Polyatomic Species: Hybrid & Molecular Orbitals

Polyatomic species like methane, CH4, can be described in terms of molecular orbital theory, however, the diagrams can be very difficult to visualise. However, structures built up from hybrid atomic orbitals are much easier comprehend.

Introduction: Methane, CH4

Using the carbon and hydrogen atomic orbitals, methane, CH4, is constructed by overlapping the carbon's one 2s and three 2p AOs with the four hydrogen 1s AOs.

Methane's MOs have a topology similar to the AOs of carbon, but the structure can be very difficult to visualise, so the methane MO construction diagrams A, B and C (below) are shown with the AOs and MOs superimposed upon line structures of the methane. But, remember that the lines are "not there", it is the bonding MOs that hold the molecule together.

It is possible to devise an infinite number of MO methane constructs depending upon how the five nuclei (which are tetrahedral with respect to each other, with the carbon at the centre) are positioned in

space with respect to the x, y and z Cartesian axes defined px, py and pz orbitals. On other words, the four hydrogen atoms can be regarded as sitting on a hypothetical spherical shell which is able to freely rotate with respect to the p orbitals.

We shall examine three methane LCAO MO constructions: A, B and C below.

MO Methane: Construction A

In construction A, the three MOs derived from the carbon's three p orbitals are degenerate, as are the four hydrogen AOs, but it is not entirely obvious how any of the hydrogen AOs are overlapping with the carbon AOs.

The hydrogens appear to be avoiding overlap with the p orbitals.

Note, however, that these LCAO diagrams are schematic, drawn to emphasise LCAO construction rather than attempting to represent the orbital overlap integral.

MO Methane: Construction B

In construction B, the 2px and 2pz orbitals overlap (are bonding) with respect to only two on the hydrogens, the other two hydrogens are on a nodal plane.

The nodal plane is non-bonding with respect to these two hydrogens.

The 2py AO is bonding with respect to all four hydrogen AOs.

MO Methane: Construction C

In construction C, the 2px AO has maximum overlap with one of the hydrogen

atoms, but at the expense of overlap with the other hydrogens and the other p orbitals.

The three p orbitals also appear to be non-equivalent and therefore non-degenerate.

Many authors explain the s H-C bonding in methane in terms of 1s + 2px (construction c) overlap.

The apparent difficulties with each of the LCAO constructions disappear when it is realised that the net carbon-hydrogen orbital overlap integral in methane is the same for any spatial orientation of the hydrogen nuclei with respect to the 2px, 2py and 2pz orbitals.

Molecular orbital calculations using software such as Spartan or Gaussian are performed by assigning the five nuclei positions in space, and then constructing wavefunctions for (or adding electrons into) the molecular orbitals, where the MOs are constructed as a linear combination of the contributing atomic orbitals, the LCAO approximation. Inter-nuclear geometries are varied until the energy is minimised:

Ethane

For symmetry reasons, the MOs of ethane, CH3CH3, are rather different to those of methane as the -bonds and *-antibonds are present.

Ethane possesses 8 atoms, 14 (valence) electrons and 7 MOs (four bonding and three antibonding). These larger numbers mean that it is easier to visualise ethane with preconstructed MOs, rather than attempting to show the full AO to MO construction.

• Carbon-carbon sigma bonding primarily arises from an (end-on/end-on) px + px -> sigma MO. This is the first (and lowest energy) ethane MO, the 1sigma MO.

• This C–C sigma bonding is counteracted by an antibonding 1sigma* MO, although this MO is bonding with respect to the hydrogen atoms.

• Superimposed upon the 1sigma and 1sigma* MOs are a degenerate pair of bonding y and z MOs, but these are counteracted (again along the C-C bond axis) by a degenerate pair of antibonding y* and z* MOs.

• Sitting (in energy) between the and * MOs is the px/px sigma MO.

• Thus – as a first approximation – the only MO which net bonds the two carbon atoms together is the sigmax MO formed by the end on interaction of the two px carbon AOs. The carbon-carbon bond of ethane is a typical carbon-carbon single bond and can rotate freely.

Ethene (Ethylene)

Ethene, H2C=CH2, has six atoms and 12 valence electrons in six MOs.

There are three sigma MOs (two bonding and one antibonding) which are similar to the equivalent MOs in ethane, but the MOs are rather different.

• The z bonding MO is counteracted by a z* antibonding MO.

• This leaves the highest energy bonding MO occupied with electrons (the alkene's HOMO) as the py + py –> MO. This alkene MO dominates the chemistry of alkenes

by inhibiting bond rotation, acting as an electron rich Lewis base centre, allowing addition reaction to occur, etc.

• Also important is the antibonding * antibonding MO, the alkene's LUMO.

Ethyne (Acetylene)

Ethyne, HCCH, has four atoms and 10 valence electrons in four molecular orbitals:

Epichlorohydrin

Epichlorohydrin is an organic molecule of intermediate complexity (and low symmetry). The MOs include:

Arrrrrrrgh... Molecular Orbitals Make My Head Hurt!

Mine too.

Richard Feynman said (here): "I think I can safely say that nobody understands quantum mechanics."

Put another way:

Electron interactions obey various selection rules and as a result give rise to

involved patterns with an intrinsic beauty and sometimes extraordinary complexity.

The electron interaction patterns can be calculated/predicted/modelled using advanced mathematical routines running on fast computers.

However, that that does not mean that electron interactions – quantum mechanics – can be fully known and understood in an intellectual sense. Ultimately, it is just how our world works.

Actually matters are even more involved than implied above. As discussed elsewhere in the chemogenesis web book wavefunctions are mathematically complex entities, and atomic and molecular orbitals cannot be directly observed in principle.

An analogy from engineering: Finite Element Analysis

Finite element analysis (FEA) is a technique for modelling mechanical structures such as: bridges, cranes and con-rods. A mathematical model of the object is created in a CAD program, which is then subjected to virtual loads. The displacement of the structure – the degree of bending – is determined by treating the object as millions of tiny, interconnected triangles, the finite elements. Overall stress is determined by considering the stresses of each finite element. The results can be presented as stress diagrams:

Image captured from L N Engineering.

Finite elements analysis is conceptually simple, unlike quantum mechanics, but it plays a similar role for the engineer as computational chemistry software does for the chemist.

Engineers can design a con-rod and predict the stresses in the solid, physical item. Likewise, a chemist can design a molecule in a software package like Spartan and predict bond-lengths, bond angles, van der Waals surfaces, conformers, rotormers, etc. with great confidence.

An engineer can zoom in and view each hypothetical finite element, if necessary. Likewise, the chemist can look inside the calculations and visualise the contributing molecular orbitals, should she so wish.

There is more on finite element analysis in Wikipedia

Hybridization

Atomic and molecular orbitals are derived from the Schrödinger wave equation, and are actually wavefunctions (waves).

Waves are well understood mathematically, and can be added together or subtracted from each other. Consider two sine waves, the product is a superposition, here:

It transpires that atomic orbitals, being waves, can be added together. The arithmetic can be carried out in various ways.

In molecular orbital theory, atomic orbitals on adjacent atoms are added together to give a linear combination of atomic orbitals (the LCAO approximation) which are used to construct molecular orbitals.

However, there is an alternative approach: The s, p and d atomic orbitals can be added together (mixed or hybridized) to produce hybrid atomic orbitals. Then, atoms in various compatible hybridization states can be joined together into polyatomic molecules. Hybridization, part of valance bond (VB) theory, was presented by Linus Pauling in 1930.

Actually, the MO and VB approached are, in principle, exactly equivalent. It is just that the LCAO approximation is very much easier to implement mathematically and it

methodology has been extensively developed over the past 80 years.

On the other hand, VB theory and hybrid atomic orbitals are much easier for the human brain to conceptualise and understand.

The s and p atomic orbitals can be added together in various ways:

2s + 2p + 2p + 2p               sp3 + sp3 + sp3 + sp3   =>   sp3 hybridised centre

2s + 2p + 2p + 2p               sp2 + sp2 + sp2 + p   =>   sp2 hybridised centre

2s + 2p + 2p + 2p               sp + sp + p + p   =>   sp hybridised centre

sp3, sp2 and sp hybridisation can be associated with carbon, nitrogen, oxygen atomic centres with various charge states. (Only these atoms are discussed here, there are many types of hybridisation.)

The various centres can be associated with well known functional groups, reactive intermediates and fragments.

They can be associated with equivalent VSEPR centres, even though these

are constructed in an entirely different way, not involving waves.

Atomic

Centre

FunctionFormu

la

Hybridisation

State

HybridisationDiagram

VSEPRCentre

VSEPRDiagram

C Alkane CH4 sp3Tetrahed

ralAX4

C– Carbanion H3C– sp3

Trigonalpyramida

lAX3E

C Alkene(part of) H2C= sp2

Trigonal planarAX3

C+ Carbenium ion H3C+ sp2

Trigonal planarAX3

C Alkyne(part of) sp Linear

AX2

N Amine NH3 sp3Trigonalpyramida

lAX3E

N+ Ammonium ion [NH4]+ sp3

Tetrahedral

AX4

N Imine(part of) HN= sp2 Angular

AX2E

N Nitrile(part of) sp Terminal

O Water H2O sp3 AngularAX2E2

O+ Oxonium ion

[H3O]+

sp3

Trigonalpyramida

lAX3E

O Carbonyl(part of) O= sp2 Terminal

O+Acyl

cation(part of)

sp Terminal

The various hybridised atomic centres can be "plugged together" to produce larger polyatomic structures. The rule is that like-joins-with-like:

sp3 + sp3 centres plug together to give structures like: CH3CH3, CH3NH2, CH3OH, CH3CH2

– and CH3NH3+.

sp2 + sp2 centres plug together to give structures like: CH2=CH2, CH2=NH and CH2=O. Notice that the p-orbitals overlap to produce a -molecular orbital, often called a -bond or even just a 'double bond'.

sp + sp centres plug together to give structures like acetylene (ethyne) and hydrogen cyanide. Notice that the two p-orbitals overlap to produce two -molecular orbitals, often called a 'triple bond'.

