ATOMIC ORBITALS
This article explains s and p orbitals in some detail, including
their shapes and energies. d orbitals are described only in terms of their
energy, and f orbitals only get a passing mention through a unique spinning status.
Orbitals and orbits
When a planet moves around the sun, we can plot a definite path for it which is
called an orbit. A simple view of the atom looks similar and we may have pictured the electrons as
orbiting around the nucleus. The truth is different, and electrons in fact
inhabit regions of space known as orbitals.
Orbits and orbitals sound similar, but they have quite different
meanings. It is essential that we
understand the difference between them.
The impossibility of drawing orbits for
electrons
To plot a path for something we need to know exactly where the object is
and be able to work out exactly where it's going to be an instant later. we can't do this for electrons.
The Heisenberg
Uncertainty Principle says - loosely - that we can't know with certainty both where an
electron is and where it's going next. (What it actually says is that it is
impossible to define with absolute precision, at the same time, both the
position and the momentum of an electron.)
That makes it impossible to plot an orbit for an electron around a
nucleus. Is this a big problem? No. If something is impossible, we have to accept it and find a way around
it.
Hydrogen's electron - the 1s orbital
Suppose we had a single hydrogen atom and at a
particular instant plotted the position of the one electron. Soon afterwards, we do the same thing, and find that it is in
a new position. we have no idea how it
got from the first place to the second.
we keep on doing this over
and over again, and gradually build up a sort of 3D map of the places that the
electron is likely to be found.
In the hydrogen case, the electron can be found anywhere within a
spherical space surrounding the nucleus. The diagram shows a cross-section through
this spherical space.
95% of the time (or any other percentage we choose), the electron will be found within
a fairly easily defined region of space quite close to the nucleus. Such a
region of space is called an orbital. we can think of an orbital as being the
region of space in which the electron lives.
What is the electron doing in the orbital? We don't know, we can't
know, and so we just ignore the problem! All we can say is that if an electron is in a
particular orbital it will have a particular definable energy.
Each orbital has a name.
The orbital occupied by the hydrogen electron is called a 1s orbital. The "1" represents the
fact that the orbital is in the energy level closest to the nucleus. The "s" tells we about the shape of the orbital. s orbitals
are spherically symmetric around the nucleus - in each case, like a hollow ball
made of rather chunky material with the nucleus at its centre.
The orbital on the left is a 2s orbital. This is similar to a
1s orbital except that the region where there is the greatest chance of finding
the electron is further from the nucleus - this is an orbital at the second
energy level.
If we look carefully, we will notice that there is another region
of slightly higher electron density (where the dots are thicker) nearer the
nucleus. ("Electron density" is another way of talking about how
likely we are to find an electron at a
particular place.)
2s (and 3s, 4s, etc) electrons spend some of their time closer to
the nucleus than we might expect. The
effect of this is to slightly reduce the energy of electrons in s orbitals. The
nearer the nucleus the electrons get, the lower their energy.
3s, 4s (etc) orbitals get progressively further from the nucleus.
p orbitals
Not all electrons inhabit s
orbitals (in fact, very few electrons live in s orbitals). At the first energy
level, the only orbital available to electrons is the 1s orbital, but at the
second level, as well as a 2s orbital, there are also orbitals called 2p orbitals.
A p orbital is rather like 2 identical balloons tied together at
the nucleus. The diagram on the left is a cross-section through that
3-dimensional region of space. Once again, the orbital shows where there is a
95% chance of finding a particular electron.
Unlike an s orbital, a p orbital points in a particular direction
- the one drawn points up and down the page.
At any one energy level it is possible to have three absolutely
equivalent p orbitals pointing mutually at right angles to each other. These
are arbitrarily given the symbols px, py and pz. This is simply for
convenience - what we might think of as
the x, y or z direction changes constantly as the atom tumbles in space.
The p orbitals at the second energy
level are called 2px, 2py and 2pz. There
are similar orbitals at subsequent levels - 3px, 3py, 3pz,
4px, 4py, 4pz and so on.
All levels except for the first level have p orbitals. At the
higher levels the lobes get more elongated, with the most likely place to find
the electron more distant from the nucleus.
d and f orbitals
In addition to s and p orbitals, there are two other sets of
orbitals which become available for electrons to inhabit at higher energy
levels. At the third level, there is a set of five d orbitals (with complicated shapes and names) as well as the
3s and 3p orbitals (3px, 3py, 3pz). At the
third level there are a total of nine orbitals altogether.
At the fourth level, as well the 4s and 4p and 4d orbitals there
are an additional seven f orbitals
- 14 orbitals in all. s, p, d and f orbitals are then available at all higher
energy levels as well.
For the moment, we need to
be aware that there are sets of five d orbitals at levels from the third level
upwards, but we probably won't be
expected to draw them or name them. Apart from a passing reference, we won't come across f orbitals at all.
Fitting electrons into orbitals
we can think of an atom as
a very bizarre house (like an inverted pyramid!) - with the nucleus living on
the ground floor, and then various rooms (orbitals) on the higher floors
occupied by the electrons. On the first floor there is only 1 room (the 1s orbital);
on the second floor there are 4 rooms (the 2s, 2px, 2pyand
2pz orbitals); on the third floor there are 9 rooms (one 3s
orbital, three 3p orbitals and five 3d orbitals); and so on. But the rooms
aren't very big . . . Each orbital
can only hold 2 electrons.
A convenient way of showing the orbitals that the electrons live
in is to draw "electrons-in-boxes".
"Electrons-in-boxes"
Orbitals can be represented as boxes with the electrons in them
shown as arrows. Often an up-arrow and a down-arrow are used to show that the
electrons are in some way differ
A
1s orbital holding 2 electrons would be drawn as shown on the right, but it can
be written even more quickly as 1s2. This is read as "one s
two" - not as "one s squared".
we mustn't confuse the two
numbers in this notation:
The order of filling orbital’s indicates
the Aufbau Principle
Aufbau is
a German word meaning building up or construction.
We imagine that as we go from one atom
to the next in the Periodic Table, we
can work out the electronic structure of the next atom by fitting an extra
electron into the next available orbital.
Electrons fill low energy orbitals (closer to the nucleus) before
they fill higher energy ones. Where there is a choice between orbitals of equal
energy, they fill the orbitals singly as far as possible.
This filling of orbitals singly where possible is known as Hund's rule. It only applies where
the orbitals have exactly the same energies (as with p orbitals, for example),
and helps to minimize the repulsions between electrons and so makes the atom
more stable.
The diagram (not to scale) summarizes the energies of the orbitals
up to the 4p level that we will need to
know when we are using the Aufbau
Principle.
It is noticed that the s orbital always has a slightly lower
energy than the p orbitals at the same energy level, so the s orbital always
fills with electrons before the corresponding p orbitals.
