1. 2013-1 BTE3510 - 01 Computational Chemistry
Tue. 2:00~3:50PM, Thur. 11:00AM
Engineering A577
NO, Kyoung Tai (盧敬泰)
Department of Biotechnology

2. Chapter 13 Atomic Structure Hydrogenic Atoms

3. The Spectra of Hydrogenic Atoms
Balmer, Lyman, Paschen, Brackett Series
Rydberg

4. The line spectra of several elements
Note: prism separates the different ’s, and red light bends least

5. R is the Rydberg constant = 1.096776 m-1
Johan Rydberg ( ), a Swedish mathematician 1 l 1
n12 1
n22 = R -
Rydberg empirically fit the observed wavelengths of the lines to this equation.
R is the Rydberg constant = m-1
Three series of spectral lines of atomic hydrogen
for the visible series, n1 = 2 and n2 = 3, 4, 5, ...

6. Results (not derivation) of Bohr Theory
RH=2.18x10-18 J
See page 265
Planck/Einstein: or
all known constants
Bohr predicted Rydberg equation (empirical) from a model (theoretical)!
(But… the equation works only for single-electron species: H, He+(Z=2),Li2+ (Z=3))

7. N = 1 (ground state), n > 1 (excited state)
The Bohr explanation of the three series of spectral lines.
Unlike the energy level diagram, the orbital diagram is not drawn to scale
E between any excited state and the ground state (n = 1) is big. The emission and absorption lines fall for such electronic transitions appear in the uv (high energy) region.
N = 1 (ground state), n > 1 (excited state)
Note: spacings are uneven due to 1/n2 dependency (eqn. 7.4)

8. Hydrogenic Atoms The Schrödinger Equation for Hydrogenic Atoms
Ze+ e- r mN me
The Schrödinger Equation for Hydrogenic Atoms
n : principle quantum number
Z=1, RH~R with Reduced mass

9. Energy Level of Hydrogen Atom
Electron
Affinity
Ionization Potential
Electron Affinity
Electronegativity
Ionization
Energy

10. Cartesian Coordinates
Hydrogen Like Atom
(x,y,z)
Cartesian Coordinates
(x, y, z)
Polar Coordinates
(r, q, f)

11. Quantum Numbers Theoretical description of the hydrogen atom:
Schrödinger equation: HY = EY
Hamiltonian operator (kinetic + potential energy of an electron)
Y - wave function (eigenfunction);
YY* is proportional to the probability of finding a particle in a given point of space
E – energy (eigenvalue)

12. Solution of the Schrödinger equation for the hydrogen atom.
Separation of variables r, q, f
For the case of the hydrogen atom the Schrödinger equation can be solved exactly when variables r, q, f are separated like
Y = R(r) Q(q) F(f)
Three integer parameters appear in the solution, n, l, m
n = 1, 2, … while solving R-equation
l = n-1, n-2, …, while solving Q-equation
ml = -l, -l+1, …, 0, …, l-1l while solving F-equation

13. The Hydrogen Atom orbitals
Radial components of the hydrogen atom eigenfunctions
(a0 = , the first orbit radius, Z = 1)
for n = 1, l = 0, ml = 0 1s orbital
for n = 2, l = 0, ml = 0 2s orbital
for n = 2, l = 1, ml = pz orbital
Exponential decay, slower for larger n
Presence of nodes (Y=0), n-l-1 in total

14. Angular Component
Angular components of the hydrogen atom eigenfunctions
QF = for l = 0, ml = 0 S orbitals
QF = for l = 1, ml =0 pz orbitals
QF = for l = 2, ml = 0 dz2 orbitals

15. Quantum Numbers
In solving Schrodinger equation for hydrogen-like atoms, there
are three boundary conditions.
r : wave function must converge to zero
q : must take values between 0~360
f : must take values between 0~180
Quantum no.
Symbol
Allowed values
Principle n
1,2,3,…..
Orbital l
0,1,2,…., n-1
Magnetic ml
l, l-1, l-2,…., -l+1, -l
Spin ms
+1/2, -1/2

16. Quantum Numbers of Hydrogenic Atoms
Orbital is specified by all the three quantum numbers.
Only for hydrogenic atoms, the energy depends only on the principle quantum number.
All orbitals of same value of n but different values of l and ml have the same energy.
Wave functions are Degenerate
n ……….
K L M N
l ……….
s p d f
no. of ml Degeneracy

17. The average distance of an electron from the nucleus of a hydrogenic atom of atomic number Z
n = 2, l = 0, ml = 0
n = 2, l = 1, ml = 0

18. Shells of Electrons

19. Hydrogen Atom

20. Hydrogenic Wave Functions

21. Wave Functions; Electron Density
Radial dependence of the
wavefunction of a 1s orbital
and corresponding probability
density

22. Wave Functions; Electron Density
Radial Distribution Function

23. Wave Functions: R(r)
Nodes

24. Wave Functions: P Orbitals
The l=1 hydrogen-like wave function has three values of m associated with it, m=-1, 0, 1. The m=0 orbital is easily recognized as the pz orbital. The -1 and +1 orbitals have a torroidal shape.

