1. 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 )
Diatomic molecules (Ch )
Polyatomic molecules (Ch. 14)
Computational chemistry (Ch. 16)

2. 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

3. An atom has translational and electronic degrees of freedoms.
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
+ Helec(elec)
Eigenvalue (energy) = sum
Htrans(rCM) N
Enuc,elec  
Eigenfunction (wave function) = product
relec (rCM as origin)  
rCM
relec (rCM as origin) N
elec(elec)
rCM (~rN)

4. Separation of Internal Motion from External Motion
H atom made up of proton and electron (2-body problem)
Potential energy V in H atom
Electron
coordinate
Nucleus
Hamiltonian
(2-body)
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!

5. V depends only on r. (”Central Potential”)
where
(3-dim.) 
Hamiltonian in spherical coordinates (r,,) 
Schrödinger equation in spherical coordinates
 r2
r is not coupled with (,).  Separation of variables

7. 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
spherical harmonics
constant radius

8. Separation of Variables -  
where
solved
Angular part (spherical harmonics)
Radial part (Radial equation) 
principal quantum no.
, n-1
angular momentum quantum no.
magnetic quantum no.
(Laguerre polynom.)

9. Separation of variables Radial Schrödinger equation
is spherical harmonic functions. Only function not known is
Radial Schrödinger equation 
Effective potential Veff(r)

10. Effective potential 

12. Separation of Variables -  
where
Angular part (spherical harmonics)
Radial part (Radial equation) 
principal quantum no.
, n-1
angular momentum quantum no.
magnetic quantum no.
(Laguerre polynom.)

13. Eigenvalues (AO Energy levels) & Ionization energy
Total energy eigenvalues are
negative by convention.
(Bound states)  
IE (1 Ry for H)
length
Minimum energy required to remove
an electron from the ground state
atomic units 
energy 1 Ry
depend only on the
principal quantum number.

14. Atomic Units (a.u.)

15. 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 ( )

16. 1885 – Johann Balmer – Line spectrum of hydrogen atoms
1913 – Niels Bohr – Theory of atomic spectra

17. Shells, subshells, and AO energy diagram
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), …, 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?

18. Review: Separation of Variables -  
where
Angular part (spherical harmonics)
Radial part (Radial equation) 
principal quantum no.
, n-1
angular momentum quantum no.
magnetic quantum no.
(Laguerre polynom.)

19. Eigenfunctions (Atomic orbitals): Electronic states
wave functions (eigenfunctions)
nlm nl
Radial wave functions Rnl
Bohr Radius

20. Review: particle-on-a-sphere solutions (MS5118)

21. Review: particle-on-a-sphere solutions (MS5118)

22. Review: particle-on-a-sphere solutions (MS5118)

23. Eigenfunctions (Atomic orbitals): Electronic states
shell shape
symmetry

24. Shells, subshells, and AO energy diagram
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), …, 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?

25. Three quantum numbers, nlm
n: Principal quantum number (n = 1, 2, 3, …)
Determines the energies of the electron
Shells
l: Angular momentum quantum number (l = 0, 1, 2, …, n1)
Determines the angular momentum of the electron
Subshells
Ll =
(s, p, d, f,…)
m: magnetic quantum number (m = 0, 1, 2, …, l)
Determines z-component of angular momentum of the electron
Lz, m =

26. Let’s focus on the radial wave functions Rnl.
How far the shell is apart from the nucleus 1s 2s 2p 3s 3p 3d 1 1
1 radial
node
2 radial nodes 1 1 1 1
*Bohr Radius
*Reduced distance
1 radial node
(ρ = 4, )

27. Radial wave functions (l = 0, m = 0): s orbitals

29. Wave function
Probability density

30. Radial distribution function (RDF)
Probability density.
Probability of finding an electron at a point (r,θ,φ)
Integral over θ and φ
Radial distribution function.
Probability of finding an electron at any radius r
Wave function (1s)
Radial distribution function (1s)
Q: Derive it!
Bohr radius

31. Atomic Units (a.u.)

32. Question:

34. 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 )

35. Radial wave functions (l  0)
Probability density

37. Separation of variables Radial Schrödinger equation
is spherical harmonic functions. Only function not known is
Radial Schrödinger equation 
Effective potential Veff(r)

38. Effective potential 

40. Spectroscopic transitions and Selection rules
Transition (Change of State)
hcRH
n1, l1,m1
Photon
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

41. Review: Shells, subshells, and AO energy diagram
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), …, 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

42. Spectra of hydrogen atom (or hydrogen-like atoms)
Balmer, Lyman and Paschen Series (J. Rydberg)
n1 = 1 (Lyman), 2 (Balmer), 3 (Paschen)
n2 = n1+1, n1+2, …
RH = cm-1 (Rydberg constant)
hcRH
Electric discharge is passed through gaseous hydrogen.
H2 molecules and H atoms emit lights of discrete frequencies.

44. 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 ( )

45. 1885 – Johann Balmer – Line spectrum of hydrogen atoms
1913 – Niels Bohr – Theory of atomic spectra