Hamiltonian Hartree-Fock level

Table 5. The NMR shielding constant and shielding polarizabilities of the xenon atom calculated at the Hartree-Fock level using the Drrac-Coulomb Hamiltonian (SR + SO), its spin-free version (SR) as well as the non-relativistic Levy-Leblond Hamiltonian. The shielding constant is given in ppm and shielding polarizabilities in ppm/(au field2) (1 a.u. field = 5.14220642X 10" V...

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

This Hamiltonian can be treated at various levels of sophistication. In the simplest approximation, the width A, which can be a function of the electronic energy co, is taken as constant (wide-band approximation),67 and the Coulomb interaction is treated at the restricted Hartree-Fock level, so that both spin states have the same occupation probability, na,a) = naro) = ). In this case, the density of states of the adsorbate takes the form of a Lorenzian ... 

The supennolecule approach is used to study the linear and second-order nonlinear susceptibilities of the 2-methyl-4-nitroaniline ciystal. The packing effects on these properties, evaluated at the time-dependent Hartree-Fock level with the AMI Hamiltonian, are assessed as a function of the size and shape of the clusters. A simple multiplicative scheme is demonstrated to be often reliable for estimating the properties of two- and three-dimensional clusters from the properties of their constitutive one-dimensional arrays. The electronic absorption spectra are simulated at the ZINDO level and used to rationalize the linear and nonlinear responses of the 2-methyl-4-nitroaniline clusters. Comparisons with experiment are also provided as well as a discussion about the reliability of the global approach. 

The derivation of the MCP formalism was detailed in the work of Hdjer and Chung [59] and one of the present authors (TZ) gave a comprehensive discussion of their derivation in the second chapter of his Ph.D. thesis [60], Interested readers should refer to those two works for a thorough exposition of this method while here, in the interest of brevity, we provide only the necessary information about MCP and focus on its level-shift operator, whose physical meaning needs more clarification. Following the same philosophy as the one behind the ECP, the MCP is naturally derived from all-electron atomic calculations at the Hartree-Fock level, and the final expression of the effective hamiltonian for the closed-shell valence electrons is... 

Despite the demands presented by such a calculation, a number of researchers have used ab initio models to treat the electronic and nuclear degrees of freedom for the quantum motif in molecular mechanics, energy minimization studies. Examples of this include the self-consistant reaction field methods developed by Tapia and coworkers [42-44], which represent only the quantum motif explicitly and use continuum models for the environmental effects (classical and boundary regions), and the methods implemented by Kollman and coworkers [45] in their studies of condensed phase (chemical and biochemical) reaction mechanisms. In both of these implementations the expectation value of the quantum motif Hamiltonian, defined in Eqs. (11) and (14) above, is treated at the Hartree Fock level with relatively small basis sets. 

Abstract. The 1/Z expansion will first be used to discuss the scaling properties of the ground-state energy of heavy (non-relativistic) neutral atoms with atomic number Z. The question will be addressed as to what order in Z electron correlation first enters the expansion. The density functional theory (DFT) invoked above will be utilized then to treat, but now inevitably more approximately, the correlation energy in a variety of molecules. Finally, recent studies at Hartree-Fock level on almost spherical B and C cages will be reviewed. For buckminsterfullerene, the role of electron correlation will then be assessed using the Hubbard Hamiltonian, as in the study of Flocke et al. 

The study of B and C cages, having almost spherical symmetry, has then been reported [8], [9]. These investigations were at Hartree-Fock level, and aspects of electron correlation in the particular case of buckminsterfullerene have been pointed out, which emerge from the study of Flocke et al [10] using the Hubbard Hamiltonian. For this special case of Ceo, Hartree-Fock studies are largely vindicated, but the same should not be assumed necessarily to apply to C50, C70 and Cs4 also treated at Hartree-Fock level. 

All the calculations reported in this work were done on a DEC 20-60 using a modified version of the GAUSSIAN 80 series of program (6). Standard ST0-3G minimal basis set (7) was considered. Polarizabilities were calculated by the finite-field SCF method of Cohen and Roothaan (8) which is virtually equivalent to the analytic Coupled Hartree-Fock scheme. A term yf, describing the interaction between the electric field, E, and the molecule is added to the unperturbed molecular Hamiltonian, H y is the total dipole moment of the molecule. At the Hartree-Fock level, the electric field appears explicitly in the one-electron part of the modified Fock operator, F( ),... 

If one has a disordered polymer, one expects that some or all of the states are localized, depending on the degree of disorder. This Anderson localization, which was shown with the help of a tight-binding Hamiltonian, also takes place at the Hartree-Fock level (see Section 4.2 and the work of Day et starting at the band edges and... 

In the PTE coupled-cluster approximation the electronic distribution is computed with PCM at the Hartree-Fock level. This approximation is easily obtained from the above PTDE equations by neglecting all the contributions related to the coupled-cluster solvent operator Qw(A, T) Vn. Specifically, the PTE Hamiltonian is the Hamiltonian H(0)n, which contains the fixed reaction potential of the solute at the HF level, and the free energy functional is given by... 

The SCRF approach has been also implemented for the treatment of solrrte-contin-uum solvent systems at the ab initio Hartree-Fock level of theory. hr addition, a general SCRF (GSCRF) approach has been proposed to accoimt for the interaction of the solvent reaction field with the arbitrary charge distribution of the solirte molecule. According to this theory, the effective Hamiltonian of the solute in the solvent has the following form... 

Electron correlation effects can be defined as the difference between results obtained from the exact solution of a Schrodinger equation with a specific Hamiltonian, and the results obtained at the imcorrelated level, e.g., at the Hartree-Fock or Dirac-Hartree-Fock level. Since for all but the simplest problems the exact solution of the Schrodinger equation is not accessible and usually approximate correlated wavefunctions are used instead. Sometimes experimental values are used rather than the results for the exact solution, which is reasonable as long as the Hamiltonian used for the uncorrelated solution includes all important terms, e.g., with regard to relativistic contributions, influence of the environment of the studied system, etc. As for relativistic effects, the magnitude of electron correlation effects depends to some extent on the details of their evaluation [23]. 

The indices are all defined in terms of the Hiickel molecular orbital method. This has been described on many occasions, and need not be discussed in detail here, but a brief statement of the basic equations is a necessary foundation for later sections. The method utilizes a one-electron model in which each tt electron moves in a effective field due partly to the a-bonded framework and partly to its averaged interaction with the other tt electrons. This corresponds conceptually to the Hartree-Fock approach (Section IX) but at this level no attempt is made to define more precisely the one-electron Hamiltonian h which contains the effective field. Instead, each 7r-type molecular orbital (MO) is approxi-... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

---