Astrophysik - Max Planck Society · 2005. 8. 31. · Astrophysik Ga rching, Germany rching.mpg.de...

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.de

Spe trum ofthe Hydrogen AtomThe Gymnasiallehrer J. J. Balmer from Basel,Switzerland, was the �rst to re ognize thatthe spe trum of the hydrogen atom onsistsof series of spe tral lines and that the fre-quen ies of the lines an be al ulated bythe formula � = �0 � Rn2R: Rydberg onstant.Referen e: J. J. Balmer, Notiz �uber die Spek-trallinien des Wassersto�s, Verhandlungen derNaturfors henden Gesells haft in Basel, 7,552 (1885)1

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deEnergy Level Diagramof the Hydrogen Atom6E

l = 0 l = 1 l = 2 l = 3n=1

n=2n=3 �HJJLLLn=4n=5n=6n=7

n: prin ipal quantum numberl: orbital angular momentum quantum number2

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deSpe tra ofMany-ele tron AtomsThe Swedish physi ist J. R. Rydberg(1854{1919) was the �rst to re ognize that\the spe tra of [many-ele tron℄ atoms anbe understood mu h in the same way as thespe trum of the Hydrogen atom"and that the frequen ies of the spe tral lines an be omputed by the formula:� = �0 � R(n� Æ)2R: Rydberg onstant.Referen e: J. R. Rydberg, On the stru tureof the line-spe tra of the hemi al elements,The London, Edinburgh and Dublin Philo-sophi al Magazine and Journal of S ien e,29, 331 (1890) 3

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deEnergy Level and RadiiDiagram of Rydberg States6E

l = 0n=1

n=2n=3 �HJJLLLn=4n=5n=6n=7

� protonn=1 1.5 Bohrn=2 6.0 Bohrn=3 13.5 Bohrn=4 24.0 Bohrn=5 37.5 Bohrn=6 54.0 Bohr...n=100 15000 Bohrn=200 60000 Bohr 6hri

maximum ele tron densityThe expe tation value for the radius of aRydberg state is given by the formulahri = 3n2 � l(l+1)2Z a0 4


Quantum Defe t Orbital(QDO) TheoryOne-ele tron S hr�odinger equation[�12 �2�r2 + V (r)℄nl� = Enlnl�QDO wavefun tionnl� = (r; �; �)nl� = 1r R(r)nl y(�; �)l�R(r)nl: radial fun tiony(�; �)l�: point group adapted ombinationsof spheri al harmoni s Y (�; �)l�n: prin ipal quantum numberl: orbital angular momentum quantum number�: magneti quantum number

5

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deQuantum Defe t Orbital(QDO) TheoryE�e tive entral �eld potentialV (r) = �(�+1)� l(l+1)2r2 � Znetr� = l� Æ � Æ: quantum defe t : parameter de�ning the nodal stru tureof the radial wavefun tion = 0 for hydrogeni orbitalsThe quantum defe t Æ is determined from thefollowing formula:Enl = T � Z2net2(n� Ænl)2Enl: dis rete spe tral energyT : ionisation energyZnet: e�e tive harge of the ore 6


Quantum Defe t Orbital(QDO) TheoryRadial transition momentRab = hRa(r)jQ(r)jRb(r)iQ(r) = rQ(r) = r[1� �r3(1� e�r=r )3℄�: dipole polarisability of the orer : ut-o� parameterThe radial transition moment may be ex-pressed in losed analyti al form. 7

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deGoal of this Investigation

to study thegeometry, stability, andele troni stru tureof small mole ulesin ex ited ele troni statesThis is done by al ulating the energy(and possibly other properties)as a fun tion of the geometri al parameters(distan es and angles)The results an be displayed as:potential energy hypersurfa es(2D: surfa e, 1D: urve)energy density distributions 8

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deTheoreti al Studies ofHydrogen - Rare-Gas andLithium - Rare-GasRydberg Mole ules

Eri h W. S hreiner andGeerd H. F. Dier ksenMax-Plan k-Institut f�ur AstrophysikKarl-S hwarzs hild-Stra�e 1D-85740 Gar hing, Germany

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Properties ofRydberg Mole ules ��

Chemi ally unbound ground stateChemi ally bound ex ited statesLarge energy gapbetween ground and ex ited states10


Choi e of Basis Fun tionsforDes ribing Rydberg StatesThe basis fun tions have to des ribe/modelat the Rydberg enter:the ele tron distribution of the positive ionthe ele tron distribution of Rydberg statesat the ligands:the ele tron distribution of the neutral sys-temthe ele tron polarisability 11

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe Cations RgH+

ReHeH = 1:3a0ReNeH = 2:0a0ReArH = 2:6a0ReKrH = 2:8a0

Total energy of the ationsRgH+, Rg = He, Ne, Ar, Kras fun tion of the internu lear distan e RRgH12


ReNeH = 2:0a0ReArH = 2:6a0ReKrH = 2:8a0Total energy of the ationsRgH+, Rg = Ne, Ar, Kras fun tion of the distan e RRgHwith and without using pseudo-potentials13


