Wavefunction correlated

Cade and Huo [21] carried out similar calculations for OH at 13 other internuclear distances and also for the united atom (fluorine) and the separated atoms in the stales with which the Hartree-Fock wavefunction correlates. Some of their data are reproduced in Table 11-6. A plot of the electronic-plus-nuclear repulsion energies is given in Fig. 11 -5 along with the experimentally derived curve. It is evident that the near HF curve climbs too steeply on the right, leading to too tight a potential well for nuclear motion and too small an equilibrium internuclear separation. This comes about because, as mentioned earlier, the HF solution dissociates to an incorrect mixture of states, some... 

Woon D E 1994 Benchmark calculations with correlated molecular wavefunctions. 5. The determination of accurate ab initio intermolecular potentials for He2, Ne2, and A 2 J. Chem. Phys. 100 2838... 

For separable initial states the single excitation terms can be set to zero at all times at this level of approximation. Eqs. (32),(33),(34) together with the CSP equations and with the ansatz (31) for the total wavefunction are the working equations for the approach. This form, without further extension, is valid only for short time-domains (typically, a few picoseconds at most). For large times, higher correlations, i.e. interactions between different singly and doubly excited states must be included. 

Fig. 2. The contribution c<, of the CSP approximation to the Cl wavefunction and the correlation coefficients d jaj/0 versus time.

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

The most commonly employed tool for introducing such spatial correlations into electronic wavefunctions is called configuration interaction (Cl) this approach is described briefly later in this Section and in considerable detail in Section 6. 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

We assume that standard Coulomb-correlated models for luminescent polymers [11] properly described the intrachain electronic structure of m-LPPP. In this case intrachain photoexcitation generate singlet excitons with odd parity wavefunctions (Bu), which are responsible for the spontaneous and stimulated emission. Since the pump energy in our experiments is about 0.5 eV larger than the optical ran... 

The total wavefunction, , is an antisymmetrized product of the one-electron functions i/q (a Slater determinant). The i/tj are called one-electron functions since they depend on the coordinates of only one electron this approximation is embedded in all MO methods. The effects that are missing when this approximation is used go under the general name of electron correlation. 

To ensure this, the-many-body wavefunction can be written as a Slater determinant of one particle wavefunctions - this is the Hartree Fock method. The drawbacks of this method are that it is computationally demanding and does not include the many-body correlation effects. 

For Q = Q , this density function describes electronic motions for given nuclear positions, while for Q = Q it describes the quantal correlation of nuclear positions at time f, which should be small for classical-like variables. The equation of motion for the density function could be derived from the original LvN equation. Instead, it is more convenient to construct it from the wavefunctions. The phase factor and the preexponential factor are trial functions to be determined from the TDVP. The procedure followed here parallels that in ref. (23). 

In order to systematically remedy the previous drawbacks, we recently proposed to perform a perturbation treatment, not on a wavefunction built iteratively, but on a wavefunction that already contains every components needed to properly account for the the chemistry of the problem under investigation [34], In that point of view, we mean that this zeroth-order wavefunction has to be at least qualitatively correct the quantitative aspects of the problem are expected to be recovered at the perturbation level that will include the remaining correlation effects that were not taken into account in the variational process any unbalanced error compensations or non-compensations between the correlation recovered for different states is thus avoided contrary to what might happen when using any truncated CIs. In this contribution, we will report the strategy developed along these lines for the determination of accurate electronic spectra and illustrate this process on the formaldehyde molecule H2CO taken as a benchmark. 

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