Title: ETH_Sem6 / PCV

Modes of angular frequency between 0 and \(\omega\):

Electric field in a black box of length L:

Wave vector of a radiation field in a black box of length L:

Q: Rayleigh-Jeans law and when it holds:
A: The Rayleigh-Jeans law only holds for low frequencies (hv << kT) and long wavelengths (infrared region).

Q: UV catastroph:
A: According to the Rayleigh-Jeans law, at high frequencies and short wavelengths the energy density becomes infinite, which violates the law of energy conservation.

Q: What resolved the UV catastrophe:
A: Max Plank introduced the « photon », which means that the radiation field can only emit or absorb energy in integer multiples of a fundamental energy quantum hv.

Energy of a photon:

Maxwell-Boltzmann distribution:

Q: Planck’s law:
A: Characterises the spectral energy distribution of a radiation field that is in thermal equilibrium with its environment at a temperature T.

Q: What are non-thermal radiation sources?
A: F.ex. lasers and gas-discharge lamps. They do not follow Planck’s law because they are not in thermal equilibrium with their environment.

Kinetic formalism for the rate of change of populations in the two-state system:

Different possible interactions of a two-level system with the radiation field:

Q: Spontaneous emission kinetic equation for two-state system:
A: Does not depend on the radiation field!

Q: Induced emission and kinetic equation:
A: It occurs along the direction imposed by the incoming radiation.

Q: Two-state system in thermal equilibrium relative populations and mode density:
A: From the Boltzmann distribution we have:
And from the equilibrium condition \(\frac{dN}{dt}=0\)

Q: Two-state system at thermal equilibrium and the relationship between the different kinetic coefficients and implications:
A: The rate of spontaneous emission is proportional to \(\nu^3\), which means that the lifetime of an excited state is much longer in the case of pure vibrational excitation than in the case of pure electronic excitation. This also directly explains why it is so difficult to build lasers in the UV or X-ray wavelength range because the rate of spontaneous emission becomes so large that it becomes difficult or impossible to maintain a population inversion.

B_12 expression within the dipole approximation:

Experimental scheme of an absorption measurement:

Intensity of a beam of light (photons):

Lambert-Beer law:

Q: Bimolecular rate law for the decrease of photons upon absorption:
A: With \(\sigma_{12}\) being the absorption cross-section.

Q: Considerations on the linewidth of a transition between two states:
A: There is a finite linewidth due to the lifetime of the excited state. An experimentally measured line is subject to additional broadening mechanisms like the Doppler broadening and power broadening.

Q: Rate constants for stimulated emission and absorption considering all frequencies that lead to the transition between the two states:
A: Excitation source varies little with its frequency.

Q: What is so-called « white-light » excitation?
A: When the photon density of the excitation source varies little with its frequency.

Integrated absorption cross-section:

Relationship between kinetic parameters and the integrated absorption cross-section:

Line strength:

Q: Relationship between kinetic parameters of the two-state system and the line strength and when this needs to be used:
A: Needs to be used when the photon density varies a lot with the frequency.

Q: Abbreviation of Laser:
A: Light Amplification by Stimulated Emission of Radiation.

Q: Basic principle of laser operation:
A: In order to generate a laser from a two state system, you need to generate a population inversion.
One also needs a feedback structure that will make light pass through the gain medium several times. This is usually done by having a totally reflecting mirror on one side of the cavity and a partially reflecting mirror on the other end.

Q: Maser
A: Same as laser but operating in the microwave region.

Q: By what is a laser beam characterised:
A: Very low divergence and a high degree of temporal and spatial coherence.

Operation principles of the three most common types of lasers (abstract):

Q: What is the problem with the two-level laser?
A: It tends to turn itself off because the amplification is suppressed when strong stimulated emission occurs.

Q: How does the ammonia maser avoid the usual problem for two-state lasers/masers?
A: The ground state molecules can be pumped away, which conserves the population inversion. However, this is of course only possible for gas lasers.

Q: What is different about the three level laser that helps avoid the problems of the two-level laser?
A: The pumping is spectrally distinct from the emitted radiation due to an extra level. Strong stimulated emission will not counteract the effect of pumping, allows the laser to run with a much higher yield.

Q: Typical example for a three state laser:
A: Ruby laser

Q: Four state lasers and their application:
A: Most practically useful lasers: high-power CO_2 and the titanium-sapphire laser

Laser equation:

In a two-state laser the solution to the laser equation when N_1 and N_2 reach a stationary regime:

Q: What leads to an additional term in the laser equation and how does this look like?
A: Losses from the cavity like scattering or a partially transmitting mirror.

What is the condition for laser operation?

Q: In which case can the electromagnetic field be treated classically and thus described by the Maxwell equations?
A: If the number of quanta per field mode is extremely large, on the order of 10^10- 10^20.

Q: What condition greatly simplifies the treatment of light-matter interaction?
A: When the wavelength of the radiation is much larger then the extension d of the particle.
This is typically true for UV and IR range but no longer for X-rays or gamma-rays.