A molecule like morphine is constructed form hydrogen, carbon, nitrogen and oxygen with the C, O & N atoms in sp3, sp2 and sp states of hybridization:

The VB/VSEPR atomic centres are available as plastic plug-together-atomic-centres in numerous molecular model kits:

Understanding Molecular Structure: A Smörgåsbord of Theories

In this author's opinion it is usually too difficult to understand the molecular orbital structure of low symmetry multi-atom species like epichlorohydrin. We have computers to run sophisticated software that cans keep track of the multiple arrays of complex numbers and calculate energies and equilibrium geometries.

Chemists mix & match theories and make conceptual simplifications when constructing models of molecules.

Polyatomic organic molecules are constructed by plugging sp3, sp2 and sp hybridized centres together. This is literally true when building plastic models of molecules!

The functional group approach, discussed on the next page of this chemogenesis web book, considers large organic molecules as consisting of functional groups (FGs): esters, aldehydes, ketones, carboxylic acids, aromatic rings, etc. When separated by –CH2– (methylene) groups each FG is deemed to behave as a discrete entity.

The Hückel approximation of sigma-pi separability, or - separability, assumes that as electrons are at a much higher energy than the -skeleton electrons and that the and electrons have no influence upon each other. The skeleton of an organic molecule is described using hybridized-VSEPR atoms with the Hückel -system functional groups superimposed 'on top'.

Frontier molecular orbital (FMO) theory models the -system functional groups. For

example, the triene ester, below has a diene HOMO and an alkene LUMO that are able to undergo Diels-Alder cycloaddition to give a bicyclic structure (more here):

Atoms and the Chemical Bond

An ATOM consists of a nucleus, which itself consists of positively charged particles, the protons , and of electrically neutral particles, the neutrons. Most of the mass of the atom is concentrated in the nucleus. In the vicinity of the nucleus -- the nucleus being very small with respect to the volume of the whole atom -- are situated the electrons . These are negatively charged, very small entities. Their number is the same as the number of protons of the given atom and because of this an atom is generally electrically neutral. These electrons are situated at several energy levels in the electric field generated by the nucleus. These energy levels are called shells. An atom is thus a configuration of nucleus and electrons and this configuration is such that the whole system (the atom) is in a state of lowest total energy. As such it is then as stable as possible. The atom accordingly is as it were a stable end-product of a ' congregation ' of the relevant constituents. Which configuration is necessary, in order for an atom to be as stable as possible (i.e. with the lowest possible energy), is prescribed by the Schrödinger Wave Equation of Quantum Mechanics, and this for every species of atom.When, starting with the Hydrogen atom (this is the simplest kind of atom), which consists of a proton (nucleus) and an electron, we want to ascend towards larger (and consequently other) species of atom, we must add protons (and neutrons) to the nucleus, and at the same time a corresponding number of electrons (with the number of protons and electrons being the same, the atoms remain electrically neutral). With respect to the electrons this is not possible just like that. The atom displays 'shells' (around the atomic

nucleus) which can be filled successively with electrons (giving rise, together with the addition of protons and neutrons in the nucleus, to other new species of atom). The Schrödinger Wave Equation dictates that this filling up of shells must take place in a determined way, in order to guarantee a maximal stability of the (new) atomic species. This goes as follows :The first shell (the lowest energy level) can accomodate two electrons only. When besides these still more electrons are added, then these latter must be recieved by a second shell (a second -- higher -- energy level). This second shell can accomodate maximally eight electrons. Yet further addition of electrons demands a third shell which also can accomodate electrons till a certain maximum.Every species of atom represents a Chemical Element.We shall confine ourselves mainly to (an exposition of) the lighter Elements (i.e. to atoms with, say, only two shells). These are the atoms Hydrogen (H), Helium (He), Lithium (Li), Beryllium (Be), Borium (B), Carbon (C), Nitrogen (N), Oxygen (O), Fluorine (F) and Neon (Ne).

In fact we only consider Hydrogen and Carbon, but with respect to that consideration the Elements Helium and Neon are, concerning stable states, also important. The other Elements from the following list are merely meant as illustration with respect to the increasing saturation of the outer electron shell.

The electron configurations of the above mentioned atomic species are as follows (if your browser does not support images then click here ) :

The first shell is saturated (i.e. totally filled) when it contains two electrons.The second shell is saturated when it contains eight electrons. When there are more shells, then all shells up to the last (= outer) one (but this one not included) are saturated. This outer shell sometimes (i.e. in the case of some elements) is saturated, sometimes not. For an atom the most stable state means that the outer shell is saturated. This is, for example, the case with Helium (He) and Neon (Ne). That is why these gases are totally inert, they do not 'want' to change anymore ("To change " here means that the outer shell takes in or gives up one or more electrons. But because in such a process -- taking place in reality -- the number of protons in the nucleus remains the same, the species of atom also remains the

same.). In order to obtain more stability, atoms, having an outer shell which is not saturated, try to give up electrons from the outer shell to another (species of) atom, or take in electrons from another (species of) atom, or try to obtain one or more pairs of electrons in common with other atoms. And all these cases result in obtaining a saturated outer shell. Giving up or gaining of electrons gives rise to ions, these are therefore electrically charged because a difference has been generated between the amount of positive charge in the atomic nucleus and the amount of negative charge of the area surrounding the atomic nucleus. The common possession of pairs of electrons with other atoms gives rise to molecules.These ions and molecules are entities which are more stable than the relevant atoms. A transition from atoms to molecules (which, it is true, in most cases needs a small energy boost), to which we, first of all, confine ourselves, can, provided that some external conditions are satisfied, proceed spontaneously. Because of the lower energy state of the generated molecule energy is given off into the environment. When the situation is energetically the other way round, energy must be supplied, and certain conditions must be such that the reaction does not reverse spontaneously.This is possible.From all this it is clear that the chemical properties of an (species of) atom are determined almost entirely by the condition of the 'outer' shell, i.e. by the configuration of the electrons in the highest energy level.

Until now we considered the distribution of the electrons among the shells. It remains to consider in what way the electrons are distributed within each shell. Only then we will obtain an adequate picture of the state of affairs of the area around the atomic nucleus. But this picture can be really adequate only when we abandon the image of an electron as a particle which performs revolutions around the nucleus.Every material particle, big or small, shows, according to Quantum Mechanics, a Wave-Particle Duality. This means that such a 'particle' (i.e. the same individual particle) sometimes behaves like a (continuous) wave, somtimes as a (discrete) particle. But this phenomenon becomes significant only in the case of very small entities. The electron is effectively the biggest particle for which the wave-particle duality must be taken into consideration. And this determines the behavior of atoms, especially the way they can form bonds with each other resulting in molecules. The wave-like properties of complete atoms or molecules can be ignored for almost all practical purposes.An electron in an atom can be most appropriately imagined as a diffuse cloud spreading itself out over almost the whole atom. Such a cloud, corresponding with a single electron, is at some locations denser than at other locations. When we still think, for a moment, about an electron as a particle, then we could say that the chance to find the particle (electron) is greatest in the denser regions of the cloud. But after this it is best to again forget this particle-image and interpreting the diffuse cloud as the 'real' electron. This diffuse cloud -- (sometimes only) representing one electron -- is in fact something like a ' standing wave ' around the atomic nucleus. Electrons of an atom are standing waves, trapped in the electrical potential field of the atomic nucleus. Because the image of an electron as a cloud is still a little vague we must specify it a little more.The electrons are, as has been said, divided over shells. These shells should not be taken

too litterally, but should be understood as energy levels . More accurately expressed : shells, but not shells alone, determine the energy levels. The division of electrons within an energy level implies a location of electrons in the form of the clouds mentioned. These clouds are called atomic orbitals . An atomic orbital is a cloud with a determined form and orientation, and can maximally accomodate two electrons. When such an orbital contains two electrons then those two electrons are not in the same state, but each of them in a different state. This state is called the spin of the electron. Electron-spin does not at all show any similarity with the rotation (spinning) of a top, but is an abstract state of the electron. Two spin-states are possible, up and down. When there are two electrons in an orbital then one of them has a spin up and the other a spin down. An orbital with two such electrons represents the most stable state. Just like the atom strives for a totally saturated outer shell, because this is a very stable state, the atom also strives for having two such paired electrons in every orbital (in the relevant shell) permitted by the Schrödinger Wave Equation. These two electrons are waves which 'mix' themselves up in such a way that a state of lowest energy results.If we confine our discussion to the lighter elements (i.e. the lighter atomic species) then we find two main types of orbitals. In the lowest energy levels we find spherical orbitals (with the atomic nucleus in the center), and a little higher up on the energy ladder we find dumbbell-shaped orbitals (with the atomic nucleus in the ' waist '). The spherical orbitals are called s-orbitals, and the dumbbell-shaped orbitals are called p-orbitals.

