# Nanotechnology/Modelling Nanosystems#The Schrödinger equation

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# Modelling Nanosystems

# The Schrödinger equation

 the Schrödinger equation

$$H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{\partial}{\partial t} \left| \psi \left(t\right) \right\rangle$$

where $\mathrm{i}$ is the imaginary unit,
$t$ is time, $\partial / \partial t$ is the partial
derivative with respect to $t$,
$\hbar$ is the reduced Planck\'s
constant (Planck\'s constant divided by
$2\pi$), $\psi (t)$ is the wave function,
and $H\!\left(t\right)$ is the
Hamiltonian "wikilink") operator.

# Hartree-Fock (HF) or self-consistent field (SCF)

In computational physics and
chemistry, the **Hartree--Fock**
(**HF**) method is a method of approximation for the determination of
the wave function and the energy of a
quantum many-body system in a
stationary state.

The Hartree--Fock method often assumes that the exact, *N*-body wave
function of the system can be approximated by a single Slater
determinant (in the case where the
particles are fermions) or by a single
permanent (in the case of
bosons) of *N*
spin-orbitals. By invoking the
w:variational method, one can derive
a set of *N*-coupled equations for the *N* spin orbitals. A solution of
these equations yields the Hartree--Fock wave function and energy of the
system.

Especially in the older literature, the Hartree--Fock method is also
called the **self-consistent field method** (**SCF**). In deriving what
is now called the Hartree equation as
an approximate solution of the Schrödinger
equation,
Hartree required the final field as
computed from the charge distribution to be \"self-consistent\" with the
assumed initial field. Thus, self-consistency was a requirement of the
solution. The solutions to the non-linear Hartree--Fock equations also
behave as if each particle is subjected to the mean field created by all
other particles (see the Fock
operator below) and hence
the terminology continued. The equations are almost universally solved
by means of an iterative method,
although the fixed-point iteration
algorithm does not always converge.[^1] This solution scheme is not the
only one possible and is not an essential feature of the Hartree--Fock
method.

The Hartree--Fock method finds its typical application in the solution
of the Schrödinger equation for
atoms, molecules, nanostructures[^2] and solids but it has also found
widespread use in nuclear physics. (See
Hartree--Fock--Bogoliubov
method for a discussion of its
application in nuclear
structure
theory). In atomic structure theory,
calculations may be for a spectrum with many excited energy levels and
consequently the Hartree--Fock method for
atoms assumes the wave
function is a single configuration
state function with
well-defined quantum numbers and that
the energy level is not necessarily the ground
state.

For both atoms and molecules, the Hartree--Fock solution is the central
starting point for most methods that describe the many-electron system
more accurately.

The rest of this article will focus on applications in electronic
structure theory suitable for molecules with the atom as a special case.
The discussion here is only for the Restricted Hartree--Fock method,
where the atom or molecule is a closed-shell system with all orbitals
(atomic or molecular) doubly occupied.
Open-shell systems, where some of the
electrons are not paired, can be dealt with by one of two Hartree--Fock
methods:

-   w:Restricted open-shell
    Hartree--Fock
    (ROHF)
-   w:Unrestricted
    Hartree--Fock (UHF)

## history

The origin of the Hartree--Fock method dates back to the end of the
1920s, soon after the discovery of the w:Schrödinger
equation in 1926. In 1927 D. R.
Hartree introduced a procedure, which he
called the self-consistent field method, to calculate approximate wave
functions and energies for atoms and ions. Hartree was guided by some
earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B.
Lindsay, and himself) set in the
w:old quantum theory of Bohr.

In the w:Bohr model of the atom, the energy
of a state with w:principal quantum
number n is given in atomic
units as $E = -1 / n^2$. It was observed from atomic spectra that the
energy levels of many-electron atoms are well described by applying a
modified version of Bohr\'s formula. By introducing the w:quantum
defect *d* as an empirical parameter, the
energy levels of a generic atom were well approximated by the formula
$E = -1/(n+d)^2$, in the sense that one could reproduce fairly well the
observed transitions levels observed in the
w:X-ray region (for example, see the empirical
discussion and derivation in w:Moseley\'s
law). The existence of a non-zero quantum
defect was attributed to electron-electron repulsion, which clearly does
not exist in the isolated hydrogen atom. This repulsion resulted in
partial screening of the bare nuclear
charge. These early researchers later introduced other potentials
containing additional empirical parameters with the hope of better
reproducing the experimental data.

Hartree sought to do away with empirical parameters and solve the
many-body time-independent Schrödinger equation from fundamental
physical principles, i.e., ab
initio. His first
proposed method of solution became known as the **Hartree method**.
However, many of Hartree\'s contemporaries did not understand the
physical reasoning behind the Hartree method: it appeared to many people
to contain empirical elements, and its connection to the solution of the
many-body Schrödinger equation was unclear. However, in 1928 J. C.
Slater and J. A. Gaunt independently
showed that the Hartree method could be couched on a sounder theoretical
basis by applying the w:variational
principle to an
w:ansatz (trial wave function) as a product of
single-particle functions.

In 1930 Slater and V. A. Fock
independently pointed out that the Hartree method did not respect the
principle of antisymmetry of the wave
function. The Hartree method used the w:Pauli exclusion
principle in its older
formulation, forbidding the presence of two electrons in the same
quantum state. However, this was shown to be fundamentally incomplete in
its neglect of w:quantum statistics.

It was then shown that a w:Slater
determinant, a
w:determinant of one-particle orbitals first
used by Heisenberg and Dirac in 1926, trivially satisfies the
antisymmetric property of the exact
solution and hence is a suitable w:ansatz for
applying the w:variational
principle. The original Hartree
method can then be viewed as an approximation to the Hartree--Fock
method by neglecting exchange. Fock\'s
original method relied heavily on w:group
theory and was too abstract for contemporary
physicists to understand and implement. In 1935 Hartree reformulated the
method more suitably for the purposes of calculation.

The Hartree--Fock method, despite its physically more accurate picture,
was little used until the advent of electronic computers in the 1950s
due to the much greater computational demands over the early Hartree
method and empirical models. Initially, both the Hartree method and the
Hartree--Fock method were applied exclusively to atoms, where the
spherical symmetry of the system allowed one to greatly simplify the
problem. These approximate methods were (and are) often used together
with the w:central field
approximation, to impose that
electrons in the same shell have the same radial part, and to restrict
the variational solution to be a spin eigenfunction. Even so, solution
by hand of the Hartree--Fock equations for a medium sized atom were
laborious; small molecules required computational resources far beyond
what was available before 1950.

## Hartree--Fock algorithm

The Hartree--Fock method is typically used to solve the time-independent
Schrödinger equation for a multi-electron atom or molecule as described
in the w:Born--Oppenheimer
approximation. Since there
are no known solutions for many-electron systems (hydrogenic
atoms and the diatomic hydrogen cation
being notable one-electron exceptions), the problem is solved
numerically. Due to the nonlinearities introduced by the Hartree--Fock
approximation, the equations are solved using a nonlinear method such as
w:iteration, which gives rise to the name
\"self-consistent field method.\"

### Approximations

The Hartree--Fock method makes five major simplifications in order to
deal with this task:

-   The w:Born--Oppenheimer
    approximation is
    inherently assumed. The full molecular wave function is actually a
    function of the coordinates of each of the nuclei, in addition to
    those of the electrons.
-   Typically, relativistic effects
    are completely neglected. The momentum
    operator is assumed to be completely
    non-relativistic.
-   The variational solution is assumed to be a w:linear
    combination of a finite number of
    basis functions "wikilink"), which are
    usually (but not always) chosen to be
    w:orthogonal. The finite basis set is
    assumed to be approximately
    complete.
-   Each w:energy eigenfunction is
    assumed to be describable by a single w:Slater
    determinant, an antisymmetrized
    product of one-electron wave functions (i.e., orbitals).
-   The mean field approximation is
    implied. Effects arising from deviations from this assumption, known
    as w:electron correlation, are
    completely neglected for the electrons of opposite spin, but are
    taken into account for electrons of parallel spin.[^3][^4] (Electron
    correlation should not be confused with electron exchange, which is
    fully accounted for in the Hartree--Fock method.)[^5]

Relaxation of the last two approximations give rise to many so-called
w:post-Hartree--Fock methods.

!Greatly simplified algorithmic flowchart illustrating the
Hartree--Fock
method{width="325"}

### Variational optimization of orbitals

The variational
theorem "wikilink") states
that for a time-independent Hamiltonian operator, any trial wave
function will have an energy w:expectation
value that is greater than or equal to
the true w:ground state wave function
corresponding to the given Hamiltonian. Because of this, the
Hartree--Fock energy is an upper bound to the true ground state energy
of a given molecule. In the context of the Hartree--Fock method, the
best possible solution is at the *Hartree--Fock limit*; i.e., the limit
of the Hartree--Fock energy as the basis set approaches
completeness.
(The other is the *full-CI limit*, where the last two approximations of
the Hartree--Fock theory as described above are completely undone. It is
only when both limits are attained that the exact solution, up to the
Born--Oppenheimer approximation, is obtained.) The Hartree--Fock energy
is the minimal energy for a single Slater determinant.

The starting point for the Hartree--Fock method is a set of approximate
one-electron wave functions known as
*w:spin-orbitals*. For an w:atomic
orbital calculation, these are typically
the orbitals for a hydrogenic atom (an atom with only one electron, but
the appropriate nuclear charge). For a w:molecular
orbital or crystalline calculation, the
initial approximate one-electron wave functions are typically a
w:linear combination of atomic
orbitals (LCAO).

The orbitals above only account for the presence of other electrons in
an average manner. In the Hartree--Fock method, the effect of other
electrons are accounted for in a w:mean-field
theory context. The orbitals are
optimized by requiring them to minimize the energy of the respective
Slater determinant. The resultant variational conditions on the orbitals
lead to a new one-electron operator, the w:Fock
operator. At the minimum, the occupied
orbitals are eigensolutions to the Fock operator via a w:unitary
transformation between themselves.
The Fock operator is an effective one-electron Hamiltonian operator
being the sum of two terms. The first is a sum of kinetic energy
operators for each electron, the internuclear repulsion energy, and a
sum of nuclear-electronic Coulombic
attraction terms. The second are Coulombic repulsion terms between
electrons in a mean-field theory description; a net repulsion energy for
each electron in the system, which is calculated by treating all of the
other electrons within the molecule as a smooth distribution of negative
charge. This is the major simplification inherent in the Hartree--Fock
method, and is equivalent to the fifth simplification in the above list.

