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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
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Chapter 13 Atomic Structure Hydrogenic Atoms
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The Spectra of Hydrogenic Atoms
Balmer, Lyman, Paschen, Brackett Series Rydberg
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The line spectra of several elements
Note: prism separates the different ’s, and red light bends least
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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, ...
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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))
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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)
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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
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Energy Level of Hydrogen Atom
Electron Affinity Ionization Potential Electron Affinity Electronegativity Ionization Energy
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Cartesian Coordinates
Hydrogen Like Atom (x,y,z) Cartesian Coordinates (x, y, z) Polar Coordinates (r, q, f)
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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)
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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
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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
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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
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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
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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
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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
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Shells of Electrons
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Hydrogen Atom
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Hydrogenic Wave Functions
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Wave Functions; Electron Density
Radial dependence of the wavefunction of a 1s orbital and corresponding probability density
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Wave Functions; Electron Density
Radial Distribution Function
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Wave Functions: R(r) Nodes
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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.
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Wave Functions: d Orbitals
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Angular Momentum and its Components
Magnitude of angular momentum Component of angular momentum
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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
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Fermion and Boson Fermion spin ½ particles (s=1/2)
electron, proton, neutron Boson spin 1 particles (s=1) photon
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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
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Chapter 13 Atomic Structure The Structures of Many-electron Atoms
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Above three-body problem can not be exactly solved
both with Classical Mechanics and Quantum Mechanics
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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.
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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
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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.
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Penetration and Shielding
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Penetration and Shielding
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Penetration and Shielding
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Penetration and Shielding
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Building-up Principle
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Chapter 13 Atomic Structure Periodic Trends in Atomic Properties
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Atomic Radius and Ionization E.
Both atomic radius and ionization energy are correlated with effective atomic charge.
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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
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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
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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
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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
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Van der Waals equation
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Ionization Potential
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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.
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Electron Affinity
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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.
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Ionization E, Electron Affinity, Electronegativity
+1 +2 +3 -1 -2 Net Atomic Charge Energy 2nd IP 1st IP 1st EA
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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
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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))
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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
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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.
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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.
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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.
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Electronegativity
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Total Energy
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Mulliken-Jaffe Electronegativity
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Electronegativity Equalization
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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
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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
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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.
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Electronegativity Equalization
XLUAO,A> XHOAO,A LUAO HOAO, B LUAO,A HOAO
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Electronegativity Equalization
LUAO XLUAO,A>= XHOAO,A LUAO,A HOAO, B HOAO
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Electrostatic Interaction
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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)
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Chemiosmotic coupling
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ATP Synthesis at Mitochondrion
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How electrons are donated by NADH
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Biological Oxidation
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Ferry electrons from one complex to the next
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H.W. Proteins – Redox process
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Chapter 13 Atomic Structure The Spectra of Complex Atoms
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States and Orbitals ……… ? E5 - E4 = E2 - E1 Orbitals State1 State2
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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
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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
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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
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Spin-Orbit Coupling
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Selection Rule
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Discussion Questions
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Exercises 13-9, 13-10, 13-20, 13-27, 13-30, 13-32, 13-34, 13-35,
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