Wavefunction method

An important conceptual, or even philosophical, difference between the orbital/wavefunction methods and the density functional methods is that, at least in principle, the density functional methods do not appeal to orbitals. In the former case the theoretical entities are completely unobservable whereas electron density invoked by density functional theories is a genuine observable. Experiments to observe electron densities have been routinely conducted since the development of X-ray and other diffraction techniques (Coppens, 2001).18... 

The results of the previous section suggest that a good hybrid method might treat antiparallel spin using a GGA, while using a wavefunction treatment for parallel-spin [54]. Since the parallel-spin hole has no cusp and is of greater spatial extent, this contribution should be more accessible to a wavefunction method beginning from one-particle wave functions, such as Cl. 

Figure 1. The shape of the potential curve for nitrogen in a correlation-consistent polarized double-zeta basis set is presented for the variational 2-RDM method as well as (a) single-reference coupled cluster, (b) multireference second-order perturbation theory (MRPT) and single-double configuration interaction (MRCl), and full configuration interaction (FCl) wavefunction methods. The symbol 2-RDM indicates that the potential curve was shifted by the difference between the 2-RDM and CCSD(T) energies at equilibrium.

Figure 3. The shapes of the potential energy curves of the OH radical from the 2-RDM methods with DQG and DQGT2 conditions as well as the approximate wavefunction methods UMP2 and UCCSD are compared with the shape of the FCl curve. The potential energy curves of the approximate methods are shifted by a constant to make them agree with the FCl curve at equilibrium or 1.00 A. The 2-RDM method with the DQGT2 conditions yields a potential curve that within the graph is indistinguishable in its contour from the FCl curve.

Figure 4. Ground-state potential energy curves of Hs from 2-RDM and wavefunction methods are shown. MP2 and MP4 denote second- and fourth-order perturbation theories, while CCSD and CCSD) represent coupled cluster methods.

Ground-State Energies from the ACSE with V, NY, and M 3-RDM Reconstructions Compared with the Energies from Several Wavefunction Methods, Including Hartree-Fock (HF), Second-Order Many-Body Perturbation Theory (MP2), Coupled-Cluster Singles-Doubles (CCSD), and Full Configuration Interaction (FCI), for Molecules in Valence Double-Zeta Basis Sets."... 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

Standard wavefunction methods (i.e., other than DFT), which have been extensively applied both to the computation of vertical (i.e., at ground state equilibrium geometry) excitation energies and excited state reaction paths are the current preferred method for applications in this field. Wavefunction methods that are used in studying photochemical mechanisms are limited to those that can describe excited states correctly. Unfortunately, standard methods for the evaluation of the ground state PES such as SCF and DFT cannot describe excited states because they are restricted to the aufbau principle. 

Since these hypothetical electrons are noninteracting v /,. can be written exactly (for a closed-shell system) as a single Slater determinant of occupied spin molecular orbitals (Section 5.2.3.1). For a real system, the electrons interact and using a single determinant causes errors due to neglect of electron correlation (Section 5.4), the root of most of our troubles in wavefunction methods. Thus for a four-electron system... 

Bianco et al. [23] proposed a direct VB wavefunction method combined with a PCM approach to study chemical reactions in solution. Their approach is based on a Cl expansion of the wavefunction in terms of VB resonance structures, treated as diabatic electronic states. Each diabatic component is assumed to be unchanged by the interaction with the solvent the solvent effects are exclusively reflected by the variation of the coefficients of the VB expansion. The advantage of this choice is related to its easy interpretability. The method has been applied to the study of the several SN1/2 reactions. 

One of the major advances in recent years in attempts to calculate more exact wavefunctions, including electron correlation, has been the implementation by Boys and Handy,46 47 of a computational scheme based on the method of moments, called the transcorrelated wavefunction method. In this method, a correlation factor is built into the wavefunction, which is written in the form (1), where 0 is a Slater... 

Table 1 Computed SCF and correlation energies by the transcorrelated wavefunction method.43 Exact correlation energy in parentheses...

Bemardi and Boys49 have examined the problem of the accuracy of the energy and other variables in this method, and give explicit formulae for improving the calculations. The original formulation of the method to cover the calculation of expectation values was given by Handy and Epstein in 1970.50 Armour 51 has examined the method of moments and the transcorrelated wavefunction method (which is a particular form of the method of moments) in some detail. Several expectation values were evaluated in the course of applications of the former method to H2, and in general fairly accurate results were obtained, but numerical problems can occur, and further study is needed. 

There is no longer any inherent difficulty in computing reasonably accurate electron densities for small molecules by either of the two main computational approaches wavefunction methods [85,86] and density functional methodologies [87-89]. With the introduction of the MEDLA technique [67,70], ab initio quality electron densities can be computed for virtually any macromolecule, including... 

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