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. 

There 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 provide a brief summary and 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 Energy ordering of MOs
How are bonds made? σ and π bonds Lobe size
Why are bonds made? Resonance HOMO & LUMO 
 

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...  (more than 75 years ago) a young French guy trying to get his Ph.D. suggested that electrons behave as waves. Well, photons behave as waves and  particles, 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.

The blue line is the square of the red function (plotted on a different scale to show it 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 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 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 3D 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 (a 3D function) 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 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 one 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 90% probability of finding an electron within the enclosed volume.  These spaces are called 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 90%; 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 often we really plot its square), because it is the wavefunction (and its sign) that is important when two 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 (90%) of their time, with "skins" differently "painted" with colors corresponding to the sign of the 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.
 
7. Atomic orbitals (s and p for us) are centered on atoms.  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. These are the familiar by now s and p orbitals.  Other atoms have similar 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.
 

H  		         
 
B 
C 
N 
O 
F 
       
S 
Cl 
 

For illustration purposes the relative sizes of several atomic 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 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.

8. The molecules are constructed from atoms, and the "new" volumes for electrons in the resulting molecules are constructed from the atomic orbitals of atoms participating in bonding. Two basic set of rules used are known as the Valence Bond (VB) Theory and the Molecular Orbital (MO) Theory.  The theories use different language, but are equivalent.

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 the same cake at the end; we mean the same 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 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 tells us the p/s ratio in the hybrid orbital.  Remember, each new orbital must be normalized (i.e. the probability of finding an electron within it should be one), and the number of orbitals must be preserved (see below). 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, see below) these hybrid or unchanged atomic orbitals on two adjacent atoms (one orbital on each) within the molecule.

  

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 that are composed of (fractions) of many atomic orbitals on different atoms. Again, each new orbital must be normalized (probability of finding an electron within it is unity), and the number of orbitals must be preserved.

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) is 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, see above). 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.

13. In the "additive" interaction of (for example) two atomic orbitals the new orbital volume encloses both atoms.  The electrons in that volume (valence bond orbital or molecular orbital) interact with both nuclei (the electrons are shared).  The new orbital is of lower energy than the atomic orbitals used to generate it.  To preserve the number of orbitals, the "additive" mode is accompanied by the "destructive" ("subtractive") mode where the new orbital volumes do not encompass both nuclei.  The electrons in such an orbital cannot be shared, i.e. the orbital has a node (or more precisely: a nodal plane) between the nuclei.  This (antibonding) orbital is of higher energy than the atomic orbitals from which it was generated.   To the very crude approximation, the lowering of energy of the bonding orbital as compared to the constituent atomic orbitals is the same as the corresponding rise in energy of the antibonding orbital.

The addition of two lobes of the same sign or "color" (now you know why keeping that info was important) leads to a new space that encompasses all the lobes involved. 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 demo for 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).  This unconnected lobes are separated by regions where the probability of finding electrons is zero, called nodal planes (or just nodes). In such an orbital electrons cannot get close to both the nuclei involved in the bonding. They are "imprisoned" in the space around just one atom  The energy of that orbital  is increased (in comparison to separate atoms). Notice different color scales for the two orbitals.

14. The interaction of two orbitals (atomic, hybridized or molecular) is stronger (more energy lowering, see the point above) 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).

The overlap is measured by the amount of the common space occupied by the interacting building-block orbitals that are of the same sign minus the common space occupied by the building-block orbitals of the opposite sign.  Brrrr...   It is best explained using a couple of illustrations.  Consider the interaction of two  pz orbitals of equal energy making a σ bond.  To see it better we gave the two orbitals different colors (black and red) and marked the algebraic sign of their lobes with shading (let's say the shade corresponds to "+").

 

If the distance between the atoms is too large (r1) the overlap is small: there is only small common part in the center.  At a certain distance (r2) the overlap will be at the maximum: the positive lobes overlap fully.  But if we push the atoms too close (r3) orbitals will interpenetrate too much: the positive lobes will overlap with negative lobes.  This "mixed-sign" interpenetration  (+/−) subtracts from the same sign interpenetration (+/+, or −/−).  Or in different words, the overlap is characterized by the amount (how much) of the common space) and the sign (+/− is negative; +/+ or −/− is positive).  The net results is what counts.

Here, look at  2s and 2p orbitals on the same atom (for example, on oxygen). The s orbital has (let us say) a negative sign, and the p orbital has the top lobe positive (shaded) and the bottom lobe negative. overlap of s and p orbitalsThe common space of the two orbitals is shown in color in the second picture (green and purple, remember it is in 3D).  The green space is a positive contribution (−/−) and the purple space is a negative contribution (+/−). 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 orbitals discussed here are both orthogonal and normalized, or orthonormal).

 

Continued... 

 
Quantum Primer Last updated 08/29/07 Copyright 1997-2008
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