We just said that the dumbbell-shaped orbitals appear at a next higher energy level. This is true for all chemical elements except Hydrogen. Here the energy of the electron levels depends only on the ' shell number ' (meaning in which shell the hydrogen electron is situated), wich implies that in Hydrogen the s-orbital of the second shell (= the 2s-orbital) lies in the same energy level as the p-orbital of the second shell (= the 2p-orbital). In all other elements the s- and p-orbitals lie in different energy levels. The first shell has (in all the elements) only a s-orbital (= the 1s-orbital). When energy is supplied to the atom, their electrons, also the one electron of the Hydrogen atom, can 'jump' from one energy level to another.

(BALL, Ph., 1994, Designing the Molecular World, p. 24, fig. 1.7)

Because the p-orbitals can differ in orientation, three types of p-orbitals are possible, namely px-orbitals, which are oriented along the X-axis of the 3-dimensional carthesian coordinate system, the py-orbitals, which are oriented along the Y-axis, and the pz-orbitals, which are oriented along the Z-axis (consequently posited vertically with respect to the other two).

Figure 1. Shapes of s and p orbitals.(After MOORE, J. & BARTON, Th., 1978, Organic Chemistry : An Overview )

By means of a number in front of the " s " of " s-orbital ", and in front of the "p" of " p-orbital ", we can denote the shell (energy level) to which the relevant orbital belongs. With a number as superscript we can denote the number of electrons in the relevant orbital.The innermost shell always contains an s-orbital only, and in all cases of the existence of more shells containing electrons, this s-orbital consists of two electrons (this is the maximum number of electrons, not only for the orbital, but also for the first shell). Thus we can denote this s-orbital as : 1s2. Now the Schrödinger Wave Equation demands (i.e. predicts) the following state of affairs at the next (higher) energy level (the next shell) :First of all there is again an s-orbital, which can be written as 2s2, but along with this one we find three other standing-wave patterns, with almost the same energy as the electrons in the just mentioned s-orbital, the px, py and pz orbitals, which, as has been said, are dumbbell-shaped, posited perpendicular to each other. Thus the second shell contains four orbitals : 2s2, 2px

2, 2py2 and 2pz

2. And when each orbital of this shell has (received) the maximum number of two electrons -- as indicated in the notation just given -- then the saturated second shell contains eight electrons.This state of affairs becomes still more complicated at higher energy levels (thus -- in this context -- in cases of heavier atoms (atomic species)). But the quantum mechanical Wave Equation exactly predicts how many electrons fit in each shell. Also the form and orientation of the orbitals is predicted by the mathematics of the Quantum Theory.

Having this knowledge it is now possible to precisely set out what the covalent chemical bond exactly is, and this means in what way the first Totalities (molecules) are formed from atoms.

The Hydrogen molecule

We'll start our exposition with the formation of the smallest (above the atomic level) Totality, the Hydrogen molecule . The Hydrogen molecule (H2) is a chemical compound of an hydrogen atom with another hydrogen atom, and in this case the bonding-type is a covalent (chemical) bond :

H+H becomes H--H (= H2)

A covalent bond between two atoms is formed by an overlap of two atomic orbitals (one from each atom) giving rise to a molecular orbital. The two atomic orbitals each contain one electron which is a less stable state. The resulting molecular orbital now contains two electrons (these become, as a pair, a common possession of the two atoms), and this is a stable state. Each hydrogen atom has an 1s-orbital, containing one electron. At the formation of the hydrogen molecule from two atoms of hydrogen these orbitals overlap and now form one molecular orbital with two electrons. Such a bonding molecular orbital now comprises both hydrogen atoms, it has a cylindrical symmetry, and is called a sigma-bond. But things are a little more complicated than this : A rigorous quantum mechanical description of the bonding in the H2 molecule demands that the total number of orbitals must be conserved. We started with two atomic orbitals, so we must end up with two molecular orbitals. One such orbital we have just described, it is called the bonding orbital in which the electron pair resides. The other molecular orbital is empty, it does not contain any electron. The energy of this empty orbital, i.e. the energy of an electron, if it were situated in -- or boosted up to -- this orbital, is higher, than the energy in the atomic orbitals. Putting electrons into this orbital will weaken the bond, since the total energy is then greater than that when electrons occupy just the molecular bonding orbital. The orbital is therefore called an antibonding orbital (BALL, 1994, op. cit.).Because H2 is a more stable configuration than two (freely existing) hydrogen atoms, energy is released to the environment during the formation of H2 from two hydrogen atoms. This is called the bonding energy. The same amount of energy is required to break the bond (here the covalent bond between two hydrogen atoms) up again, in order to get back again the two hydrogen atoms from a hydrogen molecule.So we now have described the formation of the smallest molecular Totality, namely a Hydrogen molecule.Each hydrogen atom in itself is -- considered after the fact -- a result of the gathering of a proton and an electron. This whole is in the most stable state when the electron finds itself in a spherical orbital, a 1s-orbital. This state of affairs is dictated by the Schrödinger Wave Equation. But atoms show a strong propensity to saturate their outer electron shell, and for a hydrogen atom this means that its (only) shell will strive to obtain two electrons (In the case of representatives of the other atomic species, all (except Helium) having more than one shell, the second shell is saturated not

until it contains eight electrons ). The hydrogen atom can achieve this by, for example, entering into a covalent bond with another hydrogen atom. The resulting hydrogen molecule is, according to the mathematics of Quantum Mechanics, more stable than the two freely existing atoms. A H2 molecule is accordingly the stable end-state of the interaction process of the two hydrogen atoms. Expressed alternatively, the dynamical system, consisting of two system elements, resulted (under appropriate external conditions) in the generation of a real intrinsical, but still microscopical, stable Totality, a new being. The dynamical law which has operated in this case is immanent in the two hydrogen atoms, and can be understood from the quantum math. This quantum math describes the energy states of atoms and molecules. And every system strives -- when not interfered with -- for a lowest energy state, because that guarantees the highest stability.

The Methane molecule

Methane is an ' organic ' molecule, it is moreover the simplest organic molecule.This terminology dates back to the time in which one was convinced that such molecules (such compounds) could be fabricated only by organisms. This turned out not to be true. "Organic compounds" nowadays means : Carbon compounds, with the exception of some very simple ones, as Carbon dioxide (CO2), and the like.

Methane (CH4) is a compound of one carbon atom (C) with four hydrogen atoms (H). Also in this case I am not concerned with the actual production of Methane (for example in an industrial process), but in what way Methane -- after the fact -- is composed from its constituents.The electron configuration of Carbon (C) is : 1s2, 2s2, 2px, 2py, i.e. an inner, first shell with an s-orbital, containing two electrons (1s2), and a second (and last) shell containing an s-orbital, containing two electrons (2s2), and three (dumbbell-shaped) p-orbitals (2px, 2py and 2pz, perpendicular to each other. 2px and 2py contain one electron each, 2pz is empty. Bonding with four hydrogen atoms -- each with one 1s-orbital, containing one electron -- could be viewed as an overlap of the four orbitals of the outer electronshell of Carbon with the four times one = four orbitals of the four hydrogen atoms. In this way we should obtain four covalent bonds, three of which are perpendicular to each other, namely there where we have overlap with the 2px, 2py and 2pz orbitals, and a fourth covalent bond which does not have a specific orientation, because here we have an overlap of two spherical orbitals (1s of the hydrogen and 2s of the carbon).But it is improbable that Methane is structured that way, because the four bonds would not have a symmetrical position with respect to each other. Bonds consist of electrons, and because like electrical charges repel each other, these bonds try to be as far away as possible from each other. The four bonds (in Methane) will diverge maximally from each other, and this means that the complete spatial structure of Methane should be a regular tetrahedron with a carbon atom in the center and the four hydrogen atoms at the vertices of the tetrahedron. Observations show that this is indeed the case (concerning the Methane molecule). OK, but what must happen to the orbitals in order to realize this?An electron from the 2s-orbital of the carbon atom is transferred to the empty 2pz orbital, which results in now having four orbitals each containing one electron : 2s, 2px, 2py and 2pz. These four orbitals are mixed up in such a way that four new, totally equivalent orbitals appear, each still having one electron. Such an orbital is called an sp3 hybrid orbital , and we now have four of them. And so we have obtained sp3 hybridized carbon. An overlap of each sp3 orbital with a 1s-orbital of a hydrogen atom is now possible, resulting in four totally equivalent covalent bonds, which are, just as in the case of the hydrogen molecule, sigma-bonds. Because they are equivalent, these bonds can now form the axes (reaching out from the center to the vertices) of a regular tetrahedron. Each molecular orbital (in the case of Methane) contains two electrons, which is a stable configuration (i.e. more stable than the orbitals possessing only one electron before the bond was generated), and these two electrons are symmetrically distributed around the axis between the two bonded atoms, and because of this it is called a sigma bond.

Figure 2. Orbital construction of Methane (After MOORE & BARTON, 1978)

Also in this case (of Methane) the configuration is more stable than the case of four free hydrogen atoms and a carbon atom before the bonding took place. So also here we can speak of a dynamical system, consisting of five system elements, namely four hydrogen atoms and a carbon atom. This system generates a stable Totality -- a Methane molecule -- when certain external conditions are satisfied. Expressed in another way : When these external conditions prevail, then the dynamical law for (the generation of) Methane (i.e. the generation of a Methane molecule) will ' start to operate ', and, consequently, the Totality, the Methane molecule, will appear. This dynamical law can be understood on the basis of the quantum math, which predicts the energy states.

