Constants and Conversions:

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Iris Hicks
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1 EXAM INFORMATION Radial Distribution Function: P 2 ( r) RDF( r) Br R( r ) 2, B is th normalization constant. Ordr of Orbital Enrgis: Homonuclar Diatomic Molculs * * * * g1s u1s g 2s u 2s u 2 p g 2 p g 2 p u 2 p Sphrical Polar Coordinats dv = r 2 sin()drdd r < p 2p Constants and Convrsions: h = J s ħ = h/2 = J s k = J/K c = m/s = m/s NA = mol -1 m = kg mp = kg 1 Å = 1-1 m 1 V = J 1 amu = kg 1 J = 1 kg m 2 /s 2 1 N = 1 kg m/s 2 1 au (hartr) = 225 kj/mol 1 au (hartr) = V INTEGRALS n a d n n! a d 24 d 283 d 87 d 153 d 159 d 347

2 CHEM 521 Eam 3 April 24, 215 Nam (33) 1. Short Answr Qustions (4) (a) Th primntal lctron affinity of an oygn atom is -142 kj/mol. Eplain why you would pct th Hartr-Fock stimat of an oygn atom to b mor or lss ngativ than th primntal valu. (4) (b) Writ down th prssion for th Echang Intgral [K1s2s] for th chang intractions btwn lctrons in 1s and 2s orbitals. Us atomic units, and writ th intgrals in standard doubl intgral form. (5) (c) Writ th Hamiltonian for th two lctrons in th HH + molcular ion, in SI (MKS) units. (2) (d) Considr th function, [f(1)g(2) - g(1)f(2)][12-12] (f and g ar spatial functions). This function is (i) symmtric, (ii) antisymmtric, (iii) nithr symmtric nor antisymmtric with rspct to lctron chang. No planation ncssary

3 1. Short Answr Qustions (Cont'd.) () () Considr an citd stat of th B + ion with th configuration: 1s 2 2s 1 3s 1. A simpl product wavfunction for a lithium atom in this configuration is: =1s(1)11s(2)22s(3)33s(4)4 Writ th normalizd Slatr dtrminant for th appropriatly antisymmtric wavfunction with this configuration. () (f) Writ th complt lctron configuration and giv th bond ordr of th N2 molcul in th lowst citd lctronic stat. Is th bond lngth in th citd stat highr or lowr than th ground stat bond lngth (plain your rasoning)?. (3) (g) As discussd in class, th Scular Dtrminant is th dtrminant of th cofficints of th homognous linar quations which rsult from us of th Variational Thorm to minimiz th nrgy. Eplain why th Scular Dtrminant must b qual to zro. (3) (h) Sktch th pg * 2p orbital in a homonuclar diatomic molcul AND stat what th lttr, "g", stands for.

4 For #2 -#3: Th nrgy lvls of hydrognlik atoms ar givn by: Z 1 Z En kj 2 24 a n 2 n Not: Both qustions us th abov formula, but ar not rlatd to ach othr. (5) 2. Calculat th wavlngth of th mittd light (in nm) whn an lctron in th n = 5 lvl of th Li 2+ ion drops to th n = 3 lvl. (8) 3. Th Hartr-Fock nrgy of HH + is a.u. Calculat th Homolytic* Dissociation Enrgy, D, of HH +, in lctron volts (V). *This mans that HH + braks apart into H + + H.

5 (2) 4. On of th hydrogn atom wavfunctions is: Ar 3 r/4ao i sin( ) (1) (a) St up th prssion to calculat th avrag valu of cos 2 (), <cos 2 ()>. Your 2 2 answr should b of th form: cos ( ) AI II, whr IR, I, and I ar intgrals ovr dr, d and d. R Nots: (1) It is NOT ncssary for you to valuat th intgrals. (2) Th volum lmnt and intgration limits in sphrical polar coordinats ar givn in th information pag

6 4. (Cont'd) For part (b), us th Radial Distribution Function for this wavfunction, which is givn on th information pag. For this problm, th normalization constant of th RDF is: 1 1 B 432 2a o (1) (b) Calculat th probability that th distanc of th lctron from th nuclus is btwn ao and 12ao.. You should gt a numrical valu for this probability. 9

7 (8) 5. A hypothtical wavfunction for a two lctron systm is: N 1(1)2(2) s s Assuming that th individual spatial functions, 1s and 2s ar individually normalizd, dtrmin th normalization constant, N, of th complt wavfunction abov. Show your work to rciv crdit. (1). Th thr primntal ionization nrgis of th lithium atom ar: IE1=52. kj/mol, IE2=73. kj/mol, IE3=11,81 kj/mol. Th corrlation nrgy of th lithium atom is kj/mol. Th Hartr-Fock nrgis of th Lithium +1 and -1 ions ar: EHF(Li + )= au and EHF(Li - )= au. (4) a) Writ th Hamiltonian for th lithium atom in SI Units.

8 . (Cont'd) () b) Us th data at th bginning of th problm to calculat th Hartr-Fock stimat of th first ionization nrgy, in kj/mol. () c) Us th data at th bginning of th problm to calculat th Hartr-Fock stimat of th lctron affinity, in kj/mol.

9 (1) 7. Considr a trial variational wavfunction for a hypothtical multilctron atom, which is of th form: = (), whr is a positiv Variational paramtr (i.. > ). It was found that th pctation valu for th nrgy, in a.u., using this trial wavfunction is: E H Calculat th bst valu of th hlium atom nrgy which can b obtaind using this trial wavfunction.

Hydrogen Atom and One Electron Ions

Hydrogen Atom and One Electron Ions Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial