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