Hydrogen Atom: A hydrogen atom contains only two subatomic particles: (1) an electron and (2) a proton. It is the only atom that exists in the universe with no neutron in the atomic nucleus. The energy levels or the energy eigenvalues. Of the hydrogen atom depend only on. N, which is called the. Principal quantum number. Since the energy levels and radial decay rate depend only on the. Number this number is used to identify an. For each energy. The basic structure of the hydrogen energy levels can be calculated from the Schrodinger equation. The energy levels agree with the earlier Bohr model, and agree with experiment within a small fraction of an electron volt.
Given more energy, the electron becomes unbound with some kinetic energy. For example, giving 15.0 eV to an electron in the ground state of hydrogen strips it from the atom and leaves it with 1.4 eV of kinetic energy. The hydrogen atom represents the simplest possible atom, since it consists of only one proton and one electron. The electron is bound, or confined. Its potential energy function U (r) expresses its electrostatic potential energy as a function of its distance r from the proton. U (r) = -q 2 / (4πε 0 r) = -e 2 /r.
Atoms make up ordinary matter. In 1897 J. J. Thomson discovered the electron, a negatively charged particle more than two thousand times lighter than a hydrogen atom. In 1906 Thomson suggested that each atom contained a number of electrons roughly equal to its atomic number. Since atoms are neutral, the charge of these electrons must be balanced by some kind of positive charge. Thomson proposed a 'plum pudding' model, with positive and negative charge filling a sphere ~10-10 meter across. This plum pudding model was generally accepted. Even Thomson's student Rutherford, who would later prove the model incorrect, believed in it at the time.
But in 1911 Ernest Rutherford proposed that each atom has a massive nucleus containing all of its positive charge, and that the much lighter electrons are outside this nucleus. The nucleus has a radius about ~ 10-14 - 10-15 m, ten to one hundred thousand times smaller than the radius of the atom. Rutherford arrived at this model by doing experiments. He scattered alpha particles off fixed targets and observed some of them scattering through very large angles. Scattering at large angles occurs when the alpha particles come close to a nucleus. The reason that most alpha particles are not scattered at all is that they are passing through the relatively large 'gaps' between nuclei.
Links: (animations)
The Rutherford Experiment
Thomson Model of an Atom
Rutherford Model
Rutherford revised Thomson's 'plum pudding' model, proposing that electrons orbit a positively charged nucleus, like planets orbit a star. But orbiting particles continuously accelerate, and accelerating charges produce electromagnetic radiation. According to classical physics the planetary atom cannot exist. Electrons quickly radiate away their energy and spiral into the nucleus.
In 1915 Niels Bohr adapted Rutherford's model by saying that the orbits of the electrons were quantized, meaning that they could exist only at certain distances from the nucleus. Bohr proposed that electrons did not emit electromagnetic radiation when moving in those quantized orbits.
What do our instruments reveal today? Here are 3 examples.
All the examples reveal the probability distribution of atomic electrons, i.e. the probability of finding an electron at certain positions near an atomic nucleus.
The field ion microscope and the STM look at atomic cores that are fixed in the crystal structure of a conductor. By different means they supply barely enough energy to remove the electron from the conductor's surface The electron is most likely removed from positions near the atomic cores. The removed electron is replaced by a small current flowing to the ground. So electrons can bee repeatedly removed from the same atoms and plotting the number of electrons removed or the current flowing versus position maps the electron probability distribution.
| Field Ion microscope | Scanning-Tunneling microscope (STM) |
The quantum microscope works with a beam of atoms. Electrons are removed from different, identically prepared atoms. A electron optics system images their removal position onto a detector. Each atom only contributes one lectron. but after a sufficient number of electrons has been detected, a very detailed probability distribution emerges.,
Quantum mechanics now predicts what measurements can reveal about atoms. The hydrogen atom represents the simplest possible atom, since it consists of only one proton and one electron. The electron is bound, or confined. Its potential energy function U(r) expresses its electrostatic potential energy as a function of its distance r from the proton.
U(r) = -qe2/(4πε0r).
In SI unit 1/(4πε0) = 9*109 Nm2/C2, and qe = 1.6*10-19 C.
The figure on the right shows the shape of U(r) in a plane containing the origin. The potential energy is chosen to be zero at infinity. The electron in the hydrogen atom is confined in the potential well, and its total energy is negative.
Confinement leads to energy quantization.
The allowed energies of the electron in the hydrogen atom are
En = -13.6 eV/n2.
Here n is called the principle quantum number. The values En are the possible value for the total electron energy (kinetic and potential energy) in the hydrogen atom. The average potential energy is -2*13.6 eV/n2 and the average kinetic energy is +13.6 eV/n2. The electron has four degrees of freedom, the three spatial degrees of freedom and one internal degree of freedom, called spin. To completely determine its initial wave function, we, in general, have to make four compatible measurements.
Some observables that are compatible with energy measurements and compatible with each other are
- the magnitude of the electron's orbital angular momentum, labeled by the quantum number l,
L = (l(l + 1))1/2ħ, - the projection of the electron's orbital angular momentum along one axis, for example the z-axis, labeled by the quantum number m,
Lz = mħ, - and the projection of the electron's spin along one axis, labeled by by the quantum number ms,
Sz = msħ.