Electric dipole interaction of a monochromatic linearly polarised wave of circular frequency with a system of charged particles:

Definition of the electric dipole moment:

Interaction of a dipole with a magnetic field:

Q: Which dipole interaction dominates in the optical domain?
A: The electric dipole interaction.

Q: Where does the magnetic dipole interaction dominate?
A: NMR and EPR!

Time-dependent Schrödinger equation:

Q: How can the stationary solutions of the time-dependent Schrödinger equation be written?
A: If \(\Psi(\vec r)\) is a solution to the time-independent SE with eigenvalue \(E\) and \(\Psi(\vec r, t=0) = \Psi(\vec r)\). Otherwise the wavefunction is first expanded in energy eigenfunctions.

Time-independent Schrödinger equation:

Probability density of a pure state:

Q: Superposition principle and corresponding equation:
A: Generally, any state can be written as a superposition of the different Eigenstates of a given system:
In this case, the probability density is no longer time independent!

Hamiltonian of a quantum mechanical system with radiation:

Q: What is often done to get the Eigenfunctions of a quantum mechanical system with radiation?
A: The Eigenfunctions are expanded in the Eigenfunctions of the purely molecular Hamiltonian:
Use:
1. Orthonormality of the Eigenfunctions

Q: Rabi equations:
A: From a two-level system which interacts with a coherent electromagnetic field.

Near resonant excitation and solution of this case for the Rabi equations:

Q: Off resonance excitation and consequences on the solutions to the Rabi equations:
A: The time-averaged excitation fraction amounts to:
This is actually a Lorentz function with FWHM of 2V and also corresponds to the mean absorbed energy for excitation at circular frequency w.
This behavior is also known as intensity- or power broadening of spectral transitions that is frequently observed in laser-based spectroscopic experiments.

Q: What are the two fundamentally different but complementary approaches to spectroscopy?
A: 1. Frequency-domain spectroscopy
2. Time-domain spectroscopy

Q: Frequency-domain spectroscopy and application:
A: Relies on tunable sources of radiation that are designed to be as narrow as possible in frequency. These sources are used to measure energy intervals in atoms or molecules to the highest possible precision. The measured energy intervals are then used to construct or refine the molecular hamiltonian of the system.

Q: Time-domain spectroscopy and its applications:
A: Relies on ultrashort laser pulses which have a very broad spectrum. The large coherent bandwidth of the excitation pulse is used to prepare a coherent superposition of molecular Eigenstates whose time evolution is studied. This provides information on the molecular hamiltonian which is complementary to the frequency-domain approach and may be simpler to interpret.

Hamiltonian of the Harmonic oscillator:

Eigenvalues of the harmonic oscillator:

Q: In which case can pure vibrational transitions be observed in the harmonic oscillator approximation?
A: Only if the molecular dipole moment varies with r.

Q: What is a coherent superposition of states also called?
A: A wave packet!

What is the period of oscillation of a wave packet consisting of two states?

What is the Hamiltonian of a linear rigid rotor in 3D? What are the Eigenvalues?

Q: Under which conditions can a purely rotational transition be observed?
A: Molecule also needs a permanent dipole moment.

Hamiltonian for one-electron atoms:

Mass corrected Rydberg constant:

Solution for the one-electron atoms: (Energy and decomposition)

Expression for the energies of one-electron atoms relative to the ground state:

Spherical harmonics, what they describe, how they relate to the probability density and implications:

"Basic" orbital nomenclature, which quantum number is used and what are the names?

The spherical harmonics are Eigenfunctions of which operators? What are the Eigenvalues?

What happens if we have several particles and they are no longer non-interacting but nearly so?

Q: What is the general Hamiltonian for atoms with more than one electron?
A: It cannot be solved analytically anymore!

How can we introduce back the electron repulsion term into the Hamiltonian for an atom with more than one electron to get a realistic estimation of the Eigenfunctions and Eigenvalues?

Q: How can the sequence of orbitals in the Pauli-Aufbauprinciple be qualitatively explained?
A: By taking into consideration the shielding of the nuclear charge by electrons in inner shells and the decrease, with the increasing value of l, of the penetrating character of the orbitals.

Q: Why do we need Slater determinants?
A: We need them because electrons are Fermions and polyelectronic wave functions must be antisymmetric with respect to the exchange (permutation) of coordinates of any pair of electrons --> Pauli-Principle for Fermions.

Q: General representation of a Slater-Determinant:
A: These determinants fulfill the Pauli-Principle because permutation of any two colums leads to a change of sign in the determinant. Further, if two rows of the determinant were identical, this would immediately lead to it being zero.

Q: What is the procedure known as Pauli-Aufbauprinciple?
A: The ground state configuration of an atom can be obtained by filling the orbitals in order of increasing energy with two electrons (m_s +1/2 and m_s -1/2).

Q: What is the Coulomb integral?
A: It can be interpreted as the energy arising from the repulsion between the electron clouds of the two electrons. The term arises when the contribution of the electron repulsion term in the Hamiltonian is evaluated in the first order perturbation theory.

Q: What is the exchange intergral?
A: It represents the repulsion arising between the two electrons having "exchanged" their orbitals. The term arises when the contribution of the electron repulsion term in the Hamiltonian is evaluated in the first order perturbation theory.

