Atomic and Molecular Physics: Molecules

Born-Oppenheimer approximation

Treats the nucleus as fixed in position. The mass of the nucleus is much larger than the mass of an electron, and so the nuclei move much more slowly than electrons.

When can the Born-Oppenheimer approximation be applied?

When solving the TISE for a molecule for which the motion of the electron(s) is much faster than the motion of the nuclei.

(When the electronic wavefunction changes slowly at small changes of nuclear positions.)

This is generally the case in low energy electronic states when the nuclei are much heavier than the electrons, and far from crossings between electronic potential energy curves where the electronic wavefunctions change significantly at very small changes of nuclear positions.

The total wavefunction of a diatomic molecule under the Born-Oppenheimer approximation.

The total wavefunction of a diatomic molecule can be expressed as Ψ(\vec R; \vec r_i) = ν(\vec R)ψ(\vec R; \vec r_i), where \vec R is the internuclear distance, and \vec r_i are the position vectors of individual electrons.

Here ν(\vec R) is the nuclear wavefunction which only depends on the internuclear distance, and ψ (\vec R; \vec r_i) is the electronic wavefunction, which depends on the internuclear distance and the positions of the electrons.

Linear Combination of Atomic Orbitals (LCAO) approximation

Represents the spatial components of molecular orbitals as a linear combination of atomic orbitals \phi_i^{AO} usually centered on atoms: \psi=\sum^N_{i=1}C_i\phi_i^{AO}.

The AOs here are called basis functions.

This approximation assumes that in a molecule electrons are well described by the wave functions of the corresponding atoms.

Most stable nuclear geometry

Occurs at the nuclei separation where potential energy is a minimum.

Given by the minimum energy principle: For a closed system with fixed entropy, the total energy is minimised at equilibrium.

Overlap integral

A measure of the overlap of atomic orbitals centered on two different atoms.

Defined as S(R)=\int\phi^*(\vec r_A)\phi(\vec r_B)\,\mathrm{d}r_A\,\mathrm{d}r_B extending over all of space. When R→∞, the overlap integral tends to 0 and when R →0, it tends to 1.

Lowest energy electronic states of H^+_2

Given by the linear combination of the two atomic orbitals: \psi_+=\frac{1}{\sqrt2}[\phi_{1s}(\vec r_A)+\phi_{1s}(\vec r_B)],\quad \psi_-=\frac{1}{\sqrt2}[\phi_{1s}(\vec r_A)-\phi_{1s}(\vec r_B)] , where \psi_+ is symmetric, and \psi_- anti-symmetric.

Lowest energy electronic states of H₂

Given by the linear combination of the two atomic orbitals: \psi^T=\frac{1}{\sqrt 2}\left[\phi_{1s}(\vec r_{A1})\phi_{1s}(\vec r_{B2})-\phi_{1s}(\vec r_{A2})\phi_{1s}(\vec r_{B1})\right]\,\chi^T, \quad\psi^S=\frac{1}{\sqrt 2}\left[\phi_{1s}(\vec r_{A1})\phi_{1s}(\vec r_{B2})+\phi_{1s}(\vec r_{A2})\phi_{1s}(\vec r_{B1})\right]\,\chi^S, where T and S represent the triplet and singlet states.

What effect do the overlap, Coulomb and exchange integrals have on the stability of the H₂ molecule?

The overlap integral decreases the energy of the ψ^S state with its symmetric spatial wavefunction, while it increases the energy of the ψ^T state with its antisymmetric spatial wavefunction. (Its effect on the total electronic energy of the states depends on the symmetry of their spatial wavefunctions.)

The Coulomb integral has a positive value for both the ψ^S and ψ^T states at all internuclear distances, and therefore increases the energy of both states.

The exchange integral is (in general) less than zero for the ψ^S and ψ^T states. It decreases the energy of the symmetric ψ^S state, and increases the energy of the antisymmetric ψ^T state. (Its effect on the total electronic energy of the states depends on the symmetry of their spatial wavefunctions.)

Exchange integral

Represents the exchange of the electron between the two nuclei,

|K(R)|=\int\phi^*_{1s}(\vec r_B)\frac{1}{r_A}\phi_{1s}(\vec r_A)\,\mathrm{d}\tau

Coulomb integral

Represents the interaction of the electron charge density around proton A with proton B,

|J(R)|=\int\phi^*_{1s}(\vec r_A)\frac{1}{r_B}\phi_{1s}(\vec r_A)\,\mathrm{d}\tau

Why is the ionisation energy for H₂⁺ larger than that of H₂ and that of the H atom?

The ionisation potential of H₂ is smaller due to of the Coulomb repulsion between the two electrons and the ionization potential of H is smaller because of the extra Coulomb attraction of an electron to the second proton in H2+.

Write the electronic Hamiltonian of the H₂ molecule in atomic units.

