Advance Quantum Chemistry : Atomic Structure and Spectra
FMIPA UGM – 10 March 2020
Niko Prasetyo Outline
Atomic structure and spectra – Hydrogenic atom – Atomic orbital – Spectra of electronic transitions and selection rules :( Structures of many electron system – The approximations – SCF orbitals Spectra of complex system Molecular structure – VBT, MO theory – Hückel approximation
Lecture material : http://ugm.id/1ND 2 Atomic spectra ElectronicElectronic spectrumspectrum – In this chapter, we will discuss the application of QM to describe the electronic structure of atom • Only talk about electrons! • Simplest system is one electron system (hydrogenic) → H+, He+, Li2+ etc… – Warning ! It becomes super complicated when deal with 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 ElectronicElectronic spectrumspectrum – Where is comes from ?
• Dissociation of H2 → emits series of discrete frequencies – Remember : in QM, the energy is discrete
n1 : lyman n2 : balmer n3 : paschen
5 Atomic spectra LetLet seesee thethe SchrödingerSchrödinger equationequation (again)(again)
6 Atomic spectra LetLet seesee thethe SchrödingerSchrödinger equationequation (again)(again)
7 Atomic spectra LetLet seesee thethe SchrödingerSchrödinger equationequation (again)(again) – If we have a ‘ball’ system – We can separate the function into • Radial • angular
x
8 Atomic spectra LetLet seesee thethe SchrödingerSchrödinger equationequation (again)(again) – The first term : Coulomb – The second term : centrifugal forces
9 WhenWhen ll == 0,0, thethe electronelectron hashas nono angularangular momentum,momentum, andand thethe effectiveeffective potential energy is purely Coulombic and attractive at all radii When l ≠ 0, the centrifugal term gives a positive (repulsive) contribution to the effective potential energy WhenWhen thethe electronelectron isis closeclose toto thethe nucleusnucleus (r(r ≈≈ 0),0), thisthis repulsiverepulsive term,term, which is proportional to 1/r2, dominates the attractive Coulombic component, which is proportional to 1/r, and the net result is an effective repulsion of the electron from the nucleus.
10 Atomic spectra LetLet seesee thethe SchrödingerSchrödinger equationequation (again)(again) – In close distance to n, radial wavefunction is proportional to rl – Far from nucleus, all of the radial wavefunction approach zero exponentially – Again, we can compute the energy for each conditions
11 Atomic spectra LetLet seesee thethe SchrödingerSchrödinger equationequation (again)(again) – We can also write the ‘general formula’
12 13 14 Atomic spectra AtomicAtomic orbitalsorbitals – Solution of Schrödinger equation leads to • 3 quantum numbers (n, l and m) • Wavefunction of one electron is called orbital • In chemistry, we deal with s, p, d and (sometimes) f orbitals
15 Atomic spectra AtomicAtomic orbitalsorbitals – n : principle quantum number • Determine the energy of electron • n = 1, 2, 3 ... – l : angular quantum number • describes the subshell, and gives the magnitude of the orbital angular momentum through the relation • l = 0, 1 , 2 .. n-1 – m : magnetic quantum number • describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis • m= 0, ±1, ±2,.. ±l 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 S along the specified axis:
17 Atomic spectra SS orbitalorbital – n = 1, l = 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 TransitionTransition andand selectionselection rulerule – Energy of radial solution of Schrödinger equation
– Which means, the quantization of energy is exists (energy is depend on n) – So, the electron can go ‘up’ and ‘down’ • For example : if an electron go down to lower energy state, the excess of energy is discarded via electromagnetic radiation with frequency v.
22 Selection rule TransitionTransition andand selectionselection rulerule – But, the sad thing is : not all of transition is permissible. • Leads to a complicated concept of selection rule • Angular momentum is conserved – Selection rule talks about allowed transition of electron • Very useful in UV-Vis spectroscopy analysis of inorganic complexes
23 Selection rule TransitionTransition andand selectionselection rulerule – But, the sad thing is : not all of transition is permissible. • Leads to a complicated concept of selection rule • Angular momentum is conserved – Selection rule talks about allowed transition of electron • Very useful in UV-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 – Stasionary systems • The variation of in time is not considered • All of the nuclei are fixed and together with its
30 Many electron system TheThe approximationsapproximations – No special effect of relativistic • Remember this picture • Electron closest to the nucleus has highest velocity (in fraction of speed of light) because they feel the most negative potential • Resulting in a lot of effects – Contraction in electron density » Shorter bond length » Shifted energy level » Weakened interaction
31 Many electron system Relativistic effect – Shorter bond length – The contraction of bond lengths does not require the contraction of the orbitals
32 Many electron system RelativisticRelativistic effecteffect – Color of gold
33 Many electron system RelativisticRelativistic effecteffect – Weakened interactions
– Hg2(g) does not form because the 6s2 orbital is contracted by relativistic effects and may therefore only weakly contribute to any bonding – in fact Hg–Hg bonding must be mostly the result of van der Waals forces, which explains why the bonding for Hg–Hg is weak enough to allow for Hg to be a liquid at room temperature
34 Many electron system RelativisticRelativistic effecteffect
35 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Total probability of n , N includes electron (n) and nuclei (N) – The nuclei is much heavier than electron, they do not show important quantum effect
– The nuclei move slower than electron, thus when nuclei change its position, electron will adjust instantly – And we can separate the wavefunction into
36 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Thus, we only consider the electronic subsystem – Potential energy of nuclei is treated via classical Coulombic
37 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Potential energy of nuclei is treated via classical Coulombic
38 Many electron system BondBond OppenheimerOppenheimer approximationapproximation – Born Oppenheimer is valid for ground state – When dealing with the system which is character of nuclei is dominant (e.g proton transfer) a sophisticated method is required.
