In chemistry and physics, atomic orbitals are a function called wave, which describes the properties characteristic of no more than two electrons in the vicinity or system of nuclei, as in a molecule. An orbital is often depicted as a three-dimensional region within which there is a 95 percent chance of finding an electron.
Orbitals and orbits
As a planet moves around the Sun, it traces a path called an orbit. Similarly, an atom can be represented as electrons circling in orbit around the nucleus. In reality, things are different, and electrons are found in regions of space known as atomic orbitals. Chemistry is content with a simplified model of the atom to calculate the Schrödinger wave equation and, accordingly, determine the possible states of the electron.
Orbits and orbitals sound similar, but they have completely different meanings. It is extremely important to understand the difference between the two.
Inability to image orbits
To plot the trajectory of something, you need to know exactly where the object is and be able to determine where it will be in an instant. This is impossible for an electron.
According to this, it is impossible to know exactly where the particle is at the moment and where it will end up later. (In fact, the principle says that it is impossible to determine its moment and momentum simultaneously and with absolute accuracy).
Therefore, it is impossible to construct an orbit of electron motion around the nucleus. Is this a big problem? No. If something is impossible, you should accept it and find ways around it.
Hydrogen electron - 1s orbital
Suppose there is one hydrogen atom and at a certain moment in time the position of one electron is graphically captured. Soon after this, the procedure is repeated and the observer discovers that the particle is in a new position. How she got from the first place to the second is unknown.
If you continue to act in this way, a kind of 3D map of the probable location of the particle will gradually form.
In the case, the electron can be located anywhere within the spherical space surrounding the nucleus. The diagram shows a cross section of this spherical space.
95% of the time (or any other percentage, since only the size of the Universe can provide one hundred percent certainty) the electron will be located within a fairly easily defined region of space, fairly close to the nucleus. This area is called an orbital. Atomic orbitals are regions of space in which an electron exists.
What is he doing there? We don’t know, we can’t know, and therefore we simply ignore this problem! We can only say that if an electron is in a particular orbital, then it will have a certain energy.
Each orbital has a name.
The space occupied by a hydrogen electron is called a 1s orbital. A unit here means that the particle is at the energy level closest to the nucleus. S talks about the shape of the orbit. S orbitals are spherically symmetrical about the nucleus - at least like a hollow ball of fairly dense material with the nucleus at its center.
2s
The next orbital is 2s. It is similar to 1s, except that the region where the electron is most likely to be found is located further from the nucleus. This is an orbital of the second energy level.
If you look closely, you'll notice that closer to the nucleus there is another region of slightly higher electron density ("density" is another way of referring to the likelihood that that particle is present at a particular location).
2s electrons (and 3s, 4s, etc.) spend part of their time much closer to the center of the atom than would be expected. The result of this is a slight decrease in their energy in the s orbitals. The closer electrons get to the nucleus, the lower their energy becomes.
The 3s, 4s orbitals (etc.) are located further and further from the center of the atom.
P-orbitals
Not all electrons inhabit s orbitals (in fact, very few of them do). On the first, the only available location for them is 1s, on the second, 2s and 2p are added.
Orbitals of this type are more like 2 identical balloons, connected to each other at the nucleus. The diagram shows a cross section of a 3-dimensional region of space. Again, the orbital only shows the region with a 95 percent chance of finding an individual electron.
If we imagine a horizontal plane that passes through the nucleus in such a way that one part of the orbit will be above the plane and the other below it, then there is a zero probability of finding an electron on this plane. So how does a particle get from one part to another if it can never pass through the plane of the nucleus? This is due to its wave nature.
Unlike the s-orbital, the p-orbital has a specific orientation.
At any energy level, it is possible to have three absolutely equivalent p orbitals located at right angles to each other. They are arbitrarily denoted by the symbols p x, p y and p z. This is done for convenience - what is meant by the X, Y or Z directions is constantly changing as the atom moves randomly in space.
P orbitals at the second energy level are called 2p x, 2p y and 2p z. There are similar orbitals in subsequent ones - 3p x, 3p y, 3p z, 4p x, 4p y, 4p z and so on.
All levels, with the exception of the first, have p-orbitals. At higher levels, the “lobes” are more elongated, with the most likely location of the electron being further away from the nucleus.
d- and f-orbitals
In addition to the s and p orbitals, there are two other sets of orbitals available to electrons at higher energy levels. In the third, the existence of five d-orbitals (with complex shapes and names), as well as 3s and 3p orbitals (3p x, 3p y, 3p z) is possible. In total there are 9 of them here.
