Excitation, two-electron

Figure 9.1. Potential energy diagram for the electronic states of ethylene N, ground state (w)2 r(3Biu) first excited triplet state (mr ) V, first excited singlet state (wir ) Z, two-electron excitation (w )2. For the ion C2H + R and R, Rydberg states, /, / ground and excited states. [From Ref. 2(c).]...

Figure 3. Two electrons in three orbitals system. Configuration (a) is the reference configuration. Single electron excitations with spin-flip produce configurations (bf(g). Two-electron excitations with a single spin-flip produce configurations (h)-(j). Note that non-spin-flipping excitations or excitations that flip the spin of two electrons produce M = l configurations, which do not interact through the Hamiltonian with the final M =0 states, and thus are not...

Because the A -particle Hamiltonian H) contains only two-electron excitations, the expectation value of H yields a formula for the energy involving just the 2-RDM,... 

It is instructive to discuss the MP2-R12 method [37] before going into more involved CC-R12. As in MP2, the wave function of MP2-R12 (IT1)) is a linear combination of the reference HF determinant ( o)) and doubly excited determinants produced by the action of a two-electron excitation operator T ... 

Figure 6 shows the process in a schematic way. Baer describes the two electron excitations of direct and inverse photoemission as two-step processes in which first Ep is reached and then the emission of an electron (direct) or of a photon (inverse photoemission) to vacuum occurs from Ep. A and A+ are the energies associated with the two first steps. They are counted from Ep, and they are to be considered as the minimum energies necessary to create the f and f final state. If the localized level response is a final state multiplet, therefore, Uh = A+ -I- A is given by the sum of the smallest measured energies of the multiplet. 

FIGURE 3.8 Potential energy curves for the ground state and two electronically excited states in a hypothetical diatomic molecule. Predissociation may occur when the molecule is excited into higher vibrational levels of the state E and crosses over to repulsive state R at the point C (from Okabe, 1978). 

Calculations of PN are more challenging than that of PN because it is, of course, an excited state of phenylnitrene. The first two electronically excited singlet states of PN are both of Ai symmetry and are calculated to be at 1610 and 765 nm. Neither of these transitions have been detected, since both of these states have zero oscillator strength due to symmetry considerations, and they lie outside the wavelength range accessible to our spectrometer. ... 

Electronic absorption spectra have been recorded for a large number of oxomolybdenum(IV) species but, in addition to d-d transitions, there is the possibility of two-electron excitations, charge transfer bands and splitting of degenerate energy levels due to low symmetry. These factors make it difficult to interpret or even to compare spectra.5... 

The experimental data presented in Ref. 79 unambiguously prove that the SES are formed in two-electron excitations. According to the measurements of the energy-absorption and ionization probabilities, 17( ) is below unity for CH4 at energies from 27 to 80 eV. The scheme of occupation of energy levels in a CH4 molecule is featured in Fig. 6. Since the highest ionization potential of valence electrons is 23.1 eV, and the next potential, corresponding to ionization of the K shell, is around 290 eV, the SES in this case correspond to excitation of two or more electrons. 

Direct calculation from equation (29) requires a knowledge of all excited-state wavefunctions. If these are known, equation (29) becomes a sum over all discrete states and an integration over continuum states. For systems of more than two electrons, excited-state wavefunctions are difficult to come by Even less is known about the continuum states, but for the diamagnetic susceptibility their contribution is thought to be of the same order of magnitude as that of the discrete states. For other systems the direct use of equation (29) is clearly a non-starter. 

An alternative would be the hyperspherical coordinate system, introduced into the study of the ground-state helium by Gronwall [86], developed for nuclear reactions by Delves [87, 88], adopted in molecular reactive collisions by Smith [89], and initiated applications to two-electron excited QBSs by Macek [90]. The hyperradius p and one of the hyperangles, the radial hyperangle a, are defined by... 

The correlated wavefunction which incorporates ISCI follows in analogy to equ. (1.25a) as an expansion into independent-particle wavefunctions for the ground state and contributions from virtual two-electron excitations ... 

Assuming a model wherein a molecule possesses only two electronically excited states, Schellman has proposed three mechanisms would lead to the observation of optical activity in the electronic transitions among these [9]. The three models may be summarized as ... 

An accurate solution for the problem can be found by a complete multiconfiguration SCF treatment, in which the expansion coefficients of Eqs. (1) and (2) are determined simultaneously using the SCF techniques, with the usual trial and error procedure. This formulation can be developed along the lines of that given by Veil-lard and Clementi (1967) for closed-shell systems with inclusion of only two-electron excitations. 

An important implication of Brillouin s theorem is the guarantee of no first-order errors for all one-electron operators provided the exact Hartree-Fock wavefunctions are used.426 This follows because the first-order correction to the correlation effect is derived from two-electron excitations. 

The second approximation often used to truncate Cl wavefunctions is to neglect configurations formed by more than two-electron excitations. It is easily seen that they cannot mix into the Cl wavefunction in first order. The third method comes from perturbation theory. If one takes the Hartree-Fock energy as the zeroth-order approximation to the energy... 

The 5-term involves two electronic excited states (e and 5) and provides a mechanism for resonance-enhancement of non-totally symmetric vibrations. The 5-term, can be expressed as... 

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