Main page   QUANTUM  CHEMISTRY  PRIMER:   PART  I 
 

Molecules are groups of atoms, which by sharing their electrons lower their energy by attaining noble-gas configurations (in comparison to staying  as separate, individual atoms).  You can think about them as a collection of positively charged nuclei  (spheres hanging in vacuum) surrounded by a cloud of electrons.  The nuclei repel each other and the electrons repel each other, but the electrons are attracted by the nuclei.  The balance of these Coulombic forces and the fact that electrons can occupy only certain  volumes of space (that's quantum weirdness...) gives molecules their shapes (how do they look like) and reactivity (what do they do).  We  will use only qualitative and descriptive quantum chemistry to get some ideas of what's going on.  In fact we want to avoid math altogether and use only "pictorial" approach in manipulating atomic orbitals, our building blocks, to construct the electronic picture of organic molecules. 

Here are some important concepts, rules, and terms that you should be familiar with.  We divided this primer into three parts (all bad things come in sets of three ). To help you with that material, we also offer several examples in the Orbital and Bonding corner of the Molecular Gallery.

 
 
 
 
Part I:   What are orbitals? Part II: Valence Bond Theory   Part III: Molecular Orbital Theory 
Atomic orbitals Hybridized orbitals Molecular orbitals 
Representations of orbitals Electronic structure HOMO & LUMO
How are bonds made? σ and π bonds  Energy ordering of MOs
Why are bonds made? Resonance   Lobe size
 

ABOUT ORBITALS AND BONDING

1. Orbitals are volumes of space where electrons are allowed to spend their time.  These volumes are described by mathematical functions (wavefunctions) that have algebraic signs (plus or minus - do not confuse it with charge!).
 
 

OK, let's start from the beginning. Almost...  Long time ago, a young French guy trying to get his Ph.D. suggested that electrons behave as waves. Well, electromagnetic radiation behave as particles (photons) and as waves, why not electrons? Physicists liked that idea.  Luckily for them, it was confirmed experimentally (for example, by observation of electron diffraction).  It was only a small step from there to propose that electrons in atoms (and molecules) also behave as waves (this time it was an Austrian and a Brit), and the quantum mechanics was rocking.  But how are we to describe the electron wave? It is simple, use a mathematical function (wavefunction!).  But what does it mean?

water wave

First, let's look at a simple wave created when a water drop hits water surface, as caught in the picture on the left.  Here is our wave: tall in the middle, then dropping below the level of the undisturbed water surface, to climb again, and drop again in a series of concentric hills and valleys, frozen (standing still) by the camera to be examined in detail. And, if we trace the water surface on a graph paper, representing height of the water surface on the vertical axis and the separation from the center on the horizontal axis, we get a ... surprise, surprise... a mathematical function looking like a wave.

We can use that picture to describe electron "waves" in atoms.  Let's start with a simple 2D representation.  We plot here in red the value of the wavefunction as it changes with the distance (r) from the nucleus (everything here is in atomic units). We have selected 3s atomic orbital as an illustration, because it looks more  like a water wave than 1s orbital does.

The red function starts high at the nucleus, falls rapidly below zero near the nucleus (becomes negative), and then rises in a shape of a small and broad hill (it's positive again) to finally level off (at the near zero value) at larger distances. It is like a water wave "radiating" from the center.

Highslide JS
2D pictures of hydrogen 3s orbital: wavefunction, electron density, and radial density (the verical scale is different for each function).

The blue line is the square of the red function (plotted on a different scale to show it more clearly), and it represents the probability of finding an electron in a volume of space, i.e. electron density (a fraction of an electron per unit volume, see p. 2). 

The green line (again on a different scale) is the radial probability, i.e. volume-corrected electron distribution  (electron density multiplied by volume, check it under Atomic Orbitals), and it represents the fraction of the electron to be found in a spherical shell at different distances (r) from the nucleus. In this case, the most probable distance of the electron from the nucleus is about 13 atomic units. And you can easily count the nodes (places where the function is zero). Yes, 3s orbital has two nodes!

The electronic wavefunctions are in fact multidimensional beasts.  Let's add one dimension at a time: Just imagine that we have taken the red function from above and spun it around the vertical axis centered at the nucleus. What we would get is shown in two different views here. For reference, we have redrawn the red line from above (look on the left).  So now we have a better representation of the wave. It looks very much like the water wave frozen in the picture above.  It is in fact a standing wave:  it does not spread  like a "standard" traveling wave, such as one in water.

Now, in an atom, the wavefunction stretches in all directions from the nucleus (like concentric spheres in the case illustrated here, i.e. it has a value that is the same for all points that are equal distance from the center).  We have a small problem, however. We need an additional (4th) dimension to illustrate that.  Since we do not know how to do it, we have to rely on lower dimension analogies (cross-sections).