Since the Fock operator depends on the orbitals used to construct the
corresponding w:Fock matrix, the
eigenfunctions of the Fock operator are in turn new orbitals which can
be used to construct a new Fock operator. In this way, the Hartree--Fock
orbitals are optimized iteratively until the change in total electronic
energy falls below a predefined threshold. In this way, a set of
self-consistent one-electron orbitals are calculated. The Hartree--Fock
electronic wave function is then the Slater determinant constructed out
of these orbitals. Following the basic postulates of quantum mechanics,
the Hartree--Fock wave function can then be used to compute any desired
chemical or physical property within the framework of the Hartree--Fock
method and the approximations employed.

## Mathematical formulation

### The Fock operator

Because the electron-electron repulsion term of the w:electronic
molecular Hamiltonian
involves the coordinates of two different electrons, it is necessary to
reformulate it in an approximate way. Under this approximation,
(outlined under Hartree--Fock
algorithm), all of
the terms of the exact Hamiltonian except the nuclear-nuclear repulsion
term are re-expressed as the sum of one-electron operators outlined
below, for closed-shell atoms or molecules (with two electrons in each
spatial orbital).[^6] The \"(1)\" following each operator symbol simply
indicates that the operator is 1-electron in nature.

$$\hat F\{\phi_j\} = \hat H^{\text{core}}(1)+\sum_{j=1}^{N/2}[2\hat J_j(1)-\hat K_j(1)]$$

where

$$\hat F\{\phi_j\}$$

is the one-electron Fock operator generated by the orbitals $\phi_j$,
and

$$\hat H^{\text{core}}(1)=-\frac{1}{2}\nabla^2_1 - \sum_{\alpha} \frac{Z_\alpha}{r_{1\alpha}}$$

is the one-electron core
Hamiltonian "wikilink"). Also

$$\hat J_j(1)$$

is the w:Coulomb operator, defining the
electron-electron repulsion energy due to each of the two electrons in
the *j*th orbital.[^7] Finally

$$\hat K_j(1)$$

is the w:exchange operator, defining
the electron exchange energy due to the antisymmetry of the total
n-electron wave function. [^8] This (so called) \"exchange energy\"
operator, K, is simply an artifact of the Slater determinant. Finding
the Hartree--Fock one-electron wave functions is now equivalent to
solving the eigenfunction equation:

$$\hat F(1)\phi_i(1)=\epsilon_i \phi_i(1)$$

where $\phi_i\;(1)$ are a set of one-electron wave functions, called the
Hartree--Fock molecular orbitals.

### Linear combination of atomic orbitals

Typically, in modern Hartree--Fock calculations, the one-electron wave
functions are approximated by a w:linear combination of atomic
orbitals. These
atomic orbitals are called w:Slater-type
orbitals. Furthermore, it is very
common for the \"atomic orbitals\" in use to actually be composed of a
linear combination of one or more Gaussian-type
orbitals, rather than Slater-type
orbitals, in the interests of saving large amounts of computation time.

Various basis sets "wikilink") are used in
practice, most of which are composed of Gaussian functions. In some
applications, an orthogonalization method such as the w:Gram--Schmidt
process is performed in order to
produce a set of orthogonal basis functions. This can in principle save
computational time when the computer is solving the Roothaan--Hall
equations by converting the w:overlap
matrix effectively to an w:identity
matrix. However, in most modern computer
programs for molecular Hartree--Fock calculations this procedure is not
followed due to the high numerical cost of orthogonalization and the
advent of more efficient, often sparse, algorithms for solving the
w:generalized eigenvalue
problem, of which the
Roothaan--Hall equations are an
example.

## Numerical stability

w:Numerical stability can be a
problem with this procedure and there are various ways of combating this
instability. One of the most basic and generally applicable is called
*F-mixing* or damping. With F-mixing, once a single electron wave
function is calculated it is not used directly. Instead, some
combination of that calculated wave function and the previous wave
functions for that electron is used---the most common being a simple
linear combination of the calculated and immediately preceding wave
function. A clever dodge, employed by Hartree, for atomic calculations
was to increase the nuclear charge, thus pulling all the electrons
closer together. As the system stabilised, this was gradually reduced to
the correct charge. In molecular calculations a similar approach is
sometimes used by first calculating the wave function for a positive ion
and then to use these orbitals as the starting point for the neutral
molecule. Modern molecular Hartree--Fock computer programs use a variety
of methods to ensure convergence of the Roothaan--Hall equations.

## Weaknesses, extensions, and alternatives

Of the five simplifications outlined in the section \"Hartree--Fock
algorithm\", the fifth is typically the most important. Neglecting
electron correlation can lead to large deviations from experimental
results. A number of approaches to this weakness, collectively called
w:post-Hartree--Fock methods, have
been devised to include electron correlation to the multi-electron wave
function. One of these approaches, w:Møller--Plesset perturbation
theory, treats
correlation as a perturbation of the
Fock operator. Others expand the true multi-electron wave function in
terms of a linear combination of Slater determinants---such as
w:multi-configurational self-consistent
field,
w:configuration interaction,
w:quadratic configuration
interaction, and
complete active space SCF
(CASSCF).
Still others (such as variational quantum Monte
Carlo) modify the Hartree--Fock
wave function by multiplying it by a correlation function (\"Jastrow\"
factor), a term which is explicitly a function of multiple electrons
that cannot be decomposed into independent single-particle functions.

An alternative to Hartree--Fock calculations used in some cases is
w:density functional theory,
which treats both exchange and correlation energies, albeit
approximately. Indeed, it is common to use calculations that are a
hybrid of the two methods---the popular B3LYP scheme is one such
w:hybrid functional method. Another
option is to use w:modern valence
bond methods.

## Software packages

For a list of software packages known to handle Hartree--Fock
calculations, particularly for molecules and solids, see the w:list of
quantum chemistry and solid state physics
software.

## Sources

-
-
-
## Slater determinant

### Two-particle case

The simplest way to approximate the wave function of a many-particle
system is to take the product of properly chosen
orthogonal "wikilink") wave
functions of the individual particles. For the two-particle case with
spatial coordinates $\mathbf{x}_1$ and $\mathbf{x}_2$, we have

$$\Psi(\mathbf{x}_1,\mathbf{x}_2) = \chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2).$$

This expression is used in the w:Hartree--Fock
method as an
w:ansatz for the many-particle wave function and
is known as a w:Hartree product.
However, it is not satisfactory for w:fermions
because the wave function above is not antisymmetric, as it must be for
w:fermions from the w:Pauli exclusion
principle. An antisymmetric
wave function can be mathematically described as follows:

$$\Psi(\mathbf{x}_1,\mathbf{x}_2) = -\Psi(\mathbf{x}_2,\mathbf{x}_1)$$

which does not hold for the Hartree product. Therefore the Hartree
product does not satisfy the Pauli principle. This problem can be
overcome by taking a w:linear
combination of both Hartree products

$$\Psi(\mathbf{x}_1,\mathbf{x}_2) = \frac{1}{\sqrt{2}}\{\chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2) - \chi_1(\mathbf{x}_2)\chi_2(\mathbf{x}_1)\}$$

$$= \frac{1}{\sqrt2}\begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) \end{vmatrix}$$

where the coefficient is the w:normalization
factor. This wave function is now
antisymmetric and no longer distinguishes between fermions, that is: one
cannot indicate an ordinal number to a specific particle and the indices
given are interchangeable. Moreover, it also goes to zero if any two
wave functions of two fermions are the same. This is equivalent to
satisfying the Pauli exclusion principle.

### Generalizations

The expression can be generalised to any number of fermions by writing
it as a w:determinant. For an *N*-electron
system, the Slater determinant is defined as [^9]

$$\Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) =
\frac{1}{\sqrt{N!}}
\left|
   \begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\
                      \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\
                      \vdots & \vdots & \ddots & \vdots \\
                      \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N)
   \end{matrix} \right|\equiv \left| \begin{matrix}
   \chi _1 & \chi _2 & \cdots  & \chi _N  \\
\end{matrix}
\right|,$$

where in the final expression, a compact notation is introduced: the
normalization constant and labels for the fermion coordinates are
understood -- only the wavefunctions are exhibited. The linear
combination of Hartree products for the two-particle case can clearly be
seen as identical with the Slater determinant for *N* = 2. It can be
seen that the use of Slater determinants ensures an antisymmetrized
function at the outset; symmetric functions are automatically rejected.
In the same way, the use of Slater determinants ensures conformity to
the w:Pauli principle. Indeed, the
Slater determinant vanishes if the set {χ~i~ } is w:linearly
dependent. In particular, this is the
case when two (or more) spin orbitals are the same. In chemistry one
expresses this fact by stating that no two electrons can occupy the same
spin orbital. In general the Slater determinant is evaluated by the
w:Laplace expansion. Mathematically, a
Slater determinant is an antisymmetric tensor, also known as a w:wedge
product.

A single Slater determinant is used as an approximation to the
electronic wavefunction in Hartree--Fock
theory. In more accurate theories
(such as w:configuration
interaction and
w:MCSCF), a linear combination of Slater
determinants is needed.

The word \"**detor**\" was proposed by S. F. Boys to describe the Slater
determinant of the general type,[^10] but this term is rarely used.

Unlike w:fermions that are subject to the Pauli
exclusion principle, two or more w:bosons can
occupy the same quantum state of a system. Wavefunctions describing
systems of identical w:bosons are symmetric under
the exchange of particles and can be expanded in terms of
w:permanents.

## Fock matrix

In the w:Hartree--Fock method of
w:quantum mechanics, the **Fock
matrix** is a matrix "wikilink") approximating
the single-electron w:energy operator of
a given quantum system in a given set
of basis vectors.[^11]

It is most often formed in w:computational
chemistry when attempting to
solve the w:Roothaan equations for an
atomic or molecular system. The Fock matrix is actually an approximation
to the true Hamiltonian "wikilink")
operator "wikilink") of the quantum system.
It includes the effects of electron-electron
repulsion only in an average way.
Importantly, because the Fock operator is a one-electron operator, it
does not include the w:electron
correlation energy.

The Fock matrix is defined by the *Fock operator*. For the restricted
case which assumes w:closed-shell
orbitals and single-determinantal
wavefunctions, the Fock operator for the *i*-th electron is given
by:[^12]

$$\hat F(i) = \hat h(i)+\sum_{ j=1 }^{n/2}[2 \hat J_j(i)-\hat
K_j(i)]$$

where:

$$\hat F(i)$$ is the Fock operator for the *i*-th electron in the
system,

$${\hat h}(i)$$ is the w:one-electron
hamiltonian for the *i*-th
electron,

$$n$$ is the number of electrons and $\frac{n}{2}$ is the number of
occupied orbitals in the closed-shell system,

$$\hat J_j(i)$$ is the w:Coulomb
operator, defining the repulsive force
between the *j*-th and *i*-th electrons in the system,

$$\hat K_j(i)$$ is the w:exchange
operator, defining the quantum effect
produced by exchanging two electrons.

