©, 2017 Uwe Burghaus, Fargo, ND, USA
Electronic spectroscopy with molecules
CHEM 761
/https://spec.jpl.nasa.gov
Definition Molecular spectroscopy is the study of absorption of light by molecules ….
…. well … what else would it be …
Stokes and AntiStokes were not cusins but ...
4
3
2
1
0
Vib
ratio
nal
stat
es
Virt
ual
stat
es
IR absorption
excitation Rayleigh classical scattering
Anti-Stokes Raman scattering
excitation
Stokes Raman scattering
excitation
14
Surface Chemistry Raman spectroscopy
Sir Chandrasekhara Venkata Raman (1888 – 1970) Indian physicist
• Spectra of Atoms and Molecules, 3rd Ed., Peter F. Bernath, Oxford University Press,
Chapter 9
• Just look what you have on “introduction to quantum chemistry” books in your office, e.g. • M. Karplus, R.N. Porter, ch. 5-7 • A. Szabo, N.S. Ostlund, Modern Quantum chemistry • D.B. Cook, Quantum Chemistry • I.N. Levine, Quantum Chemistry, ch. 13-17 • Etc.
Most of these also have chapters about rotations/vibrations of diatomic (see last classes)
CHEM 761
CHEM 761
one electron
atom
multi-electron atoms
molecules
©, 2017 Uwe Burghaus, Fargo, ND, USA
Electronic spectroscopy with molecules • Hamiltonian for diatomic • Theoretical
• LCAO • Idea • Examples • Term symbols • Angular momentum • Selection rules • Intensities
• Experimental • Case studies
CHEM 761
•
•
•
•
• •
•
•
•
• two-nuclei n-electrons
Born Oppenheimer
Approx.
• Schrödinger for e- in field of fixed nuclei • Schrödinger for nuclei
Spherical potential (assumption) • Center of mass motion • Relative motion
kinematicelectronic HHH ˆˆˆ +=
intˆˆˆ HHH nstranslatiokinematic +=
Small vibration rotationsvibrations HHH ˆˆˆ
int +=
Particle-in-box
Harmonic oscillator Heteronuclear Homonuclear
Born-Oppenheimer approximation
See e.g. [Levine]
CHEM 761
LCAO
https://en.wikipedia.org/wiki/Robert_S._Mulliken
Robert Sanderson Mulliken (1896 - 1986)
CHEM 761
LCAO
Robert Sanderson Mulliken (1896 - 1986)
Approximate molecular wave function by molecular orbitals (MOs) that are each a
linear combination (LC) of atomic orbitals (AO), i.e., LCAO.
Idea
Approximate solution for molecules - LCAO “rules”
LCAO
Each MO is assumed being a linear combination of two AOs.
Only AOs of approx. similar energies contribute significantly to a MO.
Nonbonding MOs have no charge depletion and no charge buildup between the nuclei.
Linear Combination of Atomic Orbitals
Bonding MOs have a charge buildup between the nuclei
Antibonding MOs have a charge depletion between the nuclei.
Atkins version
CHEM 761
Bonding / Antibonding
bonding
antibonding
Atkins version
CHEM 761
Bonding / Antibonding
antibonding
An orbital that contributes to a reduction in the cohesion between two atoms contributes to raise the energy of the molecule relative to the separate atoms Antibonding electrons pull the nuclei apart.
Atkins version
CHEM 761
Bonding / Antibonding
antibonding
An orbital that contributes to an enhancement in the cohesion between two atoms contributes to decreasing the energy of the molecule relative to the separate atoms AOs have positive amplitude in the internuclear region
Atkins version
CHEM 761
Example H2 - LCAO
overlap integral
ungerade
gerade Engel/Reid
CHEM 761
MO energy diagrams
AO AO
AO AO
MO
MO
Example H2 - LCAO CHEM 761
Molecules
H2 molecule general case
Set of φi is called the basis set
cij are determined with e.g. the Hartree-Fock method
Results include: Dipole moment, bond dissociation energy, bond length, ionization energies, …
CHEM 761
A covalent bond is a form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms, or between atoms and other covalent bonds.
Covalent bonds
A noncovalent bond is a type of chemical bond, typically between macromolecules, that does not involve the sharing of pairs of electrons, but rather involves more dispersed variations of electromagnetic interactions.
Non-covalent bonds
http://en.wikipedia.org/wiki/Winnipeg
The octet rule is a rule of thumb that states that atoms tend to combine that they each have eight electrons in their valence shells, giving them the same electronic configuration as a noble gas.
In chemistry, a lone pair is a valence electron pair which is not shared with another atom.
