# 9.5: Quantum-Mechanical Orbitals and Electron Configurations

Learning Objectives

• Represent the organization of electrons by an electron configuration and orbital diagram.

The flight path of a commercial airliner is carefully regulated by the Federal Aviation Administration. Each airplane must maintain a distance of five miles from another plane flying at the same altitude and 2,000 feet above and below another aircraft (1,000 feet if the altitude is less than 29,000 feet). So, each aircraft only has certain positions it is allowed to maintain while it flies. As we explore quantum mechanics, we see that electrons have similar restrictions on their locations.

## Orbitals

We can apply our knowledge of quantum numbers to describe the arrangement of electrons for a given atom. We do this with something called electron configurations. They are effectively a map of the electrons for a given atom. We look at the four quantum numbers for a given electron and then assign that electron to a specific orbital below.

### S Orbitals

For any value of (n), a value of (l=0) places that electron in an (s) orbital. This orbital is spherical in shape: Figure (PageIndex{1}): (s) orbitals have no orientational preference and resemble spheres.

### P Orbitals

For the table below, we see that we can have three possible orbitals when (l=1). These are designated as (p) orbitals and have dumbbell shapes. Each of the (p) orbitals has a different orientation in three-dimensional space. Figure (PageIndex{2}): (p) orbitals have an orientational preference and resemble dumbbells.

### D Orbitals

When (l=2), (m_l) values can be (-2, : -1, : 0, : +1, : +2) for a total of five (d) orbitals. Note that all five of the orbitals have specific three-dimensional orientations. Figure (PageIndex{3}): (d) orbitals have an orientational preference and exhibit complex structures.

### F Orbitals

The most complex set of orbitals are the (f) orbitals. When (l=3), (m_l) values can be (-3, : -2, : -1, : 0, : +1, : +2, : +3) for a total of seven different orbital shapes. Again, note the specific orientations of the different (f) orbitals. Figure (PageIndex{4}): (f) orbitals have an orientational preference and exhibit quite complex structures.

Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number. Some of the allowed combinations of quantum numbers are compared in Table (PageIndex{1}).

Table (PageIndex{1}): Electron Arrangement Within Energy Levels

Principal Quantum Number (left( n ight))Allowable SublevelsNumber of Orbitals per SublevelNumber of Orbitals per Principal Energy LevelNumber of Electrons per SublevelNumber of Electrons per Principal Energy Level
1(s)1122
2(s)1428
(p)36
3(s)19218
(p)36
(d)510
4(s)116232
(p)36
(d)510
(f)714

## Electron Configurations

Can you name one thing that easily distinguishes you from the rest of the world? And we're not talking about DNA—that's a little expensive to sequence. For many people, it is their email address. Your email address allows people all over the world to contact you. It does not belong to anyone else, but serves to identify you. Electrons also have a unique set of identifiers in the quantum numbers that describe their location and spin. Chemists use an electronic configuration to represent the organization of electrons in shells and subshells in an atom. An electron configuration simply lists the shell and subshell labels, with a right superscript giving the number of electrons in that subshell. The shells and subshells are listed in the order of filling. Electrons are typically organized around an atom by starting at the lowest possible quantum numbers first, which are the shells-subshells with lower energies.

For example, an H atom has a single electron in the 1s subshell. Its electron configuration is

[ce{H}:, 1s^1 onumber]

He has two electrons in the 1s subshell. Its electron configuration is

[ce{He}:, 1s^2 onumber]

The three electrons for Li are arranged in the 1s subshell (two electrons) and the 2s subshell (one electron). The electron configuration of Li is

[ce{Li}:, 1s^22s^1 onumber]

Be has four electrons, two in the 1s subshell and two in the 2s subshell. Its electron configuration is

[ce{Be}:, 1s^22s^2 onumber]

Now that the 2s subshell is filled, electrons in larger atoms must go into the 2p subshell, which can hold a maximum of six electrons. The next six elements progressively fill up the 2p subshell:

• B: 1s22s22p1
• C: 1s22s22p2
• N: 1s22s22p3
• O: 1s22s22p4
• F: 1s22s22p5
• Ne: 1s22s22p6

Now that the 2p subshell is filled (all possible subshells in the n = 2 shell), the next electron for the next-larger atom must go into the n = 3 shell, s subshell.

### Second Period Elements

Periods refer to the horizontal rows of the periodic table. Looking at a periodic table you will see that the first period contains only the elements hydrogen and helium. This is because the first principal energy level consists of only the (s) sublevel and so only two electrons are required in order to fill the entire principal energy level. Each time a new principal energy level begins, as with the third element lithium, a new period is started on the periodic table. As one moves across the second period, electrons are successively added. With beryllium (left( Z=4 ight)), the (2s) sublevel is complete and the (2p) sublevel begins with boron (left( Z=5 ight)). Since there are three (2p) orbitals and each orbital holds two electrons, the (2p) sublevel is filled after six elements. Table (PageIndex{1}) shows the electron configurations of the elements in the second period.