Q: How do the different term values arise? What is the most important consideration?
A: It is necessary to understand how the different orbital and spin angular momenta in an atom are coupled by electromagnetic interactions and in which sequence the angular momentum vectors are added to form the total angular momentum vector J. This can be achieved by ordering the different interactions according to their relative strengths and by adding the angular momentum vectors that are most strongly coupled first.

How can each angular momentum vector be quantum mechanically described and what happens if they start to couple with each other?

Q: LS coupling: When is it used?
A: It adequatly describes the ground state of almost all atoms except the heaviest ones.

Q: JJ coupling: When is it used?
A: It is less frequently used than the LS-coupling and becomes important in the description of the heaviest atoms and of electronically excited states.

The LS-coupling hierarchy:

Q: How are the terms for the LS-coupling obtained?
A: 1. Add all orbit angular momenta l_i together to from L.
2. Add all spin angular momenta s_i together to form S
3. Finally the total angular momentum J is determined by adding vectorially S and L
The terms are compacted in the form of term symbols:
However, not all of the theoretically derived term symbols are actually allowed by the Pauli-Principle.

Q: What are Hund's rules and what are they used for?
A: They can be used for the prediction of the lowest energy term of the ground electronic configuration of almost all atoms if they are applied in the following sequence:

Q: When can Hund's rules not be used?
A: For electronically excited states!

How can the energy level splittings of the components of a term ^2S+1L be described?

What does the third rule of Hund predict about the spin orbit coupling constant?

The jj coupling hierarchy:

Q: How are the terms of the jj coupling obtained?
A: 1. First add vectorially the orbital angular momentum vector l_i and the electron spin vector s_i of each electron to from a resultant electronic angular momentum j_i.
2. The total angular momentum J results from the vectorial addition of all j_i
The total orbital and spin angular momentum quantum numbers L ans S are no longer defined. Instead the terms are now specified by a different set of angular momentum quantum numbers: the total angular momentum j_i of all alactrons in partially filled subshells and the total angular momentum quantum number J  of the atom.

Q: What is a correlation diagram?
A: A diagram which represents how the energy level structure of a given system evolves as a function of one or more system parameters.

Q: Where do magnetic moments arise?
A: In systems of charged particles with nonzero angular momenta to which they are proportional.

Magnetic moment arising from the orbital motion of an electron in an atom:

Hyperfine interaction:

Equation for the Landé g factor:

Total angular momentum vector resulting from taking into account the hyperfine interaction:

What are the hyperfine energy shifts of a given state (IJF)?

To what exactly relates the intensity of a transition between an initial state of an atom or a molecule and a final state?

What is a selection rule and when is a transition "forbidden"?

Q: Why are atomic spectra usually simpler to treat than molecular spectra?
A: When studying spectra of atoms, the laboratory-fixed reference frame is the only relevant frame, because it can always be chosen to coincide with an internal, "atom-fixed" reference frame. Indeed, the point-like nature of the nucleus implies that there are no rotations of the nuclear framework.

Q: Within the dipole approximation, the photon can only exchange up to...
A: one unit of angular momentum with an atom or molecule:

Angular momentum selection rules for the LS coupling:

What is the parity operator and does it commute with the H.O. Hamiltonian?

The parity (Laporte) selection rule:

Q: When does the energy level structure of the hydrogen atom have a high level of degeneracy?
A: If we neglect fine, hyperfine and quantum-electrodynamic effects.

Balmer formula:

Q: Neglecting the fine and hyperfine structures, one-photon transitions for the alkali-metal atoms can be determined using:
A: The deviation from hydrogenic behaviour is accounted for by the l-dependent quantum defects.

Q: How are the transitions from or to the lowest energy levels grouped?
A: They are grouped into Rydberg series with the names sharp, principal, diffuse and fundamental. From this arise the names for the orbitals s, p, d and f.

How can stationary states of molecules be well approximated and why?

Types of motion, typical periods and typical energy intervals:

Q: Quickly explain the principle of the Born-Oppenheimer approximation
A: Through this approximation, we are able to reduce the dimensionality of the problem.

Q: How are we able to reduce the dimensionality of the Hamiltonian more even after the BO approximation?
A: We can separate the overall rotation of the molecule from the center-of-mass motion.

How does the molecular wave function look after the BO approximation and which fact justifies it?

Q: What is the first step of solving the problem after the BO approximation?
A: First, the electronic problem is solved. For this, the nuclear coordinated are used as parameters and stay fixed. As a result the kinetic energy vanishes. We thus then get the electronic energies  as a function of the nuclear coordinates.

How do we visualise the solutions from the first step after the BO approximation?

Q: What do we do in the second step after applying the BO approximation?
A: We solve the Schrödinger equation describing the nuclear motion on the n-th potential hypersurface.

How do we get to the equation describing only the nuclear motion in the second step of the BO approximation?

What justifies the separation of the center-of-mass motion?

Q: How can the overall translational motion of the atom or molecule be treated if we can apply the separation of center-of-mass motion?
A: The overall translational motion of the atom or molecule can then be treated as that of a free particle of mass M in a three dimensional box.