25. Wave Functions: d Orbitals

26. Angular Momentum and its Components
Magnitude of angular momentum
Component of angular momentum

27. Electron Spin (Intrinsic Spin)
The spin of an electron is an intrinsic angular momentum that every electron possesses and that can not be changed or eliminated.
The intrinsic spin is a purely quantum mechanical phenomenon and has no classical counterpart.
Electron spin is described by a spin quantum number, s, ½
the spin can be clockwise and counterclockwise
spin magnetic quantum number, ms, +1/2, -1/2
Stern-Gerlach experiment

28. Fermion and Boson Fermion spin ½ particles (s=1/2)
electron, proton, neutron
Boson spin 1 particles (s=1)
photon

29. Spectral Transitions & Selection Rules
Not all transitions between all available orbitals are possible
Allowed transition Forbidden transition
Selection Rule
No restriction on n,
Transition probability
Grotrian diagram of atomic hydrogen

30. Chapter 13 Atomic Structure The Structures of Many-electron Atoms

31. Above three-body problem can not be exactly solved
both with Classical Mechanics and Quantum Mechanics

32. Born-Oppenheimer Approximation
The nuclei (nucleus) of atoms (atom) are considerably heavier
and slower than electrons and to a good approximation,
i) the electrons can be considered to be moving in a field of
fixed nuclei
ii) the nuclei can be considered to be moving in a averaged
field produced by fast moving electrons.
This allows us to separate the Schrödinger equation into a
nuclear and electronic part.

33. The Orbital Approximation
simplify the problem by using one electron functions, or orbitals,
to approximate the full wave function.
The motions of the electrons are assumed to be independent,
and each electron is assigned its own spin orbital (a product of
a spatial function and a spin function).
The wave function of atom with n electron can be written as
Orbital approximation allows us to express electronic
structure of an atom by reporting its electron configuration

34. Pauli Exclusion Principle
No more than two electrons may occupy any given orbital, and if two electrons do occupy one orbital, then their spins must be paired.

35. Penetration and Shielding

36. Penetration and Shielding

37. Penetration and Shielding

38. Penetration and Shielding

39. Building-up Principle

40. Chapter 13 Atomic Structure Periodic Trends in Atomic Properties

41. Atomic Radius and Ionization E.
Both atomic radius and ionization energy are correlated with effective atomic charge.

42. Effective Nuclear Charge
Effective Nuclear charge: Z* = Z – σ (σ = Screening Constant)
How to determine Z*?
If the electron resides in s or p orbital
Electrons in principal shell higher than the e- in question
contribute 0 to σ
Each electron in the same principal shell contribute 0.35 to σ
Electrons in (n-1) shell each contribute 0.85 to σ
Electrons in deeper shell each contribute 1.00 to σ
Example: Calculate the Z* for the 2p electron
Fluorine (Z = 9) 1s2 2s2 2p5
Screening constant for one of the outer electron (2p):
6 (six) (two 2s e- and four 2p e-) = 6 X 0.35 = 2.10
2 (two)1s e- = 2 X 0.85 = 1.70
σ = = then Z* = = 5.20

43. Effective Nuclear Charge
If the e- resides in a d or f orbital
All e-s in higher principal shell contribute 0
Each e- in same shell contribute 0.35
All inner shells in (n-1) and lower contribute 1.00

44. Atomic Radius
The METALLIC RADIUS is half of the experimentally determined distance between the nuclei of nearest neighbors in the solid
The COVALENT RADIUS of a non-metallic element is half of the experimentally determined distance between the nuclei of nearest neighbors in the solid
The IONIC RADIUS of an element is related to the distance between the nuclei of neighboring cations and anions

45. The Ideal Gas Law “compressibility factor” Ideal gas: z = 1
Deviations from Ideal Gas Behavior
“compressibility factor”
Ideal gas: z = 1
z < 1: attractive intermolecular forces dominate
z > 1: repulsive intermolecular forces dominate

