Sep 10, 2008 - simulations. Introduction ... atom and the concept of bonding. Lecture 1 .... Bonding in solids. Types of bonds: 1. Ionic. 2. Covalent. 3. Metallic. 4.
Introduction to electronic structure simulations Lecture 1, part II
Some quantum mechanics, hydrogen atom and the concept of bonding Arkady Krasheninnikov, University of Helsinki and Helsinki University of Technology September 10, 2008
Introduction
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Outline A quantum mechanical approach to the quantum world Quantum wells Atomic units Hydrogen atom Symmetry of the electron orbitals in the H-type atom Bonding in solids Bonding in a diatomic molecule Bond order September 10, 2008
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Quantum mechanics… Rings any bells? The first quarter of the XX century:
The time of new concepts in: Art Music Literature, and Physics and Chemistry
September 10, 2008
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Quantum well and discrete energy spectrum An atom can be though of as a potential well (an attractive potential for electrons generated by the nucleus) The unlimited square well
V= ∞
n=3 n=2 n=1
h2 π 2 2 En = n 2 2m a
V=0
a
September 10, 2008
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The square well of a finite depth
For E0, the solution exists for any energy. The solutions are symmetric or antisymmetric with reference to x=0. There is a finite probability to find a particle with negative energies for |x| >a. It can easily be shown that in one-dimensional systems there is always at least one bound state. (Not true for 3d systems). September 10, 2008
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Atomic units and energy conversion factors The atomic unit system:
Length:
e = 4πε 0 = 1 Energy:
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Hydrogen atom: why do we care? The simplest system; can be solved analytically Electronic states in other atoms have the same symmetry Solutions can be used as basis functions for calculating the characteristics of molecules and solids atomic units
How do we do it in practice?
m
There is only one electron; U(r) = -1/r; We can separate the variables by introducing spherical coordinates.
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Hydrogen atom (2) r
1 me e 4 1 1 1 = E= 2 2 2 h n 2 n2
n=... n=3 n=2
U( r)
ψ = Rn ,l (r )Ylm (θ , ϕ )
n=1
n = 1, 2, 3,... is the principal quantum number. Energy levels depend on n only (degeneracy over other quantum numbers)! Radial wave functions depend on l l = 0, 1, 2, ... n-1 is the angular quantum number. States with l = 0,1,2,3 are called s, p, d, f states respectively. m = -l, -l+1,... ,l-1,l is the magnetic quantum number. This quantum number (m ) gives the component of the angular momentum vector in a particular direction which is arbitrary designed as the z-axis. September 10, 2008
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Hydrogen atom (3). The radial functions R Note the variations in the spatial distribution of the electron!
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Hydrogen atom (4). The spherical functions Y
s-orbital
Jumping ahead: Electronic states in other atoms have the same symmetry
p-orbital In many solids the electron orbitals can be represented as linear combinations Example: Carbon, graphite/diamond of H-like orbitals September 10, 2008
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Hydrogen atom (5). Hydrogen-like orbitals
http://chemistry.beloit.edu/ Stars/pages/orbitals.html
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Hydrogen-like atoms and the ‘aufbau’ principle There are many electrons 2s 2s, 2p 2p Energy level order
Hund’s rule
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s, p, d,... classification works for atomic shells Pauli exclusion principle Electron-electron interaction removes the degeneracy of states with the same principal quantum number but different angular momenta. 1s < 2s form an adequate basis set to expand the ground state of the molecule, that is The states are orthonormal: We can neglect the Coulomb interaction between the electrons Substituting the wave function into the Schrödinger equation and projecting to the states |1> and |2>
≠ as
For a non-trivial solution we require:
September 10, 2008
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Bonding in a homonuclear molecule (4) The sign of t: as |1> , |2> are s – functions (always positive), V1 < 0 (an attractive potential), hence t < 0:
Ea = Es+|t| Es
Es Eb = Es-|t|
Bonding
Ψb(r)
ρb(r)
r
r
ρa(r)
Antibonding Ψa(r)
r
r September 10, 2008
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Bond order
Bond order: One half of the difference between the number of electrons in the bonding and antibonding states in the corresponding bond
Tells us how “strong” the bond is Can be generalized to molecules and solids
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Bond order (2) This approach can easily be generalized to heteronuclear molecules (for example, HF). It is also possible to account for the nonorthegonality of basis functions
In covalently-bonded solids with many electrons bonding is due to charge accumulation in particular regions of space (where the “bonds” are). If we know the solution of the Schroedinger equation, we can apply this approach (as discussed in the future) to analyze how strong the particular bonds are.
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