The Coulomb operator is multiplied by two since there are two electrons
in each occupied orbital. The exchange operator is not multiplied by two
since it has a non-zero result only for electrons which have the same
spin as the *i*-th electron.

For systems with unpaired electrons there are many choices of Fock
matrices.

 Hartree-Fock (HF) or self-consistent field
(SCF)

# Density Functional Theory

## Connection to quantum state symmetry

The Pauli exclusion principle with a single-valued many-particle
wavefunction is equivalent to requiring the wavefunction to be
antisymmetric. An antisymmetric two-particle state is represented as a
sum of states in which one
particle is in state $\scriptstyle |x \rangle$ and the other in state
$\scriptstyle |y\rangle$:

$$|\psi\rangle = \sum_{x,y} A(x,y) |x,y\rangle$$

and antisymmetry under exchange means that
.
This implies that ,
which is Pauli exclusion. It is true in any basis, since unitary changes
of basis keep antisymmetric matrices antisymmetric, although strictly
speaking, the quantity  is
not a matrix but an antisymmetric rank-two
w:tensor.

Conversely, if the diagonal quantities
 are zero *in every basis*,
then the wavefunction component:

$$A(x,y)=\langle \psi|x,y\rangle = \langle \psi | ( |x\rangle \otimes |y\rangle )$$

is necessarily antisymmetric. To prove it, consider the matrix element:

$$\langle\psi| ((|x\rangle + |y\rangle)\otimes(|x\rangle + |y\rangle))
\,$$

This is zero, because the two particles have zero probability to both be
in the superposition state $\scriptstyle |x\rangle + |y\rangle$. But
this is equal to

$$\langle \psi |x,x\rangle + \langle \psi |x,y\rangle + \langle \psi |y,x\rangle + \langle \psi | y,y \rangle
\,$$

The first and last terms on the right hand side are diagonal elements
and are zero, and the whole sum is equal to zero. So the wavefunction
matrix elements obey:

$$\langle \psi|x,y\rangle + \langle\psi |y,x\rangle = 0
\,$$.

or

$$A(x,y)=-A(y,x)
\,$$

### Pauli principle in advanced quantum theory

According to the w:spin-statistics
theorem, particles with integer
spin occupy symmetric quantum states, and particles with half-integer
spin occupy antisymmetric states; furthermore, only integer or
half-integer values of spin are allowed by the principles of quantum
mechanics. In relativistic w:quantum field
theory, the Pauli principle follows
from applying a rotation operator in imaginary time to particles of
half-integer spin. Since, nonrelativistically, particles can have any
statistics and any spin, there is no way to prove a spin-statistics
theorem in nonrelativistic quantum mechanics.

In one dimension, bosons, as well as fermions, can obey the exclusion
principle. A one-dimensional Bose gas with delta function repulsive
interactions of infinite strength is equivalent to a gas of free
fermions. The reason for this is that, in one dimension, exchange of
particles requires that they pass through each other; for infinitely
strong repulsion this cannot happen. This model is described by a
quantum w:nonlinear Schrödinger
equation. In momentum
space the exclusion principle is valid also for finite repulsion in a
Bose gas with delta function interactions,[^13] as well as for
interacting spins "wikilink") and
w:Hubbard model in one dimension, and for
other models solvable by w:Bethe ansatz.
The ground state in models solvable by
Bethe ansatz is a Fermi sphere.

 Density Functional Theory

# References

See also notes on editing this book about how to add references
w:Nanotechnology/About#How_to_contribute.

```{=html}
<references />
```

------------------------------------------------------------------------

[^1]:

[^2]:

[^3]:

[^4]:

[^5]:

[^6]: Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Englewood
    Cliffs, New Jersey: Prentice Hall. p. 403.
    .

[^7]:

[^8]:

[^9]: Molecular Quantum Mechanics Parts I and II: An Introduction to
    QUANTUM CHEMISTRY (Volume 1), P.W. Atkins, Oxford University Press,
    1977,

[^10]: {{ cite journal\| title=Electronic wave functions I. A general
    method of calculation for the stationary states of any molecular
    system\|author-link=S. Francis Boys\|first=S. F.
    \|last=Boys\|page=542\| volume=A200 \|year=1950\|journal=
    Proceedings of the Royal Society}}

[^11]:

[^12]: Levine, I.N. (1991) *Quantum Chemistry* (4th ed., Prentice-Hall),
    p.403

[^13]: A. Izergin and V. Korepin, Letter in Mathematical Physics vol 6,
    page 283, 1982


# Nanotechnology/Modelling Nanosystems#Transport phenomena

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# Modelling Nanosystems

# The Schrödinger equation

 the Schrödinger equation

$$H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{\partial}{\partial t} \left| \psi \left(t\right) \right\rangle$$

where $\mathrm{i}$ is the imaginary unit,
$t$ is time, $\partial / \partial t$ is the partial
derivative with respect to $t$,
$\hbar$ is the reduced Planck\'s
constant (Planck\'s constant divided by
$2\pi$), $\psi (t)$ is the wave function,
and $H\!\left(t\right)$ is the
Hamiltonian "wikilink") operator.

# Hartree-Fock (HF) or self-consistent field (SCF)

In computational physics and
chemistry, the **Hartree--Fock**
(**HF**) method is a method of approximation for the determination of
the wave function and the energy of a
quantum many-body system in a
stationary state.

The Hartree--Fock method often assumes that the exact, *N*-body wave
function of the system can be approximated by a single Slater
determinant (in the case where the
particles are fermions) or by a single
permanent (in the case of
bosons) of *N*
spin-orbitals. By invoking the
w:variational method, one can derive
a set of *N*-coupled equations for the *N* spin orbitals. A solution of
these equations yields the Hartree--Fock wave function and energy of the
system.

Especially in the older literature, the Hartree--Fock method is also
called the **self-consistent field method** (**SCF**). In deriving what
is now called the Hartree equation as
an approximate solution of the Schrödinger
equation,
Hartree required the final field as
computed from the charge distribution to be \"self-consistent\" with the
assumed initial field. Thus, self-consistency was a requirement of the
solution. The solutions to the non-linear Hartree--Fock equations also
behave as if each particle is subjected to the mean field created by all
other particles (see the Fock
operator below) and hence
the terminology continued. The equations are almost universally solved
by means of an iterative method,
although the fixed-point iteration
algorithm does not always converge.[^1] This solution scheme is not the
only one possible and is not an essential feature of the Hartree--Fock
method.

The Hartree--Fock method finds its typical application in the solution
of the Schrödinger equation for
atoms, molecules, nanostructures[^2] and solids but it has also found
widespread use in nuclear physics. (See
Hartree--Fock--Bogoliubov
method for a discussion of its
application in nuclear
structure
theory). In atomic structure theory,
calculations may be for a spectrum with many excited energy levels and
consequently the Hartree--Fock method for
atoms assumes the wave
function is a single configuration
state function with
well-defined quantum numbers and that
the energy level is not necessarily the ground
state.

For both atoms and molecules, the Hartree--Fock solution is the central
starting point for most methods that describe the many-electron system
more accurately.

The rest of this article will focus on applications in electronic
structure theory suitable for molecules with the atom as a special case.
The discussion here is only for the Restricted Hartree--Fock method,
where the atom or molecule is a closed-shell system with all orbitals
(atomic or molecular) doubly occupied.
Open-shell systems, where some of the
electrons are not paired, can be dealt with by one of two Hartree--Fock
methods:

-   w:Restricted open-shell
    Hartree--Fock
    (ROHF)
-   w:Unrestricted
    Hartree--Fock (UHF)

## history

The origin of the Hartree--Fock method dates back to the end of the
1920s, soon after the discovery of the w:Schrödinger
equation in 1926. In 1927 D. R.
Hartree introduced a procedure, which he
called the self-consistent field method, to calculate approximate wave
functions and energies for atoms and ions. Hartree was guided by some
earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B.
Lindsay, and himself) set in the
w:old quantum theory of Bohr.

In the w:Bohr model of the atom, the energy
of a state with w:principal quantum
number n is given in atomic
units as $E = -1 / n^2$. It was observed from atomic spectra that the
energy levels of many-electron atoms are well described by applying a
modified version of Bohr\'s formula. By introducing the w:quantum
defect *d* as an empirical parameter, the
energy levels of a generic atom were well approximated by the formula
$E = -1/(n+d)^2$, in the sense that one could reproduce fairly well the
observed transitions levels observed in the
w:X-ray region (for example, see the empirical
discussion and derivation in w:Moseley\'s
law). The existence of a non-zero quantum
defect was attributed to electron-electron repulsion, which clearly does
not exist in the isolated hydrogen atom. This repulsion resulted in
partial screening of the bare nuclear
charge. These early researchers later introduced other potentials
containing additional empirical parameters with the hope of better
reproducing the experimental data.

Hartree sought to do away with empirical parameters and solve the
many-body time-independent Schrödinger equation from fundamental
physical principles, i.e., ab
initio. His first
proposed method of solution became known as the **Hartree method**.
However, many of Hartree\'s contemporaries did not understand the
physical reasoning behind the Hartree method: it appeared to many people
to contain empirical elements, and its connection to the solution of the
many-body Schrödinger equation was unclear. However, in 1928 J. C.
Slater and J. A. Gaunt independently
showed that the Hartree method could be couched on a sounder theoretical
basis by applying the w:variational
principle to an
w:ansatz (trial wave function) as a product of
single-particle functions.

In 1930 Slater and V. A. Fock
independently pointed out that the Hartree method did not respect the
principle of antisymmetry of the wave
function. The Hartree method used the w:Pauli exclusion
principle in its older
formulation, forbidding the presence of two electrons in the same
quantum state. However, this was shown to be fundamentally incomplete in
its neglect of w:quantum statistics.

It was then shown that a w:Slater
determinant, a
w:determinant of one-particle orbitals first
used by Heisenberg and Dirac in 1926, trivially satisfies the
antisymmetric property of the exact
solution and hence is a suitable w:ansatz for
applying the w:variational
principle. The original Hartree
method can then be viewed as an approximation to the Hartree--Fock
method by neglecting exchange. Fock\'s
original method relied heavily on w:group
theory and was too abstract for contemporary
physicists to understand and implement. In 1935 Hartree reformulated the
method more suitably for the purposes of calculation.

