Correlated calculations reference state

It is possible to construct a Cl wave function starting with an MCSCF calculation rather than starting with a HF wave function. This starting wave function is called the reference state. These calculations are called multi-reference conhguration interaction (MRCI) calculations. There are more Cl determinants in this type of calculation than in a conventional Cl. This type of calculation can be very costly in terms of computing resources, but can give an optimal amount of correlation for some problems. 

Electron propagator theory generates a one-electron picture of electronic structure that includes electron correlation. One-electron energies may be obtained reliably for closed-shell molecules with the P3 method and more complex correlation effects can be treated with renormalized reference states and orbitals. To each electron binding energy, there corresponds a Dyson orbital that is a correlated generalization of a canonical molecular orbital. Electron propagator theory enables interpretation of precise ab initio calculations in terms of one-electron concepts. 

Both enthalpy and entropy can be calculated from an equation of state to predict the deviation from ideal gas behavior. Having calculated the ideal gas enthalpy or entropy from experimentally correlated data, the enthalpy or entropy departure function from the reference state can then be calculated from an equation of state. 

It is reasonable to expect P3 calculations with open-shell reference states to be less accurate than their closed-shell counterparts. Unfortunately, there is no obvious correlation between errors and multiplicity. 

A system of linear equations as in Eq. (1) and (2) is employed. Rather than the value B of the bonding indicators in each actinide metal, AB variations are calculated with respect to the configuration of a reference state. The reference state configuration is inspired by the Engel-Brewer correlations, amply used for transition metals and alloys It is seen that the system of equations contains one equation less than the number of unknowns, so that only a range of the Ah solutions can he determined. However, this range can be shown, by a simple iterative procedure, to be limited. 

The multireference results of Table 5.12 were all based on full valence CASSCF calculations with eleven electrons in twelve active orbitals. This produces a large configuration expansion (about 85 000 CSFs) so it is not possible to perform MRCI(CAS) calculations. Reference configuration lists were selected at the cyclic and linear geometries (taken from MP2 optimized structures) and then merged. The core electrons were not correlated in any of the calculations. One complication in the CASSCF calculations should be pointed out. Since the cyclic state arises as the 2B component, in C3v symmetry, of a 2E state in the D3h symmetry (equilateral triangular) structure, it would be desirable to obtain MOs with D3h symmetry and equivalence restrictions... 

The em quantities shift the spin-free excitation energies (Em1 - E ), calculated at a lower level of correlation treatment, to the exact values (Em - Ex) or at least to higher accuracy estimates. Herein, X denotes a common reference state, in general the electronic ground state. 

The major limitation of the x(30) values is the fact that they cannot be measured for acidic solvents such as carboxylic acids. Addition of traces of an acid to solutions of (44) or (45) immediately changes the colour to pale yellow due to protonation at the phenolic oxygen atom of the dye. The protonated form no longer exhibits the long-wavelength solvatochromic absorption band. The excellent linear correlation between iix(30) and Kosower s Z values, which are available for acidic solvents, allows the calculation of t(30) values for such solvents [174]. A further limitation has been the fact that it has not been possible to measure the absorption maximum of the standard betaine dye (44) in the gas phase as a reference state. 

Orbitals were generated in the single-configuration SCF approximation and dynamical correlation was treated in one-reference state Contracted Cl (CCI) calculations (15). The Davidson correction (16) was finally added to the CCI energies to account for unlinked cluster contributions. 

The present trend in calculations with correlated wave functions is to include higher than double excitations. Feasibility of CEPA calculations and their success in chemical applications belong certainly to factors which benefited development in this direction. Explicit inclusion of certain contributions due to quadruple excitations, viz, those that are due to disconnected wave function clusters of double excitations, becomes now free of complications also in MB-RSPT through fourth order. It is therefore every reason to expect that, besides Cl-SD and CEPA, MB-RSPT will soon become a method commonly used in chemical applications, A fourth order MB-RSPT approach outlined in Section 4.0. disregards triple excitations, which, however, are hardly amenable to any existing effective method. Another topical problem is a possible extension of MB-RSPT, so that it would permit convenient treatment of the correlation problem for the multiconfiguration reference state. This is difficult with MB-RSPT, but the problem is tract-... 

The above-mentioned items are mainly valid for the ground state. Calculation of excited and ionized states with geminals has only been marginally dealt with in the literature, though it is not ab ovo evident that such wave functions were less appropriate for that purpose. Formally, the APSG wave function may serve as a correlated reference state for various excited-state methods, though the localized features of the wave function will be almost certainly lost. This, again, reflects the objective fact that excited (or ionized) states of molecules have, in most cases, delocalized characters. 

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