The real oddity is the position of the 3d orbitals. They are at a
slightly higher level than the 4s - and so it is the 4s orbital which we fill first, followed by all the 3d
orbitals and then the 4p orbitals.
Similar confusion occurs at higher levels, with so much overlap
between the energy levels that we don't
fill the 4f orbitals until after the 6s, for example.
For UK-based exam purposes, we simply have to remember that when we are using the Aufbau Principle, we fill the 4s orbital before the 3d
orbitals. The same thing happens at the next level as well - we fill the 5s orbital before the 4d
orbitals. All the other complications are beyond the scope of this site.
Knowing the order of filling is central to understanding how to
write electronic structures. Follow the link below to find out how to do this.
atomic physics,
the spin quantum number is a quantum number that parameterizes the
intrinsic angular momentum (or spin angular momentum, or
simply spin) of a given particle. The spin quantum number is
the fourth of a set of quantum numbers (the principal quantum number, the azimuthally, the magnetic quantum number, and the spin quantum number),
which completely describe the quantum state of
an electron. It is designated by the letter s. It describes the energy,
shape and orientation of orbitals.
Atomic
orbital
The shapes of the first five atomic orbitals are: 1s, 2s, 2px,
2py, and 2pz. The two colors show the phase
or sign of the wave function in each region. These are graphs of ψ(x, y, z) functions which depend on the coordinates of
one electron. To see the elongated shape of ψ(x, y, z)2 functions that show probability densitymore directly, see the graphs of
d-orbitals below.
In quantum
mechanics,
an atomic orbital is a mathematical function that describes the
wave-like behavior of either one electron or
a pair of electrons in an atom.[1] This
function can be used to calculate the probability of
finding any electron of an atom in any specific region around the atom's nucleus. The term, atomic orbital, may also refer to the physical
region or space where the electron can be calculated to be present, as defined
by the particular mathematical form of the orbital.[2]
Each
orbital in an atom is characterized by a unique set of values of the three quantum numbers n, â„“, and m,
which respectively correspond to the electron's energy, angular
momentum,
and an angular momentum vector
component (the magnetic quantum number). Each such orbital can be
occupied by a maximum of two electrons, each with its own spin quantum number s. The
simple names s orbital, p
orbital, d orbital and f
orbital refer to orbitals with angular momentum quantum
number â„“ = 0, 1, 2 and 3 respectively.
These names, together with the value of n, are used to
describe the electron configurations of atoms. They are
derived from the description by early spectroscopists of certain series of alkali metal spectroscopic
lines as sharp, principal, diffuse,
andfundamental. Orbitals for â„“ > 3 continue
alphabetically, omitting j (g, h, i, k, …).[3][4][5]
Atomic
orbitals are the basic building blocks of the atomic orbital model (alternatively
known as the electron cloud or wave mechanics model), a modern framework for
visualizing the submicroscopic behavior of electrons in matter. In this model
the electron cloud of a multi-electron atom may be seen as being built up (in
approximation) in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The
repeating periodicity of the blocks of 2,
6, 10, and 14 elements within
sections of the periodic
table arises naturally from the total number of electrons
that occupy a complete set of s, p, d and f atomic
orbitals, respectively, although for higher values of the quantum number n, particularly when the atom in question bears a positive
charge, the energies of certain sub-shells become very similar and so the order in
which they are said to be populated by electrons (e.g. Cr = [Ar]4s13d5 and
Cr2+ = [Ar]3d4) can only be rationalized somewhat
arbitrarily.
Electron properties[edit]
With
the development of quantum
mechanics and experimental findings (such as the two slits
diffraction of electrons), it was found that the orbiting electrons around a
nucleus could not be fully described as particles, but needed to be explained
by the wave-particle duality. In this sense, the electrons
have the following properties:
Wave-like
properties:
1.
The electrons do not orbit the nucleus in the sense of a planet
orbiting the sun, but instead exist as standing
waves. The lowest possible energy an electron can take is therefore
analogous to the fundamental frequency of a wave on a string. Higher energy
states are then similar to harmonics of the fundamental frequency.
2.
The electrons are never in a single point location, although the
probability of interacting with the electron at a single point can be found from
the wave function of the electron.
Particle-like
properties:
1.
There is always an integer number of electrons orbiting the
nucleus.
2.
Electrons jump between orbitals in a particle-like fashion. For
example, if a single photon strikes the electrons, only a single electron
changes states in response to the photon.
3.
The electrons retain particle-like properties such as: each wave
state has the same electrical charge as the electron particle. Each wave state
has a single discrete spin (spin up or spin down).this can depend up on its superposition
Thus, despite the obvious analogy to planets revolving around the Sun, electrons cannot be described simply as solid particles. In addition, atomic orbitals do not closely resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the atomic nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when a single electron is present in an atom. When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection (sometimes termed the atom’s “electron cloud”[6]) tends toward a generally spherical zone of probability describing where the atom’s electrons will be found. The is due to the uncertainty principal.
Formal quantum mechanical definition[edit]
Atomic
orbitals may be defined more precisely in formal quantum
mechanical language.
Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, is approximated by
an expansion (see configuration interaction expansion
and basis set) into linear combinations of anti-symmetrized
products (Slater determinants) of one-electron functions. The spatial components
of these one-electron functions are called atomic orbitals. (When one considers
also their spin component,
one speaks of atomic spin orbitals.) A state is actually a
function of the coordinates of all the electrons, so that their motion is correlated,
but this is often approximated by this independent-particle model of
products of single electron wave functions.[7] (The London dispersion force, for example, depends on the
correlations of the motion of the electrons.)
In atomic physics, the atomic spectral lines correspond to transitions
(quantum leaps) between quantum
states of an atom. These states are labeled by a set of quantum numberssummarized in the term symbol and
usually associated with particular electron configurations, i.e., by occupation
schemes of atomic orbitals (for example, 1s2 2s2 2p6 for
the ground state ofneon—term symbol: 1S0).
This
notation means that the corresponding Slater determinants have a clear higher
weight in the configuration interaction expansion.
The atomic orbital concept is therefore a key concept for visualizing the
excitation process associated with a given transition. For example, one can say for a given transition
that it corresponds to the excitation of an electron from an occupied orbital
to a given unoccupied orbital. Nevertheless, one has to keep in mind that
electrons are fermions ruled
by the Pauli exclusion principle and
cannot be distinguished from the other electrons in the atom. Moreover, it
sometimes happens that the configuration interaction expansion converges very
slowly and that one cannot speak about simple one-determinant wave function at
all. This is the case when electron correlation is large.
Fundamentally,
an atomic orbital is a one-electron wave function, even though most electrons
do not exist in one-electron atoms, and so the one-electron view is an
approximation. When thinking about orbitals, we are often given an orbital
vision which (even if it is not spelled out) is heavily influenced by this Hartree–Fock approximation, which
is one way to reduce the complexities of molecular orbital theory.