Element NameSymbolAtomic NumberElectron Configuration

Table (PageIndex{2}): Electron Configurations of Second-Period Elements

Lithium(ce{Li})3(1s^2 2s^1)
Beryllium(ce{Be})4(1s^2 2s^2)
Boron(ce{B})5(1s^2 2s^2 2p^1)
Carbon(ce{C})6(1s^2 2s^2 2p^2)
Nitrogen(ce{N})7(1s^2 2s^2 2p^3)
Oxygen(ce{O})8(1s^2 2s^2 2p^4)
Fluorine(ce{F})9(1s^2 2s^2 2p^5)
Neon(ce{Ne})10(1s^2 2s^2 2p^6)

### Aufbau Principle

Construction of a building begins at the bottom. The foundation is laid and the building goes up step by step. You obviously cannot start with the roof since there is no place to hang it. The building goes from the lowest level to the highest level in a systematic way. In order to create ground state electron configurations for any element, it is necessary to know the way in which the atomic sublevels are organized in order of increasing energy. Figure (PageIndex{5}) shows the order of increasing energy of the sublevels.

The lowest energy sublevel is always the (1s) sublevel, which consists of one orbital. The single electron of the hydrogen atom will occupy the (1s) orbital when the atom is in its ground state. As we proceed with atoms with multiple electrons, those electrons are added to the next lowest sublevel: (2s), (2p), (3s), and so on. The Aufbau principle states that an electron occupies orbitals in order from lowest energy to highest. The Aufbau (German: "building up, construction") principle is sometimes referred to as the "building up" principle. It is worth noting that in reality atoms are not built by adding protons and electrons one at a time, and that this method is merely an aid for us to understand the end result. Figure (PageIndex{5}): Electrons are added to atomic orbitals in order from low energy (bottom of the graph) to high (top of the graph) according to the Aufbau principle. Principle energy levels are color coded, while sublevels are grouped together and each circle represents an orbital capable of holding two electrons.

As seen in the figure above, the energies of the sublevels in different principal energy levels eventually begin to overlap. After the (3p) sublevel, it would seem logical that the (3d) sublevel should be the next lowest in energy. However, the (4s) sublevel is slightly lower in energy than the (3d) sublevel and thus fills first. Following the filling of the (3d) sublevel is the (4p), then the (5s) and the (4d). Note that the (4f) sublevel does not fill until just after the (6s) sublevel. Figure (PageIndex{6}) is a useful and simple aid for keeping track of the order of fill of the atomic sublevels. Figure (PageIndex{6}): The arrow leads through each subshell in the appropriate filling order for electron configurations. This chart is straightforward to construct. Simply make a column for all the s orbitals with each n shell on a separate row. Repeat for p, d, and f. Be sure to only include orbitals allowed by the quantum numbers (no 1p or 2d, and so forth). Finally, draw diagonal lines from top to bottom as shown.

Video (PageIndex{1}): Energy levels, sublevels and orbitals.

Example (PageIndex{1}): Nitrogen Atoms

Nitrogen has 7 electrons. Write the electron configuration for nitrogen.

Solution:

Take a close look at Figure (PageIndex{5}), and use it to figure out how many electrons go into each sublevel, and also the order in which the different sublevels get filled.

1. Begin by filling up the 1s sublevel. This gives 1s2. Now all of the orbitals in the red n = 1 block are filled.

Since we used 2 electrons, there are 7 − 2 = 5 electrons left

2. Next, fill the 2s sublevel. This gives 1s22s2. Now all of the orbitals in the s sublevel of the orange n = 2 block are filled.

Since we used another 2 electrons, there are 5 − 2 = 3 electrons left

3. Notice that we haven't filled the entire n = 2 block yet… there are still the p orbitals!

The final 3 electrons go into the 2p sublevel. This gives 1s22s22p3

The overall electron configuration is: 1s22s22p3.

Example (PageIndex{2}): Potassium Atoms

Potassium has 19 electrons. Write the electron configuration code for potassium.

Solution

This time, take a close look at Figure (PageIndex{5}).

1. Now the n = 1 level is filled.

Since we used 2 electrons, there are 19 − 2 = 17 electrons left

2. This gives 1s22s2

Since we used another 2 electrons, there are 17 − 2 = 15 electrons left

3. Next, fill the 2p sublevel. This gives 1s22s22p6. Now the n = 2 level is filled.

Since we used another 6 electrons, there are 15 − 6 = 9 electrons left

4. Next, fill the 3s sublevel. This gives 1s22s22p63s2

Since we used another 2 electrons, there are 9 − 2 = 7 electrons left

5. Next, fill the 3p sublevel. This gives 1s22s22p63s23p6

Since we used another 6 electrons, there are 7 − 6 = 1 electron left

Here's where we have to be careful – right after 3p6!

Remember, 4s comes before 3d

6. The final electron goes into the 4s sublevel. This gives 1s22s22p63s23p64s1

The overall electron configuration is: 1s22s22p63s23p64s1

Exercise (PageIndex{1}): Magnesium and Sodium Atoms

What is the electron configuration for Mg and Na?

