524
11 Biomolecular Structure and Interactions
The valence electrons dominate atomic interactions which create bonding. In
chemical interactions, atoms have change of energies less than about 20 eV. Under
these circumstances, the ‘core’ of atoms remain largely inert.
11.5 Pauli Exclusion Effects
In quantum theory, the wave function for a set of identical fermions must be
antisymmetric in the exchange of a pair of those fermions 9 :
(1, 2) = −(2, 1) .
(11.19)
This exchange antisymmetry means the wave function will vanish if two of the
fermions are in identical quantum states. In this way, no two electrons can occupy
the same quantum state. The Pauli exclusion principle is a result of quantum
behavior. As the fermions approach a single quantum state, their combined wave
function approaches zero. Since electrons are fermions (being ‘spin 1/2 particles’),
if two valence electrons from two atoms are in similar quantum states which
approach each other as the atoms are pushed together, the probability density for
these electrons will diminish in the region between the atoms.
Because of Pauli exclusion, and because of the relatively small electron magnetic
interactions between the electrons and the two protons, the associated energy of
a confined two-electron system with anti-aligned spins will be lower than the
aligned system. If the electrons are in distinctly different quantum states, the
electron probability density may grow in the region between the atoms as the atoms
approach, leading to covalent bonding. For example, the dominant part of the wave
function for the two electrons bound to two protons in the hydrogen molecule has
the electrons in a total spin singlet state, antisymmetric in electron exchange, and a
total electron orbital state with L = 0, symmetric in the exchange of the electrons.
One might ask how the combination of an even number of fermions to make
a boson seems to obviate the Pauli exclusion principle for the combined system.
‘Cooper pairs’ of electrons in superconductors and atoms in boson condensates
are examples. The answer is that when such systems of fermions pair up to act
as bosons, the wave function for the boson system is spread over distances much
larger than the Compton wavelengths (h/(mc)) of the internal fermions. The internal
fermions in these boson states are not being forced to be very close together.
9 More generally Eugene Wigner showed in 1939 that if two identical particles with spin quantum
number s, then, under particle exchange, (1, 2) = (−1) 2s (2, 1) , a relation which follows from
how particle waves transform in Special Relativity.
11 Biomolecular Structure and Interactions
The valence electrons dominate atomic interactions which create bonding. In
chemical interactions, atoms have change of energies less than about 20 eV. Under
these circumstances, the ‘core’ of atoms remain largely inert.
11.5 Pauli Exclusion Effects
In quantum theory, the wave function for a set of identical fermions must be
antisymmetric in the exchange of a pair of those fermions 9 :
(1, 2) = −(2, 1) .
(11.19)
This exchange antisymmetry means the wave function will vanish if two of the
fermions are in identical quantum states. In this way, no two electrons can occupy
the same quantum state. The Pauli exclusion principle is a result of quantum
behavior. As the fermions approach a single quantum state, their combined wave
function approaches zero. Since electrons are fermions (being ‘spin 1/2 particles’),
if two valence electrons from two atoms are in similar quantum states which
approach each other as the atoms are pushed together, the probability density for
these electrons will diminish in the region between the atoms.
Because of Pauli exclusion, and because of the relatively small electron magnetic
interactions between the electrons and the two protons, the associated energy of
a confined two-electron system with anti-aligned spins will be lower than the
aligned system. If the electrons are in distinctly different quantum states, the
electron probability density may grow in the region between the atoms as the atoms
approach, leading to covalent bonding. For example, the dominant part of the wave
function for the two electrons bound to two protons in the hydrogen molecule has
the electrons in a total spin singlet state, antisymmetric in electron exchange, and a
total electron orbital state with L = 0, symmetric in the exchange of the electrons.
One might ask how the combination of an even number of fermions to make
a boson seems to obviate the Pauli exclusion principle for the combined system.
‘Cooper pairs’ of electrons in superconductors and atoms in boson condensates
are examples. The answer is that when such systems of fermions pair up to act
as bosons, the wave function for the boson system is spread over distances much
larger than the Compton wavelengths (h/(mc)) of the internal fermions. The internal
fermions in these boson states are not being forced to be very close together.
9 More generally Eugene Wigner showed in 1939 that if two identical particles with spin quantum
number s, then, under particle exchange, (1, 2) = (−1) 2s (2, 1) , a relation which follows from
how particle waves transform in Special Relativity.