Types of orbitals[edit]
False-color density images of some hydrogen-likeatomic
orbitals (f orbitals
and higher are not shown)
Atomic
orbitals can be the hydrogen-like "orbitals" which are exact
solutions to the Schrödinger equation for a hydrogen-like "atom" (i.e.,
an atom with one electron). Alternatively, atomic orbitals refer to functions
that depend on the coordinates of one electron (i.e., orbitals) but are used as
starting points for approximating wave functions that depend on the
simultaneous coordinates of all the electrons in an atom or molecule. The coordinate
systems chosen for atomic orbitals are usually spherical coordinates (r, θ, φ) in
atoms and cartesians(x, y, z) in
polyatomic molecules. The advantage of spherical coordinates (for atoms) is
that an orbital wave function is a product of three factors each dependent on a
single coordinate: ψ(r, θ, φ) = R(r) Θ(θ) Φ(φ). The angular factors of atomic orbitals Θ(θ) Φ(φ) generate
s, p, d, etc. functions as real combinations of spherical harmonics Yℓm(θ, φ) (where ℓ and m are
quantum numbers). There are typically three mathematical forms for the radial
functions R(r) which
can be chosen as a starting point for the calculation of the properties of
atoms and molecules with many electrons:
1.
The hydrogen-like
atomic orbitals are
derived from the exact solution of the Schrödinger Equation for one electron
and a nucleus, for a hydrogen-like atom. The part of the function that
depends on the distance from the nucleus has nodes (radial nodes) and decays as e−(constant × distance).
2.
The Slater-type orbital (STO) is a form without radial nodes but
decays from the nucleus as does the hydrogen-like orbital.
3.
The form of the Gaussian
type orbital (Gaussians)
has no radial nodes and decays as e(−distance squared).
Although
hydrogen-like orbitals are still used as pedagogical tools, the advent of
computers has made STOs preferable for atoms and diatomic molecules since
combinations of STOs can replace the nodes in hydrogen-like atomic orbital.
Gaussians are typically used in molecules with three or more atoms. Although
not as accurate by themselves as STOs, combinations of many Gaussians can
attain the accuracy of hydrogen-like orbitals.
History[edit]
The
term "orbital" was coined by Robert
Mulliken in 1932 as an abbreviation for one-electron
orbital wave function.[8] However,
the idea that electrons might revolve around a compact nucleus with definite
angular momentum was convincingly argued at least 19 years earlier by Niels Bohr,[9] and
the Japanese physicist Hantaro Nagaoka published an
orbit-based hypothesis for electronic behavior as early as 1904.[10] Explaining
the behavior of these electron "orbits" was one of the driving forces
behind the development of quantum
mechanics.[11]
Early models[edit]
With J.J. Thomson's discovery of the electron in 1897,[12] it
became clear that atoms were not the smallest building blocks of nature, but
were rather composite particles. The newly discovered structure within atoms
tempted many to imagine how the atom's constituent parts might interact with
each other. Thomson theorized that multiple electrons revolved in orbit-like
rings within a positively charged jelly-like substance,[13] and
between the electron's discovery and 1909, this "plum pudding model" was the most widely accepted explanation of
atomic structure.
Shortly
after Thomson's discovery, Hantaro Nagaoka predicted a different
model for electronic structure.[10] Unlike
the plum pudding model, the positive charge in Nagaoka's "Saturnian
Model" was concentrated into a central core, pulling the electrons into
circular orbits reminiscent of Saturn's rings. Few people took notice of
Nagaoka's work at the time,[14] and
Nagaoka himself recognized a fundamental defect in the theory even at its
conception, namely that a classical charged object cannot sustain orbital
motion because it is accelerating and therefore loses energy due to
electromagnetic radiation.[15] Nevertheless,
the Saturnian model turned out to have
more in common with modern theory than any of its contemporaries.
Bohr atom[edit]
In
1909, Ernest
Rutherford discovered that the
bulk of the atomic mass was tightly condensed into a nucleus, which was also
found to be positively charged. It became clear from his analysis in 1911 that
the plum pudding model could not explain atomic structure. In 1913 as
Rutherford's post-doctoral student, Niels Bohr proposed
a new model of the atom, wherein electrons orbited the nucleus with classical
periods, but were only permitted to have discrete values of angular momentum,
quantized in units h/2Ï€.[9] This
constraint automatically permitted only certain values of electron energies.
The Bohr model of
the atom fixed the problem of energy loss from radiation from a ground state
(by declaring that there was no state below this), and more importantly
explained the origin of spectral lines.
After
Bohr's use of Einstein's explanation of the photoelectric effect to relate energy
levels in atoms with the wavelength of emitted light, the connection between
the structure of electrons in atoms and the emission and absorption spectra of atoms became an
increasingly useful tool in the understanding of electrons in atoms. The most
prominent feature of emission and absorption spectra (known experimentally
since the middle of the 19th century), was that these atomic spectra contained
discrete lines. The significance of the Bohr model was that it related the
lines in emission and absorption spectra to the energy differences between the
orbits that electrons could take around an atom. This was, however, not achieved
by Bohr through giving the electrons some kind of wave-like properties, since
the idea that electrons could behave as matter
waves was not suggested until eleven years later. Still,
the Bohr model's use of quantized angular momenta and therefore quantized energy
levels was a significant step towards the understanding of electrons in atoms,
and also a significant step towards the development of quantum
mechanics in suggesting that quantized restraints must account
for all discontinuous energy levels and spectra in atoms.
With de Broglie's suggestion of the
existence of electron matter waves in 1924, and for a short time before the
full 1926 Schrödinger equationtreatment of hydrogen-like atom, a Bohr electron "wavelength" could be
seen to be a function of its momentum, and thus a Bohr orbiting electron was
seen to orbit in a circle at a multiple of its half-wavelength (this physically
incorrect Bohr model is still often taught to beginning students). The Bohr
model for a short time could be seen as a classical model with an additional
constraint provided by the 'wavelength' argument. However, this period was
immediately superseded by the full three-dimensional wave mechanics of 1926. In
our current understanding of physics, the Bohr model is called a semi-classical
model because of its quantization of angular momentum, not primarily because of
its relationship with electron wavelength, which appeared in hindsight a dozen
years after the Bohr model was proposed.
The
Bohr model was able to explain the emission and absorption spectra of hydrogen. The energies of electrons in the n =
1, 2, 3, etc. states in the Bohr model match those of current physics. However,
this did not explain similarities between different atoms, as expressed by the
periodic table, such as the fact that helium (two
electrons), neon (10 electrons), and argon(18 electrons)
exhibit similar chemical inertness. Modern quantum
mechanics explains this in terms of electron shells and subshells which
can each hold a number of electrons determined by the Pauli exclusion principle. Thus the n =
1 state can hold one or two electrons, while the n =
2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all n =
1 states are fully occupied; the same for n =
1 and n = 2 in neon. In argon
the 3s and 3p subshells are similarly fully occupied by eight electrons;
quantum mechanics also allows a 3d subshell but this is at higher energy than
the 3s and 3p in argon (contrary to the situation in the hydrogen atom) and
remains empty.
