Summery: introduction to quantum chemistry
Chapter 1: from classical to quantum mechanics
New theories: when experiment and theory agree the theory and its model = accepted
Of not => theory modified
2 key properties of QC:
1) Quantization
2) Wave-particle duality
Shown in following experiments:
1) Black body radiation (bbr)= thermal em radiation emitted
a. Thermodynamic equilibrium with its environment
b. Specific continuous spectrum of wavelengths
c. spectral density shows which frequencies are
radiated= E in EM field of BB at certain frequency /V/cm^-1
d.
e. Combining c and d =
Measure spectral density: broad maximum
(T rises; f rises too)
Classical theory : bb emits infinite E at high F
Not possible!
Solution Max Planck: E ~F
E=nhv
o For a given v: energy is quantized
E can take set of discrete values at given f
and
v/T is weak => Taylorexpansion E=kT (classical theory)
v/T is high => E→0 radiation intensity→0
2) The photoelectric effect:
a. Light on a metal plate = absorbed → excitation of elektrons
b. Elektrons leave the metal => photo-elektrons
c. Absorbed energy = energy needed to eject an electron at eq and the kinetic
energy of emitted electron energy of system = CONSTANT
d. CT cannot predict this
e. Einstein: E~F => relation of energy light to energy electron
f.
g.
1
, h. Energy of electron= energy of photon- amount of
energy used to bound to solid
i. Omega= binding energy of e-= ionization E = work
function
j. In other words: K of e-<photon E by binding E
k. Beta: fitting data point = h! => E=hv
l.
wave like behaviour:
Broglie relationship: confirmed by Davisson and Gremer experiment
A moving particle has waves associated with it
1) If particles have wave nature under appropriate condition they should exhibit
diffraction
2) Shows wavelength of electrons
= double slit experiment
1) E- goes through slit one OR two
2) Don’t know through which slit the electron went
3) Inconsistent with the diffraction pattern => through both slits!
4) Result inconsistent with logic or classical physics
5) QM: electron = wave function = superposition of wavefunction going through 1 and 2
Explanation:e-wave = superposition of wave function through slit 1 AND 2= going through
both slits.
Bohr model:
Positive nucleus and elektrons orbiting around nucleus
➔ E- constantly accelerates=> crashes into the nucleus
➔ Orbits are stationary states
o Coulomb attraction to nucleus = centrifugal force
➔ Jump from high to lower by emitting a photon
De Broglie: orbits= standing electron waves with certain patterns
➔ Only certain frequencies
➔ Only certain energies E=hc/λ
Chapter 2: the Schrodinger equation
1) Time independent
a. ^H= operator
b. Psi = eigenfunction of operator; wavefunction depends on CO
c. E = eigenvalue
2
, When an operator operates on its eigenfunction it generates the eigenfunction multiplied by a
constant namely the eigenvalue.
2) Time dependent:
General principle in QM:
1) Operator for every physical observable
2) physical observable = everything that can be measured
3) if the wavefunction that describes a system is an eigenfunction of the operator then
the value of the observable is extracted by this operation
4) The value of the observable = eigenvalue and the system is in eigenstate.
a. Eigenstate=stationary state of a system
In QC Hermitian operators: eigenfunctions of HO = orthogonal and eigenvalue = real
Mathematical tools:
Particle described by wave-function y = single value when:
1) Position defined
2) Time
Time independent = stationary : energy, position coordinates stays the SAME
Wave function
Y = finite at 0 and infinity!
QM: particle not localised = PROBABILITY to be in a given volume (e- moves too fast to be
localised)
• dP=Y*YdV = finding particle in V
• density probability = dP/dV = Y*Y
• integration of Y over the total space = 1
o Y=normalized
operators
any physical quantity has an operator
O transforms Y -> Y’
OY = Y’
OY = oY (operators scales Y) => Y = eigenfunction of O and o = eigenvalue
=o = observable of state Y
Example: parity (exam) = symmetrie-eigenschappen van een system, system has parity
symmetrie when it looks the same after changing the x-co. If there is an inversion then there
is odd parity
Last function one term changes sign
other terms stays the same
No parity can be defined
Operators = linear
EF = EF when multiplied by a constant => normalize functions
3
Chapter 1: from classical to quantum mechanics
New theories: when experiment and theory agree the theory and its model = accepted
Of not => theory modified
2 key properties of QC:
1) Quantization
2) Wave-particle duality
Shown in following experiments:
1) Black body radiation (bbr)= thermal em radiation emitted
a. Thermodynamic equilibrium with its environment
b. Specific continuous spectrum of wavelengths
c. spectral density shows which frequencies are
radiated= E in EM field of BB at certain frequency /V/cm^-1
d.
