João Pedro Malhado Flashcards
What is the independent particle approximation?
Its where the Hamiltonian is approximated as a sum of terms, eahc of which are only dependent on single electrons. It assumes there is no itneraction between electrons and the wavefunction is a product of single electron wavefunctions.
What is the electronic energy of the system in independent particle approximation thought of as?
A sum of all the individual energies of the independent electrons E = e1 + e2 + e3 etc.
What is the orbital approximation?
Orbital approximation is an extension on the independent particle approximation where we include the interactions between electrons in the Hamiltonian. The wavefunction is now a linear combination of the molecular orbitals instead of just the single product of independent electrons
What is the aim of orbital approximation?
To enforce electron indistinguishibility, the idea that two eelctrons cannot be identified due to their identical properties in mass charge spin etc. This ensures the wavefunction stays the same if you swap two indentical particles positions.
It’s aim is to also now take into account electron-electron interactions which due to their same charge and motion, have effects on the system, such as its potential energy and electron density (spatial distribution).
What is LCAO?
The linear combination of atomic orbitals, a method used to contstruct molecular orbitals by combining wavefunctions A and wavefunctions B for their respective atoms A and B
What are the key points of LCAO?
molecular orbital = ψi = cAϕA + cBϕB (a combination of two atomic orbitals from different atoms)
Coefficients (cA and cB): The coefficients determine the contributions of individual atomic orbitals to the molecular orbital shape.
How do we use secular equations to determine energies and shapes of these molecular orbitals?
by solving secular equations derived from the system’s Hamiltonian and overlap matrices.
The equations arise from setting the determinant of a matrix formed from the Hamiltonian (H) and overlap (S) matrices minus the energy (E) to zero.
Hamiltonian matrix is the operator that corresponds to thetotal energy of the system and the S matrix corresponds to the overlap of the orbitals.
(H_AA - E)(H_BB - E) - (H_AB - S_AB)^2 = 0 determines the energy eigenvalues (E) for the molecular orbital, the allowed energies of the system
We use variational theorem by setting the secular determinant to 0 resulting in in approximate energy values
What is variational theorem? What are the key points/rules? What is its application?
Variational theorem provides a framework to yield approximate solutions to the Shcrodinger equation.
For a trial wavefunction, the energy always has to be greater or equal to the true ground state energy of the system
Application- By minimising the energy function with repsect to the parameters, the coefficients of the LCAO, we can approximate the energy of the ground state of the system.
How do we combine two degenerate orbitals?
we are saying e = HAA = HBB
we will have two combinations, one bonding, in phase combination stabilised with respect to the original orbitals and one out of phase anti-bonding orbital destabilised with resect to the original. Stabilisation effects occur always due to HAB, interaction of the two interacting orbitals.
Why doesn’t He2 form?
The helium molecule (He2) doesn’t form despite the existence of bonding and anti-bonding combinations.
The stabilisation energy resulting from the bonding combination is always smaller compared to the destabilisation of anti-bonding
System less energetically stable as the energy penalty from anti-bonding outweighs the energy gain from bonding combo
Therefore, the lack of stabilisation means that He2 is unikely to form under standard conditions
How can we estimate HAB from SAB?
We can estimate the interaction integral from the overlap integral as they are proportional HAB∝ -SAB except for short internuclear distances. overlap integral more easy to estimate so we use it as proxy to estimate interaction integral, when SAB is 0 HAB will also be 0 and no stabilisation will occur or orbital interaction
How do we combine non-degenerate orbitals?
Non-degenerate orbitals are orbitals that have different energies (HAA ≠ HBB) within a molecular system.
Energetic Difference: HAA ≠ HBB implies that the energies of the orbitals from different atoms are distinct within the system.
Component Distribution: In-phase combinations have a larger component of the lower energy orbital, while out-of-phase combinations have a larger component of the higher energy orbital.
How does Electronegativity and Orbital Energy affect the combination of orbitals?
Electronegativity Impact: Orbitals associated with more electronegative atoms tend to have lower energies compared to those of less electronegative atoms.
Stabilization and Destabilization: In-phase combinations (bonding) are stabilized below the energy of the lower-energy orbital, while out-of-phase combinations (anti-bonding) are destabilized above the higher-energy orbital.
How does the energy difference between orbitals affect interaction?
The larger the energy difference between orbitals to start with, the smaller the stabilisation and destabilisation energy will be. At too large energy differences, there will be no stabilisation and destabilisation energy and there will be no interaction
What is the overlap of atomic orbitals on the same atom?
Overlap integral of distinct atomic orbitals on the same atom is always 0, they’re orthogonal.
Can be proved through symmetry, orbital A would be even and symmetric to 0, like a 1s orbital, orbital B would be like a P orbital and odd when combined we would make an odd as parts would cancel etc.
For the overlap integral to be non-zero, the functions
must have the same symmetry.