Carbon--Carbon Bonds. The Ethane molecule

Carbon plays an important role in the formation of molecules that make Life possible, because the orbitals of two carbon atoms can overlap resulting in strong carbon--carbon bonds. In this way a chain of carbon atoms of unlimited length can be generated, making possible the formation of very complex molecules. Being "complex molecules" means that such molecules can perform FUNCTIONS, very special functions, as is the case with enzymes. This capacity to form long stable chains is unique for carbon. Other chemical elements, for instance Silicon, can form only short chains, because the bonding-energy is low (so only relatively little energy is needed to break up these bonds again). Thus carbon is the very element making possible the formation of very large complex biomolecules, and consequently the very element for the formation of Life. The bio-liquid in which the life processes take place is Water. Because of the very special electrical properties of water, it is, in spite of the fact of the smallness of its molecules, at normal temperatures a liquid. In living things this liquid serves as a medium of transport.

The special electrical properties of water just mentioned consists of the following : The propensity to the formation of hydrogen bonds between the molecules. Because of this the molecules are less free and

water remains a liquid at normal conditions.

With the help of the theory of orbitals we can show the structure of the carbon--carbon bonds (of course (the explanation of) this bond is also based on the quantum math).When we have a sp3 hybridized carbon atom, thus with four sp3 orbitals, then three of them can overlap with, say, hydrogen (Here we expound the formation of the Ethane molecule). In this way we obtain a so-called Methyl radical, CH3. This still contains a left-over sp3 orbital. We now can couple two such methyl radicals together by letting these left-over sp3 orbitals overlap each other, resulting accordingly in a sigma-bond between the two carbon atoms (The bonds with the hydrogen atoms also are sigma-bonds).What we obtain is an Ethane molecule, H3CCH3 (= C2H6).

Figure 3. Orbital construction of Ethane (After MOORE & BARTON, 1978)

Also in this case a more stable (with respect to the free hydrogen atoms and free carbon atoms) Totality is generated, an Ethane molecule, in which all atoms possess a totally saturated outer electronshell, because of having electron pairs in common. Moreover each orbital contains two electrons and so it is stable too.The dynamical system, with system elements H, H, H, H, H, H, C and C, generated the Totality H3CCH3, or, denoted in a more elaborated fashion :

H HCH

HCH H

It generated an Ethane molecule. This is accordingly a pattern, generated from the system elements (initially randomly located and not bonded) mentioned. This pattern is (in the present case of an Ethane molecule) 3-dimensional, meaning that it is not the case that everything of the structure lies in one and the same plane as the diagram might suggest. Also here the bonds of the carbon atoms show a tetrahedral orientation.

Also the formation of this Totality is dictated by the quantum formalism which predicts the energy states.

The Ethylene molecule

In order to understand the formation of an Ethylene molecule, and consequently the formation of a double bond, we must dig a little deeper in the phenomenon of the hybridization of atomic orbitals. In the sp3 hybridization, considered above, all three p-orbitals, 2px, 2py and 2pz were used. These were mixed up with the 2s-orbital, resulting in four equivalent orbitals, which, after overlapping with the hydrogen orbitals, form the four equivalent bonds (in the case of Methane).But it is also possible that hybridization (of, say, carbon) involves only two p-orbitals, resulting accordingly in three sp2 hybrid orbitals. Because of the mutual repellence these orbitals order themselves in a plane and include angles of 120 degrees between each other. It is also possible that only one p-orbital is involved in the hybridization, resulting in two sp hybrid orbitals. Because of their mutual repellence they will orientate themselves in each others direction, i.e. they will be in line with each other, making an angle of 180 degrees with each other. The remaining p-orbitals, not involved in the hybridization, orient themselves perpendicular to the plane of the sp2 hybrid orbitals and perpendicular to the axis of the sp hybrid orbitals.

Carbon atoms can also form double bonds with each other. Such a covalent double bond has a particular structure, which we shall explain with the (help of the) formation of an Ethylene molecule.When we have sp2 hybridized carbon (which as such is generated in the case of certain bondings), then we have three sp2 hybrid orbitals. And a p-orbital was still left over, oriented perpendicular to the plane of the three sp2orbitals. When we now couple two hydrogen atoms to two (out of the three) sp2 orbitals of the carbon atom (consequently everytime an overlap of an 1s-orbital -- containing one electron -- of hydrogen, with an sp2 orbital -- containing one electron), then we obtain a H2C unit in which there are still two orbitals left, having both only one electron, namely an sp2 orbital and the left-over p-orbital.We can now couple two such units together by letting their sp2 orbitals (which have not participated in the bonding with hydrogen) overlap (resulting also in this case in a maximally filled -- with two electrons -- molecular orbital). Consequently the carbon atoms will then be bonded to each other with a sigma-bond. But both H2C units also still have the mentioned left-over p-orbital, i.e. two p-orbitals perpendicular to the other

orbitals. These can overlap sideways with each other resulting in a stable molecular orbital. But because this molecular orbital is perpendicular to the remaining orbitals, the bonding is still, it is true, a covalent one -- because it is based on a common possession of one (or more) electronpairs -- but is of a different type (within the covalent bonds) than the sigma-bond. Such a bond is called a   pi-bond.While the bonding-orbital in the case of a sigma-bond lies in the axis (conceptually) connecting the two bonded atoms, it lies in the case of a pi-bond above and below (but not in) that axis. The resulting molecule is an Ethylene molecule, H2C==CH2. It reveals the following pattern :

H H C C H H

It is a flat molecule with the sigma-bond between the two C-atoms lying in that plane (The same is the case with the sigma-bonds with the four hydrogen atoms) and the pi-bond between these two C-atoms lying above and below that plane.The double bond between the carbon atoms thus consists of a sigma-bond and a pi-bond.

Figure 4. Construction of the double bond (After MOORE & BARTON, 1978)

Also in this case a stable Totality has been formed from the system elements H, H, H, H, C and C.

In a comparable manner also the generation of a triple bond between two carbon atoms is possible. When they saturate themselves subsequently with hydrogen then we have Acetylene, HCCH, where the C-atoms are bonded with each other by means of three bonds.

Isomerism

It often occurs that a same number and type of atoms, i.e. one and the same collection of atoms, can, in each case, form different (possible) compounds, each consisting of all the atoms of that collection, thus (can form) different configurations of those constituent atoms, for example in the case of the hydrocarbon Buthane.Buthane consists of four C-atoms (= carbon atoms) and ten H-atoms (= hydrogen atoms), C4H10.   A possible structure of this set of atoms is a straight C-chain, another one is a branched C-chain :

H H H H H C C C C H H H H H

and

H H H C H H H C C C H H H H

Thus a same collection of four C-atoms and ten H-atoms can give two different patterns of those same atoms. These are two different -- concerning content -- Totalities, but having the same empirical formula C4H10. They are called isomers .Because such molecules have different forms (patterns) they also have different physical properties, for instance different boiling-points, melting-points and solubilities. Accordingly the straight chain isomer of C4H10 has a boiling-point of 0 degrees Celcius and a melting-point of minus 138 degrees Celcius, while the branched isomer of C4H10 has a boiling-point of minus 12 degrees Celcius and a melting-point of minus 159 degrees Celcius.Here we see how properties (in this case per se properties) eventually must originate from concrete structural aspects of the Totality (the uniform thing) in question.The boiling point, melting point and solubility are, it is true, 'bulk-properties', and so it does not make sense to speak about the boiling point etc. of a Buthane-molecule. But such bulk-properties are rooted in the properties of the individual molecules, for instance their form,

their electrical properties, their weight.

When the chain lenght increases, the number of isomers grows very rapidly.When the carbon is bonded not only with hydrogen but also with other chemical

elements, for instance with oxygen (O) then there are even more possibilities for isomerism. Let's look to two compounds with the formula C2H6O, where one of them contains two C--O bonds and the other a C--O bond and a O--H bond. These isomers possess very different chemical properties, one of them is an unreactive ether and the other is a very reactive alcohol:

H H H C O C H H H

Dimethylether

H H C H H C H O H

Ethylalcohol

The ether is almost totally insoluble in water, while the alcohol is infinitely soluble in water.

The fact that the extremities of a carbon chain can bond with each other, resulting in a cyclic structure, offers new possibilities for isomerism.The formula C5H10 can represent a 5-carbon chain, straight or branched, with a double bond, or a ring structure with single bonds only.

A very interesting recently discovered (and synthesized) family of molecules are the Fullerenes. These are molecules consisting solely of carbon atoms. Until this discovery only two types of carbon 'molecules' were known : Diamond and Graphite. Both are crystalline structures consisting of an indefinite number of carbon atoms, arranged in periodic arrays. So these are crystals -- and so not molecules in the strict sense.But the recently discovered carbon structures are real molecules, because in them a definite number of individual carbon atoms is involved. Such a molecule consists of carbon atoms arranged, not in a periodic way, but forming a closed cage of relatively few atoms.The first discovered (in the 1980's) Fullerene is a molecular cage consisting of 60 carbon atoms. The cage is a closed hollow ball of 60 carbon atoms arranged in the form of a soccer ball. This ball is in fact a polyhedron consisting of twelve pentagons and twenty hexagons, just like the patches of a soccer ball. The carbon atoms reside at the vertices

(one atom at each vertex). Because in such a structure each atom is connected to three others, one of those three bonds must be a double bond, making four bonds for each carbon atom (so no bonds are unsatisfied dangling bonds). The inspiration that the new molecule should have this cage-like structure consisting of polygons, came from the structure of the geodesic dome at the Montreal Expo of 1967, designed by the American architect Richard Buckminster Fuller. So the new molecule was christined buckminsterfullerene.

