Why does a covalent bond occur

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Chapter 17
Molecules

Only a small number of elements naturally exist in the form of individual atoms. Most elements exist in bound form as individual molecules in liquids or in solids. A molecule is defined as a compound of two or more atoms to form a unit. To be precise, it is the smallest unit of a chemical compound that still has its properties. The term molecule itself comes from Latin and means small mass. In general, there are many possible combinations of atoms to molecules. It is to this fact that we owe our wealth of substances in our world. Examples of the simplest molecules are the hydrogen molecule H., the nitrogen molecule N and the oxygen molecule O. Since these molecules are composed of two identical atoms, they are called homonuclear. Diatomic and heteronuclear (composed of two different atoms) molecules would be, for example, lithium fluoride LiF, hydrochloric acid HCl or copper oxide CuO. In this chapter we deal with the description of the formation and properties of molecules.

17.1 Bonds in Molecules

In general, molecules are formed when the total energy is reduced when they are formed from individual atoms, i.e. the sum of the energies of the individual, separated atoms must be greater than the total energy of the molecule formed. There are basically two types of bonds between molecules. On the one hand there is the covalent bond, also called the homeopolar bond, and on the other hand the ionic bond, also called the heteropolar bond. It should be noted that the explanation of the formation of both bonds is only possible with the help of quantum mechanics.

17.1.1 Covalent Bond

The covalent bond occurs in bonds between non-metals. The creation of the bond is to be imagined in such a way that an attractive force acts between two atoms that form a molecule, which is mediated by electrons, which are very likely to be between the two atoms. In other words, the two atoms are held together by electrons, which are bound to both atoms at the same time or jump back and forth between the two atoms.

An example of a covalently bound molecule is the simplest existing molecule, the hydrogen molecule ion H., which consists of two hydrogen nuclei (protons) and one electron. The bond can be thought of as the electron holding the two protons together by jumping back and forth between them. Or to put it in quantum mechanics: between the wave functions that describe the position of the electron with one or the other proton, there is a positive interference, which increases the probability of the electron being between the two protons and creates a binding state. On the hydrogen molecule ion H we will go into more detail in Section 17.2.

It should be noted at this point that (covalent) bonds are not formed if the Pauli principle forbids electrons to reside in a common quantum state or if the Pauli principle can only be fulfilled by electrons being in higher excited states of the Atoms are located.

17.1.2 Ionic Bond

Ionic bonds occur when a metal and a non-metal bond. The creation of the bond is to be imagined in such a way that electrons from one atom are transferred to the other atom and thus positively and negatively charged ions arise, between which an attractive Coulomb force acts. Of course, the question still arises as to why electrons pass from one atom to another. This question can only be answered with the help of quantum mechanics, which says that it can be energetically more favorable for electrons to leave an only partially filled shell of one atom and to completely fill a shell of the other atom. In other words: electrons tend to have noble gas configurations.

Probably the most prominent example of an ionic bond is table salt (NaCl). This bond comes about when an electron from the Na atom transfers to the Cl atom. Between the positively charged NaIon and the negatively charged Cl-Ion there is consequently an electrical attraction and the NaCl molecule is formed.

17.2 The hydrogen molecular ion

As mentioned earlier, the hydrogen molecule ion is H the simplest existing molecule. For this reason, it is very suitable for illustrating the formation of a molecule (by means of covalent bonds)1.

The hydrogen molecular ion is observed as a bound state during gas discharges in a hydrogen atmosphere. In such a gas discharge, an electron is torn from the hydrogen molecule. As a result, a hydrogen molecular ion consists of two hydrogen nuclei (protons) and one electron (see Fig. 17.1).



Fig. 17.1: The hydrogen molecule ion consists of two hydrogen nuclei (protons) a and b and one electron. We denote the distance between the two cores with , the distance of the electron to the two nuclei with or. .

We now imagine the formation of the molecule in such a way that first the two nuclei are far apart and then slowly come closer. In the starting position, where the nuclei are far apart, the electron is located either around one or the other nucleus. The wave function of the electron thus corresponds to that of the hydrogen ground state or , in which

In addition, the wave functions solve and the time-independent Schrödinger equation

where for the energy values and applies .

