Advance Quantum Chemistry : Atomic Structure and Spectra · 2020. 3. 11. · Atomic spectra...

Advance Quantum Chemistry :

Atomic Structure and Spectra

FMIPA UGM – 10 March 2020

Niko Prasetyo

Outline

Atomic structure and spectraAtomic structure and spectra– Hydrogenic atom– Atomic orbital– Spectra of electronic transitions and selection rules :(

Structures of many electron systemStructures of many electron system– The approximations– SCF orbitals

Spectra of complex systemSpectra of complex systemMolecular structureMolecular structure

– VBT, MO theory– Hückel approximation

Lecture material : http://ugm.id/1NDLecture material : http://ugm.id/1ND

Atomic spectraElectronic spectrumElectronic spectrum

– 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...

Atomic spectra

Where is the origin of this picture ? Why it happens ? What is the permitted energy ?

Atomic spectraElectronic spectrumElectronic spectrum

– 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

Atomic spectraLet see the Schrödinger equation (again)Let see the Schrödinger equation (again)

– If we have a ‘ball’ system– We can separate the function into

• Radial• angular

– The first term : Coulomb– The second term : centrifugal forces

When l = 0, the electron has no angular momentum, and the effective When l = 0, the electron has no angular momentum, and the effective potential energy is purely Coulombic and attractive at all radii potential energy is purely Coulombic and attractive at all radii

When l ≠ 0, the centrifugal term gives a positive (repulsive) contribution When l ≠ 0, the centrifugal term gives a positive (repulsive) contribution to the effective potential energyto the effective potential energy

When the electron is close to the nucleus (r ≈ 0), this repulsive term, When the electron is close to the nucleus (r ≈ 0), this repulsive term, which is proportional to 1/rwhich is proportional to 1/r22, dominates the attractive Coulombic , dominates the attractive Coulombic component, which is proportional to 1/r, and the net result is an component, which is proportional to 1/r, and the net result is an effective repulsion of the electron from the nucleus.effective repulsion of the electron from the nucleus.

– In close distance to n, radial wavefunctionis proportional to rl

– Far from nucleus, all of the radial wavefunction approach zero exponentially– Again, we can compute the energy foreach conditions

– We can also write the ‘general formula’

Atomic spectraAtomic orbitalsAtomic orbitals

– 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

– 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

– 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:

Atomic spectraS orbitalS orbital

– n = 1, l = 0, m = 0– Independent of angle

Atomic spectrap orbitalp orbital

Atomic spectrad orbitald orbital

Selection ruleTransition and selection ruleTransition and selection rule

– 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.

– 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

– 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

Many electron systemMany electron systemMany electron system

– 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

Many electron systemMany electron systemMany electron system

Many electron systemThe approximationsThe approximations

– Stasionary systems• The variation of in time is not considered

• All of the nuclei are fixed and together with its <E> are computed for particular configuration

• If the positions are changed, we have to compute new using new positions of nuclei

Many electron systemThe approximationsThe approximations

– No special effect of relativistic • Remember this picture• Electron closest to the nucleushas highest velocity (in fractionof 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

Many electron systemRelativistic effectRelativistic effect

– Shorter bond length– The contraction of bond lengths does not require the contraction of the

orbitals

– Color of gold

– 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

Many electron systemBond Oppenheimer approximationBond Oppenheimer approximation

– 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

– Thus, we only consider the electronic subsystem– Potential energy of nuclei is treated via classical Coulombic

– Potential energy of nuclei is treated via classical Coulombic

– 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.

Many electron systemIndependent particle approximationIndependent particle approximation

– Douglas Hartree– Hartree product

Many electron system

Pauli principlePauli 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.

Pauli principlePauli principle– 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

Many electron systemPauli principlePauli principle

– 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

Many electron systemApproximations in QMApproximations in QM

– 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 ….

Many electron systemPenetration and ShieldingPenetration and Shielding

– 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

– 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)

– 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)

– Filling up to 3p is not problem, just put it..– But, again 3d orbital makes trouble

• The energy is close to 4s orbital

– 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)

Many electron systemIonization Energy and electron affinityIonization Energy and electron affinity

– 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)

Many electron systemIonization Energy and electron affinityIonization Energy and electron affinity

– 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

Back to Schrödinger equation (again)Back to Schrödinger equation (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)

Many electron systemRoadmap of HF method (HF (not) in nutshell)Roadmap of HF method (HF (not) in nutshell)

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

Many electron systemNow we need computation ...

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

● Reference :Reference :● Thomas Hofer, Advance Quantum Chemistry LecturesThomas Hofer, Advance Quantum Chemistry Lectures● Atkins, P. W., De Paula, J., & Keeler, J. (2018). Atkins' physical Atkins, P. W., De Paula, J., & Keeler, J. (2018). Atkins' physical

chemistry. Oxford university press.chemistry. Oxford university press.● Pyykkö, P. (2012). Relativistic effects in chemistry: more common Pyykkö, P. (2012). Relativistic effects in chemistry: more common

than you thought. Annual review of physical chemistry, 63, 45-64.than you thought. Annual review of physical chemistry, 63, 45-64.

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