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NANO266 - Lecture 1 - Introduction to Quantum Mechanics

UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.

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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 …
NANO266
2
Eψ(r) = −
h 2
2m
∇2
ψ(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), 1798-1807V. 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.
3.2
V
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
NANO266
5
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
NANO266
6
ih
∂
∂t
ψ(r,t) = −
h 2
2m
∇2
+V(r,t)
$
%
&
'
(
)ψ(r,t) = Eψ(r,t)

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NANO266 - Lecture 1 - Introduction to Quantum Mechanics

  • 1. A Gentle Introduction to Quantum Mechanics Shyue Ping Ong
  • 2. 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 … NANO266 2
  • 3. Eψ(r) = − h 2 2m ∇2 ψ(r)+V(r)ψ(r) Material Properties First Principles Computational Materials Design Quantum Mechanics Generally applicable to any chemistry Some Approximation
  • 4. Many properties can now be predicted with quantum mechanics Diffusivity Phase equilibriaVoltages S. P. Ong, et al., Chem. Mater. 2008, 20(5), 1798-1807V. 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. 3.2 V 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
  • 5. Number of papers having DFT or ab initio in their titles over the past two decades NANO266 5
  • 6. 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 NANO266 6 ih ∂ ∂t ψ(r,t) = − h 2 2m ∇2 +V(r,t) $ % & ' ( )ψ(r,t) = Eψ(r,t)
  • 7. TheTrade-OffTrinity NANO266 7 Choose two (sometimes you only get one) Accuracy Computational Cost System size
  • 8. 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: NANO266 8 − h 2 2m ∇2 +V(r) # $ % & ' (ϕ(r) = Eϕ(r) ψ(r,t) =ϕ(r) f (t) ih ∂ ∂t f (t) = Ef (t) f (t) = e −i E h t Stationary Schrödinger Equation
  • 9. Stationary Schrödinger Equation for a System of Atoms where NANO266 9 Eψ = Hψ H = − h 2 2me ∇i 2 i ∑ − h 2 2mk ∇k 2 − e2 Zk rikk ∑ i ∑ + e2 rijj ∑ i ∑ k ∑ + Zk Zle2 rkll ∑ k ∑ KE of electrons KE of nuclei Coulumbic attraction between nuclei and electrons Coulombic repulsion between electrons Coulombic repulsion between nuclei
  • 10. Two broad approaches (and a shared Nobel Prize) to solving the Schrödinger equation NANO266 10
  • 11. 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 NANO266 11
  • 12. 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 NANO266 12 ψiψj dr∫ =δij ψiHψj dr∫ = ψiEψj dr∫ = Eδij
  • 13. TheVariational Principle Let us define a guess wave function that is a linear combination of the real wave functions It can be shown that NANO266 13 φ = ciψi i ∑ φ2 dr∫ = ci 2 i ∑ φHφ dr∫ = ci 2 i ∑ Ei
  • 14. TheVariational Principle,contd Let us define the lowest Ei as the ground state E0 Since the RHS is always positive, we have NANO266 14 φHφ dr∫ − E0 φ2 dr∫ = ci 2 i ∑ (Ei − E0 ) φHφ dr∫ − E0 φ2 dr∫ ≥ 0 φHφ dr∫ φ2 dr∫ ≥ E0 We 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.
  • 15. References Essentials of Computational Chemistry: Theories and Models by Christopher J. Cramer NANO266 15