46. Van der Waals equation

47. Ionization Potential

48. Electron Affinity Electron Attachment Enthalpy, DH°ea
The enthalpy change for the gain of an electron, E(g) + e-  E-(g)
Electron Affinity: EA = -DH°ea + 5/2 RT EA = -DH°ea
Cl(g) + e-  Cl-(g) DH°ea = -349 kJ/mol
O(g) + e-  O-(g) DH°ea = -142 kJ/mol
first attachment is usually exothermic
O-(g) + e-  O2-(g) DH°ea = 844 kJ/mol
second attachment is usually endothermic
O(g) + 2e-  O2-(g) DH°ea = 702 kJ/mol
Other factors favour the presence of O2- when it is found in molecules and ionic solids.

49. Electron Affinity

50. Reduction-Oxidation (RedOx) Reactions
The gain and loss of electrons drives some of the most powerful forms of chemical reactions.
Reduction – gain of electrons
Oxidation – loss of electrons
DE°, the standard potential for an equilibrium, gives access to DG° through the following relationship:
DG° = - nFDE°
where,
n = number of electrons involved
F = Faraday’s constant = kJ mol-1 V-1 (e-)-1
Note: if DG° < 0, then must be DE° > 0
So favorable reactions must have DE° > 0
This is more important for Analytical chemistry, but we might talk more about redox reactions later.

51. Ionization E, Electron Affinity, Electronegativity+1 +2 +3 -1 -2
Net Atomic Charge
Energy
2nd IP
1st IP
1st EA

52. Electronegativity Linus Pauling
Electronegativity, c, The ability of an atom in a molecule to attract electrons in a bond to itself.
X increases
X decreases
Linus Pauling
Traditional scale goes from 0 to 4 with c of F set to 4.
First Year rule
Dc > 2 : ionic
2 > Dc > 0.5 : polar
Dc < 0.5 : covalent

53. D’(A-B) is the ionic resonance energy in kJ/mol (D(A-B) is in eV)
Electronegativity
Pauling’s definition:
Pauling reasoned that the dissociation energy of a purely covalent bond A-B should be the mean of the dissociation energies for the homonuclear bonds A-A and B-B. Any additional energy must be caused by electrostatic attraction between A and B (attributed to ionic character in a bond). The ionic character must be related to the difference in the electronegativities of A and B. He calculated this difference as follows:
D’(A-B) is the ionic resonance energy in kJ/mol (D(A-B) is in eV)
A-B A+ B-
D(A-B),theory = ½ (D(A-A) + D(B-B))
D’(A-B) = D(A-B),experimental - D(A-B),theory
XA – XB = (D’(A-B))½
0.102 is a conversion from kJ/mol to eV
(note: DHd (A-B) = D(A-B))

54. An example calculation for H-F
D(H-F),theory = ½ (D(H-H) + D(F-F)) = ½ ( ) = 297 kJ/mol
D’(H-F) = D(H-F),experimental - D(H-F),theory = 566 – 297 = 269 kJ/mol
XF – XH = (D’(H-F))½ = (269)½ = 1.67
Pauling set XF = 4.0 so: XH = 4.0 – 1.67 = 2.32
Note: is different than the value of 2.2 you see in tables because Pauling used the geometric mean instead of the arithmetic mean.
Similar calculations were used to determine X for the other elements.
(D(H-Cl) )½ = 0.98 eV relative to H so XCl  3.2
(D(H-Br) )½ = 0.73 eV relative to H so XBr  2.9
(D(H-I) )½ = 0.25 eV relative to H so XI  2.5

55. Mulliken’s Electronegativity,
Mulliken figured that the electronegativity of an element must be related to the energies of gaining and losing electrons. Specifically an atom that binds it’s electrons stongly (large DH°ie) and gains other electrons readily (very positive EA or very negative DH°ea) should do the same in molecules. Thus Mulliken calculated the electronegativity of an atom as the mean of the ionization potential and the electron affinity. For A-B, the electronegativity difference between A+B- and A-B+ is given by:
XA – XB = ½ ([IPA + EAA] – [IPB + EAB])
XA = ½ ([IPA + EAA])
(these are then scaled to fit the  0-4 scale)
Robert Mulliken
(note: DHie A = IPA)
This method makes a lot of sense, but is not used because values of DH°ea have not been accurately determined for many elements.