Structure of BuckminsterfullereneTop :   The structure of the sixty-carbon-atom cluster, called buckminsterfullerene.Left :   The pattern of single and double bonds which allows all of the atoms to form four bonds.Right :   The pattern of hexagons and pentagons in this highly symmetrical shape is the same as that in a soccer ball.Each carbon atom in the cage is equivalent.(After BALL, 1994)

Because of mathematical reasons such closed molecular cages can only be formed when, in addition to a number of hexagons, there are precisely twelve pentagons present in the structure ( NOTE 4 ). A second restriction is that the pentagons should not be located adjacent to each other -- this would give rise to an unstable bonding pattern, which would rapidly rearrange. All pentagons must be isolated from one another. This clearly is a chemical restriction. For C60 (buckminsterfullerene) there are 1812 possible ways of arranging the pentagonal and hexagonal rings, but only one in which no pentagons are adjacent. So C60 has no isomers.But also larger fullerenes have become known, and these do have isomers. But their

number is limited because of the restriction mentioned. So C76 has just two, it seems, and C78 perhaps eight or so. The giant fullerenes C120, C240 and C540 are expected to have some particularly symmetrical isomers, but these molecules haven't yet been isolated in sufficient quantities to put this to the test.

So it is clear from the first examples that isomers, having, accordingly, a same empirical formula (= a same number and type of atoms), can strongly differ from each other. Such a group of isomers represents a collection of possible combinations (configurations) of a same mumber and type of atoms.Seen after the fact it is clear that the same dynamical systems (in which " the same " means : consisting of a same number and type of system elements) can result in the generation of different (with respect to content) Totalities. This implies that those dynamical systems turn out to be not the same, with respect to content, only their pool of elements was the same. It follows that the Dynamical Law for such a Totality (generated thing) is not equivalent to the complete collection of system elements, because in that case only one dynamical law would be possible. In the case of isomerism many Dynamical Laws are possible, relating to a same pool of system elements. And in the case of the generation of the several Fullerene species we see that a large enough collection of specifically the same elements can give rise to several possible species of Totality. Now it is also clear that the nature of such a dynamical law (and, in general, every dynamical law) is abstract.

Let's give a brief exposition about the seat of a dynamical law, and how the above mentioned phenomena are thought to be possible.(See for more details the Essay on Dynamical Systems ) The Dynamical Law of a dynamical system must reside in some way in the system elements, more precisely in some of the properties of the system elements.The actually existing properties relating to each freely existing system element (i.e. as such distinguished from the Essence of such a system element) either are actual and constant structural configurations, or an interaction (result) with external factors. These last mentioned actually existing properties of a system element are as such dependent on the outside world. In this context the " outside world " not only comprises the remaining system elements but also the other entities not belonging to the dynamical system as such (not being system elements). Abiding in all these actually existing properties (including the constant properties) of all those system elements is the Dynamical Law relating to the generation of a certain Totality (a uniform thing), predetermined by this law. Altered external conditions result in a change of some of these properties (i.e. resulting in an exchange of some properties) implying that a part of the actually existing properties is now different, and this new garnish of actually existing properties of system elements now harbours a new (i.e. different with respect to content) Dynamical Law, resulting in the generation of a different, with respect to content, Totality.Of course the effects of environmental changes on (the alternation of) actually existing (i.e. actually being present) properties of system elements are partially rooted in the nature of those system elements themselves (i.e. in certain constant structures in each of those system elements).

Molecules are Totalities, each consisting of a definite and fixed number and proportion of individual atoms [ in contrast to crystals, where the proportion is, it is true, also constant (except in the case of mixed crystals, that constitute a special problem, discussed earlier), but where the number of individual atoms participating in the crystal structure is not fixed :   a single crystal consists of the periodic stacking of microscopic units in three directions, resulting in the fact that its size is not

fixed ]. A molecule consists of a definite number of parts giving rise to a definite shape and size. Such a structure we call tectological (in contrast to that of single crystals). This tectological structure we also encounter in certain twinned crystals, that can be considered as consisting of several single crystals grown together in a regular and repeatable way, according to one or another twin law. And almost all organisms, and their parts, show this structure too, which allows them to have certain definite imaginary body axes, that, together with the quality of their poles (namely being homopolar or heteropolar) allow for their stereometric basic form to be determined. On the basis of this we can classify such organisms as Axonia, while the few organisms that do not admit of such a stereometric basic form to be determined, can be called Anaxonia (The Axonia can be further classified into Homaxonia, Polyaxonia, Monaxonia, Stauraxonia and Spiraxonia, depending on the differentiation and nature of their axes and poles).

On the second (part of this) website, accessible by clicking on the last entry of the contents frame Continuation of this Series, we have laid down the complete theories about the tectological structure (TECTOLOGY) and stereometric basic form (PROMORPHOLOGY) in organisms.

Many molecules, especially the smaller ones, are Axonia, they all have a definite stereometric basic form. A nice example is the Methane molecule, CH4, that has the shape of a regular tetrahedron. And of course such a form is one of the effects of the relevant Dynamical Law.

In all the above we have demonstrated how the very small Totalities generally look like, and in what way their structure is dictated by energy-states.Because the carboncompounds can attain any degree of complexity, they can constitute the machinery of Life and Consciousness.

Crystals

In addition to molecules chemical bonds also occur in Crystals.In many crystals these are ionic bonds. For example the constituents of a crystal of common salt (NaCl) are held together by such ionic bonds. These bonds originate by reason of the fact that atoms (as we now know from the foregoing) strive for a fully saturated outer shell. Sodium (Na) possesses one electron in the outer electronshell.

Saturation of this shell would demand seven more atoms. But it is easier just to give up that one electron, resulting in the last shell but one becoming the last (= outer) shell, and this one was already saturated (otherwise the atom would never have started to fill a next shell). Because of the giving up of this electron the sodium atom becomes positively charged, it now is a sodium ion. Chlorine has seven electrons in the outer shell. It saturates this shell by taking up the electron that was given up by the sodium atom. Because of that the chlorine atom becomes negatively charged, it has become a chlorine ion. The positively charged sodium ions attract (with an electrostatical force) the negatively charged chlorine ions, resulting, when they form a supersaturated solution in water, in a crystal lattice, in which each sodium ion tries to collect as many chlorine ions as possible, and vice versa.In several kinds of other crystals (crystal species) the constituents are held together by covalent bonds. Also other bonding types (for instance the hydrogen bond) can play a role in the formation of crystals.

So in this Essay we have learned that the Chemical Bond is crucial for the possibility of the formation of whatever Totality above the atomic level.In Organisms this means the formation of complex biomolecules, which together can form a macroscopical pattern. The latter will involve concentrations of such, and other, molecules, especially concentration differences and aggregations steered by intrinsic dynamical laws. In addition to the resulting, generally dynamical, structures (which we can already encounter in non-living dissipative structures), organisms also display a never-ceasing and coherent chemism. Important for all this is the capacity of an organism to extract energy from the environment, by which it can maintain its far-from-equilibrium state (in thermodynamic respect), and thus can continue to function.In the case of crystals matters are different. They are almost completely statical Totalities without a metabolism. They are, just like the molecules, equilibrium structures.

Atomic orbitals

So far, we've seen that we can explain some experimentally observed properties using simple models like Lewis Dot structure and VSEPR. These models still do not explain many chemical properties. In this section, we will develop a new model that explains further some of the chemical reactivities we observe and also ties in our knowledge of atomic orbitals and expands it further.

So far, we've been saying that molecular bonding involves atomic orbitals in some way. We will now explore atomic orbitals and develop an understanding of their involvement in molecular orbitals (e.g. bonds).

Lets look at the VSEPR models identify the places where they are lacking in explanations. We'll then look at the model of localized molecular orbitals and see how this new model is better.

To make a molecular orbital (bond), we must use an atomic orbital from each atom involved in the bond.  What do these atomic orbitals look like?

In the molecule BeCl2 , we saw that there are two electron domains on opposite sides of the beryllium atom involved in the bonds to the two chlorines.  How do these domains arise? Our atom of beryllium has atomic orbitals for its valence shell (n = 2) of type s and p. If we try to make bonds using two of these orbitals, we find that we cannot come up with two orbitals that are identical and have lobes on opposite sides of the Be.

Image of probability domains of electrons in orbitals 2s and 2p. The color indicates the phase of the wave-function (the wave-like

properties of the electron).  In-phase functions add positively (constructive interference) and out of phase functions add

negatively (destructive interference).

If we use one of these orbitals for one bond and the other for the other bond, we will obviously not have two identical bonds. We obviously need another model for the atomic orbitals in order to explain their participation in bonding.  One thing many students forget when thinking of this process is that the orbitals them selves are not really entities and are not really the thing that matters. What matters is the overall energy of the system and its symmetry. An atom in free space is symmetrical (as much as its electrons will permit. The atomic orbitals s, p, d, f,... are merely the basis functions that we use to add up to the overall space occupied by the electrons.  Note that any sub shell always adds up to be spherically symmetrical.  We can easily choose a different set of basis functions (orbitals) to describe the space occupied by the electrons. Whatever basis set we use must exactly define the same space as our spdf model.   We are free to divide up the space any way we want in order to more easily understand and calculate what we need.

What we need to do is use different (linear) combinations of the spdf domains to create new domains (orbitals).