Q: What does the separation of center-of-mass motion mean for spectroscopy?
A: Only due to the separation of center-of-mass motion we get this reduction in dimensionality!

Q: How can the internal motion of a diatomic molecule be described?
A: It can be described as the motion of a fictional particle with mass \(\mu\) and relative position Q = Q_2-Q_1.

Spherical coordinates (illustration):

Q: How do we get to the harmonic oscillator and rigid-rotor approximation for a diatomic molecule?
A: After separating the spherical harmonics from the nuclear Hamiltonian of a diatomic molecule, we get:

How does one get a better approximation of the exact solutions of the nuclear Hamiltonian in a diatomic molecule?

Schematic of the rovibronic energy levels of a diatomic molecule:

Q: Under which condition does the harmonic oscillator represent a good approximation to the vibrational motion of a molecule?
A: Only in the vicinity of R_e.

Morse potential formula:

Morse potential vs. harmonic oscillator (picture):

Q: Under which limit does the vibrational energy spectrum become continuous?
A: When the molecule has dissociated.

Q: How are electronic states labeled?
A: Diatomic GS: \(X\)
Polyatomic GS: \(\tilde X\)
Excited increasing in Energy \(A,B,C\) (for same total spin)
\(a,b,c\) for different spin

Q: How do the BO potential energy surfaces look like for different electronic states?
A: They can look very different --> have very different shapes, because they reflect different binding mechanisms.

Q: Valence states vs. Rydberg states vs. ion pair states vs. van der Waals states:
A: - Valence: Electrons are in MO @ multiple atoms. Can be repulsive if in antibonding orbitals
- Rydberg: Excited to difuse orbital around molecular ion core. Can be labeled by \(n,l\)
- Ion-Pair: \(A^+ \) and \(B^-\) connected by coulomb
- VdW: Shallow potential, looks like GS of Noble gas dimers

Q: Symmetry labels of molecular orbitals and electronic states of diatomic molecules:
A: - MO: \(\sigma^+, \pi, \delta, \ldots\) (\(\sigma_u,\sigma_g\) for homonuc)
- States: \(\Sigma^+, \Sigma^-, \ldots\) (u,g for homonuc)

Q: What do symmetry labels for molecules also contain as information just like in atoms?
A: They also contain information of the electronic angular momentum.

Q: Good quantum numbers for diatomic molecules:
A: The symmetry of the potential is reduced to cylindrical symmetry, which means that only the projection of the total angular momentum on the internuclear axis is conserved. But also this only holds as long as spin-orbit coupling can be neglected.

Q: Qualitative limits in the representation of molecular orbitals:
A: There are two qualitative limits:
1. The united atom limit
2. The separated atom limit

United-atom limit vs. separated atom limit:

Corrleation diagram from the two 1s atomic orbitals of two identical separated atoms:

What is LCAO and what is it used for?

What are the conditions that must be fulfilled to form molecular orbitals that are distinct from the original atomic orbitals?

MO diagram of the lighter second row elements for homonuclear diatomic molecules:

MO diagram for the homonuclear diatomic molecules of the heavier second row elements:

Q: How are electronic configurations of molecules obtained?
A: By filling the spatial orbitals according to the Pauli principle with no more than two electrons. As in the case of atoms, an electronic configuration leads in general to several electronic terms and several electronic states.

Q: Which shells and subshells must be considered for the determination of possible electronic terms for diatomic molecules?
A: Only shells and subshells with partially filled orbitals, because filled shells and subshells are of highest possible symmetry.

How can electronic states be derived from an electronic configuration for a diatomic molecule?

Q: Which three cases need to be distinguished for the determination of the multiplicity of an electronic term for the case of a two-electron wave-function?
A: 1) Different Shells -> Sym & Antisym -> Triplet & Singlet
2) Same shell & Same orbital -> Sym -> No triplet state (Pauli principle)
3) Same shell but different orbital -> No simple rule, needs to consider \(\sigma_v\) symmetry

Q: Classification of electronic terms: on which assumption does the most often used classification depend and in which cases is it justified?
A: Electrostatic (incl exchange) interaction dominate over SO coupling (LS scheme)
Ok as long as \(^{2S+1}\Lambda\) terms are different in energy by more than SO coupling.

Selection rule for the vibrational transitions of a diatomic molecule under the BO approximation:

Q: What are the selection rules for the purely vibrational transitions in diatomic molecules under the harmonic oscillator approximation?
A: \(\Delta v = \pm1 \gg \Delta v = \pm 2 \gg \ldots\)

Which condition must be fulfilled for the observation of a purely rotational spectrum?

What are the angular momentum selection rules for vibrational transitions?

Q: What are the so-called P- and R-branch (Q-branch)?
A: The P-, R- and Q-branch refer to distinc regions in the spectrum of a given vibrational transition of a molecule. The R-branch refers to the region where J'-J'' = 1, the P-branch where J' - J'' = -1 and the Q-branch (if observed) to J'-J'' = 0.

Selection rules for the purely electronic transitions in a diatomic molecule:

What is the so-called Franck-Condon factor for the vibronic structure of electronic transitions?