56. Allred-Rochow Electronegativity,
The assumption is that the force that will draw an electron toward an atom is proportional to the effective nuclear charge of that atom and related to the distance of the electron from the nucleus.
Z* = effective nuclear charge
e = charge of electron
e0 = permittivity of a vacuum
r = atomic radius
The equation: X = (Z*/r2)
puts the calculated values on the Pauling scale. This definition is useful because it can be applied to many more atoms and is one of the most used scales.

57. Electronegativity
The trends in electronegativities and ionization enthalpies explain many features of chemistry such as the “diagonal relationship” (X) and the position of the metallic and non-metallic elements (DH°ie). Electronegativity also lets us predict the polarity of bonds and chemical reactivity.

58. Electronegativity

59. Total Energy

60. Mulliken-Jaffe Electronegativity

61. Electronegativity Equalization

62. Electronegativity Equalization
XA > XB XA XB
qA=0 XA
qB=0 XB XA XB
q’A<0
XA > X’A
q’B>0
XB < X’B XA XB
q’A>q’’A
X’A > X’’A
q’B<q’’B
X’B < X’’B
XA = XB

63. Polarizability
The polarizability,, of an atom is its ability to be distorted by the presence of an electric field (such as a neighbouring ion). The more easily the electron cloud is distorted, the higher . This happens primarily with large atoms and anions that have closely spaced frontier orbitals (HOAO and LUAO).
 decreases
 increases

64. Hard and Soft Atoms
The hardness,, of an atom is a related quantity. Hard atoms (high ) bind their electrons tightly and are not easily polarized. Soft atoms (low ) bind their electrons loosely and have a higher .
= ½ ([IPA - EAA]) in eV Si  3.4 F  7.0
Sn  3.0 I  3.7
The hardness,, of an atom or ion can also provide us with information about the chemistry that will happen between different reagents. In general, hard acids tend to form compounds with hard bases and soft acids tend to bind to soft bases.

65. Electronegativity Equalization
XLUAO,A> XHOAO,A
LUAO
HOAO, B
LUAO,A
HOAO

66. Electronegativity Equalization
LUAO
XLUAO,A>= XHOAO,A
LUAO,A
HOAO, B
HOAO

67. Electrostatic Interaction

68. Energy Conversion- the Mitochondrion
Chemiosmotic coupling
Mitochondria (and chloroplasts) use chemiosmotic coupling to harness energy Reflects a link between chemical bond-forming reactions that generate ATP (‘chemi’) and membrane transport (‘osmotic)

69. Chemiosmotic coupling

70. ATP Synthesis at Mitochondrion

71. How electrons are donated by NADH

72. Biological Oxidation

73. Ferry electrons from one complex to the next

74. H.W.
Proteins – Redox process

75. Chapter 13 Atomic Structure The Spectra of Complex Atoms

76. States and Orbitals ……… ? E5 - E4 = E2 - E1 Orbitals State1 State2

77. Atomic Spectra L 0 1 2 3 4 ……… S P D F G
Spectra of many electron atom is very complicated
Need the notation to specify the states of atoms
Total orbital angular momentum quantum number, L
L ………
S P D F G
To find L, we identify the orbital angular momentum quantum number of the electron, for two electrons in valance orbital, l1 and l2
L = l1+l2, l1+l2-1, …, |l1-l2|
The highest total orbital angular momentum occurs when two electrons are orbiting in the same direction; the lowest occurs when they are orbiting in opposite direction

78. Atomic Spectra S=s1+s2, s1+s2-1, …, |s1-s2|
Spin angular momentum quantum number, S
S=s1+s2, s1+s2-1, …, |s1-s2|
for two electrons,
S=1/2+1/2, 1/2+1/2-1, …, |1/2-1/2| =1,0
Multiplicity 2S+1
Total angular momentum quantum number, J
J=L+S, L+S-1, ….., |L-S| Degeneracy 2J+1

79. Term Symbol of Excited Carbon
[He] 2s22p13p1
l1=1, l2=1
L = 1+1, 1+1-1, 1+1-2, …, |1-1|=2,1,0 D, P, S
S=1,0 S=1, 2S+1=3 :Triplet S=0, 2S+1=1 :Singlet
Term symbols are
Triplet terms: 3D, 3P, 3S Singlet: 1D, 1P, 1S
3D -> L=2, S=1 J=2+1, 2+1-1, 2+1-2,…., |2-1| = 3,2,1
3P -> L=1, S=1 J=2,1,0
3S -> L=0, S=1 J=1
3D3,2, P2,1,0 3S1

80. Spin-Orbit Coupling

81. Selection Rule

82. Discussion Questions

83. Exercises
13-9, 13-10, 13-20, 13-27, 13-30, 13-32, 13-34, 13-35,