We divide up the space (sphere) in such a way that each domain (orbital) of the sphere is involved in only one bond. This makes it much easier for any calculations we might have to do.

Let's first consider what happens in the BeCl2 case:

First, we overlap the 2s and 2p in two different ways. (remember, we're just creating new functions, not moving electrons)

Here, we see that the two different ways of adding up the orbitals result in the phases interacting indifferent ways. In the 2s+2p case, the left-hand side will be smaller since the phases of the original s and p are opposite and vice versa for the 2s-2p case.

The two new orbitals we get will look like the following picture.

Two sp hybridized orbitals made from

     

As an analogy, let's consider a point in 2 dimensional cartesian space.

If the point is represent in the normal set of cartesian coordinates (basis set), we need an x value and a y value. On the other hand we can represent the same point using the alternate (rotated set) coordinate system (alternate basis set) In the rotated system, notice that one of the coordinates is zero (It still exists though.) and we can more easily ignore it in any calculations involving the point.  We do the same with the space occupied by the atom's electrons. Redefine the basis set we use so that we can simplify the mathematics.

 

 

   

an s and a p orbital. The two hybrid orbitals have axes that are 180º apart.

Now, we can see that the two different orbitals (coloured blue and green) still have properties that look like the p orbitals (two lobes, of different phase) and some properties like an s orbital (large bulbous, surrounding the nucleus (a bit). Most importantly, the two will sum up to make the same shaped space as has using the spdf model so these two orbitals will exactly describe the same space as the original s and p orbitals did.  Now, we can look at using these to make molecular bonds.

In the case of BF3, we have three fluorenes bonded to the central boron with what are experimentally observed to be identical bonds.  Again, we cannot find a way of using the spdf model to easily describe the orbitals used for this so we will re-slice the sphere to make use of the space in a new way.  First, we need only use three orbitals to create three bond so lets use s,p,p and leave one p orbital untouched on the atom.

An s and two p orbitals add up their shapes to give a fat disk (note the color coding is used here to indicate different orbitals, not different phases). The

three orbitals are 120º apart.

In the diagram here, we show the electron densities only resulting from each of three orbitals, s, p, p. colored differently If we add up the three sets of densities we get a

picture of the space we need to divide up.  It is a thick disk, sort of like a ball what is somewhat squashed in one direction.  We now divide this up into three equal domains.

Three sp 2 hybridized orbitals are made from an s and two p orbitals. Each hybrid has some p

character and some s character.

and see that these three also add up to regenerate the original fat disk the the spp did.

Here, we see the three orbital spaces added back together.  Color coding helps remind us of

the original sp2 hybridized orbitals.

The next set of hybrid orbitals is not so easy to show using these electron-cloud diagrams since the resulting set of hybrids will be evenly distributed in three dimensions rather than in just two.  Hence, I will resort to stick diagrams to represent these orbitals.

In the molecule CH4 there are four equal electron domains on the carbon each involved in one bond with a hydrogen.  How do we explain these using localized atomic orbitals.

First of all, recall that each sub shell will sum up to be a sphere.  That means that if we add all the p orbitals with the s orbital for the n=2 level (or any n level), we still get a sphere.  We divide the sphere into four equal parts and come up with domains that are 109.5º apart.

Here are the four sp 3 hybrid orbitals after dividing up the space created by the s and the three p orbitals. The four new orbitals are all equal and have axes that are 109.5º

apart.

More complicated hybridization can be found for atoms in the n > 3 shell. In these shells, there are d orbitals that can participate in the hybridizing. Thus, if an atom forms 5 bonds with 5 other atoms, it will need 5 atomic orbitals to do this.   We take s, p, p, p, d of our n = 3 available orbitals and combine them.   The resulting set of sp3d hybrids will have a trigonal bipyramidal arrangement exactly as describe in the VSEPR section. Finally, if there are six bond to six atoms, the hybridization will be sp3d2 (6 atomic orbitals needed) and the geometric arrangement of the electron domains will be octahedral (see VSEPR section).

Molecular orbitals (Bonds and antibonds)

Valence Bond approach: (Localized Bonding)

In the Valence-bond approach, we assume that the molecules are made up by simply overlapping the atomic orbitals (hybridized if necessary) from the individual atoms.  In this approach, the electrons from one atomic orbital AO don't interact with those of other AOs.  We get great three-dimensional pictures of what the molecules and bonds may look like but the information we can get from this approach is still not perfect.  We'll see MO theory later that can be used in parallel to obtain other information that this approach does a poor job at calculating.

We've seen various ways of visualizing atomic orbitals. It is important to remember that these orbitals are only models of what happens on an atom in the gaseous state (unattached to anything).  How we deal with bonding these atoms together to form molecules is a whole different story again.

To form a bond between two atoms, we must combine atomic orbitals from the two atoms in such a way that the energy level of the combination (molecular orbital, MO) is lower than the original atomic orbital (AO). This is most easily visualized using the s orbitals of two hydrogen atoms as they approach each other.  Consider, for a moment the following diagrams.

 

Two H atoms approach each other. Their Atomic Orbitals begin to overlap.

Overall space is shaped like a rugby ball.  We now divide this space into two new (Molecular) orbitals.

The best way to do this is such that the energy of the resulting molecule is lowest. In the case of hydrogen, there are two electrons to place in the new orbitals.  If we define them such that one has a much lower energy than the other then we will see the stable molecule form when both electrons are situated in the low-energy orbital while the high-energy orbital is empty.

The division that best accomplishes this is one where the orbitals are added in phase (with normalization) and out of phase as is shown below.

in-phase                          out-of-phase

Molecular orbitals created by taking linear combinations of atomic orbitals (LCAO). On the left is the in-phase

combination (addition) of the two atomic s orbitals ( orbital; Bonding orbital) and on the right is the out-of-phase

combination (subtraction) of the same two orbitals (* orbital; anti-bonding). These two add up to give the same

space as the two original atomic orbitals but now each is a single 'localized' orbital spanning the whole molecule.

We see here the combination of two atomic s orbitals on Hydrogen atoms to form an H2 molecule.  Just like in the atomic case, we always end up with the same number of orbitals that we started with.

If we look at the energy of these two orbitals we come up with a correlation diagram like this

The Molecule H2 has an electron configuration 1s)2.  It has an overall bond order of 1.

The two electrons from the hydrogen have a low-energy orbital and a high-energy orbital available to them.  Provided they can loose energy (collision with the walls or a third atom or emission of light photon) the electrons will settle into the lower energy level and hence a net release of energy is observed. This released energy is called the bond energy.

The molecular electronic configuration can be named using molecular orbitals just as we saw in individual atoms.  In this case, the only occupied orbital in the hydrogen atom is the s-orbital that was created from the 1s orbitals of the individual atoms.  This molecular orbital (MO) is thus called a 1s orbital.  Thus, since both electrons are in this MO, we have a very simple molecular electron configuration 1s)2.

If we try the same thing using Helium. we can quickly see from the correlation diagram below why the helium does not form a He2 diatomic molecule.

The He2 molecule will not remain stable since it's overall bond order is zero.  The MO electronic configuration is 1s)2*

1s)2, i.e., one bond and one antibond.

There are four electrons in the molecular orbitals which means that while two electrons can go to a lower energy level, the other two must go to the higher one.  The total energy is unchanged.  Hence, the two atoms don't form a bond at all since there is no net release of energy.

We will revisit this further.  First, let's look at a different type of bond.   The bond we just looked at is a bond that has it's major lobe on the axis joining the nuclei of the two atoms.  This is in contrast to a bond where the overlap is above and below (or on opposing sides) of the axis.

The two types of bond theory we're discussing in this section, VB and MO theories give virtually identical results for the H2 or He2 cases.  Both predict the shapes and energies to be the same.  The difference comes into play when there are more than only one orbital on each atom as in n>2.  We'll look at MO theory later.  For now, let's continue with the VB discussion.

Here, we see that the overlap occurs off axis.  It may be noticeable that the overlap is not as great as it could be if the lobes were overlapping end-to-end (  bond)

We now see that the definition of and bonds does not depend on the type of orbitals used in their creation.  They refer to the location of the overlap region with respect to the bond axis.

There are two different models that Chemists use when describing bonding in molecules.   The one they choose depends on their purpose.  For some purposes, VB theory is easiest;  hybridized orbitals coming together to form and the left-over atomic orbitals (p and d) form bonds.   This is most easily seen in organic molecules.  The simplest of these is, of course, CH4 in which we've already seen the carbon atom to be sp3 hybridized and the carbon sp3 orbitals form bonds with the hydrogen s orbital.

What about a molecule with double bonds?  Let's look at the simplest of these;  

C2H4.

If we draw these orbitals, using solid lines to represent the shape of the orbitals rather than the electron-cloud images, we get something like the following.

First, we hybridize an s and two p orbitals on each carbon to form trigonal planer sp2 orbitals which we use in making bonds.

The bonds on the ethene molecule.

Now we look at the p orbital from each carbon that we didn't yet touch and use it to form a bond.

The bonds on the ethene molecule

The complete picture looks like the following

The and bonds on the ethene molecule

This scheme does a good job of helping us understand the properties of this molecule.   We can see that the molecule is rigid and planar. It cannot rotate about the central C=C double bond because of the way the p orbitals overlap. We also see that the bond angle should be ca. the trigonal planar angle of 120º. the HCH angle is actually slightly smaller while the CCH angle is larger.