We can know the values of these observables, labeled by n, l, m, and ms, simultaneously.
For the hydrogen atom, the energy levels only depend on the principal quantum number n. The energy levels are degenerate, meaning that the electron in the hydrogen atom can be in different states, with different wave functions, labeled by a different set of quantum numbers, and still have the same energy.
The electron wave functions however are different for every different set of quantum numbers.
- For each principal quantum number n, all smaller positive integers are possible values for the quantum number l, i.e. l = 0, 1, 2, ..., n - 1. The quantum number l is always smaller than the quantum number n. Only states with high energy can have large angular momentum.
- The quantum number m can take on all integer values between -l and l.
Examples of hydrogen atom probability densities.
As n increases, |
|
Note: Energy eigenfuctions characterize stationary state. We cannot track the electron and know its energy at the same time. If we know its energy, we can only predict probabilities for where we might find it if we tried to measure its position. If we determine the position of the electron, we lose the energy information.
| l = 1 | s |
| l = 2 | p |
| l = 3 | d |
| l = 4 | f |
| l = 4 | g |
Often texts use a different notation to refer to the energy levels of the hydrogen atom. Letters of the alphabet are associated with various values of l.
| Spectroscopic notation | Quantum number n of the state | Quantum number l of the state | Possible values of the quantum number m |
|---|---|---|---|
1s | 1 | 0 | 0 |
2s | 2 | 0 | 0 |
2p | 2 | 1 | -1, 0, 1 |
3s | 3 | 0 | 0 |
3p | 3 | 1 | -1, 0, 1 |
3d | 3 | 2 | -2, -1, 0, 1, 2 |
4s | 4 | 0 | 0 |
4p | 4 | 1 | -1, 0, 1 |
4d | 4 | 2 | --2, -1, 0, 1, 2 |
4f | 4 | 3 | -3, -2, -1, 0, 1, 2, 3 |
The hydrogen line spectrum
When an electron changes from one energy level to another, the energy of the atom must change as well. It requires energy to promote an electron from a lower energy level to a higher one. This energy can be supplied by a photon whose energy E is given in terms of its frequency E = hf or wavelength E = hc/λ.
Since the energy levels are quantized, only certain photon wavelengths can be absorbed. If a photon is absorbed, an electron will be promoted to a higher energy level and will then fall back down into the lowest energy state (ground state) in a cascade of transitions. Each time the energy level of the electron changes, a photon will be emitted and the energy (wavelength) of the photon will be characteristic of the energy difference between the initial and final energy levels of the atom in the transition. The energy of the emitted photon is just the energy difference between the initial (ni) and the final (nf) state.
The set of spectral lines for a given final state nf are generally close together. In the hydrogen atom they are given special names. The lines for which nf = 1 are called the Lyman series. These transitions frequencies correspond to spectral lines in the ultraviolet region of the electromagnetic spectrum. The lines for which nf = 2 are called the Balmer series and many of these spectral lines are visible. The spectrum of hydrogen is particularly important in astronomy because most of the Universe is made of hydrogen. The Balmer series, which is the only hydrogen series with lines in the visible region of the electromagnetic spectrum, is shown in the right in more detail.
The Balmer lines are designated by H with a Greek subscript in order of decreasing wavelength. Thus the longest wavelength Balmer transition is designated H with a subscript alpha, the second longest H with a subscript beta, and so on.
Problem:
What is the wavelength of the least energetic line in the Balmer series?
Solution:
- Reasoning:
The transition from ni = 3 to nf = 2 is the lowest energy, longest wavelength transition in the Balmer series. - Details of the calculation:
∆E = -13.6 eV(1/9 - 1/4) = 1.89 eV = 3*10-19 J. λ = hc/∆E = 658 nm.
Problem:
What is the shortest wavelength in the Balmer series?
Solution:
- Reasoning:
The transition from ni = ∞ to nf = 2 is the highest energy, shortest wavelength transition in the Balmer series. - Details of the calculation:
∆E = -13.6 eV(1/∞ - 1/4) = 13.6/ 4 eV = 3.4 eV = 5.44*10-19 J. λ = hc/∆E = 365 nm.
Problem:
Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.
Solution:
- Reasoning:
From the previous problem we know that the shortest-wavelength Balmer line has λ = 365 nm.
The transition from ni = 2 to nf = 1 is the lowest energy, longest wavelength transition in the Lyman series. - Details of the calculation:
∆E = -13.6 eV(1/2 - 1) = 13.6/ 2 eV = 6.8 eV. λ = hc/∆E = 182 nm.
The two series do not overlap.
Module 11, Question 1
- Hydrogen gas can only absorb EM radiation that has an energy corresponding to a transition in the atom, just as it can only emit these discrete energies. When a spectrum is taken of the solar corona, in which a broad range of EM wavelengths are passed through very hot hydrogen gas, the absorption spectrum shows all the features of the emission spectrum. But when such EM radiation passes through room-temperature hydrogen gas, only the Lyman series is absorbed. Explain the difference.