Modern conceptions and connections to the Heisenberg
Uncertainty Principle[edit]
Immediately
after Heisenberg discovered
his uncertainty relation,[16] it
was noted by Bohr that
the existence of any sort of wave packet implies
uncertainty in the wave frequency and wavelength, since a spread of frequencies
is needed to create the packet itself.[17] In
quantum mechanics, where all particle momenta are associated with waves, it is
the formation of such a wave packet which localizes the wave, and thus the
particle, in space. In states where a quantum mechanical particle is bound, it
must be localized as a wave packet, and the existence of the packet and its
minimum size implies a spread and minimal value in particle wavelength, and
thus also momentum and energy. In quantum mechanics, as a particle is localized
to a smaller region in space, the associated compressed wave packet requires a
larger and larger range of momenta, and thus larger kinetic energy. Thus, the
binding energy to contain or trap a particle in a smaller region of space,
increases without bound, as the region of space grows smaller. Particles cannot
be restricted to a geometric point in space, since this would require an
infinite particle momentum.
In
chemistry, Schrödinger, Pauling, Mulliken and
others noted that the consequence of Heisenberg's relation was that the
electron, as a wave packet, could not be considered to have an exact location
in its orbital. Max Born suggested
that the electron's position needed to be described by a probability distribution which was connected
with finding the electron at some point in the wave-function which described
its associated wave packet. The new quantum mechanics did not give exact
results, but only the probabilities for the occurrence of a variety of possible
such results. Heisenberg held that the path of a moving particle has no meaning
if we cannot observe it, as we cannot with electrons in an atom.
In
the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom
number n for each orbital became known as an n-sphere[citation needed] in a three
dimensional atom and was pictured as the mean energy of the probability cloud
of the electron's wave packet which surrounded the atom.
Orbital names[edit]
Orbitals
are given names in the form:
{\displaystyle X\,\mathrm {type} ^{y}\ }
where X is
the energy level corresponding to the principal quantum number n, type is
a lower-case letter denoting the shape or subshell of the orbital and it
corresponds to the angular quantum number â„“, and y is
the number of electrons in that orbital.
For
example, the orbital 1s2 (pronounced "one
ess two") has two electrons and is the lowest energy level (n = 1)
and has an angular quantum number of â„“ = 0.
In X-ray
notation,
the principal quantum number is given a letter associated with it. For n =
1, 2, 3, 4, 5, …, the letters associated with those
numbers are K, L, M, N, O, … respectively.
Hydrogen-like orbitals[edit]
The
simplest atomic orbitals are those that are calculated for systems with a
single electron, such as the hydrogen
atom.
An atom of any other element ionized down
to a single electron is very similar to hydrogen, and the orbitals take the
same form. In the Schrödinger equation for this system of one negative and one
positive particle, the atomic orbitals are the eigenstates of
the Hamiltonian operator for the energy. They
can be obtained analytically, meaning that the resulting orbitals are products
of a polynomial series, and exponential and trigonometric functions. (see hydrogen atom).
For
atoms with two or more electrons, the governing equations can only be solved
with the use of methods of iterative approximation. Orbitals of multi-electron
atoms are qualitatively similar to those of
hydrogen, and in the simplest models, they are taken to have the same form. For
more rigorous and precise analysis, the numerical approximations must be used.
A
given (hydrogen-like) atomic orbital is identified by unique values of three
quantum numbers: n, â„“, and mâ„“. The rules restricting the
values of the quantum numbers, and their energies (see below), explain the
electron configuration of the atoms and the periodic
table.
The
stationary states (quantum states) of the hydrogen-like atoms are its atomic
orbitals.[clarification needed] However,
in general, an electron's behavior is not fully described by a single orbital.
Electron states are best represented by time-depending "mixtures" (linear combinations) of multiple orbitals. See Linear
combination of atomic orbitals molecular orbital method.
The
quantum number n first appeared in the Bohr model where it determines
the radius of each circular electron orbit. In modern quantum mechanics
however, n determines the mean
distance of the electron from the nucleus; all electrons with the same value of n lie
at the same average distance. For this reason, orbitals with the same value of n are
said to comprise a "shell". Orbitals with
the same value of n and
also the same value of â„“ are
even more closely related, and are said to comprise a "subshell".
Quantum numbers[edit]
Because
of the quantum mechanical nature of the electrons around a nucleus, atomic
orbitals can be uniquely defined by a set of integers known as quantum numbers.
These quantum numbers only occur in certain combinations of values, and their
physical interpretation changes depending on whether real or complex versions
of the atomic orbitals are employed.
Complex orbitals[edit]
In
physics, the most common orbital descriptions are based on the solutions to the
hydrogen atom, where orbitals are given by the product between a radial
function and a pure spherical harmonic. The quantum numbers, together with the
rules governing their possible values, are as follows:
The principal quantum number n describes the energy
of the electron and is always a positive
integer.
In fact, it can be any positive integer, but for reasons discussed below, large
numbers are seldom encountered. Each atom has, in general, many orbitals associated
with each value of n; these orbitals
together are sometimes called electron
shells.
The azimuthal quantum number â„“ describes the orbital
angular momentum of each electron and is a non-negative integer. Within a shell
where n is some integer n0, â„“ ranges across all
(integer) values satisfying the relation {\displaystyle
0\leq \ell \leq n_{0}-1}. For instance, the n = 1 shell has only orbitals with {\displaystyle \ell =0}, and the n = 2 shell has only orbitals with {\displaystyle \ell =0}, and {\displaystyle \ell =1}. The set of orbitals
associated with a particular value of â„“ are sometimes
collectively called a subshell.
The magnetic quantum number, {\displaystyle m_{\ell }}, describes the
magnetic moment of an electron in an arbitrary direction, and is also always an
integer. Within a subshell where {\displaystyle
\ell } is
some integer {\displaystyle \ell _{0}}, {\displaystyle m_{\ell }}ranges thus: {\displaystyle -\ell _{0}\leq m_{\ell }\leq \ell _{0}}.
The
above results may be summarized in the following table. Each cell represents a
subshell, and lists the values of {\displaystyle
m_{\ell }} available
in that subshell. Empty cells represent subshells that do not exist.
|
â„“ = 0
|
â„“ = 1
|
â„“ = 2
|
â„“ = 3
|
â„“ = 4
|
…
|
|
|
n = 1
|
{\displaystyle
m_{\ell }=0}
|
|||||
|
n = 2
|
0
|
−1, 0, 1
|
||||
|
n = 3
|
0
|
−1, 0, 1
|
−2, −1, 0, 1,
2
|
|||
|
n = 4
|
0
|
−1, 0, 1
|
−2, −1, 0, 1,
2
|
−3, −2, −1, 0,
1, 2, 3
|
||
|
n = 5
|
0
|
−1, 0, 1
|
−2, −1, 0, 1,
2
|
−3, −2, −1, 0,
1, 2, 3
|
−4, −3, −2,
−1, 0, 1, 2, 3, 4
|
|
|
…
|
…
|
…
|
…
|
…
|
…
|
…
|
Subshells
are usually identified by their {\displaystyle n}- and {\displaystyle \ell }-values. {\displaystyle n} is
represented by its numerical value, but {\displaystyle
\ell } is
represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by
'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with {\displaystyle n=2} and {\displaystyle \ell =0} as
a '2s subshell'.
