# 3.2: Quantum Numbers for Atomic Orbitals

- Page ID
- 338950

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives

- To apply the results of quantum mechanics to chemistry.

Although quantum mechanics uses sophisticated mathematics, you do not need to understand the mathematical details to follow our discussion of its general conclusions. We focus on the properties of the *wavefunctions* that are the solutions of Schrödinger’s equations.

**Describing the electron distribution as a standing wave leads to sets of***quantum numbers*that are characteristic of each wavefunction.**Each wavefunction is associated with a particular energy.**The energy of an electron in an atom is quantized; it can have only certain allowed values.

## Quantum Numbers

Schrödinger’s approach uses three quantum numbers (*n*, *l*, and *m*_{l}) to specify any wavefunction. The quantum numbers provide information about the spatial distribution of an electron. Although *n* can be any positive integer, only certain values of *l* and *m*_{l} are allowed for a given value of *n*.

### The Principal Quantum Number

The **principal quantum number** (n) tells the average relative distance of an electron from the nucleus:

\[n = 1, 2, 3, 4,… \label{6.5.1}\]

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 wavefunctions that have the same value of *n* are said to constitute a principal shell because those electrons have similar average distances from the nucleus. As you will see, 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

The second quantum number is often called the **azimuthal quantum number (l)**. The value of *l* describes the *shape* of the region of space occupied by the electron. The allowed values of *l* depend on the value of *n* and can range from 0 to *n* − 1:

\[l = 0, 1, 2,…, n − 1 \label{6.5.2}\]

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 wavefunctions 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 Magnetic Quantum Number

The third quantum number is the magnetic quantum number (\(m_l\)). The value of \(m_l\) describes the *orientation* of the region in space occupied by an electron with respect to an applied magnetic field. The allowed values of \(m_l\) depend on the value of *l*: *m*_{l} can range from −*l* to *l* in integral steps:

\[m_l = −l, −l + 1,…, 0,…, l − 1, l \label{6.5.3}\]

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 wavefunction 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.

Example\(\PageIndex{1}\): n=4 Shell Structure

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:**

- Given
*n*= 4, calculate the allowed values of*l*. From these allowed values, count the number of subshells. - For each allowed value of
*l*, calculate the allowed values of*m*_{l}. 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 *m*_{l}.

**B** For *l* = 0, *m*_{l} can be only 0, and thus the *l* = 0 subshell has only one orbital. For *l* = 1, *m*_{l} can be 0 or ±1; thus the *l* = 1 subshell has three orbitals. For *l* = 2, *m*_{l} 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 *m*_{l} 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 *n*^{2} = 16

Exercise \(\PageIndex{1}\): n=3 Shell Structure

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 historical terms for corresponding spectroscopic characteristics: *s*harp, *p*rinciple, *d*iffuse, and *f*undamental.) A 1*s* orbital has *n* = 1 and *l* = 0; a 2*p* subshell has *n* = 2 and *l* = 1 (and has three 2*p* orbitals, corresponding to *m*_{l} = −1, 0, and +1); a 3*d* subshell has *n* = 3 and *l* = 2 (and has five 3*d* 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 6.5.1):

- Each principal shell has
*n*subshells. For*n*= 1, only a single subshell is possible (1*s*); for*n*= 2, there are two subshells (2*s*and 2*p*); for*n*= 3, there are three subshells (3*s*, 3*p*, and 3*d*); 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 2*d*subshell would require both*n*= 2 and*l*= 2, which is not an allowed value of*l*for*n*= 2, a 2*d*subshell does not exist. - Each subshell has 2
*l*+ 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.

Each principal shell has

nsubshells, and each subshell has 2l+ 1 orbitals.

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 |

## Summary

Wavefunctions have important properties:

- the wavefunction uses three variables (Cartesian axes
*x*,*y*, and*z,*or \(r, \theta,\phi\)) to describe the position of an electron; - describing electron distributions as standing waves leads naturally to the existence of sets of
**quantum numbers**characteristic of each wavefunction; and - each spatial distribution of the electron described by a wavefunction with a given set of quantum numbers has a particular energy.

Quantum numbers provide important information about the energy and spatial distribution of an electron. The **principal quantum number** *n* can be any positive integer; as *n* increases for an atom, the average distance of the electron from the nucleus also increases. All wavefunctions with the same value of *n* constitute a **principal shell** in which the electrons have similar average distances from the nucleus. The **azimuthal quantum number** *l* can have integral values between 0 and *n* − 1; it describes the shape of the electron distribution. wavefunctions that have the same values of both *n* and *l* constitute a **subshell**, corresponding to electron distributions that usually differ in orientation rather than in shape or average distance from the nucleus. The **magnetic quantum number** *m*_{l} can have 2*l* + 1 integral values, ranging from −*l* to +*l*, and describes the orientation of the electron distribution. Each wavefunction with a given set of values of *n*, *l*, and *m*_{l} describes a particular spatial distribution of an electron in an atom, an **atomic orbital**.

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## Contributions and Attributions

Adapted by Valeria D. Kleiman