ReHeH = 1:3a0ReNeH = 2:0a0ReArH = 2:6a0ReKrH = 2:8a0

Total energy of the ationsRgH+, Rg = He, Ne, Ar, Kras fun tion of the internu lear distan e RRgH14


ReNeH = 2:0a0ReArH = 2:6a0ReKrH = 2:8a0Total energy of the ationsRgH+, Rg = Ne, Ar, Kras fun tion of the distan e RRgHwith and without using pseudo-potentials15


ReHeH = 1:3a0ReNeH = 2:0a0ReArH = 2:6a0ReKrH = 2:8a0ReXeH = 3:0a0

Total energy of the ationsRgH+, Rg = He, Ne, Ar, Kr, Xeas fun tion of the internu lear distan e RRgHusing pseudo-potentials 16

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe Cations RgLi+

ReHeLi = 3:6a0ReNeLi = 3:8a0ReArLi = 4:35a0ReKrLi = 4:3a0

Total energy of the ationsRgLi+, Rg = He, Ne, Ar, Kras fun tion of the internu lear distan e RRgLi17

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deBond Energies andBond Lengths of theCations RgH+ and RgLi+Cal ulated binding energies(in mHartree = 0.68 k al/mol = 27 meV)He Ne Ar KrH HeH+ 74 NeH+ 91 ArH+ 150 KrH+ 207Li HeLi+ 2.8 NeLi+ 8.6 ArLi+ 9.8 KrLi+ 38Cal ulated bond lengths(in a0 = 0.52 �A)RRg He (2.3) Ne (3.0) Ar (3.6) Kr (3.7)H HeH+ 1.48 NeH+ 1.84 ArH+ 2.42 KrH+ 2.60Li HeLi+ 3.60 NeLi+ 3.78 ArLi+ 4.47 KrLi+ 4.27RH � 0a0, RLi = 1:1a0 18

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deEnergy Level DiagramsEx itation energies for the hydrogen and raregas atoms (in Hartrees = 27 eV)

0.00.5H He Ne Ar Kr

Ex itation energies for the lithium and raregas atoms (in Hartrees = 27 eV)0.00.5

Li He Ne Ar Kr 19

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System HeH

ground state

n = 2n = 3n = 4n = 5 ation

Total energy of various ele troni states of HeHas fun tion of the internu lear distan e RHeH20

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System HeH

n = 2n = 3n = 4n = 5 ation

Total energy of various ele troni states of HeH asfun tion of the internu lear distan e RHeH21

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deEle tron Densities in HeH

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Ground state (top), 2s�-state (bottom)of HeH at RHeH = [1:0;1:5;5:0℄ a0with hydrogen at the enter of the ube22


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2p�-state (top), and 3dÆ-state (bottom)of HeH at RHeH = [1:0;1:5;5:0℄ a0with hydrogen at the enter of the ube23


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l-mixing of the 3s�-, 3p�- and 3d�-statesof HeH at RHeH = [1:5;2:0;3:0;5:0℄ a0with hydrogen at the enter of the ube24

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System ArH

Total energy of various ele troni states of ArHas fun tion of the internu lear distan e RArH25

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System ArH

Total energy of various ele troni states of ArH asfun tion of the internu lear distan e RArH26

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System KrH

Total energy of various ele troni states of KrH asfun tion of the internu lear distan e RKrH27

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System KrH

Total energy of various ele troni states of KrH withthe internu lear distan e RKrH 28

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deEle tron Densities in KrH

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First (top), se ond, and third ex ited stateof KrH at RHeH = [3;5;10℄ a0 with � symmetry29

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.deEle tron Densities in KrH

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Fifth (top), seventh, and eighth ex ited stateof KrH at RHeH = [3;5;10℄ a0 with � symmetry30


Potential Energy Curves forthe System XeH

Total energy of various ele troni states of XeHas fun tion of the internu lear distan e RXeHusing pseudo-potentials31

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe Systems HeH and KrH

Comparison of total the energyof various ele troni states of HeH and KrHas fun tion of the internu lear distan e32


Potential Energy Curves forthe Systems HeLi and ArLi

Comparison of the total energyof various ele troni states of HeLi and ArLias fun tion of the internu lear distan e33

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System He2H

s� statesp� statesp� states ationTotal energies of various ele troni states of He-H-Heat linear symmetri geometriesas fun tion of RHe1H = RHe2H 34


Total energies of various ex ited states of He-H-Heat bent symmetri geometrieswith RHe1H = RHe2H �xed at 2a0 and 4a0as fun tion of the bond angle35


Total energies of various ex ited states of He-H-Heat linear geometriesas fun tion of RHe1H and RHe2H 36

Max-Plan k-Institut f�ur AstrophysikGar hing, Germanyhttp://www.mpa-gar hing.mpg.dePotential Energy Curves forthe System He2H ��

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Total energies of various ex ited states of He-H-Heat linear geometriesas fun tion of RHe1H and RHe2H 37

Astrophysik - Max Planck Society · 2005. 8. 31. · Astrophysik Ga rching, Germany rching.mpg.de...

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