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Chap 5: From Stars in Galactic Clusters to Stars in Fireworks

Chap 5: From Stars in Galactic Clusters to Stars in Fireworks. New Office hours Tuesdays 1:00 to 2:30 pm 2033 YH Exam on Friday Chapters 3, 4, 5 and 6.0-6.6 One page of notes Review Session Thursday Lec (1) 5:00 - 5:50 and Lec (2) 6:00 - 6:50 FRANZ 1178. Atomic Structure of Matter.

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Chap 5: From Stars in Galactic Clusters to Stars in Fireworks

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  1. Chap 5: From Stars in Galactic Clusters to Stars in Fireworks New Office hours Tuesdays 1:00 to 2:30 pm 2033 YH Exam on Friday Chapters 3, 4, 5 and 6.0-6.6 One page of notes Review Session Thursday Lec (1) 5:00 - 5:50 and Lec (2) 6:00 - 6:50 FRANZ 1178

  2. Atomic Structure of Matter • Dalton matter constructed from atoms • Einstein: for matter of mass (m) E=mc2 • JJ Thomson: Heavy Positive Charge-protons Light Negative Charge-electrons • Rutherford: protons and neutrons; small massive nucleus surrounded by electrons bound together by the coulomb potential V(r) = -Ze2/4πe0 r • Sun’s Black Body Radiation Spectrum shows: DiscreteSpectra: bound electron/proton-Atoms Continuous Spectra: unbound electron/proton Ionized Atoms

  3. Atomic Structure of Matter • Planck Quantized the Radiation field: E=hn=hw waves move like particles with fixed energy • Rydberg, Lyman, Balmer, Franck and Hertz Experiments: discrete spectra for atoms and Ions • Bohr: Decree Spectra for one electron atoms and ion L=nh: H, He+, Li++, etc, for Z=1, 2, 3 … respectively En = -{e42h2)} (Z2/n2) n=1, 2, 3, 4 …..∞ • De Broglie : matter moving freely with constant moment p=mv moves like waves with wavelength p=h/l • Schrödinger: matter confined can only have discrete energies For an infinite Square well En=(h2/8m)(n2/L2) n(x)= Bsin(nπx/L) n=1, 2, 3, …………….∞

  4. Chap. 4: Electron Diffraction off the Surface of a Solid Chap. 4: 1-D Infinite Square Well Potential n = quantum number Probability density Amplitude + + - # of nodes n-1 nodes + - + Eigen functions n(x)=√2/L sin(nx/L) √2/L normalization Const Probability per unit length n(x)|2=2/L {sin(nx/L)}2 Eigen Values En= (h2/8meL2)n2 n=1,2,3,4,….∞.

  5. xpx≥ h /2 The linear momentum px and the position x cannot be measured Simultaneously with arbitrary precisionx and px Therefore px and x cannot be stated without an Uncertainty. That is, results of the simultaneous measurement of x and px must be given as: x ± x and px ± px and xpx≥ h/2 Heisenberg Uncertainty Principle: xpx≥ h /2 Ex. For mass (m) in a Square well potential of width L, The position of the electron is known within ~ L Therefore x~ L, the electron is said to be localized in L With an uncertainty in px, px ≥ h /2L, the minimum px~ h /2L The minimum energy: Emin~ (px)2/2m=(h /2L)2/2m= h2/8mL2 The ground state energy E1=h2/8mL2 ~Emin This is the Zero-Point Energy: every bound system must have an Emindue the Uncertainty Principle

  6. Emission of a photon at frequency  and E=h Energy V(r) 0 r 2nd excite state Ei 1st excited state E=h Ef Ground State, State of lowest energy A*(Ei)  A(Ei) + h Ef – E i= h

  7. Chap 5: Quantum Mechanical Energy levels The Energy Eigen Values are Independent of l and m! Therefore each n level has n, l levels (0,1, ..n-1), each with (2l+1) states and is therefore n2 degenerate. Each (nl) level has (2l+1) m-states and is (2l+ 1) degenerate

  8. =Zr/a0 • Recall that the Bohr radii are rn= n2/Za0 Table 5-2, p. 175

  9. Chap 5: Energy Eigen Values due to screening and e-e repulsion The effective nuclear charge Zeff results in an effective coulomb potential Vneff( r )~ Zeff(n)/r electrons in different nl energy levels (ns and np) have different energy eigen-values Enl: Ens<Enp<End Electrons closer to the nucleus “screen” outer electrons from the full Z of the nucleus and electron-electron repulsion further lowers the Zeff

  10. The Aufbau Principle: building multi-electron atoms Hund’s Rule: The ground state electron configuration with parallel spins has the lowest energy C:1s22s22px2py=C:[He]2s22px2py Building the ground state electronic configuration of multi-electron atoms using the concept of one electron per quantum states {n, l, m, ms}; The Pauli Exclusion Principle Since electrons are identical and, therefore, indistinguishable we can’t say which electron has spin up or down. Carbon Z=6

  11. Chap 5:, 2pz~R20( r )Ypz(, 3pz R31( r )Ypz( Ypz ~ cos(=f( 0.9 0.7 f( 0.5 30 45 60 |Ypz|2~ |cos( r  2pz~R21( r )Ypz( One electron orbital

  12. Chap 5: Classical Magnetic dipole moment   magnetic dipole moment <L> Attracted to high B-field ml < 0, <S> = r2 m S ~ -<S> L~ -<L> Repelled from high B-field ml > 0

  13. Chap 5: Aufbau Process; Atomic Ground State Electron Configuration P D P D P P P P P D

  14. Absorption: A + h A* Ei+ h = Ef or h =Ef – Ei Emission A* A + h Ei = Ef + h or h =Ei – Ef A Unexcited atom/molecule with energy Ei A* Excited Atom/Molecule with energy Ef after the absorption of a photon of energy h .