Discuss this with your fellow students in the discussion forum!
Hydrogenic atoms
Atoms with all but one electron removed are called hydrogenic atoms.
- If the charge of the nucleus is Z times the proton charge, then U(r) = -Zqe2/(4πε0r).
- The energy levels of such atoms are obtained by simply scaling the the solutions for the hydrogen atom.
The energy levels scale with Z2, i.e. En = -Z2*13.6 eV/n2. It takes more energy to remove an electron from the nucleus, because the attractive force that must be overcome is stronger. - The average size of the wave functions scales as 1/Z, i.e. the electron, on average, stays closer to the nucleus, because the attraction is stronger.
Problem:
Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is C+5 , a carbon atom with only a single electron.
(a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen?
(b) What is the wavelength of the first line in this ion's Paschen series?
(c) What type of EM radiation is this?
Solution:
- Reasoning:
The energy levels of hydrogenic atoms scale with Z2, the wave functions scales as 1/Z. - Details of the calculation:
(a) The charge of the carbon nucleus is 6 times the proton charge. All energy level increase by a factor of Z2 = 36.
(b) λ = hc/∆E, all wavelengths of the Paschen series of carbon decrease by a factor of 36 compare to the corresponding wavelengths of hydrogen.
For hydrogen the first line of the Paschen series has ∆E = -13.6 eV(1/16 - 1/9) = 0.66 eV, λ = hc/∆E = 1876 nm.
For carbon the first line of the Paschen series has λ = hc/∆E = 1875/36 nm = 52.1 nm.
(c) This is far UV radiation.
The Bohr Atom
In 1913 Bohr's model of the atom revolutionized atomic physics. The Bohr model consists of four principles:
- Electrons assume only certain orbits around the nucleus. These orbits are stable and called 'stationary' orbits. Electrons in these orbits do not radiate their energy away.
- Each orbit is associated with a definite value of the energy and the angular momentum. Bohr assumed that the angular momentum could only take on values that were integer multiples of ħ.
Angular momentum = mr2ω = mrv = nħ, n = 1, 2, 3, ... .
A classical electron with a definite angular momentum in an orbit about a proton also has a definite energy E .
If angular momentum = mrv = nħ, then En = -me4/(2ħ2n2) = -13.6 eV/n2.
The orbit closest to the nucleus has an energy E1, the next closest E2 and so on.
A definite angular momentum also implies a definite orbital radius.
If angular momentum = mrv = nħ, then rn = n2ħ2/(me2) = n2a0 = n2 * (52.92 pm).
a0 is called the Bohr radius. - A photon is emitted when an electron jumps from a higher energy orbit to a lower energy orbit and absorbed when it jumps from a lower energy orbit to higher energy orbit. The photon energy is equal to the energy difference ∆E = hf = Ehigh - Elow.
With these conditions Bohr was able to explain the stability of atoms, as well as the emission spectrum of hydrogen. According to Bohr's model only certain orbits are allowed, which means only certain energies are possible. These energies naturally lead to the explanation of the hydrogen atom spectrum.
Bohr's model was so successful that he immediately received world-wide fame. Unfortunately, Bohr's model worked only for hydrogen and hydrogenic atoms, such as any atom with all but one electron removed.
The Bohr model is easy to picture, but we now know that it is wrong. Any planetary model of the atom, so often seen in pictures and so easy to picture, is wrong. Energy and position are incompatible observables. We cannot track an electron with a known energy inside an atom.
Atomic Spectra
When gaseous hydrogen in a glass tube is excited by a (5000)-volt electrical discharge, four lines are observed in the visible part of the emission spectrum: red at (656.3) nm, blue-green at (486.1) nm, blue violet at (434.1) nm and violet at (410.2) nm:
Other series of lines have been observed in the ultraviolet and infrared regions. Rydberg (1890) found that all the lines of the atomic hydrogen spectrum could be fitted to a single formula
[ dfrac{1}{lambda} = mathcal{R} left( dfrac{1}{n_1^{2}} - dfrac{1}{n_2^{2}} right), quad n_1 = 1, : 2, : 3..., : n_2 > n_1 label{1}]
where (mathcal{R}), known as the Rydberg constant, has the value (109,677) cm-1 for hydrogen. The reciprocal of wavelength, in units of cm-1, is in general use by spectroscopists. This unit is also designated wavenumbers, since it represents the number of wavelengths per cm. The Balmer series of spectral lines in the visible region, shown in Figure (PageIndex{1}) , correspond to the values (n_1 = 2, : n_2 = 3, : 4, : 5) and (6). The lines with (n_1 = 1) in the ultraviolet make up the Lyman series. The line with (n_2 = 2), designated the Lyman alpha, has the longest wavelength (lowest wavenumber) in this series, with (1/ lambda = 82.258) cm-1 or (lambda = 121.57) nm.
