Calculation of vibronic couplings

In summary, the conversation discusses the derivation of linear vibronic coupling parameters in second quantization notation and the elimination of dependence of fermionic ladder operators on nuclear coordinates. The resulting Hamiltonian includes terms involving the lambdas, which can be expressed as a simple taylor expansion. The conversation ends with a request for further help in understanding the math involved.
  • #1
slawa
1
0
Hi all,

I'm referring to the paper Chem. Phys. v.26 pp.169-177 y.1977 from Cederbaum, Domcke and Köppel where the expressions for linear vibronic coupling parameters are derived in second quantization notation. I'm stuck on the point, where the dependence of the fermionic ladder operators on the nuclear coordinates is eliminated. To be more specific:

H = ∑ii(0)aiai +∑i (0.50.5∂∊i/∂Q)[aiai-ni]Q

is the part of the Hamiltonian I'm interested in. Here the fermionic ladder operators are still dependend on the nuclear coordinates Q (for simplicity let there be just one of it at this time). Now by projecting the ladder operator on an Q-independend basis

ai(Q) = ∑ji(Q)|ϕj(0)>aj(0)

these parts of the hamiltonian should look like:

ii(0)aiai +∑i 0.50.5Qi[aiai-ni]Q + ∑i,j λi,j[aiaj+ajai]Q

where the lambdas are given by

λi,j = lim Q→0[∊i(Q)-∊j(Q)]<ϕj(Q)|∂Qi(Q)>

Up to now I'm pretty shure that I understand how to get the first two summands of the new hamiltonian but the off diagonal coupling term with the lambdas...My last point was, that from the last term in the old hamiltonian leftover terms like this one

-∂Q1(0)[<ϕ1(0)|ϕ1(Q)>a11(Q)|ϕ2(0)>a2+<ϕ2(0)|ϕ1(Q)>a21(Q)|ϕ1(0)>a1]Q-∂Q2(0)[<ϕ1(0)|ϕ2(Q)>a12(Q)|ϕ2(0)>a2+<ϕ2(0)|ϕ2(Q)>a22(Q)|ϕ1(0)>a1]Q

have to become the third summand in the new hamiltonian...Here some link is missing.

Further they state, that from going from the general expression for the lambdas to a two state system, the lambdas can be written as:

λ1,2=0.75*(∂Q22[∊1(Q)-∊2(Q)]2)00.5

So here I'm obviously missing some math ;)

I'd appreciate some help and sorry for the load of equations :|

Greetings
Slawa
 
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  • #2
I think the step you are missing is a simple taylor expansion of their eq 6: ## \langle \phi_i(Q)|\phi_j(0)\rangle=\langle \phi_i(0)| \phi_j(0)\rangle +\langle \partial \phi_i(Q)/\partial Q|_{Q=0} |\phi_j(0) \rangle Q ##.
 

FAQ: Calculation of vibronic couplings

1. What are vibronic couplings?

Vibronic couplings refer to the interactions between electronic and vibrational states of a molecule. These couplings play a crucial role in determining the energy levels and transitions of a molecule and are important in various fields such as spectroscopy, photochemistry, and photobiology.

2. How are vibronic couplings calculated?

Vibronic couplings can be calculated using various theoretical methods such as density functional theory (DFT), time-dependent density functional theory (TD-DFT), and coupled cluster theory. These methods involve solving mathematical equations to determine the electronic and vibrational energy levels and their interactions.

3. What factors affect vibronic couplings?

The strength of vibronic couplings is influenced by several factors, including the nature and number of atoms in the molecule, the distance between electronic and vibrational states, and the strength of the electronic transition. Additionally, environmental factors such as temperature and solvent can also affect vibronic couplings.

4. How are vibronic couplings used in research?

Vibronic couplings play a crucial role in understanding and predicting the behavior of molecules in various chemical and biological processes. They are commonly used in research related to spectroscopy, photochemistry, and photobiology to analyze and interpret experimental data and to design new experiments.

5. What are some applications of vibronic couplings?

Vibronic couplings have numerous applications in different fields. They are used in spectroscopy techniques to study the electronic and vibrational properties of molecules, in photochemistry to understand the mechanisms of light-induced reactions, and in photobiology to investigate the effects of light on biological systems. They are also important in the design of new materials for applications in electronics and optoelectronics.

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