11.10: The Schrödinger Wave Equation for the Hydrogen Atom

Introduction (Schrödinger equation for 1 e- system)

The hydrogen atom, consisting of an electron and a proton, is a two-particle system, and the internal motion of two particles around their center of mass is equivalent to the motion of a single particle with a reduced mass (\(m=\frac{m_1 m_2}{m_1 + m_2}\)). This reduced particle is located at \(\vec{r}\), where \(\vec{r}\) is the vector specifying the position of the electron relative to the position of the proton. The length \(r\) of \(\vec{r}\) is the distance between the proton and the electron, and the direction of \(r\) is given by the orientation of the vector pointing from the proton to the electron. Since the proton is much more massive than the electron, we will assume that the reduced mass equals the electron mass and the proton is located at the center of mass.

Figure \(\PageIndex{1}\): a) The proton (\(p^+\)) and electron (\(e^-\)) of the hydrogen atom. b) Equivalent reduced particle with reduced mass μ at distance r from center of mass.

Starting with the general Schrödinger equation:

\[\dfrac{-\hbar^2}{2m}\nabla^2\Psi(\vec{r}) +V(\vec{r})\Psi(\vec{r}) = E\Psi(\vec{r}) \label{genSchr}\]

we need an expression for the potential. This is just Coulomb attraction and depends only on the separation distance r:

\[ V(r) = - \frac {e^2}{4 \pi \epsilon _0 r } \label {Pot}\]

where \(r\) is the distance between the electron and the proton. The Coulomb potential energy depends inversely on the distance between the electron and the nucleus and does not depend on any angles. Such a potential is called a central potential. Substituting in to equation \(\ref{genSchr}\) this gives us:

\[\frac{-\hbar^2}{2m}\nabla^2\Psi(\vec{r}) + \frac {e^2}{4 \pi \epsilon _0 r }\Psi(\vec{r}) = E\Psi(\vec{r}) \label{1eSchr}\]

Because of the spherical symmetry of the equation it turns out to be better to switch from Cartesian coordinates \(x, y, z\) to spherical coordinates in terms of a radius \(r\), as well as angles \(\phi\), which is measured from the positive x axis in the xy plane and may be between 0 and \(2\pi\), and \(\theta\), which is measured from the positive z axis towards the xy plane and may be between 0 and \(\pi\).

Figure \(\PageIndex{2}\): Spherical Coordinates

This generates a much more complicated, but easier to solve equation:

\[ \left \{ -\dfrac {\hbar ^2}{2 \mu r^2} \left [ \dfrac {\partial}{\partial r} \left (r^2 \dfrac {\partial}{\partial r} \right ) + \dfrac {1}{\sin \theta } \dfrac {\partial}{\partial \theta } \left ( \sin \theta \dfrac {\partial}{\partial \theta} \right ) + \dfrac {1}{\sin ^2 \theta} \dfrac {\partial ^2}{\partial \varphi ^2} \right ] - \dfrac {e^2}{4 \pi \epsilon _0 r } \right \} \psi (r , \theta , \varphi ) = E \psi (r , \theta , \varphi ) \label {6.1.4}\]

The second two terms in the square brackets of this equation only depend on the angles. This lead people to try separating the equation into two parts, one that depends on the angles (\(\theta\) and \(\varphi\)) and one that depends on the distance (r). This does work, just as the wavefunction for a particle in a multidimensional box can be represented as the product of wavefunctions that depend on each coordinate separately. 

Solutions

The solutions to this equation take the form of products of two functions as shown in equation \(\ref{6.1.8}\).

\[ \psi_{nlm_l} (r , \theta , \varphi ) = R_n (r) Y_{lm} (\theta , \varphi ) \label {6.1.8}\]

The \(Y_{lm_l} (\theta , \varphi )\) are a the Spherical Harmonic functions (Table M4), which were well known before quantum mechanics was developed. The Spherical Harmonic functions provide information about the angular probability distribution of the electron around the proton, and the radial function \(R_n(r)\) describes how far the electron is away from the proton.

The state of the electron is specified by the three quantum numbes \(n\), \(l\), and \(m_l\). For this case of a single electron atom the energy \(E_n\) depends only on n, although the other quantum numbers are required for the full description of the wavefunction:

\[ E_n = -\dfrac {m_e e^4}{8\epsilon_0^2 h^2 n^2}\label{6}\]

with \(n=1,2,3 ...\infty\).

