Multi-determinant

A generic multi-determinant trial wave function can be written as... 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

Specifically, if T] < 0.02, the CCSD(T) metliod is expected to give results close the full Cl limit for the given basis set. If is larger than 0.02, it indicates that the reference wave function has significant multi-determinant character, and multi-reference coupled cluster should preferentially be employed. Such methods are being developedbut have not yet seen any extensive use. 

Exact or multi-determinant wave function (general, electronic, nuclear)... 

More accurate multi-determinant configuration-interaction (Cl) wavefunctions are described by specifying the types of substitutions ( excitations ) from the starting HF... 

The relative stabilities of singlet and triplet electronic states of three different oxyallyl systems and the closed form of cyclopropanone, bicyclo[1.1.0]butanone, and bicyclo[2.1.0]pentan-5-one have been smdied by density-functional calculations (B3LYP/6-31G ). The results of these calculations are in good agreement with predictions of calculations based on multi-determinant methods. 

In this framework, we restrict ourselves to single-determinant wave functions employing one Slater determinant, but all derivations can be extended to multi-determinant wave functions (125-127). 

In order to obtain expressions for the local spin expectation values, different decomposition schemes exist. One may either partition the total spin expectation value (S2) (122,124) as suggested by Mayer or the total spin operator S2 (113,114) as proposed by Clark and Davidson. The corresponding decomposition schemes for multi-determinant wave functions may be found in Refs. (125-127). 

In principle, transition-metal clusters may best be treated with multi-determinant wave-function methods (139), but in practice due to their size often only DFT calculations are feasible and method-inherent errors have to be taken care of, e.g., the problem of spin contamination and the approximate nature of the exchange-correlation functionals available. 

GLC is used increasingly as a rapid procedure for multi-determination laboratories. While the equipment is relatively expensive, careful operators get good results rapidly. No doubt it will be used more in the future because many other determinations can be performed nearly simultaneously, particularly in research work Lie et al. (29) used a GLC procedure requiring only three minutes per sample. 

If a molecule with no-bond homoaromaticity is investigated, the system in question possesses a non-classical structure with an interaction distance typical of a transition state rather than a closed-shell equilibrium structure. One can consider no-bond homoconjugative interactions as a result of extreme bond stretching and the formation of a singlet biradical, i.e. a low-spin open-shell system. Normally such a situation can only be handled by a multi-determinant description, but in the case of a homoaromatic compound the two single electrons interact with adjacent rc-electrons and form together a delocalized electron system, which can be described by a single determinant ab initio method provided sufficient dynamic electron correlation is covered by the method. 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

The most challenging and therefore the most telling example for excitation theories is C2, whose ground state has a severe multi-determinant wave function. It is known that, to obtain quantitative results (errors < 0.1 eV), one must resort to EOM-CCSDTQ [134], Figure 2-11 compares EOM-CCSD, CCSDT, and various perturbation corrections to EOM-CCSD with FCI for three excited states of C2 [126], EOM-CCSD, which is usually highly accurate, is inadequate for the two states A and B with errors approaching 2 eV. All variants of the perturbation corrections are... 

The above results show that application of the one-determinant approximation to ozone in the ab initio methods results in significant errors and an abnormal spin squared. The DFT approximation is more efficient. The validity of the results may be verified in terms of the multi-determinant approximation. 

In the effective potential approximation Mg is isoelectronic with Be. But, as can be seen in Figure 2, the Mg REP is composed of three terms (s, p and d) with the s and p both repulsive. As a result, even though the correlation correction is almost as large as in Be the multi-determinant correction resulting from Equation 3 is only a tenth as big (see Table II). The discrepancy between values from references 2) and ( ) is due to large statistical or extrapolation error. Note that unlike Be one cannot make comparisons with experimental results without first taking core-valence... 

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-determinant reference wave function. Formulating... 

SCF calculations with multi-determinant wave functions including double excitations from valence molecular orbitals. 

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