Chem3D Users Manual - CambridgeSoft

the same wave function and a constant. The wave
function is called an eigenfunction, and the constant, an
eigenvalue.
nuclear energy by an electronic Hamiltonian, which
can be solved at any set of nuclear coordinates. The
electronic version of the Schrödinger equation is:
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• Exact solutions to the Schrödinger equation
are possible only for the simplest 1 electron-1
nucleus system. These solutions, however,
yield the basis for all of quantum mechanics.
• The solutions describe a set of allowable states
for an electron. The observable quantity for
these states is described as a probability
function. This function is the square of the
wave function, and when properly normalized,
describes the probability of finding an electron
in that state.
∫Ψ 2 ()r r d = 1
where r = radius (x, y, and z)
H elec
Ψ elec
=E elec
Ψ elec
Another approximation assumes that electrons act
independently of one another, or, more accurately,
that each electron is influenced by an average field
created by all other electrons and nuclei. Each
electron in its own orbital is unimpeded by its
neighbors.
The electronic Hamiltonian is thus simplified by
representing it as a sum of 1-electron Hamiltonians,
and the wave equation becomes solvable for
individual electrons in a molecule once a functional
form of the wave function can be derived.
• There are many solutions to this probability
function. These solutions are called atomic
orbitals, and their energies, orbital energies.
• For a molecule with many electrons and nuclei
the aim is to be able to describe molecular
orbitals and energies in as analogous a fashion
to the original Schrödinger equation as
possible.
Approximations to the Hamiltonian
The first approximation made is known as the
Born-Oppenheimer approximation, which allows
separate treatment of the electronic and nuclear
energies. Due to the large mass difference between
an electron and a nucleus, a nucleus moves so much
more slowly than an electron that it can be regarded
as motionless relative to the electron. In effect, this
approximation considers electrons to be moving
with respect to a fixed nucleus. This allows the
electronic energy to be described separately from
H elec
=
∑
H eff ψ =εψ
H i
eff
For a molecular system, a matrix of these 1-electron
Hamiltonians is constructed to describe the
1-electron interactions between a single electron
and the core nucleus. The following represents the
matrix for two atomic orbitals, φ µ and φ ν .
H uv ∫ =φ H eff µ v
φdτ
i
However, in molecular systems, this Hamiltonian
does not account for the interaction between
electrons with 2 or more different interaction
centers or the interaction of two electrons. Thus,
the Hamiltonian is further modified. This
modification renames the Hamiltonian operator to
the Fock operator.
Fψ=Eψ
The Fock operator is composed of a set of 1-
electron Hamiltonians that describe the 1-electron,
1 center interactions and is supplemented by terms
144•Computation Concepts
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Quantum Mechanics Theory in Brief