The Hartree--Fock method, despite its physically more accurate picture,
was little used until the advent of electronic computers in the 1950s
due to the much greater computational demands over the early Hartree
method and empirical models. Initially, both the Hartree method and the
Hartree--Fock method were applied exclusively to atoms, where the
spherical symmetry of the system allowed one to greatly simplify the
problem. These approximate methods were (and are) often used together
with the w:central field
approximation, to impose that
electrons in the same shell have the same radial part, and to restrict
the variational solution to be a spin eigenfunction. Even so, solution
by hand of the Hartree--Fock equations for a medium sized atom were
laborious; small molecules required computational resources far beyond
what was available before 1950.

## Hartree--Fock algorithm

The Hartree--Fock method is typically used to solve the time-independent
Schrödinger equation for a multi-electron atom or molecule as described
in the w:Born--Oppenheimer
approximation. Since there
are no known solutions for many-electron systems (hydrogenic
atoms and the diatomic hydrogen cation
being notable one-electron exceptions), the problem is solved
numerically. Due to the nonlinearities introduced by the Hartree--Fock
approximation, the equations are solved using a nonlinear method such as
w:iteration, which gives rise to the name
\"self-consistent field method.\"

### Approximations

The Hartree--Fock method makes five major simplifications in order to
deal with this task:

-   The w:Born--Oppenheimer
    approximation is
    inherently assumed. The full molecular wave function is actually a
    function of the coordinates of each of the nuclei, in addition to
    those of the electrons.
-   Typically, relativistic effects
    are completely neglected. The momentum
    operator is assumed to be completely
    non-relativistic.
-   The variational solution is assumed to be a w:linear
    combination of a finite number of
    basis functions "wikilink"), which are
    usually (but not always) chosen to be
    w:orthogonal. The finite basis set is
    assumed to be approximately
    complete.
-   Each w:energy eigenfunction is
    assumed to be describable by a single w:Slater
    determinant, an antisymmetrized
    product of one-electron wave functions (i.e., orbitals).
-   The mean field approximation is
    implied. Effects arising from deviations from this assumption, known
    as w:electron correlation, are
    completely neglected for the electrons of opposite spin, but are
    taken into account for electrons of parallel spin.[^3][^4] (Electron
    correlation should not be confused with electron exchange, which is
    fully accounted for in the Hartree--Fock method.)[^5]

Relaxation of the last two approximations give rise to many so-called
w:post-Hartree--Fock methods.

!Greatly simplified algorithmic flowchart illustrating the
Hartree--Fock
method{width="325"}

### Variational optimization of orbitals

The variational
theorem "wikilink") states
that for a time-independent Hamiltonian operator, any trial wave
function will have an energy w:expectation
value that is greater than or equal to
the true w:ground state wave function
corresponding to the given Hamiltonian. Because of this, the
Hartree--Fock energy is an upper bound to the true ground state energy
of a given molecule. In the context of the Hartree--Fock method, the
best possible solution is at the *Hartree--Fock limit*; i.e., the limit
of the Hartree--Fock energy as the basis set approaches
completeness.
(The other is the *full-CI limit*, where the last two approximations of
the Hartree--Fock theory as described above are completely undone. It is
only when both limits are attained that the exact solution, up to the
Born--Oppenheimer approximation, is obtained.) The Hartree--Fock energy
is the minimal energy for a single Slater determinant.

The starting point for the Hartree--Fock method is a set of approximate
one-electron wave functions known as
*w:spin-orbitals*. For an w:atomic
orbital calculation, these are typically
the orbitals for a hydrogenic atom (an atom with only one electron, but
the appropriate nuclear charge). For a w:molecular
orbital or crystalline calculation, the
initial approximate one-electron wave functions are typically a
w:linear combination of atomic
orbitals (LCAO).

The orbitals above only account for the presence of other electrons in
an average manner. In the Hartree--Fock method, the effect of other
electrons are accounted for in a w:mean-field
theory context. The orbitals are
optimized by requiring them to minimize the energy of the respective
Slater determinant. The resultant variational conditions on the orbitals
lead to a new one-electron operator, the w:Fock
operator. At the minimum, the occupied
orbitals are eigensolutions to the Fock operator via a w:unitary
transformation between themselves.
The Fock operator is an effective one-electron Hamiltonian operator
being the sum of two terms. The first is a sum of kinetic energy
operators for each electron, the internuclear repulsion energy, and a
sum of nuclear-electronic Coulombic
attraction terms. The second are Coulombic repulsion terms between
electrons in a mean-field theory description; a net repulsion energy for
each electron in the system, which is calculated by treating all of the
other electrons within the molecule as a smooth distribution of negative
charge. This is the major simplification inherent in the Hartree--Fock
method, and is equivalent to the fifth simplification in the above list.

Since the Fock operator depends on the orbitals used to construct the
corresponding w:Fock matrix, the
eigenfunctions of the Fock operator are in turn new orbitals which can
be used to construct a new Fock operator. In this way, the Hartree--Fock
orbitals are optimized iteratively until the change in total electronic
energy falls below a predefined threshold. In this way, a set of
self-consistent one-electron orbitals are calculated. The Hartree--Fock
electronic wave function is then the Slater determinant constructed out
of these orbitals. Following the basic postulates of quantum mechanics,
the Hartree--Fock wave function can then be used to compute any desired
chemical or physical property within the framework of the Hartree--Fock
method and the approximations employed.

## Mathematical formulation

### The Fock operator

Because the electron-electron repulsion term of the w:electronic
molecular Hamiltonian
involves the coordinates of two different electrons, it is necessary to
reformulate it in an approximate way. Under this approximation,
(outlined under Hartree--Fock
algorithm), all of
the terms of the exact Hamiltonian except the nuclear-nuclear repulsion
term are re-expressed as the sum of one-electron operators outlined
below, for closed-shell atoms or molecules (with two electrons in each
spatial orbital).[^6] The \"(1)\" following each operator symbol simply
indicates that the operator is 1-electron in nature.

$$\hat F\{\phi_j\} = \hat H^{\text{core}}(1)+\sum_{j=1}^{N/2}[2\hat J_j(1)-\hat K_j(1)]$$

where

$$\hat F\{\phi_j\}$$

is the one-electron Fock operator generated by the orbitals $\phi_j$,
and

$$\hat H^{\text{core}}(1)=-\frac{1}{2}\nabla^2_1 - \sum_{\alpha} \frac{Z_\alpha}{r_{1\alpha}}$$

is the one-electron core
Hamiltonian "wikilink"). Also

$$\hat J_j(1)$$

is the w:Coulomb operator, defining the
electron-electron repulsion energy due to each of the two electrons in
the *j*th orbital.[^7] Finally

$$\hat K_j(1)$$

is the w:exchange operator, defining
the electron exchange energy due to the antisymmetry of the total
n-electron wave function. [^8] This (so called) \"exchange energy\"
operator, K, is simply an artifact of the Slater determinant. Finding
the Hartree--Fock one-electron wave functions is now equivalent to
solving the eigenfunction equation:

$$\hat F(1)\phi_i(1)=\epsilon_i \phi_i(1)$$

where $\phi_i\;(1)$ are a set of one-electron wave functions, called the
Hartree--Fock molecular orbitals.

### Linear combination of atomic orbitals

Typically, in modern Hartree--Fock calculations, the one-electron wave
functions are approximated by a w:linear combination of atomic
orbitals. These
atomic orbitals are called w:Slater-type
orbitals. Furthermore, it is very
common for the \"atomic orbitals\" in use to actually be composed of a
linear combination of one or more Gaussian-type
orbitals, rather than Slater-type
orbitals, in the interests of saving large amounts of computation time.

Various basis sets "wikilink") are used in
practice, most of which are composed of Gaussian functions. In some
applications, an orthogonalization method such as the w:Gram--Schmidt
process is performed in order to
produce a set of orthogonal basis functions. This can in principle save
computational time when the computer is solving the Roothaan--Hall
equations by converting the w:overlap
matrix effectively to an w:identity
matrix. However, in most modern computer
programs for molecular Hartree--Fock calculations this procedure is not
followed due to the high numerical cost of orthogonalization and the
advent of more efficient, often sparse, algorithms for solving the
w:generalized eigenvalue
problem, of which the
Roothaan--Hall equations are an
example.

## Numerical stability

w:Numerical stability can be a
problem with this procedure and there are various ways of combating this
instability. One of the most basic and generally applicable is called
*F-mixing* or damping. With F-mixing, once a single electron wave
function is calculated it is not used directly. Instead, some
combination of that calculated wave function and the previous wave
functions for that electron is used---the most common being a simple
linear combination of the calculated and immediately preceding wave
function. A clever dodge, employed by Hartree, for atomic calculations
was to increase the nuclear charge, thus pulling all the electrons
closer together. As the system stabilised, this was gradually reduced to
the correct charge. In molecular calculations a similar approach is
sometimes used by first calculating the wave function for a positive ion
and then to use these orbitals as the starting point for the neutral
molecule. Modern molecular Hartree--Fock computer programs use a variety
of methods to ensure convergence of the Roothaan--Hall equations.

## Weaknesses, extensions, and alternatives

Of the five simplifications outlined in the section \"Hartree--Fock
algorithm\", the fifth is typically the most important. Neglecting
electron correlation can lead to large deviations from experimental
results. A number of approaches to this weakness, collectively called
w:post-Hartree--Fock methods, have
been devised to include electron correlation to the multi-electron wave
function. One of these approaches, w:Møller--Plesset perturbation
theory, treats
correlation as a perturbation of the
Fock operator. Others expand the true multi-electron wave function in
terms of a linear combination of Slater determinants---such as
w:multi-configurational self-consistent
field,
w:configuration interaction,
w:quadratic configuration
interaction, and
complete active space SCF
(CASSCF).
Still others (such as variational quantum Monte
Carlo) modify the Hartree--Fock
wave function by multiplying it by a correlation function (\"Jastrow\"
factor), a term which is explicitly a function of multiple electrons
that cannot be decomposed into independent single-particle functions.

An alternative to Hartree--Fock calculations used in some cases is
w:density functional theory,
which treats both exchange and correlation energies, albeit
approximately. Indeed, it is common to use calculations that are a
hybrid of the two methods---the popular B3LYP scheme is one such
w:hybrid functional method. Another
option is to use w:modern valence
bond methods.