Mg: 1s22s22p63s2
Na: 1s22s22p63s1

### Pauli Exclusion Principle

When we look at the orbital possibilities for a given atom, we see that there are different arrangements of electrons for each different type of atom. Since each electron must maintain its unique identity, we intuitively sense that the four quantum numbers for any given electron must not match up exactly with the four quantum numbers for any other electron in that atom.

For the hydrogen atom, there is no problem since there is only one electron in the (ce{H}) atom. However, when we get to helium we see that the first three quantum numbers for the two electrons are the same: same energy level, same spherical shape. What differentiates the two helium electrons is their spin. One of the electrons has a (+frac{1}{2}) spin while the other electron has a (-frac{1}{2}) spin. So the two electrons in the (1s) orbital are each unique and distinct from one another because their spins are different. This observation leads to the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers. The energy of the electron is specified by the principal, angular momentum, and magnetic quantum numbers. If those three numbers are identical for two electrons, the spin numbers must be different in order for the two electrons to be differentiated from one another. The two values of the spin quantum number allow each orbital to hold two electrons. Figure (PageIndex{7}) shows how the electrons are indicated in a diagram. Figure (PageIndex{7}): In an orbital filling diagram, a square represents an orbital, while arrows represent electrons. An arrow pointing upward represents one spin direction, while an arrow pointing downward represents the other spin direction.

### Hund's Rule

The last of the three rules for constructing electron arrangements requires electrons to be placed one at a time in a set of orbitals within the same sublevel. This minimizes the natural repulsive forces that one electron has for another. Hund's rule states that orbitals of equal energy are each occupied by one electron before any orbital is occupied by a second electron and that each of the single electrons must have the same spin. The figure below shows how a set of three (p) orbitals is filled with one, two, three, and four electrons. Figure (PageIndex{8}): The (2p) sublevel, for the elements boron (left( Z=5 ight)), carbon (left( Z=6 ight)), nitrogen (left( Z=7 ight)), and oxygen (left( Z=8 ight)). According to Hund's rule, as electrons are added to a set of orbitals of equal energy, one electron enters each orbital before any orbital receives a second electron.

### Orbital Filling Diagrams

An orbital filling diagram is the more visual way to represent the arrangement of all the electrons in a particular atom. In an orbital filling diagram, the individual orbitals are shown as circles (or squares) and orbitals within a sublevel are drawn next to each other horizontally. Each sublevel is labeled by its principal energy level and sublevel. Electrons are indicated by arrows inside of the circles. An arrow pointing upwards indicates one spin direction, while a downward pointing arrow indicates the other direction. The orbital filling diagrams for hydrogen, helium, and lithium are shown in the figure below. Figure (PageIndex{9}): Orbital filling diagrams for hydrogen, helium, and lithium.

According to the Aufbau process, sublevels and orbitals are filled with electrons in order of increasing energy. Since the (s) sublevel consists of just one orbital, the second electron simply pairs up with the first electron as in helium. The next element is lithium and necessitates the use of the next available sublevel, the (2s).

The filling diagram for carbon is shown in Figure (PageIndex{10}). There are two (2p) electrons for carbon and each occupies its own (2p) orbital. Figure (PageIndex{10}): Orbital filling diagram for carbon.

Oxygen has four (2p) electrons. After each (2p) orbital has one electron in it, the fourth electron can be placed in the first (2p) orbital with a spin opposite that of the other electron in that orbital. Figure (PageIndex{11}): Orbital filling diagram for oxygen.

If you keep your papers in manila folders, you can pick up a folder and see how much it weighs. If you want to know how many different papers (articles, bank records, or whatever else you keep in a folder), you have to take everything out and count. A computer directory, on the other hand, tells you exactly how much you have in each file. We can get the same information on atoms. If we use an orbital filling diagram, we have to count arrows. When we look at electron configuration data, we simply add up the numbers.

Example (PageIndex{3}): Carbon Atoms

Draw the orbital filling diagram for carbon and write its electron configuration.

Solution

Step 1: List the known quantities and plan the problem.

Known

• Atomic number of carbon, Z=6

Use the order of fill diagram to draw an orbital filling diagram with a total of six electrons. Follow Hund's rule. Write the electron configuration.

Step 2: Construct the diagram. Orbital filling diagram for carbon.

Electron configuration 1s22s22p2

Following the 2s sublevel is the 2p, and p sublevels always consist of three orbitals. All three orbitals need to be drawn even if one or more is unoccupied. According to Hund's rule, the sixth electron enters the second of those p orbitals and has the same spin as the fifth electron.

Exercise (PageIndex{2}): Electronic Configurations

Write the electron configurations and orbital diagrams for

1. Potassium atom: (ce{K})
2. Arsenic atom: (ce{As})
3. Phosphorus atom: (ce{P})

Potassium: (1s^2 2s^2 2p^6 3s^2 3p^6 4s^1) Arsenic: (1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^3) Phosphorus (1s^2 2s^2 2p^6 3s^2 3p^3) The Atom Neighborhood Figure (PageIndex{12}): The atom neighborhood. Source: Dr. Binh Dao, Sacramento City College.

