The atomic orbital is a complex mathematical function called a wave function that determines the energy, angular momentum, and location of an electron. A better way to define the atomic orbital is the space around the nucleus, which has a high probability of finding the electron.

The simplest orbital of all is the 1s orbital, which is spherical in shape (see image below). The nucleus is at the center of the orbital and the electron revolves around the nucleus.

The region around the core is the probabilistic region, not the discrete one. The probability of finding the electron in this area is high (about 90%) and the electron can exist beyond that. Therefore, an orbital is an electronic probabilistic cloud around the nucleus.

## wave function

In quantum mechanics, a wave function is a complex mathematical description of a quantum state. It gives the probability distribution of an electron and is denoted by Ψ. Each orbital has its own Ψ. |Ψ|^{2}represents the probability density function. The probability of an electron in the position*x*between a and b is given by the following equation.

We can normalize the equation.

Thus, when the probability of finding an electron is 100%, the size of the orbital is equal to the size of the universe, that is, −∞ to ∞. In practice, we define the orbital based on 90% probability, not 100%.

### hydrogen atom

The hydrogen atom has only one electron and is the only atom for which the Schrödinger equation can be solved exactly. The wave function for the hydrogen atom in the spherical coordinate system is as follows:

*R*(*R*) is the radial function that determines the distance from the nucleus*Y ^{M}_{eu}*(

*EU*,

*ϕ*)is a spherical harmonic that determines the direction of the orbital. Here,

*N*,

*eu*, E

*M*are the principal, azimuthal, and magnetic quantum numbers.

*R*(

*R*) is expressed below.

Here,*A*_{0}is the Bohr radius,, E*eu*^{2eu+1}_{N−eu−1}is a generalized Laguerre polynomial.

As you can see above, the quantum numbers*N*,*eu*, E*M*Decide on the equation for the Ψ wave function. For each set of these three quantum numbers, we get a new wave function. From this we can conclude that these three quantum numbers represent an orbital.

The main quantum number*N*is a set of positive integers;*N*=1, 2, 3… The azimuthal quantum number*eu*is a set of non-negative integers in the range 0 to (*N*−1);*eu*=0, 1, 2 … (*N*−2), (*N*−1). Finally, the magnetic quantum number*M*is a set of integers from −*eu*For*eu*;*M*=−*eu*, −(*eu*−1) … −1, 0, 1 … (*eu*−1),*eu*.

## quantum numbers

An atom can have a large number of orbitals. The three quantum numbers mentioned in the above section are used to identify an orbital in an atom.

In an atom, electronic space is divided into layers. Each layer is represented by a principal quantum number value.*N*. For example,*N*=1 is the lowest energy shell, called the K shell,*N*=2 is the second shell, L shell and so on.

Within each shell are subshells. The number of subshells in a shell is determined by the azimuthal quantum number*eu*. For each value of*N*, the possible values of*eu*are 0, 1, 2...*N*−2,*N*−1. Thus, for layer K (*N*=1),*eu*=0, para L-shell (*N*=2),*eu*=0, 1; para M Shell (*N*=3),*eu*=0, 1, 2 and so on. any value of*eu*corresponds to a sublevel.*eu*=0 is a sublevel s,*eu*=1 is a p-subshell,*eu*=2 is a sublevel d,*eu*=3 is a subshell f, etc. The following table summarizes the same.

principal quantum numberN | manga | Azimuthal Quantum Numbereu | lower shell | number of subshells |
---|---|---|---|---|

1 | k | 0 | S | 1 |

2 | eu | 0, 1 | s, s | 2 |

3 | M | 0, 1, 2 | s, p, d | 3 |

4 | N | 0, 1, 2, 3 | s, p, d, f | 4 |

5 | Ö | 0, 1, 2, 3, 4 | s, p, d, f, g | 5 |

6 | P | 0, 1, 2, 3, 4, 5 | s, p, d, f, g, h | 6 |

7 | Q | 0, 1, 2, 3, 4, 5, 6 | s, p, d, f, g, h, i | 7 |

8 | R | 0, 1, 2, 3, 4, 5, 6, 7 | s, p, d, f, g, h, i, k | 8 |

… | … | 0, 1, 2, 3, 4, 5, 6, 7… | s, p, d, f, g, h, i, k… | … |

Note: When naming subshells, the letter j is ignored. |

From the table above, the*N*-th shell tem*N*sublayers of. Each of these subshells consists of a number of orbitals determined by the magnetic quantum number*M*. For each*eu*, the possible*M*are 0, ±1, ±2 … ±(*eu*−1), ±*eu*. any value of*M*corresponds to an orbital.

