Multi Configuration Hartree Approach

Figure 4.29. Difference between TST and Multi Configuration Hartree Approach. Adapted from Ref. [118].

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

In the multi-configuration Hartree-Fock (MCHF) approach, the Af-electron wave function aLSM Ms is a linear combination of M configuration state functions (CSFs) OiLSM Ms which are eigenfunctions of the total angular momentum L, the spin momentum and their projections and S, with eigenvalues ft L(L+1), f S S+ 1), KMl and hMs, respectively... 

At this point, it is appropriate to draw a parallel with the straightforward MO explanations for the aromaticity of benzene using approaches based on a single closed-shell Slater determinant, such as HMO and restricted Hartree-Fock (RWF), which also have no equivalent within more advanced multi-configuration MO constructions. The relevance of this comparison follows from the fact that aromaticity is a primary factor in at least one of the popular treatments of pericyclic reactions Within the Dewar-Zimmerman approach [4-6], allowed reactions are shown to pass through aromatic transition structures, and forbidden reactions have to overcome high-energy antiaromatic transition structures. 

Meyer, H.-D., Manthe, U., and Cederbaum, L.S. (1990). The multi-configurational time-dependent Hartree approach, Chem. Phys. Lett. 165, 73-78. 

UV spectra usually involve electronic state transitions, so that simple Hartree-Fock and DFT calculations often are not sufficient PCM has been recently extended also to multi-configurational (MC-SCF) calculations [113] and to time-dependent approaches, allowing for the description of excited states and then the prediction of the so-called solvatochromic effects on these spectra. Nuclear magnetic resonance (NMR) and electron spin resonance (EPR) spectra are even more influenced by solute-solvent interactions moreover, the interpretation of experimental data is often very difficult without the support of reliable ab initio calculation, especially for EPR which is usually applied to unstable radical species. 

The Cl and INO calculations discussed above yield very unwieldy wave-functions which do not provide any simple qualitative picture of the bonding. An alternative approach is to pick a reasonably small number of configurations and to try to find, not only the best coefficients CK in the Cl wavefunction of (15), but also to vary the molecular orbitals so as to obtain the optimum orbitals for the chosen form of the wavefunction. This method is known as the multi-configuration SCF method (MC-SCF). The configurations chosen will normally include the Hartree-Fock configuration, plus those additional configurations which add the most important types of... 

The Hamiltonian in Eq. (4.1) has an almost product-like form since the majority of coordinates are treated as harmonic oscillators. This makes it rather suitable for quantum dynamics simulations, either in the time-dependent Hartree approximation [31] or using the more general multi-configuration time-dependent Hartree approach [36, 37]. 

Thus, the method described above allows us to obtain a number of new physical results partially presented in this communication. These calculations are carried out in the Hartree-Fock approximation for multi-electron systems and are exact solutions of the Schrodinger equation for the single-electron case. As the following development of the method we plan to implement the configuration interaction approach in order to study correlation effects in multi-electron systems both in electric and magnetic fields. 

In the most recent version of the energy-consistent pseudopotential approach the reference data is derived from finite-dilference all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian. As an example the first parametrization of such a potential,... 

In the following it will be outlined, how the parity violating potentials are computed within a sum-over-states approach, namely on the uncoupled Hartree-Fock (UCHF) level, and within the configuration interaction singles approach (CIS) which is equivalent to the Tamm-Dancoff approximation (TDA), that avoids, however, the sum over intermediate states. Then a further extension is discussed, namely the random phase approximation (RPA) and an implementation along similar lines within a density functional theory (DFT) ansatz, and finally a multi-configuration linear response approach is described, which represents a systematic procedure that... 

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