Investigate how excited states are calculated

  • We generally study the calculation of excited states in the following three idealized transition modes:

     

    (1) Vertical absorption: The electron absorbs photons from the ground state and is excited to the excited state, and the structure maintains the minimal point structure of the ground state. The calculation of vertical absorption energy consists of the following two steps

    1. Optimize the ground state geometry
    2. Calculate the excitation energy from the ground state to the excited state of interest under the structure of 1

     

    (2) Vertical emission: the electrons emit photons from the excited state and de-excited to the ground state, and the structure maintains the minimal point structure of the excited state. Calculating the vertical emission energy includes the following two steps

    1. Optimize the geometric structure of the excited state (it doesn’t matter if you guess the structure at first, if you don’t know what the structure of the excited state is, generally use the minimal structure of the ground state as the initial guess)
    2. Calculate the excitation energy from the ground state to the excited state under the structure of 1 (this is the energy that this excited state emits vertically to the ground state)

     

    (3) Adiabatic absorption: The electron is excited from the ground state to the excited state, and the structure changes from the ground state minimum point structure to the excited state minimum point structure. The calculation of adiabatic absorption energy consists of the following three steps

    1. Optimize the geometric structure of the ground state, and obtain the minimum point energy of the ground state in the last step
    2. Optimize the geometric structure of the excited state, and obtain the minimum point energy of the excited state in the last step
    3. Calculate the difference between the energy obtained in the second step and the energy obtained in the first step

    Adiabatic emission is the reverse process of adiabatic absorption, and the transition energy is the same.

     

    The above transition methods are idealized and useful for theoretical research and practical problems. The maximum peak positions of the actual absorption and emission spectra are generally relatively close to the vertical absorption energy and the vertical emission energy, respectively. The 0-0 transition, that is, the transition energy between the vibrational ground states of the two electronic states, is relatively close to the adiabatic excitation energy.

     

    When the ground state is a singlet state (S0), the launch process is divided into two categories:

     

    (1) Fluorescence emission: de-excitation of the emitted photon from the singlet excited state to the ground state. According to the kasha rule, most of the fluorescence emission is de-excited from S1 (the singlet excited state with the lowest energy) to S0. Therefore, when calculating fluorescence according to the above-mentioned vertical emission method, the excited state is generally selected as the S1 state. In a few cases, the kasha rule is not satisfied. For example, Azulene is mostly de-energized from S2 instead of S1 to S0.

     

    (2) Phosphorescence emission: de-excitation of emitted photons from the triplet excited state to the ground state. The phosphorescence emission is de-excited from T1 (the triplet excited state with the lowest energy) to S0. Therefore, when calculating phosphorescence according to the above-mentioned vertical emission method, the excited state is selected as the T1 state.

     

     

    Transition electric dipole moment and oscillator strength

     

    Assuming that the initial state electron wave function is |i> and the final state is |j>, the transition electric dipole moment between the two states is. Here r is the coordinate vector, and the negative sign is because electrons are negatively charged. The transition dipole moment generally refers to the transition electric dipole moment, but there are also transition magnetic dipole moments, transition speed dipole moments, and so on. Note that the difference between the transition electric dipole moment and the electric dipole moment between the two states, namely -, is not directly related at all, and many beginners confuse it.

     

    With the transition electric dipole moment and the energy difference ΔE between the two states, the oscillator strength f corresponding to the transition between the two states can be obtained: (2/3)ΔE*||^2, which is a dimensionless quantity.

     

    The stronger the oscillator between the ground state system and a certain excited state, the easier it is to absorb the electromagnetic wave of the corresponding frequency and transition to that excited state, so the corresponding absorption peak in the absorption spectrum is also stronger. In general, the vibrator intensity is less than 1, but it can also be greater than 1, and the vibrator intensity with a very strong absorption peak can even reach close to 7 or 8. The vibrator intensity less than 0.01 can generally be considered as a transition forbidden.

     

    The excited state lifetime can be calculated according to Einstein's formula through the vibrator intensity and the energy difference between the two states, which is suitable for both fluorescence and phosphorescence emission, and often corresponds well to the experiment. If it is very different from the experimental value, it is possible that significant internal conversion, external conversion and other non-radiative processes have occurred. The calculation formula is: life τ=3/(2*f*ΔE^2). The unit of life here is seconds, and ΔE is in cm-1. The reciprocal of life is the rate of spontaneous radiation.

     

    The triplet state and the singlet state are spin-forbidden. Only when the spin-orbit coupling is considered, the vibrator intensity is not zero, but the value is still small, so the phosphorescence lifetime is much longer than the fluorescence lifetime.

     

     

    The generation of electronic spectra

     

    Excitation energy and vibrator intensity are pure theoretical data. To convert it into UV-Vis (ultraviolet-visible absorption spectrum), fluorescence, and phosphorescence spectra that can be compared with experiments, the transition needs to be broadened. Here is a brief talk about the calculation process.

     

    (1) UV-Vis spectrum: Calculate the excitation energy and oscillator intensity from the ground state to a batch of excited states under the optimized structure of the ground state, and then broaden each transition with Gaussian function based on these two quantities, and then superimpose all transitions. Get the UV-Vis spectrum. Programs such as Gaussian also output the rotor intensity from the ground state to each excited state by the way. If you use it to replace the vibrator intensity for broadening, you will get an electronic circular dichroism (ECD).

    (2) Fluorescence spectrum: Assuming that the kasha rule is satisfied, calculate the vertical emission energy of S1->S0 and the corresponding oscillator intensity under the optimized S1 structure, and then use the Gaussian function to broaden the peak shape, which is the fluorescence spectrum.

    (3) Phosphorescence spectrum: Calculate the vertical emission energy of T1->S0 and the corresponding vibrator intensity under the optimized T1 structure (the spin-orbit coupling needs to be considered, otherwise it must be 0), and then use the Gaussian function to broaden the peak shape, which is phosphorescence Spectrum.

     

    Since electronic excitation actually involves transitions between different vibration states, the electronic spectrum has a fine structure (but it cannot be observed in polar solvents, only band spectra can be observed). If nuclear vibrations are not considered, that is, using the calculation method introduced above, the theoretical spectrum obtained has no fine structure, especially the fluorescence and phosphorescence spectra have only one Gaussian peak shape. If you want to obtain electronic spectra with fine structure, that is, vibrationally resolved (Vibrationally-resolved) electronic spectra

     

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