For a sublevel s (*eu*=0),*M*=0. Therefore, it has only one orbital called an s orbital. For a p-subshell (*eu*=1),*M*=0, ±1. Therefore, a p subshell has three orbitals: p_{z}(*M*=0) e p_{x}e P_{j}(*M*=±1).

In a subshell d (*eu*=2), there are five orbitals: d_{z2}(*M*=0), d_{xz}e d_{e}(*M*=±1) e d_{xy}e d_{x2−j2}(*M*=±2).

In an f-subshell (*eu*=3), there are seven orbitals: f_{z3}(*M*=0), f_{xz2}e f_{e2}(*M*=±1), f_{xyz}e f_{z(x2−j2)}(*M*=±2) e f_{x(x2−3j2)}e f_{j(3x2−j2)}(*M*=±3).

lower shells | Azimuthal Quantum Numbereu | magnetic quantum numberM | Orbital |
---|---|---|---|

S | 0 | 0 | S |

P | 1 | 0 | P_{z} |

±1 | P_{x}e P_{j} | ||

D | 2 | 0 | D_{z2} |

±1 | D_{xz}e d_{e} | ||

±2 | D_{xy}e d_{x2−j2} | ||

F | 3 | 0 | F_{z3} |

±1 | F_{xz2}e f_{e2} | ||

±2 | F_{xyz}e f_{z(x2−j2)} | ||

±3 | F_{x(x2−3j2)}e f_{j(3x2−j2)} |

Other higher orbitals, viz. g, h, i etc. are very complicated and rarely encountered.

As can be seen from the previous table, the number of orbitals strangely increases with the azimuthal quantum number. The number of orbitals of a die*eu*is 2*eu*+1.

For a given value of the principal quantum number*N*, the number of orbitals is given by*N*^{2}.

#### proof for*N*^{2}

The possible values of*eu*per dice*N*are 0, 1, 2...*N*−2,*N*−1 and for all*eu*there are 2*eu*+1 orbitals. The number of orbitals for a given value*N*:

An orbital can hold the top two electrons, so given the number of electrons for a*N*is 2*N*^{2}.

## Nomenclatura orbital

Usually, the orbital is named with a combination of numbers and letters. consider 4d_{z2}orbital; the first number represents the principal (or shell) quantum number. Here is 4 (n shell). The number is followed by an alphabet, where d represents a subshell. The third is the alphabet index. Indicates the orientation of an orbital. then 4d_{z2}Orbital is oriented along the d-sublevel of the 4th level*z*-Axle.

Orbitals can also be simply represented by the three quantum numbers, for example 4d_{z2}is 420 (*N*=4,*eu*=2,*M*=0).

Let's go to another example:*N*=5,*eu*=3,*M*=±1 corresponds to 5f_{xz2}and 5f_{e2}orbitals. Both are at sublevel f of 5th level. 5f_{xz2}rests on*xz*level for 5f_{e2}rests on*e*Level.

## orbital shapes

Each orbital has a unique shape, and the shape becomes more complex and difficult to follow as we move to higher orbitals.

The s orbital is spherical; The nucleus is at the center of the sphere. It is not oriented in any direction. In other words, it is undirected.