Franck-Condon principle for an electronic transition between two electronic states having almost identical BO potential energy surfaces vs. very different BO potential energy surfaces:

Q: What is the approximation contained in the Franck-Condon factors and what are the limitations of it?
A: \(<V(R)> = <V> = const\)

What is the equation for the rotational energy taking into consideration centrifugal distortion effects and also the variation of the rotational and centrifugal distortion constants that results from the anharmonicity of the vibrational motion:

For which case is the equation:
adequate to describe the rotational structure and where is it not?

Q: Where does the angular momentum vector R, describing the rotation of the nuclei lie in a diatomic molecule?
A: It lies perpendicular to the internuclear axis, because of the vanishing moment of inertia along the internuclear axis.

What is the effect called Lambda-doubling?

In which case is the a/s label valid?

In which case is the g/u label valid?

A further selection rule for homonuclear diatomic molecules due to the conservation of nuclear spin symmetry:

Q: What are ortho and para forms of homonuclear diatomic molecules?
A: The conservation of nuclear spin symmetry means that rovibronic states of "a" and "s" symmetry are not connected by transitions induced by electromagnetic radiation and can therefore be considered as belonging to two distinct forms of homonuclear diatomic molecules, calles para and ortho proms.

How can group theory be used as a tool in spectroscopy?

Q: What limites the usefullness of group theory as a tool for spectroscopy?
A: It can't make any quantitative predictions.

Q: Which two groups are particularly important in molecular physics and in molecular spectroscopy?
A: The point groups and the permutation-inversion groups.

Q: Which set of operations (and the multiplication operation) defines a symmetry group?
A: The set of symmetry operations that commute with the Hamiltonian and thus have a common set of eigenfunctions. Eigenvalues of these symmetry operations can be used as the labels of the eigenfunctions.

Q: What does a point group adequately describe?
A: Rigid molecules

Q: What are the symmetry operations of a point group (five):
A: 1. identity
2. proper rotation
3. reflection
4. inversion
5. improper rotation or rotation-reflection

Describe the symmetry operation identity:

Describe the symmetry operation proper rotation:

Describe the symmetry operation reflection:

Describe the symmetry operation inversion:

Describe the symmetry operation improper rotation or rotation-reflection:

Q: How do we immediately know if a molecule is achiral?
A: \(S_n\) \(\implies\) Achiral

Q: What do we need to use, if we want to describe a "floppy system" i.e. a system which cannot be regarded as rigid on the time-scale of a spectroscopy experiment?
A: We need to use the complete nuclear permutation inversion (CNPI) groups.Their concept relies on the fact that the symmetry operations, i.e. the permutation inversion operations leave the Hamiltonian unchanged.

Q: What are the five symmetry operations of the CNPI groups?
A: 1. Permutation
2. Cyclic permutation
3. inversion
4. permutation followed by inversion
5. cyclic permutation followed by inversion

Describe the symmetry operation permutation:

Q: Name an example of a "floppy system" which can't be described by point groups but needs to be described by CNPI groups:
A: The (HF)_2 system:

Describe the symmetry operation cyclic permutation:

Describe the symmetry operation of inversion in the context of the CNPI groups:

Describe the symmetry operation of permutation followed by inversion:

Describe the symmetry operation of cyclic permutation followed by inversion:

Why do permutation operations not affect the Hamiltonian of a molecule?

Q: Can CNPI groups only be applied to "floppy" systems?
A: No! They represent a very general description that can also be applied to rigid systems. Each point group is isomorphous to a CNPI group although the symmetry operations are not identical.

Q: What is a disadvantage of the CNPI groups and how is this problem usually solved?
A: In the case of rigid molecules, most of the time point groups are used.

What is the order of a group and which two "types of orders" exist?

What is a multiplication table and what is the origin behind them?

Under which condition is a group called abelian?

Under what condition are two elements of a group called conjugated? What do all conjugated elements of a group together form?

What is the meaning of the order of an element in a group? For which types of groups does it exist? What is the order of a rotation C_n or a reflection?

On what does the matrix represenation of a symmetry operation in a group depend?

What is the character of a symmetry operation?

Reducible vs. irreducible representations:

What is the character table of a group?

What is the character of the unity operator (E) always equal to?

Q: How can reducible representations be reduced to irreducible representations?
A: Reducible representations can be described as linear combinations of irreducible representations:

How can group theory be used to determine the symmetrised linear combinations of atomic orbitals?

Q: Which orbitals can be combined to form bonding or antibonding molecular orbitals?
A: Only orbitals of the same symmetry.

How can the representations of the rovibrational modes of a molecule be deduced?

Q: Only which atoms can contribute to the character of a symmetry operation?
A: Only the atoms that are not exchanged by the symmetry operation.

What is the nomenclature to label the vibrational states of a polyatomic molecule?

How does one find the overall symmetry of the vibrational wavefunction:

How is the overall symmetry of an electronic wavefunction obtained?

What is the point group of all homonuclear diatomic molecules?

How is the "g" or "u" label determined in direct products?

