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 ∇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 equilibria Voltages 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. 3.2 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)

The Trade-Off Trinity NANO266 7 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: 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

Stationary Schrödinger Equation for a System of Atoms where NANO266 9 Eψ = Hψ H = − h 2 2m e ∇i 2 i ∑ − h 2 2m k ∇k 2 − e2Z k r ik k ∑ i ∑ + e2 r ij j ∑ i ∑ k ∑ + Z k Z l e2 r kl l ∑ 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 NANO266 10

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

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 ψi ψj dr ∫ =δij ψi Hψj dr ∫ = ψi Eψj 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 NANO266 13 φ = c i ψi i ∑ φ2 dr ∫ = c i 2 i ∑ φHφ dr ∫ = c i 2 i ∑ E i

The Variational 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 ∫ − E 0 φ2 dr ∫ = c i 2 i ∑ (E i − E 0 ) φHφ dr ∫ − E 0 φ2 dr ∫ ≥ 0 φHφ dr ∫ φ2 dr ∫ ≥ E 0 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.

References Essentials of Computational Chemistry: Theories and Models by Christopher J. Cramer NANO266 15