Each electron also has a spin quantum number, s, which describes
the spin of each electron (spin up or spin down). The number s can
be +1/2 or −1/2.
The Pauli exclusion principle states
that no two electrons can occupy the same quantum state: every electron in an
atom must have a unique combination of quantum numbers.
The
above conventions imply a preferred axis (for example, the z direction
in Cartesian coordinates), and they also imply a preferred direction along this
preferred axis. Otherwise there would be no sense in distinguishing m = +1 from m = −1. As such, the model is most useful when applied to
physical systems that share these symmetries. The Stern–Gerlach experiment— where an atom is exposed to
a magnetic field — provides one such example.[18]
Real orbitals[edit]
An
atom that is embedded in a crystalline solid feels multiple preferred axes, but
no preferred direction. Instead of building atomic orbitals out of the product
of radial functions and a singlespherical harmonic, linear combinations of spherical harmonics are
typically used, designed so that the imaginary part of the spherical harmonics
cancel out. These real orbitals are
the building blocks most commonly shown in orbital visualizations.
In
the real hydrogen-like orbitals, for example, n and â„“ have
the same interpretation and significance as their complex counterparts, but m is
no longer a good quantum number (though its absolute value is). The orbitals
are given new names based on their shape with respect to a standardized
Cartesian basis. The real hydrogen-like p orbitals are given by the
following[19][20]
The
equations for the px and py orbitals
depend on the phase convention used for the spherical harmonics. The above
equations suppose that the spherical harmonics are defined by {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi
}P_{\ell }^{m}(\cos {\theta })}. However some
quantum physicists[21][22] include
a phase factor (-1)m in these definitions,
which has the effect of relating the px orbital to a difference of
spherical harmonics and the py orbital to the
corresponding sum.
Shapes of orbitals
Cross-section of computed hydrogen atom orbital (ψ(r, θ, φ)2) for the 6s (n =
6, â„“ = 0, m = 0) orbital.
Note that s orbitals, though spherically symmetrical, have radially placed
wave-nodes for n > 1. However, only s orbitals invariably have a center anti-node;
the other types never do.
Simple
pictures showing orbital shapes are intended to describe the angular forms of
regions in space where the electrons occupying the orbital are likely to be
found. The diagrams cannot, however, show the entire region where an electron
can be found, since according to quantum mechanics there is a non-zero
probability of finding the electron (almost) anywhere in space. Instead the
diagrams are approximate representations of boundary or contour surfaces where the probability
density | ψ(r, θ, φ) |2 has a constant value,
chosen so that there is a certain probability (for example 90%) of finding the
electron within the contour. Although | ψ |2 as
the square of an absolute
value is everywhere non-negative, the sign of the wave functionψ(r, θ, φ) is often indicated in
each subregion of the orbital picture.
Sometimes
the ψ function will be
graphed to show its phases, rather than the | ψ(r, θ, φ) |2 which
shows probability density but has no phases (which have been lost in the
process of taking the absolute value, since ψ(r, θ, φ) is
a complex number). | ψ(r, θ, φ) |2 orbital
graphs tend to have less spherical, thinner lobes than ψ(r, θ, φ) graphs,
but have the same number of lobes in the same places, and otherwise are
recognizable. This article, in order to show wave function phases, shows mostly ψ(r, θ, φ) graphs.
The
lobes can be viewed as standing
wave interference patterns
between the two counter rotating, ring resonant travelling
wave "m" and "−m" modes, with the projection of the orbital onto the
xy plane having a resonant "m"
wavelengths around the circumference. Though rarely depicted the travelling
wave solutions can be viewed as rotating banded tori, with the bands
representing phase information. For each m there are two standing
wave solutions ⟨m⟩+⟨−m⟩ and ⟨m⟩−⟨−m⟩. For the case where m = 0 the
orbital is vertical, counter rotating information is unknown, and the orbital
is z-axis symmetric. For the case where â„“ = 0 there
are no counter rotating modes. There are only radial modes and the shape is
spherically symmetric. For any given n, the
smaller â„“ is, the more radial
nodes there are. Loosely speaking n is
energy, â„“ is analogous to eccentricity, and m is orientation. For
the record, in the classical case, a ring resonant travelling wave, for example
in a circular transmission line, unless actively forced, will spontaneously
decay into a ring resonant standing wave because reflections will build up over
time at even the smallest imperfection or discontinuity.
Generally
speaking, the number n determines the size
and energy of the orbital for a given nucleus: as n increases, the size
of the orbital increases. However, in comparing different elements, the higher
nuclear charge Z of heavier elements
causes their orbitals to contract by comparison to lighter ones, so that the
overall size of the whole atom remains very roughly constant, even as the
number of electrons in heavier elements (higher Z)
increases.
Experimentally imaged 1s and
2pcore-electron orbitals of Sr, including the effects of atomic thermal
vibrations and excitation broadening, retrieved from energy dispersive x-ray
spectroscopy (EDX) in scanning transmission electron microscopy (STEM).[23]
Also
in general terms, â„“ determines an
orbital's shape, and mâ„“ its orientation.
However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on mâ„“ also.
Together, the whole set of orbitals for a given â„“ and n fill
space as symmetrically as possible, though with increasingly complex sets of
lobes and nodes.
The
single s-orbitals ({\displaystyle \ell =0}) are shaped like
spheres. For n = 1 it is roughly a solid ball (it is most dense at
the center and fades exponentially outwardly), but for n = 2 or
more, each single s-orbital is composed of spherically symmetric surfaces which
are nested shells (i.e., the "wave-structure" is radial, following a
sinusoidal radial component as well). See illustration of a cross-section of
these nested shells, at right. The s-orbitals for all n numbers
are the only orbitals with an anti-node (a region of high wave function
density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have
angular momentum, and thus avoid the nucleus (having a wave node at the
nucleus). Recently, there has been an effort to experimentally image the 1s and
2p orbitials in a SrTiO3 crystal using
scanning transmission electron microscopy with energy dispersive x-ray
spectroscopy.[23]Because the imaging
was conducted using an electron beam, Coulombic beam-orbital interaction that
is often termed as the impact parameter effect is included in the final outcome
(see the figure at right).