Lone pair
Hydroxide Dots: lone pair
In chemistry, sigma bonds (σ bonds) are the strongest type of covalent chemical bond. They are formed by head-on overlapping between atomic orbitals. Sigma bonding is most clearly defined for diatomic molecules using the language and tools of symmetry groups. In this formal approach, a σ-bond is symmetrical with respect to rotation about the bond axis. By this definition, common forms of sigma bonds are s+s, pz+pz, s+pz and dz
2+dz2
(where z is defined as the axis of the bond). Quantum theory also indicates that molecular orbitals (MO) of identical symmetry actually mix. As a practical consequence of this mixing of diatomic molecules, the wavefunctions s+s and pz+pz molecular orbitals become blended. The extent of this mixing (or blending) depends on the relative energies of the like-symmetry MO's.
https://en.wikipedia.org/wiki/Sigma_bond
CHEM 761
In chemistry, pi bonds (π bonds) are covalent chemical bonds where two lobes of one involved atomic orbital overlap two lobes of the other involved atomic orbital. Each of these atomic orbitals is zero at a shared nodal plane, passing through the two bonded nuclei. The same plane is also a nodal plane for the molecular orbital of the pi bond. The Greek letter π in their name refers to p orbitals, since the orbital symmetry of the pi bond is the same as that of the p orbital when seen down the bond axis.
https://en.wikipedia.org/wiki/Pi_bonds
CHEM 761
Symmetry and nomenclature of MOs
Arrow indicates (x,y,z) (-x,-y,-z) transformation to figure the symmetry u vv. g. Engel/Reid
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Ground state MO configurations
LCAO Levine ?
CHEM 761
Ground state MO configurations
LCAO
Look at periodic table of elements O: 1s22s22p4
O2
16 electrons
Note that Hund’s rules and the Pauli principle applies
also to molecules.
CHEM 761
F2
Engel/Reid
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H2+
ground state – contour plot of constant probability density
H2+
1sA 1sB
σu*1s
σg1s bonding
antibonding
Levine ?
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CHEM 761
PChem – Quantum mechanics
n 2S+1 L J
principal quantum number
(defines the energy)
multiplicity (number of possible
different wave functions)
L+S
angular momentum L=0 s state L=1 p state L=2 d state
If we neglect spin-orbit coupling the total energy is independent
of MS and ML
Term: same L and S but different ML and MS
Spectroscopy nomenclature: term symbols
PChem – Quantum mechanics
Spectroscopy nomenclature: term symbols
n 2S+1 L J
principal quantum number
(defines the energy)
multiplicity (number of possible
different wave functions)
L+S
angular momentum L=0 s state L=1 p state L=2 d state
If we neglect spin-orbit coupling the total energy is independent
of MS and ML
Term: same L and S but different ML and MS
n 2S+1 L J
Multi-electron atoms Molecules
n 2S+1 Λ Ω
Spectroscopy nomenclature: term symbols for molecules CHEM 761
Term symbols
for atoms (Tab. 21.5 in Engel)
for molecules (Tab. 26.1 in Engel)
CHEM 761
Angular momenta of diatomic
CHEM 761
Bernath, p. 341, 343
How to obtain these symbols here is explained in example problem 26.2 (page 577) and ch. 26.11.see also question Q26.1
Engel/Reid Bernath, p. 345
CHEM 761
X: ground state
A, B, C: excited states with same multiplicity as ground state
a, b, c: excited states with different multiplicity as ground state
CHEM 761
Atkins Quanta Oxford University Press
IR spectroscopy – rotations & vibrations
Vibration-rotation – EXAMPLE – HCl/DCl
HCl/DCl IR “data” from NDSU Pchem lab class 2008
2600 2800 3000
60
70
80
90
inensity
(a .u.)
wave number in 1/cm
10
21
32
43
01
12
23
P branch R branch
v = 1
v = 0
j = 3
j = 2 j = 1 j = 0
j = 3
j = 2
j = 1 j = 0
2900 2920 2940 2960 2980 3000
60
70
80
90
inens
ity (a
.u.)
wave number in 1/cm
isotope splitting
IR spectroscopy – rotations & vibrations
today – this class What is the
difference?
Electronic spectroscopy with molecules
ground state
v = 1
v = 0
j = 3
j = 2 j = 1 j = 0
j = 3
j = 2
j = 1 j = 0
similar to class 10
Singlet-singlet transitions
0=∆Λ−−
++
Σ→Σ
Σ→Σ11
11for example
P, R branch ∆J = ±1
1±=∆Λ−
+
Σ→Π
Σ→Π11
11for example
Q, P, R branch ∆J = 0, ±1
Non-singlet-singlet transitions complicated by spin-orbit coupling
See also Bernath, ch. 9.3 (15 pages)
Spectra are more complex than for ground state excitation; P, R branch are not separated
• Intrinsic strength of electronic transition Transition dipole moment
• Overlap integral of vibrational wave functions
• Population of levels
Franck-Condon
Boltzmann
neveve ddRrrRre ττψψµ ),(),(112221 ∫−=
)()(),( RrRr veev ψψψ = Born-Oppenheimer
electronic-vibrational wave function
nvveee dRRdrrre τψψτψψµ )()()()(121221 ∫∫−=
vibrational overlap integral
Franck-Condon principle • Vertical lines • Maximize overlap
abso
rptio
n
ener
gy
distance
τψµψ dP ∫= *
transition probability
Frank-Condon principle
Fig. 26.3 -Engel/Reid http://en.wikipedia.org/wiki/Franck-Condon_principle
The principle states that during an electronic transition, a change from one vibrational energy level to another will be more likely to happen if the two
vibrational wave functions overlap more significantly.
