State wavefunctions

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

Some of the ways in which excitai-state wavefunctions can be included in a configuration interaction Illation (Figure adapted from Hehre W ], L Roikiin, P zi R Schleyer and ] A Hehre 1986. Ab initio Molecular aital Theory. New York, Wiley.)... 

The eigenfunctions of the zeroth-order Hamiltonian are written with energies. ground-state wavefunction is thus with energy Eg° To devise a scheme by Lch it is possible to gradually improve the eigenfunctions and eigenvalues of we write the true Hamiltonian as follows ... 

To obtain expressions that permit properties other than the energy to be evaluated in terms of the state wavefunction P, the following strategy is used ... 

In the case of the hydrogen molecule-ion H2" ", we defined certain integrals Saa, Taa, Tab, Labra- The electronic part of the energy appropriate to the Heitler-London (singlet) ground-state wavefunction, after doing the integrations... 

Don t confuse the state wavefunction with a molecular orbital we might well want to build the state wavefunction, which describes all the 16 electrons, from molecular orbitals each of which describe a single electron. But the two are not the same. We would have to find some suitable one-electron wavefunctions and then combine them into a slater determinant in order to take account of the Pauli principle. 

In Chapter 6, I discussed the open-shell HF-LCAO model. 1 considered the simple case where we had ti doubly occupied orbitals and 2 orbitals all singly occupied by parallel spin electrons. The ground-state wavefunction was a single Slater determinant. I explained that it was possible to derive an expression for the electronic energy... 

Excited-state wavefunction analyses arc carried out in the framework of the Intermediate Neglect of Differential Overlap/Single Configuration Interaction (INDO/ SCI) technique to characterize the properties of the photogenerated electron-hole pairs. The SCI wavefunction writes ... 

The hierarchy of shells, subshells, and orbitals is summarized in Fig. 1.30 and Table 1.3. Each possible combination of the three quantum numbers specifies an individual orbital. For example, an electron in the ground state of a hydrogen atom has the specification n = 1, / = 0, nij = 0. Because 1=0, the ground-state wavefunction is an example of an s-orbital and is denoted Is. Each... 

Suppose each state wavefunction in Eq. (4.3) can be written as a simple product of space-only and spin-only parts then Q is given by... 

Here, /j f and Xi,f are the initial and final state electronic and vibration-rotation state wavefunctions, respectively, and i,f are the respective state enei gies which are connected via a photon of energy boo. For a particular electronic transition (i.e., a specific choice for /i and /f and for a specific choice of initial vibration-rotation state, it is possible to obtain an expression for the total rate Rj of transitions fi om this particular initial state into all vibration-rotation states of the final electronic state. This is done by first using the Fourier representation of the Dirac 5 function ... 

If the initial ground-state wavefunction (/(q is nondegenerate, the first-order term (i. e., the second term) in Eq. (1) is nonzero only for the totally-symmetrical nuclear displacements (note that g, and (dH/dQi) have the same symmetry). Information about the equilibrium nuclear configuration after the symmetrical first-order deformation will be given by equating the first-order term to zero. 

In the present paper, we propose the use of the HPHF approximation for the direct calculation of excited states, in which M5=0,just as Berthier [11], and Pople and Nesbet [12] did for the determination of states in which Ms 0. We give some examples of such calculations, either when the excited state wavefunction is orthogonal or not by symmetry to that of the ground state. 

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same marmer as it is customary when developing some scalar perturbation theory scheme by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is ... 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

Electronic Spectra Excited-State Wavefunctions and Configuration Interaction (Cl)... 

Figure 15. Snapshots of the two frontier excited-state natural orbitals (computed using the HF-OA-CAS(4/4) S wavefunction) of the excited-state trajectory of cyclobutene shown in Fig. 13. Left panels Before the onset of disrotatory motion, the excited-state wavefunction can be described using a single determinant with one electron in a tt-like orbital (4>a) and one in a 7t -like orbital (4>b). Middle panels During the disrotatory motion the simplest description of the electronic wavefunction requires two determinants. In one determinant both electrons are in the (j)a orbital, and in the other they are both in the (j)b orbital. Both orbitals (<()a and 4>b) show significant cj—it mixing, which is a consequence of the significant disrotatory motion. Right panels When the disrotatory motion is completed, the excited-state wavefunction is described by a single determinant in which both electrons are in the <()b orbital. Note how the shape of the orbitals changes as the initial bonds are broken and the two new tc bonds are formed.

---