 
2. The wavefunctions themselves do not have  a physical equivalent, but the values of their squares correspond to the probability of finding an electron in given volumes of space (called electron density).
 

Here you go... The wavefunction has a certain value in each point of space (x, y, z,  if you like Cartesian coordinates). The square of this function (sometimes with a twist that can wait till a quantum-chemistry course) represents probability of finding an electron in a minuscule amount of space around that point (the point itself has no volume, so we use this volume element, dx · dy · dz, for those mathematically inclined).  Thus, orbitals, as represented by wavefunctions' squares, are places (volumes of space) where probabilities of finding an electron are non-zero, i.e. places where the electrons are allowed to be. The probability of finding an electron in a volume of space is called electron density; it is a fraction of an electron per unit volume.

Here is the cross-section of the electron density of the 3s orbital corresponding to the wavefunction shown above (p. 1; i.e. we plot the square of the wavefunction). The highest electron density is in the center (small blue dot corresponding to the sharp peak), and in the blue ring slightly removed from the nucleus.  (This orbital has two nodes that are not visible in this presentation because of the low picture resolution. How would they look like in this picture?)

 
3. Orbitals (but not their graphical representations, see below) are constructed in such a way that the probability of finding one electron within their volume is unity (100%).
 

That is just a reality check. We want to be sure that each electron is somewhere within that orbital volume. And that we have all of our electrons... (For those who like to keep it sophisticated: we call such orbitals normalized.)

 
4. An orbital may not accommodate more than two electrons.  These electrons must have opposite spins.
 

Another physicist's contribution: No two electrons with the same quantum numbers can occupy the same space. Within an orbital the electrons have all the same quantum numbers (remember freshman chemistry?), except the spin quantum number, which has only two allowed values +½ and −½. Thus, no more than two electrons can be in the same orbital, and the two must have different spins.  OK... what is spin? Uh.... Got us here! It is a quantum property that does not have a macroscopic equivalent.  It is usually represented as a magnetic moment (the arrow) of an electron (the yellow ball) spinning in one of the two possible directions (counterclockwise and clockwise around its own axis).

 

 

 

 
5. Each orbital has some energy value associated with it; i.e. electrons within that orbital have the specified energy.
 

It is convenient to think of orbitals without electrons (we just add them later and stir gently), but what we really mean is that an electron within given orbital will have certain energy, that will depend on its average separation from the nucleus (the farther, the higher the energy), or the number of nodes (nodes are places where the electron density is zero; the more nodes the higher the energy).  Transitions of electrons between orbitals of different energy must be accompanied by the appropriate amounts of energy being given off , or absorbed from outside.

 
6. The graphical representations of orbitals are usually drawn to show "spaces" corresponding to 95% probability of finding an electron within the enclosed volume.  These spaces are called orbital lobes, or teasingly for descriptive purposes: blobs, cages, boxes, etc.  All the points on the drawn surfaces have identical (and usually quite low) electron density. They represent  sort of  "borders" or "skins" for spaces in which electrons are allowed.
 

Finally something graphical.  When we visualize the orbitals we usually make several simplifications. For example, we show only borders drawn at certain value (usually quite small) of electron density in such a way that the probability of finding an electron within the enclosed volume is high (usually 95%; well, one has to cut it off somewhere).  We usually do not show values of electron density within (or outside) the border surface. And that's because doing it in a graphically-readable form in 3D is not easy.  We color-code (or use other graphical labels) the border surface with the sign of the wavefunction itself (despite the fact that we really plot its square).. It is the wavefunctions (and their signs) that are important when orbitals start interacting.  Happily, at the graphical-level of doing quantum chemistry these simplified pictures are all we need.  The resulting representations of orbitals, such as those p orbitals shown below, (yes, those blobs, cages or whatever other name you prefer) are just demarcated spaces where electrons spend most (95%) of their time, with "skins" differently "painted" with colors corresponding to the sign of their wavefunctions. What's going inside these spaces (i.e. the details of electron density distribution) is only for those who want to do the math.  Oh... and the final note, everything we have discussed so far (1-6) applies to all kinds of orbitals, atomic, hybrid or molecular.

 
7Atomic orbitals (s and p for us) are centered on atomic nuclei.  They serve as "building blocks" for orbitals found in molecules.
 

And at the beginning was hydrogen... Yes, the shapes of orbitals were first calculated for the hydrogen atom. Among them are the familiar by now s and p orbitals.  Other atoms have similarly shaped orbitals. What changes is their size and energy.  Eventually, we also get d and f orbitals with really funny shapes, but  they are not so important in organic chemistry where we are dealing mainly with the elements from the first two rows of the periodic table.