Bernath, p. 347 Search Journal of Molecular Spectroscopy for more examples
sequences
Transitions with the same ∆v Example: 0-0; 1-1; 2-2 have all ∆v=0 0-1; 1-2; 2-3; have all ∆v=-1
progressions Example: 3-1; 2-1; 1-1; 0-1
band Example: Low resolution spectrometers may not resolve the rotational levels of each vibrational levels, i.e., only bands are seen
Gas discharge forms metastable Ar which forms excited N2
Also N2 ions can form by direct electron impact.
Emission spectrum is formed by deexcitation
Spectrum recorded with UV vis spectrometer.
∆
Different versions are in use for different applications. The basic idea is the same for all of them. The simplest version is shown on this slide.
area
)(
... ...)()()(
01
321
231201
=
−=
+∆+∆+∆=+−+−+−=
∑∞
=+
nnn
diss
EE
EEEEEEEEEE
dissE
E0
E1
E2
quantum number v’
∆E
area
Since one cannot actually measure all the lines up to the convergence limit an interpolation is used. The convergence limit corresponds to large quantum numbers.
This is a different potential energy than what we use for the
experiment, but same idea.
area
)(
...)()()( ...)()()(
01
321
231201
=
∆−∆∆=
+∆∆+∆∆+∆∆=+∆−∆+∆−∆+∆−∆=
∑∞
=+
nnn
diss
vvv
νν
ννννννν
dissE
E0
E1
E2
quantum number v’
∆ν
area
I + I
I + I*
∆ν V”
V’
E*
area
E starti
i
=
∆∆+∆∆= ∑
)()(* νν
∆ν0
∆ν
quantum number v’
∑
convergence limit
convergence limit
V”=0
Energy transfer of vibrational levels of different electronic states.
How to distinguish directly fluorescence / phosphorescence from the processes shown here?
same spin multiplicity of final and initial states
S1
S0
ener
gy
distance
different spin multiplicity of final and initial states
S1
S0
T1
ener
gy
distance
Is/was a most popular research topic • A single molecule reaction = rather simple • Photon that starts dissociation can be used as a start signal to monitor
the dynamics of the process
ener
gy
distance
direct dissociation
Excitation to purely repulsive potential
ener
gy
distance
direct dissociation
Excitation above dissociation level
Ekin
νex
ener
gy
distance
indirect dissociation
Excitation to bound level but affected by level crossing
pre-dissociation
cf., Bernath, ch. 9.6
v=0
v=1
v=2
v=3
ener
gy
v=0
v=1
v=2
v=3
νex v=0v=2
• tunable cw LASER νex excites molecules
AB AB*
νLIF
• fluorescence relaxation of these excited molecules
What fluorescence spectrum do we see
when we change νex ?
fluor
esce
nce
wavelength
line spectra reflecting
rot/vib population of ground state
vi vf
j
Fluorescence intensity ~ ground state population
fluor
esce
nce
wavelength
line spectra reflecting
rot/vib population of ground state
vi vf
j
Fluorescence intensity ~ ground state population
popu
latio
n
rotation quantum number j
v = 1 v = 2
Zare, R. N. (2012). "My Life with LIF: A Personal Account of Developing Laser-Induced Fluorescence". Annual Review of Analytical Chemistry 5: 1–14.
first reported by Zare et al. in 1968
Stanford University Richard N. Zare
• Born 1939, Cleveland, Ohio, USA • mentored over 150 PhD students and postdoctoral
researchers
https://en.wikipedia.org/wiki/Richard_Zare
v=0
v=1
v=2
v=3
ener
gy
v=0
v=1
v=2
v=3
νex v=0v=2
• tunable LASER νex excites molecules
AB AB*
# of ions, AB+, ~ vib/rot population of ground state
νex
• E.g. same LASER νex ionizes the molecule
AB* AB+ + e-
process has large cross-section
What do we need this
for?
Are used as a detection scheme together with
pump & probe techniques
(see class fs chemistry)
one electron
atom
multi-electron atoms
molecules
n 2S+1 Λ Ω
Search Journal of Molecular Spectroscopy for more examples
• Femtosecond Chemistry, VCH-Verlag (1995), Ed.: J. Manz, L. Wöste • A.H. Zewail, Femtochemistry, World Scientific publications (2000) • C.V. Shank, Generation of Ultrashort Laser pulses, in Topics in Applied Physics Vol. 60 (Ed.: W. Kaiser)
• http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1999/zewail-lecture.pdf • Review, Femtochemistry: Atomic-Scale Dynamics of the Chemical Bond Using Ultrafast Lasers (Nobel Lecture) ,
Ahmed H. Zewail, Angewandte, Volume 39, Issue 15, Pages 2586–2631
• Femtosecond LASER pulses, C. Rullier (Ed.), Publisher springer, Chapter 8
• Tutorials in Molecular Reaction Dynamics, RSC Publishing, 2010, chapter 11 ISBN 9780854041589
