Configuration coefficient

Inlet configuration Outlet configuration Coefficient of entry, C,... 

Run a final state-averaged calculation at the fuUy-optimized conical intersection using the 4-31G basis set and P to predict the energies of the two states and view the configuration coefficients. (This step will not be necessary if you chose to use P for the final conical intersection optimization job you ll find the relevant output in the CAS output for the final optimization step, preceding the table giving the stationary point geometry.)... 

An example of a multireference technique is the multiconfigurational SCF (MCSCF) approach, where the wave function is obtained by simultaneously optimizing both the molecular orbitals and the configuration coefficients, thereby blending the different resonance structures together. [28] Historically, the MCSCF approach has been used extensively to provide qualitatively accurate representations of surfaces however, this method still suffers two primary drawbacks (1) the ambiguous choice of configurations and (2) the lack of dynamical correlation. 

The most interesting corollary of the results of the previous chapter is, that using basic vector operations and features of vectors, inequalities relating the elements of density matrix can be formulated. Vectors D are completely determined by the configurational coefficients of the underlying full-CI type wave function, but we do not need the knowledge of these coefficients when deriving the inequalities. 

In the above equation, Nc is the fixed, but arbitrary, number of configurations included in the wavefunction, da the a-th configuration coefficient and... 

To determine the virtual orbitals which minimise the variational energy (15), the derivatives with respect to the basis set expansion, see Eq. (13), and the configuration coefficients, Eq. (11) are computed. Analytic gradients and second derivatives, including mixed terms, are computed and inserted into the Newton-Raphson stabilised algorithm [19]. The detailed expression of the derivatives... 

MCSCF wave lunction optimizations are therefore nomrally canied out by expanding the energy to second order in the vaiiational parameters (orbital and configurational coefficients), analogously to the second order SCF procedure described in Section 3.8.1,... 

The deviation of the CASSCF curve from the FCI curve in Fig. 2 is caused by nonstatic or dynamical correlation [1]. Although dynamical correlation is usually less geometry-dependent than static correlation, it must be included for high accuracy (see Sec. 4). One might think that it is possible to include the effects of dynamical correlation simply by extending the active space. For small molecules, this is, to some extent true, in particular when using the techniques of restricted active space SCF (RASSCF) theory [46]. Nevertheless, because of the enormous number of determinants needed to recover dynamical correlation, the simultaneous optimization of orbitals and configuration coefficients as done in MCSCF theory is not a practical approach to the accurate description of electronic systems. 

As discussed above for H2, a must be mixed with a Og function to allow the electrons to localize on the two separated atoms in a model that, unlike UHF, preserves spin and spatial symmetry. Furthermore, the GVB result requires that the form of the orbitals 2pOg and 2po-u be determined together with the configuration coefficients Cj and Q. GVB is the simplest example of a multiconfiguration (MCSCF) calculation. The GVB result for F2, shown in Figure 7 is qualitatively correct as a function of internuclear distance R. 

Quadruply excited configuration Coefficient from 37-configuration wave function by full C.I. Unlinked four-cluster coefficient calculated from pair correlation coefficients ... 

The configuration coefficients of those higher excitations are products of amplitudes of lower level excitations. For example, the main contributions of the triple excitations in the CCSD theory are given as products of amplitudes of single excitations and double excitations, tf (see Eq. 3.6), and the main contributions of the quadruple excitations are given as products of two double-excitation amplitudes, (see Eq. 3.7). 

The linear, semi-linear, and fully exponential forms of the CASCC wave function are equivalent, if the following relations between the Cl configuration coefficients... 

In a sense, CIS is a certain counterpart of the (1-electron) Hartree-Fock method for excited state. Indeed, the Hartree-Fock (HF) variational parameters can be packed into an 1-electron matrix, C, of the conventional occupied MO expansion coefficients. In turn, the CIS configurational coefficients comprise the matrix which is just the first variational derivative of the C-matrix. More exactly, given the ground state tV-electron Slater determinant ) = (1... A)). This ) is the antisymmetrized product of the spin-orbitals, the latter being the standard spinless spatial MOs, (p°), equipped with spin variables. As usually, MOs °) are... 

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