It is found that the possible values of \(l\) are constrained by \(n\) and the possible values of \(m_l\) are constrained by \(l\). We will see when we consider multi-electron atoms that these constraints explain the features of the Periodic Table. In other words, the Periodic Table is a manifestation of the Schrödinger model and the physical constraints imposed to obtain the solutions to the Schrödinger equation for the hydrogen atom.

The Three Quantum Numbers

Schrödinger’s approach requires three quantum numbers (\(n\), \(l\), and \(m_l\)) to specify a wavefunction for the electron. The quantum numbers provide information about the spatial distribution of an electron. Although \(n\) can be any positive integer (NOT zero), only certain values of \(l\) and \(m_l\) are allowed for a given value of \(n\).

The principal quantum number (n): This quantum number indicates the energy of the electron (equation \(\ref{6}\)) and the average distance of an electron from the nucleus. \(n\) is restricted to the positive integers:

\[ n = 1,\;2,\;3,\;4,\;.\;.\;.\; \label{allowed_n} \]

As \(n\) increases for a given atom, so does the average distance of an electron from the nucleus. A negatively charged electron that is, on average, closer to the positively charged nucleus is attracted to the nucleus more strongly than an electron that is farther out in space. This means that electrons with higher values of \(n\) are easier to remove from an atom. All wave functions that have the same value of \(n\) are said to constitute a principal shell, because those electrons have similar average distances from the nucleus. The principal quantum number \(n\) corresponds to the \(n\) used by Bohr to describe electron orbits and by Rydberg to describe atomic energy levels.

The Azimuthal Quantum Number (l): This quantum number indicates the shape of the region of space occupied by an electron and the electrons total orbital angular momentum. The allowed values of \(l\) are integers that depend on the value of \(n\) and can range from 0 to n − 1:

\[ l = 0,\;1,.,2,\;3,\;.\;.\;\left (n-1 \right ) \label{allow_l} \]

For example, if \(n\) = 1, \(l\) can be only 0; if \(n\) = 2, \(l\) can be 0 or 1; and so forth. For a given atom, all wave functions that have the same values of both \(n\) and \(l\) form a subshell. The regions of space occupied by electrons in the same subshell usually have the same shape, but they are oriented differently in space.

The total angular momentum of the electron has the value:

\[J = \sqrt{l(l+1)}\hbar\label{Jtot}\]

Thus the angular momentum of the electron is quantized.

The Magnetic Quantum Number (\(m_l\)): This quantum number indicates the orientation of the region of space occupied by an electron with respect to an applied magnetic field. The allowed integer values of \(m_l\) depend on the value of \(l\) and can range from \(−l\) to \(l\):

\[ m = -l,\;-l+1,\,.\;.\;.0,\;\,.\;.\;.l-1,l \label{2.2.4} \]

For example, if \(l = 0\), \(m_l\) can be only 0; if \(l\) = 1, \(m_l\) can be −1, 0, or +1; and if \(l\) = 2, \(m_l\) can be −2, −1, 0, +1, or +2.

Each wave function with an allowed combination of \(n\), \(l\), and \(m_l\) values describes an atomic orbital, a particular spatial distribution for an electron. For a given set of quantum numbers, each principal shell has a fixed number of subshells, and each subshell has a fixed number of orbitals.

Because the electron is charged and has an angular momentum it can interact with a magnetic field. If the direction of the magnetic field is taken to be along the z-axis then the component (projection) of the angular momentum along the z-axis is:

\[J_z = m_l \hbar\label{Jz}\]

This is the same quantization of \(J_z\) as seen for the particle on a ring.

Example \(\PageIndex{1}\): Number of subshells

How many subshells and orbitals are contained within the principal shell with n = 4?

Given: value of n

Asked for: number of subshells and orbitals in the principal shell

Strategy:

A. Given n = 4, calculate the allowed values of l. From these allowed values, count the number of subshells.

B. For each allowed value of l, calculate the allowed values of ml. The sum of the number of orbitals in each subshell is the number of orbitals in the principal shell.