What is actually the generalised Pauli Principle in the context of point groups?

How can the symmetry of the total wave function be determined?

How is the relative orientation of the space-fixed and the molecular-fixed frame given?

Q: Equations leading to the selection rules for the electric dipole transitions:
A: Taking into consideration the BO approximation:

Selection rule for spin conservation upon electric dipole transitions:

Angular momentum selection rules for the electric dipole transitions:

Q: What is the vanishing integral theorem? When can it be used?
A: It can be used for vibrational, rotational and electronic spectroscopy.

Vibrational spectroscopy and the application of the vanishing integral theorem:

Q: What is the condition for a vibrational transition to be observable?
A: A change of dipole has to occur when the vibration is excited.

Application of the vanishing integral theorem on electronic transitions:

What is the Franck-Condon factor for electronic transitions and what does it mean?

Under which condition can electronically forbidden transitions become weakly allowed?

Q: Which approximations in particular can be used for both diatomic as well as polyatomic molecules?
A: - the approximate separability of motion
- Born-Oppenheimer approximation

Q: What connects the classical and quantum mechanical description of a system?
A: The correspondence principle. (but only for phenomena which are know classically f.ex. spin does not have a classical analogon)

Q: Angular momentum of a rigid rotor:
A: If we diagonalise the matrix I, we get by convention the diagonal terms, I_a, I_b, I_c where I_a <= I_b <= I_c.

Q: Which five classifications of rotors exist?
A: 1. linear molecule
2. prolate symmetric top
3. oblate symmetric top
4. spherical top
5. asymmetric top

What is the condition for a linear molecule in the rigid rotor approximation in terms of moment of inertia?

What is the relationship between the different diagonal components of the rigid rotor for the prolate symmtric top? Give an example and draw a scheme

What is the relationship between the diagonal elements of the moment of inertia for the oblate symmetric top? Give an example and draw a scheme

Give the relationship between the different diagonal elements of the moment of inertia for the sperical top and give an example:

What is the relationship between the diagonal elements of the moment of inertia in the case of an asymmetric top? What is an example for an asymmetric top?

Give an additional relationship between the diagonal elements of the moment of inertia for planar molecules:

Q: What is the classical kinetic energy of a rigid rotor and give the Hamiltonian obtained through the correspondence principle?
A: Classical:
Hamiltonain:

How many quantum numbers are required to describe the rotational motion of a rotor and what do they describe?

How are the Eigenfunctions of the rigid rotor denoted and what are the operators of which they are the Eigenfunctions?

What are the rotational constants and how are they defined?

What are the Eigenvalues and the degeneracies of the spherical top?

What are the Eigenvalues and the degeneracies for the oblate symmetric top?

What are the Eigenvalues and the degeneracies for the prolate symmetric top?

What are the Eigenvalues and degeneracies of the diatomic molecule in the rogid rotor approximation?

What is unfortunate about the case of the asymmetric top and how can we get the energy levels?

Q: What is the number of vibrational degrees of freedom for a non-linear molecule?
A: f = 3N - 6 (N = number of atoms)

Q: What is the number of vibrational degrees of freedom for a linear molecule?
A: f = 3N - 5 (N = number of atoms of the molecule)

Q: What is an important step in the treatment of the vibrational motion of polyatomic molecules?
A: The near-separability of the nuclear Schrödinger Equation into the normal modes.

How does the total vibrational energy and the vibrational wave function look after we were able to apply the near-separability to the different normal modes?

How does the potential and the energy for the vibration of a polyatomic molecule look after taking into account the anharmonicity and potential degeneracies in vibrational modes?

With the example of CO_2, quickly explain how the degeneracy of a vibrational mode affects the rotational energies and the total angular momentum:

Schematicaly show how the vibrational angular momentum of CO_2 arises:

What two classes of HAH molecules exist and what are the corresponding point groups?

What is a good scheme to determine the molecular orbitals of HAH molecules?

What are the linear combinations of the atomic orbitals of linear HAH molecules?

Q: What are the most important differences between the linear and the bent HAH molecules?
A: - different point groups ;-) (D_inf.h vs. C_2v)
- the degernacy of the two non-bonding pi orbitals on the A atoms is lifted as the molecule bends --> one of them can now even interact with the linear combinations of the H 1s orbitals and becomes bonding, the extent to which it becomes bonding and thus the energy of this orbital depends strongly on the angle to which the molecule is bent
- all other orbitals (except the newly bonding orbital) become destabilised upon bending of the molecule

Q: On what does it depens whether a HAH molecule is linear or bent?
A: This depends only on the occupation of the molecular orbitals --> energy is always minimised :)

Q: How do we obtain the symmetries of the electronic states of a polyatomic molecule?
A: By forming the direct product of the irreducible representations of the occupied molecular orbitals.

Q: Approximately draw/describe the correlation diagram of the linear and bent HAH molecules? What are these types of correlation diagrams called?
A: Walsh diagrams.

Q: For what are Walsh diagrams also useful?
A: In the discussion of vibronic interactions because they enable one to see how the degeneracy of pi orbitals and of the pi, sigma, phi ... states are lifted, and how the electronic character changes, when the molecules are bent out of their linear structures.