 

         
 




       

Cl 
 

For illustration purposes the relative sizes of several atomic valence orbitals are presented in the table above. In general, the orbital size increases going down in the periodic table (the increase in the main quantum number).  Within the rows, the orbital size decreases going to the right. The effective positive nuclear charge increases in this direction; the electrons are, therefore, attracted more strongly to the nucleus and get closer to it (on average). This effect is especially pronounced in the size of s orbitals. The electrons in these orbitals, however, screen the nuclear charge, and the electrons in p orbitals are influenced less (smaller size change).

In a qualitative sense, the energies of these orbitals follow the same trend as their sizes.  Within the same row, s orbitals have lower energies than p, and the smaller the orbital the lower its energy. In other words, more electronegative atoms have lower energy orbitals.

8Molecules are constructed from atoms, and the "new" volumes for electrons in the resulting molecules are constructed from the atomic valence orbitals of atoms participating in bonding. The two most commonly applied set of rules for constructing molecules out of atoms are known as the Valence Bond (VB) Theory and the Molecular Orbital (MO) Theory.  The theories use different "languages", but if carried out properly (at the quantitative level) give essentially the same results.

Molecules are collections of atoms. The positions of the nuclei give the molecule its 3D shape, and the new spaces for electrons are constructed from the atomic orbitals. We use two recipes to do that; these are just like baking instructions. Regardless which one we use, we get very similar cakes at the end; we mean similar final electronic structure.

9. In VB, atomic orbitals on a given atom are premixed (hybridized), if necessary, and then used to form bonds, pairwise between atoms, one bond at a time.

In this procedure (here an American chemist was responsible) we first "prepare" all the atoms for bonding. We take all (or just some) of the atomic valence orbitals of a given atom and make new combinations out of them (like making a cocktail from several pure drinks).  So in addition to pure atomic orbitals (s or p, "pure drinks") we can also have variety of hybrids ("mixed drinks"), spx, where x (the index of hybridization) tells us the p/s ratio in the hybrid orbital.  Remember, each new orbital is normalized, and in most cases each orbital involved in bond making holds one electron, indeed.  Then we use these "prepared" atoms with their hybrid or atomic orbitals to make new bonds, one at at a time.  Each bond is made by superimposing (overlapping) these hybrid or unchanged atomic orbitals on two adjacent atoms (one orbital on each) within the molecule. In the process electrons from these orbitals are paired in the new bond, i.e. end up with opposite spins.

 

10. In MO, atomic orbitals of all atoms are mixed to form molecular orbitals that span many atoms (or even the whole molecule).

No premixing in here. Just take the atomic orbitals as they are, and make "super cocktails"; i.e. directly form molecular orbitals (MOs) that are composed of (fractions) of many atomic orbitals on different atoms. Again, each new orbital must be normalized,. Importantly, in this case, the number of orbitals must be preserved, as to have "space" for the same number of electrons before and after the mixing.  After the MOs are constructed, all available electrons are then placed in lowest-energy orbitals, with no more than two of them (with paired spins) per orbital.

11. The driving force for bonding is to achieve a particularly stable electronic configuration  (see noble gases) when the atoms have two (for H), or eight electrons (for B, C, O, N, F) in their valence shells.  That configuration can be achieved by totally giving up electrons to the bonding partner (or taking them away).  In such a case we deal with ionic bonds that are rather uncommon in organic chemistry.  In most cases the octet configuration (or doublet for H) is achieved by sharing of electrons.  This kind of bonding is called covalent.

Nothing to add here. Sharing is the key term!

12. The sharing (bonding) is accomplished via superposition (overlap) of atomic or hybrid orbitals to form bonds (VB) or molecular orbitals (MO).  The overlap (see below) can be accomplished in an "additive" manner (bonding) or in a "destructive" (subtractive) manner (antibonding).

To share electrons new orbitals within the molecule must allow the electrons to get close to the nuclei involved in sharing (bonding). We construct these orbitals using atomic or hybrid orbitals as building blocks.  One simple procedure would just add all the volumes of building orbitals together. This generates some problems, however.  Let's say we start with two orbitals on two neighboring atoms. These orbitals may accommodate up to four electrons. The new orbital we have just made can accept at most two electrons (of different spins).  So in the process,  we would have "lost" space for two electrons.  This example immediately tell us that we must produce two new orbitals not just one! Or, in general, the number of new orbitals must be equal the number of the building-block orbitals. 

The water-wave analogy becomes handy again: when two waves encounter each other, they superimpose in a "predictable" fashion. If they are in the same phase (in-sync) they reinforce (or add to) each other, and if they are in the opposite phase (out-of-sync) they cancel (subtract from) each other. So, to get two new orbitals we "add" or "subtract" the wavefunctions of the building-block orbitals and look at the results. We do it pictorially here, remembering that signs of orbital wavefunctions are encoded with colors or other markings.

