Why are p orbitals so unintuitive

The Molecular orbital theory (MO theory for short) is one of two complementary ways to describe the structure of atomic bonds, the other is the VB theory. At the MO procedure the atomic orbitals of the atoms involved are 'mixed'. In doing so, they split into binding and antibonding molecular orbitals.

The VB method is based on localized bonds (see valence structure theory). It is an extension of the procedure with which Walter Heitler and Fritz London first used the H2-Molecule were able to approximately calculate and thus established quantum chemistry. A little later, Friedrich Hund and Robert S. Mulliken developed the MO method, which is used today for most quantum chemical calculations.

Physical explanation

An n-electron wave function, if the spin is ignored, has the general form . The product with the complex conjugate function gives the probability of the first electron at the point , the 2nd at the point etc. to be found.

The true wave function cannot be found analytically. A useful simplification is to view the electrons as statistically independent. Mathematically, that means using a product approach . The indicate the residence areas for the individual electrons. They are called molecular orbitals. In order to adhere to the Pauli principle, the wave function is used as a Slater determinant (a sum of n products). Then the electrons are indistinguishable and change between all orbitals.

The function composed of MOs remains an approximation, no matter how good the calculation. The difference to the true energy is called the correlation energy.

Apart from the fact that MO schemes can generally not reflect the true situation, it should be noted that they are not clearly defined in MO theory. The only decisive factor is the sum over all squared orbitals, the electron density (this is also the basis for density functional theory). Mathematically speaking, the wave function is invariant to a unitary linear transformation. An example of this are the two models given for describing the double bond.

Mathematical basics

We are looking for solutions to the Schrödinger equation of a molecule. However, the calculations are much more difficult to carry out than with an isolated atom. In the normal case, when more than one electron is considered, there are no analytically specifiable exact solutions in the sense of a three-body problem. Therefore approximation methods have to be used. The VB method and the MO method, which lead to similar results, are suitable for this.

The calculation of variations is used for the approximate determination of the molecular orbitals. This is based on the fact that, because of the hermiticity of the Hamilton operator, the energy expectation value of any function is higher than the lowest energy eigenvalue. So you have to select the function with the lowest expected energy value in an extreme value problem. This is then probably the best approximation.

However, it is not sufficient to carry out a one-off calculation of variations. Orbitals are determined iteratively using the Hartree-Fock self-consistent-field method. The principle is that for each electron, the field of the other electrons is taken into account, which is not exactly known, but gets better and better as the iteration progresses.

The Born-Oppenheimer approximation is also important, according to which the electron and nuclear motion can be viewed in isolation. Thus, electron distribution and vibration can be treated separately.

VB procedure

The valence structure theory (VB theory or VB method from engl. valence bond) according to Walter Heitler, Fritz London, John C. Slater and Linus Carl Pauling assumes localized ties. It is represented by the usual structural formulas (possibly with mesomers). Historically, it was crucial to understanding chemical bonds. Today's calculations are more likely to be carried out using the MO method.

Mathematically, the electrons are distributed in different ways to the atomic orbitals and the linear combination of these valence structures is formed, which has the lowest energy and is therefore the best approximation according to the principle of variation.

A key principle in VB theory is promotion of electrons and hybridization of orbitals.

MO procedure

The MO process molecular orbital) according to Friedrich Hund and Robert Sanderson Mulliken assigns all electrons of the molecule to a set of molecular orbitals. The illustration takes place through electron clouds, which are mostly delocalized over the entire molecule.

Molecular orbitals can be used as linear combinations on a finite basis. Then the molecular orbitals are determined in an extended eigenvalue problem. As suggested by Lennard-Jones, the atomic orbitals of the isolated atoms in the sense of the LCAO approximation (Linear Combination of Atomic Orbitals) be used.

In principle, any functions could be used as a basis. Good solutions with little computational effort are obtained if physically meaningful functions are used. As Lennard-Jones was the first to establish, the atomic orbitals, which correctly describe the electrons in isolated atoms, are suitable for this. One then speaks of LCAO. For improvement, the atomic orbitals can also be varied or further functions can be included in the basic set.