We are now bringing the kernels closer together. In this case, the electron that was previously located around one of the two nuclei feels the attraction of both nuclei. The wave function in the bound state accordingly fulfills the time-independent Schrödinger equation2

The goal is now the wave function and the energy for the electron to be determined. As mentioned in Section 17.1.1, we can imagine the formation of the bond in the hydrogen molecule ion in such a way that the electron holds the two protons together by jumping back and forth between them. For this reason we choose as an approach for the wave function a linear combination of the two wave functions and i.e.

in which and are two coefficients yet to be determined. Inserting in (17.5) yields

where we use the two Hamilton operators and have introduced. We can now simplify this equation with the help of the Schrödinger equations (17.3) and (17.4). It surrenders

where we are the energy difference between the ground state energy and the energy of the electron in the bound state With have designated. We now multiply this equation by and integrate via the position coordinate of the electron

where we use the constants for the various integrals , and have introduced. With these constants this equation can be expressed in the simple form

write. Similarly, we multiply (17.8) by and integrate in turn via the position coordinate of the electron . It surrenders

where, for reasons of symmetry, we can replace the integrals with the same constants S, C, and D as before and thus the equation in the form

can write. With equations (17.10) and (17.12) we now have two equations for the two coefficients and found. This system of equations can be represented in matrix notation

A system of equations of this kind only has a nontrivial solution if the determinant of the matrix vanishes, i.e. if holds

and thus

Insertion into (17.13) gives for the coefficients and the solutions

From this we get for the wave function an antisymmetric and a symmetrical one solution3

The corresponding energy values and ring

The question now arises as to whether both of these states are realized, only one of the two or neither, i.e., in short, whether a bond is established at all. The answer to this question results from the calculation of the binding energy, i.e. from the comparison of the total energy for nuclei that are far apart with the total energy for nuclei that are close together. Only when the binding energy is negative at a certain distance between the nuclei does binding occur.

The total energy in the case of nuclei that are far apart corresponds to the ground state energy of the hydrogen atom. The total energy in the case of nuclei that are close to each other results from the sum of the energy of the electron or and the Coulomb repulsion energy between the protons

With (17.20) and (17.21) it follows for the binding energy of the hydrogen molecular ion

The representation of the binding energy depending on the distance between the nuclei (see Fig. 17.2) shows that in the antisymmetric state the binding energy is always positive and therefore no binding occurs. This state is therefore called the antibonding state. Conversely, for the symmetrical state, the binding energy becomes negative in a certain distance range and thus binding occurs. The symmetrical state is therefore called the binding state. According to our model, the binding energy is eV at a distance Å between the nuclei (see Fig. 17.2).



Fig.17.2: The binding energy of the hydrogen molecule ion for the antisymmetric (antibonding) and the symmetric (binding) state depending on the distance between the cores.

In order to better imagine the formation of a bond in the symmetrical (binding) state or the non-formation of a bond in the antisymmetrical (antibonding) state, we take a look at the corresponding wave functions and . In the symmetrical state (see Fig. 17.3 (a)) is due to the overlap of the wave functions and the probability of the electron being between the two nuclei increases. As a result, the electron often stays between the nuclei, where it feels the attraction of both nuclei. This lowers the potential energy of the overall system, which leads to bonding. In the antisymmetric state (see Fig. 17.3 (b)), on the other hand, the probability of being between the nuclei is small, in the middle it is even zero. This means that the electron seldom or never stays between the nuclei and mostly only gets to feel the attraction of one nucleus.



Fig. 17.3: The probability of the electron being located for the hydrogen molecule ion for (a) the symmetric and (b) the antisymmetric state. For the sake of simplicity, we consider the probability of stay along one dimension (x-axis).

At the end of the consideration of the hydrogen molecule ion, we compare the values ​​for the binding energy eV and the distance Å between the nuclei from our model with the experimentally determined values eV and Å. It can be seen that the deviation in the values ​​is relatively large. Nevertheless, with our model we got a good idea of ​​how the hydrogen molecular ion binds.

17.3 The hydrogen molecule

After the hydrogen molecule ion, we extend our consideration to molecules with several electrons. The simplest such molecule is the hydrogen molecule, which consists of two hydrogen nuclei (protons) and two electrons (see Fig. 17.4).



Fig. 17.4: The hydrogen molecule consists of two hydrogen nuclei (protons) a and b and two electrons 1 and 2. The distances between the electrons and protons are labeled accordingly.

The wave function this two-electron system satisfies the Schrödinger equation

where the Hamilton operator is given by

As with the hydrogen molecule ion, the goal is now the wave function and the energy to determine. To do this, we use the Heitler-London solution method, which is named after Walter Heitler and Fritz London and is based on the so-called principle of variation.

Multiplication of the Schrödinger equation (17.24) by and integration via the location coordinates and results for the energy values the expression

The variation principle now says that if instead of solving the Schrödinger equation Another function is used, one an approximate solution for the energy values receives or more precisely an upper limit.