13. Let's consider a molecule of H2. In the "additive" interaction of two atomic orbitals the new orbital volume encloses both atoms.  The electrons in that volume (valence bond orbital or molecular orbital) electrostatically interact with both nuclei (the electrons are shared).  The new orbital is of lower energy than the atomic orbitals from which it was generated.  To preserve the number of orbitals, the "additive" mode is accompanied by the "subtractive" mode where the new orbital volumes do not encompass both nuclei.  This orbital has a node between the nuclei.  This antibonding orbital is of higher energy than the atomic orbitals from which it was generated.  The lowering of energy of the bonding orbital (as compared to the constituent atomic orbitals) is a bit less than the corresponding rise in energy of the antibonding orbital.

The addition of two lobes of the same sign or "color" leads to a new space that encompasses all the lobes involved. Now it becomes clear why keeping that orbital sign info was important. The electrons may move around the nuclei enclosed within that new volume and interact with both of  them (Coulombic interaction). That is what we call sharing.  It lowers the orbital energy, as compared to the separate atom situation.  All being said and done, now every atom can pretend to be a noble ...gas. Here is a simple visualization for our H2 molecule: on the left we have B&W 3D representations of the orbitals (note that lobe signs are marked with continuous or dotted lines), and on the right we have the cross sections of the two orbitals with wavefunctions mapped out in color (the black dots approximate the position of the nuclei, although they are out of proportion in size).

  
antibonding (high energy) 
 
 
bonding (low energy) 
 
 

The subtraction of two lobes of the same sign (or addition of lobes of different sign or "color") gives two separate, "unconnected" volumes (of different signs or colors).  These unconnected lobes are separated by regions where the probability of finding electrons is zero, called nodes.  In such an orbital electrons density in the space between the nuclei is diminished. The energy of that orbital is increased (in comparison to separate atoms). Notice different color scales for the two orbitals.

The pictures we have developed here represent MO theory of H2 molecule with its two molecular orbitals (MOs). In the classical VB theory we would limit our considerations to the bonding orbital only (we would call it a bond) and would neglect the unoccupied antibonding one.  In a modified VB theory, we will acknowledge the "existence" of the unoccupied (antibonding) orbitals and even put them to a good use later on.  

14. The interaction of any two orbitals (atomic, hybridized or molecular) is stronger (more energy lowering) if the orbitals are closer in energy, and if the overlap between them (interpenetration of their spaces) is larger.

Now we are talking about the quality of sharing. We measure that by how much the energy is lowered by the bonding process. It mainly depends on the relative energies of the building-block orbitals and their overlap.  The strongest interaction is between the orbitals of equal energy; orbitals of very different energies will not interact (or interact very weakly).

Graphically, the overlap is measured by the amount of the common spaces occupied by the interacting (building-block) orbitals that are of the same sign minus the common spaces occupied by the interacting (building-block) orbitals of the opposite sign.  Brrrr...   That is almost incomprehensible.  It may help to recall that we are talking about interacting waves. If the waves are in-phase (+/+ or –/–-) in a certain region of space, they will add and the square of the new wave (i.e. the electron density) will increase in that region.  On the other hand, if the waves are out-of-phase (+/–), they will subtract or even cancel, and the square of the new wave will be less (or zero) in that region. 

Perhaps It is best explained on a few simple examples. Consider the interaction of 2s and 2p orbitals on the same atom (for example, on oxygen). The s orbital has (let us say) a positive sign (marked in red outline), and the p orbital has the top lobe positive (also red outline) and the bottom lobe negative (marked in blue outline). The common space of the two orbitals is shown in tomato-red for (+/+) overlap and in aqua for the (+/) overlap (remember it is all in 3D).  The two contribution are equal in size. Thus, the net overlap is zero!  Indeed, for those who like to keep it sophisticated we call such orbitals orthogonal, and all atomic orbitals discussed here are both orthogonal and normalized, or orthonormal.

  Another illustration involves two p orbitals on different atoms. In the perpendicular arrangement the (+/+) overlap (tomato-red) is "cancelled" by the (+/) overlap (aqua). No bonding interaction is possible in such geometry.  On the other hand, if one of the p orbitals is shifted off the nodal plane, the bonding interaction is possible (only +/+ overlap), although not optimal. Such bonds are called "bent bonds".  With the same internuclear distance (r), a better overlap may be obtained in the head-on geometry. This situation would describe a "normal" bond. But head-on overlap does not lead to a bonding interaction, if the signs of the lobes are mismatched as shown on the extreme right.  In this case the wavefunctions will "destruct" and a node will form in the middle of the internuclear axis. The result  will be an antibonding combination.

Continued... 

 
Quantum Primer Last updated 08/29/12 Copyright 1997-2013
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