Solution

A We know that l can have all integral values from 0 to n − 1. If n = 4, then l can equal 0, 1, 2, or 3. Because the shell has four values of l, it has four subshells, each of which will contain a different number of orbitals, depending on the allowed values of ml.

B For l = 0, ml can be only 0, and thus the l = 0 subshell has only one orbital. For l = 1, ml can be 0 or ±1; thus the l = 1 subshell has three orbitals. For l = 2, ml can be 0, ±1, or ±2, so there are five orbitals in the l = 2 subshell. The last allowed value of l is l = 3, for which ml can be 0, ±1, ±2, or ±3, resulting in seven orbitals in the l = 3 subshell. The total number of orbitals in the n = 4 principal shell is the sum of the number of orbitals in each subshell and is equal to n2:

\( \mathop 1\limits_{(l = 0)} + \mathop 3\limits_{(l = 1)} + \mathop 5\limits_{(l = 2)} + \mathop 7\limits_{(l = 3)} = 16\; {\rm{orbitals}} = {(4\; {\rm{principal\: shells}})^2} \)​

Exercise \(\PageIndex{1}\)

How many subshells and orbitals are in the principal shell with n = 3?

Answer: three subshells; nine orbitals

Rather than specifying all the values of n and l every time we refer to a subshell or an orbital, chemists use an abbreviated system with lowercase letters to denote the value of l for a particular subshell or orbital:

\(l\)	0	1	2	3
Designation	s	p	d	f

The principal quantum number is named first, followed by the letter s, p, d, or f as appropriate. These orbital designations are derived from corresponding spectroscopic characteristics of lines involving them: sharp, principle, diffuse, and fundamental. A 1s orbital has n = 1 and l = 0; a 2p subshell has n = 2 and l = 1 (and has three 2p orbitals, corresponding to m_l = −1, 0, and +1); a 3d subshell has n = 3 and l = 2 (and has five 3d orbitals, corresponding to m_l = −2, −1, 0, +1, and +2); and so forth.

We can summarize the relationships between the quantum numbers and the number of subshells and orbitals as follows (Table \(\PageIndex{1}\)):

• Each principal shell has n subshells. For n = 1, only a single subshell is possible (1s); for n = 2, there are two subshells (2s and 2p); for n = 3, there are three subshells (3s, 3p, and 3d); and so forth. Every shell has an ns subshell, any shell with n ≥ 2 also has an np subshell, and any shell with n ≥ 3 also has an nd subshell. Because a 2d subshell would require both n = 2 and l = 2, which is not an allowed value of l for n = 2, a 2d subshell does not exist.
• Each subshell has 2l + 1 orbitals. This means that all ns subshells contain a single s orbital, all np subshells contain three p orbitals, all nd subshells contain five d orbitals, and all nf subshells contain seven f orbitals.

Note

Each principal shell has n subshells, and each subshell has 2l + 1 orbitals.

Table \(\PageIndex{1}\): Allowed values of \(n\), \(l\), and \(m_l\) through n = 4

\(n\)	\(l\)	Subshell Designation	\(m_l\)	Number of Orbitals in Subshell	Number of Orbitals in Shell
1	0	1s	0	1	1
2	0	2s	0	1	4
	1	2p	−1, 0, 1	3
3	0	3s	0	1	9
	1	3p	−1, 0, 1	3
	2	3d	−2, −1, 0, 1, 2	5
4	0	4s	0	1	16
	1	4p	−1, 0, 1	3
	2	4d	−2, −1, 0, 1, 2	5
	3	4f	−3, −2, −1, 0, 1, 2, 3	7

The Radial Component

The first six radial functions are provided in Table \(\PageIndex{2}\). Note that the functions in the table exhibit a dependence on \(Z\), the atomic number of the nucleus. Other one electron systems have electronic states analogous to those for the hydrogen atom, and inclusion of the charge on the nucleus allows the same wavefunctions to be used for all one-electron systems. For hydrogen, \(Z = 1\) and for helium, \(Z=2\).

Table \(\PageIndex{2}\): Hydrogen-like atomic wavefunctions for \(n\) values \(1,2,3\): \(Z\) is the atomic number of the nucleus, and \(\rho = \dfrac {Zr}{a_0}\), where \(a_0\) is the Bohr radius and r is the radial variable.

