Penfold L6 Flashcards
(24 cards)
What does the variational principle allow us to do
variational principle enables us to solve the Schrödinger equation knowing we have two unknowns
Outline the variational principle
the energy obtained by solving the Schrödinger equation using an approximate trial wave function is always greater than or equal to the true ground state energy
How can we improve a trial wave function
we can improve a trial wave function by varying the coefficients until the total energy reaches a minimum
What do we mean by using an approximate trial wave function
taking a guess at the coefficients
What does the variational principle mean
variational principle means whatever we do we cannot ever get an energy from the Schrödinger equation which is lower than the true energy
How do electrons in multielectron systems interact
repulsive electron-electron interactions are strong
Describe what we need to consider for electron interactions
consider how every pair of electrons affect the position of all the other electrons and how the pair of electrons affects the behaviour of the third electron
Describe the number of interactions changes with number of electrons
the number of interactions we have to describe, increases exponentially with number of electrons
Describe how we calculate wavefunction for many electron systems
1) computer has to guess coefficients and therefore a trial wavefunction
2) then generate the total electron density
3) find the hartree potential
4)add the hartree potential to the Schrödinger equation and solve to find the molecular orbitals of the systems wavenfunction
How do we find the hartree potential
find the electrostatic potential between and electron and the average probabilty density of all the other electrons
What does the hartree potential pretend
hartree potential pretends that one electron does not interact individually with all the other electrons in the system instead it interacts with an averaged cloud of the remaining n-1 electrons.
What does SCF stand for
self consistent field
Describe the self consistent field
take a guess at wave funcition, from this we compute all interactions between electrons. The target is to find the ground state wave function and energy. If the total energy changed when we compute the energy then we need to repeat the cycle using new wave function
What is the advantage of the variational energy
energy can never be lower than exact result so we can use it to assess how good the calculations are. If we run two different calculations on the same molecule the lowest energy is the best calculation
What is the hartree Fock method
It assumes one electron interacts with a smeared electron density, Uses a Slater determinant to build the total wavefunction (so it obeys the Pauli exclusion principle)
What are the two different types of hartree fock calculation
restricted hartree Fock
Unrestricted hartree fock
Describe when restricted hartree fock is used
used where there are an even number of paired electrons
Not used for H atom or O2 molecule
describe unrestricted hartree fock
is the more general case it is identical to RHF if electrons are paired but it can also handle systems with a net spin
How do we calculate the energy from hartree fock
EHF=Ehartree + Ex
Ex - exchange energy come from slater determinant
Ehartree - energy from solving the Schrödinger equation subject to the hartree approximation
Define correlation
the wavefunction should depend on the electron electron distance and not just the electron nucleus distance
What is correlation more important for
It is more important for elements further down the periodic table where electrons approach more closely (e.g. in 3d orbitals) and for van der Waals complexes.
Describe electron correlation for hartree fock theory
In hartree fock theory we basically have zero electron correlation as one electron does not see specifically any other electron there is no correlation between them
How do we calculate the true energy for a multielectron system
Etrue = EHF +Ec
Where EHF is the hartree fock energy
Ec is the correlation energy and is negative
Why is the correlation energy negative
because the energy determined from hartree fock is higher than the actual energy so to make the true energy the correlation energy needs to be negative
