Wave functions, approximate

The electron densities of the separate atoms are described by Hartree-Fock wave functions approximated by analytic extended (Slater-type) basis sets. 

Prior to choosing the wave-function approximation it is, however, necessary to set up the electronic Hamiltonian H that describes all interactions of elementary particles. Therefore, we start with the derivation of the full semi-classical many-electron Hamiltonian describing all interactions relevant for chemical problems and subsequently discuss approximations to this full-fledged quantum chemical Hamiltonian. 

Consider each boron atom to be sp hybridized.jhe two terminal B—H bonds on each boron atom presumably are simple cr bonds involving a pair of electrons each. This accounts for eight of the total of twelve electrons available for bonding. Each of the bridging B—H—B linkages then involves a delocalized or three-center bond as follows. The appropriate combinations of the three orbital wave functions. (approximately sp hybrids), and d> (an s orbital) result in three molecular orbitals ... 

In the equation above, the symbol q designates determinants that lie outside the principal projection space used in the CC equations [Eq. (7)]—that is, triples, quadruples, and so on, for CCSD. Note that the first two terms strongly resemble those in the exact expression [Eq. (10)]. In fact, to the extent that the right and left CC wave functions approximate the exact wave function and its Hermitian conjugate and that the excited states of the system are represented by the... 

The variation method has the great drawback of giving only an upper limit to the energy, with no indication of how far from the true energy that limit is. (In Section 26e we shall discuss a closely related method, which is not, however, so easy to apply, by means of which both an upper and a lower limit can be obtained.) Nevertheless, it is very useful because there arise many instances in which we have physical reasons for believing that the wave function approximates to a certain form, and this method enables these intuitions to be utilized in calculating a better approximation to the energy than can be easily obtained with the use of perturbation theory. 

However, one may immediately observe that the entire action eikonal may be explicitly written once the successive iterative equations are solved moreover there is noted that the zero-th order in action corresponds entirely to the classical Hamilton-Jacobi equation the restriction to the first order in h makes nonetheless the Wentzel, Kramers and Brillouin (WKB) framework for semiclassical wave-function approximation ... 

To demonstrate the effect of different orders in DKH calculations. Table 16.4 presents results obtained for SnO and CsH (different ansatze for the electronic wave function were employed). As can be seen from the table, all spectroscopic parameters converge fast with increasing DKH order. The accuracy is mostly determined by the quality of the wave-function approximation. Note that DKHn denotes the scalar-relativistic variant, which has also been called the spin-averaged (i.e., the spin-free) DKH approach. Also, the two-electron terms have not been transformed. Since the nuclear charge numbers of Sn and Cs are not very high, these elements are not ultrarelativistic cases. However, also for Au2 it was found that the spectroscopic parameters are already converged with the second-order DKH2 Hamiltonian [1127], which is typical for such valence-shell dominated properties. 

In the case of the wave functions approximated by single Slater determinant, these matrices given by eq. (108)... 

CH3)2N). In order to ensure consistency of results, the wave functions approximation for all studied molecules was performed using the DFT(B2PLYP) method [37]. Only for the (CF3)2NONO and (CH3>2NONO molecules previously published the DFT(B3LYP) results [27] were used. 

As a result of the resonance tunneling between the equivalent wells, the zeroth-order energy is split into many energy levels. Since for the interaction of nondegenerate systems all of these levels, except for one, correspond to states violating the Pauli exclusion principle, the wave function approximating the physical state can be obtained by antisymmetrization of any of the equivalent resonance structures, in particular by antisymmetrization of 0Q. Thus, the antisymmetrized function Azeroth-order wave function than 00 itself. 

In Table 14.6, we have listed the order in the fluctuation potential to which the amplitudes of the different wave-function approximations are correct. We have also listed the overall order of accuracy of each wave function, as well as the number of electrons for which each approximation recovers the FCI solution. The CC2 wave function is intermediate in quality between the MPl and CCSD wave functions. In Table 14.7, we compare the energies of the various approximations (in a perturbational sense) as well as the overall cost of the approximation and the number of electrons for which the energy reproduces the FCI value. The CC2 energy is similar to the MP2 energy in both cost and accuracy. 

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