39 Many electron system IndependentIndependent particleparticle approximationapproximation – Douglas Hartree – Hartree product
40 Many electron system
Pauli principle – has to be antisymmetric, i.e it has to change sign under permutation:
– The reason for this lies in the spin of an electron being 1⁄2 and that the electrons are – indistinguishable. 41 Many electron system
PauliPauli principleprinciple – John C Slater proposed an elegant way to write the Pauli principle as a sum of Hartree product of the system including the correct sign
42 Many electron system PauliPauli principleprinciple – Shorter way to write is via Slater Determinant – Slater determinant approximates all of the possible Hartee product of all ne system. – The correct sign is automatically included
43 Many electron system ApproximationsApproximations inin QMQM – All of these approximations lead to a method so-called Hartree-Fock (HF) • But we will not discuss this today :( – Lets continue to another topic ….
44 Many electron system PenetrationPenetration andand ShieldingShielding – We already knew that there are interactions between nuclei and electron via Coulombic potential – However, not all of the electrons feel 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 ShieldingShielding – The shielding effect resulting to an effect so called effective nuclear
charge (Zeff) and we have a new constant of shielding constant (σ ). – Every orbital gives different shielding constant or effect because electron will penetrate differently.. for 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 ShieldingShielding – So far we have 2 ‘type’ of electron, inner and outer electron • Because in chemistry, most reactions only involve the outer electron (valence electron), we will discuss a lot of about this • If we deal with the heavy elements, the inner electrons can be replaced by pseudo-potential – Another question is how to fill the orbital ? • Remember, the energy is quantized in QM..so we have to follow this by the lowest first
• Aufbau principle – Again, we have to consider the repulsion of each electron.. so by nature they will in unpaired states first and then paired (Hund’s rule)
47 Many electron system PenetrationPenetration andand ShieldingShielding – Filling up to 3p is not problem, just put it.. – But, again 3d orbital makes trouble • The energy is close to 4s orbital
48 Many electron system PenetrationPenetration andand ShieldingShielding – The most probable distance of a 3d electron from the nucleus is less than that for a 4s electron, so two 3d electrons repel each other more strongly than two 4s electrons. – As a result, Sc has the configuration [Ar]3d14s2 rather than the two alternatives, for then the strong electron–electron repulsions in the 3d orbitals are minimized – Penetration and shielding play important role in ionization energy • Ionization energy is the minimum amount of energy to eject one electron from atom in gas phase (first IE)
Go 49 Many electron system IonizationIonization EnergyEnergy andand electronelectron affinityaffinity – Penetration and shielding play important role in ionization energy • Ionization energy is the minimum amount of energy to eject one electron from atom in gas phase (first IE)
Go 50 Many electron system IonizationIonization EnergyEnergy andand electronelectron affinityaffinity – Li → outermost electron is well shielded from the nucleus by the core – If we go to the right of periodic table, the EA will increase → atom tends to form anion
51 Many electron system BackBack toto SchrödingerSchrödinger equationequation (again)(again)
– To solve the Schrödinger equation for many electron system we need an iterative method so-called self consistent field (SCF) – But before it, we have ^to understand why we have to use SCF – As I talked before, the approximations in QM lead to a computational method so-called Hartree-Fock • Derivation of HF is tedious and need separate talks (but if you want we can do this) 52 Many electron system RoadmapRoadmap ofof HFHF methodmethod (HF(HF (not)(not) inin nutshell)nutshell)
53 Many electron system
54 Many electron system
55 Many electron system
In general the HF equation for electron 2p can be written as
The first term on the left is the contribution of the kinetic energy and the attraction of the electron to the nucleus, just as in a hydrogenic atom.
The second term takes into account the potential energy of the electron of interest due to the electrons in the other occupied orbitals.
The third term is an exchange correction that takes into account the spin correlation effects
56 Many electron system Now we need computation ...
57 Many electron system
58 Many electron system
59 Many electron system
The original HF method is too simple (leads to a lot of problems such as convergence issues, too many cycles, oscillation of the energy etc)
In modern QM package, the convergence aids is often employed (such as direct inversion of iterative sub-space (DIIS) or pulay mixing
60 Many electron system
61 ● ReferenceReference :: ● ThomasThomas Hofer,Hofer, AdvanceAdvance QuantumQuantum ChemistryChemistry LecturesLectures ● Atkins,Atkins, P. P. W., W., De De Paula, Paula, J., J., & & Keeler, Keeler, J. J. (2018). (2018). Atkins' Atkins' physical physical chemistry.chemistry. OxfordOxford universityuniversity press.press. ● Pyykkö,Pyykkö, P. P. (2012). (2012). Relativistic Relativistic effects effects in in chemistry: chemistry: more more common common thanthan youyou thought.thought. AnnualAnnual reviewreview ofof physicalphysical chemistry,chemistry, 63,63, 45-64.45-64.
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