Other atomic species have line spectra, which can be used as a 'fingerprint' to identify the element. However, no atom other than hydrogen has a simple relation analogous to Equation (ref{1}) for its spectral frequencies. Bohr in 1913 proposed that all atomic spectral lines arise from transitions between discrete energy levels, giving a photon such that
[ Delta E = h nu = dfrac{hc}{lambda} label{2}]
This is called the Bohr frequency condition. We now understand that the atomic transition energy (Delta E) is equal to the energy of a photon, as proposed earlier by Planck and Einstein.
The Bohr Atom
The nuclear model proposed by Rutherford in 1911 pictures the atom as a heavy, positively-charged nucleus, around which much lighter, negatively-charged electrons circulate, much like planets in the Solar system. This model is however completely untenable from the standpoint of classical electromagnetic theory, for an accelerating electron (circular motion represents an acceleration) should radiate away its energy. In fact, a hydrogen atom should exist for no longer than (5 times 10^{-11}) sec, time enough for the electron's death spiral into the nucleus. This is one of the worst quantitative predictions in the history of physics. It has been called the Hindenberg disaster on an atomic level. (Recall that the Hindenberg, a hydrogen-filled dirigible, crashed and burned in a famous disaster in 1937.)
Bohr sought to avoid an atomic catastrophe by proposing that certain orbits of the electron around the nucleus could be exempted from classical electrodynamics and remain stable. The Bohr model was quantitatively successful for the hydrogen atom, as we shall now show.
We recall that the attraction between two opposite charges, such as the electron and proton, is given by Coulomb's law
[F = begin{cases} -dfrac{e^{2}}{r^{2}} quad mathsf{(gaussian : units)} -dfrac{e^{2}}{4 pi epsilon_0 r^{2}} quad mathsf{(SI : units)} end{cases} label{3}]
We prefer to use the Gaussian system in applications to atomic phenomena. Since the Coulomb attraction is a central force (dependent only on r), the potential energy is related by
[F = -dfrac{dV(r)}{dr} label{4}]
We find therefore, for the mutual potential energy of a proton and electron,
[V(r) = -dfrac{e^2}{r} label{5}]
Bohr considered an electron in a circular orbit of radius (r) around the proton. To remain in this orbit, the electron must be experiencing a centripetal acceleration
[a = -dfrac{v^{2}}{r} label{6}]
where (v) is the speed of the electron. Using Equations (ref{4}) and (ref{6}) in Newton's second law, we find
[dfrac{e^{2}}{r^{2}} = dfrac{mv^{2}}{r} label{7}]
where (m) is the mass of the electron. For simplicity, we assume that the proton mass is infinite (actually (m_p approx 1836 m_e)) so that the proton's position remains fixed. We will later correct for this approximation by introducing reduced mass. The energy of the hydrogen atom is the sum of the kinetic and potential energies:
[E = T + V = dfrac{1}{2} mv^{2} - dfrac{e^{2}}{r} label{8}]
Using Equation (ref{7}) , we see that
[T = -dfrac{1}{2} V qquad mathsf{and} qquad E = dfrac{1}{2} V = -T label{9}]
This is the form of the virial theorem for a force law varying as (r^{-2}). Note that the energy of a bound atom is negative, since it is lower than the energy of the separated electron and proton, which is taken to be zero.
For further progress, we need some restriction on the possible values of (r) or (v). This is where we can introduce the quantization of angular momentum (mathbf{L} = mathbf{r} times mathbf{p}). Since (mathbf{p}) is perpendicular to (mathbf{r}), we can write simply
[L = rp = mvr label{10}]
Using Equation (ref{9}), we find also that
[r = dfrac{L^{2}}{me^{2}} label{11}]
We introduce angular momentum quantization, writing
[L = nhbar, qquad n = 1, : 2... label{12}]
excluding (n = 0), since the electron would then not be in a circular orbit. The allowed orbital radii are then given by
[r_n = n^{2} a_0 label{13}]
where
[a_0 equiv dfrac{hbar^{2}}{me^{2}} = 5.29 times 10^{-11} : mathsf{m} = 0.529Å label{14}]
which is known as the Bohr radius. The corresponding energy is
[E_n = -dfrac{e^{2}}{2a_0n^{2}} = -dfrac{me^{4}}{2hbar^{2}n^{2}}, qquad n = 1, : 2... label{15}]
Rydberg's formula (Equation (ref{1})) can now be deduced from the Bohr model. We have
[ dfrac{hc}{lambda} = E_{n_2} - E_{n_1} = dfrac{2pi^{2}me^{4}}{h^{2}} left( dfrac{1}{n_1^{2}} - dfrac{1}{n_2^{2}} right) label{16}]
and the Rydbeg constant can be identified as
[mathcal{R} = dfrac{2pi^{2}me^{4}}{h^{3}c} approx 109,737 : mathsf{cm}^{-1} label{17}]
The slight discrepency with the experimental value for hydrogen ( (109,677) ) is due to the finite proton mass. This will be corrected later.
