Post on 13-Dec-2015
Quantum Chemistry: Our Agenda (along with Engel)
• Postulates in quantum mechanics (Ch. 3)• Schrödinger equation (Ch. 2)
• Simple examples of V(r) Particle in a box (translation, etc.) (Ch. 4-5) Harmonic oscillator (vibration) (Ch. 7-8) Particle on a ring or a sphere (rotation) (Ch. 7-8)
• Extension to chemical systems (electronic structure)
Hydrogen(-like) atom (one-electron atom) (Ch. 9) Many-electron atoms (Ch. 10-11) Diatomic molecules (Ch. 12-13) Polyatomic molecules (Ch. 14) Computational chemistry (Ch. 16)
References for Part 1 (Atoms)
• Quantum Chemistry, Engel (3rd ed. 2013)• Quantum Mechanics in Chemistry, Ratner & Schatz (2001)• Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005) • Quantum Chemistry, D. A. McQuarrie• Elementary Quantum Chemistry, F. L. Pilar (2003)• Introductory Quantum Mechanics, R. L. Liboff (4th ed. 2004)
• A Brief Review of Elementary Quantum Chemistryhttp://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
Lecture 1. The Simplest Chemical System. Hydrogen Atom. Part 1.
• An atom has translational and electronic degrees of freedoms.
• Eigenfunction (wave function) = product
• To good approximation, degrees of freedom are not coupled. separation of variables!
• Eigenvalue (energy) = sum
+ Helec(elec)
elec(elec)
NN N N
Enuc,elec
N
N
rCM (~rN)
Htrans(rCM)
relec (rCM as origin)relec (rCM as origin)
rCM
A molecule has translational, vibrational, rotational, and electronic degrees of freedoms. In the case of
atoms…
H atom made up of proton and electron (2-body problem)
Full Schrödinger equation can be separated into two equations: 1. Atom as a whole through the space (rCM ~ rNucleus);
2. Motion of electron around the nucleus (relec with nucleus at origin).
“Electronic” structure (1-body problem): Forget about nucleus!
r
ZeH
0
22
2
42
eNe mmm
1111
Separation of Internal Motion from External Motion
Potential energy V in H atom
Hamiltonian(2-body) r
Ze
mmVEEH N
Ne
enucleusKelectronK
0
22
22
2
,, 422ˆˆ
Electron coordinate
Nucleus coordinate
r
Ze
mmH N
Ne
e 0
22
22
2
422
Hamiltonian in spherical coordinates (r,,)
V depends only on r. (”Central Potential”)
r
Ze
0
22
2
42
where (3-dim.)
r
Ze
0
2
4
Schrödinger equation in spherical coordinates
r is not coupled with (,). Separation of variables
r2
spherical harmonics
Compare with a particle-on-a-sphere case (MS5118)
constant radius
constant radius r0 from the origin (“rigid” rotor)
constant (r0)
Schrödinger equation in cartesian coordinates
Schrödinger equation in spherical polar coordinates
Spherical polar coordinates in 3D
Separation of Variables
-
where
angular momentum quantum no.
magnetic quantum no.
Angular part (spherical harmonics)Radial part (Radial equation)
...3,2,1 with 32 222
02
42
nne
eZEn
, n1principal quantum no.
nln
l
lnln eLn
NrR 2/,,, )()(
(Laguerre polynom.)
solved
Separation of variables
is spherical harmonic functions. Only function not known is
Radial Schrödinger equation
Effective potential Veff(r)
Effective potential
Separation of Variables
-
where
angular momentum quantum no.
magnetic quantum no.
Angular part (spherical harmonics)Radial part (Radial equation)
...3,2,1 with 32 222
02
42
nne
eZEn
, n1 principal quantum no.
nln
l
lnln eLn
NrR 2/,,, )()(
(Laguerre polynom.)
length
energy
Eigenvalues (AO Energy levels) & Ionization energy
2
20
0
4
ema
e
Total energy eigenvalues are negative by convention. (Bound states)
...3,2,1 with 32 222
02
42
nne
eZEn
depend only on the principal quantum
number.
1Ry
Minimum energy required to remove an electron from the ground state
IE (1 Ry for H)
atomic units
Atomic Units (a.u.)
Engel says in p. 12 that “Bohr (1913) next introduced wave-particle duality which is equivalent to asserting that the electron had the de Broglie wavelength (1924).”
Bohr Atom Model (1911-1913)
1885 – Johann Balmer – Line spectrum of hydrogen atoms
1913 – Niels Bohr – Theory of atomic spectra
Shells:n = 1 (K), 2 (L), 3 (M), 4(N), …
Sub-shells (for each n):
l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1m = 0, 1, 2, …, l
Number of orbitals in the nth shell: n2
(n2 –fold degeneracy)
Examples : Number of subshells (orbitals) n = 1 : l = 0 → only 1s (1) → 1 n = 2 : l = 0, 1 → 2s (1) , 2p (3) → 4 n = 3 : l = 0, 1, 2 → 3s (1), 3p (3), 3d (5) → 9
Shells, subshells, and AO energy diagram
...3,2,1 with 32 222
02
42
nne
eZEn
Question: Is this AO energy diagram the same as what you have known?
Review: Separation of Variables
-
where
angular momentum quantum no.
magnetic quantum no.
