Molecular orbitals 1 9 21 AM Thursday January 7 2016 Why do these molecular orbitals in allylic system look the way they do and where do they come from That s the question this document will try to answer Let s start with something simple s and p atomic orbitals Just a reminder Atomic orbitals are mathematical descriptions in a form of a wave function that describe electronic states in an atom It turns out that electrons cannot just randomly occupy space around the nucleus Instead they can occupy only very well defined trajectories of certain shape and certain energies These trajectories are referred to as states So when we say 1s orbital we are referring to a state of an electron in a molecule These states are described by 3 quantum numbers principle n angular l and magnetic m quantum number So for 1s orbital these are n 1 defines the size of the orbital and the proximity to the nucleus l 0 defines the shape 0 is a sphere and m 0 m describes the orientation Sphere has only one possible orientation so m can only be 0 for l 0 The only thing that is not defined is the spin quantum number which can be 1 2 and 1 2 Given that no two electrons can be the same or occupy the same space at the same time they cannot have all 4 quantum numbers the same As a result each orbital defined by 3 quantum numbers can accept only 2 electrons with different spin quantum numbers Overall the 1s orbital very strictly defines the state of an electron in an atom It describes the energy of the electron its momentum space that electrons can occupy and so on For us the focus is often on the probability of finding electrons at a certain point in space And often when we say an orbital we think of it as a physical space around a nucleus where electrons reside The image of a 1S orbital as a sphere around the nucleus comes from the fact that if we calculate the probability of finding an electron in 1s state remember the probability of finding an electron in a given point of space is given by the square of the wave function describing the electron we find that the there is a 95 probability of finding the electron within a relatively small sphere around the nucleus That s why we say that 1s orbital has a shape of a sphere Moving to a 1p orbital Now n 1 l 1 so the orbital is dumbbell shaped Also for given size n and shape l of an orbital m can have integer values between l and l including 0 So with l 1 we have 3 p orbitals of the same energy that are different only because of the direction they have the p orbitals are perpendicular to each other The p orbital also has another feature The sign of the function that describes the p orbital changes when you move from one side to the other We denote that as a shading Shaded part has let s say a sign of a function and the unshaded bottom part has a sign The sign is largely irrelevant for us except when two orbitals interact with each other and we ll cover that later Sign is irrelevant because we usually talk only about the probability of finding an electron somewhere and that is related to the square of the wave function so the sign of the function does not matter that s a refresher about atomic orbitals When two atoms react to give a molecules they can form a covalent or an ionic bond Most bonds in organic chemistry are covalent and we ll talk about that now When two atoms make a covalent bond they end up sharing a pair of electrons The simplest picture of that is the formation of hydrogen molecule For export Page 1 For export Page 2 Molecular orbitals 2 9 21 AM Thursday January 7 2016 two H atoms H2 molecule Two H atoms with 1 electron in 1s orbital get together and their atomic orbitals interact with each other to make molecular orbitals We say the orbitals overlap They start occupying the same space The result of the interaction is the formation of molecular orbitals Those are new defined trajectories that are available to electrons in the new molecules Electrons on those trajectories have certain energies momentum and so on and again we think of them as defined states of electrons We quantum physics figured out that the interaction of the orbitals and the resulting molecular orbitals can be represented as addition and subtraction of the atomic orbitals So when 2 1s orbitals partially overlap in the molecule of H2 the nuclei do not overlap so the spheres of 1s orbitals only partially overlap at the places where they overlap we just add the values and as a result we get a bigger blob with greatest probability of finding electrons now being in between the two nuclei so that electron can interact with both of them This new blob is called sigma molecular orbital and it is lower in energy than 1s orbitals in hydrogen atoms as shown on the diagram on the left It is lower simply because electrons in this orbital feel both nuclei and not just one Rules of the conservation say that from two atomic orbitals when you make a molecule you have to make two molecular orbitals if you have two states at the beginning you have to have two at the end So the other molecular state or molecular orbital is sigma star This one is obtained when we subtract 1s orbital from another 1s orbital 1s orbitals have the same phase sign to begin with but because we are doing the subtraction they will have a different phase same values for any given spot around the nucleus just different sign Now because atomic orbitals have opposite signs partial overlap in H2 leads to essentially annihilation the values of the resulting function in the overlap region become smaller and in some areas of the overlap becomes 0 And that is why sigma star of a hydrogen molecule has a node in the middle between the two nuclei of hydrogen the shape of the sigma star and the node in the middle make it higher in energy than the starting atomic 1s orbitals So now we have 2 electrons from 2 hydrogen atoms and they have to go into new molecular orbitals they go first to the one of the lowest energy sigma which can accommodate both of them Remember every orbital can have 2 electrons no more Having electrons in sigma instead of 1s lowers the energy of the system that s the delta E on the above graph That difference in energy is actually what keeps a molecule together and that is the strength of a covalent bond that is why we call sigma orbital bonding orbital Putting electrons from 1s into sigma star makes the system less stable and we call that orbital antibonding You can see …