The Bohr model can be readily extended to hydrogenlike ions, systems in which a single electron orbits a nucleus of arbitrary atomic number (Z). Thus (Z = 1) for hydrogen, (Z = 2) for (mathsf{He}^{+}), (Z = 3) for (mathsf{Li}^{++}), and so on. The Coulomb potential (ref{5}) generalizes to
[V(r) = -dfrac{Ze^{2}}{r}, label{18}]
the radius of the orbit (Equation (ref{13})) becomes
[r_n = dfrac{n^{2}a_0}{Z} label{19}]
and the energy Equation (ref{15}) becomes
[E_n = -dfrac{Z^{2}e^{2}}{2a_0n^{2}} label{20}]
De Broglie's proposal that electrons can have wavelike properties was actually inspired by the Bohr atomic model. Since
[L = rp = nhbar = dfrac{nh}{2pi} label{21}]
we find
[2pi r = dfrac{nh}{p} = nlambda label{22}]
Therefore, each allowed orbit traces out an integral number of de Broglie wavelengths.
Wilson (1915) and Sommerfeld (1916) generalized Bohr's formula for the allowed orbits to
[oint p , dr = nh, qquad n =1, : 2... label{23}]
The Sommerfeld-Wilson quantum conditions Equation (ref{23}) reduce to Bohr's results for circular orbits, but allow, in addition, elliptical orbits along which the momentum (p) is variable. According to Kepler's first law of planetary motion, the orbits of planets are ellipses with the Sun at one focus. Figure (PageIndex{2}) shows the generalization of the Bohr theory for hydrogen, including the elliptical orbits. The lowest energy state (n = 1) is still a circular orbit. But (n = 2) allows an elliptical orbit in addition to the circular one; (n = 3) has three possible orbits, and so on. The energy still depends on (n) alone, so that the elliptical orbits represent degenerate states. Atomic spectroscopy shows in fact that energy levels with (n > 1) consist of multiple states, as implied by the splitting of atomic lines by an electric field (Stark effect) or a magnetic field (Zeeman effect). Some of these generalized orbits are drawn schematically in Figure (PageIndex{2}).
The Bohr model was an important first step in the historical development of quantum mechanics. It introduced the quantization of atomic energy levels and gave quantitative agreement with the atomic hydrogen spectrum. With the Sommerfeld-Wilson generalization, it accounted as well for the degeneracy of hydrogen energy levels. Although the Bohr model was able to sidestep the atomic 'Hindenberg disaster,' it cannot avoid what we might call the 'Heisenberg disaster.' By this we mean that the assumption of well-defined electronic orbits around a nucleus is completely contrary to the basic premises of quantum mechanics. Another flaw in the Bohr picture is that the angular momenta are all too large by one unit, for example, the ground state actually has zero orbital angular momentum (rather than (hbar)).
The assumption of well-defined electronic orbits around a nucleus in the Bohr atom is completely contrary to the basic premises of quantum mechanics.
Quantum Mechanics of Hydrogenlike Atoms
In contrast to the particle in a box and the harmonic oscillator, the hydrogen atom is a real physical system that can be treated exactly by quantum mechanics. In addition to their inherent significance, these solutions suggest prototypes for atomic orbitals used in approximate treatments of complex atoms and molecules.
For an electron in the field of a nucleus of charge (+Ze), the Schrӧdinger equation can be written
[left{ -dfrac{hbar^{2}}{2m} nabla^{2} - dfrac{Ze^{2}}{r} right} psi(r) = Epsi(r) label{24}]
It is convenient to introduce atomic units in which length is measured in bohrs:
[a_0 = dfrac{hbar^{2}}{me^{2}} = 5.29 times 10^{-11} : mathsf{m} equiv 1 : mathsf{bohr} ]
and energy in hartrees:
[dfrac{e^2}{a_0} = 4.358 times 10^{-18} : mathsf{J} = 27.211 : mathsf{eV} equiv 1 : mathsf{hartree} ]
Electron volts ((mathsf{eV})) are a convenient unit for atomic energies. One (mathsf{eV}) is defined as the energy an electron gains when accelerated across a potential difference of (1 : mathsf{volt}). The ground state of the hydrogen atom has an energy of (-1/2 : mathsf{hartree}) or (-13.6 : mathsf{eV}). Conversion to atomic units is equivalent to setting
[hbar = e = m = 1]
in all formulas containing these constants. Rewriting the Schrӧdinger equation in atomic units, we have
[left{ -dfrac{1}{2} nabla^{2} - dfrac{Z}{r} right} psi(r) = Epsi(r) label{25}]
Since the potential energy is spherically symmetrical (a function of (r) alone), it is obviously advantageous to treat this problem in spherical polar coordinates (r, : theta, : phi). Expressing the Laplacian operator in these coordinates [cf. Eq (6-20)],
[ -dfrac{1}{2} left{ dfrac{1}{r^{2}} dfrac{partial}{partial r} r^{2} dfrac{partial}{partial r} + dfrac{1}{r^{2}sintheta} dfrac{partial}{partial theta} sintheta dfrac{partial}{partial theta} + dfrac{1}{r^{2}sin^{2}theta} dfrac{partial^{2}}{partialphi^{2}} right} times psi(r, : theta, : phi) - dfrac{Z}{r} psi(r, : theta, : phi) = Epsi(r, : theta, : phi) label{26}]
