11.6 Binding Interactions Between Atoms
525
11.6 Binding Interactions Between Atoms
In quantum theory, the interaction energy, also called the potential energy, plays a
more fundamental role than force. The potential energy of a particle enters directly
into the wave equation for the particle. We solve the wave equation to get the
probability of finding the particle at various locations at selected times.
Force in quantum theory is a derived concept coming from the negative gradient
of the interaction energy V (r) that a particle has at position r because of nearby
particles: F = −∇V . This statement is just the inverse of the definition of work
done by the force: V = −
F · dr. If you plot the potential energy of one atom
relative to a second as a function of the separation between the atoms, the negative
slope of the curve for that function is the force on the first atom due to the second
atom.
Quantum effects and electromagnetic interactions determine the nanoscale
behavior of bound and unbound atoms in biological systems. The valence electrons,
i.e. those found in the outer region of atoms, largely mold chemical binding of
atoms into molecules. The ‘core’ electrons prevent atoms from collapsing, because
the Pauli exclusion principle allows only one electron per quantum state, and in
an atom, there are only a few such states near the nucleus. The electron state most
tightly held to the nucleus is distributed over a size comparable to the Bohr radius
( ¯
h 2 /(m e c 2 ), about half an ångström, divided by Z, the charge on the nucleus.
This size is far greater than the diameter of the atomic nucleus, which is about
1.75 × 10 −5 of an ångström for a proton in hydrogen and about 15 × 10 −5 of an
ångströms for the Uranium nucleus with 239 protons and neutrons.
For simplicity in some quantum calculations for bound atoms and pairs of
molecules, the electron Pauli exclusion effect as well as short-range electromagnetic
repulsion between electrons and the long-range van der Waals interaction is
represented by the behavior of a single effective interaction. The radial dependence
of interaction energy is taken to be a simple analytic function which facilitates
solving the wave equation for the pair of atoms or molecules, assuming they each act
rigidly. Two popular choices are the Lennard-Jones potential energy and the Morse
potential energy.
The Lennard-Jones potential energy is given by 10
U LJ = ε
(r e /r)
12
− 2(r e /r)
6
,
(11.20)
where ε is the depth of the potential energy and r e is the radius to the minimum of
U LJ (Fig. 11.1). Small vibrations between two atoms with reduced mass of μ and
whose interaction is described by the Lennard-Jones potential energy will have a
frequency of f = (3/2πr e )
√
2ε/μ.
10 J.E. Lennard-Jones, On the Determination of Molecular Fields, Proc R. Soc Lond A 106:738,
463–477 (1924).
525
11.6 Binding Interactions Between Atoms
In quantum theory, the interaction energy, also called the potential energy, plays a
more fundamental role than force. The potential energy of a particle enters directly
into the wave equation for the particle. We solve the wave equation to get the
probability of finding the particle at various locations at selected times.
Force in quantum theory is a derived concept coming from the negative gradient
of the interaction energy V (r) that a particle has at position r because of nearby
particles: F = −∇V . This statement is just the inverse of the definition of work
done by the force: V = −
F · dr. If you plot the potential energy of one atom
relative to a second as a function of the separation between the atoms, the negative
slope of the curve for that function is the force on the first atom due to the second
atom.
Quantum effects and electromagnetic interactions determine the nanoscale
behavior of bound and unbound atoms in biological systems. The valence electrons,
i.e. those found in the outer region of atoms, largely mold chemical binding of
atoms into molecules. The ‘core’ electrons prevent atoms from collapsing, because
the Pauli exclusion principle allows only one electron per quantum state, and in
an atom, there are only a few such states near the nucleus. The electron state most
tightly held to the nucleus is distributed over a size comparable to the Bohr radius
( ¯
h 2 /(m e c 2 ), about half an ångström, divided by Z, the charge on the nucleus.
This size is far greater than the diameter of the atomic nucleus, which is about
1.75 × 10 −5 of an ångström for a proton in hydrogen and about 15 × 10 −5 of an
ångströms for the Uranium nucleus with 239 protons and neutrons.
For simplicity in some quantum calculations for bound atoms and pairs of
molecules, the electron Pauli exclusion effect as well as short-range electromagnetic
repulsion between electrons and the long-range van der Waals interaction is
represented by the behavior of a single effective interaction. The radial dependence
of interaction energy is taken to be a simple analytic function which facilitates
solving the wave equation for the pair of atoms or molecules, assuming they each act
rigidly. Two popular choices are the Lennard-Jones potential energy and the Morse
potential energy.
The Lennard-Jones potential energy is given by 10
U LJ = ε
(r e /r)
12
− 2(r e /r)
6
,
(11.20)
where ε is the depth of the potential energy and r e is the radius to the minimum of
U LJ (Fig. 11.1). Small vibrations between two atoms with reduced mass of μ and
whose interaction is described by the Lennard-Jones potential energy will have a
frequency of f = (3/2πr e )
√
2ε/μ.
10 J.E. Lennard-Jones, On the Determination of Molecular Fields, Proc R. Soc Lond A 106:738,
463–477 (1924).