\(n\)	\(l\)	\(m_l\)	Radial Component
1	0	0	\(\psi_{100} = \dfrac {1}{\sqrt {\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} e^{-\rho}\)
2	0	0	\(\psi_{200} = \dfrac {1}{\sqrt {32\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} (2-\rho)e^{\dfrac {-\rho}{2}}\)
2	1	0	\(\psi_{210} = \dfrac {1}{\sqrt {32\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} \rho e^{-\rho/2} \cos(\theta)\)
2	1	\(\pm 1\)	\(\psi_{21\pm\ 1} = \dfrac {1}{\sqrt {64\pi}} \left(\dfrac {Z}{a_0}\right)^{\dfrac {3}{2}} \rho e^{-\rho/2} \sin(\theta) e^{\pm\ i\phi}\)
3	0	0	\(\psi_{300} = \dfrac {1}{81\sqrt {3\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} (27-18\rho +2\rho^2)e^{-\rho/3}\)
3	1	0	\(\psi_{310} = \dfrac{1}{81} \sqrt {\dfrac {2}{\pi}} \left(\dfrac {Z}{a_0}\right)^{\dfrac {3}{2}} (6r - \rho^2)e^{-\rho/3} \cos(\theta)\)
3	1	\(\pm 1\)	\(\psi_{31\pm\ 1} = \dfrac {1}{81\sqrt {\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} (6\rho - \rho^2)e^{-r/3} \sin(\theta)e^{\pm\ i \phi}\)
3	2	0	\(\psi_{320} = \dfrac {1}{81\sqrt {6\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} \rho^2 e^{-\rho/3}(3cos^2(\theta) -1)\)
3	2	\(\pm 1\)	\(\psi_{32\pm\ 1} = \dfrac {1}{81\sqrt {\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} \rho^2 e^{-\rho/3} \sin(\theta)\cos(\theta)e^{\pm\ i \phi}\)
3	2	\(\pm 2\)	\(\psi_{32\pm\ 2} = \dfrac {1}{162\sqrt {\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} \rho^2 e^{-\rho/3}{\sin}^2(\theta)e^{\pm\ 2i\phi}\)

Visualizing the variation of an electronic wavefunction with r, \(\theta\), and \(\phi\) is important because the absolute square of the wavefunction depicts the charge distribution (electron probability density) in an atom or molecule. The charge distribution is central to chemistry because it is related to chemical reactivity. For example, an electron deficient part of one molecule is attracted to an electron rich region of another molecule, and such interactions play a major role in chemical interactions ranging from substitution and addition reactions to protein folding and the interaction of substrates with enzymes.

Plots of the radial functions, R(r), for the 1s, 2s, and 2p orbitals plotted in figure \(\PageIndex{2}\) (left). The quantity \(R(r)^* R(r)\) gives the radial probability density; i.e., the probability density for the electron to be at a point located the distance r from the proton. Radial probability densities for three types of atomic orbitals are plotted in figure \(\PageIndex{2}\) (middle). The probability of finding an electron at a particular distance from the nucleus is the integral of \(R(r)^* R(r)\) over the surface of the sphere of radius r, \(P(r) = 4\pi r^2 R(r)^* R(r)\), which is plotted in figure \(\PageIndex{2}\) (right).

Figure \(\PageIndex{2}\): (left) Radial function, R(r), for the 1s, 2s, and 2p orbitals. (middle) Radial probability densities for the 1s, 2s, and 2p orbitals.(right) Radial distribution functions for the 1s, 2s, and 2p orbitals. Units on the x-axis are Bohrs = 0.529 Å = a_o.

For the hydrogen atom, the peak in the radial probability plot (right colum in figure \(\PageIndex{2}\)) occurs at r = 0.529 Å (52.9 pm), which is exactly the radius calculated by Bohr for the n = 1 orbit. Thus the most probable radius obtained from quantum mechanics is identical to the radius calculated by classical mechanics. In Bohr’s model, however, the electron was assumed to be at this distance 100% of the time, whereas in the Schrödinger model, it is at this distance only some of the time. The difference between the two models is attributable to the wavelike behavior of the electron and the Heisenberg uncertainty principle.

