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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)
Lecture 1. The Simplest Chemical System.
Hydrogen Atom. Part 1.
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 Chemistry
http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
A molecule has translational, vibrational, rotational, and
electronic degrees of freedoms. In the case of atoms…
• An atom has translational and electronic degrees of freedoms.
• To good approximation, degrees of freedom are not coupled. 
separation of variables!
N
N
N
N
Htrans(rCM)
• Eigenvalue (energy) = sum
N
Enuc,elec
• Eigenfunction (wave function) = product
N
elec(elec)
+ Helec(elec)


rCM
relec (rCM as origin)
 r (r as origin)
elec
CM
rCM (~rN)
H atom made up of proton and electron
(2-body problem)
Potential energy V in H atom
Hamiltonian
(2-body)
2
2
2


Ze
2
2
H  Eˆ K ,electron  Eˆ K ,nucleus  V  
e 
N 
2me
2mN
40 r
Electron
coordinate
Nucleus
coordinate
Separation of Internal Motion from External Motion
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!
2 2
2
Ze 2
2
H 
e 
N 
2me
2mN
40 r
2 2
Ze 2
H 
 
2
40 r
1


1
1
1


me mN me
V depends only on r. (”Central Potential”)
2 2
Ze 2

 
2
40 r

where
(3-dim.)
Hamiltonian in spherical coordinates (r,,)

Ze 2

40 r
Schrödinger equation in spherical coordinates
 r2
r is not coupled with (,).  Separation of variables
Compare with a particle-on-a-sphere case (MS5118)
constant radius r0 from the origin (“rigid” rotor)
Schrödinger equation in cartesian coordinates
constant (r0)
Spherical polar coordinates in 3D
Schrödinger equation in spherical polar coordinates
constant radius
spherical harmonics
Separation of Variables



-
where
solved
Angular part (spherical harmonics)
Radial part (Radial equation)

, n1
angular momentum quantum no.
magnetic quantum no.
Z 2 e 4
En  
32 2 e02 2 n 2
with n  1,2,3...
principal quantum no.

Rn ,l (r )  N n ,l   Ln ,l (  )e   / 2 n (Laguerre polynom.)
n
l
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 part (spherical harmonics)
Radial part (Radial equation)

, n1
angular momentum quantum no.
magnetic quantum no.
Z 2 e 4
En  
32 2 e02 2 n 2
with n  1,2,3...
principal quantum no.

Rn ,l (r )  N n ,l   Ln ,l (  )e   / 2 n (Laguerre polynom.)
n
l
Eigenvalues (AO Energy levels) & Ionization energy
Total energy eigenvalues are
negative by convention.
(Bound states)
Z 2 e 4
En  
32 2e02 2 n 2
with n  1,2,3...
40  2
a0 
 ee2
m
length

atomic units
energy
1
Ry
depend only on the
principal quantum number.
IE (1 Ry for H)
Minimum energy
required to remove
an electron from
the ground state
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, subshells, and AO energy diagram
Shells:
n = 1 (K), 2 (L), 3 (M), 4(N), …
En  
Z 2 e 4
32 2 e02 2 n 2
with n  1,2,3...
Sub-shells (for each n):
l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1
m = 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
Question: Is this AO energy diagram the same as what you have known?
Review: Separation of Variables



-
where
Angular part (spherical harmonics)
Radial part (Radial equation)

, n1
angular momentum quantum no.
magnetic quantum no.
Z 2 e 4
En  
32 2 e02 2 n 2
with n  1,2,3...
principal quantum no.

Rn ,l (r )  N n ,l   Ln ,l (  )e   / 2 n (Laguerre polynom.)
n
l
Eigenfunctions (Atomic orbitals): Electronic states
1-electron
wave functions
(eigenfunctions)
nlm
nl

(r )  N n ,l   Ln ,l (  )e   / 2 n
n
l
Radial wave functions Rnl
Bohr Radius
40  2
a0 
me e 2
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, subshells, and AO energy diagram
Shells:
n = 1 (K), 2 (L), 3 (M), 4(N), …
En  
Z 2 e 4
32 2 e02 2 n 2
with n  1,2,3...
Sub-shells (for each n):
l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1
m = 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
Question: Is this AO energy diagram the same as what you have known?
Three quantum numbers, nlm
n: Principal quantum number (n = 1, 2, 3, …)
Determines the energies of the electron
Z 2 e 4
En  
32 2 e02 2 n 2
Shells
with n  1,2,3...
l: Angular momentum quantum number (l = 0, 1, 2, …, n1)
Determines the angular momentum of the electron
Ll =
l(l  1)1/2 
Subshells
with l  0,1,.., n  1 (s, p, d, f,…)
m: magnetic quantum number (m = 0, 1, 2, …, l)
Determines z-component of angular momentum of the electron
Lz, m = ml with ml  0,1,2,...,l
Let’s focus on the radial wave functions Rnl.
How far the shell is apart from the nucleus
1s
2 radial nodes
1 radial
node
1
1
2s
1
2p
1
3s
1
3p
1
3d
*Bohr Radius
40  2
a0 
me e 2
*Reduced distance
1 radial node
(ρ = 4, r  2a0 / Z )

2 Zr
a0
Radial wave functions (l = 0, m = 0): s orbitals
Wave function
Probability density
Radial distribution function (RDF)

2
Probability density.
Probability of finding an electron at a point (r,θ,φ)
Integral over θ and φ
P(r )  4r 2 2 Radial distribution function.
Probability of finding an electron at any radius r
Wave function (1s)
  e  2 Zr / a
2
Radial distribution function (1s)
0
4Z 3 2 2 Zr / a 0
P(r )  3 r e
a0
Q: Derive it!
Bohr radius
40  2
a0 
me e 2
Atomic Units (a.u.)
Question:
Radial wave functions for l  0 :
np orbitals (l = 1) and nd orbitals (l = 2)
p orbital for n = 2, 3, 4, …
( l = 1; ml = 1, 0, 1 )
2p
3p
3d
d orbital for n = 3, 4, 5, …
(l = 2; ml = 2, 1, 0, 1, 2 )
Radial wave functions (l  0)
Wave function
Probability density
Separation of variables
is spherical harmonic functions. Only function not known is
Radial Schrödinger equation

Effective potential Veff(r)
Effective potential

Spectroscopic transitions and Selection rules
Transition (Change of State)
n1, l1,m1
En  
Photon
E  hv
4
 1
1 


2
2 
 n1 n2 
~  RH 
n2, l2,m2
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.)
Selection rule for hydrogen atom
l  1
hcRH
Z e
32 2 e02 2 n 2
2
ml  0,1
with
Review: Shells, subshells, and AO energy diagram
Shells:
n = 1 (K), 2 (L), 3 (M), 4(N), …
En  
Z 2 e 4
32 2 e02 2 n 2
with n  1,2,3...
Sub-shells (for each n):
l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1
m = 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
Selection rule for hydrogen atom
l  1
ml  0,1
Spectra of hydrogen atom (or hydrogen-like atoms)
Balmer, Lyman and Paschen Series (J. Rydberg)
 1
1 


2
2 
 n1 n2 
~  RH 
En  
hcRH
Z e
32 2 e02 2 n 2
2
4
with n  1,2,3...
n1 = 1 (Lyman), 2 (Balmer), 3 (Paschen)
n2 = n1+1, n1+2, …
RH = 109667 cm-1 (Rydberg constant)
Electric discharge is passed through gaseous hydrogen.
H2 molecules and H atoms emit lights of discrete frequencies.
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