2. • Many atoms can combine to form particular molecules, e.g. Chlorine (Cl) and Sodium (Na)
atoms form molecules NaCl
• Bonding between ions, as in the negative charged chlorine ion and the positively charged sodium
ion, could be understood in the light of coulomb interaction (attraction) between oppositely
charged bodies.
• But atoms of the same type can also form bonds, as for example in the case of H2
• It remained, however, inexplicable that two similar atoms, which are electrically neutral, could
form a bound state.
• We will try to understand the formation of molecule from atom in the quantum mechanical
framework.
3. The H2
+ Molecular Ion It has two identical nuclei (protons) and one electron
Interaction potential, Epot
because RA = −RB. With R = rA − rB we can replace rA and rB by
The Schrödinger equation for this three body problem is:
the first two terms → the kinetic energy of the nuclei,
the third term → KE of the electron.
The nuclei and the electron move in the potential Epot(r, R).
4. The Exact Solution for the Rigid H2
+ Molecule
• The Schrödinger equation cannot be solved analytically.
• Use approximations
• M/m ≈ 1836
• Nuclear kinetic energyEkin = p2/2M is much smaller than that of the electron.
• Hence neglect the nuclear KE
In this approximation of the rigid H2
+ the Schrödinger equation becomes:
where rA and rB depend on the coordinates both of the electron and of the two nuclei.
The internuclear distance R must be treated as a fixed parameter, not as variable.
Solve above eqn analytically. PE is no longer spherically symmetric, but has cylindrical symmetry, it is
convenient to use elliptical coordinates:
the location of the two nucleiare the focalpoints of the ellipsoid with cylindrical symmetry and the z-axis as symmetry
axis is chosen to coincide with the line between the two nuclei.
Azimuthal angle
Confocal coordinates
5. In elliptical coordinates the wave function, which is dependent on three coordinates, is separable into the product
M(μ) : constant for μ = const., i.e. for rA + rB = constant. The value of the
constant dependson the quantum number associated with M(μ). This is
fulfilled on the surface of an ellipsoid with rotational symmetry around the z-
axis.
N(ν) : constant for ν = const., for instance for rA = rB. This is the symmetry
plane z = 0, perpendicularto the z-axis.
f (φ) constant for φ = const. ⇒ y = c · x, which gives planes through the z-
axis.
The nodal surfaces M(μ) = 0 are ellipsoids, N(ν) = 0 are planes
perpendicular to the z-axis, f (φ) = 0 planes through the z-axis.
6. The functionsE(R) can monotonouslydecrease with increasing R (repulsive unstable energy states) or they can show
minima at a certain value of R (stable energy states)
Potentialenergy surfaces.
7. |l| depends generally on the internuclear separation R.
Its projection onto the z-axis has for a stationary state a well-defined eigen value.
the integer m = 0, ±1, ±2, . . . ±l and
lz for a given potential curve E(R) independent of R. The reason
for this is that the operator
depends solely on φ and not on R.
The difference between the atomic H in magnetic field and molecular H is that in the axial electric field of
a nonrotating diatomic molecule the energy of a level does not depend on the direction of the field.
This means, that levels with lz = ±m ·ℏ have the same energy. Therefore the molecular levels are described
by the quantum numberλ = |m|. So, the topmost eqn becomes
Analogous to the nomenclature in atoms, electrons in diatomic molecules are called
→ σ-electrons for λ = 0;
→ π-electrons for λ= 1;
→ δ-electrons for λ = 2 etc.
The Latin letters used in atoms are just replaced by Greek letters for molecules.
8. If the electron spin s is considered, the magnetic moment μs can have two differentorientations in this field, similar to the
situation for atoms in the Stern–Gerlach experiment. The electron spin precesses around the magnetic field direction (which is
the z-direction) and only its z-component has the definite eigen value.