The
shapes of p, d and f-orbitals are described verbally here and shown graphically
in the Orbitals table below. The three
p-orbitals for n = 2 have
the form of two ellipsoids with
a point of
tangency at the nucleus (the
two-lobed shape is sometimes referred to as a "dumbbell"—there are two
lobes pointing in opposite directions from each other). The three p-orbitals in
each shell are
oriented at right angles to each other, as determined by their respective
linear combination of values of mâ„“. The overall result is a lobe
pointing along each direction of the primary axes.
Four
of the five d-orbitals for n = 3 look
similar, each with four pear-shaped lobes, each lobe tangent at right angles to
two others, and the centers of all four lying in one plane. Three of these
planes are the xy-, xz-, and yz-planes—the lobes are between the pairs of
primary axes—and the fourth has the centres along the x and y axes themselves.
The fifth and final d-orbital consists of three regions of high probability
density: a torus with
two pear-shaped regions placed symmetrically on its z axis. The overall total
of 18 directional lobes point in every primary axis direction and between every
pair.
There
are seven f-orbitals, each with shapes more complex than those of the d-orbitals.
Additionally,
as is the case with the s orbitals, individual p, d, f and g orbitals with n values
higher than the lowest possible value, exhibit an additional radial node
structure which is reminiscent of harmonic waves of the same type, as compared
with the lowest (or fundamental) mode of the wave. As with s orbitals, this
phenomenon provides p, d, f, and g orbitals at the next higher possible value
of n (for example, 3p
orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher
values of n further increase the
number of radial nodes, for each type of orbital.
The
shapes of atomic orbitals in one-electron atom are related to 3-dimensional spherical harmonics. These shapes are not unique, and any linear
combination is valid, like a transformation to cubic
harmonics,
in fact it is possible to generate sets where all the d's are the same shape,
just like thepx, py, and pz are
the same shape.[24][25]
Orbitals table[edit]
This
table shows all orbital configurations for the real hydrogen-like wave
functions up to 7s, and therefore covers the simple electronic configuration
for all elements in the periodic table up to radium. "ψ"
graphs are shown with − and + wave function phases shown in two
different colors (arbitrarily red and blue). The pz orbital is the same
as the p0 orbital, but the px and py are
formed by taking linear combinations of the p+1 and p−1 orbitals
(which is why they are listed under the m = ±1 label).
Also, the p+1 and p−1 are
not the same shape as the p0,
since they are pure spherical harmonics.
|
s (â„“ = 0)
|
p (â„“ = 1)
|
d (â„“ = 2)
|
f (â„“ = 3)
|
|||||||||||||
|
m = 0
|
m = 0
|
m = ±1
|
m = 0
|
m = ±1
|
m = ±2
|
m = 0
|
m = ±1
|
m = ±2
|
m = ±3
|
|||||||
|
s
|
pz
|
px
|
py
|
dz2
|
dxz
|
dyz
|
dxy
|
dx2−y2
|
fz3
|
fxz2
|
fyz2
|
fxyz
|
fz(x2−y2)
|
fx(x2−3y2)
|
fy(3x2−y2)
|
|
|
n = 1
|
 |
|||||||||||||||
|
n = 2
|
 |
 |
 |
 |
||||||||||||
|
n = 3
|
 |
 |
 |
 |
 |
 |
 |
 |
 |
|||||||
|
n = 4
|
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
|
n = 5
|
 |
 |
 |
 |
 |
 |
 |
 |
 |
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
|
n = 6
|
 |
 |
 |
 |
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
|
n = 7
|
 |
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
. . .
|
Qualitative understanding of shapes[edit]
The
shapes of atomic orbitals can be understood qualitatively by considering the
analogous case of standing waves on a circular drum.[26] To
see the analogy, the mean gravitational displacement of each bit of drum membrane
from the equilibrium point over many cycles (a measure of average drum membrane
velocity and momentum at that point) must be considered relative to that
point's distance from the center of the drum head. If this displacement is
taken as being analogous to the probability of finding an electron at a given
distance from the nucleus, then it will be seen that the many modes of the
vibrating disk form patterns that trace the various shapes of atomic orbitals.
The basic reason for this correspondence lies in the fact that the distribution
of kinetic energy and momentum in a matter-wave is predictive of where the
particle associated with the wave will be. That is, the probability of finding
an electron at a given place is also a function of the electron's average
momentum at that point, since high electron momentum at a given position tends
to "localize" the electron in that position, via the properties of
electron wave-packets (see the Heisenberg uncertainty principle for
details of the mechanism).
This
relationship means that certain key features can be observed in both drum
membrane modes and atomic orbitals. For example, in all of the modes analogous
to s orbitals (the top row in the animated
illustration below), it can be seen that the very center of the drum membrane
vibrates most strongly, corresponding to the antinode in
all s orbitals in an atom. This antinode means the
electron is most likely to be at the physical position of the nucleus (which it
passes straight through without scattering or striking it), since it is moving
(on average) most rapidly at that point, giving it maximal momentum.
A
mental "planetary orbit" picture closest to the behavior of electrons
in s orbitals, all of which have no angular
momentum, might perhaps be that of a Keplerian orbit with the orbital eccentricity of 1 but a finite
major axis, not physically possible (because particles were
to collide), but can be imagined as a limit of orbits with equal
major axes but increasing eccentricity.
Below,
a number of drum membrane vibration modes and the respective wave functions of
the hydrogen atom are shown. A correspondence can be considered where the wave
functions of a vibrating drum head are for a two-coordinate system ψ(r, θ) and
the wave functions for a vibrating sphere are three-coordinate ψ(r, θ, φ).
|
s-type drum
modes and wave functions
|
||
Drum
mode {\displaystyle u_{01}}
|
Drum
mode {\displaystyle u_{02}}
|
Drum
mode {\displaystyle u_{03}}
|
Wave
function of 1s orbital (real part, 2D-cut, {\displaystyle
r_{max}=2a_{0}})
|
Wave
function of 2s orbital (real part, 2D-cut, {\displaystyle r_{max}=10a_{0}})
|
Wave
function of 3s orbital (real part, 2D-cut, {\displaystyle
r_{max}=20a_{0}})
|
None
of the other sets of modes in a drum membrane have a central antinode, and in
all of them the center of the drum does not move. These correspond to a node at
the nucleus for all non-s orbitals in an atom.
These orbitals all have some angular momentum, and in the planetary model, they
correspond to particles in orbit with eccentricity less than 1.0, so that they
do not pass straight through the center of the primary body, but keep somewhat
away from it.