Triple bonds are even harder to draw. Consider the molecule ethyne C2H2.   The carbons on this molecule are sp hybridized, leaving two p orbitals on each to form two bonds. The final picture looks something like the following diagram.

The and and bonds on the ethene molecule.

Again, we see this model does a reasonably good job of modeling the shape of the real molecule.  Ethyne is linear and has a CC triple bond.

In both the cases previously, the bonds used are localized orbitals.

Molecular Orbital  approach: (Distributed orbitals)

Another model chemists studying inorganic compounds use is a more complete molecular-orbital approach.  The previous model, for example is not useful at all in describing diatomic molecules like B2, C2, N2 and O2. For this, we need a more complete Molecular Orbital approach where all atomic orbitals can interact simultaneously to form molecular orbitals.  In this approach, we don't bother with the hybridization.  All Atomic orbitals in the valence shell are involved in producing a molecular orbital scheme that mimics the experimentally observed properties.  The four molecules mentioned above all use their n=2 valence shell to do the bonding.  Let's look at the correlation diagram that result when the 2s and 2p orbitals from two such atoms interact.  It is important to note that the MO energy levels 2p) and 2p) may not remain in this sequence for all atomic pairs.  For example, the 2p) levels are higher then the 2p) for O2 and F2.  For our purposes, however, it will be sufficient to always use this one correlation diagram for any n=2 diatomic molecule, recognising that it is not necessarily an exact representation of the energy levels for certain molecules.

First, we'll consider the Boron molecule.  It has three electrons from each atom for a total of 6 electrons that we must place in the molecular orbitals.  So the final scheme will look like the diagram below.

In this diagram, we can easily see that the MO electronic configuration is 2s)2

2s)22p)2 for an overall bond order of 1  ( (1 bond + 2 half bonds – 1 antibond)) but where the 2p) orbitals contain unpaired electrons (Hund's Rule) because they are degenerate (Identical energies).

This model predicts that the boron diatomic molecule is paramagnetic (two unpaired electrons).  This is exactly what is observed experimentally.  Note also that the bond order is essentially 1.  The bond and * anti-bond cancel out, leaving the two bonds each half occupied. How does this correlate with the Lewis dot structure?

. ..B:B.

The molecule of C2 is different. Experimentally, it is diamagnetic (no unpaired electrons) and has a bond order of 2.

In this diagram, we can easily see that the MO electronic configuration is 2s)2

2s)22p)4 for an overall bond order of 2 (3 bonds – 1 antibond).

 

We see that there are no bonds in carbon dimer.  It is held together with bonds only.  This contradicts the other model where hybridization always predicts the first bond in a multiple bond is a bond.

Nitrogen molecules are triple bonded as we can see in the following diagram.

In this diagram, we can easily see that the MO electronic configuration is 2s)2

2s)22p)42p)2 for an overall bond order of 3 (4 bonds – 1 antibond).

We see that there is a 2p) bond and two 2p) bonds holding the molecule together in this model.  The 2s) and *2s) cancel each other out and don't contribute to the bonding.

Finally, Oxygen is paramagnetic with a bond-order of two.  Lewis dot and VB theory doesn't properly predict the paramagnetic properties.  A good Lewis dot diagram of O2 shows a double bond with two lone pairs on each oxygen.

 

In this diagram, we can easily see that the MO electronic configuration is 2s)2

2s)22p)42p)22p)2 for an overall bond order of 2 (4 bonds – 2

antibonds).  Again, the upper two electrons are unpaired in the 2p

degenerate orbitals.  This molecule is paramagnetic because of Hund's Rule.

Hund's rule says we must put the last two electrons to into the orbitals one at a time. This gives us two unpaired electrons (Paramagnetic) and lowers the overall bond order to 2.

We can use these same diagrams for diatomic molecules or ions for any combination of n=2 elements.  for elements of n=3 or n=4, there is no guarantee that the sequence of the MOs will remain the same.  It will always remain as pictured above for n=2 valence levels.

Let's Try the NO molecule:

In this diagram, we can easily see that the MO electronic configuration is 2s)2

2s)22p)42p)22p)1 for an overall bond

order of 2 ½ (4 bonds – 1½ antibonds).  Here, the upper electron is unpaired in one of the two

2p orbitals.  They are no longer degenerate since one is empty.  This molecule is paramagnetic, containing an odd number of electrons.

CHEMISTRY OF ETHYNE    

INTRODUCTION     Molecular formula = C2H2

   Empirical formula = CH   Molecular mass = 26   Empirical mass = 13   Common name = Acetylene   Homologous series = AlkynesORBITAL STRUCTURE

OF ETHYNE  

   COMPOSITION OF ETHYNE MOLECULE:   Ethyne molecule consists of two C-atoms and two H-atoms (C2H2).

   NATURE OF HYBRIDIZATION:   In ethyne molecule, each carbon atom is Sp-hybridized. Due to Sp-hybridization each carbon atom    generates two Sp-hybrid orbitals. In this way there exists four Sp-orbital in ethyne. These Sp-orbital    are arranged in linear geometry and 180o apart. Remaining py and pz unhybrid orbitals of each carbon    atom lie perpendicular to the plane of Sp-orbitals.   SIGMA BOND FORMATION:   One Sp-hybrid orbital of each carbon atom overlaps to produce one sigma bond between two     C-atoms.The remaining one Sp-orbital of each C-atom overlaps with one H-atom to produce sigma    bond.   Pi-BOND FORMATION:   Py and Pz orbital of two carbon atoms are un-hybrid and make parallel overlapping to produce pi-bond.   BOND LENGTH:   The C--H bond is 1.09A and C-C is 1.2A.o.   BOND ANGLE:   HCC bond angle is 180o.

METHODS OF PREPARATION  

   BY THE HYDROLYSIS OF CALCIUM CARBIDE:          CaC2 + 2H2O C2H2 + Ca(OH)2

   BY TETRA CHLOROETHANE:          Cl2CH-CHCl2 + 2Zn C2H2 + 2ZnCl2                BY 1,2-DIBROMOETHANE:          Br-CH2-CH2-Br + KOH C2H2 + 2KBr + 2H2O

CHEMICAL PROPERTIES  

    COMBUSTION REACTION:          2C2H2 + 5O2 4CO2 + 2H2O + heat

    ADDITION REACTIONS:   Addition of Hydrogen:          C2H2 CH2=CH2 C2H6

   Addition of Halogen:          C2H2 +Cl2 Cl-CH=CH-Cl Cl2CH-CHCl2   Addition of Hydrogen Halide:          C2H2 + HBr CH2=CH-Br + HBr CH3-CHBr2

   Addition of HCN:          C2H2 + HCN CN-CH=CH-CN   Addition of Water:          C2H2 + HOH CH2=CH-OH [rearrangement] CH3CHO (Ethanal)   Oxidation of Ethyne:   In cold solution          C2H2 + HOH + 3[O] 2HCOOH (Formic acid)   In hot solution,          C2H2 +4[O] (COOH)2 (Oxalic acid)

    SUBSTITUTION REACTIONS:         CONSULT YOUR TEXT BOOK

PHYSICAL PROPERTIES  

    Ethyne is a colourless gas with a characteristic smell.

    It has a melting point -81oC.

    It has a boiling point -84oC.

    It is significantly soluble in water. However readily soluble in organic solvents. USES OF ETHYNE  

    It is used in welding to produce oxy-acetylene flame of temperature about 3000oC.

    It is used in the synthesis of useful compounds such as ethanol, ethanoic acid, PVC, acetaldehyde.   For latest information , free computer courses and high impact notes visit : www.citycollegiate.com

Orbital Hybridization

We've learned how constructive and destructive interference of atomic orbitals explains the formation of bonding and anti-bonding orbitals. We also leaned about two types of bonding: σ and π bonding. So you might expect that for polyatomic molecules, all you need to do is put the atoms of the molecule near each other in the right geometry and then see what σ or π bonds form between all the atomic orbitals.

Well, it is almost that simple. The only problem is that for most molecular geometries the atomic orbitals on an atom do not point in the right direction for a σ or π bond to form. Let's look at BF3 as an example. From VSEPR we know the geometry around the Boron atom should be trigonal planar.

But for a Boron atom all the valence elelctrons are in the 2s, 2px, 2py, 2pz orbitals. Recall their shapes:

The problem you'll find is that there's no way you can put three Fluorine atoms around the s and p orbitals of Boron in a trigonal planar configuration and form 3 equivalent σ or 3 equivalent π bonds. Yet, we know the B-F bonds are all equivalent because they all have the same bond dissociation energy.

Actually, what happens is that as you bring the three Fluorine atoms near Boron, the atomic orbitals on Boron change (or hybridize) so that they can form σ bonds in a trigonal planar shape.

In this example, one s, and two p orbitals, i.e., px, and py, hybridize to form 3 new orbitals that point along the correct direction to form σ bonds with all 3 Fluorines. This is called sp2 hybridization;

Let's look at another example, BeF2. From the VSEPR model we know its structure is

In Be, the s and px orbitals hybridize to give two similar sp hybrid orbitals.

Remember that the atomic orbitals are standing waves associated with the electrons bound to a nucleus. When you bring atoms together the boundary conditions for these standing waves change and so the standing waves which were the atomic orbitals change. That is all hybridization is. It's analogous to holding down and releasing a violin string while you're playing. There's one standing wave (one frequency) while you're holding down the string, and another standing wave (another frequency) when you release the string.

Let's conside another example, CH4.

To get these bonds you hybridize one s and three p orbitals. These are called sp3 hybrid orbitals.