In the fourth, along with 4s and 4p and 4d, 7 additional f orbitals appear - a total of 16, also available at all higher energy levels.
Placing electrons in orbitals
The atom can be thought of as a very fancy house (like an inverted pyramid) with the nucleus living on the ground floor and various rooms on the upper floors occupied by electrons:
- on the ground floor there is only 1 room (1s);
- on the second there are already 4 rooms (2s, 2p x, 2p y and 2p z);
- on the third floor there are 9 rooms (one 3s, three 3p and five 3d orbitals) and so on.
But the rooms are not very big. Each of them can only contain 2 electrons.
A convenient way to show the atomic orbitals in which given particles are found is to draw "quantum cells".
Quantum cells
Atomic orbitals can be represented as squares with the electrons in them depicted as arrows. Often up and down arrows are used to show that these particles are different from each other.
The need for different electrons in an atom is a consequence of quantum theory. If they are in different orbitals, that's great, but if they are in the same orbital, then there must be some subtle difference between them. Quantum theory gives particles a property called “spin,” which is what the direction of the arrows denotes.
The 1s orbital with two electrons is depicted as a square with two arrows pointing up and down, but it can also be written even more quickly as 1s 2 . This is read as "one s two" rather than "one s squared". The numbers in these designations should not be confused. The first denotes the energy level, and the second the number of particles in the orbital.
Hybridization
In chemistry, hybridization is the concept of mixing atomic orbitals into new hybrid ones capable of pairing electrons to form chemical bonds. Sp hybridization explains the chemical bonding of compounds such as alkynes. In this model, the 2s and 2p atomic orbitals of carbon mix to form two sp orbitals. Acetylene C 2 H 2 consists of an sp-sp intertwining of two carbon atoms to form a σ bond and two additional π bonds.
The carbon atomic orbitals in saturated hydrocarbons have identical hybrid sp 3 orbitals, shaped like a dumbbell, one part of which is much larger than the other.
Sp 2 hybridization is similar to the previous ones and is formed by mixing one s and two p orbitals. For example, in an ethylene molecule three sp 2 and one p orbital are formed.
Atomic orbitals: filling principle
By imagining the transitions from one atom to another in the periodic table of chemical elements, it is possible to establish the electronic structure of the next atom by placing an additional particle into the next available orbital.
Electrons, before filling higher energy levels, occupy lower ones, located closer to the nucleus. Where there is a choice, they fill the orbitals separately.
This filling order is known as Hund's rule. It is used only when the atomic orbitals have equal energies and also helps to minimize the repulsion between electrons, which makes the atom more stable.
It should be noted that the s orbital always has slightly less energy than the p orbital at the same energy level, so the former are always filled before the latter.
What's really strange is the position of the 3d orbitals. They are at a higher level than 4s, and so the 4s orbitals are filled first, followed by all the 3d and 4p orbitals.
The same confusion occurs at higher levels with more intertwining between them. Therefore, for example, 4f atomic orbitals are not filled until all the places in 6s are occupied.
Knowing the filling order is central to understanding how to describe electronic structures.
According to Heisenberg's uncertainty principle, the position and momentum of an electron cannot be simultaneously determined with absolute accuracy. However, despite the impossibility of accurately determining the position of an electron, it is possible to indicate the probability of an electron being in a certain position at any given time. The region of space in which there is a high probability of finding an electron is called an orbital. The concept of "orbital" should not be identified with the concept of orbit, which is used in Bohr's theory. In Bohr's theory, an orbit refers to the trajectory (path) of an electron around the nucleus.
Electrons can occupy four different types of orbitals, called s-, p-, d-, and f-orbitals. These orbitals can be represented by three-dimensional surfaces bounding them. The regions of space bounded by these surfaces are usually chosen so that the probability of finding a single electron within them is 95%. In Fig. Figure 1.18 schematically shows the shape of the s- and -orbitals. The s-orbital is spherical, and the -orbital is dumbbell-shaped.
Since an electron has a negative charge, its orbital can be considered as some kind of charge distribution. This distribution is usually called an electron cloud (Fig. 1.19).
Rice. 1.18. Shape of s- and p-orbitals.
Rice. 1.19. Electron cloud in cross section. The circle represents the area around the nucleus within which the probability of finding an electron is 95%.
When discussing the chemical properties of atoms and molecules - structure and reactivity - an idea of the spatial form of atomic orbitals can be of great help in the qualitative solution of a particular issue. In the general case, AOs are written in complex form, but using linear combinations of complex functions related to the same energy level with the principal quantum number P and with the same value of the orbital momentum /, it is possible to obtain expressions in real form that can be depicted in real space.