In
addition, the drum modes analogous to p and d modes
in an atom show spatial irregularity along the different radial directions from
the center of the drum, whereas all of the modes analogous to s modes
are perfectly symmetrical in radial direction. The non radial-symmetry
properties of non-s orbitals are
necessary to localize a particle with angular momentum and a wave nature in an
orbital where it must tend to stay away from the central attraction force,
since any particle localized at the point of central attraction could have no
angular momentum. For these modes, waves in the drum head tend to avoid the
central point. Such features again emphasize that the shapes of atomic orbitals
are a direct consequence of the wave nature of electrons.
|
p-type drum
modes and wave functions
|
||
Drum
mode {\displaystyle u_{11}}
|
Drum
mode {\displaystyle u_{12}}
|
Drum
mode {\displaystyle u_{13}}
|
Wave
function of 2p orbital (real part, 2D-cut, {\displaystyle r_{max}=10a_{0}})
|
Wave
function of 3p orbital (real part, 2D-cut, {\displaystyle
r_{max}=20a_{0}})
|
Wave
function of 4p orbital (real part, 2D-cut, {\displaystyle
r_{max}=25a_{0}})
|
d-type
drum modes
·
Mode {\displaystyle u_{21}} (3d
orbital)
·
Mode {\displaystyle u_{22}} (4d
orbital)
·
Mode {\displaystyle u_{23}} (5d
orbital)
Orbital energy[edit]
In
atoms with a single electron (hydrogen-like atoms), the energy of an orbital (and, consequently, of
any electrons in the orbital) is determined exclusively by {\displaystyle n}. The {\displaystyle n=1} orbital
has the lowest possible energy in the atom. Each successively higher value of {\displaystyle n} has
a higher level of energy, but the difference decreases as {\displaystyle n} increases.
For high {\displaystyle n}, the level of energy
becomes so high that the electron can easily escape from the atom. In single
electron atoms, all levels with different {\displaystyle
\ell } within
a given {\displaystyle n} are
(to a good approximation) degenerate, and have the same energy. This
approximation is broken to a slight extent by the effect of the magnetic field
of the nucleus, and by quantum electrodynamics effects. The latter
induce tiny binding energy differences especially for s electrons
that go nearer the nucleus, since these feel a very slightly different nuclear
charge, even in one-electron atoms; see Lamb shift.
In
atoms with multiple electrons, the energy of an electron depends not only on
the intrinsic properties of its orbital, but also on its interactions with the
other electrons. These interactions depend on the detail of its spatial
probability distribution, and so the energy
levels of orbital’s depend not only on {\displaystyle n} but
also on {\displaystyle \ell }. Higher values of {\displaystyle \ell } are
associated with higher values of energy; for instance, the 2p state is higher
than the 2s state. When {\displaystyle \ell =2}, the increase in
energy of the orbital becomes so large as to push the energy of orbital above
the energy of the s-orbital in the next higher shell; when {\displaystyle \ell =3} the
energy is pushed into the shell two steps higher. The filling of the 3d orbital’s
does not occur until the 4s orbitals have been filled.
The
increase in energy for subshells of increasing angular momentum in larger atoms
is due to electron–electron interaction effects, and it is specifically related
to the ability of low angular momentum electrons to penetrate more effectively
toward the nucleus, where they are subject to less screening from the charge of
intervening electrons. Thus, in atoms of higher atomic number, the {\displaystyle \ell } of
electrons becomes more and more of a determining factor in their energy, and
the principal quantum numbers {\displaystyle n} of
electrons becomes less and less important in their energy placement.
The
energy sequence of the first 35 sub shells (e.g., 1s, 2p, 3d, etc.) is
given in the following table. Each cell represents a subshell with {\displaystyle n} and {\displaystyle \ell } given
by its row and column indices, respectively. The number in the cell is the
subshell's position in the sequence. For a linear listing of the subshells in
terms of increasing energies in multielectron atoms, see the section below.
|
s
|
p
|
d
|
f
|
g
|
h
|
|
|
1
|
1
|
|||||
|
2
|
2
|
3
|
||||
|
3
|
4
|
5
|
7
|
|||
|
4
|
6
|
8
|
10
|
13
|
||
|
5
|
9
|
11
|
14
|
17
|
21
|
|
|
6
|
12
|
15
|
18
|
22
|
26
|
31
|
|
7
|
16
|
19
|
23
|
27
|
32
|
37
|
|
8
|
20
|
24
|
28
|
33
|
38
|
44
|
|
9
|
25
|
29
|
34
|
39
|
45
|
51
|
|
10
|
30
|
35
|
40
|
46
|
52
|
59
|
Note:
empty cells indicate non-existent sublevels, while numbers in italics indicate
sublevels that could (potentially) exist, but which do not hold electrons in
any element currently known.
Electron placement and the periodic table[edit]
Electron atomic
and molecular orbitals. The chart of orbitals (left) is
arranged by increasing energy (see Madelung
rule).Note that atomic orbits are functions of three variables (two
angles, and the distance r from the
nucleus). These images are faithful to the angular component of the orbital,
but not entirely representative of the orbital as a whole.
Atomic orbitals and periodic table construction
Several
rules govern the placement of electrons in orbitals (electron configuration). The first dictates that no
two electrons in an atom may have the same set of values of quantum numbers
(this is the Pauli exclusion principle). These quantum
numbers include the three that define orbitals, as well as s, or spin quantum number. Thus, two electrons may occupy a single orbital,
so long as they have different values of s. However, only two
electrons, because of their spin, can be associated with each orbital.
Additionally,
an electron always tends to fall to the lowest possible energy state. It is
possible for it to occupy any orbital so long as it does not violate the Pauli
exclusion principle, but if lower-energy orbitals are available, this condition
is unstable. The electron will eventually lose energy (by releasing a photon) and drop into the lower orbital. Thus, electrons fill orbitals
in the order specified by the energy sequence given above.
This
behavior is responsible for the structure of the periodic
table.
The table may be divided into several rows (called 'periods'), numbered
starting with 1 at the top. The presently known elements occupy seven periods.
If a certain period has number i, it consists of
elements whose outermost electrons fall in the ith shell. Niels Bohr was the first to
propose (1923) that the periodicityin the properties of
the elements might be explained by the periodic filling of the electron energy
levels, resulting in the electronic structure of the atom.[27]
The
periodic table may also be divided into several numbered rectangular 'blocks'. The elements belonging to a given block have this common
feature: their highest-energy electrons all belong to the same â„“-state (but the n associated with that â„“-state depends upon the period). For instance, the
leftmost two columns constitute the 's-block'. The outermost electrons of Li and Be respectively belong
to the 2s subshell, and those of Na and Mg to the
3s subshell.
