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represents the probability of the particle being at a given location. (You cannot
know for sure where a particle is; rather, you have a probability distribution for its
existence in various locations in space. To the best of our knowledge so far, this is
the reality, but it might sound counterintuitive because this reality reveals itself
only at small scales, and so we are not used to it in everyday life.) Note that the
Schro¨ dinger equation is, in general, a time-dependent differential equation. Here
we have shown the time-independent case just for the sake of introduction. At the
root of modeling a nanoscale device is the problem of writing the above equation
for the system by including the proper terms in the Hamiltonian and solving it to
find C and e . From C and e one can proceed to finding other parameters of
interest in the system. Thus, for nanoscale modeling, learning quantum mechanics,
at least at an introductory level, is the first necessary step. The reader is referred to
[71, 72] as good examples of quantum mechanics textbooks. Finding a solution to
the above equation without invoking empirical parameters is what ab initio
approaches try to accomplish.
S LATER D ETERMINANTS AND THE H ARTREE -F OCK M ETHOD . In one simplifica-
tion, exact instantaneous electron-electron interactions are neglected in studying a
system of atoms. The problem is reduced to that of a single electron moving in the
''average'' field created by the rest of the system. This approximation is very useful
in many practical cases and captures the basic physics of the problem while greatly
simplifying the computational complexity. When trying to solve the Schro ¨ dinger
equation in this case, one can represent the overall wave function C of the system
as products of individual wave functions for electrons; this is basically the
technique of separation of variables in solving differential equations. In addition,
Pauli's exclusion principle that no two electrons with the same spin can reside in
the same state must be satisfied. This, together with the fact that electrons are
indistinguishable particles from one another, requires that the system wave
function be antisymmetric with respect to simultaneous change of spin and space
coordinates of electrons. A general form for the many-body wave function that
satisfies this condition is the so called Slater determinant [73, 74]:
f 1 ðÞ f 1 ð Þ
f 1 ðÞ
f 2 ðÞ f 2 ð Þ
f 2 ðÞ
C /
.
.
.
.
. .
f N ðÞ f N ð Þ
f N ðÞ
In the above each f represents a single-electron wave function (including the
spin part). Since swapping two rows or columns in a determinant changes the
sign, this form satisfies the antisymmetry condition. In the Hartree-Fock method,
the wave function of the system is obtained in the form of a Slater determinant.
More on this method can be found in [74, 75]. Although this method neglects
electron-electron correlation (i.e., the instantaneous nature of electron-electron
 
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