Approximation multiconfigurational

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

Time-dependent Hartree-Fock (TDHF) approximation. See also Multiconfiguration time-dependent Hartree (MCTDH) method... 

These single reference-based methods are limited to cases where the reference function can be written as a single determinant. This is most often not the case and it is then necessary to use a multiconfigurational approach. Multrreference Cl can possibly be used, but this method is only approximately size extensive, which may lead to large errors unless an extended reference space is used. For example, Osanai et al. [8] obtained for the excitation energy in Mn 2.24 eV with the QCISD(T) method while SDCI with cluster corrections gave 2.64 eV. Extended basis sets were used. The experimental value is 2.15 eV. 

In the MCSCF linear response theory [48], also called multiconfigurational RPA [54], the reference state is approximated hy a MCSCF wavefunction... 

One of the drawbacks of DFT is its inability to treat multiconfigurational problems properly. In these cases, some type of configuration interaction (Cl) method is needed [60], Ideally, one would like to carry out full Cl calculations, but this is generally not possible for practical reasons. Thus, some kind of approximation is needed and, in this respect, the so-called complete active self-consistent field (CASSCF) procedure is often used. 

An analysis in terms of VB structures (see exercise 3) shows that this configurational mixing corresponds to approximately 40% diradical character in the wave function for ozone. The RHF wave function, on the other hand, contains only 12% of the diradical VB structure (the result was obtained using Hiickel values for the coefficients of the orbitals (2 11)). It is clear from these considerations that a correct treatment of the electronic structure for the ozone molecule must be based on a multiconfigurational wave function. 

Similar iterative schemes were used to determine the MO s for multiconfigurational wave functions, in the early implementations. Fock-like operators were constructed and diagonalized iteratively. The convergence problems with these methods are, however, even more severe in the MCSCF case, and modem methods are not based on this approach. The electronic energy is instead considered to be a function of the variational parameters of the wave function - the Cl coefficients and the molecular orbital coefficients. Second order (or approximate second order) iterative methods are then used to find a stationary point on the energy surface. 

Kotler, Z., Nitzan, A., and Kosloff, R. (1988). Multiconfiguration time-dependent self-consistent field approximation for curve crossing in presence of a bath. A fast Fourier transform study, Chem. Phys. Lett. 153, 483-489. 

Reaction field theory with a spherical cavity, as proposed by Karlstrom [77, 78], has been applied to the calculation of the ECD spectrum of a rigid cyclic diamide, diazabicyclo[2,2,2]octane-3,6-dione, in an aqueous environment [79], In this case, the complete active space self-consistent field (CASSCF) and multiconfigurational second-order perturbation theory (CASPT2) methods were used. The qualitative shape of the solution-phase spectrum was reproduced by these reaction field calculations, although this was also approximately achieved by calculations on an isolated molecule. 

Since a CASSCF calculation is faster than a direct SC calculation, owing to the advantages associated with orbital orthogonality in CASSCF, it is practical to extract an approximate SC wave function (or another type of VB function, e.g., a multiconfigurational one) from a CASSCF wave function. The conversion from one wave function to the other relies on the fact that a CASSCF wave function is invariant under linear transformations of the active orbitals. Based on this invariance principle, two different procedures were developed and both share the same name CASVB . Thus, CASVB is not a straightforward VB method, but rather a projection method that bridges between CASSCF and VB wave functions. 

Configuration Interaction by Perturbation with Multiconfigurational Zero-Order Wave Function Selected by Iterative Process Configuration Interaction with Singles and Doubles Density Functional Theory Effective Core Potential Generalized Gradient Approximation Hartree-Fock... 

Finally, we should mention some approximate calculations on H2. Jug77 has developed a semi-empirical version of the multiconfiguration SCF (MCSCF) method, using CNDO- and INDO-type approximations, and has reported the results of a double-configuration approach to Ha. It was shown that the eigenvalues of the EHF operator have physically interpretable characteristics and follow dissociation properly. Further results of this method should be very interesting. 

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