The
following is the order for filling the "subshell" orbital’s, which
also gives the order of the "blocks" in the periodic table: 1s, 2s,
2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p
The
"periodic" nature of the filling of orbital, as well as emergence of
the s, p, d and f "blocks",
is more obvious if this order of filling is given in matrix form, with
increasing principal quantum numbers starting the new rows
("periods") in the matrix. Then, each subshell (composed of the first
two quantum numbers) is repeated as many times as required for each pair of
electrons it may contain. The result is a compressed periodic table, with each
entry representing two successive elements:
1s
2s 2p 2p 2p
3s 3p 3p 3p
4s 3d 3d 3d 3d 3d 4p 4p 4p
5s 4d 4d 4d 4d 4d 5p 5p 5p
6s 4f 4f 4f 4f 4f 4f 4f 5d 5d 5d 5d 5d 6p 6p 6p
7s 5f 5f 5f 5f 5f 5f 5f 6d 6d 6d 6d 6d 7p 7p 7p
|
Although
this is the general order of orbital filling according to the Madelung rule,
there are exceptions, and the actual electronic energies of each element are
also dependent upon additional details of the atoms (see Electron configuration Atoms: Aufbau principle and Madelung rule). The number of
electrons in an electrically neutral atom increases with the atomic number. The electrons in the outermost shell, or valence
electrons,
tend to be responsible for an element's chemical behavior. Elements that
contain the same number of valence electrons can be grouped together and
display similar chemical properties. For elements with high atomic number Z, the effects of relativity become more pronounced, and
especially so for s electrons, which move at relativistic velocities as
they penetrate the screening electrons near the core of high-Z atoms.
This relativistic increase in momentum for high speed electrons causes a
corresponding decrease in wavelength and contraction of 6s orbital’s relative
to 5d orbital (by comparison to corresponding s and d electrons in lighter
elements in the same column of the periodic table); this results in 6s valence
electrons becoming lowered in energy.
Examples
of significant physical outcomes of this effect include the lowered melting
temperature of mercury (which
results from 6s electrons not being available for metal bonding) and the golden
color of gold and caesium (which results from
narrowing of 6s to 5d transition energy to the point that visible light begins
to be absorbed).[28]
In
the Bohr Model, an n = 1 electron has a velocity given by ,
where Z is the atomic number, is
the fine-structure constant, and c is
the speed of light. In non-relativistic quantum mechanics, therefore, any atom
with an atomic number greater than 137 would require its 1s electrons to be
traveling faster than the speed of light. Even in the Dirac equation, which accounts for relativistic effects, the wave function
of the electron for atoms with Z > 137 is
oscillatory and unbounded. The significance of
element 137, also known as untriseptium, was first pointed
out by the physicist Richard
Feynman.
Element 137 is sometimes informally called feynmanium (symbol
Fy)[citation needed]. However, Feynman's approximation fails to predict
the exact critical value of Z due to the non-point-charge
nature of the nucleus and very small orbital radius of inner electrons,
resulting in a potential seen by inner electrons which is effectively less than Z. The critical Z value
which makes the atom unstable with regard to high-field breakdown of the vacuum
and production of electron-positron pairs, does not occur until Z is
about 173. These conditions are not seen except transiently in collisions of
very heavy nuclei such as lead or uranium in accelerators, where such
electron-positron production from these effects has been claimed to be
observed. See Extension
of the periodic table beyond the seventh period.
There
are no nodes in relativistic orbital densities, although individual components
of the wavefunction will have nodes.[29]
Transitions between orbitals[edit]
Bound
quantum states have discrete energy levels. When applied to atomic orbitals,
this means that the energy differences between states are also discrete. A
transition between these states (i.e., an electron absorbing or emitting a
photon) can thus only happen if the photon has an energy corresponding with the
exact energy difference between said states.
Consider
two states of the hydrogen atom:
State 1) n = 1, â„“ = 0, mâ„“ = 0 and s = +1/2
State 2) n = 2, â„“ = 0, mâ„“ = 0 and s = +1/2
By
quantum theory, state 1 has a fixed energy of E1, and state 2 has a fixed energy of E2. Now, what would happen if an electron in state 1
were to move to state 2? For this to happen, the electron would need to
gain an energy of exactly E2 − E1. If the electron receives energy that is less than or
greater than this value, it cannot jump from state 1 to state 2. Now,
suppose we irradiate the atom with a broad-spectrum of light. Photons that
reach the atom that have an energy of exactly E2 − E1 will
be absorbed by the electron in state 1, and that electron will jump to
state 2. However, photons that are greater or lower in energy cannot be
absorbed by the electron, because the electron can only jump to one of the
orbitals, it cannot jump to a state between orbitals. The result is that only
photons of a specific frequency will be absorbed by the atom. This creates a
line in the spectrum, known as an absorption line, which corresponds to the
energy difference between states 1 and 2.
The
atomic orbital model thus predicts line spectra, which are observed
experimentally. This is one of the main validations of the atomic orbital
model.
The
atomic orbital model is nevertheless an approximation to the full quantum
theory, which only recognizes many electron states. The predictions of line
spectra are qualitatively useful but are not quantitatively accurate for atoms
and ions other than those containing only one electron.
There is yet another
way to writing electron configurations. It is called the "Box and
Arrow" (or circle and X) orbital configuration. Sublevels can be broken down into regions called "orbitals". An orbital is defined as the most probable location for finding an electron. Each orbital holds 2 electrons.
|
sublevel
|
No of
electrons in each sublevel
|
No. of
orbitals
|
Names of
each orbital
|
|
s
|
2
|
1
|
s
|
|
p
|
6
|
3
|
pz px
py
|
|
d
|
10
|
5
|
dz2 dxz
dyz dxy dx2-y2
|
|
f
|
14
|
7
|
fz3 fxz2
fyz2 fxyz fz(x2-y2)
fx(x2-3y2) fy(3x2-y2)
|
|
g
|
18
|
9
|
|
This sublevel configuration can be broken down into orbitals (boxes).
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6........
Video below on how the sublevels build.
There are a few rules for the box and arrow configurations.
Aufbau Principle - electrons fill orbitals starting at the lowest available energy state before filling higher states (1s before 2s).
Pauli Exclusion Principle
An orbital can hold 0, 1, or 2 electrons only, and if
there are two electrons in the orbital, they must have opposite (paired) spins.
When we draw electrons, we
use up and down arrows. So, if an electron is paired up in a box, one arrow is
up and the second must be down.
(Therefore, no two electrons in the same atom can have the
same set of four Quantum
Numbers. )
incorrect; electrons must spin in opposite directions
correct; the electrons have opposite spins
Hund’s Rule
When filling sublevels
other than s, electrons are placed in individual orbitals before they are
paired up.
Electrons fill like people do on a bus. You would never
sit right next to someone you did not know if there are free seats available,
unless of course all the seats are taken then you must pair up.
So when working with the p sublevel, electrons fill like
this....up, up, up...down, down, down...take a look
|
atom
|
orbital box diagram
|
||||||
|
B
|
 1s |
2s |
  2p |
||||
|
C
|
1s |
2s |
2p |
||||
|
N
|
1s |
2s |
2p |
||||
|
O
|
1s |
2s |
2p |
||||
|
F
|
1s |
2s |
2p |
||||
|
Cl
|
1s |
2s |
2p |
3s |
3p |
|
|
|
Mn
|
1s |
2s |
2p |
3s |
3p |
4s |
3d |
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