nano266 - lecture 1 - introduction to quantum mechanics
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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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