Atomic Orbital ~ God of Chemistry (2023)

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|²represents the probability density function. The probability of an electron in the positionxbetween 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 nucleusY^M
_eu(EU,ϕ)is a spherical harmonic that determines the direction of the orbital. Here,N,eu, EMare the principal, azimuthal, and magnetic quantum numbers.R(R) is expressed below.

Here,A₀is the Bohr radius,, Eeu^2eu+1
_N−eu−1is a generalized Laguerre polynomial.

As you can see above, the quantum numbersN,eu, EMDecide 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 numberNis a set of positive integers;N=1, 2, 3… The azimuthal quantum numbereuis 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 numberMis a set of integers from −euForeu;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 numbereu. For each value ofN, the possible values ​​ofeuare 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 ofeucorresponds 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.

From the table above, theN-th shell temNsublayers of. Each of these subshells consists of a number of orbitals determined by the magnetic quantum numberM. For eacheu, the possibleMare 0, ±1, ±2 … ±(eu−1), ±eu. any value ofMcorresponds 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_xe P_j(M=±1).

In a subshell d (eu=2), there are five orbitals: d_z²(M=0), d_xze d_e(M=±1) e d_xye d_x²−j²(M=±2).

In an f-subshell (eu=3), there are seven orbitals: f_z³(M=0), f_xz²e f_e²(M=±1), f_xyze f_z(x²−j²)(M=±2) e f_{x(x²−3j²)}e f_{j(3x²−j²)}(M=±3).

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 dieeuis 2eu+1.

For a given value of the principal quantum numberN, the number of orbitals is given byN².

proof forN²

An orbital can hold the top two electrons, so given the number of electrons for aNis 2N².

Nomenclatura orbital

Usually, the orbital is named with a combination of numbers and letters. consider 4d_z²orbital; 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_z²Orbital is oriented along the d-sublevel of the 4th levelz-Axle.

Orbitals can also be simply represented by the three quantum numbers, for example 4d_z²is 420 (N=4,eu=2,M=0).

Let's go to another example:N=5,eu=3,M=±1 corresponds to 5f_xz²and 5f_e²orbitals. Both are at sublevel f of 5th level. 5f_xz²rests onxzlevel for 5f_e²rests oneLevel.

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_zorbital is aligned with thez- Axis, S_xasx-axis and finally p_jasj-Axle.

o D_z²Orbital has two lobes on thez-Axis and a donut on thexyLevel. The remaining four d orbitals each have two dumbbells. D_xze d_elie on thexzEeplanes, while the other two, I_xye d_x²−j², They are inxyLevel. D_xze d_eare 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_xye d_x²−j²are identical and differ by an angle of 45°.

o f_z²Orbital has two lobes along thezshaft and two donuts in the middle. f_xz²e f_e²are similar to d_xze d_eexcept they have two additional bean-shaped lobes lined up on thex- that's me f_xz²E noj- that's me f_e². F_xyze f_z(x²−j²)it has eight lobes; four of them are below thatxyAirplane and the remaining four above. f_{x(x²−3j²)}e f_{j(3x²−j²)}each has six lobes; All lobes are placed in thexyLevel. 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 applyeuEMare always zero, the only onesNvaries. ifN=1, it is 1s orbital. The wave function of the 1s orbital for the hydrogen atom can be obtained by substitutionN,eu, EMas 1, 0, 0 in the generalized wave function mentioned above.

PS²is the probability density function. The probability density function is the probability of finding an electron per unit volume. it's at the maximumR=0 and approaches zero as the value increasesR(see first chart below). We get the radial probability by multiplying the area of ​​the sphere by 4πR², for the probability density. The third graph shows the radial probability versus the radius. At theR=0 eR=∞, the probability is zero. From the graph, the probability peaks atR=52.9 pm, which is also the Bohr radius (A₀).

SeN=2 eN=3 we get the 2s and 3s orbitals. Its wave functions are as follows:

The following graph is the radial probability (Ψ²R²) 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 sayNorbital 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 toeu=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_xe P_j(eu=1;M=±1).

ForN=2 the wave functions are as follows:

PS₂₁₀is the wave function for 2p_zorbital e Ψ_21±1is for 2p_xand 2p_j. OEUEϕA part of Ψ decides the orientation of the orbital. In the graphs below, only the radial part of Ψ is considered, i.e. H.EUEϕis ignored.

Unlike the s orbital, the probability density of the p orbital is not maximal atR=0. We can see in the figure above that it is at zeroR=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 thexyplane for 2p_zorbital. Likewise, the corresponding aircraft for 2p_xand 2p_jThey areeExz. 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_xe 3p_jThey are as follows:

The diagram below shows the radial node in 3p_zorbital. 2p_zit has no radial nodes.

me 4 p_zOrbital, not shown in the figure above, has two radial nodes. In short, theNp-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_z²it's unique; has two lobes on thez-axis and a donut-shaped lobe on thexyLevel. 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 theNd 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;Nf 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.

From the table above, we generalize the formula for ourselves. The angular nodes depend only on the value ofeuand it's the sameeu. On the other hand, radial nodes depend on bothNEeuand is given byN−eu−1. Therefore, the total number of nodes is the sum of both and isN−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.

Related articles

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