbt 29

It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

wbt 30

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

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Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

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It follows from the BO approximation that the total molecular wave function can be factorized into the electronic and nuclear parts

Ψ = Ψ e(qQ)Ψn(Q)where q and Q are electronic and nuclear coordinates Ψeand Ψn represent the wfs of the electronic and nuclear parts respectively Thus E = Ee + En

Eq (154)

In addition Ψn can be factorized into Ψv and Ψr Ψn = Ψ vΨr

Eq (156)

Eq (155)

It follows that En = Ev + Er Eq (157)

As a result of the BO approximation

Ψ = Ψ e Ψ v Ψr

E = E e + E v + Er

Eq (158)

Eq (159)

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135 The Rigid Rotor

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If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

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136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

135 The Rigid Rotor

wbt 31

If the bond joining the nuclei of a diatomic molecule is rigid and weightless rod the angular momentum is given by

PJ = [J(J + 1)]12ħ Eq (160)where the rot qt no J = 0 1 2 In general J is associated with total angular momentum excluding nuclear spin ie rotational + orbital + electron spin

(PJ)z = MJħ Ms = J J -1 -J Eq (161)Space quantization of the rotational angular momentum is

In the absence of an electric or magnetic field each rotationallevel is (2J + 1)-fold degenerate Solution of the Sch Eq for a rigid rotor shows that the rotational energy Er is quantized with values

Er = I

h2

2

8π Eq (162)

Where the moment of inertia I = micror2 and micro is the reduced mass

J(J + 1)

wbt 32

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

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136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 33

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

136 The Harmonic Oscillator For small displacements the stretching and compression of the bond of a diatomic molecule is assumed to obey Hookersquos law

Restoring force = -dV(x) dx = -kx Eq (163)where V is the potential energy k is the force constant whose magnitude reflects the strength of the bond and x(= r ndash re) is the displacement from the equilibrium bond length re

V(x) = frac12 kx2 Eq (164)The QM hamiltonian for a 1-d harmonic oscillator (HO) is given by

H = -[ħ22m][d2dx2] + frac12 kx2 Eq (165)

Then the Sch Eq [Eq (127)] becomesEq (166)d2Ψv(x)dx2 + (2μEħ2 ndash μkx2ħ2)Ψv(x) = 0

wbt 34

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

wbt 35

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36

Ev = hν(v + frac12 ) Eq (167)where v is the classical vibrational frequency given by

ν = (12π)(kmicro)12 Eq (168)Spectroscopists may use vibrational wavenumbers

Ev = hcω(v + frac12 ) Eq (169)

The wfs Ψv resulting from Eq (166) areΨv = [1(2vvπ12)]12Hv(y)exp(-y22)

where Hv(y) are known as Hermite polynomials andEq (170)

Eq (171)y = [(4π2νωh)]12(r - re)

wbt 36