Q: What is the problem when you apply the projection formula to a degenerate irreducible representation? How do we solve this?
A: If you use two different orbitals, you get two linearly independent, but non-orthogonal combinations of the original basis vectors. However, we usually want to work in orthonormal basis vectors also for the molecular orbitals, so we orthogonalise them using the Schmidt orthogonalisation algorithm.

What is the equation for the Schmidt orthogonalisation algorithm in case of two linearly independent but non-orthogonal vectors?

Q: What is the usual way to determine the qualitative energetic ordering of molecular orbitals?
A: The energy increases with increasing number of nodal planes.

Q: How can we get a better estimation of the energy of pi orbitals than just by ordering them with respect to increasing number of nodal planes f.ex. for benzene?
A: Using the Hückel molecular orbital (HMO model).

Q: How are the energies and expansion coefficients obatined in the HMO model?
A: By solving the secular determinant:

What are the approximations introduced by the HMO model?

If we already know the symmetry adapted basis functions of a pi system, how can we use the Hückel operator to obtain the Hückel energies?

Show the energy level diagram for the different pi orbitals of benzene taking into account the orbital coefficients:

Q: What effect does the group theoretical treatment usually have on the Hamiltonian matrix in the HMO model?
A: It normally only divides the Hamiltonian matrix in as many diagonal blocks as there are irreducible representations.

Q: What is a Frost Muslin diagram?
A: It is a diagram depicting the energetic ordering of the corresponding pi orbitals of a cyclic fully conjugated system.

Q: How are Frost-Muslin diagrams constructed?
A: 1. Draw a regular polygone representing the cyclic polyene into a cricle, placing one vortex on the lowest point
2. the vertices of the polygone then provide the energies of the pi orbitals of the polyene
3. provides also the sequence and degeneracies of the HMO of cyclic polyenes

What is different with large polyatomic molecules in terms of symmetry and what effects does this have on the electronic structure and transition?

How are electronic states in large polyatomic molecules with low symmetry labelled?

How are electronic transitions in large polyatomic molecules of low symmetry usually labelled and what qualitative predictions can be made from this?

Q: What is special about the photochemistry and photophysics of the DNA bases and how might this be important biologically?
A: Although they absorb strongly in the UV range, they hardly show any fragmentation, unlike many other molecules. They are able to convert the energy of the photon to vibrational energy. This property might be important for preserving the genetic information.

What are the two cases that need to be distinguished to determine the spin multiplicity of large polyatomic molecules of low symmetry?

Q: Which quantity determines the relative intensities of the vibrational bands in an electronic transition and how is it called?
A: Franck-Condon factor

What can lead to electronically "forbidden" transitions to becoming weakly observable and how is this mechanism called?

Q: Under which condition is the following factorisation no longer valid and what are the consequences of this?
A: --> this is the so called Herzberg-Teller intensity borrowing mechanism

Q: What does the intensity of the vibrational structure of an electronically allowed transition provide for informations on the geometry?
A: When an electronic transition is allowed, the relative intensities of the transitions to different vibrational levels of the electronically excited state approximately correspond to Franck-Condon factors, and the vibrational structure of an electronically allowed transition contains information of the relative equilibrium geometries of the two electronic states connected by the transition.

Q: Can infinite point groups be normally treated with the reduction formula?
A: No!

Q: What do square brackets in a product table indicate?
A: They indicate that this direct product has an antisymmetric spatial part and thus it needs to have a symmetric spin part --> it's a triplet state.

Q: How do we get the dipole allowed electronic transitions of the ground state of a molecule?
A: The irreducible representations of the dipole operator are the same as the linear functions and can thus be found in the character table for this analysis.

How do we derive the character of a given symmetry operation for the total reducible representation of a molecule?

Q: What alternative exists for the determination of the total reducible representation taking into account we know the irreducible representation of the unit vectors?
A: We then multiply these two to get the total reducible representation, just like in the example below:

Q: If you know the irreducible representations of the vibrational modes of a given molecule, which ones are IR active and which ones are Raman active?
A: (the irreducible representations given here are from a specific example)

Q: Which condition has to be fulfilled for a vibrational mode to be IR active?
A: Infrared spectra result from the absorption of light by a molecule. Lets assume, that infrared light, which excites the molecule, is polarised along the x axis of the coordinate system in which the position of the molecule is defined. Since the absorption intensity is proportional to the matrix element of the dipole moment component:
The selection rules establish non-zero integrals for symmetry reasons. At room temperature most molecules are in the vibrational ground state v_0 which has a totally symmetric vibrational wave function. Thus phi_1 has to belong to the same representation as the dipole moment component (are the same as for the translations!), which is the same as for x in our case. Thus, absorption active modes have the same symmetry as translational vectors. (Can be found in the character table)

Q: Which condition has to be fulfilled for a vibration to be Raman active?
A: Raman spectroscopy is based on inelastic scattering. For a mode to be Raman active, the corresponding integral:
must be non-zero. Here is a component of the polarizability tensor
Raman-active modes have the same symmetry as the quatratic functions. (Can be found in the character table!)

