Advance Quantum Chemistry : Atomic Structure and Spectra FMIPA UGM – 10 March 2020 Niko Prasetyo Outline Atomic structure and spectra – Hydrogenic atom – Atomic orbital – pectra o! electronic transitions and selection rules "# tructures o! many electron system – $he appro%imations – CF orbitals pectra o! comple% system Molecular structure – '($) M* theory – H+ckel appro%imation -ecture material " http:..ugm/id.101 2 Atomic spectra 2lectronic2lectronic spectrumspectrum – In this chapter) we will discuss the application of QM to describe the electronic structure of atom • *nly talk about electrons5 • implest system is one electron system #hydrogenic6 7 H8) He8) -i28 etc9 – :arning ! It becomes super complicated when deal 3ith many electron system (He) • Why ; We will see later... 3 Atomic spectra Where is the origin of this picture ? Why it happens ? What is the permitted energy ? 4 Atomic spectra 2lectronic2lectronic spectrumspectrum – :here is comes from ; • 1issociation of H2 7 emits series of discrete !re<uencies – =emember : in 4M, the energy is discrete n1 : lyman n2 : balmer n3 : paschen 5 Atomic spectra -et-et seesee thethe chr>dingerchr>dinger e<uatione<uation #again6#again6 6 Atomic spectra -et-et seesee thethe chr>dingerchr>dinger e<uatione<uation #again6#again6 7 Atomic spectra -et-et seesee thethe chr>dingerchr>dinger e<uatione<uation #again6#again6 – I! we ha?e a ‘ballA system – :e can separate the !unction into • =adial • angular x 8 Atomic spectra -et-et seesee thethe chr>dingerchr>dinger e<uatione<uation #again6#again6 – $he first term : &oulomb – $he second term : centrifugal forces 9 WhenWhen ll BB 0,0, thethe electronelectron hashas nono angularangular momentum,momentum, andand thethe ef!ecti?eef!ecti?e potential energy is purely &oulombic and attracti?e at all radii :hen l C 0, the centri!ugal term gives a positive #repulsi?e6 contribution to the ef!ecti?e potential energy WhenWhen thethe electronelectron isis closeclose toto thethe nucleusnucleus #r#r DD 0),0), thisthis repulsiverepulsive term,term, 3hich is proportional to 1/r2, dominates the attractive &oulombic component, 3hich is proportional to 1/r, and the net result is an e!!ective repulsion o! the electron !rom the nucleus/ 10 Atomic spectra -et-et seesee thethe chrödingerchrödinger e<uatione<uation (again6(again6 – In close distance to n) radial wavefunction is proportional to rl – Far !rom nucleus) all of the radial wavefunction approach Eero exponentially – Again) we can compute the energy for each conditions 11 Atomic spectra -et-et seesee thethe chrödingerchrödinger e<uatione<uation (again6(again6 – We can also write the ‘general formulaA 12 13 14 Atomic spectra AtomicAtomic orbitalsorbitals – olution of chr>dinger e<uation leads to • 3 <uantum numbers (n) l and m6 • Wavefunction of one electron is called orbital • In chemistry) we deal with s, p) d and #sometimes6 ! orbitals 15 Atomic spectra AtomicAtomic orbitalsorbitals – n : principle <uantum number • Determine the energy o! electron • n B 1, 2, 3 /// – l " angular <uantum number • describes the subshell, and gives the magnitude o! the orbital angular momentum through the relation • l = 0, 1 , 2 .. n-1 – m : magnetic <uantum number • describes the speci!ic orbital #or "cloud") within that subshell, and yields the proHection o! the orbital angular momentum along a speci!ied a%is • mB 0, I1, I2,// Il 16 Atomic spectra AtomicAtomic orbitalsorbitals – s : describes the spin (intrinsic angular momentum) of the electron within that orbital) and gives the projection of the spin angular momentum along the speci!ied axis" 17 Atomic spectra orbitalorbital – n B 1, l B 0, m = 0 – Independent of angle 18 Atomic spectra pp orbitalorbital s 19 Atomic spectra dd orbitalorbital 20 Atomic spectra dd orbitalorbital 21 Selection rule $ransition$ransition andand selectionselection rulerule – 2nergy of radial solution of chrödinger equation – :hich means, the <uantization of energy is exists #energy is depend on n6 – o, the electron can go ‘upA and @downA • For example " if an electron go down to lower energy state, the excess of energy is discarded ?ia electromagnetic radiation 3ith !re<uency ?