Combinations of configurations

Let us assume that the orbit-generating wavefunction for the nth state is given in the form of a configuration interaction wavefunction (i.e., a linear combination of configuration state functions) ... 

Handy (Clary and Handy, 1977). The wave function proposed by them is expanded as a linear combination of configuration [Pg.193]

Biological interactions between molecules are stereo-specific the fit in such interactions must be stereo-chemically correct. The three-dimensional structure of biomolecules large and small—the combination of configuration and conformation—is of the utmost importance in their biological interactions reactant with enzyme, hormone with its receptor on a cell surface, antigen with its specific antibody, for example (Fig. 1-22). The study of biomolecular stereochemistry with precise physical methods is an important part of modem research on cell structure and biochemical function. 

Now if a similar reaction were conducted with a chiral substrate that has a preexisting chiral center (A+). the combinations of configurations at the centers in the product molecules would be A + B+ and A + B -. These are... 

Ellison, M.E., Ng, D., Dang, H., and Garcia-Garibay, M.A. (2003) Enantiospecific synthesis of vicinal stereogenic tertiary and quaternary centers by combination of configurationally-trapped radical pairs in crystalline solids. Organic Letters, 5, 2531-2534. 

As noted above, the helium diatomic dication has been well studied11 7,17,19,321. The standard three configuration VB wave function for the "singly-bonded" (He,-Heb)2+ consists of a linear combination of configurations (1) to (3) ... 

The electronic structure of the asymptotic bT + F+ ions is a linear combination of configurations 1 and 2 (structures 1 to 4) in Table 3, with equal coefficients. Configurations 1 and 2 describe the simultaneous presence of both a o and a n (two center, spin-coupled) bond, while configuration 3 has two n and no a bonds. The calculated structure weights of the ground state at... 

Usually, quite a number of combinations of configurations of the two segments give results compatible with all experimental observations. The method thus chiefly serves to rule out possibilities. 

If the states i ) and i) belong to different symmetry manifolds, characterised by the quantum numbers, j, then the Hamiltonian matrix element is zero. It is economical to consider the diagonalisation in a particular symmetry manifold and we will begin our discussion in this way. The basis states fk) are now symmetry configurations consisting of linear combinations of configurations which have the symmetry /j of the manifold. 

The density matrix is a linear combination of configuration density... 

Finally, states of given multiplicity M = IS + 1, e.g., singlet (S = 0) and triplet (5=1) states, may be described by a linear combination of configurational functions of appropriate multiplicity and symmetry ... 

Normally these molecular orbitals are obtained as expansions in a set of atom-centred basis functions (the linear combination of atomic orbitals (LCAO) method), m being the number of such functions. Recently, two-dimensional numerical integration methods have been developed to solve the MCSCF equations for linear molecules. The dimension m is then, in principle, infinite (practice, it is determined by the size of the grid used in the numerical integration). The molecular-orbital space is further divided into three subspaces the inactive, the active and the external orbitals. The inactive and active subspaces constitute the internal (occupied) orbital subspace, while the external orbitals are unoccupied. The CASSCF wavefunction is formed as a linear combination of configuration state functions (CSFs) generated from these orbitals in the following way. 

An interesting example of a nitrone featuring both carbon and nitrogen substituents, each with a stereocenter, was reported in a cycloaddition-based synthesis of aspartame70. Unfortunately, the matching combination of configurations that promotes a totally stereoselecLive cycloaddition leads to the epimer with the undesired configuration. 

In this procedure [48, 50], one starts from an orbit generating wavefunction 3( rt, s, ) for the ground state of an /V-particle system. Such a wavefunction may be given, for example, by the linear combination of configuration state functions (cf., Eq. (56)) ... 

E and Z boronic esters (13) and (14) were made easily accessible by bromoboration of acetylene followed by esterification of the resulting dibromoboranes The Pd(0) crosscoupling reaction occurs under mild conditions at room temperature leading to a wide variety of the butadienylboronic esters (15) with all possible combination of configurations of the conjugated double bonds as related in table 1. 

In approximations commonly used to describe the true state function, 0> may depend on variational parameters C, C2,..., C, which may be expansion coefficients describing either the linear combination of configurations in 0> or the orbitals [Eq. (1.18)] appearing in these configurations. The total energy forms an energy hypersurface in these parameters E(C, C2,..., C ). We wish to determine stationary points or extrema of the energy hypersurface that, of course, occur when... 

That is, the wavefunction used in the SCI calculation (Eq. (2.70)) is identical, through first order in the parameters, to a new linear combination of configurations with the same C o coefficients but with orbitals < , that can be expressed in terms of the original orbitals as... 

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