ELEC-E3260 - Biomolecules

Quantum mechanics and modeling

Lassi Hällström 14.01.2020

Learning goals

Need to know

• Hierarchy of interactions

• Time independent Schrödinger equation

• What is modeling

Should know

• Wavefunction

• Differences between modeling techniques, approximations

• What can be modeled

Nice to know

• Time dependent Schrödinger equation

• Mathematical formulations of quantum mechanical problems

Biomolecules

https://www.youtube.com/watch?v=YO244P1e9QM

Biomolecules theory

https://upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Protein_structure_%28full%29.png/1200px-Protein_structure_%28full%29.png

• Interaction of atoms and electrons

– Fundamentally quantum problem

• Different electronic interactions

result in bonding and forming of

molecules

• Chemical reactions between

molecules drive biological

processes

Chemical reactions

https://pdb101.rcsb.org/motm/238

H2 + O2 → H2O

PDB-101 Molecule of the month October 2019

Atomic theory

https://en.wikipedia.org/wiki/File:Bohr_atom_animation_2.gif https://en.wikipedia.org/wiki/File:Niels_Bohr.jpg

Atomic theory

https://en.wikipedia.org/wiki/File:Bohr_atom_animation_2.gifhttps://upload.wikimedia.org/wikipedia/commons/e/e7/Hydrogen_Density_Plots.png

Intro to QM

• Physical model for describing nature at atomic scales (and below)

• Quantities such as energy, momentum, electric charge limited to discrete

(quantized) values

• QM is essentially a mathematical construct, ‘real’ particles such as electrons

are special cases.

Postulates of QM

• The state of a quantum mechanical system is

completely specified by a wavefunction Ψ(𝒓, 𝑡).

• Observables are described by operators

• Measurement of the observable associated with

operator መ𝐴, must result in some eigenvalue 𝑎,

which satisfies the eigenvalue equation መ𝐴Ψ = 𝑎Ψ

• Time evolution of the system is defined by the

time-dependent Schrödinger equation

𝐻 ۧ|Ψ = E| ۧΨ

𝐻 ۧ|Ψ(𝑡) = iℏ𝑑

𝑑𝑡| ۧΨ(t)

The Double slit experiment

Particle and potential • Simple case: infinite potential well

Particle and potential

• Atomic orbitals are solutions to

electron(s) with given energy

oscillating in the potential defined

by the nucleus and other

electrons.

• Why wont the electron ‘fall’ into

the nucleus?

• 𝐻 = −ℏ2

2𝑚𝑒∇2 −

𝑞2

4𝜋𝜖𝑟

Molecular orbitals

even / gerade

odd / ungerade

H2 energy diagram

https://upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Dihydrogen-MO-Diagram.svg/1600px-Dihydrogen-MO-Diagram.svg.png

https://www.dreamstime.com/stock-illustration-types-chemical-bonding-diagram-covalent-polar-nonpolar-ionic-metallic-hydrogen-bridge-bonds-models-educational-image80714653

H2 molecule

https://www.chemicool.com/definition/ground_state_of_diatomic_molecule.html

Particle interaction defined by QM

• (Full) Configuration Interaction doable

for 22 electrons, 4 atoms (2017)

https://aip.scitation.org/doi/10.1063/1.4989858

𝐻 ۧ|Ψ = E| ۧΨ

+ spin effects +1.63eV+0.98eV

O2 ionization energy ~12.2 eV

Ground state

Particle interaction defined by QM

• Approximate methods needed for

large systems

• Born-Oppenheimer: nuclei

considered static

𝐻 ۧ|Ψ = E| ۧΨ

Constant

Particle interaction defined by QM

• Approximate methods needed for

large systems

• Born-Oppenheimer: nuclei

considered static

• LCAO: no interaction between

atoms

𝐻 ۧ|Ψ = E| ۧΨ

Constant

LCAO of H2+ ion

• LCAO is variational calculated energies

always higher than true ground state

• Mostly useful as qualitative model,

Combined electron vibration-rotation

Emission spectrum of N2 molecule, J. A. Marquisee

Molecular structure

• Electron structure defines

molecule shape proteins

Alanine+ ionAlanine

~10eV difference

Length scale

Sebastian Kmiecik, Dominik Gront, Michal Kolinski, Lukasz Wieteska, Aleksandra Elzbieta Dawid, and Andrzej KolinskiChemical Reviews 2016 116 (14), 7898-7936 DOI: 10.1021/acs.chemrev.6b00163

Part 2: Modelling

Recap

• Electron interaction and atomic bonds

fundamentally quantum mechanical problem

𝐻 ۧ|Ψ = E| ۧΨ

Configuration interaction

• Numerically exact in the full CI

limit.

