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

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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

Chap 5: Hartree Single Electron Orbitals nlm(r,,)/

r12

-e

-e

r1

r2

+2e

The energy eigenvalues for a nl level enl as calculated using theHartree Self-Consistent-Field (SCF) orbital approximation.The SCF method uses a potential that is averaged over electron-electron repulsions experienced by the electron in a one electron, hydrogen like orbital nlm(r,,).

Consider a two electron atom or ion; He, Li+, etc; this case He

V( r1,r2)~(-2e2)/r1 +(-2e2)/r2 + (e2)/r12

Attraction to the Nucleus V( r1)~(-e)(2e)/r1 and V(r2) ~(-e)(2e)/r2 V( r1,r2 )~ (e)(e)/r12 and electron-electron repulsion

Chap 5: Hartree Single Electron Orbitals nlm(r,,)/

r12

-e

-e

r1

r2

+2e

The energy eigenvalues for a nl level Enl as calculated using theHartree Self-Consistent-Field (SCF) orbital approximation.The SCF method uses a potential that is averaged over electron-electron repulsions experienced by the electron in the one electron orbital nlm(r,,).

Consider a two electron atom or ion; He, Li+, etc; this case He

V( r1,r2)~(-2e2)/r1 +(-2e2)/r2 + (e2)/r12

Attraction to the Nucleus V( r1)~(-e)(2e)/r1 and V(r2) ~(-e)(2e)/r2 V( r1,r2 )~ (e)(e)/r12 and electron-electron repulsion

-e

+eZeff

r1

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 =c0 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

Chap 5: Energy Eigen Values due to screen and e-e repulsion

IE21~21

IE20~20

Koopman Approximation IEnl= - nl

Singlet configuration

1s1s =[-Zeff2/(1)2] + [- Zeff2/(1)2]

Fig. 5-14, p. 189

Chap 5: Energy Eigen Values due to sheilding and e-e repulsion

IE21

IE20

Koopman Approximation IEnl= - nl

Excited

atom

Singlet configuration

1s2s =[-Zeff2/(1)2] + [- Zeff2/(2)2]

Fig. 5-14, p. 189

Chap 5: Energy Eigen Values due to screen and e-e repulsion

IE21

IE20

Koopman Approximation IEnl= - nl

Excited

atom

Triplet Configuration

1s2s =[-Zeff2/(1)2] + [- Zeff2/(2)2]

Fig. 5-14, p. 189

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

Chap 5: Periodic Table reflects the Electron Configuration; Atomic Properties

Alkali Metals

Rare

Gases

Noble

Metals

Transition Metals

Halogens

Alkaline earth

Lanthanides

Actinides

Chap 5: multi-electron Atoms (Ar)- Pauli Exclusion Principle

One electron per state with the set of quantum numbers {n, l, m, ms}

K

L

Shell model maximum number of electrons 2n2 (nl ) sub-shells 2(2l+1) electrons K shell n=1: (1s) 2 L shell n=2: (2s 2p) 8M shell n=3: (3s 3p 3d) 18

M

Chap 6. Electronic Energy Eigen Values as a function inter-nuclear distance

Electronic Energy as a Function of R(the distance between nuclei): Born-Oppenheimer Potential Curve

Fig. 6-CO, p. 211

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

LzmRr,=mlhmRr,

Lz= mlh

m- Magnetic Quantum number

is still Good and can be used to

label the electronic eigen states

= m2

m=0,±1,±2,±3,±4,

Use the Aufbau

concept to build up the

electron configuration of

Homonucleardiatomics

+

L

+

Chap 6: H2+ Electronic Eigen States Classified by =m2 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, ±3labels 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 molecular orbital eigen states:

1g ,1u 2g2u1u3g1g3u

Chap 5: multi-electron Atoms (Ar)- Pauli Exclusion Principle

II

III

I

V=∞

V=0

V=∞

-x

0

L

+x

Potential without inversion symmetry V(x)≠V(-x)

Y1~ sin(πx/d)

Y2~ sin(2πx/d)

Potential with inversion symmetry V(x)=V(-x)

II

III

I

Y1~ cos(πx/d) even, g

Y2~ sin(2πx/d) odd, u

iYg/u=eYg/ue=±1 for g/u

V=∞

V=0

V=∞

-x

x=-d/2 x=0

+x

x=d/2

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