Angular part (spherical harmonics)Radial part (Radial equation)
...3,2,1 with 32 222
02
42
nne
eZEn
, n1 principal quantum no.
nln
l
lnln eLn
NrR 2/,,, )()(
(Laguerre polynom.)
Radial wave functions Rnl
Eigenfunctions (Atomic orbitals): Electronic states
nlm nl
nln
l
lnln eLn
NrR 2/,,, )()(
1-electron wave
functions (eigenfunctio
ns)
Bohr Radius
2
20
0
4
ema
e
Review: particle-on-a-sphere solutions (MS5118)
Review: particle-on-a-sphere solutions (MS5118)
Review: particle-on-a-sphere solutions (MS5118)
Eigenfunctions (Atomic orbitals): Electronic states shell shape
symmetry
Shells:n = 1 (K), 2 (L), 3 (M), 4(N), …
Sub-shells (for each n):
l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1m = 0, 1, 2, …, l
Number of orbitals in the nth shell: n2
(n2 –fold degeneracy)
Examples : Number of subshells (orbitals) n = 1 : l = 0 → only 1s (1) → 1 n = 2 : l = 0, 1 → 2s (1) , 2p (3) → 4 n = 3 : l = 0, 1, 2 → 3s (1), 3p (3), 3d (5) → 9
Shells, subshells, and AO energy diagram
...3,2,1 with 32 222
02
42
nne
eZEn
Question: Is this AO energy diagram the same as what you have known?
n: Principal quantum number (n = 1, 2, 3, …)Determines the energies of the electron
...3,2,1 with 32 222
02
42
nne
eZEn
Shells
Subshells
l ,...,2,1,0m with m llLz, m =
1,..,1,0 with 1)l(l 1/2 nlLl = (s, p, d, f,…)
Three quantum numbers, nlm
l: Angular momentum quantum number (l = 0, 1, 2, …, n1)
Determines the angular momentum of the electron
m: magnetic quantum number (m = 0, 1, 2, …, l) Determines z-component of angular momentum of the
electron
Let’s focus on the radial wave functions Rnl.
1s
2s
2p
3s
3p
3d
*Reduced distance
*Bohr Radius
2
20
0
4
ema
e
0
2
a
Zr1 radial node
(ρ = 4, ) Zar /2 0
2 radial nodes
1 radial node
1
1
1
1
1
How far the shell is apart from the nucleus
1
Radial wave functions (l = 0, m = 0): s orbitals
Probability densityWave function
Probability density. Probability of finding an electron at a point (r,θ,φ)
2
224)( rrP
0/2230
34)( aZrer
a
ZrP
Radial distribution function (RDF)
Integral over θ and φ
Wave function (1s) Radial distribution function (1s)
Bohr radius
Radial distribution function. Probability of finding an electron at any radius r
0/22 aZre
2
20
0
4
ema
e
Q: Derive it!
Atomic Units (a.u.)
Question:
p orbital for n = 2, 3, 4, … ( l = 1; ml = 1, 0, 1 )
Radial wave functions for l 0 :np orbitals (l = 1) and nd orbitals (l = 2)
d orbital for n = 3, 4, 5, … (l = 2; ml = 2, 1, 0, 1, 2 )
2p
3p
3d
Radial wave functions (l 0)
Probability density
Wave function
Separation of variables
is spherical harmonic functions. Only function not known is
Radial Schrödinger equation
Effective potential Veff(r)
Effective potential
All possible transitions are not permissible.Photon has intrinsic spin angular momentum : s = 1
d orbital (l=2) s orbital (l=0) (X) forbidden
(Photon cannot carry away enough angular momentum.)
n1, l1,m1
n2, l2,m2
PhotonhvE
Spectroscopic transitions and Selection rules
Selection rule for hydrogen atom 1,0 lm1l
Transition (Change of State)
22
21
11~nn
RH
hcRH
Shells:n = 1 (K), 2 (L), 3 (M), 4(N), …
Sub-shells (for each n):
l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1m = 0, 1, 2, …, l
Number of orbitals in the nth shell: n2
(n2 –fold degeneracy)
Examples : Number of subshells (orbitals) n = 1 : l = 0 → only 1s (1) → 1 n = 2 : l = 0, 1 → 2s (1) , 2p (3) → 4 n = 3 : l = 0, 1, 2 → 3s (1), 3p (3), 3d (5) → 9
Review: Shells, subshells, and AO energy diagram
...3,2,1 with 32 222
02
42
nne
eZEn
Selection rule for hydrogen atom 1,0 lm1l
Balmer, Lyman and Paschen Series (J. Rydberg)
n1 = 1 (Lyman), 2 (Balmer), 3 (Paschen)
n2 = n1+1, n1+2, …
RH = 109667 cm-1 (Rydberg constant)
Spectra of hydrogen atom (or hydrogen-like atoms)
Electric discharge is passed through gaseous hydrogen.H2 molecules and H atoms emit lights of discrete frequencies.
22
21
11~nn
RHhcRH
Engel says in p. 12 that “Bohr (1913) next introduced wave-particle duality which is equivalent to asserting that the electron had the de Broglie wavelength (1924).”
Bohr Atom Model (1911-1913)
1885 – Johann Balmer – Line spectrum of hydrogen atoms
1913 – Niels Bohr – Theory of atomic spectra