Since we now have two electrons instead of one electron in the hydrogen molecule compared to the hydrogen molecule ion, we have to set up an approach for the wave function take into account the Pauli principle, i.e. the total wave function including the spins of the electrons of the hydrogen molecule must be antisymmetric. The first possibility of fulfilling this condition arises with a symmetrical spatial wave function and an antisymmetrical spin wave function

where is the symmetrical spatial wave function according to (16.13) is given by

and the antisymmetric spin wave function by

The second possibility to construct an antisymmetric overall wave function results with an antisymmetric spatial wave function and a symmetric spin wave function

where is the antisymmetric spatial wave function according to (16.14) is given by

and the symmetric spin wave function by

According to the solution method of Heitler-London, these two wave functions are now and as an approach to the wave function to choose to insert this in (17.26) and thus to obtain an approximate solution for the energy values ​​of the hydrogen molecule in the sense of the principle of variation. We will forego this calculation at this point and consider the solution for the binding energy right away for both cases (see Fig. 17.5). As in the case of the hydrogen molecule ion, in the hydrogen molecule the wave functions with a symmetrical spatial wave function lead to binding states, whereas the wave functions with an antisymmetrical spatial wave function lead to antibonding states. According to our model, the binding energy is eV at a distance Å between the nuclei. Again are the deviations from the experimentally determined values eV and Å relatively large. This means that the wave function according to the Heitler-London model is still relatively imprecise and further effects have to be taken into account in order to improve the model.



Fig. 17.5: The binding energy of the hydrogen molecule for the antisymmetric (antibonding) and the symmetric (binding) state depending on the distance between the cores.

At this point it should be noted that - as expected - the binding energy in the hydrogen molecule is stronger than in the hydrogen molecule ion, since two electrons are responsible for the bond between the nuclei.

17.4 Complex Molecules - Hybridization

The description of more complex molecules than the hydrogen molecule ion or the hydrogen molecule is generally a very difficult task. The only helpful thing is that it is often sufficient to examine the interaction of the electrons in the outer shells (valence electrons) of the atoms involved. The reason for this is that electrons in the inner shells contribute only weakly to atom-atom interactions due to their strong bond to the atomic nucleus. In addition, X-ray spectra confirm this finding.

The reason why certain molecules like the hydrogen molecule H. or the water molecule HO and other molecules such as the helium molecule He not, is essentially based on the Pauli principle. For example, the helium atom has two electrons in the ground state -State with different spin. If it comes together with another helium atom, each helium atom will have two electrons with the same spin for part of the time. According to the Pauli principle, however, this is forbidden and that is why the helium molecule He exists Not.

We now take another look at the creation of a (covalent) bond between two atoms. From a quantum mechanical point of view, when two atoms come together, their wave functions overlap. If this overlap increases the probability density of the electrons between the atoms involved, this leads to attractive forces and ultimately to a (covalent) bond. However, the electrons involved can have probability densities that differ from those of isolated atoms. The following s and p states are important for the formation of bonds in molecules (see Fig. 17.6):



  • -Conditions ( = 1, 2, 3, ..., , ):
  • -Conditions ( = 2, 3, 4, ..., , ):
  • -Conditions ( = 2, 3, 4, ..., , ):
  • -Conditions ( = 2, 3, 4, ..., , ):

Molecular states are according to their total angular momentum quantum number (Total angular momentum along the bond axis, i.e. z-axis) with a Greek letter:

In the case of the hydrogen molecule, for example, there is one -Binding in front. General education conditions -Bonds, - and -Conditions -Bonds. It is also possible to form bonds from states belonging to different angular momenta. One example is the water molecule. The two simply occupied -States of the oxygen atom come with the -States of the hydrogen atoms together and each form one -Binding. The mutual repulsion between the hydrogen atoms increases the angle between the bond axes by 90 to 104.5.

This explanation of the formation of a bond in the water molecule cannot be applied to all molecules like the example of methane CH shows. Namely, one would expect the carbon atom to have an electronic configuration with two hydrogen atoms a CH Molecule with two -Bonds at a bond angle of just over 90 forms. The methane molecule CH also occurs in nature that is perfectly symmetrical with four exactly equal C-H bonds. The question now naturally arises, how is such a bond possible? The answer is that the carbon atom has to form superpositions from the -Condition and the three -States to four new states (States) is coming. This phenomenon is called hybridization. Such hybridized states occur when the binding energies are lower than those of the unhybridized states. It should be noted at this point that there are two further possibilities of hybridized states in the case of the carbon atom: -States (an electron remains in the -State, the remaining three occupy hybridized states) and -States (two electrons stay in the -State, the remaining two occupy hybridized states).