Equation (ref{26}) shows that the second and third terms in the Laplacian represent the angular momentum operator (hat{L}^{2}). Clearly, Equation (ref{26}) will have separable solutions of the form
[psi(r, : theta, : phi) = R(r)Y_{ell m}(theta, : phi) label{27}]
Substituting Equation (ref{27}) into Equation (ref{26}) and using the angular momentum eigenvalue Equation Equation (ref{6-34}), we obtain an ordinary differential equation for the radial function (R(r)):
[left{ -dfrac{1}{2r^{2}} dfrac{d}{dr} r^{2} dfrac{d}{dr} + dfrac{ell(ell + 1)}{2r^{2}} - dfrac{Z}{r} right} R(r) = ER(r) label{28}]
Note that in the domain of the variable (r), the angular momentum contribution (ell (ell + 1) / 2r^{2}) acts as an effective addition to the potential energy. It can be identified with centrifugal force, which pulls the electron outward, in opposition to the Coulomb attraction. Carrying out the successive differentiations in Equation (ref{29}) and simplifying, we obtain
[dfrac{1}{2}R'(r) + dfrac{1}{r}R'(r) + left[dfrac{Z}{r} - dfrac{ell(ell + 1)}{2r^{2}} + Eright]R(r) = 0 label{29}]
another second-order linear differential equation with non-constant coefficients. It is again useful to explore the asymptotic solutions to Equation (ref{29}), as (r rightarrow infty). In the asymptotic approximation,
[R'(r) - 2rlvert E rvert R(r) approx 0 label{30}]
having noted that the energy (E) is negative for bound states. Solutions to Equation (ref{30}) are
[R(r) approx mathsf{const} , e^{pmsqrt{2lvert E rvert}r} label{31}]
We reject the positive exponential on physical grounds, since (R(r) rightarrow infty) as (r rightarrow infty), in violation of the requirement that the wavefunction must be finite everywhere. Choosing the negative exponential and setting (E = -Z^{2}/2) the ground state energy in the Bohr theory (in atomic units), we obtain
[R(r) approx mathsf{const} , e^{-Zr} label{32}]
It turns out, very fortunately, that this asymptotic approximation is also an exact solution of the Schrӧdinger equation (Equation (ref{29})) with (ell = 0), just what happened for the harmonic-oscillator problem in Chap. 5. The solutions to Equation (ref{29}), designated (R_{nell}(r)), are labeled by (n), known as the principal quantum number, as well as by the angular momentum (ell), which is a parameter in the radial equation. The solution in Equation (ref{32}) corresponds to (R_{10}(r)). This should be normalized according to the condition
[int_{0}^{infty} [R_{10}(r)]^{2} , r^{2} , dr = 1 label{33}]
A useful definite integral is
[int_{0}^{infty} r^{n} , e^{-alpha r} , dr = dfrac{n!}{alpha^{n + 1}} label{34}]
The normalized radial function is thereby given by
[R_{10}(r) = 2Z^{3/2} e^{-Zr} label{35}]
Since this function is nodeless, we identify it with the ground state of the hydrogenlike atom. Multipyling Equation (ref{35}) by the spherical harmonic (Y_{00} = 1/ sqrt{4pi} ), we obtain the total wavefunction (Equation (ref{27}))
[psi_{100}(r, theta, phi) = left( dfrac{Z^{3}}{pi} right)^{1/2} e^{-Zr} label{36}]
This is conventionally designated as the 1s function (psi_{1s}(r)).
Integrals in spherical-polar coordinates over a spherically-symmetrical integrand (like the 1s orbital) can be significantly simplified. We can do the reduction
[int_{0}^{infty} int_{0}^{pi} int_{0}^{2pi} f(r) , r^{2} , sintheta , dr , dtheta , dphi = int_{0}^{infty} f(r) , 4pi r^{2} , dr label{37}]
since integration over (theta) and (phi) gives (4pi), the total solid angle of a sphere. The normalization of the 1s wavefunction can thus be written as
[int_{0}^{infty} [psi_{1s}(r)]^{2} , 4pi r^{2} , dr = 1 label{38}]
Hydrogen Atom Ground State
There are a number of different ways of representing hydrogen-atom wavefunctions graphically. We will illustrate some of these for the 1s ground state. In atomic units,
[psi_{1s}(r) = dfrac{1}{sqrt{pi}}e^{-r} label{39}]
is a decreasing exponential function of a single variable (r), and is simply plotted in Figure 3.
Figure (PageIndex{3}) gives a somewhat more pictorial representation, a three-dimensional contour plot of (psi_{1s}(r)) as a function of (x) and (y) in the (x), (y)-plane.
According to Born's interpretation of the wavefunction, the probability per unit volume of finding the electron at the point ((r, : theta, : phi) ) is equal to the square of the normalized wavefunction
[rho_{1s}(r) = [psi_{1s}(r)]^{2} = dfrac{1}{pi}e^{-2r} label{40}]
This is represented in Figure 5 by a scatter plot describing a possible sequence of observations of the electron position. Although results of individual measurements are not predictable, a statistical pattern does emerge after a sufficiently large number of measurements.