\hat H=-\frac{\nabla_1^2}2-\frac{\nabla_2^2}2-\frac{1}{r_{1A}}-\frac{1}{r_{1B}}-\frac{1}{r_{2A}}-\frac{1}{r_{2B}}+\frac{1}{r_{12}}+\frac{1}{R_{AB}}

where A and B refer to the two protons, 1 and 2 refer to the two electrons, and R is the internuclear distance.

This is within the Born-Oppenheimer approximation.

Why electronegativity is a good qualitative parameter to describe the character of chemical bonding in diatomic molecules?

An element’s electronegativity provides us with a single value that we can use to characterize the chemistry of an element.

• Elements with a high electronegativity ≥ 2.2 have very negative affinities and large ionization potentials, so they are generally nonmetals that tend to gain electrons in chemical reactions.

• Elements with a low electronegativity ≤ 1.8 have electron affinities that have either positive or small negative values and small ionization potentials. These elements are generally metals and good electrical conductors that tend to lose their valence electrons in chemical reactions.

The distinction between metals and nonmetals is one of the most fundamental we can make in categorizing the elements and predicting their chemical behaviour.

What is included in the Hamiltonian operator for a (diatomic) molecule?

The kinetic energies of the nuclei, the kinetic energies of the electrons and the total potential energy of the system.

The total potential energy terms are the Coulomb interactions: the sum over the electron-electron Coulomb repulsion, the nucleus-nucleus Coulomb repulsion, and the electron-nucleus Coulomb interaction.

Why is it not possible to simplify the TISE by separating the nuclear and electronic parts of the total wavefunction?

Since the electron-nucleus Coulomb interaction is dependent on both the positions of the nuclei and electrons.

Solutions to the TISE for each value of internuclear separation give what?

Each value of internuclear separation gives a set of eigenvalues (energies, E_{elec}(R)) and eigenfunctions \psi_{elec}(\vec r, R) (molecular orbitals).

Molecular orbital

The superposition of atomic orbitals.

What does the equilibrium distance between the nuclei in the H₂ molecule correspond to?

The distance where the force acting on the nuclei is equal to zero.

What is the difference between calculated and experimental values of the dissociation energy of H₂⁺?

The calculated value is smaller than the experimental value.

This is because the calculation (in lectures) only used 1s orbitals of the hydrogen atom when constructing the LCAOs.

When does an ionic bond form in a molecule?

When the electronegativity values of the two atoms differs by more than 2.

When do atomic orbitals contribute significantly to molecular bond formation?

When they have substantial overlap in the region between the nuclei.

(Molecular orbitals are the superposition of different atomic orbitals.)

What is the role of the two-electron exchange integral in determining the energy of electrons in the singlet state of the H₂ molecule?

It lowers the energy of singlet states via:

• Reducing the expectation value of the electron-electron repulsion,

• Allows to satisfy the antisymmetry principle: the requirement that the overall wavefunction of the system must be antisymmetric under the exchange of any two electrons' coordinates.

What does the zero-point energy of a diatomic molecule correspond to?

The lowest possible vibrational energy of the molecule

Degrees of freedom for a molecule with N nuclei

In total, 3N degrees of freedom:

• 3 translational

• 3 rotational or 2 for a linear molecule

• 3N-6 vibrational or 3N-5 for a linear molecule

What property must a diatomic molecule possess for it to undergo pure rotational transitions?

For a diatomic to undergo pure rotational transitions it must have a permanent electric dipole moment, i.e., it must be polar.

What is the rigid rotor approximation?

When the internuclear distance remains constant for all values of J and is not affected by the molecule’s rotation.

Energies of rotational states for diatomic molecules

Given by

\frac{E_{rot}}{hc} = BJ(J + 1)

where J is the rotational quantum number, which takes the values J=0,1,2,...

Spectral line spacing of a gas, e.g. CO

Spectral lines are equally spaced.

The spectra of pure rotational transitions between these states are governed by the selection rule that ΔJ = ±1. In the spectrum of a warm gas of CO in which many rotational states are populated, the lowest wavenumber transition that will be observed is the J = 0 →J' = 1 transition which lies at ΔE/hc = 2B. The next lowest wavenumber transition is the J = 1 →J' = 2 transition, which lies at ΔE/hc = 4B, followed by the J = 2 →J' = 3 transition at ΔE/hc = 6B, etc.

The pure rotational spectrum is therefore composed of this set of transitions with the corresponding spectral lines equally spaced by 2B wavenumbers.

Rotational constant, B

Inversely proportional to the reduced mass, \mu_{AB} , of the molecule, and directly proportional to the angular rotational frequency (given by a rotational quantum number J).

Antisymmetry principle

A postulate that electrons must be described by wavefunctions which are antisymmetric with respect to interchange of the coordinates (including spin) of a pair of electrons.