17.5 Molecular Spectra

The total energy of a molecule depends on its electronic state, the vibrations of the bound atoms against each other and the rotation of the molecule as a whole. With these three energies, the distances between the discrete energy levels move on different scales, which is also reflected in the molecular spectra:

  1. The distance between the energy levels for the outermost electrons in molecules is a few eV, which results in spectral lines in the visible or UV range.
  2. Oscillation states are separated by a few 0.1 eV. Transitions between these states lead to spectral lines in the infrared range.
  3. The distances between the states of rotation move on an even smaller scale. They are in the range of a few meV, which causes spectral lines in the microwave range.

Molecular spectra are useful for analyzing the structure of molecules. For example, bond lengths and bond angles can be determined from the vibrational and rotational spectra

17.5.1 Rotational Spectra

To get an idea of ​​rotational spectra we consider a diatomic molecule that rotates around the center of gravity (see Fig. 17.7).



The moment of inertia of this molecule around an axis through the center of gravity and perpendicular to the line connecting the two atoms is

where in the last step we use the reduced mass

have introduced. Hence, the rotation of a diatomic molecule corresponds to the rotation of a single particle of mass about an axis at a distance . The amount of angular momentum of the molecule is given by

in which is the angular velocity. As always, the angular momentum is quantized, i.e. it applies

where we are the rotational quantum number have introduced. This results in the rotational energy

The lines in the spectrum result from transitions between the different energy levels . For the crossover frequencies applies

However, as with atoms, the transitions that can be realized are restricted by a selection rule

In addition, only molecules with a permanent dipole moment show rotational spectra, since only such molecules can emit or absorb electromagnetic radiation. Molecules without a permanent dipole moment can only change their state of rotation through collisions. Examples of molecules without a permanent dipole moment are methane CH or carbon dioxide CO.

According to (17.43) is the transition frequency proportional to the rotational quantum number . For this reason, equidistant spectral lines can be observed in the spectrum of a molecule.

From the measurement of a rotation spectrum, the moment of inertia can be obtained with (17.43) of a molecule can be determined. Also are the masses and of the atoms involved is known, the bond length results from (17.38) of the molecule.

17.5.2 Vibration spectra

Molecules can be excited to vibrate along their bonds. For small amplitudes, the oscillations of a molecule are well described by a harmonic oscillator. The vibration energy is therefore given according to Chapter 10 by

where we are the vibrational quantum number have introduced and only the transitions that the selection rule meet, are admitted. In reality, the distance between the energy levels of the molecular oscillation is not equidistant, but decreases for high energies. The reason for this is that for high energies the description by the harmonic oscillator is no longer justified.

Polyatomic molecules have different modes of vibration. These can occur locally in only a part of the molecule or globally in the whole molecule. In particular, this fact also allows the binding structure of molecules to be examined in detail on the basis of studies of vibration spectra.

17.5.3 Fluorescence and Phosphorescence

Fluorescence and phosphorescence are two phenomena that occur in connection with rotational and vibrational spectra.

To explain fluorescence we consider a molecule that was excited by the absorption of a photon. The excited state can then go back to the ground state. This can be done in two different ways. On the one hand by emitting a photon of the same frequency or on the other hand, the molecule can first lose vibrational energy in collisions with other molecules and then emit a photon with a lower frequency. If the second variant occurs, one speaks of fluorescence. The principle of fluorescence is used, for example, with fluorescent tubes or with brighteners in detergents.

Phosphorescence occurs when a molecule is excited from a singlet ground state to an excited singlet state and then changes from the singlet state to a triplet state as a result of collisions with other molecules. The triplet state is now due to the selection rule for the spin quantum number it is very unlikely that it will return to the ground state and the molecule remains in a so-called metastable state4. Such molecules can therefore give off their energy again long after they have been excited. Phosphorescence is used, for example, in signs or luminous numbers.

17.6 (Further) literature

  • Lecture slides spring semester 2008
  • A. Beiser, Concepts of Modern Physics, McGraw-Hill, 2003
  • H. Haken and H. Wolf, Atomic and Quantum Physics, Introduction to the Experimental and Theoretical Basics, Springer, Berlin Heidelberg New York, 2004.
  • H. Haken and H. Wolf, Molecular Physics and Quantum Chemistry, Introduction to the Experimental and Theoretical Basics, Springer, Berlin Heidelberg New York, 2003.
  • F. Schwabl, Quantum Mechanics (QM I), An Introduction, Springer, Berlin Heidelberg New York, 2002.

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