A Gentle Introduction to Quantum Mechanics

Shyue Ping Ong

The development of quantum mechanics is arguably the biggest scientific revolution in the 20th century with impact on the lives of people

1900 – Max Planck suggests quantization of radiation

1905 – Albert Einstein proposes light quanta that behaves like a particle

1913 – Bohr constructs a quantum theory of atomic structure

1924 – de Broglie proposes matter has wave-like properties

1925 – Pauli formulates exclusion principle

1926 – Schrodinger develops wave mechanics

1927 – Hsienberg formulates the uncertainty principle

1928 – Dirac combines QM with special relativity

… and many more developments thereafter …

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Eψ(r) = − h2

2m∇2ψ(r)+V (r)ψ(r)

Material Properties

First Principles Computational Materials Design

Quantum Mechanics Generally applicable to any chemistry

Some Approximation

Many properties can now be predicted with quantum mechanics

Diffusivity Phase equilibriaVoltages

S. P. Ong, et al., Chem. Mater. 2008, 20(5), 1798-1807 V. L. Chevrier, et al., Phys. Rev. B, 2010, 075122. A. Van Der Ven, et al. Electrochem. and Solid-State Letters, 2000, 3(7), 301-304.

Crystal structure

G. Hautier et al., Chem. Mater., 2010, 22(12),3762 -3767

Polarons

S. P. Ong, et al. Phys. Rev. B, 2011, 83(7), 075112.

3.2V

3. 86 V

3.7 V

3.76 V

4.09 V

Surface energies

L. Wang, et al. Phys. Rev. B,2007, 76(16), 1-11. 4

Number of papers having DFT or ab initio in their titles over the past two decades

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The Schrödinger Equation: Where it all begins

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. - Paul Dirac, 1929

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ih ∂∂tψ(r, t) = −

h 2

2m∇2 +V (r, t)

$

%&

'

()ψ(r, t) = Eψ(r, t)

The Trade-Off Trinity

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Choose two (sometimes you only get

one)

Accuracy

Computational Cost

System size

Stationary Schrödinger Equation

If the external potential has no time dependence, we can write the wave function as a separable function

And show that the Schrödinger Equation can be decomposed to:

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−h 2

2m∇2 +V (r)

#

$%

&

'(ϕ(r) = Eϕ(r)

ψ(r, t) =ϕ(r) f (t)

ih ∂∂tf (t) = Ef (t) f (t) = e

−i Eht

Stationary Schrödinger Equation

Stationary Schrödinger Equation for a System of Atoms

where

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Eψ = Hψ

H = −h 2

2me

∇i2

i∑ −

h 2

2mk

∇k2 −

e2Zk

rikk∑

i∑ +

e2

rijj∑

i∑

k∑ +

ZkZle2

rkll∑

k∑

KE of electrons

KE of nuclei

Coulumbic attraction between nuclei and electrons

Coulombic repulsion between electrons

Coulombic repulsion between nuclei

Two broad approaches (and a shared Nobel Prize) to solving the Schrödinger equation

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Two broad approaches to solving the Schrödinger equation

Variational Approach

Expand wave function as a linear combination of basis functions

Results in matrix eigenvalue problem

Clear path to more accurate answers (increase # of basis functions,

number of clusters / configurations)

Favored by quantum chemists

Density Functional Theory

In principle exact

In practice, many approximate schemes

Computational cost comparatively low

Favored by solid-state community

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Solving the Schrödinger Equation

In general, there are a complete set of eigenfunctions ψi (with corresponding eigenvalues Ei.

Without loss of generality, let us assume that the wave functions are orthonormal

Hence, we have

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ψiψ j dr∫ = δij

ψiHψ j dr∫ = ψiEψ j dr∫ = Eδij

The Variational Principle

Let us define a guess wave function that is a linear combination of the real wave functions

It can be shown that

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φ = ciψii∑

φ 2 dr∫ = ci2

i∑

φHφ dr∫ = ci2

i∑ Ei

The Variational Principle, contd

Let us define the lowest Ei as the ground state E0

Since the RHS is always positive, we have

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φHφ dr∫ −E0 φ 2 dr∫ = ci2

i∑ (Ei −E0 )

φHφ dr∫ −E0 φ 2 dr∫ ≥ 0

φHφ dr∫φ 2 dr∫

≥ E0We can judge the quality of the wave functions by the energy – the lower the energy, the better. We may also use any arbitrary basis set to expand the guess wave function.

References

Essentials of Computational Chemistry: Theories and Models by Christopher J. Cramer

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