Wavefunctions excited

H. Nakatsuji, Chem. Phys. Lett., S9, 362 (1978). Cluster Expansion of the Wavefunctions. Excited States. 

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

In words, equation (Al.6.89) is saying that the second-order wavefunction is obtained by propagating the initial wavefunction on the ground-state surface until time t", at which time it is excited up to the excited state, upon which it evolves until it is returned to the ground state at time t, where it propagates until time t. NRT stands for non-resonant tenn it is obtained by and cOj -f-> -cOg, and its physical interpretation is... 

The object now will be to steer the wavefunction out of a specific exit chaimel on the ground electronic state, using the excited electronic state as an intennediate. Insofar as the control is achieved by transferring amplitude between two electronic states, all tire concepts regarding the central quantity introduced above will now come into play. 

One expresses the fmal-state wavefunction (i.e. describing the excited, cation, or anion state) in temis... 

For separable initial states the single excitation terms can be set to zero at all times at this level of approximation. Eqs. (32),(33),(34) together with the CSP equations and with the ansatz (31) for the total wavefunction are the working equations for the approach. This form, without further extension, is valid only for short time-domains (typically, a few picoseconds at most). For large times, higher correlations, i.e. interactions between different singly and doubly excited states must be included. 

For /2(Ar)i7, an extensive CI-CSP simulation was carried out, and the results were compared with those of the simple CSP approximation. Both calculations are for the ultrafast dynamics following excitation of the I2 into the B state. We found that the CI-CSP calculation, including doubly excited configurations , is close to converged for times up to t 500 fs, when 1500 configurations are included. Fig. 2 shows co(t)p, the coefficient of the CSP term and the doubly excited terms in the full CI-CSP wavefunction,... 

I la2g la2y 2a2g 2a2 and all single and double excitations relative to this (dominant) CSF, which is a very common type of Cl procedure to follow, the Bc2 wavefunction would not have contained the particular CSFs ls2 2p2 ls2 2p2 b because these CSFs are four-fold excited relative to the la2g la2y 2a2g 2a2 reference CSF. 

The first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

The calculation mixes all single determinant wavefunctions that can be obtained from the ground state by exciting electrons from a subset of the occupied orbitals (of the ground state) to a subset of the unoccupied orbitals. The subsets are specified as a fixed number (highest occupied or lowest unoccupied) or by an energy criterion associated with the energy difference between the occupied orbital and the unoccupied orbital. 

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

We will find an excitation which goes from a totally symmetric representation into a different one as a shortcut for determining the symmetry of each excited state. For benzene s point group, this totally symmetric representation is Ajg. We ll use the wavefunction coefficients section of the excited state output, along with the listing of the molecular orbitals from the population analysis ... 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

I don t mean that such a wavefunction is necessarily very accurate you saw a minute ago that the LCAO treatment of dihydrogen is rather poor. I mean that, in principle, a Slater determinant has the correct spatial and spin symmetry to represent an electronic state. It very often happens that we have to take combinations of Slater determinants in order to make progress for example, the first excited states of dihydrogen caimot be represented adequately by a single Slater determinant such as... 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

I mentioned the Brillouin theorem in earlier chapters if rpQ is a closed-shell HF wavefunction and represents a singly excited state, then... 

If we used perturbation theory to estimate the expansion coefficients c etc., then all the singly excited coefficients would be zero by Brillouin s theorem. This led authors to make statements that HF calculations of primary properties are correct to second order of perturbation theory , because substitution of the perturbed wavefunction into... 

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