  15. Ne Hg Ar

  16. Atomic Emission Spectra

  17. Chap 5: Energy Eigen Values due to screen and e-e repulsion Singlet configuration IE21~21 IE20~20 Koopman Approximation IEnl= - nl hn 1s1s =[-Zeff2/(1)2] + [- Zeff2/(1)2] Fig. 5-14, p. 189

  18. Chap 5: Hartree Single Electron Radial Orbitals Rnl for He, Li+,etc Average V( r1, r2) over positions r2 for electron (1) and avg r1 for electron (2) <V( r1,r2)>r2 ~ Vneff(r1)~ - Zeff(n)e2/r1Z(n=1)=1.69 for He: where Zeff represents the avg over e-e repulsion.The new two electron potential is now V( r1,r2) ~ - Zeff(n)e2/r1 - Zeff(n)e2/r2Separable solution: (r1, r2) =1s(1)1s(2) The ground state solution of the two electronic Schrödinger Eq. is:1s1s(1,2)= 1s(1)1s(2)  Singlet state spins are anti parallel S=0, Ms =0 1s2s(1,2)= 1s(1)2s(2)Triplet state spins are parallel S=1, Ms =0, ±1 The Hartree one electron orbital energy n=-Zeff(n)2/n2(2.18x10-18J) = Zeff(n)2/n2( 13.6 eV)The ground state Energy n1n2 obtained form the Calculations using the Hartree 1electron orbital to construct a 2 electron orbital n1n2 =[-Zeff(n1)2/(n1)2] + [- Zeff(n2)2/(n2)2] n1n2(1,2)= n1(1)n2(2)  -e +eZeff r1

  19. Chap 5: Energy Eigen Values due to screening and e-e repulsion The effective nuclear charge Zeff results in an effective coulomb potential Vneff( r )~ Zeff(n)/r electrons in different nl energy levels (ns and np) have different energy eigen-values Enl: Ens<Enp<End Electrons closer to the nucleus “screen” outer electrons from the full Z of the nucleus and electron-electron repulsion further lowers the Zeff

  20. Chap 5: Photoelectron Spectroscopy is to measure the one electron Orbital Energies h Energy (nl) of a Hartree Orbital One electron hydrogen like orbital nlmr is, approximately the Electron’s ionization energy IEnl This is the so called “Koopmans’ approximation” Stated more precisely: IEnl= - nl NeNe+ + e E=IE~ - nl KE 2p 2s Therefore a measurement of IEnl is the same as measuring nl KE= (1/2)mev2 = h- IEnl 1s Orbital Energies for Ne

  21. Chap 5: Energy Eigen Values due to screen and e-e repulsion IE21 IE20 Koopman Approximation IEnl= - nl Excited atom hn Triplet Configuration 1s2s =[-Zeff2/(1)2] + [- Zeff2/(2)2] Fig. 5-14, p. 189

  22. Chap 5: Periodic Table reflects the Electron Configuration; Atomic Properties Alkali Metals Rare Gases Noble Metals Transition Metals Halogens Alkaline earth Lanthanides Actinides

  23. Chap 6: Molecular Hydrogen Ion; H2+ the simplest Molecule (Diatomic) Due to the lack of Spherical Symmetry the angular momentum quantum number (l) is no longer Good so L2 cannot be measured. However, due to the Cylindrical Symmetry Lz can be measured and (Lz)2 LzmRr,=mlhmRr, Lz= mlh m- Magnetic Quantum number is still Good and can be used to label the electronic eigen states = |m| m=0,±1,±2,±3,±4, Use the Aufbau concept to build up the electron configuration of Homonucleardiatomics + L +

  24. Chap 6: H2+ Electronic Eigen States Classified by =|m| and Symmetry of V Since (l) is no longer useful, =m2 is now used to classify the molecular electronic Eigen States and Eigen Values E(R ) R is the internuclear distance.The magnetic quantum number m= 0, ±1, ±2, ±3labels the eigen values and eigen functions electronic states as well as the inversion symmetry eigen values: = +1(g, even, gerade) and  = -1(u, odd, ungerade) inversion symmetry In order of increasing energy (number of nodes) for the exact H2+molecular orbitals(MO)s/eigen functions: 1g 1u 2g2u1u3g1g3u