Notice that because the surface area of a sphere with radius of zero is zero, there is no probability of finding the electron at the nucleus (r = 0). See the right hand column in figure \(\PageIndex{2}\). This quantum prediction can be thought of as an explanation for why atoms do not collapse into a single uncharged particle.

Figure \(\PageIndex{3}\) compares the electron probability densities for the hydrogen 1s, 2s, and 3s orbitals in 3-D. Note that all three are spherically symmetrical. For the 2s and 3s orbitals, however (and for all other s orbitals as well), the electron probability density does not fall off smoothly with increasing r. Instead, a series of minima and maxima are observed in the radial probability plots (part (c) in Figure \(\PageIndex{3}\)). The minima correspond to spherical nodes (regions of zero electron probability), which alternate with spherical regions of nonzero electron probability.

Figure \(\PageIndex{3}\): Probability Densities for the 1 s, 2 s, and 3s Orbitals of the Hydrogen Atom. (a) The electron probability density in any plane that contains the nucleus is shown. Note the presence of circular regions, or nodes, where the probability density is zero. (b) Contour surfaces enclose 90% of the electron probability, which illustrates the different sizes of the 1s, 2s, and 3s orbitals. The cutaway drawings give partial views of the internal spherical nodes. The orange color corresponds to regions of space where the phase of the wave function is positive, and the blue color corresponds to regions of space where the phase of the wave function is negative. (c) In these plots of electron probability as a function of distance from the nucleus (r) in all directions (radial probability), the most probable radius increases asn increases, but the 2s and 3s orbitals have regions of significant electron probability at small values of r.

The Angular Component

The angular component of the wavefunction \(Y_{lm_l}(\theta,\phi)\) in Equation \(\ref{6.1.8}\) does much to give an orbital its distinctive shape. \(Y\) is typically normalized so the the integral of \(Y^2\) over the unit sphere is equal to one. In this case, \(Y^2\) serves as a probability function. The probability function can be interpreted as the probability that the electron will be found on the ray emitting from the origin that is at angles \((\theta,\phi)\) from the axes. The probability function can also be interpreted as the probability distribution of the electron being at position \((\theta,\phi)\) on a sphere of radius r, given that it is r distance from the nucleus. \(Y_{l,m_l}(\theta,\phi)\) are also the wavefunction solutions to Schrödinger’s equation for a rigid rotor consisting of rotating bodies, for example a diatomic molecule. These are called Spherical Harmonic functions (Table M4).

s Orbitals (l=0)

Three things happen to s orbitals as n increases (figure \(\PageIndex{3}\)):

1. They become larger, extending farther from the nucleus.
2. They contain more nodes. This is similar to a standing wave that has regions of significant amplitude separated by nodes, points with zero amplitude.
3. For a given atom, the s orbitals also become higher in energy as n increases because of their increased distance from the nucleus.

Orbitals are generally drawn as three-dimensional surfaces that enclose 90% of the electron density. Although such drawings show the relative sizes of the orbitals, they do not normally show the spherical nodes in the 2s and 3s orbitals because the spherical nodes lie inside the 90% surface. Fortunately, the positions of the spherical nodes are not important for chemical bonding.

p Orbitals (l=1)

Only s orbitals are spherically symmetrical. As the value of l increases, the number of orbitals in a given subshell increases, and the shapes of the orbitals become more complex. Because the 2p subshell has l = 1, with three values of m_l (−1, 0, and +1), there are three 2p orbitals).

Figure \(\PageIndex{4}\): Illustration of a slice through a p_z orbital, where the dot density indicates the relative probability of finding the electron. The colors correspond to regions of space where the phase of the wave function is positive (orange) and negative (blue).

The electron probability distribution for one of the hydrogen 2p orbitals is shown in Figure \(\PageIndex{4}\). Because this orbital has two lobes of electron density arranged along the z axis, with an electron density of zero in the xy plane (i.e., the xy plane is a nodal plane), it is a 2p_z orbital. As shown in Figure \(\PageIndex{5}\), the other two 2p orbitals have identical shapes, but they lie along the x axis (2p_x) and y axis (2p_y), respectively. Note that each p orbital has just one nodal plane. In each case, the phase of the wave function for each of the 2p orbitals is positive for the lobe that points along the positive axis and negative for the lobe that points along the negative axis. It is important to emphasize that these signs correspond to the phase of the wave that describes the electron motion, not to positive or negative charges.