## Software packages

For a list of software packages known to handle Hartree--Fock
calculations, particularly for molecules and solids, see the w:list of
quantum chemistry and solid state physics
software.

## Sources

-
-
-
## Slater determinant

### Two-particle case

The simplest way to approximate the wave function of a many-particle
system is to take the product of properly chosen
orthogonal "wikilink") wave
functions of the individual particles. For the two-particle case with
spatial coordinates $\mathbf{x}_1$ and $\mathbf{x}_2$, we have

$$\Psi(\mathbf{x}_1,\mathbf{x}_2) = \chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2).$$

This expression is used in the w:Hartree--Fock
method as an
w:ansatz for the many-particle wave function and
is known as a w:Hartree product.
However, it is not satisfactory for w:fermions
because the wave function above is not antisymmetric, as it must be for
w:fermions from the w:Pauli exclusion
principle. An antisymmetric
wave function can be mathematically described as follows:

$$\Psi(\mathbf{x}_1,\mathbf{x}_2) = -\Psi(\mathbf{x}_2,\mathbf{x}_1)$$

which does not hold for the Hartree product. Therefore the Hartree
product does not satisfy the Pauli principle. This problem can be
overcome by taking a w:linear
combination of both Hartree products

$$\Psi(\mathbf{x}_1,\mathbf{x}_2) = \frac{1}{\sqrt{2}}\{\chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2) - \chi_1(\mathbf{x}_2)\chi_2(\mathbf{x}_1)\}$$

$$= \frac{1}{\sqrt2}\begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) \end{vmatrix}$$

where the coefficient is the w:normalization
factor. This wave function is now
antisymmetric and no longer distinguishes between fermions, that is: one
cannot indicate an ordinal number to a specific particle and the indices
given are interchangeable. Moreover, it also goes to zero if any two
wave functions of two fermions are the same. This is equivalent to
satisfying the Pauli exclusion principle.

### Generalizations

The expression can be generalised to any number of fermions by writing
it as a w:determinant. For an *N*-electron
system, the Slater determinant is defined as [^9]

$$\Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) =
\frac{1}{\sqrt{N!}}
\left|
   \begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\
                      \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\
                      \vdots & \vdots & \ddots & \vdots \\
                      \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N)
   \end{matrix} \right|\equiv \left| \begin{matrix}
   \chi _1 & \chi _2 & \cdots  & \chi _N  \\
\end{matrix}
\right|,$$

where in the final expression, a compact notation is introduced: the
normalization constant and labels for the fermion coordinates are
understood -- only the wavefunctions are exhibited. The linear
combination of Hartree products for the two-particle case can clearly be
seen as identical with the Slater determinant for *N* = 2. It can be
seen that the use of Slater determinants ensures an antisymmetrized
function at the outset; symmetric functions are automatically rejected.
In the same way, the use of Slater determinants ensures conformity to
the w:Pauli principle. Indeed, the
Slater determinant vanishes if the set {χ~i~ } is w:linearly
dependent. In particular, this is the
case when two (or more) spin orbitals are the same. In chemistry one
expresses this fact by stating that no two electrons can occupy the same
spin orbital. In general the Slater determinant is evaluated by the
w:Laplace expansion. Mathematically, a
Slater determinant is an antisymmetric tensor, also known as a w:wedge
product.

A single Slater determinant is used as an approximation to the
electronic wavefunction in Hartree--Fock
theory. In more accurate theories
(such as w:configuration
interaction and
w:MCSCF), a linear combination of Slater
determinants is needed.

The word \"**detor**\" was proposed by S. F. Boys to describe the Slater
determinant of the general type,[^10] but this term is rarely used.

Unlike w:fermions that are subject to the Pauli
exclusion principle, two or more w:bosons can
occupy the same quantum state of a system. Wavefunctions describing
systems of identical w:bosons are symmetric under
the exchange of particles and can be expanded in terms of
w:permanents.

## Fock matrix

In the w:Hartree--Fock method of
w:quantum mechanics, the **Fock
matrix** is a matrix "wikilink") approximating
the single-electron w:energy operator of
a given quantum system in a given set
of basis vectors.[^11]

It is most often formed in w:computational
chemistry when attempting to
solve the w:Roothaan equations for an
atomic or molecular system. The Fock matrix is actually an approximation
to the true Hamiltonian "wikilink")
operator "wikilink") of the quantum system.
It includes the effects of electron-electron
repulsion only in an average way.
Importantly, because the Fock operator is a one-electron operator, it
does not include the w:electron
correlation energy.

The Fock matrix is defined by the *Fock operator*. For the restricted
case which assumes w:closed-shell
orbitals and single-determinantal
wavefunctions, the Fock operator for the *i*-th electron is given
by:[^12]

$$\hat F(i) = \hat h(i)+\sum_{ j=1 }^{n/2}[2 \hat J_j(i)-\hat
K_j(i)]$$

where:

$$\hat F(i)$$ is the Fock operator for the *i*-th electron in the
system,

$${\hat h}(i)$$ is the w:one-electron
hamiltonian for the *i*-th
electron,

$$n$$ is the number of electrons and $\frac{n}{2}$ is the number of
occupied orbitals in the closed-shell system,

$$\hat J_j(i)$$ is the w:Coulomb
operator, defining the repulsive force
between the *j*-th and *i*-th electrons in the system,

$$\hat K_j(i)$$ is the w:exchange
operator, defining the quantum effect
produced by exchanging two electrons.

The Coulomb operator is multiplied by two since there are two electrons
in each occupied orbital. The exchange operator is not multiplied by two
since it has a non-zero result only for electrons which have the same
spin as the *i*-th electron.

For systems with unpaired electrons there are many choices of Fock
matrices.

 Hartree-Fock (HF) or self-consistent field
(SCF)

# Density Functional Theory

## Connection to quantum state symmetry

The Pauli exclusion principle with a single-valued many-particle
wavefunction is equivalent to requiring the wavefunction to be
antisymmetric. An antisymmetric two-particle state is represented as a
sum of states in which one
particle is in state $\scriptstyle |x \rangle$ and the other in state
$\scriptstyle |y\rangle$:

$$|\psi\rangle = \sum_{x,y} A(x,y) |x,y\rangle$$

and antisymmetry under exchange means that
.
This implies that ,
which is Pauli exclusion. It is true in any basis, since unitary changes
of basis keep antisymmetric matrices antisymmetric, although strictly
speaking, the quantity  is
not a matrix but an antisymmetric rank-two
w:tensor.

Conversely, if the diagonal quantities
 are zero *in every basis*,
then the wavefunction component:

$$A(x,y)=\langle \psi|x,y\rangle = \langle \psi | ( |x\rangle \otimes |y\rangle )$$

is necessarily antisymmetric. To prove it, consider the matrix element:

$$\langle\psi| ((|x\rangle + |y\rangle)\otimes(|x\rangle + |y\rangle))
\,$$

This is zero, because the two particles have zero probability to both be
in the superposition state $\scriptstyle |x\rangle + |y\rangle$. But
this is equal to

$$\langle \psi |x,x\rangle + \langle \psi |x,y\rangle + \langle \psi |y,x\rangle + \langle \psi | y,y \rangle
\,$$

The first and last terms on the right hand side are diagonal elements
and are zero, and the whole sum is equal to zero. So the wavefunction
matrix elements obey:

$$\langle \psi|x,y\rangle + \langle\psi |y,x\rangle = 0
\,$$.

or

$$A(x,y)=-A(y,x)
\,$$

### Pauli principle in advanced quantum theory

According to the w:spin-statistics
theorem, particles with integer
spin occupy symmetric quantum states, and particles with half-integer
spin occupy antisymmetric states; furthermore, only integer or
half-integer values of spin are allowed by the principles of quantum
mechanics. In relativistic w:quantum field
theory, the Pauli principle follows
from applying a rotation operator in imaginary time to particles of
half-integer spin. Since, nonrelativistically, particles can have any
statistics and any spin, there is no way to prove a spin-statistics
theorem in nonrelativistic quantum mechanics.

In one dimension, bosons, as well as fermions, can obey the exclusion
principle. A one-dimensional Bose gas with delta function repulsive
interactions of infinite strength is equivalent to a gas of free
fermions. The reason for this is that, in one dimension, exchange of
particles requires that they pass through each other; for infinitely
strong repulsion this cannot happen. This model is described by a
quantum w:nonlinear Schrödinger
equation. In momentum
space the exclusion principle is valid also for finite repulsion in a
Bose gas with delta function interactions,[^13] as well as for
interacting spins "wikilink") and
w:Hubbard model in one dimension, and for
other models solvable by w:Bethe ansatz.
The ground state in models solvable by
Bethe ansatz is a Fermi sphere.

 Density Functional Theory

# References

See also notes on editing this book about how to add references
w:Nanotechnology/About#How_to_contribute.

```{=html}
<references />
```

------------------------------------------------------------------------

[^1]:

[^2]:

[^3]:

[^4]:

[^5]:

[^6]: Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Englewood
    Cliffs, New Jersey: Prentice Hall. p. 403.
    .

[^7]:

[^8]:

[^9]: Molecular Quantum Mechanics Parts I and II: An Introduction to
    QUANTUM CHEMISTRY (Volume 1), P.W. Atkins, Oxford University Press,
    1977,

[^10]: {{ cite journal\| title=Electronic wave functions I. A general
    method of calculation for the stationary states of any molecular
    system\|author-link=S. Francis Boys\|first=S. F.
    \|last=Boys\|page=542\| volume=A200 \|year=1950\|journal=
    Proceedings of the Royal Society}}

[^11]:

[^12]: Levine, I.N. (1991) *Quantum Chemistry* (4th ed., Prentice-Hall),
    p.403

[^13]: A. Izergin and V. Korepin, Letter in Mathematical Physics vol 6,
    page 283, 1982


# Nanotechnology/Semiconducting Nanostructures#Carbon Nanotubes

  Navigate
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# Nanotubes

Certain compounds are capable of forming nanotubes where the tube
consists of round shell of a single layer of atoms in a cylindrical
lattice. Carbon nanotubes is the most famous example, but also other
materials can form nanotubes such as Boron-nitride, Molybdenum sulfide
and others.

Nanotubes can also be made by etching the core out of an shell
structured rod, but such tubes will normally contain many atomic layers
in the wall and have crystal facets on the sides.

# Carbon Nanotubes

Carbon nanotubes are fascinating nanostructures. A sheet of graphene as
in common graphite, but rolled up in small tubes rather than planar
sheets.

Carbon nanotubes have unique mechanical properties such as high
strength, high stiffness, and low density [^1] and also interesting
electronic properties. A single-walled carbon nanotube can be either
metallic or semiconducting depending on the atomic arrangement [^2].