The MO process can be understood intuitively for small symmetrical molecules. For reasons of symmetry, the molecular orbitals result from addition or subtraction of the atomic orbitals. In the case of more complex molecules, the molecular orbitals are composed as a linear combination of different atomic orbitals. Strictly speaking, the 2s and 2p already interact in the second periodz-Orbitals, so that even there more complicated linear combinations would be obtained. In conjugated π systems, the Hückel approximation is a method for the rough determination of MOs.

A fundamental mistake of this method is that the electrons (except for compliance with the Pauli principle) are seen as statistically independent of each other. Much more complex correlated calculations, especially CI (configuration interaction), also consider the electron correlation.

Draw LCAO-MO diagrams

Qualitative LCAO-MO diagrams can also be drawn without an invoice. It should be noted that with the linear combination of two AOs, a binding MO with lower energy than the lower-lying AO and an antibonding MO with higher energy than the higher-lying AO are formed. As a first approximation, the split is determined by the overlap. For example, one can predict that a σ-bond splits more strongly than a π-bond.

σ bond

σ bond

Molecular orbital of H2




A bond that is rotationally symmetrical to the bond axis is referred to as a σ bond. In other words, orbitals with the magnetic quantum number ml = 0 combined, i.e. H. s-, px-, dz2-Orbitals and mixtures (hybrids) of these.


  • The hydrogen molecular orbital is created by the overlap of the 1s orbitals of the hydrogen atoms. The small circles correspond to the bond distance, the large circles to the atomic radius.
  • In the water molecule, the 1s orbitals of two hydrogen atoms combine with one sp3-Hybrid orbital of the oxygen atom to two σ-bonds. The four orbitals of the bonding and non-bonding electron pairs are aligned with the corners of a tetrahedron. The electron pairs drawn in red are in the atomic orbitals and the gray electron pairs in the molecular orbitals.
  • In hydrogen fluoride, the spherical 1s orbital of the hydrogen atom connects with the dumbbell-shaped px-Orbital of the fluorine atom to a molecular orbital with unequal orbital halves. (The non-binding py- and pz-Orbitals are not shown.)
  • In ethyne, two sp hybrid orbitals of the carbon atoms combine to form a molecular orbit, the other hybrid orbitals also form a molecular orbit each with the 1s orbitals of the hydrogen atoms, the py- and pz-Orbitals of the carbon atoms, which were not used to hybridize the atomic orbitals, are perpendicular to the bond axis and form two π bonds (see below). (In the figure, in addition to the molecular orbitals, the p orbitals are indicated as black lines)

The molecular orbital of the double bond

It is mirror symmetrical with respect to the bond axis.

σ-π model

π bond

Molecular orbital of the π bond

Molecular orbital of the π electrons in ethene

A double bond consists of a σ-bond and a π-bond, whereby the binding partners in the sp2hybridized state: three hybrid orbitals point into the corners of an equilateral triangle, perpendicular to this is the py-Orbital that was not used for hybridization. The σ-bond arises from the overlap of two hybrid orbitals, the π-bond arises from the overlap of the two py-Orbitals. Since both py-Orbitals have to be parallel to each other, a new molecular orbital with a nodal plane is created.

The π molecular orbital is obtained by combining orbitals with | ml| = 1 formed. It contains a nodal plane in the core axis.

  • Example ethene: The two halves of the π molecular orbital are above and below the level of the σ bonds (green, the C-C and H-C sigma bonds are only shown as black lines).
  • Example ethyne: The bond situation in ethyne (common name acetylene), which contains a triple bond, consists of a σ-bond, which is located between the core connection axis, and two π-bonds.

τ model

τ model

Molecular orbitals according to the τ model

A seldom used method to describe double bonds is the τ model. s and p orbitals are first mixed (both carbon atoms are sp³-hybridized) and the double bond is assembled from the two hybrid orbitals. The τ bonds are created by the overlap of two hybrid orbitals, two mirror-image molecular orbitals (“banana bonds”) are formed. It turns out that the τ-model reproduces bond angles and lengths appropriately.

The distinction only makes sense in the VB theory. In the case of an LCAO method, the two models merge, since the sum of the same electron density is obtained in both cases. This is the only relevant one.