Sometimes it is not necessary for all the valence electron orbitals to hybridize. For example, ethylene has the following structure:

The bonds between C and H are all σ bonds between sp2 hybridized C atoms and the s-orbitals of Hydrogen. The double bond between the two C atoms consists of a σ bond (where the electron pair is located between the atoms) and a π bond (where the electron pair occupies the space above and below the σ-bond).

You should remember that we learned about molecules where the central atom gets more than an octet of electrons.

We learned earlier that the extra bonding electron pairs are possible if we include the d-orbitals of phosphorous. This is done by forming hybrid orbitals from s, p, and now d orbitals. For trigonal bipyramidal the central atom is bonded through dsp3 hybrid orbitals.

In the case of molecules with an octahedral arrangement of electron pairs, another d-orbital is used and the hybridization of the central atom is d2sp3

In summaryTotal # of L.P. and B.P

about atomArrangement Hybridization

2 linear sp3 trigonal planar sp2

4 tetrahedral sp3

5trigonal bipyramidal

d sp3

6 octahedral d2sp3

What is the hybridization of Xe in XeF4?

Starting with the # of valence electrons = 8 + 4 ( 7 ) = 36 e- (i.e., 18 pairs), and using the VSEPR model we predict an octahedral arrangement of electron pairs about Xe:

Therefore we say that Xe has a d2sp3 hybridization.

9.4 Hybrid Orbitals

For polyatomic molecules we would like to be able to explain:

The number of bonds formed Their geometries

sp Hybrid Orbitals

Consider the Lewis structure of gaseous molecules of BeF2:

The VSEPR model predicts this structure will be linear What would valence bond theory predict about the structure?

The fluorine atom electron configuration:

1s22s22p5

There is an unpaired electron in a 2p orbital This unpaired 2p electron can be paired with an unpaired electron in the

Be atom to form a covalent bond

The Be atom electron configuration:

1s22s2

In the ground state, there are no unpaired electrons (the Be atom is incapable of forming a covalent bond with a fluorine atom

However, the Be atom could obtain an unpaired electron by promoting an electron from the 2s orbital to the 2p orbital:

This would actually result in two unpaired electrons, one in a 2s orbital and another in a 2p orbital

The Be atom can now form two covalent bonds with fluorine atoms We would not expect these bonds to be identical (one is with a 2s

electron orbital, the other is with a 2p electron orbital)

However, the structure of BeF2 is linear and the bond lengths are identical

We can combine wavefunctions for the 2s and 2p electrons to produce a "hybrid" orbital for both electrons

This hybrid orbital is an "sp" hybrid orbital

The orbital diagram for this hybridization would be represented as:

Note:

The Be 2sp orbitals are identical and oriented 180° from one another (i.e. bond lengths will be identical and the molecule linear)

The promotion of a Be 2s electron to a 2p orbital to allow sp hybrid orbital formation requires energy.

o The elongated sp hybrid orbitals have one large lobe which can

overlap (bond) with another atom more effectively o This produces a stronger bond (higher bond energy) which offsets

the energy required to promote the 2s electron

sp2 and sp3 Hybrid Orbitals

Whenever orbitals are mixed (hybridized):

The number of hybrid orbitals produced is equal to the sum of the orbitals being hybridized

Each hybrid orbital is identical except that they are oriented in different directions

BF3

Boron electron configuration:

The three sp2 hybrid orbitals have a trigonal planar arrangement to minimize electron repulsion

An s orbital can also mix with all 3 p orbitals in the same subshell

CH4

Thus, using valence bond theory, we would describe the bonds in methane as follows: each of the carbon sp3 hybrid orbitals can overlap with the 1s orbitals of a hydrogen atom to form a bonding pair of electrons

H2O

Oxygen

Hybridization Involving d Orbitals

Atoms in the third period and higher can utilize d orbitals to form hybrid orbitals

PF5

Similarly hybridizing one s, three p and two d orbitals yields six identical hybrid sp3d2 orbitals. These would be oriented in an octahedral geometry.

Hybrid orbitals allows us to use valence bond theory to describe covalent bonds (sharing of electrons in overlapping orbitals of two atoms)

When we know the molecular geometry, we can use the concept of hybridization to describe the electronic orbitals used by the central atom in bonding

Steps in predicting the hybrid orbitals used by an atom in bonding:

1. Draw the Lewis structure

2. Determine the electron pair geometry using the VSEPR model

3. Specify the hybrid orbitals needed to accommodate the electron pairs in the geometric arrangement

NH3

1. Lewis structure

2. VSEPR indicates tetrahedral geometry with one non-bonding pair of electrons (structure itself will be trigonal pyramidal)

3. Tetrahedral arrangement indicates four equivalent electron orbitals

9.5 Multiple Bonds

The "internuclear axis" is the imaginary axis which passes through the two

nuclei in a bond:

The covalent bonds we have been considering so far exhibit bonding orbitals which are symmetrical about the internuclear axis Bonds in which the electron

density is symmetrical about the internuclear axis are termed "sigma" or "" bonds

In multiple bonds, the bonding orbitals arise from a different type arrangement:

Multiple bonds involve the overlap between two p orbitals These p orbitals are oriented perpendicular to the internuclear (bond)

axis

This type of overlap of two p orbitals is called a "pi" or "" bond

In bonds:

The overlapping regions of the bonding orbitals lie above and below the internuclear axis (there is no probability of finding the electron in that region)

The size of the overlap is smaller than a bond, and thus the bond strength is typically less than that of a bond

Generally speaking:

A single bond is composed of a bond A double bond is composed of one bond and one bond A triple bond is composed of one bond and two bonds

C2H4 (ethylene; see structure above)

The arrangement of bonds suggests that the geometry of the bonds around each carbon is trigonal planar

Trigonal planar suggests sp2 hybrid orbitals are being used (these would be bonds)

What about the electron configuration?

Carbon: 1s2 2s2 2p2

Thus, we have an extra unpaired electron in a p orbital available for bonding

This extra p electron orbital is oriented perpendicular to the plane of the three sp2 orbitals (to minimize repulsion):

The unpaired electrons in the p orbitals can overlap one another above and below the internuclear axis to form a covalent bond

This interaction above and below the internuclear axis represents the single bond between the two p orbitals

Experimentally, we know that the 6 atoms of ethylene lie in the same plane. If there was a single bond between the two carbons, there would be nothing stopping the atoms from rotating around the C-C bond. But, the atoms are held rigid in a planar orientation. This orientation allows the overlap of the two p orbitals, with formation of a bond. In addition to this rigidity, the C-C bond length is shorter than that expected for a single bond. Thus, extra electrons (from the bond) must be situated between the two C-C nuclei.

C2H2 (acetylene)

The linear bond arrangement suggests that the carbon atoms are utilizing sp hybrid orbitals for bonding

This leaves two unpaired electrons in p orbitals To minimize electron replusion, these p orbitals are at right angles to

each other, and to the internuclear axis:

These p orbitals can overlap two form two bonds in addition to the single bond (forming a triple bond)

Delocalized Bonding

localized electrons are electrons which are associated completely with the atoms forming the bond in question

In some molecules, particularly with resonance structures, we cannot associate bonding electrons with specific atoms

C6H6 (Benzene)

Benzene has two resonance forms

The six carbon - carbon bonds are of equal length, intermediate between a single bond and double bond

The molecule is planar The bond angle around each carbon is approximately 120°

The apparent hybridization orbital consistent with the geometry would be sp2 (trigonal planar arrangement)

This would leave a single p orbital associated with each carbon (perpendicular to the plane of the ring)

With six p electrons we could form three discrete bonds

However, this would result in three double bonds in the ring, and three single bonds

This would cause the bond lengths to be different around the ring (which they are not)

This would also result in one resonance structure being the only possible structure

The best model is one in which the electrons are "smeared" around the ring, and not localized to a particular atom

Because we cannot say that the electrons in the bonds are localized to a particular atom they are described as being delocalized among the six carbon atoms

Benzene is typically drawn in two different ways:

The circle indicates the delocalization of the p bonds

 

back to RJC Chem Web

..:: Molyorbital™ Sets ::..

The Molymod molecular orbital system uses standard shiny colored atom parts to represent the central atom cores. Average scale is 3.5cm per Angstrom. The atomic and molecular orbital parts are represented by pastel/matte colored pieces and are color-coded according to their use.

The pink and purple pear-shaped lobes and concave pi links represent the +ve and -ve signs of the wave functions Ψ of the lobes of atomic and molecular orbitals. The gray pieces represent hybridized sigma bonds. The light brown spheres are used to indicate lone electron pair orbitals. Where the electrons are involved in hydrogen bonding, a light brown pear-shaped lobe is used.

Organic Structures

4-model collection set

Contains sufficient parts to make benzene, ethane, ethene, and ethyne. Shows sigma and pi bonding orbitals, and concepts of hybridization and delocalization.

Z54,455-8

Atomic Orbitals

14-model collection set

Contains 1s; 2s; 2p; 3d; 2s+ 3 x 2p; and sp, sp2, and sp3 hybridized models. Pink and purple lobes represent the +/- wave phases of the p and d atomic orbitals. Models come with transparent bases for display.

Z54,456-6

Electron Repulsion Theory

8-model collection set

Shows 1-6 coordinate systems: terminal, linear, bent, trigonal, pyramidal, tetrahedral, trigonal bipyramidal, octahedral. Lone pairs are represented by brown spheres or brown pear-shaped parts.

Z54,457-4