This section is a short introduction to carbon nanotubes. For a broader
overview the reader is referred to one of the numerous review articles
or books on carbon nanotubes

[^3] [^4] [^5]

## Geometric Structure

The simplest type of carbon nanotube consists of just one layer of
graphene rolled up in the form of a seamless cylinder, known as a
single-walled carbon nanotube (SWCNT) with a typical diameter of just a
few nanometers. Larger diameter nanotube structures are nanotube ropes,
consisting of many individual, parallel nanotubes closed-packed into a
hexagonal lattice, and multi-walled carbon nanotubes (MWCNTs) consisting
of several concentric cylinders nested within each other.

! Multiwall carbon nanotube (MWCNT) sample made by a CVD process using
iron containing catalystic particles. The MWCNT are adhering in
mats. sample made by a CVD process using iron containing catalystic particles. The MWCNT are adhering in mats."){width="300"}

### Single walled Carbon Nanotube

The basic configuration is thus the SWCNT. Its structure is most easily
illustrated as a cylindrical tube conceptually formed by the wrapping of
a single graphene sheet. The hexagonal structure of the 2-dimensional
graphene sheet is due to the $sp^{2}$ hybridization of the carbon atoms,
which causes three directional, in-plane $\sigma$ bonds separated by an
angle of 120 degrees.

The nanotube can be described by a chiral vector $C$ that can be
expressed in terms of the graphene unit vectors $\textbf{a}_{1}$ and
$\textbf{a}_{2}$ as $\textbf{C}=n\textbf{a}_{1}+m\textbf{a}_{2}$ with
the set of integers $(n,m)$ uniquely identifying the nanotube. This
chiral vector or \'roll-up\' vector describes the nanotube circumference
by connecting two crystallographically equivalent positions i.e. the
tube is formed by superimposing the two ends of $\textbf{C}$.

Based on the chiral angle SWCNTs are defined as zig-zag tubes
($\theta =0 deg \leftrightarrow m=0$), armchair tubes
($\theta =30 deg \leftrightarrow n=m$), or chiral tubes
($0 deg < \theta  < 30 deg$).

### Multiwalled Carbon Nanotubes

MWCNTs are composed of a number of SWCNTs in a coaxial geometry. Each
nested shell has a diameter of
$d=\sqrt{3}a_{C-C}(m^{2}+n^{2}+mn)^{1/2}/\pi$ where $a_{C-C}$ is the
length of the carbon-carbon bond which is 1.42 Å. The difference in
diameters of the individual shell means that their chiralities are
different, and adjacent shell are therefore in general non-commensurate,
which causes only a weak intershell interaction.

The intershell spacing in MWCNTs is $\sim$ 0.34 nm - quite close to the
interlayer spacing in turbostratic graphite [^6]

## Electronic Structure

The electronic structure of a SWCNT is most easily described by again
considering a single graphene sheet. The 2-D, hexagonal-lattice graphene
sheet has a 2-D reciprocal space with a hexagonal Brillouin zone (BZ).

The $\sigma$ bonds are mainly responsible for the mechanical properties,
while the electronic properties are mainly determined by the $\pi$
bands. By a tight-binding approach the band structure of these $\pi$
bands can be calculated [^7]

Graphene is a zero-gap semiconductor with an occupied $\pi$ band and an
unoccupied $\pi^{*}$ band meeting at the Fermi level at six $K$ points
in the BZ, thus it behaves metallic, a so-called semimetal.

Upon forming the tube by conceptually wrapping the graphene sheet, a
periodic boundary condition is imposed that causes only certain
electronic states of those of the planar graphene sheet to be allowed.
These states are determined by the tube\'s geometric structure, i.e. by
the indices $(n,m)$ of the chiral vector. The wave vectors of the
allowed states fall on certain lines in the graphene BZ.

Based on this scheme it is possible to estimate whether a particular
tube will be metallic or semiconducting. When the allowed states include
the $K$ point, the system will to a first approximation behave metallic.
However, in the points where the $\pi$ and the $\pi^{*}$ bands meet but
are shifted slightly away from the $K$ point due to curvature effects,
which causes a slight band opening in some cases [^8]

This leads to a classification scheme that has three types of nanotubes:

-   Metallic: These are the armchair tubes where the small shift of the
    degenerate point away from the $K$ point does not cause a band
    opening for symmetry reasons.

```{=html}
<!-- -->
```
-   Small-bandgap semiconducting: These are characterized by $n-m = 3j$
    with $j$ being an integer. Here, the wave vectors of the allowed
    states cross the $K$ point, but due to the slight shift of the
    degenerate point a small gap will be present, the size of which is
    inversely proportional to the tube diameter squared with typical
    values between a few and a few tens meV

[^9]

-   Semiconducting: In this case $n-m \neq 3j$. This causes a larger
    bandgap, the size of which is inversely proportional to the tube
    diameter: $E_{g}=k/d$ with experimental investigations suggesting a
    value of $k$ of 0.7-0.8 eV/nm

[^10]

Typically the bandgap of the type 2 nanotubes is so small that they can
be considered metallic at room temperature. Based on this it can be
inferred that 1/3 of all tubes should behave metallic whereas the
remaining 2/3 should be semiconducting. However, it should be noted that
due to the inverse proportionality between the bandgap and the diameter
of the semiconducting tubes, large-diameter tubes will tend to behave
metallic at room temperature. This is especially important in regards to
large-diameter MWCNTs.

From a electrical point of view a MWCNT can be seen as a complex
structure of many parallel conductors that are only weakly interacting.
Since probing the electrical properties typically involves electrodes
contacting the outermost shell, this shell will be dominating the
transport properties [^11] In a simplistic view, this can be compared to
a large-diameter SWCNT, which will therefore typically display metallic
behavior.

## Electrical and Electromechanical Properties

Many studies have focused on SWCNTs for exploring the fundamental
properties of nanotubes. Due to their essentially 1-D nature and
intriguing electronic structure, SWCNTs exhibit a range of interesting
quantum phenomena at low temperature [^12]

The discussion here will so far, however, primarily be limited to room
temperature properties.

The conductance $~G~$ of a 1-dimensional conductor such as a SWCNT is
given by the Landauer formula [^13]

$~G=G_{0}\sum_{i}T_{i}~$,

where $~G_{0}=2e^{2}/h = (12.9 \rm k\Omega)^{-1}$;\
$~1/G_{0}~$ is the conductance quantum;\
and $~T_{i}~$ is the transmission coefficient of the contributing
channel $~i~$.

## More information on nanotubes

-   Wikipedia  Carbon nanotubes
-   The nanotube site
-   Compiled overview of properties of carbon
    nanotubes

## Commercial suppliers of carbon nanotubes and related products

-   Nanoledge.com nanotubes and related
    products - fibers, pellets, resins and dispersions\...
-   Carbon Solution, Inc. Mainly
    SWCNT.
-   BuckyUSA Fullerens, SWCNT, MWCNT.
-   CNI
    HiPco Carbon nanotubes
-   Carbolex SWCNT,sub-gram quantities are
    sold via sigma aldric
-   Sigma-Aldrich
    SWCNT

## \"Buckyball\"

!C~60~ with isosurface of ground state electron density as calculated
with
DFT{width="160"}
!An w:association football is a
model of the Buckminsterfullerene
C~60~{width="160"}

**Buckminsterfullerene** (IUPAC name
**(C~60~-I~h~)\[5,6\]fullerene**) is the smallest fullerene molecule in
which no two pentagons share an edge (which can be destabilizing, as in
pentalene). It is also the most common in
terms of natural occurrence, as it can often be found in
soot.

The structure of C~60~ is a truncated (T = 3)
icosahedron, which resembles a
soccer ball#Association_football "wikilink") of the
type made of twenty hexagons and twelve pentagons, with a carbon atom at
the vertices of each polygon and a bond along each polygon edge.

The w:van der Waals diameter of a
C~60~ molecule is about 1 nanometer (nm). The
nucleus to nucleus diameter of a C~60~ molecule is about 0.7 nm.

The C~60~ molecule has two bond lengths. The 6:6 ring bonds (between two
hexagons) can be considered \"double bonds\"
and are shorter than the 6:5 bonds (between a hexagon and a pentagon).
Its average bond length is 1.4 angstroms.

Silicon buckyballs have been created around metal ions.

### Boron buckyball

A new type of buckyball utilizing boron atoms
instead of the usual carbon has been predicted and described by
researchers at Rice University. The B-80 structure, with each atom
forming 5 or 6 bonds, is predicted to be more stable than the C-60
buckyball.[^14] One reason for this given by the researchers is that the
B-80 is actually more like the original geodesic dome structure
popularized by Buckminster Fuller which utilizes triangles rather than
hexagons. However, this work has been subject to much criticism by
quantum chemists[^15][^16] as it was concluded that the predicted Ih
symmetric structure was vibrationally unstable and the resulting cage
undergoes a spontaneous symmetry break yielding a puckered cage with
rare Th symmetry (symmetry of a
volleyball "wikilink"))[^17]. The number of six
atom rings in this molecule is 20 and number of five member rings is 12.
There is an additional atom in the center of each six member ring,
bonded to each atom surrounding it.

### Variations of buckyballs

Another fairly common buckminsterfullerene is C~70~,[^18] but fullerenes
with 72, 76, 84 and even up to 100 carbon atoms are commonly obtained.

In mathematical terms, the structure of a
**fullerene** is a trivalent "wikilink") convex
polyhedron with pentagonal and hexagonal
faces. In graph theory, the term
**fullerene** refers to any 3-regular,
planar graph with all faces of size 5 or 6
(including the external face). It follows from Euler\'s polyhedron
formula, \|V\|-\|E\|+\|F\| = 2,
(where \|V\|, \|E\|, \|F\| indicate the number of vertices, edges, and
faces), that there are exactly 12 pentagons in a fullerene and
\|V\|/2-10 hexagons.

