Download Chemistry 2000 Review: quantum mechanics of

Chemistry 2000
Spring 2017
Review: The quantum mechanical model
of the atom
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Review: Quantum Mechanics (Section 6.4)
http://fakescience.org
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Review: Quantum Mechanics (Section 6.4)
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The Schrödinger equation – the solution for the H atom
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Austrian physicist Erwin Schrödinger used de Broglie’s insight to calculate what electron
waves might act like, using a branch of mathematics known as wave mechanics
The result is encapsulated in a complex formula known as the Schrödinger equation:
 2  2  2
8 2m
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  2 ( E  V )
2
2
2
x
y
z
b
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This equation was know to belong to a special class known as an eigenvector equation: an
operator acts on a function (ψ) and generates a scalar times the same function
Ψ is known as the wavefunction of the electron: there are an infinite number of such
wavefunctions, each of which is characterized by a precise energy En, where n is an integer
from 1,2,3,4….∞. V in the equation is a constant value: the potential energy from attraction
to the nucleus. The remaining terms are all fundamental constants
A wave function that satisfies the Schrödinger equation is often called an orbital. Orbitals are
named for the orbits of the Bohr theory, but are fundamentally different entities
An orbital is a wave function
An orbital is a region of space in which an electron is most likely to be found
The square of the amplitude of the wavefunction, ψ2, expresses the probability of finding the
electron within a given region of space, which is called the electron density. Wavefunctions
do not have a precise size, since they represent a distribution of possible locations of the
electron, but like most distributions, they do have a maximum value.
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probability distribution
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probability density
Figure 6.17
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Quantum Numbers of an Atomic Orbital
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Shells
Subshells
Table 6.2 The hierarchy of quantum numbers for atomic orbitals
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Figure 6.19 The 2p orbitals
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Figure 6.22 Energy levels of the H atom
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Multi-electron atoms:
Electron-electron repulsion - impossible to
solve exactly.
ms also enters into the theory (+½ or −½).
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The quantum mechanics of molecules: H2+
Simplest molecule
Three-body problem, no exact solutions.
e−
1H
+
1H+
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Born-Oppenheimer approximation
Treat nuclei as immobile
Separated by distance R
Orbital energy = kinetic energy + electron-nuclear attraction
H nuclei
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The Effective Potential
Figure courtesy of Prof. Marc Roussel
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The Effective Potential
Figure 8.13
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