/ 22 Selection rule $ransition$ransition andand selectionselection rulerule – But) the sad thing is : not all of transition is permissible/ • -eads to a complicated concept of selection rule • Angular momentum is conserved – election rule talks about allo3ed transition of electron • 'ery use!ul in U'-Vis spectroscopy analysis of inorganic complexes 23 Selection rule $ransition$ransition andand selectionselection rulerule – But) the sad thing is : not all of transition is permissible/ • -eads to a complicated concept of selection rule • Angular momentum is conserved – election rule talks about allo3ed transition of electron • 'ery use!ul in U'-Vis spectroscopy analysis of inorganic complexes 24 25 Many electron system ManyMany electronelectron systemsystem – Before we go deeper to selection rule) we have to understand the many electron system – Many electron system is complicated thing because all of the electrons interact with one another – Interaction via Hamiltonian operator 26 Many electron system ManyMany electronelectron systemsystem 27 Many electron system ManyMany electronelectron systemsystem 28 Many electron system ManyMany electronelectron systemsystem 29 Many electron system TheThe approximationsapproximations – tasionary systems • $he variation of in time is not considered • All of the nuclei are fi%ed and together with its K2L are computed for particular con!iguration • If the positions are changed) we have to compute new using new positions of nuclei 30 Many electron system $he$he approximationsapproximations – 0o special e!!ect o! relati?istic • =emember this picture • 2lectron closest to the nucleus has highest ?elocity (in !raction o! speed o! light) because they !eel the most negati?e potential • =esulting in a lot o! e!!ects – &ontraction in electron density » horter bond length » hi!ted energy le?el » :ea,ened interaction 31 Many electron system =elativistic ef!ect – horter bond length – $he contraction o! bond lengths does not require the contraction o! the orbitals 32 Many electron system =elativistic=elativistic e!fecte!fect – &olor of gold 33 Many electron system =elativistic=elativistic effecteffect – Weakened interactions – Hg2#g6 does not form because the 6s2 orbital is contracted by relativistic effects and may therefore only weakly contribute to any bonding – in !act Hg–Hg bonding must be mostly the result of van der Waals forces, which e%plains why the bonding for Hg–Hg is weak enough to allow for Hg to be a li<uid at room temperature 34 Many electron system =elativistic=elativistic effecteffect 35 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Total probability of n , N includes electron #n6 and nuclei #06 – The nuclei is much heavier than electron) they do not show important <uantum e!!ect – The nuclei move slower than electron) thus 3hen nuclei change its position) electron will adHust instantly – And we can separate the 3avefunction into 36 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Thus, we only consider the electronic subsystem – Potential energy of nuclei is treated via classical &oulombic 37 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Potential energy of nuclei is treated via classical &oulombic 38 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Born *ppenheimer is valid !or ground state – :hen dealing with the system which is character of nuclei is dominant (e.g proton transfer6 a sophisticated method is re<uired/ 39 Many electron system IndependentIndependent particleparticle approximationapproximation – 1ouglas Hartree – Hartree product 40 Many electron system Pauli principle – has to be antisymmetric) i/e it has to change sign under permutation" – $he reason !or this lies in the spin of an electron being 1N2 and that the electrons are – indistinguishable/ 41 Many electron system PauliPauli principleprinciple – Oohn & later proposed an elegant 3ay to write the Pauli principle as a sum of Hartree product of the system including the correct sign 42 Many electron system PauliPauli principleprinciple – horter way to write is via later 1eterminant – later determinant approximates all of the possible Hartee product of all ne system/ – The correct sign is automatically included 43 Many electron system ApproximationsApproximations inin 4M4M – All of these approximations lead to a method so-called Hartree-Fock #HF) • But we will not discuss this today "# – -ets continue to another topic 9/ 44 Many electron system PenetrationPenetration andand hieldinghielding – :e already knew that there are interactions between nuclei and electron via &oulombic potential – Howe?er) not all of the electrons !eel same potential because the inner electrons will shield the nuclei) thus the outer electron will less attract to nuclei 45 Many electron system PenetrationPenetration andand hieldinghielding – $he shielding ef!ect resulting to an ef!ect so called ef!ective nuclear charge #Pe!!) and we have a new constant of shielding constant (σ ). – 2very orbital gives di!!erent shielding constant or ef!ect because electron 3ill penetrate di!!erently// !or example // Electron in orbital s can penetrate close to nuclei → experience less shield but it is a good shield for outer electron (3s > 3p > 3d) 46 Many electron system PenetrationPenetration andand hieldinghielding