Nonrelativistic Hartree-Fock

Fig. 5 Relativistic stabilization of the ns and npi/2 orbitals and the spin-orbit splitting of the np orbitals for the noble gases Xe, Rn and element 118. The Dirac-Fock atomic energies are from [21] and the Hartree-Fock (nonrelativistic) values are from [8]...

There are also ways to perform relativistic calculations explicitly. Many of these methods are plagued by numerical inconsistencies, which make them applicable only to a select set of chemical systems. At the expense of time-consuming numerical integrations, it is possible to do four component calculations. These calculations take about 100 times as much CPU time as nonrelativistic Hartree-Fock calculations. Such calculations are fairly rare in the literature. 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

With sufficiently large basis set, the Hartree-Fock (HF) method is able to account for 99% of the total energy of the chemical systems. However, the remaining 1% is often very important for describing chemical reaction. The electron correlation energy is responsible for the same. It is defined as the difference between the exact nonrelativistic energy of the system ( 0) and Hartree-Fock energy (E0) obtained in the limit that the basis set approaches completeness [36] ... 

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [10]. The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component... 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

Anomeric effect, 82, 310-311, 305 Antarafacial, 163 examples, 164 sigma bonds, 167 Anti-Bredt olefin, 102 Approximations of MO theory Born-Oppenheimer, 22 Hartree-Fock, 222 Huckel, 35, 86 independent electron, 35 LCAO, 229 nonrelativistic, 22 SHMO, 87... 

A measure of the extent to which any particular ab initio calculation does not deal perfectly with electron correlation is the correlation energy. In a canonical exposition [79] Lowdin defined correlation energy thus The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. This is usually taken to be the energy from a nonrelativistic but otherwise perfect quantum mechanical procedure, minus the energy calculated by the Hartree-Fock method with the same nonrelativistic Hamiltonian and a huge ( infinite ) basis set ... 

On the other hand, Gold shows large relativistic effects (the Gold maximum — see eg. [21]). In fact, it has been explicitly demonstrated that for Au relativistic and arc-effects are nonadditive [22]. This is most obvious for its electron affinity While a nonrelativistic Cl-calculation [23] gives a value of 1.02 eV and a fully relativistic Coupled-Cluster calculation [22] yields 2.28 eV, the corresponding nonrelativistic and relativistic Hartree-Fock values are 0.10 eV [22] and 0.67 eV, respectively. Thus immediately the question arises to which extent the GGA s failure for metallic Au is due to the neglect of relativistic arc-contributions in Exc[n. ... 

Figure 13 Nonrelativistic 6s and 6p radial wave functions (solid) versus relativistic 6s 1/2 (dotted), 6p /2 (dashed-dotted), 6/23/2 (dashed) radial wave functions of the thallium atom calculated at the Hartree-Fock and Dirac-Fock levels, respectively.

The term "electron correlation energy" is usually defined as the difference between the exact nonrelativistic energy and the energy provided by the simplest MO wave function, the mono-determinantal Hartree-Fock wave function. This latter model is based on the "independent particle" approximation, according to which each electron moves in an average potential provided by the other electrons [14]. Within this definition, it is customary to distinguish between non dynamical and dynamical electron correlation. 

Non dynamical electron correlation is the part of the total correlation that is taken into account in a CASSCF calculation that correlates the valence electrons in valence orbitals. Physically, the non dynamical electron correlation is a Coulomb correlation that permits the electrons to avoid one another and reduce their mutual repulsion as much as possible with respect to a given zero order electronic structure defined by the Hartree-Fock wave function. In VB terms, the non dynamical correlation ensures a correct balance between the ionic and covalent components of the wave function for a given electronic system. The dynamical correlation is just what is still missing to get the exact nonrelativistic wave function. 

As an approach analogous of nonrelativistic Hartree-Fock theory, the four-component Dirac-Hartree-Fock wave function is described with a Slater determinant of one-electron molecular functions ( aX l= U Nelec, ... 

Correlation energy The difference between the Hartree-Fock energy calculated for a system and the exact nonrelativistic energy of that system. The correlation energy arises from the approximate representation of the electron-electron repulsions in the Hartree-Fock method. 

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