9. The molecular orbitals are characterized by the three quantum
numbersn, l and λ as nlλ, where n is a numberthat labels the
electronic states according to increasing energy (analogous to
the principal quantumnumber n in atoms), l is the quantum
number of the orbital angular momentum l which is, however,
only defined for large R, and λ the projection quantumnumber.
As in atoms the states with differentl-valuesfor R→∞are
labeled with Latin letters (s for l = 0, p for l = 1 . . .).
A molecular orbital with n = 1, l = 0, λ = 0 is named 1sσ and
with n = 2, l = 1, λ = 1 is a 2pπ orbital.
10. Molecular Orbitals and LCAOApproximations
The H2
+ molecular ion can be composed of an H atom and an H+-ion (= proton).For the lowest energy state of H2
+, the H
atom is in its 1s ground state. The atomic orbital of the electron in the H atom is then
The molecular wave function is the linear combination because the electron
can be found on eitherAor B and we cannot distinguish between the two.
where rA = r + R/2 and rB = r − R/2 can be substituted by the nuclear
separation R and the distance r = |r|of the electron from the center
of mass.
11. The wave function should be normalized for arbitrary values of R. This demands
where the integration is performed over the coordinates of the electron.
SAB depends on the spatial overlap of the two atomic wave functions and is called an overlap integral.
Its value depends on the internuclear separation R, because the integration occurs over the electron coordinates r =
{x, y, z} which depend on R.
12. The molecular wave function has to be either symmetric or antisymmetric with respect to the exchange of the two atomic
orbitals (which is equivalent to the exchange of the electron between nucleusAand B). This demands c1 = ±c2 and yields the
normalized molecular wave functions
The expectation value for the energy is
The energies depend on the
nuclear separation R.
13. All integrals depend on the
nuclear separation R.
The variables rA and rB in the atomic
orbitals gives the distance between the
electron and the nucleusA or B,
respectively.
These integrals represent two-center integrals, since the atomic
orbitals are expressed in coordinates with two different centers.
14. Es(R) has a minimum and corresponds to a bound state,
Ea(R) is a monotonic function, falling with increasing R.
It represents a repulsive potential and corresponds to an
unstable molecular state.
The energy E(R) → the sum of the kinetic energy of the
electron and its potential energy in the attractive force field of
the two positively charged nuclei.
In addition, the repulsive energy between the two nuclei has to
be considered.
After evaluating the integrals:
POTENTIALENERGY CURVES
15. The LCAO approximation in its simplest form is not in good agreement with the correct solutions. This gives for
limit R→ 0
Since for R → 0 a nucleus with a charge Ze with Z = 2 is formed, the energy should be −Z2EA = −4EA.
Two different effects contribute to the binding:
a) The lowering of the potential energy at the equilibrium distance Re, where the total energy E(R) has its minimum.
The electron charge distribution,with its maximum in the middle between the two nuclei, pulls the two nuclei
towards each other, due to the Coulomb force between electrons and protons. It acts like a glue that keeps the
nuclei together. This is the largest contribution to the binding.
b) The molecular wave function Φs has a larger spatial extension than the atomic 1s orbitals. The spatial uncertainty
for the electron is increased and its momentum uncertainty is smaller than for the H-atom, which means that its
mean kinetic energy Ekin = p2/2m is decreased.
16. Improvements to the LCAO ansatz
The simple LCAO approximation can be improved if instead of the unperturbed atomic orbitals modified functions are
used, where the two parameters λ and η(R) are optimized for each nuclear separation R to bringthe energy E(R) to a
minimum.
λ → considers that the charge distribution of the electron is no longer spherically symmetric in the force field of the two
protons, but showsa deformation in the z-direction, which reducesthe spherical symmetry to cylindrical symmetry.
η(R) → describes the radial distributions |Φ(rA)|2 and |Φ(rB)|2 will depend on the nuclear separation.
For η > 1 the orbitals ΦA and ΦB are contracted, which causes a lowering of the energy because the electron
charge becomes more concentrated between the two nuclei.