Let us consider sequentially a series of AOs in the hydrogen atom.
The wave function of the ground state 4^ looks most simple. It has spherical symmetry
The value of a is determined by the expression where the value
called Bohr radius. The Bohr radius indicates the characteristic sizes of atoms. The value of 1/oc determines the scale of the characteristic decay of functions in one-electron atoms
From (EVL) it is clear that the size of one-electron atoms shrinks as the nuclear charge increases in inverse proportion to the value of Z. For example, in the He + atom the wave function will decrease twice as fast as in a hydrogen atom with a characteristic distance of 0.265 A.
The dependence of *F ls on distance is shown in Fig. 3.3. The maximum of the function *Fj is at zero. Finding an electron inside a nucleus should not be too surprising, since the nucleus cannot be imagined as an impenetrable sphere.
The maximum probability of detecting an electron at some distance from the nucleus in the ground state of a hydrogen atom occurs at r = a 0 = 0.529 A. This value can be found as follows. Probability of finding an electron in some small volume A V equal to |*P| 2 DY. Volume AV we assume so small that the value of the wave function can be considered constant within this small volume. We are interested in the probability of finding an electron at a distance G from the core in a thin layer of thickness A G. Since the probability of finding an electron at a distance G does not depend on the direction and the specific direction does not interest us, then we need to find the probability of an electron staying in a very thin spherical layer of thickness A G. Since the value | V F| 2 is easy to calculate, we need
Rice. 3.3. Dependence of *F 1s on distance. The values of the function are normalized to its value in at r = O
Rice. 3.4.Scheme for calculating the volume of a spherical layer
find the volume of the spherical layer, which we denote by A K. It is equal to the difference in the volumes of two balls with radii G And g + Ar(Fig. 3.4):
Since A G little compared to G, then when calculating the value (g + Ar) 3 we can limit ourselves to the first two terms. Then for the volume of the spherical layer we obtain
The last expression can be obtained in a simpler way. Since A G little compared to G, then the volume of the spherical layer can be taken equal to the product of the area of the spherical layer and its thickness (see Fig. 3.4). The area of the sphere is 4kg 2, and thickness A G. The product of these two quantities gives the same expression (3.11).
So the probability W find the electron in this layer is equal to
The expression for *P ls is taken from Appendix 3.1. If we consider the value of A G constant, then the maximum of the reduced function is observed at G = a 0 .
If you want to know what the probability is W detect electron in volume V, then it is necessary to integrate the probability density of detecting an electron over this region of space in accordance with expression (3.6).
For example, what is the probability of detecting an electron in a hydrogen atom in a spherical region of space with a center at the nucleus and with a radius x 0. Then
Here the value d V during the calculations it was replaced by 4kg 1 dr by analogy with (3.11), since the wave function depends only on the distance and therefore there is no need to integrate over angles due to the absence of angular dependence of the integrable function.
A qualitative idea of the distribution of the wave function in space is given by the image of atomic orbitals in the form of clouds, and the more intense the color, the higher the value of the H" function. The orbital will look like this (Fig. 3.5):
Rice. 3.5.
Orbital 2p z B the form of a cloud is shown in Fig. 3.6.
Rice. 3.6. Image of the 2p g orbital of a hydrogen atom in the form of a cloud
In a similar way, the electron density distribution will look like a cloud, which can be found by multiplying the probability density I"Fj 2 by the electron charge. In this case, they sometimes talk about electron smearing. However, this in no way means that we are dealing with smearing electron across space - no real smearing of the electron across space occurs, and therefore the hydrogen atom cannot be represented as a nucleus immersed in a real cloud of negative charge.
However, such images in the form of clouds are rarely used, and much more often lines are used to create an idea of \u200b\u200bthe angular dependence of the H" functions. To do this, calculate the values of the H" functions on a sphere drawn at a certain distance from the nucleus. Then the calculated values are plotted on the radii, indicating the sign of the Ch"-functions for the most informative plane section for a given Ch"-function. For example, the Is orbital is usually depicted as a circle (Fig. 3.7).
Rice.
In Fig. 3.8 2/> r-orbital is built on a sphere of some radius. To obtain a spatial picture, it is necessary to rotate the figure relative to the z axis. The index “z” when writing a function indicates the orientation of the function along the “z” axis. The signs “+” and “-” correspond to the signs of the H"-functions. The values of the 2/? z-function are positive in the region of space where the ^-coordinate is positive, and negative in the region where the ^-coordinate is negative.