  25. The Schrödinger Equation for H2+(P2/2)r, R) + (p2/2me)r, R) + V(r, R)r, R) = Er, R) -e + r + R Relative Kinetic Energy term for the Nuclei Only depends on R This is set to Zero first, then E( R), the electronic Energy found and is used as the potential to Solve Schrödinger Eq. for motion of the nuclei Kinetic Energy term For the electron Only depends r r, R): total wave function for the electrons and nuclei E: total energy for theelectronsand nuclei Potential Energy between Electron and Nuclei, ris associated with the position of the electron and R the relative position of the nuclei. r, R)~ el(r;R)vib( R) Makes R and r separable and thus the electronic wave function el(r;R) is only parametrically dependent on R, the Inter-nuclear separation vib( R) vibrational wave function for the nuclei Born-Oppenheimer Approximation (BOA) (P2/2)r, R) ~ zero Sinceme/m ~ zero (p2/2me)el(r;R)vib( R)+V(r, R)el(r;R)vib( R)= Eel(R)el(r;R)vib( R)

  26. (p2/2me)el(r;R)vib( R)+V(r, R)el(r;R)vib( R)= Eel(R)el(r;R)vib( R) Cancel vib( R) on both sides (p2/2me)el(r;R)+V(r, R)el(r;R)= Eel(R)el(r;R) Now solve this equation for el(r;R) and Eel(R) at different inter-nuclear distances R. E(R) Eel (R ) =E(R ) el (r;R)=(r:R) H + H+ DeElectronic Well depth Eel (R ) = E(R ) el (r;R) = (r:R ) H---H+

  27. Chap 6: Approximate Electronic MO’s LCAO Destructive Interference anti-bonding |1sA|2 + |1sB |2 - 2(|1sA1sB||) Adding wavefunctions constructive Interference bonding |1sA|2 + |1sB |2 + 2(|1sA1sB|) The degree of Overlap (|1sA1sB|) Adding Probabilities |1sA|2 + |1sB |2 Fig. 6-6, p. 227

  28. Chap 6: Electronic Potential Energy Curves/Correlation Diagrams H+ + H+ + 2e H-H  H + H E= De Molecular Dissociation H2 -13.6 eV E1s E1s H+H H-H Re(H-H) Bond Length, Electronic Bond Energy De=(Re)

  29. Chap 6: Bond Energy, Bond Order and Bond Length Building First Row Diatomic Molecules/ions using the Aufbau Process:Pauli Exclusion Principle and Hund’s Rule Bond Order = (1/2) {# Bonding electrons – # Anti-bonding} Summarizes the results of the Molecular Schrödinger Equation: Bond energy De and bond length R=Re,

  30. Aufbau Process for 2nd Row Homonuclear Diatomics sp- sp+

  31. O2 is paramagnetic, ground state is a Triplet N2 is Diamagnetic, ground state is a Singlet Parallel Unpaired Spin s=1 2s+1=3 m-states Anti-Parallel Paired Spin s=0 2s+1=1 m-state

  32. Chap 6: Approximate Electronic Eigen functions ; LCAO Chap 6: Approximate Electronic Eigen functions ; LCAO (p2/2me) (r;R) + V(r,R) (r;R ) = E(R ) (r;R ) (r;R) and E(R ) are calculated at different R values using the Born Oppenheimer Approximation (BOA) where the nuclei are held at a fixed nuclear separation R! The eigen functions, Molecular Orbital(MO) Can be approximated by the Linear Combination of Atomic Orbitals LCAO (r;R) = a(1sA+1sB ) + b(2sA+ 2sB) + c(2pzA+ 2pzB) +…..

  33. Chap 6: Approximate Electronic Eigen functions ; LCAO Linear combination of atomic orbital (LCAO) is used to construct approximate MO’s AO’s contribute to bond formation if their atomic energy eigen values are very close are in value. The AO’s on different atoms contribute based on the degree of overlap between the atomic orbitals g1s= (1/√2){1sA + 1sB} +…+… u1s= (1/√2){1sA - 1sB} +…+… The First terms make up the minimum AO basis set for electronic MO Eigen functions g1s~ (1/√2){1sA + 1sB} u1s~ (1/√2){1sA - 1sB}

  34. Chap 6: Approximate Electronic Eigen functions ; LCAO ~ {1sA + 1sB} ~ {1sA - 1sB} ~{2sA + 2sB} ~ {2sA - 2sB} ~ {2pxA + 2pxB}, {2pyA + 2pyB} ~ {2pzA - 1pzB} ~ {2pxA - 1pxB}, {2pyA - 2pyB} ~ {2pzA + 2pzB} The largest contributions, i.e., the relative magnitudes of a,b,c, etc. are made by AO’s of the correct Symmetry and Energy

  35. If 2sA and 2pB Not close in energy, if close Mixing or Hybridization occurs

  36. Chap 6: E(R ) Calculations using Approximate and Exact MO’s Exact(solid lines) and LCAO approximation(dashed lines) BOA electronic energies are the potential energy curves for motion of the Nuclei The minimum of the Potential curves determine bond lengths and bond angles for molecules, i.e., their molecular structure Fig. 6-8, p. 228

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