+----------------+----------------+----------------+----------------+
| ![]            | ![]            | ![]            | ![]            |
| (Graph_of_20-f | (Graph_of_26-f | (Graph_of_60-f | (Graph_of_70-f |
| ullerene_w-nod | ullerene_5-bas | ullerene_w-nod | ullerene_w-nod |
| es.svg "Graph_ | e_w-nodes.svg  | es.svg "Graph_ | es.svg "Graph_ |
| of_20-fulleren | "Graph_of_26-f | of_60-fulleren | of_70-fulleren |
| e_w-nodes.svg" | ullerene_5-bas | e_w-nodes.svg" | e_w-nodes.svg" |
| ){width="200"} | e_w-nodes.svg" | ){width="200"} | ){width="200"} |
|                | ){width="200"} |                |                |
+----------------+----------------+----------------+----------------+
| 20-fullerene\  | 26-fullerene   | 60-fullerene\  | 70-fullerene   |
| (dodecahedral  | graph          | (truncated     | graph          |
| graph)         |                | icosahedral    |                |
|                |                | graph)         |                |
+----------------+----------------+----------------+----------------+

The smallest fullerene is the
w:dodecahedron\--the unique C~20~. There
are no fullerenes with 22 vertices.[^19] The number of fullerenes C~2n~
grows with increasing n = 12,13,14\..., roughly in proportion to n^9^.
For instance, there are 1812 non-isomorphic fullerenes C~60~. Note that
only one form of C~60~, the buckminsterfullerene alias w:truncated
icosahedron, has no pair of
adjacent pentagons (the smallest such fullerene). To further illustrate
the growth, there are 214,127,713 non-isomorphic fullerenes C~200~,
15,655,672 of which have no adjacent pentagons.

w:Trimetasphere carbon nanomaterials were
discovered by researchers at w:Virginia
Tech and licensed exclusively to w:Luna
Innovations. This class of novel
molecules comprises 80 carbon atoms (C80) forming a sphere which
encloses a complex of three metal atoms and one nitrogen atom. These
fullerenes encapsulate metals which puts them in the subset referred to
as w:metallofullerenes. Trimetaspheres
have the potential for use in diagnostics (as safe imaging agents),
therapeutics and in organic solar
cells.

# Semiconducting nanowires

Semiconducting nanowires can be made from most semiconducting materials
and with different methods, mainly variations of a chemical vapor
deposition process (CVD).

There are many different semiconducting materials, and heterosrtuctures
can be made if the lattice constants are not too incompatible.
Heterostructures made from combinations of materials such as GaAs-GaP
can be used to make barriers and guides for electrons in electrical
systems.

Low pressure metal organic vapor phase epitaxy (MOVPE) can be used to
grow III-V nanowires epitaxially on suitable crystalline substrates,
sucha s III-V materials or silicon with a reasonably matching lattice
constant.

!Low pressure metal organic vapor phase epitaxy (MOVPE) can be used to
grow III-V nanowires epitaxially on suitable crystalline substrates,
sucha s III-V materials or silicon with a reasonably matching lattice
constant. Nanowire growth is catalyzed by various nanoparticles, which
are deposited on the substrate surface, typically gold nanoparticles
with a diameter of 20-100nm.
 can be used to grow III-V nanowires epitaxially on suitable crystalline substrates, sucha s III-V materials or silicon with a reasonably matching lattice constant. Nanowire growth is catalyzed by various nanoparticles, which are deposited on the substrate surface, typically gold nanoparticles with a diameter of 20-100nm. "){width="300"}

Nanowire growth is catalyzed by various nanoparticles, which are
deposited on the substrate surface, typically gold nanoparticles with a
diameter of 20-100nm.

To grow for instance GaP wires, the sample is typically annealed at 650C
in the heated reactor chamber to form an eutectic with between the gold
catalyst and the underlying substrate.

Then growth is done at a lower temperature around 500C in the presence
of the precursor gasses trimethyl gallium and phosphine. By changing the
precursor gasses during growth, nanowire heterostructures with varying
composition can be made

!SEM image of epitaxial nanowire heterostructures grown from catalytic
gold
nanoparticles{width="400"}

## Resources

-   wikipedia  Semiconducting nanowires
-   IOFFE semiconductor physical
    properties

# Nanoparticles

-    Nanoparticles
-    Quantum dots

## Catalytic particles

-    Catalyst
-    Catalysis
-    Haber process

## Commercial suppliers of nanoparticles

-   Reade has a wide selection of
    nanoparticles
-   sigma-aldrich
    has dispersions of
    nanoparticles
-   nanophase

# Contributors and Acknowledgements

-   Jakob Kjelstrup Hansen

# References

See also notes on editing this book about how to add references
Nanotechnology/About#How_to_contribute.

```{=html}
<references />
```

------------------------------------------------------------------------

[^1]: qian2002

[^2]: hamada1992

[^3]: avouris2003

[^4]: dresselhaus2001

[^5]: saito1998

[^6]: dresselhaus2001

[^7]: saito1998

[^8]: hamada1992.

[^9]: zhou2000

[^10]: wildoer1998,odom1998

[^11]: frank1998

[^12]: nygard1999,dresselhaus2001

[^13]: datta1995

[^14]: *Bucky\'s brother \-- The boron buckyball makes its début* Jade
    Boyd April **2007**
    eurekalert.orgLink

[^15]: *The boron buckyball has an unexpected Th symmetry* G. Gopakumar,
    Nguyen, M. T., Ceulemans, Arnout, Chem. Phys. lett. 450, 175,
    2008.1&_cdi=5231&_sort=d&_docanchor=&view=c&_ct=53&_acct=C000024498&_version=1&_urlVersion=0&_userid=4466739&md5=2e8f3b9eacbc9a4b9a3dbd42fdbf2826)

[^16]: \"Stuffing improves the stability of fullerenelike boron
    clusters\" Prasad, DLVK; Jemmis, E. D.; Phys. Rev. Lett. 100,
    165504,
    2008.2

[^17]:

[^18]: Buckminsterfullerene: Molecule of the
    Month

[^19]: Goldberg Variations Challenge: Juris Meija, Anal. Bioanal. Chem.
    2006 (385)
    6-7


# Nanotechnology/Semiconducting Nanostructures#.22Buckyball.22

  Navigate
  ---------------------------------------------------------------------------------------------
   \<\< Prev: Overview of Production methods
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------------------------------------------------------------------------

# Nanotubes

Certain compounds are capable of forming nanotubes where the tube
consists of round shell of a single layer of atoms in a cylindrical
lattice. Carbon nanotubes is the most famous example, but also other
materials can form nanotubes such as Boron-nitride, Molybdenum sulfide
and others.

Nanotubes can also be made by etching the core out of an shell
structured rod, but such tubes will normally contain many atomic layers
in the wall and have crystal facets on the sides.

# Carbon Nanotubes

Carbon nanotubes are fascinating nanostructures. A sheet of graphene as
in common graphite, but rolled up in small tubes rather than planar
sheets.

Carbon nanotubes have unique mechanical properties such as high
strength, high stiffness, and low density [^1] and also interesting
electronic properties. A single-walled carbon nanotube can be either
metallic or semiconducting depending on the atomic arrangement [^2].

This section is a short introduction to carbon nanotubes. For a broader
overview the reader is referred to one of the numerous review articles
or books on carbon nanotubes

[^3] [^4] [^5]

## Geometric Structure

The simplest type of carbon nanotube consists of just one layer of
graphene rolled up in the form of a seamless cylinder, known as a
single-walled carbon nanotube (SWCNT) with a typical diameter of just a
few nanometers. Larger diameter nanotube structures are nanotube ropes,
consisting of many individual, parallel nanotubes closed-packed into a
hexagonal lattice, and multi-walled carbon nanotubes (MWCNTs) consisting
of several concentric cylinders nested within each other.

! Multiwall carbon nanotube (MWCNT) sample made by a CVD process using
iron containing catalystic particles. The MWCNT are adhering in
mats. sample made by a CVD process using iron containing catalystic particles. The MWCNT are adhering in mats."){width="300"}

### Single walled Carbon Nanotube

The basic configuration is thus the SWCNT. Its structure is most easily
illustrated as a cylindrical tube conceptually formed by the wrapping of
a single graphene sheet. The hexagonal structure of the 2-dimensional
graphene sheet is due to the $sp^{2}$ hybridization of the carbon atoms,
which causes three directional, in-plane $\sigma$ bonds separated by an
angle of 120 degrees.

The nanotube can be described by a chiral vector $C$ that can be
expressed in terms of the graphene unit vectors $\textbf{a}_{1}$ and
$\textbf{a}_{2}$ as $\textbf{C}=n\textbf{a}_{1}+m\textbf{a}_{2}$ with
the set of integers $(n,m)$ uniquely identifying the nanotube. This
chiral vector or \'roll-up\' vector describes the nanotube circumference
by connecting two crystallographically equivalent positions i.e. the
tube is formed by superimposing the two ends of $\textbf{C}$.

Based on the chiral angle SWCNTs are defined as zig-zag tubes
($\theta =0 deg \leftrightarrow m=0$), armchair tubes
($\theta =30 deg \leftrightarrow n=m$), or chiral tubes
($0 deg < \theta  < 30 deg$).

### Multiwalled Carbon Nanotubes

MWCNTs are composed of a number of SWCNTs in a coaxial geometry. Each
nested shell has a diameter of
$d=\sqrt{3}a_{C-C}(m^{2}+n^{2}+mn)^{1/2}/\pi$ where $a_{C-C}$ is the
length of the carbon-carbon bond which is 1.42 Å. The difference in
diameters of the individual shell means that their chiralities are
different, and adjacent shell are therefore in general non-commensurate,
which causes only a weak intershell interaction.

The intershell spacing in MWCNTs is $\sim$ 0.34 nm - quite close to the
interlayer spacing in turbostratic graphite [^6]

## Electronic Structure

The electronic structure of a SWCNT is most easily described by again
considering a single graphene sheet. The 2-D, hexagonal-lattice graphene
sheet has a 2-D reciprocal space with a hexagonal Brillouin zone (BZ).

The $\sigma$ bonds are mainly responsible for the mechanical properties,
while the electronic properties are mainly determined by the $\pi$
bands. By a tight-binding approach the band structure of these $\pi$
bands can be calculated [^7]

Graphene is a zero-gap semiconductor with an occupied $\pi$ band and an
unoccupied $\pi^{*}$ band meeting at the Fermi level at six $K$ points
in the BZ, thus it behaves metallic, a so-called semimetal.

Upon forming the tube by conceptually wrapping the graphene sheet, a
periodic boundary condition is imposed that causes only certain
electronic states of those of the planar graphene sheet to be allowed.
These states are determined by the tube\'s geometric structure, i.e. by
the indices $(n,m)$ of the chiral vector. The wave vectors of the
allowed states fall on certain lines in the graphene BZ.

