11.4 Molecular Interactions
523
momentum quantum number for particle quantum states. For l = 4, 5 . . . we use
g, h, . . . in the sequence of the alphabet.
Now consider the electrons in an element heavier than hydrogen. Suppose
the nucleus has a charge Ze. The electron orbitals can be found by solving
the appropriate wave equation with Z electrons bound around the nucleus:
H H(r 1 , r 2 , . . . , r Z ) = EE(r 1 , r 2 , . . . , r Z ). One technique to express the orbitals
takes advantage of the fact that the eigenstates of any observable forms a complete
set of quantum states. An arbitrary wave can be expanded into a complete set of
orthonormal waves.
The valence electrons of atoms are the electrons in orbitals the least bound to
the atom, and able to take part in chemical bonding. If the orbital for a valence
electron is expressed in terms of hydrogenic orbitals with an effective ‘screened’
central charge, then only a few such hydrogenic orbitals may be needed to describe
the chemical bonding in which this electron takes part. If several valence electrons
take part, then linear combinations of their orbitals may be sufficient to loosely
characterize their participation in a bond. The combination of orbitals which occurs
after the system settles is that which minimizes the energy of the system.
11.4 Molecular Interactions
As we have indicated, interactions between molecules of biological interest are
dominated by electrical interactions. The electrical interaction comes about because
atoms are made from charged particles: relatively heavy positive nuclei and
relatively light electrons. So far (since 1928), the application of wave mechanics
to molecular systems, when we can handle the mathematics, successfully predicts
atomic bonding and molecular interactions. Of particular interest is the nature of the
chemical bond, the structure of macromolecules, the behavior of proteins in the cell,
and the sequence of interactions and reactions within a cell that allow for life.
In principle, ‘all we have to do’ is solve the wave equation for the system: H H =
i ¯
h∂∂/∂t, with all the Coulombic interactions between each electron i and each
nuclei N , V Ni = −kZe 2 /r Ni , and every pair of electrons, V ij = +ke 2 /r ij . For
the ‘ground state’ of each molecule, the lowest energy eigenvalue and eigenstate
solution of H H = EE would be taken.
In practice, there are no ‘closed form’ solutions for atoms or molecules with
more than two charges. To find such solutions numerically by ‘brute force’ may
require excessive computer time, depending on the number of interacting particles
present and on the ingenuity of the programmer. Investigators have developed many
clever schemes to approximate and to simplify the calculation. For example, since
the nuclei are more than 1836 times the mass of the electrons, their inertia keeps
them close to stationary and near the center-of-mass of the atom of which they are
a part. One can start a calculation assuming the nuclei are fixed but with an as-yet
undetermined separation. For the electrons, one can start by taking the inner shells
as undisturbed by the valence shells.
523
momentum quantum number for particle quantum states. For l = 4, 5 . . . we use
g, h, . . . in the sequence of the alphabet.
Now consider the electrons in an element heavier than hydrogen. Suppose
the nucleus has a charge Ze. The electron orbitals can be found by solving
the appropriate wave equation with Z electrons bound around the nucleus:
H H(r 1 , r 2 , . . . , r Z ) = EE(r 1 , r 2 , . . . , r Z ). One technique to express the orbitals
takes advantage of the fact that the eigenstates of any observable forms a complete
set of quantum states. An arbitrary wave can be expanded into a complete set of
orthonormal waves.
The valence electrons of atoms are the electrons in orbitals the least bound to
the atom, and able to take part in chemical bonding. If the orbital for a valence
electron is expressed in terms of hydrogenic orbitals with an effective ‘screened’
central charge, then only a few such hydrogenic orbitals may be needed to describe
the chemical bonding in which this electron takes part. If several valence electrons
take part, then linear combinations of their orbitals may be sufficient to loosely
characterize their participation in a bond. The combination of orbitals which occurs
after the system settles is that which minimizes the energy of the system.
11.4 Molecular Interactions
As we have indicated, interactions between molecules of biological interest are
dominated by electrical interactions. The electrical interaction comes about because
atoms are made from charged particles: relatively heavy positive nuclei and
relatively light electrons. So far (since 1928), the application of wave mechanics
to molecular systems, when we can handle the mathematics, successfully predicts
atomic bonding and molecular interactions. Of particular interest is the nature of the
chemical bond, the structure of macromolecules, the behavior of proteins in the cell,
and the sequence of interactions and reactions within a cell that allow for life.
In principle, ‘all we have to do’ is solve the wave equation for the system: H H =
i ¯
h∂∂/∂t, with all the Coulombic interactions between each electron i and each
nuclei N , V Ni = −kZe 2 /r Ni , and every pair of electrons, V ij = +ke 2 /r ij . For
the ‘ground state’ of each molecule, the lowest energy eigenvalue and eigenstate
solution of H H = EE would be taken.
In practice, there are no ‘closed form’ solutions for atoms or molecules with
more than two charges. To find such solutions numerically by ‘brute force’ may
require excessive computer time, depending on the number of interacting particles
present and on the ingenuity of the programmer. Investigators have developed many
clever schemes to approximate and to simplify the calculation. For example, since
the nuclei are more than 1836 times the mass of the electrons, their inertia keeps
them close to stationary and near the center-of-mass of the atom of which they are
a part. One can start a calculation assuming the nuclei are fixed but with an as-yet
undetermined separation. For the electrons, one can start by taking the inner shells
as undisturbed by the valence shells.