Q: What is ortho or para H_2?
A: Since the nuclear spin is I = 1/2, the hydrogen atoms are Fermions and thus the total nuclear wavefunction has to be antisymmetric with respect to the permutation of the two nuclei according to the Pauli principle.

What is the formula for the moment of inertia?

Stokes vs. anti-Stokes Raman Streuung: Kurz erklärt, was ist es?

3D rotation matrix about the x-axis:

3D rotation matrix about the y-axis:

3D rotation matrix about the z-axis:

Q: What are the symmetry considerations for asymmetric and symmetric tops?
A: - if a molecule contains a C_3 or higher rotational axis (including S_4 improper rotational axis) at least two eigenvalues of I are the same and it can't be an asymmetric top
- if a molecule contains two or more non collinear C_3 or higher rotational axes, all eigenvalues have to be equal and it must be a spherical top
- for a molecule with D_2 symmetry, its inertia matrix commutes with all the three matrices for the C_2 rotations, yet it can still have three unique eigenvalues and be asymmetric top

What happens if you switch off the photon flux suddenly upon reaching steady state in a two state system? What would you observe experimentally?

Q: Using the formalism of a two-state system, give an equation that describes the condition for lasing. Can this condition be met for a simple two-level system?
A: No! It can't be met.
So, the maximum possible value for N_2 is N_tot/2 and this does obviously not fulfill the condition for lasing.

In a maser with the active medium being NH_3 (effectively a two-level system), how did Gordon and coworkers manage to achieve amplification?

Also draw a diagram with all the different levels and the conversions between them.

How does N_2/N_tot look for different A_21 vs. K_12 in a two-level system? And how does it change after switching off the photon source?

What are the two types of broadening of a spectral transition and describe what they mean?

What is the observed frequency shift dw for a molecule traveling at velocity v along the direction of observation? Consider what is usually a sufficient approximation and when it holds.

How does one find the FWHM of a Gaussian function?

When is s^-1 used and when is Hz used, what does it mean and how do we convert if we want to compare the two cases?

Q: Define spectroscopy and it‘s goal:
A: Spectroscopy is the study of absorption and emission of radiation by matter. Goal: Determine the structure and dynamic properties of matter.

Q: What are the finite linewidths of all measured lines in spectroscopy caused by? (five points)
A: - doppler broadening
- power broadening
- finite resolution of instruments
- collision broadening
- lifetime broadening

Which approximation greatly simplifies light-matter interaction and when is it actually valid?

What is the equation for the electric dipole?

What is the equation for the electric-dipole interaction of linearly polarised light with a system of charged particles?

NMR vs. optical spectroscopy, what role does spontaneous emission play on the coherence?

Q: What is meant by "coherent" in coherent light-matter interaction and practical terms?
A: In this context, the term "coherent" implies that the light field is a superposition of oscillators with the same frequency and with zero phase difference. In practical terms, this is equivalent to all of the photons in the light source having their crests and troughs aligned as they propagate. It is important to note that this must be consistent for all photons being emitted (i.e. for the total field or beam).

Q: What is the most common way to get a coherent light source?
A: Laser light sources are the most common types of coherent light sources.

Q: What is the rotating wave approximation?
A: The rotating wave approximation is the approximation that disregards any frequencies like 2w or higher, if we are considering a specific frequency and oscillator.

What is the differenct between descirbing a two-level system with the Rabi equations vs. the rate equations?

Was ist die Proportionalität einer Schwingung bezüglich der Masse eines diatomaren Moleküls?

Q: Was ist die Proportionalität von w_ex_e zur reduzierten Masse eines Oszillators?
A: Es is proportional zu (µ)^-1

Wie berechnet man die Anzahl gebundener Zustände für einen anharmonischen Oszillator?

Q: Was ist die Proportionalität der Anzahl gebundenen Zustände eines anharmonischen Oszillators zur reduzierten Masse?
A: N_gebunden α (µ)^1/2

Q: Erkläre den Begriff Fundamentalschwingung:
A: Die Fundamentalschwingung ist der Übergang aus dem Grundzustand in den ersten angeregten Zustand.

Q: Erkläre den Begriff Obertöne:
A: Obertöne sind Übergänge aus dem Grundzustand in höher angeregte Zustände. (Nicht dem ersten angeregten!)

Q: Erkläre den Begriff Kombinationsschwingungen:
A: Kombinationsschwingungen sind Anregungen von mehreren Normalmoden gleichzeitig.

Q: Erkläre den Begriff Hot Bonds:
A: Hot Bonds sind Übergänge aus angeregten Zuständen in höher angeregt Zustände, die aber meist weniger populiert sind.

Was ist die Taylorentwicklung des Dipol Moments für den Harmonischen Oszillator?

Q: Was ist gemeint mit der doppelt harmonischen Approximation?
A: 1. Wir benutzen den Hamiltonian mit dem harmonischen Potential
2. In der Taylorentwicklung des Dipol Moments brechen wir vor der zweiten Ordnung ab

Q: Wie geht das Potential eines induzierten Dipols mit einem Ion mit dem Abstand?
A: r^-4