Figure \(\PageIndex{5}\): The Three Equivalent 2 p Orbitals of the Hydrogen Atom. The surfaces shown enclose 90% of the total electron probability for the 2p_x, 2p_y, and 2p_z orbitals. Each orbital is oriented along the axis indicated by the subscript and a nodal plane that is perpendicular to that axis bisects each 2p orbital. The phase of the wave function is positive (orange) in the region of space where x, y, or z is positive and negative (blue) where x, y, or z is negative.

Just as with the s orbitals, the size and complexity of the p orbitals for any atom increase as the principal quantum number n increases. The shapes of the 90% probability surfaces of the 3p, 4p, and higher-energy p orbitals are, however, essentially the same as those shown in figure \(\PageIndex{5}\), but with additional radial nodes.

Note: Orbitals specified by m_l are not the same as those usually labeled p_x, p_y or p_z

The p_x, p_y and p_z orbitals are not the solutions to the Schrödinger equation associated with m_l = -1, 0 and 1.  They are valid wavefunctions built from linear combinations of the m_l = -1, 0, 1 wavefunctions in order to get real valued functions. See the table of Spherical Harmonics and Real Valued Spherical Harmonics. 

 

d Orbitals (l=2)

Subshells with l = 2 have five d orbitals; the first principal shell to have a d subshell corresponds to n = 3. The five d orbitals have m_l values of −2, −1, 0, +1, and +2. The surfaces enclosing the 90% probability of finding the electron for the five real-valued linear combinations of the complex orbitals specified by m_l are shown in figure \(\PageIndex{6}\).

Figure \(\PageIndex{6}\): The five equivalent 3d orbitals of the hydrogen atom. The surfaces shown enclose 90% of the total electron probability for the five hydrogen 3d orbitals. Four of the five 3d orbitals consist of four lobes arranged in a plane that is intersected by two perpendicular nodal planes. These four orbitals have the same shape but different orientations. The fifth 3d orbital, \(3d_{z^2}\), has a distinct shape even though it is mathematically equivalent to the others. The phase of the wave function for the different lobes is indicated by color: orange for positive and blue for negative.

The hydrogen 3d orbitals have more complex shapes than the 2p orbitals. All five 3d orbitals contain two nodal surfaces, as compared to one for each p orbital and zero for each s orbital. In three of the d orbitals, the lobes of electron density are oriented between the x and y, x and z, and y and z planes; these orbitals are referred to as the \(3d_{xy}\), \(3d_{xz}\), and \(3d_{yz}\) orbitals, respectively. A fourth d orbital has lobes lying along the x and y axes; this is the \(3d_{x_2−y_2}\) orbital. The fifth 3d orbital, called the \(3d_{z^2}\) orbital, has a unique shape: it looks like a \(2p_z\) orbital combined with an additional doughnut of electron probability lying in the xy plane. Despite its peculiar shape, the \(3d_{z^2}\) orbital is mathematically equivalent to the other four and has the same energy. In contrast to p orbitals, the phase of the wave function for d orbitals is the same for opposite pairs of lobes. As shown in Figure \(\PageIndex{6}\), the phase of the wave function is positive for the two lobes of the \(dz^2\) orbital that lie along the z axis, whereas the phase of the wave function is negative for the doughnut of electron density in the xy plane. Like the s and p orbitals, as n increases, the size of the d orbitals increases, but the overall shapes remain similar to those depicted in figure \(\PageIndex{6}\).

f Orbitals (l=3)

Principal shells with n = 4 can have subshells with l = 3 and m_l values of −3, −2, −1, 0, +1, +2, and +3. These subshells consist of seven f orbitals. Each f orbital has three nodal surfaces, so their shapes are complex (not shown).