Rice. 3.8. Form 2p z-orbitals. Built on a sphere of some radius
The situation is similar in the case of the remaining /orbitals. For example, 2/? The x-orbital is oriented along the x-axis and is positive in that part of space where the x-coordinate is positive, and its values are negative where the x-coordinate values are negative (Fig. 3.9).
The image of wave functions indicating the sign is important for a qualitative description of the reactivity of chemical compounds, and therefore images such as those shown in Fig. 3.9 are found most often in chemical literature.
Let us now consider d-orbitals (Fig. 3.10). Orbitals dxy, dxz, dyz, look equivalent. Their orientation and signs are determined by subscripts: index xy shows
Rice. 3.9. Form 2p x - orbitals. Built on a sphere of some radius
that the orbital is oriented at angles of 45° with respect to the x and axes at and that the sign of the Y-function is positive where the product of the indices x and at positively.
Rice. 3.10.
The situation is similar with the remaining ^/-orbitals. The image of ^/-orbitals shown in Fig. 3.10, is most often found in the literature. It can be seen that the orbitals d , d x2 _ y2 , d z2 are not equivalent. Only orbitals are equivalent d , d xz , d yz . If five equivalent ^/-orbitals are needed to describe the structure of a molecule, then they can be constructed using linear combinations of orbitals.
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Due to the fact that when describing elements they are divided into groups with different orbitals, let us very briefly recall the essence of this concept.
According to Bohr's model of the atom, electrons rotate around the nucleus in circular orbitals (shells). Each shell has a strictly defined energy level and is characterized by a certain quantum number. In nature, only certain electron energies are possible, that is, discrete (quantized) orbital energies (“allowed”). Bohr's theory assigns electron shells K, L, M, N and further in the order of the Latin alphabet, in accordance with the increasing energy level of the shells, principal quantum number n, equal to 1, 2, 3, 4, etc. Subsequently, it turned out that electron shells are split into subshells, and each is characterized by a certain quantum energy level, characterized by orbital quantum number l.
According to uncertainty principle Heisenberg, it is impossible to accurately determine the location of an electron at any given time. However, you can indicate the likelihood of this happening. The region of space in which the probability of finding an electron is highest is called an orbital. Electrons can occupy 4 orbitals of different types, which are called s- (sharp), p- (principal), d- (diffuse) and f- (fundamental) orbitals. Previously, these letters denoted the spectral lines of hydrogen, but currently they are used only as symbols, without decoding.
Orbitals can be represented as three-dimensional surfaces. Typically, the regions of space bounded by these surfaces are chosen so that the probability of detecting an electron within them is 95%. A schematic representation of the orbitals is shown in Fig. 1.
Rice. 1.
The s-orbital has a spherical shape, the p-orbital has the shape of a dumbbell, the d-opbital has the shape of two dumbbells intersecting in two nodal mutually perpendicular planes, the s-subshell consists of one s-orbital, the p-subshell has 3 p-orbitals, d-subshell - of 5 d-orbitals.
If no magnetic field is applied, all orbitals of one subshell will have the same energy; in this case they are called degenerate. However, in an external magnetic field, the subshells split (Zeeman effect). This effect is possible for all orbitals except the s orbital. It is characterized magnetic quantum number t. The Zeeman effect is used in modern atomic absorption spectrophotometers (AASP) to increase their sensitivity and reduce the detection limit in elemental analyses.
For biology and medicine, it is important that orbitals of the same symmetry, that is, with the same numbers l and m, but with a different value of the principal quantum number (for example, orbitals 1s, 2s, 3s, 4s), differ in their relative size. The volume of the internal space of electron orbitals is larger for atoms with a large n value. An increase in the volume of the orbital is accompanied by its loosening. During complex formation, the size of the atom plays an important role, since it determines the structure of coordination compounds. In table Figure 1 shows the relationship between the number of electrons and the principal quantum number.
Table 1. Number of electrons at different values of quantum number n
In addition to the three named quantum numbers, which characterize the properties of the electrons of each atom, there is one more - spin quantum number s , characterizing not only electrons, but also atomic nuclei.
Medical bioinorganics. G.K. Barashkov
m quantum numbers.The wave function is calculated using the Schrödinger wave equation within the framework of the one-electron approximation (Hartree-Fock method) as the wave function of an electron located in a self-consistent field created by the atomic nucleus with all other electrons of the atom.