## Summary

There are four different classes of electron orbitals. These orbitals are determined by the value of the angular momentum quantum number . An orbital is a wave function for an electron defined by the three quantum numbers, n, and m. Orbitals define regions in space where you are likely to find electrons. s orbitals ( = 0) are spherical shaped. p orbitals ( = 1) are dumb-bell shaped. The three possible p orbitals are always perpendicular to each other.

Electron configuration notation simplifies the indication of where electrons are located in a specific atom. Superscripts are used to indicate the number of electrons in a given sublevel. The Aufbau principle gives the order of electron filling in an atom. It can be used to describe the locations and energy levels of every electron in a given atom. Hund's rule specifies the order of electron filling within a set of orbitals. Orbital filling diagrams are a way of indicating electron locations in orbitals. The Pauli exclusion principle specifies limits on how identical quantum numbers can be for two electrons in the same atom.

## Vocabulary

principal quantum number (n)
Defines the energy level of the wave function for an electron, the size of the electron's standing wave, and the number of nodes in that wave.
quantum numbers
Integer numbers assigned to certain quantities in the electron wave function. Because electron standing waves must be continuous and must not "double over" on themselves, quantum numbers are restricted to integer values.

## 9.5: Multielectron Atoms

Atoms with more than one electron, such as Helium (He) and Nitrogen (N), are referred to as multielectron atoms. Hydrogen is the only atom in the periodic table that has one electron in the orbitals under ground state.

In hydrogen-like atoms (those with only one electron), the net force on the electron is just as large as the electric attraction from the nucleus. However, when more electrons are involved, each electron (in the nn-shell) feels not only the electromagnetic attraction from the positive nucleus, but also repulsion forces from other electrons in shells from &lsquo1&rsquo to &lsquonn&lsquo. This causes the net force on electrons in the outer electron shells to be significantly smaller in magnitude. Therefore, these electrons are not as strongly bonded to the nucleus as electrons closer to the nucleus. This phenomenon is often referred to as the Orbital Penetration Effect. The shielding theory also explains why valence shell electrons are more easily removed from the atom. Electron Shielding Effect: A multielectron atom with inner electrons shielding outside electrons from the positively charged nucleus

The size of the shielding effect is difficult to calculate precisely due to effects from quantum mechanics. As an approximation, the effective nuclear charge on each electron can be estimated by: (mathrm<>=Z&minus&sigmaZ>_ ext = mathrm &ndash sigma ), where (mathrm) is the number of protons in the nucleus and &sigmasigma is the average number of electrons between the nucleus and the electron in question. &sigmasigma can be found by using quantum chemistry and the Schrodinger equation or by using Slater&rsquos empirical formula.

For example, consider a sodium cation, a fluorine anion, and a neutral neon atom. Each has 10 electrons, and the number of nonvalence electrons is two (10 total electrons minus eight valence electrons), but the effective nuclear charge varies because each has a different number of protons:

As a consequence, the sodium cation has the largest effective nuclear charge and, therefore, the smallest atomic radius.

## 3.2 - Electron Configurations of Atoms

When electrons fill the energy levels, it fills principal energy levels, sublevels, atomic orbitals from lowest energy first. to view the order in which the sublevels are ordered according to energy. Look carefully and you will see:

1. some 4 sublevel is lower in energy than a 3 sublevel (i.e. 4s is lower in energy than 3d)
2. some 5 or 6 sublevel is lower in energy than a 4 sublevel (i.e. 5p and 6s are lower in energy than 4f )

At first glance it appears that the sequence for electrons to fill the atomic orbitals are of random order. Read on to find an easier way to remember the order of atomic orbitals according to energy.

### 3F - Filling Order of the Sublevels

How do we go about remembering the sequence in which electrons fill the sublevels?

1. Write the principal energy levels and their sublevels on separate lines (as shown on the diagram).
2. Draw arrows over the sublevels (see the red diagonal lines on the diagram by placing your mouse over the diagram).
3. Join the diagonal lines from end to end (click on the diagram to see how I have joined the red diagonal lines).
1. Follow the arrows. The sublevels are magically arranged in the correct sequence from lowest energy. compare the order of filling sublevel sequence with the energy diagram of the sublevels.

### 3G - Electron Configuration Notations

There is a way to represent precisely the electron arrangement in atoms. Let's take a look at the simplest atom, hydrogen.

A hydrogen atom has 1 electron . That electron will occupy the lowest principal energy level, n = 1, and the only sublevel, s. We denote the electron configuration of hydrogen as

• Helium has 2 electrons the 2 electrons both occupy the s sublevel in principal energy level 1.
• Helium's electron configuration is 1s 2
• Lithium's electron configuration is 1s 2 2s 1
• Beryllium's electron configuration is 1s 2 2s 2

### 3H - Electron Configuration and the Periodic Table

There is a pattern between the electron configuration for the elements and their positions on the periodic table. You should take a look at and look closely at the first 20 elements. Compare the electron configuration of an element and its position on the periodic table.

