Lecture 3: Non-adiabatic dynamics

Question 1

• Reminder from lecture 2
  
  • Saw how we can propagate wavefunctions on a … potential energy surface by solving the time dependent Schrodinger equation using … … … (…) method

Answer 1

• Reminder from lecture 2
  
  • Saw how we can propagate wavefunctions on a single potential energy surface by solving the time dependent Schrodinger equation using discrete variable representation (DVR) method

Question 2

• What part of photoexcitation must be accounted for in quantum dynamics calculations?

Answer 2

• Must be able to account for more than one electronic surface in quantum dynamic simulations
• Photon absorption can result in a change of electronic state, the excited states of which give many excited PESs
• How these PESs couple is an essential part of what define the photodynamics of a reaction

Question 3

• What method is used in photochemistry to infer the dynamics of a process?

Answer 3

• Experimentalists use pump-probe spectroscopy with two pulsed lasers
• One to initiate dynamics (GS excitation)
• One to measure adsorption (after fs-ps delay)
• Adsorption energies and decay time-scales used to infer dynamic process

Question 4

• How can pathways in photochemistry be confirmed theoretically?

Answer 4

• Non-adiabatic simulations can be used to confirm (or provide new) dynamic hypothesis

Question 5

• How does has the BOA simplified the TDSE thus far?

Answer 5

• Ignores terms which couple electronic-nuclear wavefunction evolution
• Assumed electronic state does not change when nuclear coordinates change therefore 2 end terms on KE expansion can be ignored

Question 6

• What is the problem with the BOA in excited state dynamics?

Answer 6

• When two electronic states are close in energy, ignored derivatives become large
• Breakdown of BOA means explicit treatment for influence of coupled electronic states on nuclear dynamics must be made

Question 7

• Give the general exact solution of the SEQ that describes more than one electronic state and state its origin.

Answer 7

• Before, with only 1 electronic state, had 1 product ψ comprised of nuclear and electronic part
• Not sum of product ψ forms where ψ is spread across multiple states
• Can have 90% GS and 10% ES etc

Question 8

• Wavefunction dynamics on one electronic state is coupled to the others (non-adiabatic). How can the multiple state SEQ be used to investigate this coupling between states?

Answer 8

• Substitute in to the TDSE, integrating over electronic coordinates, r
• Electronic states coupled by KE-like terms
• T⁽¹⁾_mn gives an indication of coupling strength between states m/n

Question 9

• What are the drawbacks of the adiabatic representation?

Answer 9

• Propagating nuclear wavefunction parts on each electronic surface requires evaluation NACMEs (T⁽¹⁾_mn [R]) which is very computationally expensive
• These adiabatic states (T⁽¹⁾_mn [R]) can have singularities (when two states approach one another) which also makes propagation very difficult.
• NACME - non adiabatic coupling matrix elements

Question 10

• What is an alternative representing non-adiabatic dynamics, that do not require NACMEs?

Answer 10

• Transform original adiabatic electronic states to a new set of electronic states where NACMEs are zero
• T⁽¹⁾_mn(R) ≈ 0, T⁽²⁾_mn(R) ≈ 0,
• New basis of electronic states is called a diabatic basis
• NACME - non adiabatic coupling matrix elements

Question 11

What form does the TDSE take with the new diabatic basis?

Answer 11

• Diabatic representation controlled now by potential energy matrix is smooth and well-behaved and much more appropriate for use in QD simulations

Question 12

How do diabatic states form out of an adiabatic ab intio calculation?

Answer 12

• Transformation from adiabatic to diabatic basis at nuclear configuration R is a matrix operation via a diabatization matrix
• This matrix is unique to each model system

Question 13

• Compare adiabatic and diabatic representations of PESs

Answer 13

• Away from regions of coupling, both representations of states appear similar
• Near crossing, where electronic states approach one another they avoid crossing in adiabatic but cross directly in diabatic
• Note PES is defined in terms of two important DOFs

Question 14

What are conical intersections

Answer 14

• In many dimensional molecules, configurations where two electronic states touch from conical intersections (CIs)
• CIs allow ultrafast (fs) transfer between electronic states

Question 15

Name and explain some uses of Cis in nature and industry

Answer 15

• Vision: Rhodopsin and cis/trans retinal relies on ultrafast relaxation in a conical intersection
• Photo switching: excited molecule relaxes very quickly through CI, leading to different geometry
• Molecular sunscreens: adsorb energy, dissipating as heat quickly, before harmful radicals form, via CI pathways

Question 16

• With reference to the method used to describe the wavefunction of many eigen states in lecture 2, how could we do so in a system describing many states?

Answer 16

• DVR method could used to model a system with a few nuclear coordinates with multiple electronic states