There are three dumbbell-shaped p orbitals. Each orbital has two lobes aligned on one of three axes. the p_{z}orbital is aligned with the*z*- Axis, S_{x}as*x*-axis and finally p_{j}as*j*-Axle.

o D_{z2}Orbital has two lobes on the*z*-Axis and a donut on the*xy*Level. The remaining four d orbitals each have two dumbbells. D_{xz}e d_{e}lie on the*xz*E*e*planes, while the other two, I_{xy}e d_{x2−j2}, They are in*xy*Level. D_{xz}e d_{e}are identical to each other. They have the same radial component of the wave function and differ only by an angle of 90°. In the same way d_{xy}e d_{x2−j2}are identical and differ by an angle of 45°.

o f_{z2}Orbital has two lobes along the*z*shaft and two donuts in the middle. f_{xz2}e f_{e2}are similar to d_{xz}e d_{e}except they have two additional bean-shaped lobes lined up on the*x*- that's me f_{xz2}E no*j*- that's me f_{e2}. F_{xyz}e f_{z(x2−j2)}it has eight lobes; four of them are below that*xy*Airplane and the remaining four above. f_{x(x2−3j2)}e f_{j(3x2−j2)}each has six lobes; All lobes are placed in the*xy*Level. All three pairs above are identical and have the same radial component but different angular components.

With the exception of the s orbital, all orbitals are directional – they point in a specific direction.

## orbital s

For s orbitals apply*eu*E*M*are always zero, the only ones*N*varies. if*N*=1, it is 1s orbital. The wave function of the 1s orbital for the hydrogen atom can be obtained by substitution*N*,*eu*, E*M*as 1, 0, 0 in the generalized wave function mentioned above.

PS^{2}is the probability density function. The probability density function is the probability of finding an electron per unit volume. it's at the maximum*R*=0 and approaches zero as the value increases*R*(see first chart below). We get the radial probability by multiplying the area of the sphere by 4π*R*^{2}, for the probability density. The third graph shows the radial probability versus the radius. At the*R*=0 e*R*=∞, the probability is zero. From the graph, the probability peaks at*R*=52.9 pm, which is also the Bohr radius (*A*_{0}).

Se*N*=2 e*N*=3 we get the 2s and 3s orbitals. Its wave functions are as follows:

The following graph is the radial probability (Ψ^{2}*R*^{2}) versus radius for 1s, 2s, and 3s orbitals.

As can be seen from the diagram above, the 1s orbital has only one peak, while the 2s and 3s have two and three peaks, respectively. As a result, there is a minimum probability point between every two peaks. With the exception of extremes, the probability at these points becomes zero. Those points where the probability becomes zero are called nodes. The 1s orbital has zero nodes, the 2s orbital has one node, and the 3s orbital has two nodes; in general we can say*N*orbital hat s (*N*−1) it.

Note: The different colors in the graphs indicate different phases of the orbital, as well as peaks and troughs in the wave.

Nodes in orbitals are similar to standing wave nodes, where they are defined as points with zero amplitudes. You can think of nodes as the space between spheres that are placed inside each other.

From the above diagram it can also be seen that the size of the orbital increases with the principal quantum number – that is, 3s>2s>1s.

## p-orbital

The p orbital corresponds to*eu*=1. The p orbital, unlike the s orbital, is not spherical; it is dumbbell shaped. As already mentioned, there are three p orbitals: p_{z}(*eu*=1;*M*=0) e p_{x}e P_{j}(*eu*=1;*M*=±1).

For*N*=2 the wave functions are as follows:

PS_{210}is the wave function for 2p_{z}orbital e Ψ_{21±1}is for 2p_{x}and 2p_{j}. O*EU*E*ϕ*A part of Ψ decides the orientation of the orbital. In the graphs below, only the radial part of Ψ is considered, i.e. H.*EU*E*ϕ*is ignored.

Unlike the s orbital, the probability density of the p orbital is not maximal at*R*=0. We can see in the figure above that it is at zero*R*=0. In the middle, the value reached its peak. The radial probability plot also shows similar behavior. Since the probability density in the nucleus is zero, the electron spends more time outside the nucleus in the p orbital than in the s orbital, where the probability density in the nucleus is maximum.

Also, the probability is zero in the*xy*plane for 2p_{z}orbital. Likewise, the corresponding aircraft for 2p_{x}and 2p_{j}They are*e*E*xz*. These levels, at which the probability becomes zero, are called node levels. There are three levels of nodes in p orbitals, as shown below.

These levels of nodes are called angle nodes. Thus, p orbitals have three angular nodes. The s orbital has no angular nodes; they have only radial nodes. The p orbitals have radial and angular nodes.