## Quantum Mechanical Model of the Atom Orbitals and Electron Configuration - PowerPoint PPT Presentation

### Quantum Mechanical Model of the Atom Orbitals and Electron Configuration Mrs. Hayes Chemistry Quantum Mechanical Model Orbitals and the Quantum Mechanical . &ndash PowerPoint PPT presentation

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## Examples

The electron configurations of a few elements are provided with illustrations in this subsection.

### Hydrogen

The atomic number of hydrogen is 1. Therefore, a hydrogen atom contains 1 electron, which will be placed in the s subshell of the first shell/orbit. The electron configuration of hydrogen is 1s 1 , as illustrated below.

### Oxygen

The atomic number of oxygen is 8, implying that an oxygen atom holds 8 electrons. Its electrons are filled in the following order:

Therefore, the electron configuration of oxygen is 1s 2 2s 2 2p 4 , as shown in the illustration provided below.

is it meant by the electronic configuration of an element?

The electronic configuration of an element is a symbolic notation of the manner in which the electrons of its atoms are distributed over different atomic orbitals. While writing electron configurations, a standardized notation is followed in which the energy level and the type of orbital are written first, followed by the number of electrons present in the orbital written in superscript. For example, the electronic configuration of carbon (atomic number: 6) is 1s22s22p2.

Why are electronic configurations important?

Electron configurations provide insight into the chemical behaviour of elements by helping determine the valence electrons of an atom. It also helps classify elements into different blocks (such as the s-block elements, the p-block elements, the d-block elements, and the f-block elements). This makes it easier to collectively study the properties of the elements.

is the electronic configuration of copper?

The electronic configuration of copper is [Ar]3d104s1. This configuration disobeys the aufbau principle due to the relatively small energy gap between the 3d and the 4s orbitals. The completely filled d-orbital offers more stability than the partially filled configuration.

## Second Electron Shell

The second electron shell may contain eight electrons. This shell contains another spherical s orbital and three “dumbbell” shaped p orbitals, each of which can hold two electrons . After the 1s orbital is filled, the second electron shell is filled, first filling its 2s orbital and then its three p orbitals. When filling the p orbitals, each takes a single electron once each p orbital has an electron, a second may be added. Lithium (Li) contains three electrons that occupy the first and second shells. Two electrons fill the 1s orbital, and the third electron then fills the 2s orbital. Its electron configuration is 1s 2 2s 1 . Neon (Ne), on the other hand, has a total of ten electrons: two are in its innermost 1s orbital, and eight fill its second shell (two each in the 2s and three p orbitals). Thus, it is an inert gas and energetically stable: it rarely forms a chemical bond with other atoms. Diagram of the S and P orbitalsThe s subshells are shaped like spheres. Both the 1n and 2n principal shells have an s orbital, but the size of the sphere is larger in the 2n orbital. Each sphere is a single orbital. p subshells are made up of three dumbbell-shaped orbitals. Principal shell 2n has a p subshell, but shell 1 does not.

## Atomic Orbitals

The electrons in the partially filled outermost shell (or shells) determine the chemical properties of the atom it is called the valence shell. Each shell consists of one or more subshells, and each subshell consists of one or more atomic orbitals.

The properties of an atom depend ultimately on the number of electrons in the various orbitals, and on the nuclear charge which determines the compactness of the orbitals. In order to relate the properties of the elements to their locations in the periodic table, it is often convenient to make use of a simplified view of the atom in which the nucleus is surrounded by one or more concentric spherical “shells,” each of which consists of the highest-principal quantum number orbitals that contain at least one electron these are s- and p-orbitals and can include d- or f-orbitals, which is atom dependent. The shell model, as with any scientific model, is less a description of the world than a simplified way of looking at it that helps us to understand and correlate diverse phenomena.

We will look at several visualizations of the periodic table. First, however, it would be instructive to see how it is constructed from a logical viewpoint. The table today is the result of an ongoing effort of more than 100 years of observation, measurement, prediction and proof of the relationships of chemical and physical phenomena to electron configurations and charges.

## The Phase of Orbitals

When constructing molecular orbitals, the phase of the two orbitals coming together creates bonding and anti-bonding orbitals.

### Learning Objectives

Describe how atomic orbitals combine to form molecular orbitals.

### Key Takeaways

#### Key Points

• The electron, being a quantum particle, cannot have a distinct location but the electron’s orbital can be defined as the region of space around the nucleus in which the mathematical probability threshold of finding the electron exceeds some arbitrary value, such as 90% or 99%.
• Orbitals are simply mathematical functions that describe particular standing-wave patterns that can be plotted on a graph but have no physical reality.
• Two atomic orbitals can overlap in two ways depending on their phase relationship. The phase of an orbital is a direct consequence of the wave-like properties of electrons.