Energies

As noted previously, the energy for a single electron atom depend only on \(n\), the principle quantum number. The solutions reproduce the same quantized expression for hydrogen atom energy levels that was obtained from the Bohr model of the hydrogen atom.

\[E=−\frac{Z^2}{n^2}Rhc \label{6.6.1}\]

or

\[ E_n = - \frac {Z^2 \mu e^4}{8 \epsilon ^2_0 h^2 n^2} \]

The relative energies of the atomic orbitals with n ≤ 4 for a hydrogen atom are plotted in Figure \(\PageIndex{7}\) ; note that the orbital energies depend on only the principal quantum number n and are all less than zero, indicating attraction to the nucleus. Consequently, the energies of the 2s and 2p orbitals of hydrogen are the same; the energies of the 3s, 3p, and 3d orbitals are the same; and so forth. The orbital energies obtained for hydrogen using quantum mechanics are exactly the same as the allowed energies calculated by Bohr. In contrast to Bohr’s model, however, which allowed only one orbit for each energy level, quantum mechanics predicts that there are 4 orbitals with different electron density distributions in the n = 2 principal shell (one 2s and three 2p orbitals), 9 in the n = 3 principal shell, and 16 in the n = 4 principal shell.

For a single electron atom, the energy of that electron is a function of only the principal quantum number (Equation \(\ref{6.6.1}\)).

The different values of l and m_l for the individual orbitals within a given principal shell are not important for understanding the emission or absorption spectra of the hydrogen atom under most conditions, but they do explain the splittings of the main lines that are observed when hydrogen atoms are placed in a magnetic field. As we have just seen, however, quantum mechanics also predicts that in the hydrogen atom, all orbitals with the same value of n (e.g., the three 2p orbitals) are degenerate, meaning that they have the same energy. Figure \(\PageIndex{7}\) shows that the energy levels become closer and closer together as the value of n increases, as expected because of the 1/n² dependence of orbital energies.

Figure \(\PageIndex{7}\): Orbital energy level diagram for the hydrogen atom. Each box corresponds to one orbital. Note that the difference in energy between orbitals decreases rapidly with increasing values of n.

In general, both energy and radius decrease as the nuclear charge increases. Thus the most stable orbitals (those with the lowest energy) are those closest to the nucleus. For example, in the ground state of the hydrogen atom, the single electron is in the 1s orbital, whereas in the first excited state, the atom has absorbed energy and the electron has been promoted to one of the n = 2 orbitals. In ions with only a single electron, the energy of a given orbital depends on only n, and all subshells within a principal shell, such as the p_x, p_y, and p_z orbitals, are degenerate.

Note: Bohr vs. Schrödinger

It is interesting to compare the results obtained by solving the Schrödinger equation with Bohr’s model of the hydrogen atom. There are several ways in which the Schrödinger model and Bohr model differ.

1. First, and perhaps most strikingly, the Schrödinger model does not produce well-defined orbits for the electron. The wavefunctions only give us the probability for the electron to be at various directions and distances from the proton.
2. Second, the quantization of angular momentum is different from that proposed by Bohr. Bohr proposed that the angular momentum is quantized in integer units of \(\hbar\), while the Schrödinger model leads to an angular momentum of \((l (l +1) \hbar ^2)^{\frac {1}{2}}\).
3. Third, the quantum numbers appear naturally during solution of the Schrödinger equation while Bohr had to postulate the existence of quantized energy states. Although more complex, the Schrödinger model leads to a better correspondence between theory and experiment over a range of applications that was not possible for the Bohr model.

Electron Spin: The Fourth Quantum Number

The quantum numbers \(n, \ l, \ m_l\) are not sufficient to fully characterize the physical state of the electrons in an atom. In 1926, Otto Stern and Walther Gerlach carried out an experiment that could not be explained in terms of the three quantum numbers \(n, \ l, \ m_l\) and showed that there is, in fact, another quantum-mechanical degree of freedom that needs to be included in the theory. The experiment is illustrated in the Figure \(\PageIndex{8}\). A beam of atoms (e.g. hydrogen or silver atoms) is sent through a spatially inhomogeneous magnetic field with a definite field gradient toward one of the poles. It is observed that the beam splits into two beams as it passes through the field region.

Figure \(\PageIndex{8}\): The Stern-Gerlach apparatus. from Wikipedia

The fact that the beam splits into 2 beams suggests that the electrons in the atoms have a degree of freedom capable of coupling to the magnetic field. That is, an electron has an intrinsic magnetic moment \(M\) arising from a degree of freedom that has no classical analog. The magnetic moment must take on only 2 values according to the Stern-Gerlach experiment. The intrinsic property that gives rise to the magnetic moment is called spin, \(S\); unlike position and momentum, which have clear classical analogs, spin does not. The implication of the Stern-Gerlach experiment is that we need to include a fourth quantum number, \(m_s\) in our description of the physical state of the electron. That is, in addition to give its principle, angular, and magnetic quantum numbers, we also need to say if it is a spin-up electron or a spin-down electron.

