Roleoftemperature
Temperaturecausesquantumlocalizationtodecreaseastemperatureisincreased.
Thethermalwavelength
kisBoltzmannsconstant.
Considermethane:CH4
H
H
C
1.8
H
C
H
ThefourHsareidentical.Doweneedtoconsiderthesymmetryofthewavefunctionat300k?
Thetherma
Hellman-Feynman Theorem
Consider a system with a Hamiltonian
Let
be an eigenvector of
that depends on some parameters .
with eigenvalue
(16)
We further assume that
is normalized so that
(17)
The Hellman-Feynman theorem states that
(18)
The proof of the He
HAMILTONIAN AIMD
The next few lectures will treat the problem of quantum chemistry, a subfield of
quantum mechanics also known as molecular quantum mechanics. The idea of
quantum chemistry is to use only the simple facts molecules and, indeed, all of
ordi
Ground Motion Nuclear Evolution
In many cases, non-adiabatic effects can be neglected, and we may consider
motion only on the ground electronic surface described by:
(10)
Moreover, if nuclear quantum effects can be neglected, then we may arrive at classic
electroniceigenvalueproblem:IntheBornOppenheimerapproximation,theelectronic
eigenvalueproblemtakestheform.
Let
wherexdenotestheelectroniccoordinates,andRdenotesthenuclearcoordinates.
Crudestapproximation.Assumetheequationisseparableandthattheelectronsare
Born-Oppenheimer surfaces: full internuclear potential for each electronic surface is
given by
. On each Born-Oppenheimer surface, the nuclear
eigenvalue problem can be solved, which yields a set of levels (rotational and
vibrational in the nuclear motion
The Born-Oppenheimer (named for its original inventors, Max Born and Robert
Oppenheimer) is based on the fact that nuclei are several thousand times heavier than
electrons. The proton, itself, is approximately 2000 times more massive than an
electron. In
Curvature of Wave Functions
Schrdingers equation in the form
can be interpreted by saying that the left-hand side, the rate of change of slope, is
the curvature so the curvature of the function is proportional to
This means that if E > V(x), for
negative