#### Key Terms

• molecular orbital: The quantum mechanical behavior of an electron in a molecule describing the probability of the electron occupying a particular position and energy, which is approximated by a linear combination of atomic orbitals.
• phase: Any one point or portion in a recurring series of changes, as in the changes of motion of one of the particles constituting a wave or vibration one portion of a series of such changes, in distinction from a contrasted portion, as the portion on one side of a position of equilibrium, in contrast with that on the opposite side.

### Defining Electron Orbitals

The electron is a quantum particle and cannot have a distinct location, but the electron’s orbital can be defined as the region of space around the nucleus in which the probability of finding the electron exceeds some arbitrary threshold value, such as 90% or 99%.

Because of the wave-like character of matter, the orbital corresponds to a standing-wave pattern in 3-dimensional space that we can often represent more clearly in a 2-dimensional cross section. The quantity that is varying (“waving”) is a number denoted by ψ (psi), whose value varies from point to point according to the wavefunction for that particular orbital.

Orbitals of all types are simply mathematical functions that describe particular standing-wave patterns that can be plotted on a graph but have no physical reality of their own. Because of their wavelike nature, two or more orbitals (i.e., two or more functions ψ) can be combined both in-phase and out-of-phase to yield a pair of resultant orbitals that, to be useful, must have squares that describe actual electron distributions in the atom or molecule.

### Molecular Orbitals and Their Phases

When combining orbitals to describe a bonding interaction between two species, the symmetry requirements for the system dictate that the two starting orbitals must make two new orbitals. One orbital, based on in-phase mixing of the orbitals, will be lower in energy and termed bonding. Another orbital, based on out-of-phase mixing of the orbitals, will be higher in energy and termed anti-bonding.

Hydrogen molecular orbitals: The dots here represent electrons. The in-phase combination of the s orbitals from the two hydrogen atoms provides a bonding orbital that is filled, whereas the out-of-phase combination provides an anti-bonding orbital that remains unfilled.

### Orbitals that Overlap

Two atomic orbitals can overlap in two ways depending on their phase relationship. The phase of an orbital is a direct consequence of the wave-like properties of electrons. In graphical representations of orbitals, orbital phase is depicted either by a plus or minus sign (which have no relationship to electric charge) or by shading one lobe. The sign of the phase itself does not have physical meaning except when mixing orbitals to form molecular orbitals.

### Constructive Overlap

Two same-sign orbitals have a constructive overlap forming a molecular orbital with the bulk of the electron density located between the two nuclei. This molecular orbital is called the bonding orbital and its energy is lower than that of the original atomic orbitals. A bond involving molecular orbitals that are symmetric with respect to rotation around the bond axis is called a sigma bond (σ-bond). If the phase changes, the bond becomes a pi bond (π-bond). Symmetry labels are further defined by whether the orbital maintains its original character after an inversion about its center if it does, it is defined gerade (g), German for “straight.” If the orbital does not maintain its original character, it is ungerade (u), German for “odd.”

### Destructive Overlap

Atomic orbitals can also interact with each other out-of-phase, which leads to destructive cancellation and no electron density between the two nuclei at the so-called nodal plane depicted as a perpendicular dashed line. In this anti-bonding molecular orbital with energy much higher than the original atomic orbitals, any electrons present are located in lobes pointing away from the central internuclear axis. For a corresponding σ-bonding orbital, such an orbital would be symmetrical but differentiated from it by an asterisk, as in σ*. For a π-bond, corresponding bonding and antibonding orbitals would not have such symmetry around the bond axis and would be designated π and π*, respectively.

Two p-orbitals forming a π-bond: If two parallel p-orbitals experience sideways overlap on adjacent atoms in a molecule, then a double or triple bond can develop. Although the π-bond is not as strong as the original σ-bond, its strength is added to the existing single bond.

P-orbital overlap is less than head-on overlap between two s orbitals in a σ-bond due to orbital orientation. This makes the π-bond a weaker bond than the original σ-bond that connects two neighboring atoms however the fact that its strength is added to the underlying σ-bond bond makes for a stronger overall linkage. Electrons in π-bonds are often referred to as π- electrons. They limit rotational freedom about the double bond because a parallel orientation of the p-orbitals must be preserved to maintain the double or triple bond.