• Scales exponentially with

system size

• CCSD(T) considered ‘Gold

standard’ of quantum chemistry

Booth, G., Grüneis, A., Kresse, G. et al. Towards an exact description of electronic wavefunctions in real solids. Nature 493, 365–370 (2013) doi:10.1038/nature11770

Coupled Cluster example

A. Álvarez, M. Borges, J. J. Corral-Pérez, J. G. Olcina, L. Hu, D. Cornu, R. Huang, D. Stoian, A. Urakawa, ChemPhysChem 2017, 18, 3135.

Density Functional Theory

• Instead of solving the many-body system Ψ 𝒓𝑖 explicitly, solve electron

density 𝑛(𝒓).

• Energy depends on density: ground state energy is found at ground state

density

• In theory exact, in practice, energy functional is approximate.

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

Density Functional Theory

• Allows approximate solution of

electronic interaction for systems of

~100 atoms.

• Solutions on atomic scale

– Bond lengths, relaxed geometry

HOMO of para-benzoquinone

Reaction barriers and reaction rates

https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Rxn_coordinate_diagram_5.PNG/800px-Rxn_coordinate_diagram_5.PNG

𝑘 = 𝐴𝑒−𝐸𝑎𝑅𝑇

CO oxidation catalyst

• Catalysts can lower the

energy barrier

Lopez-Acevedo, O., Kacprzak, K., Akola, J. et al. Quantum size effects in ambient CO oxidation catalysed by ligand-protected gold clusters. Nature Chem 2, 329–334 (2010) doi:10.1038/nchem.589

DFT example

Katyanna S. Bezerra, Umberto L. Fulco, Stephany C. Esmaile, José X. Lima Neto, Leonardo D. Machado, Valder N. Freire, Eudenilson L. Albuquerque, and Jonas I. N. OliveiraThe Journal of Physical Chemistry B 2019 123 (30), 6421-6429 DOI: 10.1021/acs.jpcb.9b04468

Atomic interaction approximated with MM (MD)

• Atoms (nucleus+electrons) considered as single

point-like particles.

• Forces between atoms described by a classical

potential or force field.

– Potential can be initially computed with more accurate

methods

• Much lighter than QM methods but cannot

describe electronic interactions.

– All chemical reactions are electronic interactions.

– https://www.youtube.com/watch?v=5JcFgj2gHx8

MD example

Atomistic Molecular Dynamics Simulations of Mitochondrial DNA Polymerase γ: Novel Mechanisms of Function and PathogenesisLiliya Euro, Outi Haapanen, Tomasz Róg, Ilpo Vattulainen, Anu Suomalainen, and Vivek SharmaBiochemistry 2017 56 (9), 1227-1238DOI: 10.1021/acs.biochem.6b00934

Molecular level: coarse-grained and multiscale models

• Small molecules (10s of

atoms) electronic interaction

can be included

• Large molecules such as

proteins require simplification

– Studying protein folding is one of

the largest current computational

chemistry challenges

Biological system level, multi-state modeling

• State space model

• Does not necessarily

need any information of

spatial structure

• Instead of fundamental

forces, state evolves

according to predefined

rules

• For example reaction

rates from DFT

State 1

State 4State 3

State 2

Quantum computing?

https://upload.wikimedia.org/wikipedia/en/b/b2/Quantum_Computer_Zurich.jpg

• Y. Nam et al., “Ground-state

energy estimation of the water

molecule on a trapped ion

quantum computer,”

arxiv.org/1902.10171.

• Exact calculation H2 needs 56

qubits (Gabriel Popkin, Waiting for the

Quantum Simulation Revolution, October

21, 2019, Physics 12, 112.

https://physics.aps.org/articles/v12/112)

A Kandala et al. Nature 549, 242–246 (2017)

doi:10.1038/nature23879

Conclusion

𝐻 ۧ|Ψ = E| ۧΨ

Relevant webcomic

https://www.merelyaboutstuff.com/Comics/Physics%20degree.png