Note: Spin is a Quantum Phenomenon

Unlike position and momentum, which have clear classical analogs, spin does not. Spin may be thought of as a relativistic phenomenon. Solutions to the relativistic form of the wave equation for an electron (Dirac equation) do require the inclusion of a spin quantum number.¹ This is related to the relativistic effect that leads to the generation of a magnetic field by moving charged particles.² For most situations it is not necessary to use the relativistic formulation and spin is just added as an extra quantum number with two values.

George Uhlenbeck (1900–1988) and Samuel Goudsmit (1902–1978), proposed that the splittings were caused by an electron spinning about its axis, much as Earth spins about its axis. When an electrically charged object spins, it produces a magnetic moment parallel to the axis of rotation, making it behave like a magnet. Electrons are not spinning as they are a wave, but they do have a magnetic moment. The name electron spin stuck, despite not being particularly accurate.

The magnetic moment has two possible orientations, described by the fourth quantum number, m_s = +½ (up) and m_s = −½ (down). The subscript s is for spin. An electron behaves like a magnet that has one of two possible orientations, aligned either with the magnetic field or against it. The implications of electron spin for chemistry were recognized almost immediately by an Austrian physicist, Wolfgang Pauli (1900–1958; Nobel Prize in Physics, 1945), who determined that each orbital can contain no more than two electrons. This is usually referred to as  the Pauli exclusion principle.

Pauli Exclusion Principle

No two electrons in an atom can have the same values of all four quantum numbers (\(n\), \(l\), \(m_l\), \(m_s\)).

By giving the values of n, l, and m_l, we specify a particular orbital (e.g., 1s with n = 1, l = 0, m_l = 0). Because m_s has only two possible values (+½ or −½), two electrons, and only two electrons, can occupy any given orbital, one with spin up and one with spin down. With this information, we can proceed to construct the entire periodic table, which was originally based on the physical and chemical properties of the known elements.

Example \(\PageIndex{2}\)

List all the allowed combinations of the four quantum numbers (n, l, m_l, m_s) for electrons in a 2p orbital and predict the maximum number of electrons the 2p subshell can accommodate.

Given: orbital

Asked for: allowed quantum numbers and maximum number of electrons in orbital

Strategy:

1. List the quantum numbers (n, l, m_l) that correspond to an n = 2p orbital. List all allowed combinations of (n, l, m_l).
2. Build on these combinations to list all the allowed combinations of (n, l, m_l, m_s).
3. Add together the number of combinations to predict the maximum number of electrons the 2p subshell can accommodate.

Solution:

A For a 2p orbital, we know that n = 2, l = n − 1 = 1, and m_l = −l, (−l +1),…, (l − 1), l. There are only three possible combinations of (n, l, m_l): (2, 1, 1), (2, 1, 0), and (2, 1, −1).

B Because m_s is independent of the other quantum numbers and can have values of only +½ and −½, there are six possible combinations of (n, l, m_l, m_s): (2, 1, 1, +½), (2, 1, 1, −½), (2, 1, 0, +½), (2, 1, 0, −½), (2, 1, −1, +½), and (2, 1, −1, −½).

C Hence the 2p subshell, which consists of three 2p orbitals (2p_x, 2p_y, and 2p_z), can contain a total of six electrons, two in each orbital.

Exercise \(\PageIndex{2}\)

List all the allowed combinations of the four quantum numbers (n, l, m_l, m_s) for a 6s orbital, and predict the total number of electrons it can contain.

Answer: (6, 0, 0, +½), (6, 0, 0, −½); two electrons

References

1. Leonard I. Schiff, Quantum Mechanics, 3rd Ed. (McGraw-Hill, New York, 1968) Pp. 480 - 481.

2. Paul Lorain and Dale Corson, Electromagnetic Fields and Waves, 2nd Ed. (W. H. Freeman and Co. New York, 1970) Chpt 6.

Contributors and Attributions

• David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")

• J. Gutow (UW Oshkosh)