The 3p wave functions_{z}, 3p_{x}e 3p_{j}They are as follows:

The diagram below shows the radial node in 3p_{z}orbital. 2p_{z}it has no radial nodes.

me 4 p_{z}Orbital, not shown in the figure above, has two radial nodes. In short, the*N*p-orbital hat (*N*−2) radial nodes.

## d orbital

There are five d orbitals: d_{z2},D_{xz},D_{e},D_{xy}, e d_{x2−j2}. Of these orbitals, d_{z2}it's unique; has two lobes on the*z*-axis and a donut-shaped lobe on the*xy*Level. The d orbitals emanate from the 3rd shell and their wave functions are mentioned below.

In all the above wavefunctions, the angular parts are different, while the radial parts are multiples of each other.

Like p orbitals, d orbitals have angular nodes. Each d orbital has 2 angular nodes.

The radial nodes for the*N*d are orbitals (*N*−3) us. So 3d has no radial nodes. Radial nodes in the d orbital start at 4d.

## f orbital

The f orbital is much more complex than the d orbital and is observed in heavy elements. There are seven f orbitals: f_{z3}, F_{xz2}, F_{e2}, F_{xyz}, F_{z(x2−j2)},_{x(x2−3j2)}, e f_{j(3x2−j2)}.

Each of these orbitals has 3 angular nodes. Radial nodes start at 4f;*N*f orbital hat (*N*−4) radial nodes.

## It

As discussed above, nodes are the region with zero probability of finding electrons. They can be divided into two types: angular and radial. Radial nodes are obtained from the radial component of the wave function, while angular nodes are obtained from the angular component, ie H. spherical harmonics.

orbital | N | eu | radial nodes | angular nodes | us in general |
---|---|---|---|---|---|

1s | 1 | 0 | 0 | 0 | 0 |

2s | 2 | 0 | 1 | 0 | 1 |

3 Sek | 3 | 0 | 2 | 0 | 2 |

2p | 2 | 1 | 0 | 1 | 1 |

3p | 3 | 1 | 1 | 1 | 2 |

4p | 4 | 1 | 2 | 1 | 3 |

3d | 3 | 2 | 0 | 2 | 2 |

4d | 4 | 2 | 1 | 2 | 3 |

5d | 5 | 2 | 2 | 2 | 4 |

4f | 4 | 3 | 0 | 3 | 3 |

From the table above, we generalize the formula for ourselves. The angular nodes depend only on the value of*eu*and it's the same*eu*. On the other hand, radial nodes depend on both*N*E*eu*and is given by*N*−*eu*−1. Therefore, the total number of nodes is the sum of both and is*N*−1.

## Orbit x Orbit

Orbits and orbitals are often confused with each other, especially by beginners. They sound the same, but they are completely different concepts and should not be used interchangeably. The table below explains the difference between the two.

orbits | Orbital |
---|---|

Orbits are fundamental to the Bohr model, a predecessor of the quantum mechanical model; describes the orbit as a circular path followed by an electron. Orbits are concentric circular paths of electrons. | An orbital is a space around the nucleus of an atom that has a high probability of finding an electron. |

The orbits come from the Bohr-Rutherford atomic model. | The notion of orbitals was derived from the quantum model of an atom. |

Orbits are always concentric circles. In the modified Bohr model, the orbits have an elliptical shape. | Orbitals come in multiple forms and their shapes become mind-boggling as we move to higher orbitals. The simplest of these is the s orbital, which is spherical in shape. |

The orbit is a two-dimensional figure. | Orbitals are three-dimensional. |

They are direction independent. | They are directional except for the s orbitals. |

Since orbits have a specific path, we can predict their position and momentum. This contradicts Heisenberg's uncertainty principle. | We cannot estimate the exact position of an electron in an orbital. We can only find the probability of one electron; uncertainty always remains. |

Orbits are not real; they do not exist and electrons do not revolve around the nucleus in a specific orbit. | Orbitals are realistic. |

## Related articles

- principal quantum number
- Azimuthal Quantum Number
- Bohr's Atomic Model
- Rutherfords Atommodell