The probability density is normalized such that
[int_{0}^{infty} rho_{1s}(r) , 4pi r^{2} , dr = 1 label{41}]
In some ways (rho (r)) does not provide the best description of the electron distribution, since the region around (r = 0), where the wavefunction has its largest values, is a relatively small fraction of the volume accessible to the electron. Larger radii (r) represent larger physical regions since, in spherical polar coordinates, a value of (r) is associated with a shell of volume (4pi r^{2} , dr). A more significant measure is therefore the radial distribution function
[D_{1s}(r) = 4pi r^{2} [psi_{1s}(r)]^{2} label{42}]
which represents the probability density within the entire shell of radius (r), normalized such that
[int_{0}^{infty} D_{1s}(r) , dr = 1 label{43}]
The functions (rho_{1s}(r) ) and (D_{1s}(r) ) are both shown in Figure (PageIndex{6}). Remarkably, the 1s RDF has its maximum at (r = a_0), equal to the radius of the first Bohr orbit
Atomic Orbitals
The general solution for (R_{nell}(r)) has a rather complicated form which we give without proof:
[R_{nell}(r) = N_{nell} , rho^{ell} , L_{n + ell}^{2ell + 1} , (rho) e^{-rho /2} qquad rho equiv dfrac{2Zr}{n} label{44}]
Here (L_{beta}^{alpha}) is an associated Laguerre polynomial and (N_{nell}), a normalizing constant. The angular momentum quantum number (ell) is by convention designated by a code: s for (ell = 0), p for (ell = 1), d for (ell = 2), f for (ell = 3), g for (ell = 4), and so on. The first four letters come from an old classification scheme for atomic spectral lines: sharp, principal, diffuse and fundamental. Although these designations have long since outlived their original significance, they remain in general use. The solutions of the hydrogenic Schrӧdinger equation
in spherical polar coordinates can now be written in full
[psi_{nell m}(r, : theta, : phi) = R_{nell}(r)Y_{ell m}(theta, : phi) n = 1, : 2... qquad ell = 0, : 1... : n - 1 qquad m = 0, : pm 1, : pm 2... : pm ell label{45}]
where (Y_{ell m}) are the spherical harmonics tabulated in Chap. 6. Table 1 below enumerates all the hydrogenic functions we will actually need. These are called hydrogenic atomic orbitals, in anticipation of their later applications to the structure of atoms and molecules.
| [psi_{1s} = dfrac{1}{sqrt{pi}} e^{-r}] |
| [psi_{2s} = dfrac{1}{2sqrt{2pi}} left( 1 - dfrac{r}{2} right) e^{-r/2}] |
| [psi_{2p_z} = dfrac{1}{4sqrt{2pi}} z , e^{-r/2}] |
| [psi_{2p_x}, : psi_{2p_y} qquad mathsf{analogous}] |
| [psi_{3s} = dfrac{1}{81sqrt{3pi}} (27 - 18r + 2r^{2}) e^{-r/3}] |
| [psi_{3p_z} = dfrac{sqrt{2}}{81sqrt{pi}} (6 - r) z , e^{-r/3}] |
| [psi_{3p_x}, : psi_{3p_y} qquad mathsf{analogous}] |
| [psi_{3d_{z^{2}}} = dfrac{1}{81sqrt{6pi}}(3z^{2} - r^{2}) e^{-r/3}] |
| [psi_{3d_{zx}} = dfrac{sqrt{2}}{81sqrt{pi}}zx , e^{-r/3}] |
| [psi_{3d_{yz}}, : psi_{3d_{xy}} qquad mathsf{analogous}] |
| [psi_{3d_{x^{2} - y^{2}}} = dfrac{1}{81sqrt{pi}}(x^{2} - y^{2}) e^{-r/3}] |
The energy levels for a hydrogenic system are given by
[E_n = -dfrac{Z^{2}}{2n^{2}} : mathsf{hartrees} label{46}]
and depends on the principal quantum number alone. Considering all the allowed values of (ell) and (m), the level (E_n) has a degeneracy of (n^{2}). Figure 7 shows an energy level diagram for hydrogen ((Z = 1) ). For (E geq 0), the energy is a continuum, since the electron is in fact a free particle. The continuum represents states of an electron and proton in interaction, but not bound into a stable atom. Figure (PageIndex{7}) also shows some of the transitions which make up the Lyman series in the ultraviolet and the Balmer series in the visible region.