## Electron Configuration

### Many-electron atoms

For many-electron atoms, as we have already mentioned, SE cannot be solved analytically because of the electron–electron interaction. The starting point is to ignore this part of the Hamiltonian. This leads to the orbital approximation or the independent electron model. The multielectron wave function is approximated as a product of one-electron wave functions, orbitals. This product is called electron configuration . Each electron is described with its own, hydrogen-like orbital, obtained from its own SE (obviously with modified nuclear charge Z). What we have learned in previous subsections is that the lowest energy solution of the one-electron SE is 1s orbital. Would that mean that in the ground state of a multielectron atom, all electrons are going to be described by their 1s orbitals? Obviously, this is not correct. We know that independent electrons are not really independent. It is impossible that two electrons are in the same place. Remember, electrons are moving in four-dimensional space (spatial+spin), but their position is not known exactly. Instead of the particle’s position, we need to think of electrons as waves described completely by the set of four quantum numbers. Thus two electrons cannot have all four quantum numbers identical. This is Pauli’s exclusion principle. Another way of stating this is that each orbital can accommodate up to two electrons (with “opposite spins,” i.e., one α, the other β). The last expression is commonly used while speaking about electron configuration. It may also look convenient to draw the configurations either in boxes or as line/arrow diagrams. It is also common to use an expression like “there are two electrons in d orbital,” or “1s orbital is occupied by two electrons.” However, never forget that an orbital (spin–orbital), by definition, is a one-electron wave function—it describes one, and only one electron. And strictly speaking, one cannot “put an electron in,” and an orbital cannot “accommodate an electron.” Again, orbitals are one-electron wave functions, described by four quantum numbers, used to approximate the multielectron wave function ( Autschbach, 2012 ). All the statements above are, of course, completely valid, but only if we are aware of their true meaning.

We need to admit that the orbital approximation is very crude. Electron–electron interactions cannot be so small to be neglected. But there is a simple way our model can be improved, while still ignoring electron–electron repulsion explicitly. An electron in a multielectron atom can be considered as moving in the effective field created by the nucleus and all the other electrons. That means that the electron is “feeling” the nuclear charge, but it is reduced due to the presence of other electrons. Thus electron–electron repulsion is mimicked by the so-called electron shielding. The electrons with different n and l quantum numbers will be shielded differently, that is, will feel different effective nuclear charge, Zeff. This points to one important difference compared to the hydrogen atom—the energy of the orbitals will depend both on n (as in the H-atom case), but also on l quantum number. Thus the degeneracy of s, p, d levels with the same n is removed. However, we still use the orbital approximation. The electron configuration is symbolically written as the product of all the “occupied” subshells, with a superscript indicating the number of electrons “in” it. A maximum number of electrons in each subshell is given by its degeneracy, 2(2l+1). For example, the ground state of He-atom is described as 1s 2 , of Li-atom 1s 2 2s 1 , of O-atom 1s 2 2s 2 2p 4 . Zeff can be determined by a set of semiempirical rules. More often it is a variable which can systematically improve an initial approximate wave function by the means of the variational calculations.

To understand the effect of shielding we refer to Fig. 2.2 , which depicts the radial distributions of H-like orbitals. The angular wave function of each orbital is still described by spherical harmonics. While the radial part in the multielectron system will be different, the qualitative picture will still hold. Because of the higher nuclear charge (Zeff>1) nucleus will more strongly attract electrons, and the region of maximum probability density will be closer to the nucleus. The maximum radial probability of orbitals with the same n is still on a similar distance from the nucleus regardless of different l values. The shell model of an atom gives rough features of the electron density of an atom. Within each shell, a finer picture is provided by the subshells. Electrons with lower n are closer to the nucleus. They are tightly bound to the nucleus because they are only slightly shielded from the full nuclear charge. However, these electrons will reduce the average charge of the nucleus experienced by the electrons in orbitals with higher n. Thus similarly to the H-atom, energy will be higher for orbitals with higher n. Let us now try to understand why the orbitals with the same n but different l will be shielded differently. Let us think about 2s and 2p orbitals. In the one-electron system, 2s and 2p orbitals are degenerate. 2s orbital will have higher kinetic energy in the radial part of the wave function (one radial node), but 2p orbital will have higher kinetic energy in angular part (one angular node), and the total energy will be the same in both cases (as it depends only on n). But what happens in a multielectron atom, for example, in configurations 1s 2 2s 1 or 1s 2 2p 1 , which one would have lower energy? If we look into radial distributions of 2s and 2p orbitals ( Fig. 2.2 ), we see that 2s electron is, on average, slightly further away from the nucleus than a 2p electron. From that, naively, we could expect that energy of 2s electron will be higher than that of the 2p electron. But at the same time, the maximum of the 2p distribution is closer to the region occupied by the 1s electrons. Therefore 1s and 2p electrons are, on average, closer together than the 1s and 2s electrons. Consequently, the 1s and 2p electrons will repel each other more than 1s and 2s electrons ( Cooksy, 2014 ). Here, we are approximating the electron–electron repulsion by shielding, so 2p orbitals will be more shielded than 2s, and 2s orbital will have lower energy. Similar reasoning could be applied for 3s, 3p, and 3d orbitals. Within each subshell, 2(2l+1) degeneracy is not removed. The energy order of orbitals in the ground state of multielectron atoms is typically 1s, 2s, 2p, 3s, 3p, (3d, 4s), 4p… The order of 3d and 4s orbitals will depend on atoms as they lie close in energy. Constructing the electron configuration is typically based on the Aufbau principle. The orbitals are arranged by increasing energy, and electrons are added one by one, according to this energy order, subject to the Pauli exclusion principle. A maximum of two electrons can be assigned to one orbital, and if so, then they will have paired spins. So far, we have not talked too much about the spin, but there is an additional rule that is accounted for during this build-up principle. When more than one orbital is available for occupation, which happens due to the degeneracies of subshells, electrons occupy separate orbitals (with different ml values) before entering an already half-occupied orbital. Doing so, they will have parallel spins (e.g., both with α spin). This Hund’s rule of maximum multiplicity is obviously important only if we care about the occupation of orbitals with the same l value. For example, O-atom’s configuration, 1s 2 2s 2 2p 4 , can be written as 1 s 2 2 s 2 2 p x 2 2 p y 1 2 p z 1 , where the last two electrons are both α spin, and px, py, pz indicates three electrons with ml values −1, 0, +1, respectively. Qualitatively, occupying two different degenerate orbitals gives them a greater spatial separation, hence lower energy. Maximum spin is related to the effects of so-called spin correlation, which will be discussed shortly later. However, in the orbital approximation, all different occupations of subshells, microstates, are still degenerate. Specification of subshell occupation in electron configuration imitates the real ground state of atoms, with explicit electron–electron repulsion. Completely filled shells are tightly bound to the nucleus and these electrons are called core electrons. Electrons in partially filled shells are valence electrons and are further away from the nucleus. As explained, core electrons shield valence ones from the nucleus. Core electrons usually do not contribute to the chemistry of atoms. Chemistry is often related mainly to the properties of valence electrons.