e. Combining c and d =
Measure spectral density: broad maximum
(T rises; f rises too)
Classical theory : bb emits infinite E at high F
Not possible!
Solution Max Planck: E ~F
E=nhv
o For a given v: energy is quantized
E can take set of discrete values at given f
and
v/T is weak => Taylorexpansion E=kT (classical theory)
v/T is high => E→0 radiation intensity→0
2) The photoelectric effect:
a. Light on a metal plate = absorbed → excitation of elektrons
b. Elektrons leave the metal => photo-elektrons
c. Absorbed energy = energy needed to eject an electron at eq and the kinetic
energy of emitted electron energy of system = CONSTANT
d. CT cannot predict this
e. Einstein: E~F => relation of energy light to energy electron
f.
g.
1
, h. Energy of electron= energy of photon- amount of
energy used to bound to solid
i. Omega= binding energy of e-= ionization E = work
function
j. In other words: K of e-<photon E by binding E
k. Beta: fitting data point = h! => E=hv
l.
wave like behaviour:
Broglie relationship: confirmed by Davisson and Gremer experiment
A moving particle has waves associated with it
1) If particles have wave nature under appropriate condition they should exhibit
diffraction
2) Shows wavelength of electrons
= double slit experiment
1) E- goes through slit one OR two
2) Don’t know through which slit the electron went
3) Inconsistent with the diffraction pattern => through both slits!
4) Result inconsistent with logic or classical physics
5) QM: electron = wave function = superposition of wavefunction going through 1 and 2
Explanation:e-wave = superposition of wave function through slit 1 AND 2= going through
both slits.
Bohr model:
Positive nucleus and elektrons orbiting around nucleus
➔ E- constantly accelerates=> crashes into the nucleus
➔ Orbits are stationary states
o Coulomb attraction to nucleus = centrifugal force
➔ Jump from high to lower by emitting a photon
De Broglie: orbits= standing electron waves with certain patterns
➔ Only certain frequencies
➔ Only certain energies E=hc/λ
Chapter 2: the Schrodinger equation
1) Time independent
a. ^H= operator
b. Psi = eigenfunction of operator; wavefunction depends on CO
c. E = eigenvalue
2
, When an operator operates on its eigenfunction it generates the eigenfunction multiplied by a
constant namely the eigenvalue.
2) Time dependent:
General principle in QM:
1) Operator for every physical observable
2) physical observable = everything that can be measured
3) if the wavefunction that describes a system is an eigenfunction of the operator then
the value of the observable is extracted by this operation
4) The value of the observable = eigenvalue and the system is in eigenstate.
a. Eigenstate=stationary state of a system
In QC Hermitian operators: eigenfunctions of HO = orthogonal and eigenvalue = real
Mathematical tools:
Particle described by wave-function y = single value when:
1) Position defined
2) Time
Time independent = stationary : energy, position coordinates stays the SAME
Wave function
Y = finite at 0 and infinity!
QM: particle not localised = PROBABILITY to be in a given volume (e- moves too fast to be
localised)
• dP=Y*YdV = finding particle in V
• density probability = dP/dV = Y*Y
• integration of Y over the total space = 1
o Y=normalized
operators
any physical quantity has an operator
O transforms Y -> Y’
OY = Y’
OY = oY (operators scales Y) => Y = eigenfunction of O and o = eigenvalue
=o = observable of state Y
Example: parity (exam) = symmetrie-eigenschappen van een system, system has parity
symmetrie when it looks the same after changing the x-co. If there is an inversion then there
is odd parity
Last function one term changes sign
other terms stays the same
No parity can be defined
Operators = linear
EF = EF when multiplied by a constant => normalize functions
3