The (ns) orbitals are all spherically symmetrical, being associated with a constant angular factor, the spherical harmonic (Y_{00} = 1/ sqrt{4pi} ). They have (n - 1) radial nodes—spherical shells on which the wavefunction equals zero. The 1s ground state is nodeless and the number of nodes increases with energy, in a pattern now familiar from our study of the particle-in-a-box and harmonic oscillator. The 2s orbital, with its radial node at (r = 2) bohr, is also shown in Figure (PageIndex{3}).
p- and d-Orbitals
The lowest-energy solutions deviating from spherical symmetry are the 2p-orbitals. Using Equations (ref{44}), (ref{45}) and the (ell = 1) spherical harmonics, we find three degenerate eigenfunctions:
[psi_{210}(r, : theta, : phi) = dfrac{1}{4sqrt{2pi}}re^{-r/2} costheta label{47}]
and
[psi_{21 pm 1}(r, : theta, : phi) = mp dfrac{1}{4sqrt{2pi}}re^{-r/2} sintheta e^{pm i phi} label{48}]
The function (psi_{210}) is real and contains the factor (r costheta ), which is equal to the cartesian variable (z). In chemical applications, this is designated as a 2pz orbital:
[psi_{2p_z} = dfrac{1}{4sqrt{2pi}}ze^{-r/2} label{49}]
Hydrogen Energy Level
A contour plot is shown in Figure (PageIndex{8}). Note that this function is cylindrically-symmetrical about the (z)-axis with a node in the (x), (y)-plane. The (psi_{21 pm 1}) are complex functions and not as easy to represent graphically. Their angular dependence is that of the spherical harmonics (Y_{1 pm 1}) , shown in Figure 6-4. As noted in Chap. 4, any linear combination of degenerate eigenfunctions is an equally-valid alternative eigenfunction. Making use of the Euler formulas for sine and cosine
[cosphi = dfrac{e^{iphi} + e^{-iphi}}{2} qquad mathsf{and} qquad sinphi = dfrac{e^{iphi} - e^{-iphi}}{2} label{50}]
and noting that the combinations (sinthetacosphi) and (sinthetasinphi) correspond to the cartesian variables (x) and (y), respectively, we can define the alternative 2p orbitals
[psi_{2p_x} = dfrac{1}{sqrt{2}}(psi_{21-1} - psi_{211}) = dfrac{1}{4sqrt{2pi}} xe^{-r/2} label{51}]
and
[psi_{2p_y} = -dfrac{i}{sqrt{2}}(psi_{21-1} + psi_{211}) = dfrac{1}{4sqrt{2pi}} ye^{-r/2} label{52}]
Clearly, these have the same shape as the 2pz-orbital, but are oriented along the (x)- and (y)-axes, respectively. The threefold degeneracy of the p-orbitals is very clearly shown by the geometric equivalence the functions 2px, 2py and 2pz, which is not obvious for the spherical harmonics. The functions listed in Table 1 are, in fact, the real forms for all atomic orbitals, which are more useful in chemical applications. All higher p-orbitals have analogous functional forms (x , f(r)), (y , f(r)) and (z , f(r)) and are likewise 3-fold degenerate.
The orbital (psi_{320}) is, like (psi_{210}), a real function. It is known in chemistry as the (d_{z^{2}})-orbital and can be expressed as a cartesian factor times a function of (r):
[psi_{3d_{z^{2}}} = psi_{320} = (3z^{2} - r^{2}) f(r) label{53}]
A contour plot is shown in Figure (PageIndex{9}). This function is also cylindrically symmetric about the (z)-axis with two angular nodes—the conical surfaces with (3z^{2} - r^{2} = 0). The remaining four 3d orbitals are complex functions containing the spherical harmonics (Y_{2 pm 1} ) and (Y_{2 pm 2}) pictured in Figure 6-4. We can again construct real functions from linear combinations, the result being four geometrically equivalent 'four-leaf clover' functions with two perpendicular planar nodes. These orbitals are designated (d_{x^{2} - y^{2}}, : d_{xy}, : d_{zx}) and (d_{yz}). Two of them are shown in Figure 9. The (d_{z^{2}}) orbital has a different shape. However, it can be expressed in terms of two non-standard d-orbitals, (d_{z^{2} - x^{2}}) and (d_{y^{2} - z^{2}}). The latter functions, along with (d_{x^{2} - y^{2}}) add to zero and thus constitute a linearly dependent set. Two combinations of these three functions can be chosen as independent eigenfunctions.
Summary
The atomic orbitals listed in Table 1 are illustrated in Figure (PageIndex{20}). Blue and red indicate, respectively, positive and negative regions of the wavefunctions (the radial nodes of the 2s and 3s orbitals are obscured). These pictures are intended as stylized representations of atomic orbitals and should not be interpreted as quantitatively accurate.
The electron charge distribution in an orbital (psi_{nell m}(mathbf{r})) is given by
[rho(mathbf{r}) = lvert psi_{nell m}(mathbf{r}) rvert ^{2} label{54} ]
which for the s-orbitals is a function of (r) alone. The radial distribution function can be defined, even for orbitals containing angular dependence, by
[D_{nell}(r) = r^{2} [R_{nell}(r)]^{2} label{55}]
Hydrogen Atom Energy Level
This represents the electron density in a shell of radius (r), including all values of the angular variables (theta), (phi). Figure (PageIndex{11}) shows plots of the RDF for the first few hydrogen orbitals.
Contributors and Attributions
Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor)
- Integrated by Daniel SantaLucia (Chemistry student at Hope College, Holland MI)
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