#### Periodic system of elements

The electron configuration of atoms explains the common form of the periodic system of elements ( Fig. 2.3 ). Elements are classified into “blocks” according to the subshell that is being “filled” as the atomic number increases. Each period starts with the elements whose highest energy electrons are in the ns orbital. These are s-elements, with group 1 having ns 1 and group 2 having ns 2 configuration of outermost electrons. The first period has only H and He because with n=1 only s-orbitals are available. The second period will have six more elements, p-block (groups 13–18) elements that will have 2p orbitals occupied, obviously up to 2p 6 . The third period will also have eight elements, because of the occupation of 3s and 3p orbitals. The complications start with the fourth period. Orbital energies depend on Zeff, which is important for the relative order of, for example, 3d and 4s orbitals ( Eugen Schwarz, 2010 Eugen Schwarz and Rich, 2010 ). For K and Ca the order is 4s<3d, thus these two elements belong to the s-block and start the fourth period. However, from Sc on, the order is reversed, and the 3d orbital is lower in energy. The ground state configuration of Sc is however 3d 1 4s 2 . Because the 3d orbital is much more localized than the 4s orbital, the much greater repulsion energy of the two electrons in the 3d orbital is more important than the simple energy order of the orbitals. The total energy of the atom is lower despite populating the higher energy 4s orbital. The same reasoning is generally true for atoms Sc to Zn, and first-row d-block elements in the periodic system typically have valence electron configuration 3d n 4s 2 . Exceptions are Cr (3d 5 4s 1 ) and Cu (3d 10 4s 1 ). A similar trend is observed in the fifth period. In the sixth period, filling 4f orbitals generates the f-block, rare earth elements. The energies of the 4f, 5d, and 6s orbitals are comparable, and true electronic configuration must be deduced considering effects that go beyond the one-electron picture. The ground state of cations is obtained by removing electrons from neutral atom configuration. In 3d metal cations that means removing electrons first from the 4s orbitals. Electronic configuration of TM cations, d n , is the starting point of the LF approach for understanding the properties of coordination compounds (see details in the Ligand field theory section, as well as a discussion related to the oxidation states and using ionic configuration in complexes in the previous edition of this book Neese, 2013 ). Figure 2.3 . Periodic system of elements.

We see that electronic configuration alone is not enough to describe the ground states of atoms. The true ground state is the state of lowest energy and needs to have included other effects, most notably electron–electron repulsion. Thus in principle, we cannot use electronic configuration and we must introduce terms. And spectroscopically determined ground electronic configurations are deduced from the transitions between true electronic states (terms).

## Quantum Numbers

Quantum numbers refer to electrons, so I'll assume you mean the electron number that would correspond with the atomic number of the element. Quantum numbers are basically like an address for electrons, giving us information about the location of an electron from most general to most specific.

We need to apply the Aufbau principle, Hund's rule and the Pauli exclusion principle to answer this question.

Quantum numbers are listed in the following order: n, l, #m_l# , #m_s#

n = principal number - tells us which energy level an electron is in
l = angular number - tells us which sublevel and electron is in
#m_l# = magnetic number - tells us which orbital the e- is in
#m_s# = spin number - tells us if the electron is spin up or spin down

The quantum numbers for electron 1 (hydrogen) are: 1,0,0,+1/2
2 (He): 1,0,0,-1/2
3 (Li): 2,0,0,+1/2
4 (Be): 2,0,0,-1/2

There is a lot of info you'll need to understand. - I'll recommend this video to help. Quantum numbers are discussed in the final minute but there is info through the entire video you'll use to determine quantum numbers.

Zero electrons.
Electrons don't share all four identical quantum numbers. Each electron has its own set of four quantum numbers. Something similar like people have unique fingerprint. Quantum number "is telling" us everything about electron.