Conjugated π-bond

Delocalized molecular orbital in 1,3-butadiene

Delocalization occurs when a molecule contains multiple double bonds that conjugated are. This means that there is always exactly one single bond between them. To do this, all pz-Orbitals are parallel and in close proximity to each other. Then all pz-Orbitals can be combined into a single molecular orbital, which can be proven quantum mechanically.

Examples in chemical compounds


binding antibonding

Additive superposition of the wave function

Subtractive superposition of the wave function

Binding molecular orbital

Antibonding molecular orbital
Occupation of the molecular orbitals of hydrogen and helium
hydrogen helium

Occupation with hydrogen

Occupation at the helium

The lone electrons necessary for the bond are each in the 1s orbital of the two atoms H.a and Hb, that by the eigenfunctions ψa(1s) and Ψb(1s) is described.

The addition of the wave functions ψa(1s) + ψb(1s) results in a rotationally symmetrical binding Molecular orbital (σ (1s)) with increased charge density between the nuclei of the binding partners. The attraction of the nuclei by the charge holds the molecule together.

The subtraction of the wave functions ψa(1s) - ψb(1s) gives a antibonding Molecular orbital (σ * (1s)) with a nodal plane between the nuclei of the binding partners. The resulting low electron density between the nuclei leads to a repulsion of the atoms.

The molecular orbitals (like the atomic orbitals) can be occupied by a maximum of two electrons with opposite spins. Since each hydrogen atom provides one electron, the binding molecular orbital in the lowest-energy ground state is occupied by an electron pair, while the antibonding remains empty. (In the excited state, the binding and the antibonding molecular orbital are each occupied by one electron.)

Another example is helium. Here every 1s orbital is already occupied with an electron pair. When combining these atomic orbitals, both the binding and the antibonding molecular orbital would have to be occupied by one pair of electrons each. Their effects would cancel each other out, there is no bond.


LCAO-MO scheme of oxygen

Occupation of the energy levels

The LCAO-MO scheme can be derived qualitatively as described above. Every oxygen atom has six valence electrons in the ground state on the second main energy level. The twelve valence electrons of an O2Oxygen molecule are bound to the four binding (σs, σx, πy, and πz) and three of the four antibonding molecular orbitals (σs*, πy*, πz*) distributed. Since two antibonding orbitals are occupied by only one electron (a “half bond”), a double bond results.

According to Hund's rule, dioxygen has two unpaired electrons with parallel spins with its ground state, a triplet state. The paramagnetism and diradical character of oxygen can be explained by this electron distribution. Interestingly, the diradical character lowers the reactivity, since a concerted reaction would contradict the preservation of spin. The excited diamagnetic singlet oxygen is particularly reactive.

Another consequence of the MO occupation is that for O2 it is difficult to give a correct Lewis formula. Either the diradical character is neglected or the double bond.


π system of butadiene
binding antibonding

strongly binding MO (occupied)

strongly antibonding MO (unoccupied)

weakly binding MO (occupied)

weakly antibonding MO (unoccupied)

The π system of butadiene is composed of 4 pz-Orbitals that are initially occupied by one electron each. These 4 atomic orbitals are now linearly combined to form four molecular orbitals. The coefficients are obtained by symmetry-adapted linear combination (SALC) or according to Hückel's theory. This creates the orbitals drawn on the right. The red / blue color indicates whether the orbital had a negative or positive sign before it was squared. It has no physical relevance.

Each of these orbitals can be occupied by 2 electrons. So the two lower orbitals are fully filled and the two upper ones remain empty. The orbital, in which all pz-Orbitals have the same sign and therefore the electrons can move almost freely over the whole molecule.

One recognizes the property demanded by SALC that all symmetry elements of the molecule are preserved in every molecular orbital. You can also see how the number of nodal levels increases with increasing energy.

Binding order

The bond order describes the number of effective bonds in a molecule. It is half the difference between the bonding and antibonding valence electrons (see bond order). It is easy to read because it is equal to the number of dashes in the Lewis notation of the compound.

See also

Categories: Chemical Bond | Theoretical chemistry