E. Schrödinger himself considered an electron in an atom as a negatively charged cloud, the density of which is proportional to the square of the value of the wave function at the corresponding point of the atom. In this form, the concept of an electron cloud was also accepted in theoretical chemistry.
However, most physicists did not share the beliefs of E. Schrödinger - there was no evidence of the existence of the electron as a “negatively charged cloud”. Max Born substantiated the probabilistic interpretation of the square of the wave function. In 1950, E. Schrödinger, in the article “What is an elementary particle?” I am forced to agree with the arguments of M. Born, who was awarded the Nobel Prize in Physics in 1954 with the wording “For fundamental research in the field of quantum mechanics, especially for the statistical interpretation of the wave function.”
Quantum numbers and orbital nomenclature
Radial probability density distribution for atomic orbitals at different n And l.
- Principal quantum number n can take any positive integer value, starting from one ( n= 1,2,3, … ∞) and determines the total energy of the electron in a given orbital (energy level):
- The orbital quantum number (also called the azimuthal or complementary quantum number) determines the angular momentum of the electron and can take integer values from 0 to n - 1 (l = 0,1, …, n- 1). The angular momentum is given by the relation
The letter designations for atomic orbitals come from the description of spectral lines in atomic spectra: s (sharp) - a sharp series in atomic spectra, p (principal)- home, d (diffuse) - diffuse, f (fundamental) - fundamental.
- Magnetic quantum number m l determines the projection of the orbital angular momentum onto the direction of the magnetic field and can take integer values in the range from - l before l, including 0 ( m l = -l … 0 … l):
In the literature, orbitals are denoted by a combination of quantum numbers, with the principal quantum number denoted by a number, the orbital quantum number by the corresponding letter (see table below) and the magnetic quantum number by a subscript expression showing the projection of the orbital onto the Cartesian axes x, y, z, For example 2p x, 3d xy, 4f z(x²-y²). For orbitals of the outer electron shell, that is, in the case of describing valence electrons, the main quantum number in the orbital notation is usually omitted.
Geometric representation
The geometric representation of an atomic orbital is a region of space bounded by a surface of equal density (equidensity surface) of probability or charge. The probability density on the boundary surface is chosen based on the problem being solved, but, usually, in such a way that the probability of finding an electron in a limited area lies in the range of values 0.9-0.99.
Since the electron energy is determined by the Coulomb interaction and, therefore, the distance from the nucleus, the principal quantum number n sets the size of the orbital.
The shape and symmetry of the orbital are determined by the orbital quantum numbers l And m: s-orbitals are spherically symmetrical, p, d And f-orbitals have a more complex shape, determined by the angular parts of the wave function - angular functions. Angular functions Y lm (φ, θ) - eigenfunctions of the squared angular momentum operator L², depending on quantum numbers l And m(see Spherical functions), are complex and describe in spherical coordinates (φ, θ) the angular dependence of the probability of finding an electron in the central field of an atom. The linear combination of these functions determines the position of the orbitals relative to the Cartesian coordinate axes.
For linear combinations Y lm the following notations are accepted:
| Orbital quantum number value | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
|---|---|---|---|---|---|---|---|---|---|
| Magnetic quantum number value | 0 | 0 | 0 | ||||||
| Linear combination | |||||||||
| Designation |
An additional factor sometimes taken into account in the geometric representation is the sign of the wave function (phase). This factor is significant for orbitals with an orbital quantum number l, different from zero, that is, not having spherical symmetry: the sign of the wave function of their “petals” lying on opposite sides of the nodal plane is opposite. The sign of the wave function is taken into account in the molecular orbital method MO LCAO (molecular orbitals as a linear combination of atomic orbitals). Today, science knows mathematical equations that describe geometric figures representing orbitals (depending on the coordinates of the electron versus time). These are equations of harmonic oscillations that reflect the rotation of particles over all available degrees of freedom - orbital rotation, spin,... Hybridization of orbitals is represented as interference of oscillations.
Filling of orbitals with electrons and the electronic configuration of an atom
Each orbital can contain no more than two electrons, differing in the value of the spin quantum number s(back). This prohibition is determined by the Pauli principle. The order of filling orbitals of the same level with electrons (orbitals with the same value of the principal quantum number n) is determined by the Klechkovsky rule, the order in which electrons fill orbitals within one sublevel (orbitals with the same values of the principal quantum number n and orbital quantum number l) is determined by Hund's Rule.
A brief record of the distribution of electrons in an atom over various electron shells of the atom, taking into account their principal and orbital quantum numbers n And l called