Single-configuration

Given a set of A -electron space- and spin-synnnetty-adapted configuration state fiinctions in tenns of which is to be expanded as T = S. Cj two primary questions arise (1) how to detemiine the 9 coefficients and the energy E and (2) how to find the best spin orbitals ( ). ] Let us first consider the 1 where a single configuration is used so only the question of detemiining the spin orbitals exists. 

Before moving on to discuss metliods that go beyond the single-configuration mean-field model, it is important to examine some of the computational effort that goes into carrying out an SCE calculation. 

This method [ ] uses the single-configuration SCF process to detennine a set of orbitals ( ).]. Then, using an unperturbed Flamiltonian equal to the sum of the electrons Fock operators // = 2 perturbation... 

These approaches provide alternatives to the conventional tools of quantum chemistry. The Cl, MCSCF, MPPT/MBPT, and CC methods move beyond the single-configuration picture by adding to the wavefimction more configurations whose amplitudes they each detennine in their own way. This can lead to a very large number of CSFs in the correlated wavefimction and, as a result, a need for extraordinary computer resources. 

Note that, although there is a r esemblance, this ansatz is quite differ ent from the Born representation of Eq. (A.3) due to the time dependence of the electronic functions. By taking a single configuration. 

The electrical (AC) conductivity for a single configuration of each system is obtained using the Kubo-Greenwood formula [8] ... 

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 ... 

On the basis of the optimized ground-slate geometries, we simulate the absorption speetra by combining the scmicmpirical Hartree-Fock Intermediate Neglect of Differential Overlap (INDO) Hamiltonian to a Single Configuration Interaction... 

These surfaces are all based on some combination of ab initio electronic structure calculations plus fitting. The AD and BM surfaces are based respectively in whole or in part on extended-basis-set single-configuration self-consistent-field calculations, whereas the RB and RBST calculations are based on calculations including electron correlation by Moller-Plesset fourth-order perturbation theory. For the rigid-rotator calculations R., the intramolecular internuclear distances R- and R ... 

Figure 3. Charge distributions in the S, and S, slates of the 2-3 and 3-4 perp forms (0=90°) of BMPC, as obtained by single configuration (ground state SCF and [HOMO,LUMO]) wavefunctions. Only the overall charge is reported for the methyl substituents. The global net charges of the perpendicular subunits, emphasizing localization of the charge, are also indicated.

The single-electron crystal-field picture may be a crude over-simplification because Coulomb repulsion and SOC can lead to a situation where the ground term is a composite mixture of = 1/2, 3/2, and 5/2 states that does not derive from a single configuration [71, 72] the corresponding physical description is obtained from proper quantum chemical calculations [73-75]. 

Multireference methods provide a straightforward way to treat excited states, since studying excited states requires the equivalent treatment of these states. Multireference methods are extensions of the single reference Hartree-Fock or configuration interaction (Cl) methods, where many configurations are used instead of a single configuration,... 

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

Transannular cyclizations of nine-membered cyclic sulfides provided a novel route to thioanalogues of swainsonine derivatives. Treatment of the enantiopure polyhydroxylated thiacyclononane derivative 48a with trimethylsilyl iodide yielded the bicyclic sulfonium salt 49a as a single diastereomer, as shown in Equation (16) <2003JOC3311 >. The presence of a C2-axis and the complete regio- and stereoselectivity of the transannular substitution led to a single configurationally homogeneous product irrespective of the fact that the sulfur atom would attack C-5 or C-6. 

Finally, before turning to a brief review of methodological and theoretical aspects, we mention that one of the distinguishing features of planar conjugated polyene radical cations (cf Section II.D) is that their EA spectra reveal a breakdown of the single-configuration picture for ionic excited states which had been used so successfully in interpreting their PE spectra (cf. Section II). 

Although in many cases, particularly in PE spectroscopy, single configurations or Slater determinants 2d> (M+ ) were shown to yield heuristically useful descriptions of the corresponding spectroscopic states 2 f i(M+ ), this is not generally true because the independent particle approximation (which implies that a many-electron wavefunction can be approximated by a single product of one-electron wavefunctions, i.e. MOs 4>, as represented by a Slater determinant 2 j) may break down in some cases. As this becomes particularly evident in polyene radical cations, it seems appropriate to briefly elaborate on methods which allow one to overcome the limitations of single-determinant models. 

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

Actually, two bands of quite different intensity separated by nearly 2 eV are observed Whereas the positions of the two bands could possibly be accommodated by appropriate parametrization, this is not possible for the band intensities which reveals the limitations inherent in the single-configuration approximation. 

Note 0 The CISD energies, which include Davidson correction for unlinked quadruple excitations, where obtained ot the MCSCF/6-31 G(d)optimized geometries and used a TC-SCF reference for closed-shell singlet states and a single configuration reference for triplet and open-shell singlet states. The SCF/6-31G(d) geometry was employed for the A state. See Ref. 55. 

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