Based on this scheme it is possible to estimate whether a particular
tube will be metallic or semiconducting. When the allowed states include
the $K$ point, the system will to a first approximation behave metallic.
However, in the points where the $\pi$ and the $\pi^{*}$ bands meet but
are shifted slightly away from the $K$ point due to curvature effects,
which causes a slight band opening in some cases [^8]

This leads to a classification scheme that has three types of nanotubes:

-   Metallic: These are the armchair tubes where the small shift of the
    degenerate point away from the $K$ point does not cause a band
    opening for symmetry reasons.

```{=html}
<!-- -->
```
-   Small-bandgap semiconducting: These are characterized by $n-m = 3j$
    with $j$ being an integer. Here, the wave vectors of the allowed
    states cross the $K$ point, but due to the slight shift of the
    degenerate point a small gap will be present, the size of which is
    inversely proportional to the tube diameter squared with typical
    values between a few and a few tens meV

[^9]

-   Semiconducting: In this case $n-m \neq 3j$. This causes a larger
    bandgap, the size of which is inversely proportional to the tube
    diameter: $E_{g}=k/d$ with experimental investigations suggesting a
    value of $k$ of 0.7-0.8 eV/nm

[^10]

Typically the bandgap of the type 2 nanotubes is so small that they can
be considered metallic at room temperature. Based on this it can be
inferred that 1/3 of all tubes should behave metallic whereas the
remaining 2/3 should be semiconducting. However, it should be noted that
due to the inverse proportionality between the bandgap and the diameter
of the semiconducting tubes, large-diameter tubes will tend to behave
metallic at room temperature. This is especially important in regards to
large-diameter MWCNTs.

From a electrical point of view a MWCNT can be seen as a complex
structure of many parallel conductors that are only weakly interacting.
Since probing the electrical properties typically involves electrodes
contacting the outermost shell, this shell will be dominating the
transport properties [^11] In a simplistic view, this can be compared to
a large-diameter SWCNT, which will therefore typically display metallic
behavior.

## Electrical and Electromechanical Properties

Many studies have focused on SWCNTs for exploring the fundamental
properties of nanotubes. Due to their essentially 1-D nature and
intriguing electronic structure, SWCNTs exhibit a range of interesting
quantum phenomena at low temperature [^12]

The discussion here will so far, however, primarily be limited to room
temperature properties.

The conductance $~G~$ of a 1-dimensional conductor such as a SWCNT is
given by the Landauer formula [^13]

$~G=G_{0}\sum_{i}T_{i}~$,

where $~G_{0}=2e^{2}/h = (12.9 \rm k\Omega)^{-1}$;\
$~1/G_{0}~$ is the conductance quantum;\
and $~T_{i}~$ is the transmission coefficient of the contributing
channel $~i~$.

## More information on nanotubes

-   Wikipedia  Carbon nanotubes
-   The nanotube site
-   Compiled overview of properties of carbon
    nanotubes

## Commercial suppliers of carbon nanotubes and related products

-   Nanoledge.com nanotubes and related
    products - fibers, pellets, resins and dispersions\...
-   Carbon Solution, Inc. Mainly
    SWCNT.
-   BuckyUSA Fullerens, SWCNT, MWCNT.
-   CNI
    HiPco Carbon nanotubes
-   Carbolex SWCNT,sub-gram quantities are
    sold via sigma aldric
-   Sigma-Aldrich
    SWCNT

## \"Buckyball\"

!C~60~ with isosurface of ground state electron density as calculated
with
DFT{width="160"}
!An w:association football is a
model of the Buckminsterfullerene
C~60~{width="160"}

**Buckminsterfullerene** (IUPAC name
**(C~60~-I~h~)\[5,6\]fullerene**) is the smallest fullerene molecule in
which no two pentagons share an edge (which can be destabilizing, as in
pentalene). It is also the most common in
terms of natural occurrence, as it can often be found in
soot.

The structure of C~60~ is a truncated (T = 3)
icosahedron, which resembles a
soccer ball#Association_football "wikilink") of the
type made of twenty hexagons and twelve pentagons, with a carbon atom at
the vertices of each polygon and a bond along each polygon edge.

The w:van der Waals diameter of a
C~60~ molecule is about 1 nanometer (nm). The
nucleus to nucleus diameter of a C~60~ molecule is about 0.7 nm.

The C~60~ molecule has two bond lengths. The 6:6 ring bonds (between two
hexagons) can be considered \"double bonds\"
and are shorter than the 6:5 bonds (between a hexagon and a pentagon).
Its average bond length is 1.4 angstroms.

Silicon buckyballs have been created around metal ions.

### Boron buckyball

A new type of buckyball utilizing boron atoms
instead of the usual carbon has been predicted and described by
researchers at Rice University. The B-80 structure, with each atom
forming 5 or 6 bonds, is predicted to be more stable than the C-60
buckyball.[^14] One reason for this given by the researchers is that the
B-80 is actually more like the original geodesic dome structure
popularized by Buckminster Fuller which utilizes triangles rather than
hexagons. However, this work has been subject to much criticism by
quantum chemists[^15][^16] as it was concluded that the predicted Ih
symmetric structure was vibrationally unstable and the resulting cage
undergoes a spontaneous symmetry break yielding a puckered cage with
rare Th symmetry (symmetry of a
volleyball "wikilink"))[^17]. The number of six
atom rings in this molecule is 20 and number of five member rings is 12.
There is an additional atom in the center of each six member ring,
bonded to each atom surrounding it.

### Variations of buckyballs

Another fairly common buckminsterfullerene is C~70~,[^18] but fullerenes
with 72, 76, 84 and even up to 100 carbon atoms are commonly obtained.

In mathematical terms, the structure of a
**fullerene** is a trivalent "wikilink") convex
polyhedron with pentagonal and hexagonal
faces. In graph theory, the term
**fullerene** refers to any 3-regular,
planar graph with all faces of size 5 or 6
(including the external face). It follows from Euler\'s polyhedron
formula, \|V\|-\|E\|+\|F\| = 2,
(where \|V\|, \|E\|, \|F\| indicate the number of vertices, edges, and
faces), that there are exactly 12 pentagons in a fullerene and
\|V\|/2-10 hexagons.

+----------------+----------------+----------------+----------------+
| ![]            | ![]            | ![]            | ![]            |
| (Graph_of_20-f | (Graph_of_26-f | (Graph_of_60-f | (Graph_of_70-f |
| ullerene_w-nod | ullerene_5-bas | ullerene_w-nod | ullerene_w-nod |
| es.svg "Graph_ | e_w-nodes.svg  | es.svg "Graph_ | es.svg "Graph_ |
| of_20-fulleren | "Graph_of_26-f | of_60-fulleren | of_70-fulleren |
| e_w-nodes.svg" | ullerene_5-bas | e_w-nodes.svg" | e_w-nodes.svg" |
| ){width="200"} | e_w-nodes.svg" | ){width="200"} | ){width="200"} |
|                | ){width="200"} |                |                |
+----------------+----------------+----------------+----------------+
| 20-fullerene\  | 26-fullerene   | 60-fullerene\  | 70-fullerene   |
| (dodecahedral  | graph          | (truncated     | graph          |
| graph)         |                | icosahedral    |                |
|                |                | graph)         |                |
+----------------+----------------+----------------+----------------+

The smallest fullerene is the
w:dodecahedron\--the unique C~20~. There
are no fullerenes with 22 vertices.[^19] The number of fullerenes C~2n~
grows with increasing n = 12,13,14\..., roughly in proportion to n^9^.
For instance, there are 1812 non-isomorphic fullerenes C~60~. Note that
only one form of C~60~, the buckminsterfullerene alias w:truncated
icosahedron, has no pair of
adjacent pentagons (the smallest such fullerene). To further illustrate
the growth, there are 214,127,713 non-isomorphic fullerenes C~200~,
15,655,672 of which have no adjacent pentagons.

w:Trimetasphere carbon nanomaterials were
discovered by researchers at w:Virginia
Tech and licensed exclusively to w:Luna
Innovations. This class of novel
molecules comprises 80 carbon atoms (C80) forming a sphere which
encloses a complex of three metal atoms and one nitrogen atom. These
fullerenes encapsulate metals which puts them in the subset referred to
as w:metallofullerenes. Trimetaspheres
have the potential for use in diagnostics (as safe imaging agents),
therapeutics and in organic solar
cells.

# Semiconducting nanowires

Semiconducting nanowires can be made from most semiconducting materials
and with different methods, mainly variations of a chemical vapor
deposition process (CVD).

There are many different semiconducting materials, and heterosrtuctures
can be made if the lattice constants are not too incompatible.
Heterostructures made from combinations of materials such as GaAs-GaP
can be used to make barriers and guides for electrons in electrical
systems.

Low pressure metal organic vapor phase epitaxy (MOVPE) can be used to
grow III-V nanowires epitaxially on suitable crystalline substrates,
sucha s III-V materials or silicon with a reasonably matching lattice
constant.

!Low pressure metal organic vapor phase epitaxy (MOVPE) can be used to
grow III-V nanowires epitaxially on suitable crystalline substrates,
sucha s III-V materials or silicon with a reasonably matching lattice
constant. Nanowire growth is catalyzed by various nanoparticles, which
are deposited on the substrate surface, typically gold nanoparticles
with a diameter of 20-100nm.
 can be used to grow III-V nanowires epitaxially on suitable crystalline substrates, sucha s III-V materials or silicon with a reasonably matching lattice constant. Nanowire growth is catalyzed by various nanoparticles, which are deposited on the substrate surface, typically gold nanoparticles with a diameter of 20-100nm. "){width="300"}

Nanowire growth is catalyzed by various nanoparticles, which are
deposited on the substrate surface, typically gold nanoparticles with a
diameter of 20-100nm.

To grow for instance GaP wires, the sample is typically annealed at 650C
in the heated reactor chamber to form an eutectic with between the gold
catalyst and the underlying substrate.

Then growth is done at a lower temperature around 500C in the presence
of the precursor gasses trimethyl gallium and phosphine. By changing the
precursor gasses during growth, nanowire heterostructures with varying
composition can be made

!SEM image of epitaxial nanowire heterostructures grown from catalytic
gold
nanoparticles{width="400"}

## Resources

-   wikipedia  Semiconducting nanowires
-   IOFFE semiconductor physical
    properties

# Nanoparticles

-    Nanoparticles
-    Quantum dots

## Catalytic particles

-    Catalyst
-    Catalysis
-    Haber process

## Commercial suppliers of nanoparticles

-   Reade has a wide selection of
    nanoparticles
-   sigma-aldrich
    has dispersions of
    nanoparticles
-   nanophase

# Contributors and Acknowledgements

-   Jakob Kjelstrup Hansen

# References

See also notes on editing this book about how to add references
Nanotechnology/About#How_to_contribute.

```{=html}
<references />
```

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