Lecture 2: Few dimensional systems, Discrete variable representation, SOFT

Question 1

• Reminder from Lec 1
  
  • After making the …-… approximation, the time-dependent Schrodinger equation has the form below
  • This … … … tells us how the … wavefunction changes with time on a single … …
  • We want to use the TDSE to … wavefunctions

Answer 1

• Reminder from Lec 1
  
  • After making the Born-Oppenheimer approximation, the time-dependent Schrodinger equation has the form below
  • This equation of motion tells us how the nuclear wavefunction changes with time on a single electronic state
  • We want to use the TDSE to propagate wavefunctions

Question 2

• Reminder from Lec 1 part 2
  
  • Also saw last time that if we know the … of the … , wavefunction propagation is simplified
  • If the wavefuntion is one of the … of the system, simply picks up an … … … which changes with time

Answer 2

• Reminder from Lec 1 part 2
  
  • Also saw last time that if we know the eigenstates of the PES, wavefunction propagation is simplified
  • If the wavefuntion is one of the eigenstates of the system, simply picks up an oscillating phase factor which changes with time

Question 3

What if the wavefunction being propagated is not an eigenstate, but instead a combination of states is being populated?

Answer 3

• Eigenstates of a Hamiltonian form a complete set
• Any function can be written as a linear combination of eigenstates of this Hamiltonian
• Time dependence now carried by expansion coefficients which contribute differently to each state

Question 4

• What are the two steps for propagating known eigenstates

Answer 4

Question 5

• How can eigenstates included in the nuclear SEQ be represented?

Answer 5

• Want to find states that obey nuclear SEQ
• Can expand each eigenstate in a fixed basis set of introduced functions ϕ
• a indicates how much each basis function k contribute to eigenstate i

Question 6

Our basis set is orthonormal, what does this mean?

Answer 6

• Delta function is 0 if states i/j are not the same
• Delta function is 1 if states i/j are the same

Question 7

• How is the set of equations of linear combinations of each eigenstate (linear combination of basis-functions) collected?

Answer 7

• Set of equations collected together in a matrix, which is easy to compute
• Where input is Hamiltonian matrix expressed in basis functions ϕ
• Output is a diagonal matrix of eigenvalues (energies) and coefficients associated with them, a
• MxM coefficient matrix, a indicates, for each eigenstate, how much each basis set contributes.

Question 8

• What is a good choice for basis-functions, ϕ?

Answer 8

• Represent each basis function as a collection of complex waves (this is one example)
• Each appear as delta functions centred at each grid-point
• Increase in j moves line to left of graph (v.v)

Question 9

Describe the differences and similarities between these two eigenstate representations

Answer 9

• Sum of linear combination of functions of different contributions form different eigenstates
• Same basis set in each but different set of contributions
• Delta function on a uniform grid, height of which gives a (contribution) of basis function to eigenfunction

Question 10

• What contributes to the Hamiltonian matrix elements for our chosen basis functions?

Answer 10

• H_ij = T_ij + V_ij
• T term depends on if i/j are the same or not, and can be calculated easily as know contributing components e.g. mass
• V_ij depends on delta function, works out as diagonal matrix where value at each grid point is just the V at that particular grid point
• Now Hamiltonian matrix solved in Ha = Ea, (step 1) must find coefficients of M eigenstates.
  
  • –> step 2

Question 11

• What is an example of a process that can be used to find values of eigenstates?

Answer 11

• Colbert-miller DVR used to find eigenstate numbers in matrix a

Question 12

• After eigenstates found one must move the calculated … (time dependent) associated with … in time

Answer 12

After eigenstates found one must move the calculated coefficients (time dependent) associated with wavefunction in time

Question 13

• What is the first step in calculating coefficients as they evolve in time?

Answer 13

• Work out initial coefficients
• Form a linear combination of eigenstates and coefficients i at t =0

Question 14

• What is the result of subbing in our linear combination of coefficients into the TDSEQ?

Answer 14

• Once initial coefficients known, propagation becomes simple
• Only thing changing with time is coefficients
• Can now calculate any coefficient at any time t

Question 15

• What should be the result of calculated expansion coefficients for each eigenfunction relative to the initial wavefunction

Answer 15

• Should get near exact agreement with initial wavefunction

Question 16

• Once TDSEQ solved can use … at a certain time to calculate an expectation value e.g. (…/…)

Answer 16

• Once TDSEQ solved can use wavefunction at a certain time to calculate an expectation value e.g. (momentum/position)

Question 17

• Why can DVR not be used for more degrees of freedom?

Answer 17

• For f degrees-of-freedom N^f grid-points are required
• This quickly becomes computationally demanding
• Large memory requirement for calculation of PE at each grid point
• Restricted to 5 DOFs